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Essays on Political Economy Contests

D I S S E R T A T I O N

of the University of St. Gallen,

School of Management,

Economics, Law, Social Sciences

and International Affairs

to obtain the title of

Doctor of Philosophy in Economics and Finance

submitted by

Philipp Georg Denter

from

Germany

Approved on the application of

Prof. Dr. Martin Kolmar

and

Prof. Stergios Skaperdas, PhD

Dissertation no. 4061

epubli GmbH, Berlin 2012

The University of St. Gallen, School of Management, Economics, Law,

Social Sciences and International Affairs hereby consents to the print-

ing of the present dissertation, without hereby expressing any opinion

on the views herein expressed.

St. Gallen, May 15, 2012

The President:

Prof. Dr. Thomas Bieger

Acknowledgements

During the process of writing this dissertation I have met so many people to

whom I now owe gratitude for their guidance and support, for distraction and

encouragement. Without them finishing this project would probably have been

much harder, maybe even impossible, and definitively not such a memorable

time.

First, I would like to thank my supervisors Martin Kolmar and Stergios

Skaperdas for their guidance and support. I have learned from both so much

more than just the art of conducting research. Learning from and working

with them was not only educational and enlightening but also inspirational

and entertaining. I am really grateful for having had this great opportunity.

A special thanks to Dana, who is not only my former office mate co-author

of Chapters 1, 3, and 4, but who is also my good friend for almost a decade

now. Discussing ideas and projects with her during day and night was always

very enjoyable and fruitful.

Another special thanks to John Morgan, who co-authored Chapter 3 and

who gave so many insightful comments on the other chapters.

Many thanks go to my friends and colleagues from St. Gallen, Irvine, and

Mainz, who made studying and conducting research such a pleasurable thing

to do. Fortunately, it is not possible to name them all now and I apologize to

everybody I am forgetting at this point. Thanks to Darjusch, Philip, Giovanni,

Mirco, Roland, Manuel, Kerstin, Vally, Alex, Sara, Martin, Hendrik, Ermira,

Marco, and Dirk.

A very special thanks goes to Sebastian, Christian, and Stefan for being

great friends for so many years, for always keeping me grounded, and for

always managing to put things in perspective for me. I have known them for

almost 30 years now and I am looking forward to the next 30.

Special thanks also to Barbara. Without her encouragement six years ago I

probably would never have started my PhD studies. Throughout the last years

she gave me encouragement and support and frequently reminded me that

there is more in life than just academia.

iv

Last but not least I would like to thank my family, Papa, Anne, Christoph,

Maria, Florian, Stepha, and my grandparents, for their unconditional support

throughout the years. The greatest thanks, however, goes to the only person

from this list that is not here anymore, Mama. Since it is not possible to express

in words all the things you did for us until you had to leave two years ago, and

how grateful I am and will remain, I keep it simple: Danke!

July 2012 Philipp Denter

Contents

Summary 1

Zusammenfassung 3

Introduction 5

1 The Effect of Polls in Political Campaigns 11

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 The Effect of Polls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 A Campaign Without a Poll: The Benchmark . . . . . . . 19

1.3.2 A Campaign with a Poll . . . . . . . . . . . . . . . . . . . . 21

1.4 Polls and the intensity of political competition . . . . . . . . . . . 28

1.5 Polls and candidates’ spending profiles . . . . . . . . . . . . . . . 29

1.6 Extending the basic model . . . . . . . . . . . . . . . . . . . . . . . 30

1.6.1 Functional Assumptions . . . . . . . . . . . . . . . . . . . . 30

1.6.2 Proportional Representation . . . . . . . . . . . . . . . . . 33

1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Political campaigns with specialized candidates 63

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3 When Salience has no Effect: A Benchmark . . . . . . . . . . . . . 71

2.4 When Salience Matters . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Transparency in Contests 87

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi CONTENTS

3.3 Information Acquisition . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4 Information Disclosure . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.5 Information Transmission . . . . . . . . . . . . . . . . . . . . . . . 98

3.6 More General Contest Success Function . . . . . . . . . . . . . . . 101

3.7 Mandatory Disclosure Policy . . . . . . . . . . . . . . . . . . . . . 102

3.8 Noisiness of the Contest and the Scope for Agreement . . . . . . 106

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4 Imperfect Property Rights 131

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.2.1 Basic Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.2.2 Concerning Property Rights . . . . . . . . . . . . . . . . . 137

4.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.3.1 Complete Information: A Benchmark . . . . . . . . . . . . 138

4.3.2 Incomplete Information . . . . . . . . . . . . . . . . . . . . 138

4.4 Generalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . 143

4.5 When There is a State . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Bibliography 153

Curriculum Vitae 165

List of Figures

1.1 Reactions functions and stage 1 equilibria. . . . . . . . . . . . . . 25

1.2 Parameter values that generate multiple equilibria. . . . . . . . . 26

1.3 Expected spending profiles, theory and data. . . . . . . . . . . . . 31

1.4 Anti-momentum in a polarized populace. . . . . . . . . . . . . . . 38

1.5 Appendix: Possibility of multiple equilibria in stage 1. . . . . . . 50

1.6 Appendix: Second derivatives in stage 1. . . . . . . . . . . . . . . 52

1.7 Appendix: Proof that polls decrease wastefulness of competition. 54

1.8 Appendix: Stage 2 mixed strategy equilibrium in the all-pay auc-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1.9 Appendix: Stage 1 pure strategy equilibrium in the all-pay auction. 59

2.1 Specialization and κ. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.1 Best response functions of A against both opponent types. . . . . 95

3.2 Sequence of moves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.3 Welfare comparison in the all-pay auction. . . . . . . . . . . . . . 108

3.4 Appendix: Utility differences and zone of agreement in case of a

continuous uniform distribution. . . . . . . . . . . . . . . . . . . . 129

3.5 Appendix: Welfare implications of mandatory transparency in

case of a continuous uniform distribution. . . . . . . . . . . . . . 130

4.1 Marginal utility of effort. . . . . . . . . . . . . . . . . . . . . . . . . 141

4.2 Degree of property rights protection and expected relative valu-

ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

List of Tables

2.1 Equilibria of a parameterized version of the campaign game. . . 78

Summary

My dissertation consists of four chapters, all of which are devoted to the anal-

ysis of contests in a political economy context.

Chapter 1 studies the impact public opinion polls have on candidates’ incen-

tives to compete in a political campaign. Under a plurality rule, polls always

can cause situations in which the more popular candidate campaigns more

heavily. In close campaigns also the opposite may happen if having an influ-

ence is ceteris paribus not very expensive. Under proportional representation

no clear statement about the effect is possible unless there is more information

about the electorate available.

Chapter 2 studies communication strategies of candidates for a political

office during the campaign contest, when campaigning has both a policy ad-

vertising and an issue priming component. If candidates have comparative

advantages, they specialize in different issues. However, unlike predicted by

Riker’s dominance principle, specialization is usually imperfect, meaning can-

didates address all issue during the campaign. If one candidate has a large

marginal cost advantage, he may spend more than his opponent also on issue

in which he is disadvantaged. A candidate may also spend more on an issue

he does not specialize in if voters ex-ante perceive this issue to be relatively

important.

Chapter 3 studies situations where two lobbying groups compete for a po-

litical favor. When the goal of a policy maker is to keep the wastefulness of the

competition low, under mild conditions a policy that mandates the disclosure

of private information is not optimal and dominated by a policy of decentral-

ized information sharing. The same holds true if the goal of a politician is to

increase allocative efficiency.

Chapter 4 studies self-enforcement of property rights in a contest game.

While under complete information the defender may deter the appropriator

and so achieve secure property rights, this cannot happen when there is un-

certainty about the appropriator’s type, independent of risk preferences and

relative ability in the contest. Hence, we give a new rationale for imperfectly

2

enforced property that is not based on power but on heterogeneity and the

thereby caused uncertainty faced by the defender.

Zusammenfassung

Meine Dissertation hat vier Kapitel. In allen Kapiteln benutze ich Wettkampf-

modelle um politökonomische Fragestellungen zu untersuchen.

In Kapitel 1 untersuchen ich und meine Koautorin Dana Sisak welchen

Effekt öffentliche Meinungsumfragen für das Verhalten von konkurrierenden

Parteien in einem politischen Wahlkampf hat. Ohne Umfragen haben die

Parteien unabhängig vom Wahlsystem identische Anreize im Wahlkampf. Falls

es Umfragen gibt, existiert in Demokratien mit einfachem Mehrheitswahlrecht

immer ein Gleichgewicht in welchem die bislang beliebtere Partei stärker in

den Wahlkampf investiert. Falls der Wahlkampf einen starken Einfluss auf den

Wahlausgang hat, kann auch der gegenteilige Effekt auftreten. Handel es sich

statt um Mehrheitswahlrecht um Verhältniswahlrecht, dann kann keine allge-

meingültige Aussage bezüglich des Effektes von Umfragen gemacht werden,

da die Ergebnisse explizit von den Wählerpräferenzen abhängen.

In Kapitel 2 untersuche ich wie Parteien in Wahlkämpfen verschiedene The-

men hervorheben, wenn die Sichtbarkeit eines Themas die wahrgenommene

Wichtigkeit beeinflusst. Haben Parteien komparative Vorteile, werden sie sich

im Wahlkampf spezialisieren, solange kein Kandidat einen signifikanten Vorteil

durch niedrigere Grenzkosten hat.

In Kapitel 3 untersuchen ich und meine Koautoren John Morgan und Dana

Sisak den freiwilligen und dezentralen Austausch von Informationen in einem

Wettkampf. Die Ergebnisse stelle ich einer Situation mit verpflichtender Offen-

legung von Informationen gegenüber. Verpflichtende Offenlegung senkt häu-

fig die allokative Effizienz sowie steigert die ineffiziente Ressourcennutzung

im Vergleich zur dezentralen Lösung. Transparenz hat somit oftmals negative

Nebeneffekte.

In Kapitel 4 untersuchen meine Koautorin Dana Sisak und ich die endogene

Durchsetzung von Eigentumsrechten in Wettkämpfen mit sequentieller Strate-

gienwahl und unvollständigen Informationen. Zieht die Verteidigung zuerst

und ist hinreichend stark, können bei vollständigen Informationen Eigentums-

rechte perfekt gesichert werden, was in der Literatur häufig gezeigt wurde.

4 ZUSAMMENFASSUNG

Dieses Ergebnis bricht jedoch zusammen, sobald unvollständige Informationen

berücksichtigt werden. Somit wird eine neue Erklärung für unsichere Eigen-

tumsrechte gegeben.

Introduction

In this dissertation I employ contest models to analyze political economy phe-

nomena. But what exactly is a contest? While most people do not think about

contests in their lives frequently, contests are almost omnipresent. When we

compete for jobs or promotions we participate in a contest. We frequently

watch and participate in sporting contests. Firms compete in advertising con-

tests for market shares or engage in R&D contests to get a patent. Politicians

compete in campaign contests for voters’ support and lobbyists compete in rent-

seeking contest for political favors. And throughout history, mankind engaged

in violent contests such as war or social conflict. Although all these activities

are obviously very different in many respects, they share an important com-

mon characteristic: Different actors or players spend time, money and effort in

order to get or to win something they value, and if they are not successful they

nevertheless have to pay their outlays. These are the defining characteristics of

a contest.

I will now give a short and therefore naturally non-exhaustive account of

past research in the field of contest theory and its applications. For recent

surveys of this literature see Corchón (2007), Konrad (2009), or Sisak (2009).

Although the probably first paper using a contest model to analyze an

economic problem, due to Friedman (1958), was concerned with the study of

advertising, the classic application is lobbying and rent-seeking competition.

This literature dates back at least to the seminal paper of Tullock (1967), in

which he emphasizes that the actual costs of monopolies and tariffs are likely

to be much larger than suggested by the Harberger triangle, measuring just

the distorted market equilibrium. The reason is that apart from the distortion

of the market allocation, there is also a redistribution of rents from consumers

to producers. While this redistribution of rents is in itself not harmful from a

welfare perspective, it creates incentives to engage in socially wasteful activities:

Generally governments do not impose protective tariffs on their own. They

have to be lobbied or pressured into doing so by the expenditure of resources

in political activity. One would anticipate that the domestic producers

6 INTRODUCTION

would invest resources in lobbying for the tariff until the marginal return

on the last dollar so spent was equal to its likely return producing the trans-

fer. There might also be other interests trying to prevent the transfer and

putting resources into influencing the government in the other direction.

These expenditures, which may simply offset each other to some extent, are

purely wasteful from the standpoint of society as a whole; they are spent not

in increasing wealth, but in attempts to transfer or resist transfer of wealth.

Tullock (1967), page 228.

In other words, whenever there are transfers from one party to another to be

granted by a politician, there is likely to emerge a contest. Tullock’s analy-

sis sparked a whole literature analyzing the topic of rent-seeking and lobby-

ing, with notable contributions being Krueger (1974), Tullock (1980), Bhagwati

(1982), Hillman and Riley (1989), Ellingsen (1991), Baye et al. (1993), or Epstein

and Nitzan (2004). Surveys of this literature can be found in Nitzan (1994) and

Congelton et al. (2008). The focus of this literature is mostly the inefficiency

of the political process and the wastefulness of the competition. An important

question posed is to what extent the contested rents are dissipated in the com-

petition, because this is then a natural measure of the additional social costs

caused.

Another classic application of contest models is the study of violent con-

flict or war. Classical contributions include Hirshleifer (1991, 1995), Skaperdas

(1992), or Grossman and Kim (1995). A common denominator in this literature

is that property rights are not well defined or enforced and hence a contest in

which opposing players fight for possession might emerge. Hence, among the

central questions of this literature are when will there be peace or war, what are

the implications of imperfect property rights for economic efficiency under au-

tarky and trade, and under which conditions can secure property rights emerge.

Among the most important contributions to this literature is to show that (mar-

ket) income not necessarily reflects marginal productivity, what follows for ex-

ample from Hirshleifer (1991) and Skaperdas (1992), or that absent well defined

and enforced property rights trade may lead to (expected) efficiency losses rel-

ative to autarky, as has been shown by for example Skaperdas and Syropoulos

(2002) and Garfinkel et al. (2008). Other authors are concerned with identifying

conditions for the breakout of violent conflict as opposed to peace, for exam-

ple Skaperdas (1992), Grossman and Kim (1995), Slantchev (2010), or McBride

et al. (2011), and the emergence of political institutions and elites, for example

Grossman (2001, 2002), Hafer (2006), Kolmar (2008), or Hoffmann (2010). A

INTRODUCTION 7

recent and comprehensive survey of this literature can be found in Garfinkel

and Skaperdas (2007).

Related is also a strand of literature using contests to model political cam-

paigns. Two of the classics in the field, Brams and Davis (1973) and Snyder

(1989), study how candidates that compete in a campaign for political office al-

locate funds to different electoral districts under different electoral rules. Brams

and Davis (1973) thereby develop a ’3/2’ rule, that states that if districts are

of different sizes, larger districts receive a more than proportional share of

campaign resources. Klumpp and Polborn (2006) study the allocation of cam-

paign funds during the Primaries and Caucuses determining the candidates

for the presidential race. They thereby need to take into account the sequen-

tiality of the campaigns and find a strategic explanation for the so called ’New-

Hampshire effect’, which states that early states receive a more than propor-

tional share of funds, because success in an early state creates momentum.

Skaperdas and Grofman (1995) study conditions for candidates to engage in

negative campaigning. They show, for example, that in two candidate cam-

paigns the more popular candidate engages in less negative campaigning than

his opponent. Other more recent papers in this field are for example Konrad

(2004), Iaryczower and Mattozzi (2009, 2011), or Amoros and Puy (2011).

While the aforementioned strands of literature usually take the contest

structure as given and analyze the implications for behavior and outcomes,

in the tournament literature researchers study how a principal can design an

optimal contest to further his goals. For example, a principal may have the dis-

cretionary power to change the allocation of a prize purse to different prizes,

the number and characteristics of competing players, or the informational feed-

back rule during the tournament. Among the seminal contributions in this

field are Lazear and Rosen (1981), Nalebuff and Stiglitz (1983), O’Keeffe et al.

(1984), and Rosen (1986). See Lazear (1995) and Prendergast (1999) for sur-

veys. More recent contribution to this literature include Gradstein and Konrad

(1999), Moldovanu and Sela (2001, 2006), Gershkov and Perry (2009), Gürtler

and Münster (2010), or Ederer (2010).

This thesis has four chapters on contests in a political economy framework.

The first two chapters add to the literature analyzing political campaigns. The

first paper is concerned with the effect of public opinion polls for candidates’

campaigning incentives. Given the increased availability of polls and other in-

formation sources, it is interesting to ask in what way this kind of information

influences political outcomes. When studying political outcomes (at least) two

8 INTRODUCTION

sorts of actors are of importance: Voters and politicians. Since the early 1980s a

large literature has developed that analyzes the impact of public opinion polls

on voting decisions. To mutually exclusive hypotheses have been put forward

by scientists, namely that there are bandwagon effects and that there are under-

dog effects. If there is a bandwagon effect, voters’ propensity to vote for a can-

didate increases in the candidate’s past popularity. Justifications put forward

for the existence of bandwagon effects are various and include information cas-

cades (e.g. Bikhchandani et al. (1992)), and voters’ power to influence policy

(e.g. Straffin Jr. (1977)) to uncertainty aversion (e.g. Hong and Konrad (1998))

or the desire to vote for the winner (e.g. Callander (2007)). To the contrary, if

there is an underdog effect, there is to some degree a tendency that prevents

a party or candidate to become too dominant. A recent paper explaining why

there should be underdog effects is Goeree and Grosser (2007). In their model

a group that was in the majority in a pre-election poll has stronger incentives

to free-ride than the minority group, and hence voter turnout in the stronger

group is likely to be lower than in the minority group. While there exists a

literature analyzing candidate behavior in campaigns, and another literature

analyzing the effect of polls on voting behavior, there is no paper I am aware

of studying the effect polls have on candidate behavior. However, this topic is

interesting for various reasons. First, the availability of information through

polls increased significantly in recent years, and it is unlikely that the availabil-

ity of information does not have an effect on incentives. Second, due to increas-

ing party detachment in recent years, the group of voters that is targeted by

candidates in their campaigns – the swing voters – increased and hence cam-

paigning has become more important, too. In Chapter 1 my co-author Dana

Sisak and me develop a dynamic contest model to systematically study how

polls influence political campaigns. To keep the focus of the analysis on the

effect of polls we assume candidates have identical campaign technologies, but

may have different popularity levels before the campaign starts. We compare

two different electoral systems, a simple plurality rule and proportional repre-

sentation. Without a poll, candidates always invest identically in the campaign.

If there is a poll, we show that under a plurality rule there always exist equi-

libria in which the more popular candidate invests more in the campaign. In

close campaigns also the less popular candidate may invest more if the cam-

paign is important in determining election outcomes or – equivalently – if it

is not too expensive to have a significant influence on the campaign outcome.

Under proportional representation no generally valid conclusion can be drawn

unless there is more information concerning the electorate. If the electorate is

INTRODUCTION 9

relatively centrist results resemble those from a plurality rule. However, if the

electorate is polarized, in close campaigns there exist equilibria in which the

less popular candidate invests more. Therefore, the electoral system has an

important influence on candidates’ campaign incentives.

Chapter 2 also analyzes candidate behavior in a political campaign. In all

the papers mentioned above a campaign has only one dimension. However, in

reality we see candidates competing in a variety of issues and the winner is

the candidate that is perceived best in a weighted average of those issues. I

model a campaign between two candidates that compete in two issues, and the

issues’ weights for the aggregation are also endogenous. The reason for the lat-

ter assumption is that by talking a lot about an issue a candidate increases the

issue’s salience. Endogenous salience has been discussed in a variety of papers,

but I am aware of no paper except Amoros and Puy (2011) where this has an

effect on relative weights of the discussed issues in the campaign contest. In

their paper, however, candidates can only influence the issues’ weights by mak-

ing them salient. In contrast, I develop a model of three intertwined contests,

where one contest determines the relative importance of the other two contests

for competence in an issue. I show that when candidates have comparative ad-

vantages in the campaign, they must partially specialize in equilibrium; that is,

one candidate talks more about one issue and his opponent talks more about

the other issue. This is true unless one candidate has a significant advantage in

the marginal costs of funding. The model is helpful to explain and interpret for

example the presidential campaign between Barrack Obama and John McCain

2008 in the U.S.

Chapter 3 adds to the literature on lobbying or rent-seeking competition. In

many countries lobbyists are legally obliged to frequently disclose information

about their business. For example, in the U.S. the Lobbying and Disclosure Act

of 1995 and the Honest Leadership and Open Government Act of 2007 have

the goal to increase transparency of lobbying practices and thereby make lob-

byists accountable. In this paper I analyze together with my co-authors John

Morgan and Dana Sisak the effect of mandatory disclosure policies on different

measures of efficiency. To do so we study endogenous and voluntary transmis-

sion of information in a contest and contrast this to mandatory transmission.

It is shown that under mild conditions mandating transparency is detrimental

since it increases the wastefulness of the competition and decreases allocative

efficiency in the political process. Hence, we identify two negative side-effects

of transparency policy.

Finally, Chapter 4 adds to the literature on conflict and property rights.

10 INTRODUCTION

In this literature many authors have shown that without well-defined and en-

forced property rights, that is when an outside enforcer (Hafer (2006)) is miss-

ing, there are important inefficiencies and trade may actually decrease welfare,

see for example Skaperdas and Syropoulos (2002) or Garfinkel et al. (2008).

Other authors have claimed that even when no outside enforcer defines and

enforces property rights, incumbency may give rise to secure property rights

because it enables individuals to deter appropriative activities, see for exam-

ple Grossman and Kim (1995), Grossman (2001), Hafer (2006), or Gintis (2007).

It then seems that incumbency – or, equivalently, the opportunity to commit

resources to defend property rights – might overcome some of the discussed

problems and that some of the inefficiencies related to the absence of property

rights, for example distorted production incentives, disappear. However, as

shown by Kolmar (2008) this is not true, since deterrence is at the margin and

so production incentives remain distorted. In Chapter 4 my co-author Dana

Sisak and me go into a similar direction but show that incumbency does under

mild conditions not lead to deterrence, once informational asymmetries and

uncertainty are taken into account. Hence, contrary to the above cited papers,

we show that incumbency usually does not lead to secure property rights.

Chapter 1

The Effect of Polls in PoliticalCampaigns: A Dynamic Contest Model

Philipp Denter and Dana Sisak†

†We would like to thank Johann Bauer, Stefan Bühler, Patrick Button, Benoît Crutzen, Oliver Gürtler,Martin Kolmar, Michael McBride, Giovanni Mellace, John Morgan, Hendrik Rommeswinkel, PhilipSchuster, Stergios Skaperdas, participants of seminars at UC Irvine, University of St.Gallen, Universityof Tuebingen, the 2009 SSES meeting in Geneva, the 2011 APET meeting in Bloomington, the YoungResearchers Workshop on Contests and Tournaments 2011 in Berlin as well as the Erasmus PoliticalEconomy Workshop 2012 at Erasmus University Rotterdam. The kind hospitality of UC Berkeley andUC Irvine is gratefully acknowledged. All errors remain our own. Both authors gratefully acknowledgethe financial support of the Swiss National Science Foundation.

12 CHAPTER 1. THE EFFECT OF POLLS

1.1 Introduction

While an informed electorate is generally considered essential for the well-

functioning of a democracy, one exception concerns polls of candidates’ rela-

tive standing. Critics claim that polls undermine both the incentive to vote

as well as the vote itself; thus distorting voting decisions. As a consequence

the preferences of the populace are warped by the echo chamber of opinion

polls. For this reason, many countries have imposed a ban on the publication

of pre-election polling results. This ban can range from one day before the

election, as in France, to a whole month before the election, as for example

in Luxembourg (Spangenberg, 2003). Candidates, however, are still allowed to

commission opinion polls, even if the general public is not allowed access to

the results. In this paper we study the effect of opinion polls on candidates’

incentives to invest in their campaign. These investments in turn influence the

voters’ ballot choice on election day (Erikson and Palfrey, 2000) and thus the

final election outcome.

Polls are ubiquitous. In addition to traditional providers like Gallup and

Rasmussen Reports, newspapers and TV stations conduct their own polls. Poll

aggregator websites, such as realclearpolitics.com collect the plethora of poll re-

sults and offer a structured overview to the public. In addition, prediction mar-

kets such as politicalbetting.com and oddschecker.com offer alternative sources

of voters’ approval (Berg et al., 2008). Given the relevance of polls, numerous

studies have analyzed the various channels through which information about

candidates’ relative standing might affect voters’ behavior. A prominent hy-

pothesis is the existence of a bandwagon effect, which posits that voters in-

creasingly cast their vote for candidates doing well in the polls. This suggests

that polls create momentum in the sense that the front runner improves his

position over time. However, there are also studies predicting anti-momentum,

for example due to a mobilization effect, with the consequence that the front

runner in the polls experiences a loss of support in the election.

While scholars show a strong interest in the analysis of voters’ reaction to

polls, the other side of the political market – the politicians and parties – has

been virtually neglected. During political campaigns candidates spend large

amounts of resources to increase their chances of coming out ahead. For ex-

ample, total campaign spending during the 2008 Presidential Campaign in the

United States amounted to more than USD 1.6 billion, according to the Federal

Election Commission.1 Because the fraction of partisan voters has been shrink-

1Data on individual spending can be found on the commission’s home page: www.fec.gov/

1.1. INTRODUCTION 13

ing in many countries (e.g. Dalton and Wattenberg (2001)), there are more

swing voters to be swayed in a campaign, making campaigning increasingly

important in determining the election outcome.

The availability of reliable public information about candidates’ relative

standing has undoubtedly an effect on their incentives to compete. In this chap-

ter we analyze how this information influences the incentives of candidates to

spend in the course of a political campaign. In particular, we are interested

in how feedback about relative standing changes the candidates’ relative incen-

tives to compete and consequently whether polls affect the likelihood of an

incumbent coming out ahead. We construct a model in which candidates for

political office may spend resources early and late in the campaign to gain vot-

ers’ support. If there is no poll, candidates a priori have a common belief about

the median voter’s candidate ranking and cannot update their beliefs as the

election day comes closer. If there is a poll, candidates know the median’s can-

didate ranking when making their investment decisions. Our main results are

as follows:

• Under a plurality rule, polls always give the front runner an incentive to

campaign harder than his opponent and thereby create momentum. Thus

there always exists an equilibrium where the front-runner increases his

lead in expectation.

• In an environment where candidates have similar popularity and cam-

paign expenditures are very effective in influencing voters, polls can also

create anti-momentum. In other words, the trailing candidate may run

a more costly campaign than his adversary. In this case equilibria with

momentum and anti-momentum exist.

• Polls tend to make the campaign less "wasteful", i.e. expected aggregate

expenditures decrease.

To illustrate the intuition behind our results consider two ex-ante equally

popular candidates. They campaign over a certain period of time and during

this time random (unobservable) shocks to their popularity occur. Without a

poll, candidates never learn about their current standing with the voter and

thus at any point in time incentives are completely symmetric. With polls on

the other hand, candidates receive updates about their current relative stand-

ing. This alters their incentives in the following way. A candidate who receives

DisclosureSearch/MapAppRefreshCandList.do?d-16544-p=1&d-16544-s=4&d-16544-o=1.


the information that he is ahead, now has an additional incentive to invest. The

reason is that any additional investment now will afford him an even greater

lead in expectation in the next poll. This in turn will decrease the expected

intensity and thus expected costs of future competition. A trailing candidate

on the other hand will have a weaker incentive to invest. Any additional unit of

investment now will bring him closer in expectation to his opponent in the next

poll and thus it will make future competition more fierce and costly in expec-

tation. Consequently polls drive a wedge between the investment incentives

of the trailing and leading candidate and thereby create momentum. When

campaigning is relatively effective and the candidates are relatively close, this

intuition is valid for the trailing candidate as well. A large investment will help

him overtake his opponent and at the same time defuse future competition in

expectation. In these situations both candidates have an incentive to preempt

the other with the objective to save costs in the future.

The chapter is organized as follows. In the remainder of this section we

review the relevant literature. Section 1.2 sets up and explains the basic model.

In Section 1.3 we explore the effect of providing information through polls in

a system of pure majority voting (FPTP). We discuss the effect of polls on the

intensity of political competition in Section 1.4. In Section 1.5 we study the

expected spending profile of candidates. Section 1.6 contains two important

extensions of the basic model. First we show that our technical assumptions

are not too restrictive and that in a more general model our findings will be

qualitatively preserved. In the second part of the section we study the effects

of polls under an alternative electoral system, proportional representation (PR).

Section 1.7 concludes. We relegate all formal proofs to the appendix.

Related Literature. Many scholars have directed their attention to the effect of

polls on election outcomes. The incentives to conduct polls about voters’ policy

preferences to choose a favorable campaign position are analyzed for example

in Bernhardt et al. (2009) and Jacobs and Shapiro (1994). In contrast, in our pa-

per polls inform about candidates’ relative standing and we study the effect of

polls on campaign spending. Earlier models of how candidates use their money

and time during an electoral contest were studied for example by Brams and

Davis (1973, 1974), Snyder (1989), Skaperdas and Grofman (1995), Stromberg

(2008), Iaryczower and Mattozzi (2009, 2011), and Meirowitz (2008). To be able

to study polls and the associated repercussions for electoral outcomes, we study

candidate incentives in a dynamic campaigning model. The dynamic nature of

the model is essential, since the effects we are interested in can only emerge if


candidates can learn over time. The only other paper we are aware of study-

ing dynamics in campaign spending is Klumpp and Polborn (2006). While the

authors study incentives in sequential electoral contests during the primaries,

we consider spending dynamics and the role of informational feedback within

a single contest.

Among others Straffin Jr. (1977), Bikhchandani et al. (1992), Callander (2007),

and Hong and Konrad (1998) show how polls can distort voters’ decisions at

the ballot in favor of the more popular candidate. Rationales for this bandwagon

effect include information cascades, the desire to vote for the winner, and vot-

ers’ aversion to uncertainty. Other authors, e.g. Goeree and Grosser (2007),

argue in the opposite direction, claiming that leading in the polls might actu-

ally be harmful, due to a negative mobilization effect. A number of empirical

papers have tried to falsify one or the other hypothesis, but by now evidence is

at best inconclusive (see for example Blais et al. (2006), McAllister and Studlar

(1991), Nadeau et al. (1993), or Vowles (2002)). Our paper also addresses the

effect of polls on voting decisions and election outcomes, but through an indi-

rect channel: the strategies of political candidates. Candidates vie for voters

by spending time and money on their campaign. We study how candidates’

incentives to engage in such persuasive efforts are affected by polls.

Given that in our setting we find an advantage for more popular candidates

through polls, our paper also adds to the literature identifying and explaining

an incumbency advantage in political competition. For some early empirical

evidence on the existence of an incumbency advantage see for example Erikson

(1971) or Gelman and King (1990). A textbook justification for this phenomenon

are political business cycles as studied by Nordhaus (1975) or Drazen (2001). A

consequence of these theories is that incumbents are likely to have popularity

advantages at the outset of a campaign. Our paper shows that this advantage

is likely to be amplified through opinion polls.

Finally, a literature that is related because of the class of models employed

analyzes workers’ incentives in labor market tournaments. Gershkov and Perry

(2009), Aoyagi (2010), and Ederer (2010) study the optimal feedback policy in

a dynamic promotion tournament, when the firm’s goal is the maximization of

aggregate effort. For their purposes a two-period model with ex-ante identical

individuals is sufficient. We show that asymmetries in popularity only lead

to asymmetric behavior when a future period exists where the frontrunner can

reap the benefits of reduced competition caused by his earlier investment. Thus

to study momentum in a campaign we either need a model with at least three

periods or ex-ante asymmetric candidates.


1.2 The Model

The model has three stages, t ∈ {1,2,3}, in which two candidates i ∈ {A, B}campaign to sway voters.2 In stage 1 and 2 candidates choose non-negative

campaign efforts xti ≥ 0. We denote the cumulative effort a candidate spent

in stages prior to t by Xti . Hence, X1

i = 0, X2i = x1

i , and X3i = x1

i + x2i . Effort

in each stage is costly and costs are quadratic. In particular, C(xti ) =

c2(xt

i )2,

c > 0. Convex costs may reflect both increasing marginal costs of effort or

decreasing marginal products. Similarly, c may both reflect marginal costs and

the marginal product of effort.3 In stage 3 the election takes place, in which

each voter casts a ballot for her preferred candidate. Each voter ι assigns to

each candidate utility utι,i(Xt

i , Eti ,αι,i) if he wins, where

utι,i(Xt

i , Eti ,αι,i) = Xt

i − Eti + αι,i. (1.1)

In each stage t = 1,2, after candidates chose effort, voters’ utility is muted

by a random shock ǫti , and the realization is observable not before the next

stage. Eti is the cumulative shock until t: E1

i = 0, E2i = ǫ1

i , and E3i = ǫ1

i + ǫ2i . ǫ1

i

and ǫ2i are i.i.d. random variables and, for simplicity, identical for all voters.

Hence we may interpret them as ’macro’ shocks.4 They represent changes in

the ranking that are not under the control of either candidate. For specificity

and simplicity, let ǫti be normally distributed with mean zero and variance σ/2.

This implies that the difference ǫt := ǫtA − ǫt

B is also normally distributed with

mean zero and variance σ. Throughout the paper, we shall use the symbols φ

and Φ to denote the PDF and CDF of the standard normal distribution where

needed. αι,i is the a priori value voter ι gains if i gets elected. As an example,

assume each candidate stands for a policy platform vi and voter ι has quadratic

preferences over policies of the form −(bι − pi)2. Moreover, each candidate may

2We assume campaigning is persuasive, rather than informative. Mueller and Stratmann (1994) arguedthat purely persuasive campaigning actually seems to be more in accord with most real world observa-tions than informative campaigning, that is, campaigning that reveals or discloses to the electorate theplatform of a candidate. Persuasive campaigning can, to the contrary, be defined as ’convincing a voterto vote for a particular candidate regardless of her position (p. 59)’. Hence, persuasive campaigningis very similar to persuasive advertising, see Chapter 2.2 in Bagwell (2007) for a recent survey of thisliterature.

3To see the analogy of decreasing marginal products and increasing marginal costs consider thefollowing: Assume the effective effort is f (x), which is weakly concave and increasing in effort x, andthat spending effort has costs C(x), which is weakly convex and increasing. We hence have increasingmarginal costs and decreasing marginal products. By defining effective effort x := f (x)⇔ x = f−1(x),we can also write costs in terms of effective effort, C(x) = C( f−1(x)), which is weakly convex in x.

4The assumption of such a ’macro’ shock is not necessary, but significantly simplifies the exposition.However, in general the shocks could as well have any other form that allows inferring the expectedranking of candidates in the next stage.

1.2. THE MODEL 17

have valence vi. Then, for example, αι,i =−(bι − pi)2 + vi. We do not model αι,i

explicitly but assume it is exogenously given. The example gives you an idea

what it may represent.

It is useful to define the difference of the utilities voter ι expects to get after

the election:

dtι := ut

ι,A − utι,B.

dtι is distributed according to Pt(dt

ι) on P t, with the differentiable density pt(dtι)

and median dt. The distribution may, for example, represent the distribution

of bliss points in the electorate. If d1> 0, then the median a priori prefers

candidate A over B, and vice versa if d1< 0. If the median is indifferent so

is the electorate as a whole, and both candidates receive a vote share of 50

percent.5

If there is no poll, candidates do not know the exact position of the median.

In this case they share the prior belief that d1 is uniformly distributed on [d,d],

d > d. Moreover, they do not learn the realization of ǫ1 before they campaign

in stage 2. To the contrary, if there is a poll, candidates learn d1 and, after

spending effort in t = 1, they learn d2. Note that it would only be possible

to correctly infer the median’s position from the result of a poll if pt had full

support on the real line, which follows from the assumption of normal shocks

ǫ. In particular, for all poll results stating one of the two candidate gets a

vote share of zero, both candidates are not able to tell which of the continuum

of points actually is correct. Such a situation is not extremely realistic, but

may happen in the model, which is due to the assumption of normal shocks,

which have full support. We keep this assumption in most of the paper for

analytical convenience, but note that is does not have any important qualitative

implications for our results (see Section 1.6.1). Hence, we assume a poll gives

candidates a precise estimate of the distribution of voters’ candidate ranking,

respectively of d1.6

The random shock deserves some more interpretation. A possible inter-

pretation is the one of random shocks to a voters’ candidate ranking. It is

well established how random events influence human decision making, also

in elections. Gassebner et al. (2008) show the influence of terrorist attacks on

5You may see that the assumed candidate ranking of a voter resembles a tug-of-war, as it has beenstudied by – for example – Harris and Vickers (1987), Konrad and Kovenock (2005) or Moscarini andSmith (2011). Where the analogy breaks down is how the game is decided. In the tug-of-war it is theplayer who first gets a given advantage over his opponent that wins, while in our model you need tohave the advantage exactly at t = 3.

6Morgan and Stocken (2008), Burke and Taylor (2008), and Meirowitz (2005) explore the informationcontent of polls when voters have an incentive to act strategically.


politicians’ popularity. While this may still be related to policy or qualification,

other studies show the impact of pure random events – at east from the point

of view of a politician – influence election outcomes. For example, Healy et al.

(2010) study the impact of local college football games just before an election

takes place. If the local team wins, the incumbent’s vote share on average in-

creases by more than 1.6 percentage points. Similarly, natural disasters may

have a direct influence of voters’ candidate ranking.7 Finally, even weather con-

ditions influence our decision making and ranking of alternatives, as has been

shown by for example Saunders (1993) or Kamstra et al. (2003) in the context

of stock market behavior. Apart from shocks to candidate ranking, ǫ may as

well be interpreted as a random shock muting candidates’ campaign efforts in

a given stage, so that effective effort is not xti but xt

i − ǫti . This is a standard

assumption in the literature on labor tournaments that has been pioneered by

Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983). In the context of po-

litical competition it may represent uncertainty about the effects of campaign

spending, and that campaigning might backfire as well (e.g. Dukakis in the

tank).

Finally, a word concerning the cost function. In our setting we naturally

need strictly convex cost functions to guarantee the existence of a pure strategy

equilibrium, since otherwise the second order conditions cannot hold for both

candidates simultaneously. However, strict convexity is not yet sufficient, but

costs need to be sufficiently convex. What this means will be discussed in the

relevant appendices. Throughout the paper, with the exception of our discus-

sion of the All-pay auction in Section 1.6.1, we will assume costs are sufficiently

convex.

1.3 The Effect of Polls

We now start our analysis of the campaign by studying candidate behavior in

simple plurality electoral systems (first past the post / FPTP). In a FPTP system,

the goal of a politician, a party, or a candidate is to win the simple majority of

7A recent example for such a shock is the disaster caused by the damaged Fukushima Daiichinuclear power plant in Japan that followed the earthquake and the thereby caused the tsunami ofMarch 11, 2011. In the following, in many countries around the world a shift of voters’ preferencesin favor of green or anti-nuclear movements was accounted for. In Germany, two important stateelections took place just one week after the disaster. In political polls a large increase of anti nuclearmovements and green party support was recognized. In the traditionally conservative state of Baden-Württemberg the Green Party won the election after 66 consecutive years of a conservative ChristianDemocratic government and the Green Party’s candidate, Winfried Kretschmann, is now the first greenstate governor (Ministerpräsident) in Germany. The Green Party’s success was attributed, among otherthings, to the revived anti-nuclear movement.

1.3. THE EFFECT OF POLLS 19

votes. That is, as long as a plurality is achieved, the candidate is indifferent

whether he wins by a small or by a large margin; because he is declared the

winner anyway. As an example, take the electoral college in the U.S. Candidates

for presidency compete in several states for voters’ support, and the candidate

who wins the plurality in a state wins all presidential electors, too, no matter

how close the election outcome was.8 In a second stage, it is again the abso-

lute majority in the Electoral College that is the majority of electors, which is

needed to elect the president and vice-president. Other countries with FPTP

systems are for example the United Kingdom and most of the countries that

were historically under significant British influence, such as India, or Canada.

With more than 2 billion people in 47 countries living with a FPTP electoral sys-

tem in 2005, it is together with list proportional representation (PR) the most

important electoral system worldwide (see Reynolds et al. (2005)).

Let us now start with the analysis of conducting a poll. First, we establish

a benchmark by looking at a situation without polls.

1.3.1 A Campaign Without a Poll: The Benchmark

If there is no poll the campaign protocol is as follows. In t = 1 and t = 2 both

candidates have the opportunity to spend effort, and they do not receive a

signal as to their current popularity with the electorate in either stage. Hence,

they have to base their decision on their common prior beliefs F(d1) in both

stages.

Let us conjecture what happens in equilibrium. First, because a player does

not receive a signal as to the realization of ǫ both candidates’ beliefs after t = 1

are unaltered. Given an identical technology to produce effective campaign

effort in both stages, we should expect each candidate to choose the same action

in each stage. But how does F(d1) influence effort choices? First, if d1 = 0, both

candidates are identical in expectation, and hence it is intuitive to assume there

is a symmetric equilibrium, too.9 However, what happens if one candidate

has a competitive edge over his opponent? The answer is not obvious because

there are many different effects at work and there are three different sources

8Exceptions are Nebraska and Maine, in which the winner of the popular vote receives twoelectoral votes, and the winner of each of the congressional districts, there are two in Maineand three in Nebraska, receives one electoral vote in addition. Just recently, a discussion,started by newly in power republican politicians, centered on the question whether or not tochange the electoral system also in Pennsylvania from FPTP to a mixture of winning the popu-lar vote and the single congressional districts. http://www.nytimes.com/2011/09/19/us/politics/

pennsylvania-republicans-weigh-electoral-vote-changes.html.9By a symmetric equilibrium we mean an equilibrium in which both candidates choose identical

strategies.


of uncertainty. One source is the lack of knowledge about the current state of

the world, and finds representation in F(d1). The other two sources are related

to uncertainty about future shifts in the distribution of preferences, given by ǫ1

and ǫ2. Since candidates are different it seems intuitive that they also engage

in different behavior. The leading candidate might have an incentive to be

aggressive and secure his lead by spending a lot. But since effort is costly he

may as well take a soft position to safe on costs and hence spend relatively little

compared to his opponent. In analogy to an markets it is intuitive to believe in

the first course of things, since in such a setting the laggard often loses market

share in expectation (see for example Athey and Schmutzler (2001) or Cabral

(2002)). In our case this intuition is, however, flawed:

Proposition 1.3.1. The equilibrium is unique. Both candidates choose in each stage

identical effort, and efforts are also identical across stages. Hence, absent a poll the

ex-ante expected popular vote of a candidate in t = 3 is identical to his popularity in

t = 1.

Proof. See appendix.

To see why efforts are identical in equilibrium it is instructive to look at

candidates’ marginal incentives. For a fixed median position d1, note that the

marginal increase in popular vote / probability to win of A is exactly the share

(density) of currently indifferent voters. Of course, this also holds for B. Taking

into account uncertainty about d1 this must hold also in expectation. Moreover,

since candidates have identical technologies to produce effort, they also have

identical marginal products and marginal costs of effort. Consequently, for

any prior distribution F(d1) both candidates have identical marginal incentives,

and hence the equilibrium must be symmetric. This is similar to a result by

Lazear and Rosen (1981) in their analysis of labor market tournaments, when a

candidate enjoys a head start.

A direct implication of the proposition is, that the campaign does not have

any effect on the expected winner of the election, if there is no poll. Candidates

always spend identical effort, the net effect is zero, and the winning probability

will be unaffected in equilibrium. Here we clearly see how political campaigns,

or contests in general, are similar to the Prisoner’s Dilemma: both candidates

would be better of by reducing effort by some k > 0, or by spending nothing at

all, because effort is costly. However, this is not an equilibrium.


1.3.2 A Campaign with a Poll

Now turn to the case of a campaign with a poll. Unlike before, in t = 1 both

candidates learn d1 and d2. In most modern democracies polls are an almost

pervasive element of the political landscape and this kind of information is, by

now, accessible in abundance. Apart from polls, politicians may as well learn

something about their popularity from prediction markets in the internet. This

information makes it possible for a candidate to react to changes in his pop-

ularity, and, at the same time, even if the current popularity does not change

(d1 = d2), this gives additional information to the candidates; after all, learning

about the popularity in t = 2 is confounded by less noise than it was in t = 1,

and for sure by less than absent a poll.

Look at stage 2 first. After learning d2, both candidates spend effort. This

situation is strategically similar to the situation of the candidates absent a poll;

the structure and amount of noise is, however, different. But we have seen that

structure and amount of noise were not decisive for relative incentives. Hence,

although efforts probably differ in absolute levels, the relative level of effort is

as it was before:

Proposition 1.3.2. A unique equilibrium exists in stage 2. In this equilibrium both

candidates always choose identical effort.


The intuition for identical efforts is identical to before: by marginally in-

creasing effort, the additional probability to win equals exactly his opponent’s

loss in winning probability; the marginal benefit of effort is identical for both

candidates independent of their popularity. Because marginal costs are also

identical, both must choose identical effort. Hence, we can conclude that the

expected popular vote a candidate gets in t = 3 equals his popularity in stage

2.

Now turn to the comparative statics of the equilibrium. Most importantly,

consider the effect on efforts of varying d2. Those comparative statics are of

great importance, because from the point of view of t = 1, the relative popular-

ity in stage 2 is endogenous. Since the equilibrium is symmetric, the difference

in efforts, x2A − x2

B, is zero, independent of d2. Hence, the marginal increase

in the winning probability in equilibrium is the density of the shock / noise,

evaluated at d2. The normal density is strictly quasi-concave and symmetric at

zero and the shock distribution determines the marginal product of campaign-

ing effort. By spending more, the probability that the realization of the shock


is such that the candidate wins, gets larger. The marginal product of effort is

exactly the marginal increase of this probability, and hence in the symmetric

equilibrium it must be φ(d2). We know now that φ(z) is strictly decreasing

in |z|, implying marginal effort decreases as |d2| gets larger. Put differently,

equilibrium effort in t = 2 is an inversely U-shaped function of d2 and has its

maximum at d2 = 0. More plastically, the intensity of the campaign in t = 2

increases as the popularity difference vanishes. What happens if we vary σ?

Increasing it implies φ goes up at the tails and goes down in the center, hence

becomes flatter. Therefore, if the competition is lopsided with one candidate

enjoying a big lead, increasing the variance of the shock increases equilibrium

spending. Intuitively, if the variance is large the trailing candidate has a real-

istic chance to catch up without spending overly in the campaign. This also

gives him an incentive to increase effort, since, as we have discussed before,

φ determines the marginal product of effort. To the contrary, if σ is relatively

low, catching up due to luck is relatively unlikely, and hence the trailing candi-

date has no incentive to compete anymore. As a result both candidates’ efforts

decrease. Finally, efforts strictly decrease in c.

Next turn to stage 1. Candidates take into account the expected effect their

campaign spending in t = 1 has on stage 2 through d2 = d1 + ∆1 − ǫ1. So far

we have seen that candidates choose identical efforts, no matter what is the

relative popularity. What should we expect now given the equilibrium in stage

2? If d1 = 0 and hence candidates are perfectly symmetric, it seems intuitive

that there also exists a symmetric equilibrium. It is, however, not completely

clear, intuitively, what happens if d1 6= 0. Has the trailing player an incentive

to try extra hard to “gamble for resurrection”? Or has the leading candidate

stronger incentives? As it turns out the results depend on both c and σ, more

specifically, on ρ := cσ2. We can interpret ρ as a measure of how expensive it is

to gain a substantial advantage due to effort in a close campaign. The higher

ρ, the more expensive it is. To see this note that a high variance σ implies the

density in the center of the shock distribution is low, and hence there is not

much mass. Spending a given level of effort hence has a lower impact on the

probability to win the campaign than it would have if σ was small. Hence, we

can interpret ρ as a measure of the expensiveness to influence the campaign’s

outcome. In particular, there exists a threshold, ρ := (334√

π)−1, that determines

two regimes of equilibrium outcomes. We will discuss both regimes separately.

ρ > ρ. If ρ is relatively large it is relatively expensive to influence the expected

outcome of the election when the race is close. Now what effect does this have


on the equilibrium of the campaign game in t = 1? When it is relatively costly

to influence the expected outcome in a close campaign, the leading candidate

(if d1 6=0) will probably not be able to make a decisive step forward, but the

high level of noise makes it more likely that the trailing candidate catches up.

Hence, it seems that the trailing candidate benefits from higher risk and thus

that he has stronger incentives to invest, too. However, deeper inspection of the

situation reveals that this logic is missing an important point. To see that con-

sider the decision each candidate has to make in t = 1 in more detail. There are

three different channels through which effort influences a candidate’s expected

utility. First, effort increases the probability to win the election. This effect is,

as in the previous section, identical for both, and hence cannot be the reason

for differences in campaign disbursements. Second, effort has immediate costs.

But candidates have identical cost functions, and hence marginal costs are also

identical. Therefore, this cannot be the reason for different behavior, either.

There is, however, a third channel. By increasing effort in t = 1 a candidate

changes the expected state of the campaign in t = 2 a bit in his favor, because

d2 = d1 + ∆1 − ǫ1; the marginal impact of effort in t = 1 on d2 is equal to one.

Changing d2 has consequences for spending, and hence for costs, too. From the

discussion of the comparative statics in t = 2 we know that costs are highest

when the race is tied, d2 = 0, and that costs decrease monotonically if we let |d2|grow, because spending goes down then. This implies that the leading candi-

date, by exerting effort in stage 1, locally increases |d2| in expectation and hence

decreases expected costs in stage 2 for both, while the opposite holds for the

trailing candidate. In more technical terms, this implies the leading candidate’s

efforts in stage 1 and 2 are strategic substitutes, while the trailing candidates

efforts are strategic complements (Bulow et al. (1985)). This implies the leader

has lower expected marginal costs than his opponent. As a consequence, he

will spend more in equilibrium and thereby in expectation increase the differ-

ence to his opponent; we have a situation in which the leader acts tough, while

his opponent takes a softer stance. Hence, unlike the situation without a poll,

the analogy to the increasing dominance results of Athey and Schmutzler (2001)

and Cabral (2002) is valid, although the mechanism is a different one.

Where now does the expensiveness to have an influence come into play? A

large ρ is due to either a high risk, or high marginal costs, or both. The degree

of risk or noise in the electoral contest implies the effect stage 1 effort has on

the expected stage 2 campaign is somewhat limited. Because both high risk

implies efforts are relatively low in equilibrium, so are costs. Hence, there is

not much to save in stage 2 due to spending more in stage 1. The expected


marginal effect of effort today on costs tomorrow is low. High marginal costs,

on the other hand, make it directly unattractive to spend a lot. While this is

true for both candidates, it is even more relevant for the candidate trailing. For

him it is prohibitively expensive to overinvest in stage 1 so as to gamble for

resurrection; he would have to spend a lot to safe a bit only. This, however, is

not true for the leader. Hence, if the costs of influence are large there cannot

be an equilibrium in which the trailing candidate acts tough and tries to force

a change. Rather, he adopts a ’wait and see’ posture. Before discussing the

implication of a relatively low ρ we summarize the results our results in the

following proposition:

Proposition 1.3.3. If it is relatively expensive to change the outcome in expectation,

ρ > ρ, in the unique equilibrium the leading candidate chooses greater effort than his

opponent. If the race is tied both candidates spend identical efforts.


ρ ≤ ρ. Now consider the case that ρ is relatively low, which means it is ceteris

paribus relatively cheap to increase the probability to win in expectation if the

race is close. This is so because either marginal costs are low, there is little noise

/ σ is small, or both. Then the aforementioned logic, that the trailing candidate

cannot outspend the leader in equilibrium, does not apply anymore, and hence

’gambling for resurrection’ may actually be an equilibrium. However, there is

another condition that needs to hold for the leader taking a soft stance, and his

opponent acting aggressively: the difference between candidates must not be

too large. To see why this is the case recall the aforementioned logic of strategic

substitutes and complements. A candidate’s efforts are strategic substitutes if

and only if in equilibrium he is in expectation more popular than his opponent

in stage 2. If this does not hold, by spending marginally more a candidate

decreases the expected gap and hence increases expected costs, while the oppo-

site holds for his opponent. Hence, the opponent has lower marginal costs of

effort and hence it cannot be an equilibrium to spend more than the opponent.

Consequently, the trailing candidate may spend more in equilibrium, but if he

does this he needs to spend enough to turn the state in his favor in expectation.

Of course, turning the state becomes increasingly expensive as the difference

between candidates increases, and there exists a threshold gap determining the

maximum lead that the trailing candidate may try to turn by investing heavily

in campaigning. Obviously, if the trailing candidate can take charge and act

aggressively in the campaign in equilibrium, this is also possible for the leader.


0.40 0.45 0.50 0.55 0.60 0.65 0.70

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.40 0.45 0.50 0.55 0.60 0.65 0.70

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.40 0.45 0.50 0.55 0.60 0.65 0.70

0.40

0.45

0.50

0.55

0.60

0.65

0.70

x1Ax1

Ax1A

x1Bx1

Bx1B

Figure 1.1: Reaction functions for A (gray) and B (brown, dashed) and stage 1 equi-libria for ρ = 0.242 and d1 ∈ {0,0.04,0.1}.

After all, for him it is cheaper to stay in the lead than for his opponent to turn

the electorate’s sentiment. Hence, there are multiple equilibria. It remains to

show what happens when the race is tied. As before, to conjecture that there ex-

ists a symmetric equilibrium is appealing. But we also have seen that there are

multiple equilibria if the game is not tied and no candidate dominates. Those

asymmetric equilibria also exist in the tied race. In addition, a symmetric equi-

librium exists. Hence, there are three equilibria for d1 = 0, or more generally,

when |d1| is small. However, note that the symmetric equilibrium at d1 = 0 is

asymptotically unstable, while the asymmetric equilibria are stable. In Figure

1.1 we simulated the equilibria using the best response functions of the can-

didates for ρ < ρ and d1 ∈ {0,0.01,0.1}. We summarize the results formally:

Proposition 1.3.4. If ρ is relatively small, there exist multiple equilibria, and, if we

confine ourselves to asymptotically stable equilibria, there does not exist a symmetric

equilibrium. In close races either candidate may spend more, while if one candidate has

a sufficiently large advantage, this candidate spends more than his opponent.


A final word concerning the equilibria when ρ is low is in order. A low

value of ρ implies that either marginal costs are low, the campaign is – to

borrow from the theory of contests – relatively discriminating (low σ), or both.

Either of those makes it less likely that an interior pure strategy equilibrium

actually exists, since the second order conditions are then likely to be violated.

However, there exists a small range of values for ρ for which the second order

conditions hold and ρ ≤ ρ. In Figure 1.2 you can see combinations of c and σ

for which equilibria as discussed above actually exist.


0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

Figure 1.2: Combinations of c (ordinate) and σ (abscissa) for which multiple andasymmetric interior pure strategy equilibria in stage 1 exist.

Having discussed campaigning incentives, we now explore the expected

impact a poll has on election outcomes in more detail. We are in particular

interested in the probability that the more popular candidate increases his lead.

Because the campaign is noisy, spending more is not sufficient to improve one’s

position, but makes it more likely. Note that in a campaign without any noise,

this would not change because than there is no pure strategy equilibrium and

candidates hence create noise endogenously by randomizing (see for example

Nalebuff and Stiglitz (1983), Hillman and Riley (1989), or Baye et al. (1993)).

We can now determine the probability distribution of changes in the relative

standing. Absent a poll we have seen that efforts are always identical, and

hence (i) campaigning has no effect on the outcome in equilibrium, and (ii), each

candidate improves his standing with 50 percent probability. Conducting a poll

has no effect on relative incentives in stage 2, but alters incentives significantly

in stage 1.

Assume without loss of generality that d1 ≥ 0. The probability that A in-

creases his lead is

q(ρ,d1) := Pr[d2 ≥ d1|(ρ,d1)] = Pr[∆1 ≥ ǫ1|(ρ,d1)

]= Θ

(∆1(ρ,d1)

σ

).

Not surprisingly, this probability is strictly increasing in the difference of

efforts spent in equilibrium, ∆1. We hence need to explore the incentive dif-

ferential of the two candidates, as a function of the median’s ranking d1. To

develop an intuition for the underlying mechanism, consider first the two po-

lar case, d1 = 0 and d1 → ∞. In the first case candidates are identical, and as

we have seen before, the equilibrium is then symmetric. Hence in this case we

have q(ρ,0) = 12 . In the latter case A de facto already won the campaign. As a

result no candidate has an incentive to spend effort at all, and hence we find


also q(ρ,∞) = 12 . If we increase d1 starting at zero and let it converge to infin-

ity, the incentive differential is increasing first, and so is ∆1. At a given point,

however, this stops and incentives become more and more identical again. The

differential incentives first increase because it becomes more likely for A to ac-

tually improve his position. However, as his popularity grows this becomes

less and less interesting, because spending more means incurring higher costs;

but if A is already very popular he starts to care more about the costs he can

save by spending less, because the probability that he wins is already large, and

he cannot safe much on efforts either, because in a lopsided campaign in t = 2

both candidates will not spend much effort in equilibrium. Therefore, there

exists an inversely U-shaped relation between d1 ∈ [0,∞] and q(ρ,d1).

Proposition 1.3.5. Assume ρ < ρ, that is the campaign is expensive to decide. q(ρ,d1)

is an inversely U-shaped function of d1 on [0,∞]. It attains its maximal value of

Θ(

112√

2eπρ2

)>

12 when d1 =

√32σ.


Note that the maximum is strictly decreasing in ρ, what is intuitive. If

having a significant influence on campaign outcomes is relatively expensive

we should not expect candidates to try investing a lot to ’decide’ it. It also

follows that the maximum of the maxima will be attained when ρ = ρ and is

q(ρ,√

32σ) = 57.37. This does not seem bo be a very large number, but note that

absent a poll it would have been 50 percent instead, and that the infinite sup-

port of the normal distribution naturally decreases the number. For different

distributions it will certainly be larger.

If it is not expensive to influence the expected campaign outcome, the tra-

jectory of q changes for small values of d1, and is not equal to zero for d1 = 0.

However, assuming equilibria are stable, the results are qualitatively unaltered

– the trajectory is inversely U-shaped – and hence we relegate the discussion of

this case to the appendix.

What are now the effects of conducting a poll? One relates to the role of

campaigning to decide the race. We have seen in the previous section that ab-

sent a poll campaigning has no influence whatsoever on the outcome of the

election and candidates are basically trapped in a Prisoner’s dilemma. Even

when candidates are asymmetric in the sense that one has a popularity edge,

incentives to campaign are identical and hence campaigning is only costly with-

out helping anybody to increase his expected share. This changes as soon as we


introduce a poll. The knowledge of getting feedback in a future stage and hence

having the ability to adjust campaigning efforts accordingly creates incentives

to behave differently in the earlier stage of the campaign. Hence, campaign-

ing has an effect on the expected outcome. Our second result relates to the

direction of this effect. In general there always is an equilibrium in which the

leading candidate assumes a tough posture by spending more and so increases

his expected lead. However, in a close campaign in which it is relatively cheap

to influence the probability to win, also the trailing candidate may act tough.

1.4 Polls and the intensity of political competition

So far we have focused on the effects of polls on the outcome of the election,

which is determined through relative candidate spending in the campaign. We

have neglected the actual level of expenditures. Campaign expenditures can

be quite substantial. For example, they typically exceed one billion U.S. dol-

lars in the case of presidential elections in the U.S. Thus the "wastefulness of

competition" is also an important aspect to study and relate to the existence of

polls. We are aware that not all campaign efforts are necessarily wasteful since

campaigns also inform voters about the candidates’ positions. Nevertheless,

we define wastefulness as expected aggregate spending over the course of the

campaign. Thus we implicitly assume that polls do not influence the level of

informative campaigning and candidates on the margin invest for example in

image building activities. The following proposition shows the result:

Proposition 1.4.1. When there are polls, aggregate campaign expenditures decrease in

expectation. Thus, ceteris paribus, polls make competition less wasteful.


The intuition for the result is the following. When polls create momen-

tum, initial asymmetries are amplified. This decreases the expected intensity of

competition in subsequent rounds, and hence expected wastefulness decreases,

too.

The effect of increased information on the intensity of competition has been

a subject of interest in other studies as well. A recent example is Chapter 3 of

this dissertation, where we analyze the effect of mandatory disclosure require-

ments in competitive environments such as lobbying competition and political

campaigns. They study the competitors’ incentives to share private information

about their characteristics, for example their valuation of winning, and how

mandated disclosure affects the outcome of competition. Their main result is

1.5. POLLS AND CANDIDATES’ SPENDING PROFILES 29

that mandatory disclosure increases the intensity and decreases allocative effi-

ciency of competition in expectation. In their setting, an uninformed player can

use the chance of facing a weak opponent to appease a strong opponent, and

at the same time the threat to face a strong opponent to discourage a possible

weak opponent. The lack of information about the opponent helps to commit

to strategies which would not be credible under complete information. In con-

trast, in this paper, candidates are identical except for their current popularity.

There is no private information held by the candidates, and hence they can not

use his lack of knowledge as a commitment.

1.5 Polls and candidates’ spending profiles

In this section we now take a closer look at the spending profiles of the candi-

dates when there is a poll. If there is no poll the result is straightforward, spend-

ing decreases as one candidate becomes more and more advantaged. Hence,

here the focus is on spending profiles when there are polls. We will then com-

pare our results with empirical findings and show that the model performs

relatively well in predicting spending profiles.

We know already that in a campaign that is costly to decide polls create

momentum and the more popular candidate invests more heavily in the cam-

paign. We are now interested how the two candidates’ spending varies in

absolute terms as we increase |d1| from zero. Conjecturing about how efforts

change one is tempted to say efforts unambiguously decrease, as the campaign

becomes more lopsided and hence the need to spend decreases. This also what

we observed in stage 2, where spending decreases monotonically in |d2|, and is

also what would happen in the models of for example Snyder (1989), Erikson

and Palfrey (2000), or Klumpp and Polborn (2006). However, as we have seen

already before, introducing dynamics may change results significantly and it

also does in this case:

Proposition 1.5.1. Assume ρ > ρ. Candidate A’s expected total effort increases in d1

when d1 = 0, while candidate B’s expected total effort decreases.


The trailing candidate’s campaign effort decreases monotonically as |d1|grows larger. His more popular adversary, however, has an incentive to first in-

crease effort when his lead grows larger, before he also cuts down on spending

when he becomes more and more advantaged in the campaign. Hence we can

now describe the two candidates spending profiles completely:


Corollary 1.5.1. As |d1| increases from zero the more popular candidate first increases

effort in the campaign and his spending declines only after he reaches a certain popu-

larity advantage. The trailing candidate monotonically decreases spending and spends

less than his opponent for all |d1| > 0.


In the right panel of Figure 1.3 we plotted expected total campaign spend-

ing for both candidates. As stated in the corollary, the more popular candi-

date’s spending first increase until it reaches a maximum, and monotonically

decreases thereafter. The trailing candidate’s spending directly decreases mono-

tonically in d1. An interesting question is how well the model performs in pre-

dicting candidate spending in a campaign. For a comparison we show in the

left panel of Figure 1.3 the vote-on-spending effects estimated by Erikson and

Palfrey (2000). Although our model is relatively simple it predicts the spending

profile surprisingly well. At d1 = 0, which represents a predicted incumbent

vote of 50 percent, both candidates choose identical spending. The incumbent’s

spending increases first and decreases after peaking at around 55 percent un-

til the predicted incumbent vote reaches some 80-85 percent. Then, somewhat

surprisingly, spending goes up again. The challenger’s spending almost mono-

tonically decreases as the incumbents predicted vote increases over 50 percent

and also remains below the incumbent’s spending.

1.6 Extending the basic model

In this section we discuss two extensions to the basic model. We start off show-

ing that our results from the previous section are qualitatively robust to more

general functional specifications. Next we depart from the assumption of pure

maximization of the probability to win and consider candidates that are also

interested in the popular vote per se, as it is the case in proportional represen-

tation (PR) electoral systems.

1.6.1 Functional Assumptions

So far we assumed the distribution of shocks, respectively the noise, is normally

distributed. Also, costs were quadratic. In this section we show that similar

results will emerge if we relax these assumptions.

Consider the distribution of ǫ first. While, as we argued above, the nor-

mality assumption can be justified – for example by invoking the central limit

1.6. EXTENDING THE BASIC MODEL 31

50 55 60 65 70 75 80

0.0

0.2

0.4

0.6

0.8

Figure 1.3: Total spending per candidate, depending on popularity. The left panelshows Figure 3 from Erikson and Palfrey (2000), which shows candidatespending in the U.S. The right panel shows the predictions of our model(expected total spending) when there is a unique equilibrium under FPTP.Our predictions resemble the data pretty well. When popularity is around50 percent for each candidate, spending is more or less equal. If weincrease the popularity of one candidate this candidate’s total effort in-creases first and decreases then again. For the less popular candidatespending monotonically declines. Generally, the more popular candidatespends more than his opponent.

theorem – it is nevertheless of interest to explore the robustness of our results in

more detail. Hence we relax this assumption and assume the following instead:

Assumption 1.6.1. ǫ1 and ǫ2 are independently distributed on S1 ⊆ R and S2 ⊆ R

with densities g1(ǫ) and g2(ǫ). gt(ǫ) is differentiable on S t, symmetric around zero

and g′(|ǫ|) ≤ 0.

By Assumption 1.6.1 each shock has a quasi-concave density, implying that

a large shock is less likely to occur than a small shock. Equivalently, it is less

likely to have a series of random events only in support of one of the two can-

didates than a series that is relatively balanced, i.e. each candidate gains from

some events and loses from others. The identical logic applies to stochastic

productivity of effort. A specific distribution fulfilling the assumption is for

example the uniform distribution or a symmetrically truncated normal distri-

bution. At the end of this section we also explore what happens if the variance

of ǫ converges to zero and hence there is no noise in the game.

Now turn to the cost of effort function. Instead of quadratic costs we as-

sume the following:

Assumption 1.6.2. Spending effort x implies costs C(x), where C(0) = 0, C′(0) = 0,

C′> 0 for all x > 0, C′′

> 0 and |C′′′| is finite.


This cost function is strictly convex and hence may reflect both increasing

marginal costs of funding and other campaign efforts, as well as diminishing

marginal returns of effort. Apart from this convex costs are also necessary to

guarantee the existence of a pure strategy equilibrium. That is also the purpose

of the bounded third derivative.

The protocol of the campaign game is as before. If the variance of ǫ is

positive we find the following:

Proposition 1.6.1. Absent a poll both candidates choose identical effort in stage t =

1,2. If there is a poll, both candidates choose identical effort in t = 2, but there may

only be a stable symmetric equilibrium in t = 1 if d1 = 0 and if there exists a unique

equilibrium (for all d1). If the equilibrium is unique, in all campaigns with asymmetric

candidates the leader takes a tough position and spends more in equilibrium. If there

are multiple equilibria, in a close race either candidate may spend more in equilibrium,

while if the race is sufficiently lopsided the leader spends more. If there is momentum,

polls decrease expected aggregate spending.


Now let the variance of ǫ converge to zero and hence the noisiness disap-

pears. The campaign is then a perfectly discriminating contest or – using the

language of auction theory – an All-pay auction (see Konrad (2009)). This form

of contests has been studied extensively with applications ranging from lobby-

ing contest (Hillman and Riley, 1989; Baye et al., 1993) to political campaigns

(Meirowitz, 2008), revolution or war (Polborn, 2006) and internal labor markets

(Nalebuff and Stiglitz, 1983; Moldovanu and Sela, 2001). Siegel (2009) studies

general equilibrium characteristics of All-pay auctions. An All-pay auction de-

scribes a world in which effective effort is completely decisive for the contest

outcome. Hence, we may interpret it as a limiting case in which contestants

have perfect control over their effective effort. Clearly, when E[(ǫ)2] = 0, in our

basic model we would have σ = 0 and hence ρ = 0, too. In this case there we

found stage 2 behavior is identical and in stage 1 there may be multiple equilib-

ria in close races. An equilibrium in which the more popular candidate invests

more always exists. However, we had the problem that for ρ too small the

second order conditions did not hold and hence our analysis does not directly

extend to this case. However, the following proposition shows that qualitatively

all our results are preserved:

Proposition 1.6.2. Absent a poll both candidates choose identical effort in stage t = 1,2

also if noise vanishes. If there is a poll, in stage 2 there is a unique equilibrium. If


|d2| < C−1(1) this equilibrium is in mixed strategies and both candidates choose in

expectation identical effort. Otherwise, the campaign is decided already and both candi-

dates spend zero equilibrium. In stage 1 there always exists a pure strategy equilibrium

in which the leading candidate spends positive effort while the trailing candidate stays

passive, if costs are quadratic. In close races also the opposite – an equilibrium in which

the trailing candidate acts tough and spends a positive amount, while the leader remains

passive – exists. Hence, there are multiple pure strategy equilibria if the campaign is

close in t = 1. Expected aggregate spending is weakly smaller when there are polls.


The proposition shows that our findings for ρ < ρ are robust, too. Absent a

poll campaigning has no influence on the expected outcome in equilibrium. This

changes once there is informational feedback before stage 2, and campaigning

again matters in equilibrium. There may be multiple equilibria in t = 1 if there

is no clear leader. In general, it is likely to see the leader spending more in stage

1 and so the leader is more likely to improve his position than his contender.

However, in close campaigns both candidates may adopt a tough stance. In

stage 2 we see both candidates choosing the same expected effort.

1.6.2 Proportional Representation

So far we assumed throughout the analysis that candidates compete in a FPTP

system and hence care only about the probability to come out ahead, not the

actual share of votes won. This is so because the winner of the campaign gets

all rents as long as he gets a majority; candidates’ campaign in a winner-take-all

competition. Having a relatively large share in t = 1 is a good thing, but only

because this makes it more likely to win a majority. The additional voters in

excess of the 50 percent give no additional benefit. But this is actually not a very

good description of political competition in many democratic countries, for

example when the political system is one of proportional representation (PR)

instead of FPTP. Looking at the number of countries using different electoral

systems, with 70 countries PR was the most popular system worldwide in 2005

(see Reynolds et al. (2005)). PR systems (and its close derivatives) can be found

in most European countries, and are also dominant in Latin America and Africa.

Examples include Brazil, Indonesia, or South Africa. Under PR campaigning

incentives differ from FPTP, because at least a share of the benefits is allocated

according to vote shares in the popular vote. Hence, while winning a majority

is under PR also valuable in its own right, this is not the only goal a candidate or

party pursues. One reason is that under PR the composition of the parliament is


proportional to the popular vote, and hence a party’s influence increases in the

gained vote share. Another reason is that with only a marginal majority it often

is relatively hard to pass laws and bills, since then all members of government

would have to tie the party line, which is often not the case. In many cases a

government with a slight majority only turned out to be not able to effectively

shape and pass its own policies. Thus, with a comfortable majority it is easier

to govern and lead a country.

Assume a candidate or party values winning the majority by λ ∈ [0,1]. We

can interpret this as a plurality premium as in Iaryczower and Mattozzi (2011).

In addition, candidates get utility si(1 − λ) from gaining a share of si of the

total popular vote. (1 − λ) measures the relative importance of the vote share.

For example, in many countries political parties receive financial subsidies in

proportion to their vote share. Increasing those subsidies would imply a higher

(1 − λ). Similarly, the influence of a normal member of parliament that is not

part of the government may determine (1 − λ), too. The benefit of candidate i

in the election is hence

bi =

{λ + si(1 − λ) if si >

12 ,

si(1 − λ) else.

As before, candidates maximize their expected benefit subject to costs of cam-

paigning effort. We assume the shocks and costs have the same shape as in

Section 1.3.

Now let us analyze the effect of polls, and hence we start with a situa-

tion without informational feedback. The protocol of the campaign game is as

above, both candidates spend effort in stage 1 and 2 and then the election takes

place. As under FPTP, absent a poll the information a candidate holds in stages

1 and 2 is identical. Moreover, in any stage, both candidates have the same

marginal incentives, because both the marginal increase in probability and the

marginal increase in the vote share must be identical for both. Hence, absent a

poll we find results that are qualitatively the same as they were under FPTP:

Proposition 1.6.3. The equilibrium is unique. Both candidates choose in each stage

identical effort, and efforts are also identical across stages. Hence, also under PR,

absent a poll the ex-ante expected popular vote of a candidate in t = 3 is identical to his

popularity in t = 1.


This shows that the result from before, that campaigning does not matter in

equilibrium for the election outcome, remains valid also in PR. The intuition for


this result is also straightforward and similar to the one under FPTP. Looking

at the plurality premium, as before, the marginal increase in the probability

to win is identical for both at any effort profile, and hence, so are spending

incentives. Looking at the other part of interest, the vote share, the exactly

same is also true: the marginal vote share gained from spending is identical for

both. Consequently, efforts must be identical in equilibrium.

Now we introduce polls and look at stage 2 first. As in a campaign without

a poll and following the identical intuition as under FPTP, the equilibrium must

again be unique and symmetric.

Proposition 1.6.4. Candidates choose identical efforts in stage 2 and the equilibrium

is unique.


Given the analysis to far this result is not really surprising. Also, the im-

portance of the result does not lie in the symmetry of the stage 2 equilibrium,

but in the comparative statics with respect to d2 of this equilibrium. Because,

as we have seen in the previous section, these comparative statics determine

incentives in the first stage decisively.

There are two different effects at work that may lead to differential incen-

tives: the effect of a marginal change in effort on the winner-take-all part and

the effect on the vote share part. We already know how the first effect looks

like: efforts are decreasing in |d2| and thus there always exist momentum equi-

libria. Hence, turn to the latter effect, caused by the proportionality of the

obtained rent and the vote share. And here we are immediately in trouble.

Without assuming a particular shape of the distribution Pt(dtι) – the distribu-

tion of bliss points – it is not possible to make any general statement about the

effect of varying d2. To see this, consider the logic of campaigning for vote

shares in more detail. The marginal benefit of effort is identical to the marginal

vote share gained, and this vote share is equal to the density of the bliss point

distribution. But so far we did not specify this distribution at all, what was

not a problem under FPTP. Because there all we needed to know was the ’dis-

tance’ to the median voter d1, and not how the exact distribution of bliss points

looks like. Also, it is not obvious how we should expect the distribution of

bliss points to look like. For example, a centrist distribution would probably

have a quasi-concave shape where the bulk of voters are centered around the

median and the frequency of bliss points decreases when we move away from

the median. For example, we could use a normal distribution with zero mean


to model such a case. But it is not in any way obvious that the distribution of

voters is centrist. In many countries it seems to be the case that the electorate

is rather polarized with respect to their tastes, implying when we move away

from the median the frequency (density) of bliss points is likely to increase

first, before it must at some point decrease again. This would be a reasonable

distributional assumption if the fraction of partisan voters is relatively large. It

is also not obvious that preferences are symmetrically distributed. And all this

has important consequences for comparative statics. In particular, it may be the

case that efforts increase in |d2| over a given range or that comparative statics

are not at all related to |d2| in some regions. But these comparative statics are

important for stage 1 behavior. Hence, depending on the distribution of bliss

points, the results under PR might differ significantly from those under FPTP.

To show that this is true we now look more closely at how the shape of the

distribution of bliss points determines stage 2 behavior and then show by way

of an example how this alters stage 1 incentives. To be able to do all this we

make the following assumption:

Assumption 1.6.3. The distribution of bliss points has density

pt(dt) =φ(−µp+dt

σp ) + φ(µp+dt

σp )

2,

where µp ≥ 0 and σp> 0.

pt(dt) is a mixture distribution of two normals with identical variances σp

and means −µp and µp. Hence, it is symmetric around dt = 0. We restrict

ourselves to symmetric distributions for analytical purposes and because it is

sufficient to make the general point that incentives in the different electoral

system are often different. It is easy to generalize the model to more flexible

distributions, but the additional insights gained from such a generalization are

only marginal.10

We now look at the comparative statics with respect to d2 of the equilibrium

in stage 2:

Lemma 1.6.1. If µp>

√σ2 + (σp)2, candidates’ stage 2 spending increases in |d2|

when |d2| and λ are small. An upper bound for λ is λ ≡ 2

2+e32≈ 0.308. Otherwise

spending decreases in |d2|.10For example, if we look at mixture distributions of two normals with different means and variances,

the resulting distribution may be asymmetric and skewed to the left or right. As a consequence,whether or not there is momentum can be completely independent of |d2| over a given range.



If the condition in the lemma is fulfilled, the expected stage 3 distribution of

bliss points is bimodal, and hence we could say the electorate is polarized in

expectation. Note that µp> σp is sufficient for pt to be bimodal. However, if it

is only ’marginally bimodal’, taking the expectation makes it unimodal again.

Hence, the distribution needs to be sufficiently bimodal. If λ becomes larger

the whole expected benefit distribution again becomes more unimodal, since

the plurality premium becomes more and more important. Hence, λ needs to

be sufficiently small and µp needs to be sufficiently large.

If we look now at stage 1 behavior, a similar argument as before demands

that the expected stage 3 distribution of bliss point be bimodal. This implies

the conditions from stage 2 are not yet sufficient to generate any qualitatively

different behavior in stage 1. The following proposition proves the main result

in this section:

Proposition 1.6.5. If λ is small and µp is large, there are anti-momentum equilibria

in stage 1 in close races.


What is the difference now to the anti-momentum equilibrium we have

seen under FPTP? First, the mechanism is obviously a different one, since the

result is driven by the polarized electorate. Second, unlike before, the result

does not relate to the expensiveness of influence in the campaign. The result

holds whether or not it is costly to influence the election outcome and the

equilibrium might be unique as well. But why does the result only hold in close

races? To see this note that ’close’ in Proposition 1.6.5 is not to be interpreted as

an absolute measure of tightness of competition, but relative to the competitive

environment (the parameters of the game). Hence, it might well be the case

that a candidate currently gathers two-thirds of all voters, but there is still

anti-momentum. The general point is that once the campaign is extremely

lopsided at the outset, the trailing candidate must in expectation increase future

intensity of competition. This is seen most easily by looking at a candidate who

currently does not have any voter support. His campaign spending in stage 1

decreases the gap and thereby increases his expected share in stage 2, what in

turn increases his expected costs. In our example there is never a candidate

with a current share of zero because pt has full support, but the intuition is still

right. In Figure 1.4 you can find an example where there is anti-momentum in

close races.


-4 -2 2 4

0.05

0.10

0.15

0.20

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.105

0.110

0.115

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.10

0.11

0.12

0.13

0.14

dt

pt(dt)

d1 d2

x2ix1

i

Figure 1.4: Distribution of voters’ bliss points and equilibrium efforts in stage 1 and2 of candidate A (black and solid line) and 2 (brown and dashed line),depending on dt, t = 1,2. The parameter values are mp =

√6, σp = σ = c =

1, λ = 1/10. The equilibrium is unique.

While we looked at symmetric distributions of bliss points in this section,

the results are easily generalized to more general distributions. However, by

looking at skewed distributions an additional effect has to be considered: rela-

tive stage 1 incentives may over some range be independent of |d1|. In partic-

ular, there might be a neighborhood of d1 ∈ (−o,o), o > 0, in which the same

candidate always has an incentive to spend more on campaigning than his op-

ponent, no matter if he is leading or trailing. For example, if pt is unimodal but

skewed, candidates spending incentives in stage will always differ. In particu-

lar, if the distribution is negatively skewed, candidate B has over a large range

of parameters stronger incentives to invest than candidate A, independent of d1,

and similar effects benefiting B are true if the distribution is positively skewed.

1.7 Conclusion

In this paper we explore the effect public opinion polls have on candidates’

incentives to campaign. We found that in FPTP systems there always exists

an equilibrium in which the leading candidate gains momentum, but in close

campaigns also the trailing candidate may adopt a tough stance and outspend

the leader. Hence, anti-momentum may also be an equilibrium. For this to be

the case it needs to hold that having an influence on the election outcome in the

campaign is relatively cheap. We have shown that those results are relatively

robust to functional form assumptions or the extent to which random events

influence the election outcome. Quite generally, polls lead to a decrease in

expected spending in the campaign and so decrease the expected intensity of

the electoral competition. However, these results are not robust if we look at PR

systems instead. There the distribution of voters’ candidate ranking determines

relative campaigning incentives. If the electorate is relatively polarized, in close

1.7. CONCLUSION 39

campaigns the trailing candidate has stronger incentives to campaign. If the

electorate is relatively moderate, for example if the density of bliss points is

normal, than incentives are similar to those under FPTP.

An implication of the model is that if there is an incumbency advantage at

the outset of the campaign, polls are likely to foster this advantage and hence

decrease the rate of turnover in political offices in countries with FPTP electoral

systems. Indeed, the spending profile predicted by our model fits well to the

estimated profiles in Erikson and Palfrey (2000), who show that incumbents

with an early popularity edge tend to improve their chances by spending more

than their opponents in the competition.

For future research it is interesting to empirically validate the results pro-

posed by the theory. We are currently starting this project, however, do not

have results yet. Another interesting direction for future research is to study

how polls influence not only campaigning incentives, but also candidates’ plat-

form choices.


Appendix

Proof of Proposition 1.3.1

Proof. We prove the proposition assuming the more general model discussed

in Section 1.6.1, which as a special case nests the model discussed in Section

1.3.1. If there is no poll, candidates do not observe the realization of ǫ1 and

hence their information does not change between periods 1 and 2. Hence, they

maximize

max(x1

A,x2A)∈R2

+

1d−d

∫ dd G(d1 + x1

A + x2A − x1

B − x2B)d d1 − C(x1

A)− C(x2A),

max(x1

B,x2B)∈R

2+

1 − 1d−d

∫ dd G(d1 + x1

A + x2A − x1

B − x2B)d d1 − C(x1

B)− C(x2B),

where G is the cdf of the convolution g1 ∗ g2. The system of first order condi-

tions is1

d − d

∫ d

dg(d1 + x1

A + x2A − x1

B − x2B)d d1 − C′(xt

i ) = 0,

i = A, B and t = 1,2. It is easily observed that in any interior pure strategy

equilibrium it must hold that x1A = x2

A = x1B = x2

B = x∗. In particular,

x∗ = C′−1

(1

d − d

∫ d

dg(d1)d d1

).

The second order conditions are

1

d − d

∫ d

dg′(d1)d d1 − C′′

(C′−1

(1

d − d

∫ d

dg(d1)d d1

))< 0

for A and

− 1

d − d

∫ d

dg′(d1)d d1 − C′′

(C′−1

(1

d − d

∫ d

dg(d1)d d1

))< 0

for B. If C′′(.) is sufficiently large relative to g′(.) the second order conditions

hold for all d2 and hence this is an equilibrium. We assume this to be the case.

Note that if the variance of g1 and g2 vanishes, the second order conditions

always hold, as we show later in the proof of Proposition 1.6.2. If the variance

becomes very large, the slope of g1 and g2 decreases and hence the second

order conditions also always hold.

For the parameterized model in Section 1.3.1 the second order conditions

APPENDIX 41

hold if

c > max

−

φ(

d√2σ

)− φ

(d√2σ

)

d − d,φ(

d√2σ

)− φ

(d√2σ

)

d − d

.

For example, if σ = 1, d = 2, and d = −1, it must hold that c > 0.03864. For the

remainder of the paper we assume C′′ to be sufficiently large to guarantee the

second order conditions. Hence, the proof is complete and this also proves the

first part of Proposition 1.6.1.

Proofs of Proposition 1.3.2 and 1.6.1

Proof. We again proof the proposition directly in the general version of Propo-

sition 1.6.1.

First look at stage 2 when there is a poll. In this case both candidates know

the median’s exact position in stage 2, d2. Candidates maximize

maxx2

A∈R+

G2(d2 + x2A − x2

B)− C(x2A),

maxx2

B∈R+

1 − G2(d2 + x2A − x2

B)− C(x2B).

First order conditions read

g2(d2 + x2A − x2

B)− C′(x2i ) = 0,

where i = A, B, and the assumption on C′ guarantee that there exists an x

fulfilling these. It is immediately observed that – as without a poll – both

candidates choose identical effort, x2A = x2

B. In particular,

x2A = x2

B = x∗∗(d2) = C′−1(g2(d2)).

The second order condition for A is

g2′(d2)− C′′(C′−1(g2(d2))) < 0.

Since it must hold for all d2, if this holds for A it also holds for B. We assume

this holds for all d2. The problem each candidate faces is then continuous and

concave and hence a pure strategy equilibrium exists. For example, if σ = 1 it

must hold that c > 0.242. For the remainder of the paper we assume C′′ to be

sufficiently large to guarantee the second order conditions are fulfilled.

This proves Proposition 1.3.2 and the first part of Proposition 1.6.1.


Proofs of Propositions 1.3.3, 1.3.4, and 1.6.1

Proof. Expected utility of A and B, conditional on being in state d2 in the second

stage, is

EU∗A(d

2) = G2(d2) − C(

C′−1(

g2(d2)))

,

EU∗B(d

2) = (1 − G2(d2)) − C(

C′−1(

g2(d2)))

.

Note that d2 = d1 + x1A − x1

B − ǫ1. Then we can write the optimization problemof the candidates as:

maxx1

A≥0

∫

S1G2(d1 + x1

A − x1B − ǫ1)− C

(C′−1

(g2(d1 + x1

A − x1B − ǫ1)

))g1(ǫ1)dǫ1 − C(x1

A),

maxx1

B≥0

∫

S1(1 − G2(d1 + x1

A − x1B − ǫ1))− C

(C′−1

(g2(d1 + x1

A − x1B − ǫ1)

))g1(ǫ1)dǫ1 − C(x1

B).

Taking the derivative with respect to the respective own strategy yields

∫

S1

(g2(·) −

[C′(C′−1(·))∂C′−1(·)

∂g2(·)∂g2(·)∂x1

A

])g1(ǫ1)dǫ1 − C′(x1

A),

∫

S1

(g2(·) −

[C′(C′−1(·))∂C′−1(·)

∂g2(·)∂g2(·)

∂x1B

])g1(ǫ1)dǫ1 − C′(x1

B).

Using C′(C′−1(z)) = z and ∂g2(·)/∂x1A = ∂g2(·)/∂d1 =−∂g2(·)/∂x1

B these equa-

tions simplify to

∫

S1

(g2(·) −

[g2(·)∂C′−1(·)

∂g2(·)∂g2(·)

∂d1

])g1(ǫ1)dǫ1 − C′(x1

A)

∫

S1

(g2(·) +

[g2(·)∂C′−1(·)

∂g2(·)∂g2(·)

∂d1

])g1(ǫ1)dǫ1 − C′(x1

B)

Using ∂C′−1(g2(·))/∂g2(·) = 1/(C′′(C′−1(g2(·))) and ∂g2(·)/∂d1 = g2′(·), and

letting d1 + x1A − x1

B =: κ we get

∫

S1

(g2(κ − ǫ1) −

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1 − C′(x1

A), (1.2)

∫

S1

(g2(κ − ǫ1) +

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1 − C′(x1

B). (1.3)

Before we analyze the equilibrium in detail we now need to show that an

equilibrium exists.

APPENDIX 43

Lemma 1.7.1. If C′′(x) > Γ(g1, g2) (defined below) a pure strategy equilibrium exists

for all d1.

Proof. The second derivative of A’s payoff function is

∫

S1

[g2 ′(κ − ǫ1)−

(g2 ′(κ − ǫ1)

C′′(x∗∗(·))

)2(C′′(x∗∗(·))− g2 ′(κ − ǫ1)C′′′(x∗∗(·))

C′′(x∗∗(·))

)− g2(κ − ǫ1)g2 ′′(κ − ǫ1)

C′′(x∗∗(·))

]dG(ǫ1)− C′′(x1

A).

To show strict concavity of the payoff function we need to show that this is

strictly negative for all κ and x1A. Consider

(g2′(κ − ǫ1)

C′′(x∗∗(ǫ1))

)2(C′′(x∗∗(κ − ǫ1))− g2′(κ − ǫ1)C′′′(x∗∗(ǫ1))

C′′(x∗∗(κ − ǫ1))

). (1.4)

Let the γ1 > 0 be the smallest value of C′′ for which

γ21 > g2′(κ − ǫ1)C′′′(x∗∗(κ − ǫ1))

for all κ − ǫ1. If the infimum of C′′ is at least γ1 (1.4) is non-negative. Since C′′′

is bounded by Assumption 1.6.2 and because it follows from differentiability

and quasi-concavity that g2′(κ − ǫ1) is bounded, too, there exists finite γ1 for

which this is the case. Hence assume (1.4) is zero. If this is the case and we can

show that the second derivative is negative, this is even more so the case when

(1.4) is positive. Therefore we are left with

∫

S1

[g2′(κ − ǫ1)− g2(κ − ǫ1)g2′′(κ − ǫ1)

C′′(x∗∗(·))

]dG(ǫ1)− C′′(x1

A). (1.5)

From strict quasi-concavity of g2 it follows that the expectation is bounded.

Hence, there exists γ2 > 0 such that if C′′(x) > γ2 for all x, (1.5) is strictly neg-

ative. It follows that if C′′(x) > Γ(g1, g2) := max{γ1,γ2} the second derivative

is strictly negative and hence the problem is strictly concave. Thus, assuming

this holds, and since payoffs are continuous in x1A and x2

B, existence of a pure

strategy Nash equilibrium follows from Theorem 1.2 in Fudenberg and Tirole

(1991), which is due to Debreu (1952), Fan (1952), and Glicksberg (1952).

Now let us go more into the details of the first order conditions to deter-

mine the properties an equilibrium must have. For this purpose we need the

following lemma:

Lemma 1.7.2. Let

ξ(κ) := Eǫ1

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

].


Then ξ(0) = 0, ξ(+) > 0 and ξ(−) < 0.

Proof. Define

ω(κ) :=g2(κ − ǫ1)g2(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1))),

which is the function we want to take the expectation of. Now remember

that by adding an arbitrary constant a to the argument of a function, the

graph of the function is shifted horizontally by −a. Therefore, if g2(ǫ) is

axis-symmetric across zero, g2(κ − ǫ) is axis-symmetric across −κ. As a conse-

quence, the same holds true also for functions of this function, like C′−1(g2(κ −ǫ)), C′′(C′−1(g2(κ − ǫ))), and

g2(κ−ǫ1)C′′(C′−1(g2(κ−ǫ)))

. Because g2(ǫ) is axis-symmetric

across zero, its derivative is point-symmetric across zero. By a similar argument

as above it also holds that g2′(κ − ǫ) is point-symmetric across −κ. Therefore,

we have that also the product of an axis-symmetric function across −κ and a

point-symmetric function across this point, in our case this function is ω(κ), is

point-symmetric across −κ.

We first show that ξ(0) = 0. Let f (z) be a function which is axis-symmetric

across zero and let h(z) be another function which is point-symmetric across

zero. Both functions share the same support K. Then, if we want to find∫K f (z)h(z)dz, we can split the integral into two parts:

∫

Kf (z)h(z)dz =

∫

{z∈K:z≤0}f (z)h(z)dz +

∫

{z∈K:z>0}f (z)h(z)dz.

Because of the symmetry properties f (z) = f (−z) and h(z) = −h(−z) we can

rewrite the second term as∫

{z∈K:z>0}f (z)h(z)dz = −

∫

{z∈K:z≤0}f (z)h(z)dz.

Using this substitution it is easily verified that

∫

Kf (z)h(z)dz =

∫

{z∈K:z≤0}f (z)h(z)dz −

∫

{z∈K:z≤0}f (z)h(z)dz = 0.

Now let f (z) = ω(ǫ) and h(z) = g2(ǫ) and observe that the integral we want to

calculate is the expectation of ω(ǫ) and therefore equal to ξ(0) to complete this

part of the proof.

Next, consider κ > 0. ω is shifted to the left and is point-symmetric across

−κ. Now note two things: First, to the left of −κ the values of ω are positive,

to the rights the values are negative. Second, for any shock φ leading to a real-

ization ω(κ + φ) = m there exists exactly one other shock φ′, which leads to a

APPENDIX 45

realization ω(κ + φ′) = −m and is an inversion of the former point at (−κ,0).

Moreover, this holds true for any point in the graph of ω. Accordingly, we

can define the whole graph as pairs of inversion points. Now observe, that the

probability of an outcome −m is always weakly larger than the probability of

outcome m for all m ≥ 0. To see this note that a shock generating m must be

of size −κ − c, while the shock generating −m must be −κ + c, for some con-

stant c ≥ 0. But then the shock φ that produces outcome m is in absolute value

weakly larger than φ′. As a consequence, because shocks are distributed sym-

metrically around zero, the density of φ′ is weakly larger than the density of φ,

g2(φ′) ≥ g2(φ). Note that this must hold for all m, φ and φ′, and accordingly

the expectation of ω must be negative. From a similar argument it follows that

the converse must hold if we assume κ < 0. Hence the proof is complete.

Knowing now that a pure strategy equilibrium exists we focus on interior

equilibria henceforth. In Section 1.6.1 we also discuss the case when noise van-

ishes and the contest takes the form of a fully discriminating All-pay auction.

In this case there are only corner equilibria and the results are qualitatively

identical.

From (1.11) and (1.3) it follows that the first order conditions in an interior

equilibrium are

∫

S1

(g2(κ − ǫ1) −

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1 − C′(x1

A) = 0

and

∫

S1

(g2(κ − ǫ1) +

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1 − C′(x1

B) = 0.

Simple manipulations reveal that equivalently the following must hold:

x1A = C′−1

(∫

S1

(g2(κ − ǫ1) −

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1

)

x1B = C′−1

(∫

S1

(g2(κ − ǫ1) +

[g2(κ − ǫ1)g2′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1

).

Using κ = ∆1 + d1, it follows that in equilibrium it must hold that


∆1 = Σ(κ)

:= C′−1(∫

S1

(g2(κ − ǫ1) −

[g2(κ − ǫ1)g2 ′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1

)(1.6)

− C′−1(∫

S1

(g2(κ − ǫ1) +

[g2(κ − ǫ1)g2 ′(κ − ǫ1)

C′′(C′−1(g2(κ − ǫ1)))

])g1(ǫ1)dǫ1

).

The shape of this function is now important to determine equilibrium behavior.

We now establish a few lemmata that help us to characterize equilibria.

Lemma 1.7.3. Sign[Σ(∆1 + d1)

]= Sign[∆1 + d1]. Moreover, Σ(∆1 + d1) is contin-

uous, bounded, point symmetric at −d1 in ∆1, and lim|∆1|→∞ Σ(∆1 + d1) = 0.

Proof. From C′′> 0 it follows that the inverse C′−1

is increasing. This together

with Lemma 1.7.2 directly implies Σ(0) = 0, Σ(+) = (+), and Σ(−) = (−). For

the symmetry properties look at d1 = 0 first. Then we have that

C′−1(∫

S1

(g2(∆1 − ǫ1) −

[g2(∆1 − ǫ1)g2′(∆1 − ǫ1)

C′′(C′−1(g2(∆1 − ǫ1)))

])g1(ǫ1)dǫ1

)

and

C′−1(∫

S1

(g2(∆1 − ǫ1) +

[g2(∆1 − ǫ1)g2′(∆1 − ǫ1)

C′′(C′−1(g2(∆1 − ǫ1)))

])g1(ǫ1)dǫ1

)

are mirror images of each other (in ∆1) with the reflection axis being the vertical

through zero. This follows from the first argument of C′−1being axis symmet-

ric at zero and the second being point symmetric at zero. Hence, the difference

must be point symmetric at zero. Now note that by adding an arbitrary con-

stant – for example d1 – to the argument of a function, the function is shifted

horizontally by −d1. Hence, Σ must be point symmetric at −d1. Continuity

and boundedness follow directly from all terms and C′−1being continuous

and bounded. That the limit vanishes follows from lim|x|→∞ g2(x)g2′(x) = 0,

which follows from quasi-concavity of g2. This proves the lemma.

Lemma 1.7.4. Assume d1 6= 0 and let the effort of the more popular candidate in stage

1 be xL and the effort of his opponent be xT. In any equilibrium we have that

xT /∈ (xL, xL + |d1|).

APPENDIX 47

Proof. To see this look at the first order conditions. Without loss of generality

assume d1> 0 and also assume x1

B ∈ (x1A, x1

A + |d1|). This implies κ > 0 and thus,

by Lemma 1.7.2, ξ(κ) < 0. Hence, B’s efforts are strategic complements, and

A’s strategic substitutes. If A’s first order condition holds, B’s must be strictly

negative and he hence would like to decrease effort. If B’s first order condition

holds, A’s must be strictly positive and he would like to increase effort. Hence,

this cannot be an equilibrium.

Lemma 1.7.5. There exists d ≥ 0 such that if |d1| > d the equilibrium in stage 1 is

unique and xL ≥ xT.

Proof. This follows from Lemma 1.7.4 and the fact, that any effort greater than

x := C−1(1) is strictly dominated. Thus, when |d1| becomes larger and larger,

outspending the leading candidate becomes too expensive. The inequality is

weak because we did not assume ǫ2 has full support and hence the race might

be decided if |d1| is sufficiently large. If we assume ǫ2 has full support – S2 = R

– the inequality is strict because the leading candidate always spends positive

effort.

The derivative of Σ with respect to ∆1 is

Eǫ1

[g2′(·)

]− Eǫ1

[(g2′(·))2

C′′(w)+ g2(·)g2′′(·)

C′′(w)− g2(·)(g2 ′(·))2C′′′(w)

(C′′(w))3

]

C′′(

C′−1(

Eǫ1

[(g2(·)−

[g2(·)g2′(·)

C′′(C′−1(g2(·)))

])])) (1.7)

−Eǫ1

[g2′(·)

]+ Eǫ1

[(g2′(·))2

C′′(w)+

g2(·)g2′′(·)C′′(w)

− g2(·)(g2 ′(·))2C′′′(w)(C′′(w))3

]

C′′(

C′−1(

Eǫ1

[(g2(·) +

[g2(·)g2′(·)

C′′(w)

])])) (1.8)

where w = C′−1(g(.)). The proposition we want to prove states that in close

games there might be both equilibria in which the leading candidate spends

more and the some in which the trailing candidate spends more, depending

on the distributions of ǫ1 and ǫ2 and the shape of the cost function. If one

candidate has a sufficiently large advantage, in all equilibria this candidate will

spend weakly more. If the equilibrium is unique for all d1, in this equilibrium

also the leading candidate will spend weakly more. A necessary and sufficient

condition for a unique equilibrium for all d1 is that Σ′(∆1) < 1 for all ∆1. To se

this note that if there are to be multiple equilibria, that is ∆1 = Σ(∆1) intersect

more than once, then Σ must be steeper than ∆1 somewhere. Starting from

an intersection of the two functions, if the slope is strictly smaller than 1 to

the right of the intersection Σ is strictly smaller than ∆1, and to the left strictly


larger, and hence there cannot be another equilibrium. If, however, there is

some region in which the slope is larger than 1, there exists d1 shifting Σ in a

way such that there are multiple equilibria. Hence, if and only if

Σ′(∆1) =Eǫ1

[g2′(·)

]− Eǫ1

[(g2′(·))2

C′′(w))+ g2(·)g2′′(·)

C′′(w))− g2(·)(g2′(·))2C′′′(w))

(C′′(w)))3

]

C′′(

C′−1(

Eǫ1

[(g2(·)−

[g2(·)g2 ′(·)

C′′(w)

])])) (1.9)

−Eǫ1

[g2′(·)

]+ Eǫ1

[(g2′(·))2

C′′(w))+ g2(·)g2′′(·)

C′′(w))− g2(·)(g2′(·))2C′′′(w))

(C′′(w)))3

]

C′′(

C′−1(

Eǫ1

[(g2(·) +

[g2(·)g2 ′(·)

C′′(C′−1(g2(·)))

])])) < 1

where w = C′−1(g(.)), for all ∆1, there is a unique equilibrium. It is easy to see

that the absolute value of the slope is strictly decreasing in C′′ (evaluated at

the equilibrium). Hence, if C′′(x) is sufficiently large for all x the equilibrium

is unique for all d1. For d1 = 0 the equilibrium is symmetric and ∆1 = 0. If

we now increase d1 we thereby shift Σ to the left, what because of the fact that

Σ(+) = (+) (see Lemma 1.7.3), implies the intersection is now where ∆1> 0.

This remains true for all d1> 0, and the opposite is similarly true for d1

< 0.

If

Σ′(∆1) > 1

for some ∆1 there are multiple equilibria for some d1. This follows from the

discussion above. If this is true for small |d1| it is likely that either candidate

might spend more in equilibrium. A sufficient condition for such equilibria is

that

Σ′(0) =−2Eǫ1

[(g2′(−ǫ1))2

C′′(C′−1(g(−ǫ1)))+ g2(−ǫ1)g2′′(−ǫ1)

C′′(C′−1(g(−ǫ1)))

]

C′′(

C′−1 (Eǫ1 [(g2(−ǫ1))])) > 1.

This derivative is strictly positive (follows from Lemma 1.7.2). If C′′ is suffi-

ciently small the derivative gets larger than 1. Then there is one equilibrium

∆1 = 0. Moreover, because Σ vanishes as |∆1| → ∞ (see Lemma 1.7.3) and Σ

is continuous, it follows from the intermediate value theorem that there are at

least two more equilibria, one with ∆1> 0 and one with ∆1

> 0. Because Σ

is point symmetric in ∆1 at zero (see Lemma 1.7.3) the asymmetric equilibria

are symmetric to each other. Note that for the analysis here hinges on the ap-

plicability of the first order conditions, and hence when C′′ gets too small the

second order conditions are violated. In the parameterized version below we

show that those equilibria can exist.

APPENDIX 49

From Lemma 1.7.5 it follows that in races with one dominant candidate this

candidate will always spend more effort in the campaign, or both spend zero.

Now go the specific example of a normal distribution with variance σ2 and

zero mean and a cost function C(x) = c2 x2. The marginal cost function is then

linear and the second derivative of the cost function is c. Hence, Σ simplifies

significantly:

Σ(∆1 + d1) = −Eǫ1

[2φ(∆1+d1−ǫ1

σ )φ′(∆1+d1−ǫ1

σ )

c2

]=

(∆1 + d1)e−(∆1+d1)2

3σ2

√27πc2σ4

.

Now look at the shape of Σ. Since d1 only shifts the function horizontally, we

assume d1 = 0 for now. The derivative with respect to ∆1 is

(3σ2 − 2(∆1)2)e−(∆1)2

3σ2

9√

3πc2σ6. (1.10)

This is strictly positive for |∆1| <√

32σ, negative for |∆1| >

√32σ, and zero for

|∆1| =√

32σ. The maximum of Σ = 1

3√

2eπc2σ3is attained at ∆1 + d1 =

√32σ, and

the minimum of Σ = − 13√

2eπc2σ3is attained at ∆1 = −

√32σ. Now look at the

second derivative of Σ,

2∆1(2∆1 − 9σ2))e− (∆1)2

3σ2

27√

3πc2σ8.

This is strictly negative on [−∞,− 3√2σ) ∪ (0, 3√

2σ), and hence the function is

strictly concave in this region, which also must include (and does) the max-

imum. Hence, Σ is strictly concave between zero and the maximum, and

decreases monotonically thereafter. Hence, if Σ′(∆1) ≤ 1, the slope is strictly

smaller than 1 (the slope of ∆1) for all ∆ ≥ 0. Hence, there exists a unique ∆1

fulfilling ∆1 = Σ(∆) not only for d1 = 0, but for all d1 ∈ R. Note that for given

σ the derivative becomes arbitrarily small at ∆1 = 0 if we increase c, and hence

there exists c such that for all c > c the slope is less than 1, and larger than 1

else. In Figure 1.5 we show two examples. If the derivative at D1 = 0 is larger

than 1 there exist multiple equilibria. Because Σ is strictly concave on [0,√

32σ],

and the derivative is zero at the end of this interval, it must become equal to 1

at some ∆1 ∈ [0,√

32σ]. Denote this by ∆. If we increase d1 now from zero we

thereby shift Σ to the left by d1. Hence, the two outer intersections of ∆1 and Σ

move to the right (∆1 increases), while the inner intersection moves to the left.


-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.0

-0.5

0.5

1.0

-1.5 -1.0 -0.5 0.5 1.0 1.5

-1.0

-0.5

0.5

1.0

∆1∆1

Figure 1.5: ∆1 (gray) and Σ(∆1 + d1) (brown) for σ = .5 and c ∈ {1.1,0.97}.

Hence, there are two intersections converging to each other, the ones where

∆1< 0. At d they converge to ∆, and hence there are only two equilibria left. If

we increase d1 now further this equilibrium vanishes and only one equilibrium

remains, in which d1> 0.

To complete the proof we now show by example that the second order

condition can hold in both stages when there are multiple equilibria in stage 1.

The second order condition in stage 2 for A is

φ′(d2

σ)− c < 0,

and this must hold for all d2. The second derivative has a maximum if d2 =−σ,

and if c > 1√2eπσ2

⇔ ρ >1√2eπ

this maximum is strictly negative and A’s second

order condition holds for all d2. Note that this is then also guarantees B’s

second order condition since candidates are symmetric in d2. Moreover, note

that 1√2eπ

< ρ = 133/4

√π

. Hence, if the second order condition holds marginally

in stage 2, there are multiple equilibria in stage 1.

Now look at stage 1. Assuming the second order condition in stage 2 holds

marginally, the second order condition in stage 1 is

− κe− κ2

4σ2

4√

πσ3− 1√

2eπσ2+

(3σ2 − 2κ2)e12− κ2

3σ2

9√

6πσ4,

where, as above, κ = ∆1 + d1. This seems to be strictly negative for all (σ,κ),

and is strictly negative in all our numerical calculation, see Figure 1.6.

What is left to prove Proposition 1.3.4 is to show that no stable symmetric

equilibrium exists when ρ < ρ. In the following we show that the slope of each

candidate’s best response function for d1 = 0, evaluated at the symmetric inter-

APPENDIX 51

section, is less than minus one. Being notationally a little sloppy we drop the

argument of g2(∆1 − ǫ1). Each candidate’s best response is implicitly defined

by

BR1A(x1

B) = max

{{x1

A : Eǫ1

[g2(.) −

[g2(.)g2′(.)

C′′(C′−1(g2(.)))

]]= C′(x1

A)},0

},

BR1B(x1

A) = max

{{x1

B : Eǫ1

[g2(.) +

[g2(.)g2′(.)

C′′(C′−1(g2(.)))

]]= C′(x1

B)},0

}.

Assuming a symmetric equilibrium with x1A = x1

B, it follows from Lemma 1.7.2

that the indirect effect is zero for both and hence

x1A = x1

B = C′−1(Eǫ1 g2(ǫ1))

is indeed an equilibrium. From the implicit function theorem it follows that the

slope of the best responses is

∂BR1i (x1

j )

∂x1j

=

Eǫ1

[(g2 ′(ǫ1)

C′′(x∗∗(ǫ1))

)2(

C′′(x∗∗(ǫ1))− g2(ǫ1)C′′′(x∗∗(ǫ1))

C′′(x∗∗(ǫ1))

)+

g2(ǫ1)g2′′(ǫ1)

C′′(x∗∗(ǫ1))

]

Eǫ1

[(g2 ′(ǫ1)

C′′(x∗∗(ǫ1))

)2 (C′′(x∗∗(ǫ1))− g2(ǫ1)C′′′(x∗∗(ǫ1))

C′′(x∗∗(ǫ1))

)+

g2(ǫ1)g2 ′′(ǫ1)

C′′(x∗∗(ǫ1))

]+ C′′(C′−1(Eǫ1 g2(ǫ1)))

.

If this is smaller than minus 1 the equilibrium is unstable. Note that the de-

nominator must be positive, because it is the negative of the second derivative

in equilibrium, and this has to be negative in equilibrium. This is the case

whenever

−2Eǫ1

[(g2 ′(ǫ1)

C′′(x∗∗(ǫ1))

)2(

C′′(x∗∗(ǫ1))− g2(ǫ1)C′′ ′(x∗∗(ǫ1))

C′′(x∗∗(ǫ1))

)+

g2(ǫ1)g2 ′′(ǫ1)

C′′(x∗∗(ǫ1))

]> C′′(C′−1

(Eǫ1 g2(ǫ1))).

Hence, the cost function must be sufficiently convex but not too convex. Using

the functional forms assumed in the main part of the paper, this expression

simplifies to

−2Eǫ1

[(φ′(

ǫ1

σ

))2

+ φ

(ǫ1

σ

)φ′′(

ǫ1

σ

)]> c2,

which yields

ρ := cσ2< (33/4

√π)−1 = ρ.

This is the condition for the existence of multiple equilibria in the game with

quadratic costs and normal shocks. Hence, whenever there are multiple equi-

libria in this game, there is no stable symmetric equilibrium. This completes

the proof.


0

1

2

3

-10-5

05

10

-0.2

-0.1

0.0

-4 -2 2 4

-6

-5

-4

-3

-2

-1

κ

κ

σ

Figure 1.6: Numerical calculations of A’s second derivative in stage 1, assuming sec-ond order conditions in stage 2 hold marginally, c > 1√

2eπσ2 . Right panel:

σ ∈ {1/4,1/2,1}.


Proof. After observing that the probability to increase the lead is a function

only of ∆1 and the distribution of ǫ2, the proposition follows immediately from

Lemma 1.7.3 and Equation 1.10 and the discussion thereafter.


Proof. From the proof of Proposition 1.3.1 we know that aggregate expenditures

without a poll are equal to

4

c

(1

d − d

∫ d

dg(d1)d d1

).

With a poll, second period expected aggregate expenditures are also symmetric

and equal to

2

c

1

d − d

∫ d

d

(∫

S1(g2(d2))g1(ǫ1)dǫ1

)d d1.

as the proof of Proposition 1.3.2 shows. In period 1 we unfortunately cannot

generally get a closed form solution for equilibrium efforts with poll. So we

need to take an indirect approach. Recall from the proof of Proposition 1.3.3

that the first-order conditions are given by

APPENDIX 53

∫

S1

(g2(·) −

[g2(·)∂C′−1(·)

∂g2(·)∂g2(·)

∂d1

])g1(ǫ1)dǫ1 − cx1

A = 0,

∫

S1

(g2(·) +

[g2(·)∂C′−1(·)

∂g2(·)∂g2(·)

∂d1

])g1(ǫ1)dǫ1 − cx1

B = 0.

After simple manipulations, it follows from this that aggregate effort in period

1 is2

c

∫

S1g2(d1 + ∆1 − ǫ1) g1(ǫ1)dǫ1,

and expected aggregate stage 1 spending is

1

d − d

∫ d

d

(2

c

∫

S1g2(d1 + ∆1 − ǫ1) g1(ǫ1)dǫ1

)d d1.

Thus, the difference between expected aggregate effort with and without poll

is equal to

4

c

1

d − d

∫ d

d

(∫

S1g2(κ − ǫ1) g1(ǫ1)dǫ1 − g(d1)

)d d1.

By definition of g as the convolution of g1 and g2 we can simplify the expres-

sion:4

c

1

d − d

∫ d

d

(g(d1 + ∆1)− g(d1)

)d d1.

Because g is strictly quasi-concave and symmetric around zero, and because in

a momentum equilibrium Sign[d1 ] = Sign[∆1 ], this difference is weakly nega-

tive whenever there is momentum. Hence, in this case polls decrease expected

aggregate spending.

Now consider a situation where there is anti-momentum. To prove that

polls also in this case decrease intensity of competition, it is sufficient to show

that in all anti-momentum equilibria ∆1> −2d1. Because if that is the case,

|d1 +∆1|> d1 as before and therefore spending must decrease, too. Since as we

decrease |d1| to zero ∆1 increases, it suffices to show that for the maximum |d1|that still admits multiple equilibria, d, the relation holds as we need it.

We prove that this is the case with the help of Figure 1.7. A marks the

anti-momentum equilibrium with the highest value of d1 allowing for such an

equilibrium. AB is the effort difference in this equilibrium, ∆1. CD corresponds

to d. Hence, we need to show that AB > 2CD. Note that CD = AB − BC. Hence,

AB > 2(AB − BC) ⇔ 2BC > AB ⇔ ABBC < 2. Now note that AB

BC is exactly the


D

CB

A

-0.4 -0.2 0.2 0.4

a

-0.4

-0.2

0.2

0.4

b

Figure 1.7: Appendix: Proof that polls decrease wastefulness of competition.

slope of line through A and C and we also know that the slope of the function

through both point is increasing from one as we approach C. Hence, for the

average slope to be smaller than two, it is sufficient that the function’s slope

in C is smaller than 2. It is straightforward to show that this is the case for all

ρ >1

33/4√

2π= ρ/

√2. Since the second order conditions are not guaranteed to

hold whenever ρ ≤ 1√2eπ

< ρ/√

2, we know that in all interior anti-momentum

equilibria ∆1> −2d1 when d1 = d. Whenever d1

< d this relation remains true.

As d1 decreases to zero, ∆1 increase and hence for all d1 admitting multiple

equilibria, ∆1> −2d1. This directly implies that the expected lead in stage 2

must increase, E[|d2|] > E[|d1|]. That is sufficient to complete the proof.


Proof. Also consider the situation without a poll first. In this situation the

candidates do not know the median’s exact ranking but have beliefs F(d1). For

candidate A the probability to win the election is

Pr[x1A + x2

A + d1 − x1B − x2

B > 0] = 1 − F(x1B + x2

B − x1A − x2

A),

and similarly for B we get F(x1B + x2

B − x1A − x2

A). Hence, we may write the

candidates objectives as

APPENDIX 55

max(x1A,x2

A)∈R2+

1 − F(x1B + x2

B − x1A − x2

A)− C(x1A)− C(x2

A),

max(x1B,x2

B)∈R2+

F(x1B + x2

B − x1A − x2

A)− C(x1B)− C(x2

B).

The corresponding first order condition for i in t reads

f (x1B + x2

B − x1A − x2

A)− C′(xti )

!= 0.

This is identical for all t = 1,2 and i = A, B. Hence,

xtA = C′−1( f (0)).

For this to be an equilibrium the second order conditions need to hold:

f ′(0)− C′′(C′−1( f (0))) ≤ 0.

If costs are sufficiently convex this inequality holds generally. For the assumed

uniform prior we get f ′(0) = 0 and hence the inequality always holds.

Now look at the case of a campaign with polls. Candidates have perfect

knowledge of the median’s ranking, and since there is no shock, there is no

exogenous noise left. Consider stage 2 first, and assume without loss of gener-

ality d2 ≥ 0. It is easily shown that there cannot be a pure strategy equilibrium

in this stage (see e.g. Nalebuff and Stiglitz (1983) or Hillman and Riley (1989)).

The stage game is similar to the game analyzed in Meirowitz (2008) with the

difference that we have strictly convex costs. Let x := C−1(1), the maximum ef-

fort that is not strictly dominated. Moreover, let Qi(x2i ) be candidate i’s mixed

strategy with support Si, i = A, B. The following proposition summarizes the

stage 2 equilibrium:

Proposition 1.7.1. Without loss of generality, let d2 ≥ 0. There is a unique equilibrium

in mixed strategies if d2< x with SA = [0, x − d2] and SB = {0} ∪ [d2, x]. Candidate

A randomizes according to

QA(x2A) =

0 if xA < 0,

C(d2 + xA) if xA ∈ [0, x − d2],

1 if xA > x − d2,

while B’s mixed strategy is given by


0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.2

0.4

0.6

0.8

1.0

x2Ax2

A

QA(x2A)QA(x

2A)

Figure 1.8: Effort distribution functions for the leader (dashed) and his opponent(solid). We assumed d2 = 3

10 and in the left panel c = 1, while in theright c = 7.

QB(x2B) =

0 if xB < 0,

1 − C(x − d2) if xB ∈ [0,d2],

1 − C(x − d2) + C(xB − d2) if xB ∈ [d2, x],

1 if xB > x.

Expected utilities are 1− C(x − d2) for A and 0 for B. If d2 ≥ x both candidates spend

zero effort, x2A = x2

B = 0, and expected utilities are 1 and 0 respectively.

Proof. First we show that this is an equilibrium. Consider A choosing effort

x ∈ [0, x − d2]. The associated payoff is then

Pr[x + d2> x2

B]− C(x) = QB(x + d2)− C(x)

= 1 − C(x − d2) + C(x + d2 − d2)− C(x)

= 1 − C(x − d2) + C(x)− C(x) = 1 − C(x − d2).

He is hence indifferent between all pure strategies in SA given QB. Note also,

that x > x − d2 is strictly dominated since lowering effort to x − d2 saves costs

without altering the probability to win. Hence, given QB A is indifferent be-

tween all pure strategies in SA and cannot improve on the expected payoff by

deviating. Now consider B. Given QA, choosing x ∈ SB gives the following

payoff:

Pr[x > x2A + d2]− C(x) = Pr[x − d2

> x2A]− C(x) = QA(x − d2)− C(x)

= C(d2 + x − d2)− C(x) = C(x)− C(x) = 0.

Effort x = 0 never wins but does not imply costs, either. All efforts x ∈ [0,d2]

APPENDIX 57

are strictly dominated since they cannot win but imply costs. Choosing effort

x ∈ [d2, x] also yields zero payoff, since the probability to win exactly equals the

associated effort costs. Hence, all x ∈ SB yield zero expected payoff. Note that

deviating to x > x is strictly dominated, since it would win for sure but imply

costs that are greater than the value of winning. Hence, there is no beneficial

deviation for B as well and thus QA and QB are an equilibrium.

Now consider the support SA. By elimination of strictly dominated strate-

gies it is apparent that A will never choose effort greater than x − d2, because B

will never choose effort greater than x. The maximum bid neither can be lower,

because than B would have a winning strategy and could simply overbid A

and win for sure. That A’s minimum bid must be zero and that the support

cannot have holes, or that it must be convex, follows directly from the proof in

Hillman and Riley (1989). Hence, SA is the only support a mixed strategy of

A can have in equilibrium. Now consider SB. It is apparent that all x ∈ (0,d2)

can never win but imply positive costs, and are hence strictly dominated. The

maximum possible bid is naturally x, and must also be x. Otherwise A could

overbid x2B and win for sure. Convexity of SB follows again from Hillman and

Riley (1989).

It is now easy to see that given SA and SB there do not exist other effort

distribution making the opponent indifferent between all effort in his mixed

strategy. Hence, the proof is complete.

To see which candidate has stronger incentives to invest we now need to

calculate the expected effort of each. This is, unfortunately, not possible for our

general cost function. We hence resort to the quadratic form employed before,

where C(x) = c2 x2. Candidate A’s expected effort in the campaign is

Ex2A =

∫

SA

x C′(x + d2)dx =∫

SA

xc(x + d2)dx =

[c

3x3 +

cd2

2x2

]x−d2

0

=c

3

(√2

c− d2

)3

+cd2

2

(√2

c− d2

)2

=

√8

9c− d2 +

c

6(d2)3

Similarly, for candidate B we get

Ex2B =

∫

SB

x C′(x − d2)dx =∫

SB

xc(x − d2)dx =

[c

3x3 − cd2

2x2

]x

d2

=c

3(x)3 − cd2

2(x)2 − c

3(d2)3 +

c

2(d2)3 =

√8

9c− d2 +

c

6(d2)3,


which is identical and hence the expected difference of efforts is zero. Formally:

Corollary 1.7.1. The difference in expected efforts in stage 2 is equal to zero.

This resembles our findings from a campaign with exogenous noise.

Now look at stage 1. We focus on pure strategy equilibria. Assume without

loss of generality d1 ≥ 0 and x1B = 0. A chooses effort to maximize

1 − C(x − d2)− C(x1A) = 1 − C(x − d1 − x1

A)− C(x1A).

His first order condition reads

C′(x − d1 − x1A)− C′(x1

A) = 0 ⇔ x − d1 = 2x1A.

The equivalence in the last step follows from the strict convexity of C(.), due

to which C′(.) is strictly increasing. Because the second derivative is strictly

negative,

−C′′(x − d1 − x1A)− C′′(x1

A) < 0,

we can use the first order condition to find the global maximum. A’s optimal

reaction to x1B = 0 is

x∗A =x − d1

2.

Now turn to B. He either spends zero and gets zero in expectation or maxi-

mizes

1 − C

(3

2x +

1

2d1 − x1

B

)− C(x1

B).

The first order condition reads

C′(3

2x +

1

2d1 − x1

B) = C′(x1B)⇔ 2xB =

3

2x +

1

2d1,

and hence

xB =3

4x +

d1

4.

The second order condition holds, but note that we only looked at positive

effort. For this to be optimal it needs to hold in addition that expected utility

is weakly positive:

1 − 2C

(3

4x +

d1

4

)≥ 0.

This inequality could in principle hold for small d1, depending on the curvature

of the costs function and when d1 is relatively small. For concreteness, however,

APPENDIX 59

0.0 0.5 1.0 1.5 2.0

-2

-1

0

1

2

Figure 1.9: Combinations of c (horizontal) and d1 (vertical) admitting a pure strategyNash equilibrium in which A spends more.

we stick to the functional employed mostly in the paper, C(x) = c2 x2. Then

1 − 2C

(3

4x +

d1

4

)= 1 − c

(3

4

√2

c+

d1

4

)2

< 0.

To see this note that the second term is strictly increasing in d1. Hence, if utility

is negative for d1 = 0 this is true also for all d1> 0. Using d1 = 0 we get

1 − c

(3

4

√2

c+

d1

4

)2

= 1 − c

(3

4

√2

c

)2

= 1 − 9

8= −1

8< 0.

Hence, xB cannot be an optimal choice and B chooses zero effort instead. There-

fore, we established that this is an equilibrium. Moreover, note that as long as

the utility from spending x1B is negative, (x∗A,0) is an equilibrium. This is the

case for all

d1 ≥ 4 − 3√

2√c

< 0.

Hence, as before when E[(ǫ)2] 6= 0, it may also be the case that the trailing

candidate spends more. In Figure 1.9 we show combinations of c and d1 for

which there exists a pure strategy Nash equilibrium in which A spends x1A > 0

while B stays passive.

Note that by symmetry the whole analysis also applies for B if d1 ≤ 0.

Hence, there always exists a pure strategy Nash equilibrium in stage 1 if we

assume costs to be quadratic, and there are multiple pure strategy equilibria in

close enough games:

Proposition 1.7.2. In stage 1 of the competition with quadratic costs there always

exists a pure strategy Nash equilibrium. In a close race there may also exist multiple

equilibria, and either candidate may adopt a tough stance and spend more. If one


candidate has a relatively large advantage, this candidate will also choose larger effort.

Proof. This follows immediately from the discussion above.

It follows from Corollary 1.7.1 and Proposition 1.7.2 that the campaign

game qualitatively perfectly resembles what we have seen before in the case

of a noisy competition. This proves Proposition 1.6.2.


Proof. Candidates maximize

Ed1

[λΦ

(d1 + ∆1 + ∆2

√2σ

)+ (1 − λ)Eǫ1,ǫ2

[1 − P(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)

]]

− c

2(x1

A)2 − c

2(x2

A)2

Ed1

[λ

(1 − Φ

(d1 + ∆1 + ∆2

√2σ

))+ (1 − λ)Eǫ1,ǫ2

[P(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)

]]

− c

2(x1

B)2 − c

2(x1

B)2(x2

B)2

respectively. Taking first order conditions with respect to spending in t leadsto

Ed1

[λ√2σ

φ

(d1 + ∆1 + ∆2

√2σ

)+ (1 − λ)Eǫ1,ǫ2

[p(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)

]]− cxt

A!= 0

Ed1

[λ√2σ

φ

(d1 + ∆1 + ∆2

√2σ

)+ (1 − λ)Eǫ1,ǫ2

[p(d1 + ∆1 + ∆2 − ǫ1 − ǫ2)

]]− cxt

B!= 0

It is apparent that in any interior equilibrium both spend identical effort in

both stages. Moreover, since efforts perfectly cancel out, the equilibrium must

be unique.


Proof. This follows immediately from the proof of Proposition 1.6.3. For the

stage 1 analysis we need an expression of the effort. Noticing that the noise

structure changes since only one random term is remaining, and that candi-

dates now know d2 precisely, from the FOCs is follows that

x2A = x2

B = x2 =

λσ φ(

d2

σ

)+ (1 − λ)Eǫ2

[p(d2 − ǫ2)

]

c.

APPENDIX 61

Making use of Assumption 1.6.3, we find

x2 =

λσ φ(

d2

σ

)+ (1−λ)√

σ2+(σP)2

φ

(−µP+d2√σ2+(σP)2

)+φ

(µP+d2√σ2+(σP)2

)

2

c.

Proof of Lemma 1.6.1

Proof. Stage 2 spending is

x2 =

λσ φ(

d2

σ

)+ (1−λ)√

σ2+(σP)2

φ

(−µP+d2√σ2+(σP)2

)+φ


)

2

c.

Differentiation with respect to d2 gives

∂x2

∂d2=

λσ2 φ′

(d2

σ

)+ (1−λ)

σ2+(σP)2

φ′(

−µP+d2√σ2+(σP)2

)+φ′


)

2

c.

At d2 = 0 the derivative is clearly zero. We hence take the second derivative to

see the curvature:

∂2x2

∂(d2)2=

λσ3 φ′′

(d2

σ

)+ (1−λ)

(σ2+(σP)2)3/2

φ′′(

−µP+d2√σ2+(σP)2

)+φ′′


)

2

c.

At d2 = 0 we get

∂2x2

∂(d2)2|d2=0 =

λσ3 φ′′ (0) + (1−λ)

(σ2+(σP)2)3/2

φ′′(

−µP√σ2+(σP)2

)+φ′′

(µP√

σ2+(σP)2

)

2

c

=

λσ3 φ′′ (0) + (1−λ)

(σ2+(σP)2)3/2 φ′′(

µP√σ2+(σP)2

)

c

The first term is strictly negative, implying the derivative ∂x2

∂d2 decreases and so

does effort as we increase |d2| from zero. Hence look at the second term. It is


straightforward to verify that

Sign

[φ′′(

µP

√σ2 + (σP)2

)]= Sign

[µP −

√σ2 + (σP)2

].

It follows that µP>

√σ2 + (σP)2 is necessary for effort to be increasing in d2. It

is, however, not sufficient. It is clear that when λ becomes large, the first term

dominates and the sign of the second derivative must be positive. However, if λ

is small, that is if the plurality premium is not very important, the second term

dominates. It is straightforward to show that for any λ ≥ 22+e3/2 the first term

always dominates and hence efforts decrease in |d2| (available upon request).


Proof. It is apparent from the discussion that for too small µP incentives are as

under FPTP. However, if µP>

√σ2 + (σP)2 stage 2 efforts increase over some

range in |d2|. It is easily verified that when µP = µP =√

3√

σ2 + (σP)2 the

second derivative of second stage effort with respect to d2 has its maximum.

It is easily shown that comparative statics ∂x1A/∂d1 at d1 = 0, in case a unique

equilibrium exists, are negative if µP = µP and if λ is small (available upon

request). Hence, in this region there is always anti-momentum in close games.

Chapter 2

Political campaigns with specializedcandidates

Philipp Denter†

†I would like to thank Micael Castanheira, Martin Kolmar, Michael McBride, Stergios Skaperdas,seminar participants at UC Irvine, Tuebingen as well as participants of EPEW Rotterdam 2012 forinsightful comments and suggestions. The kind hospitality of UC Irvine is gratefully acknowledged. Ialso gratefully acknowledge the financial support of the Swiss National Science Foundation.

64 CHAPTER 2. POLITICAL CAMPAIGNS WITH SPECIALIZED CANDIDATES

2.1 Introduction

“[...] there is no shortage of explanations for why issue convergence is such a rare

commodity in American campaigns. Perhaps surprisingly, though, there is a shortage

of convincing evidence that issue convergence really is a rare commodity.”

Sigelman and Buell (2004)

Obama has devoted 68 percent of his total TV advertising this year to

ads that include health care themes, and McCain has devoted 13 percent.

[...] In October, McCain spent 1.5 percent of his TV ads on health

care, while Obama upped the ante to 86 percent of his total budget.

Chris Frates on Politico.com (October 2008)1

Political competition determines which party or politician gets elected, and

therefore also which policies will be carried out during a term. Candidates

or parties develop a policy platform, which states more or less vaguely what

policies they plan to carry out if they should get elected. Then candidates

enter the campaigning stage, during which they spend considerable amounts

of time and money to convince voters that they are the right choice at the ballot.

But how does communication between politicians and the electorate look like?

Which policy issues will be important during the campaign? Which issues will

be neglected by one or both candidates? When will candidates communication

strategies converge, that is when will both emphasize the same issues during

the campaign? And under which conditions do candidates diverge? To answer

these questions is the purpose of this paper.

The campaigning literature so far is dominated by the seminal works of

Petrocik (1996) and Riker (1996). Petrocik (1996) argued that each candidate

owns certain issues, i.e. she is perceived to be more competent in this issue

than her political opponent. Such an advantage in perceived competence can

have many different sources, such as a party’s history, personal professional

experience of candidates and the like. For example, in the current campaign

contest with Barrack Obama, the Republican candidate Mitt Romney tries to

use his experience as a leader of a big business as a proof of his leadership abil-

ities. Riker (1996) developed two principals of campaigning rhetoric from issue

ownership. “When one side has an advantage on an issue, the other side ignores it;

but when neither side has an advantage, both seek new and advantageous issues” (page

106). He calls the former the dominance principle and the latter the dispersion prin-

1The article is available here: http://www.politico.com/news/stories/1008/14887.html.


ciple. The prediction of his theory is that there is a strict form of divergence in

communication strategies and that some issues will be completely neglected.

How is the predictive power of these two principles in practice? As a first

example look at the 2008 campaign between Obama and John McCain.2 Obama

published some 23 percent of all his TV ads on the issue ’the economy’, which

was almost matched by McCain who published more than 20 percent of all

his TV ads on that issue. This issue was perceived the most important during

the campaign, with the US economy struck hard by the financial crisis and

many people fearing about their jobs. 91 percent of respondents of a Gallup

/ USA Today poll said this issue is either extremely or very important. Also,

48 percent of respondents believed Obama to be the right person to tackle eco-

nomic problems, while only 32 percent believed McCain to be more competent.

Although Obama was considered more competent by the average voter, candi-

dates roughly converged in their communication strategies, that is, the fraction

of TV ads they bought on that issue. Now look at the issue ’healthcare’, which

was considered less important than ’the economy’ but still 80 percent of poll

respondents classified it as either extremely or very important. In this issue,

Obama’s supremacy over McCain was even more extreme, with 51 percent

of respondents believing him to be more competent in this issue, and merely

26 percent believing in the opposite ranking. The above quotation by Chris

Frates shows that here candidates’ communication strategies diverged signifi-

cantly, and Obama was taking the lead. Finally, look at the issue ’terrorism’.

76 percent of poll respondents classified the issue as either extremely or very

important, and now Obama was perceived to be less competent in solving en-

ergy related problems, with 33 percent believing he was more competent vs.

52 percent believing McCain to be more competent. Also in this issue there

was significant divergence in communication strategies, but now McCain was

taking the lead, since Obama remained completely mute on that issue, while

McCain published some 5 percent of all his TV ads on that issue. On most other

issues both candidates’ strategies converged again, for example ’Iraq’ or ’taxa-

tion’. Hence, there is often some degree of divergence, and in some issues this

divergence can become quite extreme, but mostly there is only a moderate di-

vergence and most important issues were addressed by both candidates. Many

empirical studies confirm this conclusion, e.g. Damore (2004) and Sigelman

and Buell (2004). Therefore, the predictive power of the dominance principle is

2All numbers on TV ads are calculated by me using data from an online archive of the Wash-ington Post: http://projects.washingtonpost.com/politicalads/issues/. Data on issues’ im-portance and candidates’ competence are taken from a Gallup / USA Today poll from June 2008:http://www.gallup.com/poll/108331/obama-has-edge-key-election-issues.aspx.


rather weak, as has been put forward by a number of authors already.

In this paper I develop a theoretical model of political communication. Can-

didates can buy TV ads to sway voters. Buying TV ads has two effects: it ad-

vertises a candidate’s policy platform in the issue and it primes the issue. The

first effect increases a candidate’s perceived competence in the issue, or makes

voters like the platform better. The second increases the issue’s salience by di-

recting attention to it, and thereby increases the perceived importance of the

issue. This simple structure is sufficient to generate the following main results:

• In a pure advertising campaign candidates perfectly converge in their com-

munication strategies if they have identical marginal costs of funding. If

one candidate has a marginal cost advantage, he spends more on both

issues.

• In an advertising and priming campaign in which candidates do not have

comparative advantages, there exists an equilibrium in which they converge

completely in their communication strategy.

• In an advertising and priming campaign in which candidates have compar-

ative advantages, there will be some form of divergence, but never com-

plete divergence.

• A candidate might publish the biggest number of TV ads on an issue in

which he is disadvantaged, if this issue is sufficiently important and the

disadvantage not too big.

The intuition for the main result is as follows. If candidates have compar-

ative advantages, it is beneficial to highlight this advantage for two reasons.

First, advertising policy strengthens the advantage. Second, priming the issue

of the comparative advantage draws attention to one’s strength and away from

one’s weakness. This is also beneficial. While the advertising effect is also ben-

eficial in issues in which a candidate does not have an advantage, the priming

effect is now detrimental. However, as long as the first effect dominates the

candidate will still publish some TV ads on that issue. However, due to the ad-

ditional cost as which we can interpret the negative priming effect, candidates

publish more ads on their comparative advantages than on the other, unless the

issue in which there is a disadvantage becomes extremely important. If this is

the case a marginal increase in perceived quality of one’s policy platform in an

issue has a relatively strong effect on a candidate overall assessment, and hence

in that issue the advertising effect becomes dominating. To the contrary, in less


important issues the priming effect is more important and there we should

hence expect more divergence.

The model can explain most features of the 2008 campaign such as the

strong form of convergence in communication strategies in the issue ’the econ-

omy’ and most other issues, and the strong form of divergence in ’healthcare’

and ’terrorism’. The issues in which candidates’ strategies diverged signifi-

cantly were not among the most important issues and in both issues the per-

ceived competence advantage of one candidate was relatively large. The model

predicts that both of these facts make divergence more likely. In issues that are

more important and in which comparative advantages are smaller, there is a

tendency to converge.

The chapter is organized as follows. Next I discuss the relevant related

literature. In Section 2.2 I present the formal model. In Section 2.3 I discuss

the benchmark of a pure policy advertising campaign. Section 2.4 studies the

general model with both policy advertising and issue priming. Section 2.5

concludes.

Related literature. Many scholars have directed their attention to the study of

political campaigns. An important question thereby is and was how candidates

allocate their time and money during an electoral contest to different states and

electoral districts, e.g. Brams and Davis (1973, 1974), Snyder (1989), Klumpp

and Polborn (2006), and Stromberg (2008), or to different forms of campaigning,

e.g. Skaperdas and Grofman (1995). Meirowitz (2008), Iaryczower and Mattozzi

(2009), and Sahuguet and Persico (2006) consider different effects of campaign

spending regulations. In contrast to all those papers, the focus of my paper is

to study candidates’ communication strategies during a campaign, that is how

different issues are addressed during a campaign when there are advertising

and priming elements of campaigning.

Petrocik (1996) argued in his influential paper that candidates and parties

have a reputation to own certain issues, i.e. that they are in the perception

of the populace more competent in this issue than their political opponent. A

consequence of owning an issue is then that candidates will focus in their cam-

paign communication on issues that they own, and mute others. This strategy

has been called the ’dominance principle’ by Riker (1996) and subsequent au-

thors, e.g. Amoros and Puy (2011). The predictions of the dominance principle

are that there is some extreme form of divergence in communication strate-

gies. However, empirical scholarship so far tends to reject the hypotheses and

finds that there is moderate divergence in some issues, but often candidates


also tend to converge in their communication, see for example Bélanger and

Meguid (2008), Damore (2004, 2005), Green and Hobolt (2008), Petrocik et al.

(2003), or Sigelman and Buell (2004).

Recently, some authors have developed theories that want to explain where

issue ownership might come from, for example Krasa and Polborn (2010) and

Aragonès et al. (2012). In contrast to this literature the focus here is not the

emergence of issue ownership, but the consequences for campaigning. Moen

and Riis (2010) develop a theoretical justification for issue trespassing, which

is based on a signaling argument. In the current paper issue trespassing can

also happen, but the reason is that policy advertising has an effect on voters’

policy assessment. Egan (2009) shows how candidates can use their perceived

advantage in the issues they own to choose policy platforms that are not as

conform with voters’ preferences. In contrast, I am not concerned with how

candidates’ choose policy platforms, but how issue ownership and policy plat-

forms influence candidates’ communication strategies during a campaign.

2.2 The Model

I model a multi issue campaign between two candidates i ∈ {1,2} in two is-

sues j ∈ {H,S}, which are mnemonics for health care and homeland security.

A candidate spends effort xji ≥ 0 to increase his competence in this issue as

perceived by the voters, and thereby his probability to win the election. This

can include positive campaigning in which the candidate highlights parts of his

policy plans to convince voters that his plan is good, or it could mean negative

campaigning, which targets the opponent’s weaknesses in the issue. Marginal

costs of effort are constant and equal to mi > for candidate i and reflect costs

to raise and spend funds as well as time. Differences in marginal costs may re-

flect, for example, different budgets. I assume that campaigning is effective in

the sense that spending increases a candidate’s perceived competence. Letting

θ j ∈ (0,1) and 1− θ j denote the electorate’s common prior concerning candidate

1’s and 2’s competence, after campaign beliefs are

cj1 =

θ j f (xj1)

θ j f (xj1) + (1 − θ j) f (x

j2)

(2.1)

and cj2 = 1 − c

j1. Competence is Hence defined as relative competence. (2.1) is a

competence ’production function’ and depends on both candidates’ efforts or

inputs. This form of contest has been used to analyze various problems such as

2.2. THE MODEL 69

lobbying, litigation, or war. Skaperdas (1996) provides an axiomatic foundation

and Skaperdas and Vaidya (2009) provide an inferential micro foundation for

this function to model persuasion, i.e. the transformation of ex-ante beliefs into

ex-post beliefs due to persuasive effort. Accordingly, I interpret θ j and 1− θ j as

the respective prior beliefs regarding competence in issue j.

The function f (.) transforms effort x into effective effort. I make the follow-

ing assumptions concerning this function:

Assumption 2.2.1. 1. f (x) is at least twice continuously differentiable.

2. f (0) > 0.

3. f ′(x) > 0.

4. f ′′(x) < 0.

5. limx→0 f ′(x) = ∞.

Part 2 means that even if a candidate does not spend anything in the cam-

paign his perceived competence remains positive. A direct implication is that

perceived competence is continuous in candidates’ efforts. Part 3 reflects that

spending effort is beneficial to a candidate. Part 4 implies marginal products of

effort are decreasing. Alternatively, it might as well be interpreted as increas-

ing marginal costs of effort, for example, because marginal costs of funding

increase. Part 5 is not a necessary assumption but is sufficient to guarantee the

existence of an interior equilibrium. Assumption 2.2.1 is not necessary but suffi-

cient. Below I will solve a numerical example when f (.) is a linear function. An

example functional fulfilling Assumption 2.2.1 is f (x) = c + xa for some c > 0

and 1 > a > 0.

Apart from increasing the perceived competence of a candidate in an issue,

effort has another effect that is of relevance: it increases the salience of the issue.

If salience does not have an influence on voters decisions this would not make a

difference. However, I assume that increasing the salience of an issue increases

the relative importance of the issue as perceived by voters. In particular, letting

θw and 1 − θw denote the prior importance weights of issues H and S, weights

after the campaign are

w =θwg(κ + xH

1 + xH2 )

θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2). (2.2)

for H and 1 − w for S. κ ≥ 0 is a parameter that determines the effectiveness

of creating salience to make the issue more important and is exogenous. It


may represent institutional features such as the degree to which the media

follow the candidates in creating salience. I make the following assumptions

concerning g(.):

Assumption 2.2.2. 1. g(x) is at least twice continuously differentiable.

2. g(0) > 0.

3. g′(x) > 0 and finite for finite x and limx→∞ g′(x) = 0.

4. g′′(x) < 0 for finite x and limx→∞ g′′(x) = 0.

5. g′′′(x) > 0.

Part 1 reflects that an issue’s importance cannot drop to zero just because

nobody talks about it. Part 2 reflects creating salience affects the relative impor-

tance of the two issues. Part 3 means there are decreasing returns of salience,

and part 4 and 5 imply that returns do not decrease very quickly. The latter two

assumptions are motivated by mathematical necessity rather than following a

real world intuition. The assumptions are also not necessary for the analysis,

but – as we will see below – sufficient to guarantee a nicely behaved prob-

lem for the candidates.3 An example function fulfilling Assumption 2.2.2 is

g(x) = (γ + x)α for some γ > 0 and 1 > α > 0.

Now it is possible to determine voters’ choice. I adopt the assumption of a

probabilistic voting decision as in Snyder (1989) or Klumpp and Polborn (2006),

but I add a particular structure.

Assumption 2.2.3. The probability that candidate 1 wins the election is

Pr1[win] = cHw + cS(1 − w),

and candidate 2 wins with probability

Pr2[win] = 1 − Pr1[win] = (1 − cH)w + (1 − cS)(1 − w).

Hereby I generalize the models of Snyder (1989) and Klumpp and Polborn

(2006) to two issues and possibly endogenous issue weights. The probability to

win is a weighted average of the relative competence measures. Ceteris paribus,

the more competent a candidate is perceived in an issue, the more likely it

is that he wins the election. Similarly, when the strong issue of a candidate

becomes more important, his winning probability also increases. Hence, in

3An alternative assumption guaranteeing a nicely behaved problem, that would allow us to dispensewith Assumptions 2.3-2.5, is g′(x) < α1 and |g′′(x)|< α2 for appropriately chosen α1 and α2.

2.3. WHEN SALIENCE HAS NO EFFECT: A BENCHMARK 71

my model being competent in one or the other issue are imperfect substitutes.

There is no lexicographic ordering of issues.

Finally, before I start with the analysis, I need to make clear what I mean

when I say candidates do or do not specialize. There are two candidates and

two issues. I shall call an equilibrium in which one candidate spends more

on one issue, while his opponent spends more on the other, an equilibrium

with specialization. This definition is intuitive and refers to specialization in

absolute terms, i.e. when a candidate spends absolutely more on one issue and

absolutely less on the other. An alternative and weaker form of specialization,

in which a candidate specializes in an issue if he spends relative to his opponent

more on this issue than on the other, is not considered in this paper.

Definition 2.2.1. If

Sign[xH1 − xH

2 ] = −Sign[xS1 − xS

2 ]

in equilibrium, I shall call this a specialization equilibrium.

2.3 When Salience has no Effect: A Benchmark

To see why specialization as discussed above cannot be reconciled with the stan-

dard model with exogenous issue weights, I now shortly discuss the standard

model and its predictions.

Assume a candidate’s competence is determined as in Assumption 2.2.1.

However, salience does not have an influence on perceived importance of an

issue, and hence wH = w = θw and wS = 1 − θw. The whole problem faced by

a candidate is then separable in two sub-contests for competence only. In each

sub-contest candidate 1 wants to maximize

θ j f (xj1)

θ j f (xj1) + (1 − θ j) f (x

j2)

wj − m1xj1,

while candidate 2 maximizes

(1 − θ j) f (xj2)

θ j f (xj1) + (1 − θ j) f (x

j2)

wj − m2xj2.

This game has been analyzed extensively, see for example Rosen (1986), Dixit

(1987), Snyder (1989), or Yildirim (2005):

Proposition 2.3.1. The game has a unique equilibrium in pure strategies. Candidate 1

spends on both issues more (less) than candidate 2 if m1 < m2 (m1 > m2). Both spend

identical effort on a given issue whenever m1 = m2.



The proposition shows that in this model there is no specialization as de-

fined above. Differences in marginal costs may explain why a candidate spends

in total more than his opponent, but do not explain specialization. Differences

in the electorate’s prior belief do not have any influence on relative spending

(but on aggregate spending). A candidate may Hence spend strictly more than

his adversary, but it may not happen that he spends more on one issue and less

on another, as it was for example the case in the Obama vs. McCain example

above.

2.4 When Salience Matters

Now consider the game when salience influences issue weights. The aim of a

candidate is again to maximize his winning probability subject to costs of effort.

Hence, following the assumptions made in Section 2.2, each candidate faces the

following problem:

max(xH

1 ,xS1 )∈R2

+

u1 = cHw + cS(1 − w)− m1(xH1 + xS

1 ) (2.3)

max(xH

2 ,xS2 )∈R2

+

u2 = (1 − cH)w + (1 − cS)(1 − w)− m2(xH2 + xS

2 ) (2.4)

Unlike before, w is now endogenously determined as defined in Assumption

2.2.2. Before going into the details of the equilibrium, I need to establish that

an equilibrium actually exists. This is done in the following proposition:

Proposition 2.4.1. For κ sufficiently large but finite, the game has an interior pure

strategy Nash equilibrium. There cannot exist a corner equilibrium, in which a candi-

date spends no effort on a given issue.


The intuition for the existence result is straightforward. The effect of the

relative importance on the second order conditions must not be too large, be-

cause then the problem is not concave anymore. By restricting the curvature

of g(.) and letting κ become relatively large, we can avoid this problem and

apply standard equilibrium existence proofs.4 That there cannot exist a corner

equilibrium follows from part 4 of Assumption 2.2.1.

4Note, however, that for this purpose Assumption 2.2.2 is sufficient, but not necessary. There arealternative assumptions guaranteeing existence. What needs to hold is that the first and second deriva-tive of g are sufficiently small.

2.4. WHEN SALIENCE MATTERS 73

Having established that an interior equilibrium exists and that there cannot

be corner equilibria, it follows that I can use first order conditions. From (2.3)

and (2.4) the following system of first order conditions follows:

∂u1

∂xH1

=∂cH

∂xH1

w + (cH − cS)∂w

∂xH1

− m1 = 0 (2.5)

∂u1

∂xS1

=∂cS

∂xS1

(1 − w) + (cH − cS)∂w

∂xS1

− m1 = 0 (2.6)

∂u2

∂xH2

= − ∂cH

∂xH2

w − (cH − cS)∂w

∂xH2

− m2 = 0 (2.7)

∂u2

∂xS2

= − ∂cS

∂xS2

(1 − w)− (cH − cS)∂w

∂xS2

− m2 = 0 (2.8)

Increasing effort influences the probability to win through two different

channels. First, the perceived competence of the candidate increases in the is-

sue he spends the effort on. Second, the relative weight voters assign to this

issue increases. While the former effect is always beneficial, the latter might be

detrimental. Assume candidate 1 is considered to be much more competent in

health care issues than candidate 2, while their relative competence in security

questions is more equal. If candidate 2 now spends effort in a health care cam-

paign, he might play his adversary in the hand; because while he decreases the

gap in perceived competence, he emphasizes his weak spot. Hence, spending

effort on one’s weak issue involves additional costs for a candidate.

From the first order conditions we can derive the following condition, which

has to hold in any interior equilibrium:

w

(∂cH

∂xH1

+ ∂cH

∂xH2

)+ ∆

∂w∂xH

1

=

(1 − w)

(∂cS

∂xS1

+ ∂cS

∂xS2

)+ ∆

∂w∂xS

1

where ∆ = m2 − m1. (2.9)

This equation is fundamental for the analysis. Since the derivatives in the

denominator have opposite signs it follows that the enumerators also must

have opposite signs or be zero. Formally:

Lemma 2.4.1. In any equilibrium it must hold that

Sign

[w

(∂cH

∂xH1

+∂cH

∂xH2

)+ ∆

]= −Sign

[(1 − w)

(∂cS

∂xS1

+∂cS

∂xS2

)+ ∆

].



What does this now mean for the equilibrium of the game? I start with the

simplest case, that is when ∆ = 0 and hence candidates face identical marginal

costs of effort. Since 1 > w > 0, the condition in Lemma 2.4.1 then boils down

to

Sign

[∂cH

∂xH1

+∂cH

∂xH2

]= −Sign

[∂cS

∂xS1

+∂cS

∂xS2

].

This tells us that either candidate 1 has a larger marginal product of effort in

equilibrium in the one issue and a lower in the other, or both candidates have

identical marginal products in both issues. I define a candidate’s comparative

advantage in the campaign in the following way:

Definition 2.4.1. A candidate has a comparative advantage in a given issue, if the

voters ex ante belief he is in this issue more competent than in the other issue. If

θH> θS, candidate 1 has a comparative advantage in issue H and candidate 2 in issue

S, and vice versa if θH< θS. If θH = θS, no candidate has a comparative advantage.

Comparative advantages, like in trade theory and other fields of economics,

are among the driving forces for specialization. To see how comparative ad-

vantages affect specialization I assume ∆ = 0 first, such that candidates have

identical marginal costs. In this case, whenever candidates have comparative

advantages, there is specialization in equilibrium:

Proposition 2.4.2. Let ∆ = 0.

i. A no specialization equilibrium exists if and only if θH = θS, that is, when candi-

dates do not have a comparative advantage.

ii. If in addition θw = 12 , a perfectly symmetric equilibrium exists in which each

candidates spends identical effort on each issue.

iii. If θH 6= θS, and hence candidates have a comparative advantage, each candidate

specializes in a given issue.


A no-specialization equilibrium may only exist if candidates have no com-

parative advantages. Then each candidates spends identical effort on issue i. If

in addition issues have identical ex ante weights, there exists a perfectly sym-

metric equilibrium. However, as soon as there are comparative advantages,


symmetric equilibria cannot exist anymore. To see why this must be the case

consider to the contrary, that a symmetric equilibrium exists in which candi-

dates spend identical effort on issue i. Then, by definition of comparative ad-

vantages and symmetric equilibria, cH − cS = θH − θS 6= 0. Hence, changing the

relative weights of the issues has an effect on the winning probabilities at the

margin, and this effect goes in opposite directions. Hence, since the marginal

effect of effort on perceived competence in an issue is identical in a symmetric

equilibrium, this cannot be an equilibrium, because there are profitable devia-

tions. Hence, there must be specialization in equilibrium.

I cannot exclude that there may be specialization even when there is no

comparative advantage. Depending on the particular contest technologies f (.)

and g(.) this might be the case. This means I cannot proof uniqueness of Nash

equilibria, either. Actually, in principle it might even happen that candidates in

equilibrium specialize against their ex ante comparative advantage. However,

if this is the case it must hold that the issue a candidate specializes in is the

issue he has an ex post comparative advantage. Hence, when a candidate has a

significant advantage in a given issue, he would have to spend a lot to turn his

comparative advantage. Hence, depending on the marginal costs, at a given

point this cannot be worthwhile and thus when marginal costs are high and

there are ex ante comparative advantages, candidates specialize accordingly.

Now let ∆ 6= 0, so that one candidate has a marginal cost advantage. As-

sume without loss of generality that ∆ > 0, such that candidate 1 has the advan-

tage because his marginal costs are lower. Intuitively, in this situation candidate

1 will increase efforts relative to his opponent. But how is specialization af-

fected? Clearly, if any candidate’s marginal costs approach zero, this candidate

will spend more on both issue than his opponent. But if differences in marginal

costs are not too pronounced, there is still specialization in equilibrium:

Proposition 2.4.3. If |∆| > 0 but sufficiently small, in equilibrium candidates spe-

cialize when there are comparative advantages. However, if |∆| becomes too large, the

candidate with lower marginal costs spends more on both issues.


How large |∆| may become in order to sustain specialization depends on

the parameters and functional forms of the game. When there is a high degree

of specialization when ∆ = 0, the differences in marginal costs of funding may

be relatively large. If specialization is only marginal, even slight differences in

marginal costs may prevent specialization. Note that ∆ is the absolute differ-

ence in marginal costs. When effort is very cheap at the margin (relative to


0 5 10 15 20 25

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

κ

xi1/xi

2

Figure 2.1: xH1 /xH

2 as well as xS1 /xS

2 for θw = 4/9, θH = 5/9, θS = 1/3, m1 = m2 = 1,and variable values of κ. As κ → ∞ both ratios converge to 1.

the prize of winning the election, which is assumed to be 1), relative marginal

costs may be very different and specialization may still occur. If marginal costs

are large, relative marginal costs must be close to one for specialization equilib-

rium.

Note that specialization does not mean that a candidate spends more on

issue H than on S or the other way around. A candidate might specialize in

issue H and still spend more on issue S. This is so because I defined specializa-

tion relative to the opponent’s spending. For example, in the 2008 presidential

campaign McCain clearly specialized on security, but nevertheless spend more

on his health campaign. This is likely to happen when one issue is much more

important than the other. For example, if θw is relatively large, the candidate

with a comparative advantage in issue S is likely to spend absolutely more on

issue H, while he still spends more on S than his opponent.

Proposition 2.4.4. A candidate may spend more on the issue he has the comparative

disadvantage in, than on the issue in which in which he has the comparative advantage,

if the issue of his disadvantage is relatively important.


For example, in Table 2.1 below in rows (vi.) and (vii.) this is the case.

Candidate 1 has a comparative advantage in S and candidate 2 in H. However,

because issue 2 is perceived to be much more important, both spend more on

S than on H.

Now look at the parameter κ and assume ∆ = 0. It determines the marginal

influence of salience on perceived importance. An interpretation of κ is a pa-

rameter determining the responsiveness of the media to the candidates’ polit-

ical advertising. If the media more than proportionally picks up issues made


salient by the candidates, then the media intensifies salience. This would be

the case when κ is relatively small and so small differences in total spending

for the different issues has a large effect. That makes it similar to the dis-

criminatory power in standard lottery contests, see for example Tullock (1980)

or Gradstein and Konrad (1999). Similarly, Amegashie (2006) interprets κ as

a noise parameter. In standard contest models increasing the discriminatory

power of the contest – i.e. in my framework lowering κ – usually has the effect

of increasing effort in the contest. This is so because a higher discriminatory

power implies increasing returns to effort. In the present framework there are

three different intertwined contests. By decreasing κ, the marginal returns of

effort in the salience contest increase relative to the returns in the competence

contests. Therefore, specialization becomes more and more attractive and thus

candidates tend to specialize more when κ becomes smaller. However, since I

cannot exclude multiple equilibria, a comparative statics analysis is problem-

atic. To avoid this in the following I will assume that the equilibrium is unique.

Define the ratio ρi := xi1/xi

2 as a measure of the degree of specialization in issue

i. If ρi> 1 candidate 1 specializes in issue i and, by Proposition 2.4.2, candi-

date 2 specializes in the other issue. Also assume ∆ = 0. Figure 2.1 shows

the ratios ρH and ρS for increasing values of κ for a parameterized version of

the model. In this version f (x) = x and g(κ + x1 + x2) = κ + x1 + x2. This

contradicts Assumptions 2.2.1 and 2.2.2, but note that these assumptions only

served the purpose to prove existence of an equilibrium in the general model.

They do not affect the comparative statics qualitatively. I chose this functional

form for computational convenience. The figure shows clearly that the degree

of specialization decreases in κ for the chosen parameter values. This was true

for all numerical solutions of the model I computed and seems to be generally

true for the functional forms I used. Note that when ∆ 6= 0 the effort ratios do

not converge to 1, as it is the case in the figure, but to some other constant.

Before I conclude, to get a better understanding of the comparative statics

of the model, I calculated equilibria for varying parameter values in Table 2.1.

There are three different regions. In the first region, rows (i.) to (iv.), no can-

didate has a comparative advantage and marginal costs of effort are identical.

Hence, there is no specialization in equilibrium. In the middle region, rows (v.)

to (viii.), candidates have comparative advantages but marginal costs are still

identical. Hence, there is always specialization. Finally, in the lower region,

rows (ix.) to (xii.), there are still comparative advantages but also differences in

marginal costs. In rows (ix.) and (x.) there is nevertheless specialization, but in

rows (xi.) and (xii.) the difference is too large and candidate 2 spends more on


θH θS θw m1 m2 κ xH1 xS

1 xH2 xS

2xH

1

xH2

xS1

xS2

(i.) 1/2 1/2 1/2 1/4 1/4 2 1/2 1/2 1/2 1/2 1 1(ii.) 1/2 1/2 1/3 1/4 1/4 2 .268 .732 .268 .732 1 1(iii.) 1/3 1/3 1/2 1/4 1/4 2 .444 .444 .444 .444 1 1(iv.) 1/3 1/3 1/3 1/4 1/4 2 .244 .645 .244 .645 1 1

(v.) 1/3 2/3 1/2 1/4 1/4 2 .330 .466 .466 .330 .708 1.412(vi.) 1/3 2/3 1/3 1/4 1/4 2 .192 .664 .256 .531 .750 1.250(vii.) 1/3 2/3 1/4 1/4 1/4 2 .137 .741 .170 .639 .806 1.160(vii.) 1/4 2/3 1/2 1/4 1/4 2 .226 .489 .355 .323 .637 1.514

(ix.) 1/3 2/3 1/2 1/4 1/5 2 .274 .521 .514 .452 .533 1.153(x.) 1/3 2/3 2/3 1/4 1/5 2 .462 .284 .752 .257 .614 1.105(xi.) 1/3 2/3 1/3 1/4 1/5 2 .159 .730 .274 .733 .580 .996(xii.) 1/2 2/3 1/2 1/4 1/5 2 .455 .478 .688 .490 .661 .976

Table 2.1: Equilibrium values for a parameterized version of the campaign contest.Pr0

1 is candidate 1’s ex ante winning probability, Pr11 his ex post probability

to win, that is after taking into account the campaign.

both issues.

2.5 Conclusion

In this paper I analyzed a multi issue campaign with endogenous issue weights.

When candidates have comparative advantages and marginal costs are similar,

there is specialization in the sense that each candidate spends more than his

opponent on one issue and less on the other. When marginal costs become

too different in absolute terms, the candidate with lower costs spends more on

both issues. The model is helpful in explaining real world campaigns and an

example can be found in the 2008 presidential campaign in the U.S. The driv-

ing force for my results is that efforts cannot be targeted precisely to promote

competence only, but there are spillovers influencing the perceived relative im-

portance of an issue. This is so because promoting a policy platform on a given

issue necessarily makes the issue more salient.

Apart from political campaigns, the model may be applied to persuasive

advertising in goods markets when there are multiple dimensions. While there

exists a large literature analyzing persuasive advertising, see for example the

recent survey by Bagwell (2007), the issue of multiple product features has not

received much attention so far. Goods with multiple features include comput-

ers, cars, or smart phones to name just a few. As an example one can think

of Apple computers vs. Microsoft Windows based computers. While in most

2.5. CONCLUSION 79

Apple commercials style and graphical features are made salient, Microsoft

Windows based computer companies often highlight the computer’s aptitude

for business purposes.

There are some immediate directions for future research that follow from

the current paper. First, it is interesting to study how the results change when

there are more issues and competing candidates. In particular, what happens

when there are more candidates than (important) issues is an interesting ques-

tion. A second immediate route for future research is to give candidates a

fixed budget instead of assuming fixed marginal costs of effort. Finally, it is in-

teresting to explicitly model voters’ preferences and to incorporate a platform

choice decision prior to the campaign. This latter generalization is a project I

am currently working on.


Appendix


Proof. Existence of an interior equilibrium follows directly from strict concavity

of the candidates’ optimization problem and has been established for example

by Yildirim (2005) and Snyder (1989). Given w is a constant, each candidate

separately optimizes utility in any issue. This yields the following first order

condition in issue i for 1:

θi(1 − θi) f ′(xi1) f (xi

2)

(θi f (xi1) + (1 − θi) f (xi

2))2

wi − m1!= 0.

The first order condition of his adversary in the same issue is

θi(1 − θi) f (xi1) f ′(xi

2)

(θi f (xi1) + (1 − θi) f (xi

2))2

wi − m2!= 0.

Now note that 1’s marginal return of effort is higher / lower than 2’s if and

only if he spends less / more on this issue. If they spend identical effort, the

marginal returns are identical, too. Since in equilibrium marginal costs must

equal marginal returns of effort, the candidate with lower marginal costs must

have lower marginal returns, too, and hence spend more effort. If candidates

have identical marginal costs, m1 = m2, they spend identical effort in equilib-

rium. (See also Comments 3.1 and 3.2 in Snyder (1989)).


Proof. To prove existence of a pure strategy Nash equilibrium it suffices to show

that strategy spaces are convex and compact and utility functions are concave

and continuous. Since any effort larger than 1 is strictly dominated the relevant

strategy space is [0,1]2, which is both convex and compact. Moreover, since

f (0) > 0 as well as g(0) > 0, utility function are also continuous. Hence, it

remains to show that they are also concave. For concavity it has to hold that

∂2u1

∂(xH1 )2

< 0 and∂2u1

∂(xH1 )2

∂2u1

∂(xS1)

2−(

∂2u1

∂xH1 ∂xS

1

)2

> 0

for all relevant effort combinations. I show that for κ sufficiently large this holds

generally, which is then stronger than needed but analytically more convenient.

Because the aim of this paper is not to show conditions that are necessary

APPENDIX 81

for an equilibrium to exist, but to analyze candidate behavior in equilibrium,

providing a sufficient condition for an equilibrium to exist is enough for my

purpose.

I will now show when the payoff functions are concave and without loss of

generality, I focus on candidate 1’s payoff. For candidate 2 the exact same steps

then would show the same.

The candidates’ first order conditions are

∂u1

∂xH1

=∂cH

∂xH1

w + (cH − cS)∂w

∂xH1

− m1 = 0 (2.10)

∂u1

∂xS1

=∂cS

∂xS1

(1 − w) + (cH − cS)∂w

∂xS1

− m1 = 0 (2.11)

∂u2

∂xH2

= − ∂cH

∂xH2

w − (cH − cS)∂w

∂xH2

− m2 = 0 (2.12)

∂u2

∂xS2

= − ∂cS

∂xS2

(1 − w)− (cH − cS)∂w

∂xS2

− m2 = 0 (2.13)

and second derivatives for 1 are

∂2u1

∂(xH1 )2

=∂2cH

∂(xH1 )2

w + 2∂w

∂xH1

∂cH

∂xH1

+∂2w

∂(xH1 )2

(cH − cS) (2.14)

∂2u1

∂(xS1 )

2=

∂2cS

∂(xS1 )

2(1 − w)− 2

∂w

∂xS1

∂cS

∂xS1

+∂2w

∂(xS1 )

2(cH − cS) (2.15)

∂2u1

∂xH1 ∂xS

1

=∂cH

∂xH1

∂w

∂xS1

− ∂cS

∂xS1

∂w

∂xH1

+ (cH − cS)∂2w

∂xH1 ∂xS

1

(2.16)

Using the specific functional forms the relevant terms as follows:

∂2cH

∂(xH1 )2

=θH(1 − θH) f (xH

2 ) f ′′(xH1 )[θH f (xH

1 ) + (1 − θH) f (xH2 )]

(θH f (xH1 ) + (1 − θH) f (xH

2 ))3(2.17)

− 2θH( f ′(xH1 ))2

(θH f (xH1 ) + (1 − θH) f (xH

2 ))3< 0

∂2cS

∂(xS1 )

2=

θS(1 − θS) f (xS2 ) f ′′(xS

1 )[θS f (xS

1 ) + (1 − θS) f (xS2 )]

(θS f (xS1 ) + (2 − θS) f (xS

2 ))3

(2.18)

− 2θS( f ′(xS1 ))

2

(θS f (xS1 ) + (2 − θS) f (xS

2 ))3< 0


∂2w

∂(xH1 )2

=θw(1 − θw)g(κ + xS

1 + xS2 )g′′(κ + xH

1 + xH2 )

(θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2))3

(2.19)

×[θwg(κ + xH

1 + xH2 ) + (1 − θw)g(κ + xS

1 + xS2)]

− 2θw(g′(κ + xH1 + xH

2 ))2

(θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2))3

∂2w

∂(xS1 )

2= − θw(1 − θw)g(κ + xH

1 + xH2 )g′′(κ + xH

1 + xH2 )

(θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2))3

(2.20)

×[θwg(κ + xH

1 + xH2 ) + (1 − θw)g(κ + xS

1 + xS2)]

− 2θw(g′(κ + xH1 + xH

2 ))2

(θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2))3

∂2w

∂xH1 ∂xS

1

=θw(1 − θw)g′(κ + xH

1 + xH2 )g′(κ + xS

1 + xS2)

(θwg(κ + xH1 + xH

2 ) + (1 − θw)g(κ + xS1 + xS

2))3

(2.21)

×[θwg(κ + xH

1 + xH2 )− (1 − θw)g(κ + xS

1 + xS2)]

From Assumption 2.2.2 it follows that (2.19), (2.20), and (2.21) vanish as κ →∞. Because (2.17) and (2.18) are strictly negative it follows that there exists

some finite κ such that for all κ > κ the problem each candidate faces is strictly

concave. Therefore, existence of a pure strategy Nash equilibrium follows from

Theorem 1.2 in Fudenberg and Tirole (1991), which is due to Debreu (1952),

Fan (1952), and Glicksberg (1952). From part 4 of Assumption 2.2.1 it follows

that a corner equilibrium cannot exist, because a deviation would always be

profitable. Hence, an interior pure strategy equilibrium must exist.


Proof. If we equate (2.10) and (2.12) and (2.11) and (2.13) and by noting that∂w

∂xH1

= ∂w∂xH

2

as well as ∂w∂xS

1

= ∂w∂xS

2

we get after simple manipulations that

−2(cH − cS) =

(∂cH

∂xH1

+ ∂cH

∂xH2

)w

∂w∂xH

1

, (2.22)

−2(cH − cS) =

(∂cS

∂xS1

+ ∂cS

∂xS2

)(1 − w)

∂w∂xS

1

. (2.23)

APPENDIX 83

Equating both yields the fundamental condition discussed above for ∆ = 0:

(∂cH

∂xH1

+ ∂cH

∂xH2

)w

∂w∂xH

1

=

(∂cS

∂xS1

+ ∂cS

∂xS2

)(1 − w)

∂w∂xS

1

(2.24)

Since ∂w∂xS

1

< 0 but ∂w∂xH

1

> 0 it is apparent that the signs of the terms in brackets

musts be different, or both bracket terms must be zero. Hence,

Sign

[∂cH

∂xH1

+∂cH

∂xH2

]= −Sign

[∂cS

∂xS1

+∂cS

∂xS2

].

The derivatives of ck with respect to xki are

∂ck

∂xk1

=θk(1 − θk) f ′(xk

1) f (xk2)

(θk f (xk1) + (1 − θk) f (xk

2))2

, (2.25)

∂ck

∂xk2

= − θk(1 − θk) f (xk1) f ′(xk

2)

(θk f (xk1) + (1 − θk) f (xk

2))2

. (2.26)

It follows that

∂cH

∂xH1

+∂cH

∂xH2

=θH(1 − θH)

[f ′(xH

1 ) f (xH2 )− f (xH

1 ) f ′(xH2 )]

(θH f (xH1 ) + (1 − θH) f (xH

2 ))2, (2.27)

∂cS

∂xS1

+∂cS

∂xS2

=θS(1 − θS)

[f ′(xS

1 ) f (xS2 )− f (xS

1 ) f ′(xS2 )]

(θS f (xS1 ) + (1 − θS) f (xS

2 ))2

. (2.28)

Sinced f ′(x)/ f (x)

dx = f ′′(x) f (x)− f ′(x)2

f (x)2 < 0 we find that

Sign

[∂ck

∂xk1

+∂ck

∂xk2

]=

+ if xk1 < xk

2,

0 if xk1 < xk

2,

− if xk1 < xk

2.

Hence, for (2.24) to hold it must be the case that

Sign[xH1 − xH

2 ] = −Sign[xS1 − xS

2 ].

Thus, unless both spend identical effort in issue H and also in issue S, can-

didates must specialize. Identical efforts in an issue can, however, only be an

equilibrium if θH = θS. To see this consider (2.22) and (2.23). If both spend iden-

tical effort on any issue, it follows from (2.27) and (2.28) that the right hand side

is zero. The left hand side, however, is then only zero if θH = θS. Hence, a no


specialization equilibrium can only occur if there are no comparative advan-

tages. To see that there can exist a no-specialization equilibrium look at the

first order conditions. If θH = θS and candidates do not specialize, cH − cS = 0.

Thus, what is left is equivalent to what we have seen when salience does not

matter for perceived importance. Existence of a symmetric equilibrium then

follows immediately. If in addition θw = 1/2, the ’prize’ in each competence

contest is identical and thus a perfectly symmetric equilibrium exists.


Proof. Consider the system of first order conditions in (2.10) - (2.13). I already

established that for m1 = m2 there is specialization whenever θH 6= θS. Note

that the comparative statics of the equilibrium can be derived by totally dif-

ferentiating the (2.10) - (2.13) and using Cramer’s rule. Since all derivatives

are continuous, a marginal change in ∆, evaluated at ∆ = 0, also must have a

marginal effect on efforts. Effort differences in the specialization equilibrium

are, however, discrete. Hence, there must exist a neighborhood of ∆ = 0 in

which there is still specialization as it was before.

To show that for ∆ too large one candidate spends more on both issues,

look at the limit when ∆ → ∞. Candidate 2 will then abstain from the contest

or spend only marginally, since the slope of f (x) at zero is also infinite. This is

independent of the effort spent by 1. Hence, candidate 2’s effort converges to

zero as m2 grows larger and larger. What now is candidate 1’s best response?

He clearly spends strictly positive effort on both issues. The issue he has a

comparative advantage in he clearly wants to promote. The other issue, since

initial spending has a very large impact on perceived competence, but his costs

are relatively low compared to candidate 2, he also spends and he spends more

than his opponent. To put it more technical, since the slope of w is finite by

assumption, the indirect cost of advertising the week issue are bounded for

either candidate. The total marginal costs are those indirect costs of making

the weak spot salient and the direct marginal costs of effort. As the direct

marginal costs of effort increase at one point the difference in marginal costs

outweighs the indirect costs of making the comparative disadvantage salient,

such that for the candidate with lower costs of effort in both issues marginal

costs are lower. Hence, this must also be true for the marginal products of

effort, implying this candidate must spend more on both issues. Hence, for ∆

too large there cannot be specialization anymore.

APPENDIX 85


Proof. Consider a symmetric, no-specialization equilibrium when θH = θS, θw>

12 , and ∆1 = 0. Then both candidates spend more effort on issue H. Now con-

sider a marginal change in θS. Since the first order conditions are continuous in

all variables this marginal change must have a marginal impact on efforts, too.

Effort differences between issues were, however, discrete, and hence a marginal

change cannot overcome this. Hence, there must exist a neighborhood around

the original value of θS for which both candidates spend more on issue H. In

this neighborhood one candidate must have a comparative advantage in S.

Chapter 3

“Where Ignorance is Bliss, ’tis Folly tobe Wise”: Transparency in Contests

Philipp Denter, John Morgan, and Dana Sisak†

†“Where ignorance is bliss, ’tis folly to be wise” is taken from from Thomas Gray’s (1768) poem“Ode on a Distant Prospect of Eaton College”. Many thanks to Stefan Bühler, Catherine Roux, ErnestoDal Bó, Qiang Fu, Martin Kolmar, Jochen Mankart, Johannes Münster, Steve Slutsky, Uwe Sunde,Steven Tadelis, Felix Várdy, Iván Werning, the participants of seminars in Berkeley, St. Gallen, andTübingen, of the 2009 SSES Congress, the 2009 Silvaplana Workshop on Political Economy, the 2009Young Researchers Workshop on Contests and Tournaments, the 2010 Public Choice Society meetingas well as the 2010 APET meeting. The kind hospitality of UC Berkeley and UC Irvine is gratefullyacknowledged. Dana Sisak and Philipp Denter both gratefully acknowledge the financial support ofthe Swiss National Science Foundation. All errors remain our own.

88 CHAPTER 3. TRANSPARENCY IN CONTESTS

3.1 Introduction

On March 20, 2009, U.S. president Barack Obama released a presidential memo-

randum on the subject of ensuring responsible spending of Recovery Act funds.

In this he promises to disclose all lobbying contacts on the distribution of Re-

covery Act funds within three business days to the public. In reaction to this,

on April 7, 2009 the Sunlight Foundation, a nonprofit, non-partisan organi-

zation promoting government openness and transparency, presented its own

proposal for real-time lobbying disclosure on their blog.1 After meeting with a

lobbyist, the government agency immediately submits a summary of the meet-

ing details through a standardized platform, and the results are accessible to

the general public on the internet. Instead of learning about them every quarter

year, journalists as well as the public will have an immediate basis to evaluate

the decisions of policymakers and the influence they were facing. But this is

not the only effect of increased transparency. Real-time disclosure also directly

informs the competing lobbyists about their opponent’s interests and doings.

In this paper we show how this can have bad consequences. Lobbying compe-

tition can become more fierce and less efficient.

This paper addresses the following questions. What information policy is

optimal, if a competitor in a contest can decide and commit to acquire relevant

information about his rival or disclose his own private information to the rival?

Do the competitors agree on information transmission? What is the effect of

mandatory disclosure policy on the outcome of competition? Our main results

are:

• Strong transparency policy in a competitive environment can have detri-

mental side effects for society. We identify conditions where it leads to

increased competition and less efficient outcomes.

• Decentralizing information disclosure instead is often beneficial. We iden-

tify conditions where the competing groups will agree to transparency

decisions, benefiting both the competitors and society at large.

• When outcomes are very sensitive to (lobbying) expenditures (e.g. luck

and outside factors become less important), decentralized agreement be-

comes unlikely. In these circumstances, neither mandatory disclosure nor

a laissez-faire transparency rule are optimal.

1http://sunlightfoundation.com/blog/2009/04/07/a-vision-of-real-time-lobbying-disclosure/.


Our main results may be illustrated through the following simple example:

Two competitors are vying for some prize. One of them (the incumbent) has a

known valuation for the prize while the valuation of the other (the newcomer)

is (potentially) unknown, and may be either high or low. The key intuition

underlying all of the results stems from the following observation: Competi-

tion is fiercest when the two rivals have similar valuations and milder when

valuations diverge. Consider first the decision to acquire information. While

better information helps the incumbent to choose an optimal effort level, if the

decision to acquire information is revealed, then the newcomer will also re-

spond. When the incumbent has a relatively high valuation, he is better off not

acquiring information since, if this information reveals that his opponent has

a high valuation, competition is sharpened while if the opponent is revealed

to have a low valuation, then the incumbent can no longer credibly commit to

deter his opponent through overinvestment. Thus, information acquisition is

unambiguously bad. On the other hand, when the incumbent has a relatively

low valuation, acquiring information is beneficial as it reduces the efforts of the

opponent regardless of valuation—in the case of high valuation, it stems from

the revealed divergence of values while in the case of low valuation, it stems

from discouragement.

Now, consider the decision of the newcomer to disclose information. If the

newcomer faces an incumbent with a relatively high valuation, competition will

be fierce if he discloses a high valuation and mild when his value is revealed to

be low. Since not disclosing leads to an intermediate level of competition, low

valuation newcomers prefer to reveal while high valuation ones do not. The

reverse is true when the newcomer faces a relatively weak incumbent: high val-

uation newcomers prefer disclosure while low valued ones prefer opacity. How

does this translate into a newcomer’s ex ante disclosure policy? His expected

payoffs are dominated by how he fares when he has a high valuation since this

raises both the benefits and chances of winning the contest. As a result, the

optimal policy is to disclose when the incumbent has a relatively high value

and to remain opaque when the incumbent has a relatively low value.

This means that the competing parties agree on disclosure when the value

of the incumbent is relatively high, and on non-disclosure otherwise. Thus, a

central insight to emerge from this analysis is that, despite the fact that the

two sides have opposing interests in that both want to win, they agree that less

“effort”, ceteris paribus, is good. Since information sharing affects the degree

of competition, there is scope for agreement. Furthermore information sharing

not only influences the degree of competition but also the efficiency in allocat-


ing the prize to the party who values winning most. Agreement on reduced

competition often also leads to greater efficiency in allocating the prize. When

information sharing is optimal, it results in greater separation in the efforts of

the two parties and, as a result, the prize is awarded to the higher valued party

more often. Likewise, when information sharing is not optimal, it again results

in greater separation of efforts. Thus, endogenous information sharing leads

to ex ante Pareto gains. In this circumstance, mandatory disclosure policies can

increase wasteful competition and distort prize allocations.

Consider some other examples of competitive environments in which trans-

parency policy is relevant. In the U.S., transparency in political campaigning

is regulated by the Federal Election Campaign Act (FECA). It requires candi-

dates to disclose sources of campaign contributions and campaign expenditure

quarterly. Not only is the public opinion affected by disclosure of this infor-

mation but also the campaign decisions of competing candidates and hence

competition. Disclosure of campaign contributions and expenditures conveys

information about the depth of financial support of a candidate and this in

turn influences the decisions of the opposing candidates and hence the election

outcome. This paper suggests mandating transparency can make candidates

compete more fiercely and thus competition more wasteful. Or consider com-

petition between firms. In the U.S., the Securities and Exchange Commission

(SEC) as well as the Federal Accounting Standards Board (FASB) regulate firms’

disclosure of financial information. This information is not only accessible by

stakeholders of a firm but also by its competitors, which has implications for

competition between firms if private information is revealed. Our results shed

light on how mandatory disclosure influences competition in winner-take-all

markets, or more generally markets where competition can be represented by

a contest. This is for example the case in advertising intensive markets, like the

market for soft drinks.

The chapter is organized as follows. Next we survey the related literature.

Section 3.2 introduces the model. Section 3.3 studies information acquisition,

Section 3.4 studies disclosure incentives. Section 3.5 puts the two decisions

together. Section 3.6 considers a more general contest success function and

Section 3.7 discusses the effect of mandatory disclosure policy. Section 3.8

studies the robustness of our findings with respect to the discriminatoriness of

the competition. Section 3.9 concludes.

Literature Review The nearest antecedent to our paper is Kovenock et al. (2010),

who study information disclosure between firms when the contest outcome is


very sensitive to contest expenditures. Our concerns are with both information

disclosure and acquisition and how they relate to the sensitivity of the contest

outcome to expenditures. Baik and Shogren (1995) study the effects of spying

and information acquisition in contest games. To gain tractability, they abstract

away from strategic considerations in the expenditures themselves – essentially,

the contest game is decision-theoretic. Our analysis, however, highlights the

importance of the strategic interaction between acquisition/disclosure and con-

test expenditures. Indeed, our main result is driven by the fact that acquisition

changes the behavior not just of the party gaining new information but also the

party whose information was disclosed.

Information acquisition and/or disclosure decisions have been studied in

three different but complementary settings to ours: Cournot and Bertrand com-

petition, auctions and agency theory. Vives (1984), Li (1985), Shapiro (1986) and

Darrough (1993) amongst others study information transmission in the context

of Cournot and Bertrand competition. With contests we add a third possible

form of competition between firms. Other papers, e.g. Persico (2000) or Eso

and Szentes (2007) have analyzed the incentives to acquire or disclose infor-

mation either about one’s private value or about a common value in auction

settings. With our analysis of an all-pay auction we complement this literature,

while adding a different dimension with the analysis of non-fully discriminat-

ing contests. One of our main results is to show that it can be optimal for a

lobbying group or firm to remain ignorant about the valuation its rival places

on “winning” the contest. The strategic value of ignorance has also been shown

in the context of agency theory. A principal may benefit from ignorance as it

alters the agent’s incentives to exert effort. The agent may benefit as well, as

ignorance may make it harder for the principal to extract rents. Papers high-

lighting these effects are for example Dewatripont and Maskin (1995), Barros

(1997) and Kessler (1998). While this literature focusses on vertical relationships

between two distinct parties, in our model the focus is on competing parties in

a horizontal relationship.

Information transmission from lobbies to the policy maker through lobby-

ing has been studied for example by Potters and van Winden (1992), Lagerlöf

(2007) and Grossman and Helpman (2001). The focus of this literature is on

the welfare implications of lobbying when lobbyists have private information

which is relevant to the policy maker and the policy maker attempts to learn

by observing lobbying expenditures. In contrast we focus on information trans-

mission between lobbyists and its implications for welfare and efficiency, and

highlight consequences for disclosure policy.


Information disclosure has also been studied in the context of goods mar-

kets, e.g. Jovanovic (1982), Milgrom (2008) and Daughety and Reinganum

(2008), where the focus is on whether markets lead to optimal incentives for

firms to disclose information about the quality of their goods. This literature

revolves around the trade-off that disclosure is beneficial for the consumer but

costly to the seller. In contrast, we show that mandatory disclosure can be

harmful even without direct monetary costs, purely through its strategic effect.

Finally, our paper is of course also related to the literature on asymmet-

ric information in contests (e.g. Hurley and Shogren (1998), Katsenos (2009),

Moldovanu and Sela (2001) or Hernandez-Lagos and Tadelis (2011)), and the

role of commitment in contests (e.g. Dixit (1987), Morgan (2003), Baik and

Shogren (1992), Morgan and Várdy (2007), Yildirim (2005) and Fu (2006)), al-

though the form of commitment typically consists of committing to a sequence

of moves. In contrast we study contests where players are able to commit to

certain informational regimes.

3.2 The Model

While we couch the model in the context of lobbying, it is easily translated

into other competitive situations.2 Consider two lobbying groups i = A, B who

vie for favorable legislation to be passed. Success yields lobby i a value vi

while failure yields zero. To affect the chances of success, each group chooses

lobbying effort xi. The chance that i is successful depends on the contest success

function (CSF):

pi

(xi, xj

)=

xi

xi + xj. (3.1)

If both groups choose zero lobbying effort (xi = 0) a coin toss determines suc-

cess. Lobbyists are risk-neutral with a constant marginal cost of effort normal-

ized to one. While each lobbying group knows its own valuation for success,

information about the other party differs. In particular, the valuation of group

A is commonly known while group B has private information about its value.

One can think of this situation arising when group A is an “incumbent” who

2We can easily reframe our model in terms of another introductory example – political campaigns.Two politicians i = A, B are campaigning for a political office. The political office yields i a value vi

while failure yields a value normalized to zero. To affect the chances of success, each politician choosessome amount of campaign expenditures xi. The chance that i is successful depends on the contestsuccess function (CSF) defined in equation 3.2. The talent of the incumbent politician is more or lesscommon knowledge and hence his value for office vA is known. For the newcomer we assume thevalue is low with probability q and high else.

3.3. INFORMATION ACQUISITION 93

has engaged in many past fights over related issues while group B is a new-

comer or, alternatively, where publicly available information makes it easy to

estimate A’s value while B’s value, perhaps being more subjective, is harder for

outsiders to estimate. For simplicity, we assume that B’s value is binary—it is

either low, vB = vL, with probability q or high, vB = vH , with the complemen-

tary probability. In Appendix 3.9, we show that qualitatively similar results are

obtained when B’s distribution of values occurs on a continuum. The payoff

functions are equal to

πB =xB

xB + xAvB − xB

πA =

(q

xA

xBL + xA+ (1 − q)

xA

xBH + xA

)vA − xA.

We focus on the case where there is uncertainty as to which lobbying group

has the higher valuation, i.e., when vA ∈ [vL,vH]. Furthermore we assume that

the policy is valuable enough for all lobbying groups to choose strictly positive

lobbying effort.

3.3 Information Acquisition

In this section we consider the incentives to acquire information about one’s

opponent before the contest. In terms of our model, suppose that it were cost-

less for group A to acquire a credible report as to B’s valuation before the start

of the contest and this decision is common knowledge. Afterwards the contest

described in Section 3.2 takes place. One might be tempted to draw an analogy

with a bargaining situation. In effect, A and B are negotiating (through their

efforts) on who will receive the valuable legislative prize. The usual advice

in such situations is to “know thy enemy”. That is, group A should gather

as much information as possible about group B, including its valuation. This

information will enable it to make the best possible decision regarding its ne-

gotiation strategy, which can now be type-specific. Since information gathering

is costless, it seems obvious that the optimal strategy is complete information

gathering.

Where the analogy breaks down is in the form of the “negotiation” be-

tween the two parties. Here, success will be determined by performance in an

imperfectly discriminating contest; thus, there is an integrative as well as dis-

tributive aspect to the “negotiation.” In particular, both lobbying groups benefit

if lobbying efforts are more muted and, since only relative lobbying efforts de-


termine the outcome, equilibrium success probabilities would be unaffected if

both sides could agree to scale down their efforts.

But how can ignorance enable the lobbying groups to scale down effort?

Consider a lobbying group A which has a valuation above the average of lob-

bying group B. If it knew for sure it faces a strong group B, competition be-

tween the similarly strong groups would be very intense. But the chance to

encounter a much weaker group B diminishes A’s investment incentive, and

hence also the strong group B’s reaction because from its view investments are

strategic complements. On the other hand, A overinvests against a weak group

B to increase its chances in case its opponent turns out to be strong. The weak

group B will react to this discouragement by lowering its investment because

its investments are strategic substitutes. By optimally choosing to remain igno-

rant about lobbying group B’s valuation, A can on the one hand discourage a

weaker rival and on the other hand appease a stronger rival, thereby softening

the competition between the two lobbies. Thus, unlike a decision-theoretic or

negotiation context, rent-seeking competition between the two parties creates a

value to ignorance.

A sharp illustration of this intuition may be seen for the case where group

A has diffuse priors (i.e. q = 1/2). Here we show that, when group A is strong

compared to B, it prefers to remain ignorant while when it is weak, it seeks

information to mitigate this disadvantage. Formally,

Proposition 3.3.1. If lobbying group A is relatively strong compared to group B (vA >√vLvH) it strictly prefers not to acquire any information about B’s value while a

relatively weak lobbying group A (vA <√

vLvH) always acquires costless information

about group B.


Figure 3.1 illustrates the intuition behind the value to ignorance graphically.

It shows the best response functions of both groups when A knows the valu-

ation of group B. Optimal lobbying expenditures under complete information

are given where the best response functions intersect. If group A’s value is

relatively high, its lobbying effort under ignorance (vertical line) is higher than

under complete information in case it faces the low value opponent (left panel),

while the opposite is true against the high value opponent (right panel). We

can directly see that this benefits A by decreasing both its opponents’ lobbying

efforts.3

3Technically speaking, our results are due to the non-monotonicity of reaction functions. This

3.3. INFORMATION ACQUISITION 95

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.5

1.0

1.5

2.0

2.5

3.0

xBxB

xAxA

xA(xB)xA(xB)

xB(xA;vH)

xB(xA;vL)

xAIAxAI

A

Figure 3.1: The left panel shows the full-information best response functions whenlobbying group A faces a weak opponent, the right panel when it faces astrong opponent. xAI

A denotes the lobbying effort of A under ignorance.Under ignorance (dot) both types of B expend less than under full infor-mation (square).

Softening competition through ignorance does not always work. If group

A’s valuation is below the geometric mean of vB, ignorance increases compe-

tition. A weak group A invests little when facing a much stronger group B

while it fights hard against the just slightly weaker group B, where competi-

tion is more equal. By staying ignorant A finds itself overinvesting in case it

faces the stronger group B, which reacts to this threat with an increase in in-

vestment. At the same time it underinvests in case it faces the weak group B,

which also reacts with an increase in investment, sensing a good opportunity.

Hence a weak lobbying group A always acquires costless information.

Note that if group A’s decision to acquire information were not observable

to group B, A would always choose to acquire information about B’s value.

Deviating from ignorance to information acquisition enables A to play a best

response while B does not change its behavior as the deviation is unobservable.

In equilibrium this is anticipated by group B and the contest always takes place

under complete information. In this sense observability is a form of commit-

ment opportunity that enables A to commit to a beneficial action which would

otherwise not be feasible, as it is not in its complete-information best response.

implies that efforts are strategic complements for the favorite while they are strategic substitutes forthe underdog, where in our set-up the favorite is the group with the higher valuation. See Dixit (1987)for a discussion.


In fact, commitment to ignorance can have a similar effect as pre-commitment

of effort. If group A had the opportunity to be a Stackelberg leader, meaning

it could pre-commit its contest effort in a way observable to B, it would choose

to overinvest relative to simultaneous moves against a lower-valued rival while

it would choose to underinvest against a higher-valued rival. Both rivals react

to this precommitment with a decrease in investment (Dixit (1987)).

3.4 Information Disclosure

Lobbying group A’s decision to stay ignorant could well be obsolete if group

B can credibly disclose its value to A. In fact, it is not clear what happens if A

and B disagree about whether B’s value should be revealed. In this section we

explore the other side of the information transmission decision and focus on

group B’s incentives to disclose its valuation to A. There are many possibilities

how disclosure could work. As a first step we assume that lobbying group B

has the opportunity to commit ex-ante, before learning its value, to a disclosure

policy. In case it chooses to disclose, it discloses its value truthfully and without

cost to A after learning it and before the start of the contest. In this sense we

give B a commitment opportunity to maximize its ex-ante welfare. At the end

of this section we discuss this assumption and analyze an alternative model

where B can only use a costly signal to signal its value to A.

Even though disclosure enables the opponent to make a more informed

decision, this does not necessarily mean that the disclosing group is hurt by

this. For example if the opponent learns that the group has a much higher

valuation it will optimally react by lowering its expenditures, as its chances of

success are so slim, and this is beneficial for both groups. On the other hand, if

the opponent learns the lobbying group has a very low valuation, it might also

find it beneficial to lower its expenditures, as not much is needed for success.

Disclosing a similar valuation on the other hand makes competition fiercer.

If the disclosure decision is made ex-ante, we find that information is only

disclosed when B faces a relatively weak group A. Formally,

Proposition 3.4.1. Assume lobbying group B does not know its value yet but is given

the opportunity to commit ex-ante to a disclosure policy. If lobbying group B expects to

be relatively weak compared to lobbying group A (√

vLvH < vA) it strictly prefers to

commit to non-disclosure. On the other hand, a lobbying group B with a high expected

valuation (√

vLvH > vA) always commits to disclose its value.


3.4. INFORMATION DISCLOSURE 97

To make the intuition behind Proposition 3.4.1 clearer let us first look at the

incentives of a high- and a low-value lobbying group B separately. A high-value

lobbying group B will prefer disclosure if it can discourage lobbying group

A from expending lobbying effort. This is the case whenever it is relatively

strong, or vA <√

vLvH . For vA ≥ √vLvH disclosing makes A more aggressive,

as it learns that its opponent is of similar strength. The opposite is true for a

weak lobbying group B. When facing a strong group A it prefers to disclose its

valuation, as A will react with lower lobbying effort. If A is weak on the other

hand, revealing its valuation makes competition stronger, as A learns that it is

facing a similarly strong opponent. The weak and the strong lobbying group

B’s incentives are never aligned. If disclosing is beneficial for one, it is harmful

to the other. From an ex-ante point of view, before learning its valuation, the

strong lobbying groups’ interests always dominate though. The reason is that

an increase in success probability in case the value is high is worth more than

in case the value turns out to be low.

Notice that the conditions for information disclosure/withholding in Propo-

sition 3.4.1 are identical to those in Proposition 3.3.1 when group A is determin-

ing whether to pursue this information. That is, despite competing with one

another, both groups agree on information revelation. We formalize this obser-

vation in Corollary 3.5.1 in Section 3.5.

Ex-ante commitment to a disclosure policy is an interesting benchmark but

might not always be feasible. Also costless and truthful revelation can be an

unrealistic assumption in some settings. To test the robustness of our results,

we consider an alternative model. Let us assume that disclosure of the lobbying

group’s value is costly and not verifiable. Instead, a lobbying group has the

option of sending a costly signal in order to try to inform A of its value. We

assume that costs of the signal si are linear, c(si) = si, i = H, L, and signaling

takes place before the start of the contest. Then we find:

Proposition 3.4.2. After lobbying group B learns its valuation and given the chance

to send a costly signal before the contest to group A, only a high-value lobbying group

credibly reveals its valuation. This is only profitable in a situation where group A is

relatively weak (√

vLvH > vA). Otherwise no information is disclosed.


The intuition for this proposition carries over from the one for Proposition

3.4.1. A lobbying group with a high valuation stands to gain more from a

decrease in A’s lobbying effort. This means that it is willing to expend more


��

��

��

��

��

��

��

��

��

��

��

t

t = 1 t = 2 t = 3

Figure 3.2: Sequence of moves.

signaling effort than a low-value group. If it is in its interest, it will always be

able to imitate a low-value group’s signal so that no information is disclosed.

Hence against a strong group A information will never be disclosed because

it is detrimental to the high-value group, while against a weak group A the

high-value group is willing to credibly disclose its valuation through the costly

signal. Our results are in line with the results in Katsenos (2009) who analyzes

costly signaling in a lottery contest with two-sided asymmetric information

and two possible types of valuations, vH and vL for both parties. He finds

that separating equilibria only exist, when the probability to face a strong op-

ponent is sufficiently low. Our result complements this finding in a one-sided

asymmetric information setting where vA can be different from vH and vL.

3.5 Information Transmission

So far we have analyzed the lobbying groups’ disclosure and acquisition deci-

sions separately. Now we combine these analyses to find out, how lobbying

groups exchange information voluntarily. In Section 3.7 we then compare our

findings to lobbying under mandatory disclosure policy.

The game proceeds as follows: Prior to the start of lobbying, each lobbying

group engages in information disclosure/acquisition decisions; that is, group

A decides whether to pursue credible information about B’s valuation while

group B simultaneously decides on its disclosure policy. Following information

acquisition/disclosure, both lobbying groups simultaneously choose lobbying

efforts and payoffs are resolved. Figure 3.2 illustrates the flow of the game.

We assume that lobbying group B has not learned its valuation when decid-

ing on information disclosure. In Proposition 3.4.2 we showed that our results

extend to an alternative set-up where B has learned its valuation and has the

possibility to send a costly signal to group A. Then if both lobbying groups

agree that information should be exchanged (B prefers disclosure and A acqui-

sition) A will learn the value of group B. If on the other hand both lobbying

3.5. INFORMATION TRANSMISSION 99

groups agree not to disclose (B prefers non-disclosure and A ignorance), no

information is transmitted. What is not so clear is what happens if A and B do

not agree. For example A might want to acquire information about B’s value,

but B might not be willing to disclose it. Or B might want to disclose its value

while A does not want to acquire it. The payoff in these situations which we

denote by πDi , could be equal to πCI

i , or πAIi or anything in between depending

on how exactly information transmission works. For our results in this section

we do not need to make any assumption as to what exactly will happen in

these cases as long as πDi ≤ max

{πCI

i ,πAIi

}.

Consider again the case with group A having diffuse priors (i.e. q = 1/2).

Then the lobbying groups always agree on information transmission between

them. Formally,

Corollary 3.5.1. If lobbying group B expects to be relatively weak compared to lobbying

group A (√

vLvH < vA) both lobbying groups agree not to transfer any information

while if lobbying group B expects to have a high valuation compared to A (√

vLvH >

vA) both agree on disclosure.

Proof. This follows from the proof of Propositions 3.3.1 and 3.4.1. There we

found that πCIi > πAI

i for vA <√

vHvL and πCIi < πAI

i for vA >√

vHvL, i = A, B.

For vA =√

vHvL both groups are indifferent. We have the following payoff

matrix.B discloses B doesn’t disclose

A acquires πCIA ,πCI

B πDA ,πD

B

A doesn’t acquire πDA ,πD

B πAIA ,πAI

B

Depending on πDi , multiple Nash equilibria are possible. For example even

though πCIi > πAI

i , i = A, B in case vA <√

vHvL, staying ignorant and not dis-

closing is a Nash equilibrium when information is only transferred if both

parties agree(πD

i = πAIi

). This equilibrium though is Pareto dominated by the

one where A acquires information and B discloses. In this sense the lobbying

parties, given a chance to coordinate, would always agree on the Pareto supe-

rior equilibrium. This is also the only trembling hand perfect equilibrium. Note

that this is also the unique equilibrium when parties can decide sequentially

on information transmission.

We find the lobbying groups’ incentives to be always aligned.The reason

for this is that there exist gains from coordination in the form of reduced com-

petition. By coordinating, both parties can save on lobbying expenditures.

This finding can also be related to the literature on sequential moves and

pre-commitment of effort in contests. Corollary 3.5.1 is in a sense analogous to


the findings in Baik and Shogren (1992) and Leininger (1993), who analyze the

choice of the order of moves in sequential rent-seeking contests. They find that

it is in the interest of both lobbying groups to choose the sequence of moves

where the least efforts are expended. This means that both groups always pre-

fer the weak group to go first and pre-commit contest effort. It chooses a low

lobbying effort and the strong group reacts with lower lobbying effort as well.

Even though the weak group ends up winning less often, it is compensated

by lower lobbying costs. When choosing whether to disclose a similar logic

applies. Staying ignorant can have a similar effect as moving first, if it enables

A to move closer to its Stackelberg point. As we have shown, this is the case

for a relatively strong lobbying group A. By staying ignorant it can credibly re-

duce its investment against the high-valuation lobbying group B who will react

by reducing its expenditures as well. Interestingly in this set-up the strategic

complementarities from facing a high-valued rival always dominate, and hence

agreement is possible, even though efforts are strategic substitutes for the low-

valued lobbying group B.

Our results require very little structure in determining how exactly infor-

mation transmission works. The only essential prerequisite is some form of

commitment opportunity. In reality, this could take many forms. For exam-

ple, one purpose of trade associations is to facilitate information exchange (e.g.

Kirby (1988) or Vives (1990)). Members commit themselves to share their pri-

vate information with the help of the trade association, while for non-members

it will be much harder to reveal and receive credible information. Another

example of institutionalized information exchange are strategic marriages. A

strategic marriage policy was pursued by many houses of European rulers dur-

ing the Renaissance and thereafter. The probably best known example is the

House of Habsburg’s strategic marriage to Spain and Italy. Among other things

these strategic marriages can serve as commitments to disclose credible infor-

mation to and acquire credible information about other empires. Another nice

historical example about the voluntary exchange of credible information can be

found in Schelling (1960), “[t]he ancients exchanged hostages, drank wine from the

same glass to demonstrate the absence of poison, met in public places to inhibit the mas-

sacre of one by the other, and even deliberately exchanged spies to facilitate transmittal

of authentic information”. Our analysis provides a rationale for this: exchanging

authentic information can decrease the fierceness of conflict, something that is

good for both parties.

3.6. MORE GENERAL CONTEST SUCCESS FUNCTION 101

3.6 More General Contest Success Function

So far we have assumed that the lobbying process can be represented by a

simple lottery contest. In order to show the robustness of our results, in this

section we assume the political process can be represented by a more general

CSF of the following form:

pi

(xi, xj

)=

f (xi)

f (xi) + f(

xj

) (3.2)

where f ′ > 0 and f ′′ ≤ 0.4

As we have seen in the previous section, whether ignorance is bliss for the

lobbying groups is determined by whether or not group A’s value is above the

average of group B’s valuations. Proposition 3.3.1 shows though, that it is not

the arithmetic average; rather the decision to acquire or disclose information

turns on the geometric mean of B’s value. Next we show that such a criti-

cal value of lobbying group A’s valuation, let us denote it by vA, exists more

generally.

Lemma 3.6.1. For every q, there exists a value vA ∈ [vL,vH] such that, if vA = vA,

lobbying group A is indifferent between acquiring information or not, and lobbying

group B is indifferent between disclosing information or not.


To illustrate the intuition for the proof of this lemma, assume A knows

its opponent. When A faces a weak opponent B, a relatively small lobbying

effort will basically guarantee success for A. With an increase in B’s value,

A increases its optimal lobbying effort until both groups have an equal value.

Here competition is at its fiercest. Now an increase in B’s value will start to

discourage A from investing, until at one point B becomes so strong that A

invests barely anything. This logic implies that there will always be two possi-

ble values of group B, one larger than A’s, one smaller, such that A expends

exactly the same lobbying effort. If group B has exactly these values, vL and

vH , A’s behavior will be unchanged whether it knows B’s value or not.

It is tempting to reason from Lemma 3.6.1 that Propositions 3.3.1 and 3.4.1

hold for more general prior probabilities of B’s values vL and vH and more

general lobbying technologies. Indeed, we can generalize Propositions 3.3.1

and 3.4.1 as well as Corollary 3.5.1 locally around the critical value vA.

4This is a standard contest success function, see Skaperdas (1996) for an axiomatization.


Proposition 3.6.1. In a neighborhood of vA, if lobbying group B expects to be relatively

weak compared to lobbying group A (vA < vA) both lobbying groups agree not to

transfer any information while if lobbying group B expects to have a high valuation

compared to A (vA > vA) both agree on disclosure.


Is there a reason why Proposition 3.6.1 might not always hold globally, as

does Corollary 3.5.1? It can be shown that under certain circumstances there

can be disagreement between the lobbying groups. The reason is that the criti-

cal value vA for Lemma 3.6.1 is not always the only critical value for group A.

To illustrate, take a very strong lobbying group A with a value close to vH and

assume that the probability of facing a strong group B is small. Then group

A’s lobbying effort under ignorance is similar to the lobbying effort knowing it

is facing a weak group B. But if B happens to be strong and A were ignorant,

it would underinvest by a large amount. Even though this leads the strong

group to reduce its effort, this is not optimal for group A. In fact, there is

an optimal degree of underinvestment against a stronger opponent. If A had

the opportunity to precommit lobbying effort, this would be the effort level it

would optimally choose, the so-called Stackelberg point. Ignorance can enable

lobbying group A to move closer to this optimal effort in certain situations. In

other situations A will surpass the Stackelberg point under ignorance, as in

the example above. If A surpasses the Stackelberg point by too much, acquir-

ing information is the optimal strategy. Consequently, there exist situations

like the one described above where the two lobbying groups will not agree on

information transmission.

3.7 Mandatory Disclosure Policy

Transparency policy is a topic of high relevance in many political debates

around the world. For example in the U.S., transparency laws have been passed

regulating lobbying, political campaigning or financial accounting of firms. A

large part of the U.S. economy is hence affected through transparency laws.

Thus it is important to understand all possible consequences of mandatory dis-

closure policy. In competitive environments like the ones mentioned above,

transparency policy can affect the nature and outcome of competition. Here

we take a closer look at exactly this effect. In the previous section we saw that

typically the competitors agree on whether to disclose information between

themselves. In many cases they agree not to disclose any information to their

3.7. MANDATORY DISCLOSURE POLICY 103

mutual benefit. Transparency policy, on the other hand, forces the competing

parties to disclose certain information to the public, and hence also to their

competitors.

We focus our analysis on two outcome variables: expected aggregate lob-

bying efforts and expected allocative efficiency. It is typically in the interest of

a society to keep lobbying efforts low, since lobbying activities are not directly

productive but serve only to influence policy. In our model this is captured

by the fact that by scaling down efforts proportionally both groups still win

the contest with identical probability. This decrease in lobbying investment can

be used for directly productive activities. Of course, in a frictionless world

one could argue that markets would always allocate these funds efficiently. In

reality, this is certainly not always the case. Furthermore there is also a misal-

location of non-monetary resources, as for example human capital, and hence

reducing lobbying efforts seems a reasonably aim. It is also in the interest of

a society to have the probability that a law or bill which has a relatively high

social value be passed as large as possible. This social value is represented in

our analysis by the lobbying groups’ valuations. We implicitly assume that all

individuals affected by the policy are part of one of the two lobbying groups,

for example a “pro” and a “contra” group. Inside each group there are no trans-

action costs and no externalities, and thus the groups’ valuations for the policy

perfectly reflect societal preferences. This can be seen as an approximation for a

situation where both groups face similar free-rider problems.5 Consequently it

is in society’s interest that a higher valued lobbying group has the best chances

to succeed. We will refer to this as expected allocative efficiency henceforth.

Mandatory disclosure policy can take many different forms, ranging from

disclosure of information about actions (e.g. expenditures or efforts) to disclo-

sure of characteristics (e.g. valuations, costs or productivity), or any mix thereof.

Depending on its form, mandatory disclosure will impact competition to differ-

ent degrees. In this paper we consider disclosure policy about the competitors’

characteristics. Transparency about actions can reveal something about char-

acteristics, but does not necessarily have to (in technical terms there can be

pooling or separating equilibria). Thus our analysis also applies to disclosure

about actions, whenever information about characteristics is revealed.

Let us resume the example of lobbying introduced in Section 3.2 and first

assume that we are interested in keeping the expected wastefulness of the lob-

bying competition low and it is irrelevant for society which lobbying group is

5Free-rider problems in group contests with public goods prizes are discussed for example in Este-ban and Ray (2001) or Kolmar and Rommeswinkel (2010).


successful. This could for example be the case in rent-seeking contests. We

then get the following result.

Proposition 3.7.1. Expected aggregate effort is lower under

• information disclosure if lobbying group A is relatively weak (vA ≤√vHvL),

• asymmetric information if lobbying group A is relatively strong (vA >√

vHvL).


As foreshadowed in Section 3.5 we find that if the uninformed lobbying

group is relatively strong, mandatory information disclosure makes the lobby-

ing process more wasteful in expectation. In addition, we have shown in Corol-

lary 3.5.1 and Proposition 3.6.1 that in many situations the lobbying groups

voluntarily agree not to transfer any information. In these cases a “laissez-

faire” policy leads to less wasteful competition. If we assume that information

can only be transferred when it is in lobbying group B’s interest to disclose its

information, we can conclude the following.

Corollary 3.7.1. If society is interested in keeping lobbying expenditures low a “laissez-

faire” policy is preferable to a policy of mandatory disclosure.

Next we consider expected allocative efficiency. We define expected al-

locative efficiency as the probability that the lobbying group with the highest

valuation wins the lobbying contest. Then we can show

Proposition 3.7.2. Expected allocative efficiency is greater under

• information disclosure if lobbying group A is relatively weak (vA ≤√vHvL),

• asymmetric information if lobbying group A is relatively strong (vA >√

vHvL).


This finding also relates to the literature on sequential contests. As we

discussed in Section 3.3, asymmetric information enables the uninformed lob-

bying group to act similar to a Stackelberg leader when it is sufficiently strong

relative to the informed lobbying group. Morgan (2003) finds that sequential

rent-seeking contests dominate simultaneous ones in terms of efficiency. Hence

if asymmetric information enables A to get closer to its Stackelberg point, which

is true for vA >√

vHvL, it also improves efficiency. Together with the results in

Corollary 3.5.1 and Proposition 3.7.1 we find the following.

3.7. MANDATORY DISCLOSURE POLICY 105

Corollary 3.7.2. Assume that society is interested in increasing expected allocative

efficiency and keeping expected wastefulness of the lobbying competition low. Then a

“laissez-faire” policy is always weakly superior, independent of the relative weights the

policy maker places on the two goals.

With a completely altruistic policy maker, transparency is clearly beneficial

for efficiency. Only if it is known which policy is the best, can it be chosen by

the policy maker. If the policy maker follows his self-interests and bases his

decision on lobbying efforts, transparency will have a differential effect on the

lobbying groups, sometimes favoring the “weaker”, sometimes the “stronger”

one. As we have shown, this can lead to another undesirable side-effect of trans-

parency policy, a decrease in expected allocative efficiency. At the same time,

our result has the potential to explain the emergence of mandatory disclosure

policies, even though shown to be inefficient. A policy maker interested in max-

imizing his rent-seeking revenues always weakly prefers mandatory disclosure

to voluntary disclosure.

Furthermore, note that disclosure policy which does not affect current lob-

bying competition, in other words disclosure with a sufficient time lag or with

“soft” disclosure requirements which do not reveal anything about the com-

petitors’ characteristics, does not have these detrimental effects. At the same

time it can still afford possible benefits through increased accountability and

better informed voters. In this respect our findings help evaluate calls for an

increase in transparency, as for example by the Sunlight Foundation in the U.S..

Coming back to our introductory example, the demand for real-time lobbying

disclosure, our findings imply that even apart from the direct costs of increased

transparency such as bureaucratic expenses, this policy is likely to have indi-

rect costs in terms of an increase in expected wastefulness and a decrease in

expected allocative efficiency of lobbying competition, which have to be traded

off against the additional benefits.6

6There may be another negative effect of transparency, not captured in our model. Higher trans-parency makes direct transfers of funds from lobbying groups to policy makers less likely, because thiswould be considered bribery or corruption, which is typically illegal. Of course, this does not meanthat lobbying groups stop exerting pressure. Rather they (partially) substitute away from transfers tolegal sources of effort, which are usually labor intensive. But this has direct negative consequencesfor efficiency and wastefulness of the competition. While bribing is purely distributive and thereforefunds are not “wasted”, labor intensive lobbying directly wastes resources and hence is an allocativeproblem. Therefore, it can be argued from a wastefulness perspective that bribery has an advantageover lobbying, what is in line with for example Lambsdorff (2002). Consequently, transparency may notonly increase lobbying effort, but is likely to influence the composition of lobbying effort in a sociallyundesirable way.


3.8 Noisiness of the Contest and the Scope for Agree-

ment

So far we have implicitly assumed that lobbying expenditures do not perfectly

determine the outcome of the competition. By spending more in the contest a

lobbying group can increase its chances to succeed, but there always remains

some uncertainty. Put differently, the lobbying group with the lower expendi-

tures still has a non-zero chance of success – the lobbying process is at least

somewhat noisy. There are different reasons this might be true. For example,

policy makers may have preferences over political outcomes unknown to the

lobbying groups, or face imperfectly observable constraints. Another reason

for a noisy lobbying process from the lobbying groups’ perspective is that lob-

bying efforts are only imperfectly observable by the policy maker. This could

be due to the complexity of the subject so that it is difficult for lobbyists to com-

municate their concerns properly, or because it is not clear ex-ante what the

best strategy to approach a political decision maker is and which consequences

of the favored bill to highlight.

We have captured this uncertainty by using a non-deterministic CSF of the

ratio form, as defined in equation (3.2). We now consider a CSF which can

be interpreted as the limiting case when noise vanishes completely, the all-pay

auction. It represents a situation where the political process is very sensitive

to lobbying effort and where the lobbying group with the highest expenditure

wins with certainty.7 This higher sensitivity implies higher marginal returns to

lobbying effort and therefore increases the fierceness of the competition. It is

interesting to consider this situation as an extreme case, because it is implicitly

assumed that policy makers do not have any private preferences about the po-

litical outcomes, do not face any constraints and the process of communication

between the lobbying groups and the policy maker is free of misunderstand-

ings and noise. In short, the policy maker bases his decision solely on lobbying

expenditures. The next proposition shows how an absence of noisiness influ-

ences the incentives to coordinate on information transmission.

Proposition 3.8.1. When the political process takes the form of an all-pay auction

1. disclosing information is weakly dominated for lobbying group B,

2. staying ignorant is weakly dominated for lobbying group A,

7The standard references analyzing all-pay auctions are Hillman and Riley (1989), Baye et al. (1993,1996), and Krishna and Morgan (1997).

3.8. NOISINESS OF THE CONTEST AND THE SCOPE FOR AGREEMENT 107

3. the lobbying groups’ incentives are never aligned and therefore they will never

agree on transferring information voluntarily.

Proof. See appendix.8

This result reveals that the contest’s degree of sensitivity to rent-seeking ef-

forts influences when the lobbying groups agree on information transmission.

In contrast to ratio form contests, in a fully discriminating contest the lobbying

groups’ incentives are never aligned. The informed group never discloses its

information while the uninformed group always takes an opportunity to ac-

quire information. Because of the fierceness of competition there is no scope

for agreement.

Consider the lobbying groups’ incentives separately. Why does lobbying

group B never benefit from disclosing its valuation? Under a noisy political

process, by disclosing its value, a strong group B discourages a weak group A

from investing. This does not work when the political process is fully discrim-

inating. By disclosing information, a strong lobbying group will only secure

itself a payoff equal to the difference in valuations between itself and its op-

ponent. All other rents are dissipated through competition. With asymmetric

information competition is less fierce and it can in addition earn informational

rents. In fact, it can secure itself the exact same payoff with one-sided asym-

metric information (by marginally overbidding group A’s valuation) and might

even do better. Technically speaking, in all-pay auctions both reaction functions

are monotonically increasing until the valuation of the weakest lobbying group

so there will be no discouragement effect in the relevant range.

Why is there no value to ignorance? When policy makers are perfectly re-

sponsive to lobbying expenditures, there is no advantage to pre-committing lob-

bying expenditures, as has been shown for example in Konrad and Leininger

(2007). In fact, a low-valuation lobbying group is indifferent with respect to tim-

ing while a high-valuation group prefers to decide after its opponent chooses

its expenditures. Hence the advantage from ignorance highlighted under an

imperfectly discriminating political process does not apply this setting — igno-

rance cannot dampen competition to the benefit of both parties, it only benefits

the opponent. Hence lobbying group A always acquires information.

What are the consequences for disclosure policy? First of all, Proposition

3.8.1 shows that lobbying groups don’t agree on disclosure and hence it is no

8A proof for part 2 of the Proposition has first been given in Kovenock et al. (2010) for two-sidedasymmetric information and a continuous distribution of types.


a.) b.)

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.2

1.4

1.6

1.8

2.0

q

vA

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.2

1.4

1.6

1.8

2.0

q

vA

Figure 3.3: Comparison of aggregate effort (panel a.)) and efficiency (panel b.)).

longer clear what happens under a laissez-faire transparency rule. Further-

more, a reduction in expected aggregate effort and an increase in expected

allocative efficiency, two possible objectives of society, are no longer necessar-

ily compatible as we show now in an example. We find that expected aggre-

gate effort is typically smaller under complete information when A’s value is

not too close to either vH or vL and under asymmetric information else. Ex-

pected allocative efficiency is typically greater under asymmetric information

except if vA is relatively small and q is relatively large. The reason is the fol-

lowing. Asymmetric information has two effects on allocative efficiency when

the policy maker is perfectly responsive to lobbying expenditures. On the one

hand it stratifies the range of efforts of lobbying group B. A low-valuation

group chooses its investment from an interval of the form [0, x] while the high-

valuation group chooses from [x, x]. In contrast, under complete information

they choose from the interval [0, xi], i = H, L. This is beneficial for efficiency. On

the other hand we showed that lobbying group B benefits from informational

rents. Especially when A is very likely to face a low-valuation opponent and vA

is close to vL, this becomes important for efficiency. B’s informational advan-

tage will lead to a low-valuation type winning too often, decreasing efficiency.

In theses cases the detrimental effect of asymmetric information dominates and

expected allocative efficiency is higher under complete information.

Figure 3.3 illustrates this for vL = 1 and vH = 2. In darkgray regions com-

plete information is optimal while in lightgray regions asymmetric information

is preferred. So decreasing expected aggregate effort often implies decreasing

expected allocative efficiency. We can draw the following conclusions regarding

mandatory and voluntary disclosure policy.

Corollary 3.8.1. Policy makers who are perfectly responsive to the influence of lob-

3.9. CONCLUSION 109

byists make decentralized agreement impossible. In these circumstances, neither a

laissez-faire transparency rule nor mandated disclosure is optimal in our framework.

Furthermore, achieving an increase in expected allocative efficiency and a decrease in

expected aggregate effort through disclosure policy becomes unlikely as these two goals

are often in conflict.

Summarizing our results, we find differential effects of transparency policy

on lobbying competition depending on the noisiness of the political process.

While under a sufficiently noisy political process a laissez-faire policy leads to

the best outcome in terms of expected aggregate effort as well as allocative effi-

ciency, this need not be true under a perfectly discriminating political process.

Here the effect of transparency policy is ambiguous and no general results can

be obtained to guide policy decisions.

3.9 Conclusion

How do we evaluate the recent proposals for more transparency in U.S. lobby-

ing? If transparency were free to implement, would more transparency always

be better for society? Even though we cannot give a conclusive answer to these

questions, our analysis highlights a side-effect of transparency policy which has

been absent from the policy debate so far. We show how an increase in trans-

parency can lead to an increase in the wastefulness of lobbying competition and

at the same time to a decrease in the probability that the lobbying group with

the most pressing interests succeeds. Furthermore we show that in the absence

of mandatory disclosure policy, competitors often agree whether or not to share

information and this decision reduces wastefulness and increases allocative ef-

ficiency. Our results have implications beyond lobbying. These considerations

hold weight for the analysis of transparency policy in other competitive settings

like political campaigning or financial accounting of rival firms.

While we focused in our assessment of the welfare implication of trans-

parency on an environment in which effort is considered wasteful, there are

other environments in which effort is considered (socially) beneficial. An im-

mediate example is student’s effort in school or at university. Higher effort

generates better educated graduates, which is beneficial for society as a whole.

Typically grades are based on relative performance (grading on a curve), so

students’ competition for grades is a contest and we can apply our results. We

know from Section 3.7 that transparency leads in expectation to increased effort.

Consequently, to increase students’ efforts a transparent studying environment


is likely to be helpful. This can be achieved by promoting studying in groups

or by testing students frequently over the term and publicizing the test scores.

An interesting extension of our analysis would be to allow for common

values. This can be relevant in many settings. In our lobbying example the

lobbyists might posses relevant information about the value of the policy at

stake, as for example when lobbying for a monopoly position and each firm has

done market research. Lobbying groups learn not only about their opponent’s

interest, but also about their own. Most importantly, to draw more precise

policy conclusions a more general model of all affected parties is needed to

evaluate all the possible effects of transparency policy and their interactions.

For example transparency policy in lobbying will also affect the relationship

between the policy maker and the general public. To combine these factors

into one model is an important avenue for future research and will allow a

more thorough evaluation of transparency policy.

APPENDIX 111

Appendix

Proof of Propositions 3.3.1, 3.4.1 and 3.4.2

Equilibrium under Full- and Asymmetric Information. Equilibrium efforts, prob-

ability of success and utility under complete information are equal to (see Nti

(1999))

xCIi

(vi,vj

)=

v2i vj

(vi + vj

)2(3.3)

pCIi

(vi,vj

)=

vi

vi + vj

πCIi

(vi,vj

)=

v3i(

vi + vj

)2.

It is easily verified that A will invest more against a high-value opponent than

against a low-value one iff vA >√

vHvL. Let ∆v :=√

vH − √vL. Under one-

sided asymmetric information, effort, probability of success, and utility in an

interior solution are:

xAIA (vA,vL,vH) =

vLvHv2A ((1 − q)

√vL + q

√vH)

2

(vHvL + vA ((1 − q)vL + qvH))2

xAIH (vA,vL,vH) =

((1 − q)√

vL + q√

vH)vAvH√

vLvH


× (√

vHvL + qvA∆v)

xAIL (vA,vL,vH) =

((1 − q)√

vL + q√

vH)vAvL√

vLvH


× (√

vLvH − (1 − q)vA∆v)

pAIA (vA,vL,vH) =

vA ((1 − q)√

vL + q√

vH)2

(vHvL + vA ((1 − q)vL + qvH))

pAIH (vA,vL,vH) = 1 − vA ((1 − q)

√vL + q

√vH)


√vL

pAIL (vA,vL,vH) = 1 − vA ((1 − q)

√vL + q

√vH)


√vH

πAIA (vA,vL,vH) =

v3A (q

√vH + (1 − q)

√vL)

2 (qvH + (1 − q)vL)

(vA (qvH + (1 − q)vL) + vLvH)2


πAIH (vA,vL,vH) =

(qvAvH∆v + v3/2

H vL

)2

(qvA(vH − vL) + vL(vA + vH))2

πAIL (vA,vL,vH) =

(v3/2

L (vA(1 − q) + vH)− (1 − q)vA√

vHvL

)2

(qvA(vH − vL) + vL(vA + vH))2.

Acquiring Information. Let us consider lobbying group A’s incentives to ac-

quire information. The difference in expected utility is equal to

∆πA =(1 − q)qv3

A (√

vH −√vL)

2 (vA −√vH

√vL)

(vA + vH)2(vA + vL)2(qvA (vH − vL) + vAvL + vHvL)2

×((vH − vL)q

(v3

A − 3v2A

√vLvH − vAvHvL − v3/2

H v3/2L

)

−2qvA√

vLvH

(v2

H − v2L

)+ v3

AvL − 3v2A

√vHv3/2

L − 4vAv3/2H v3/2

L

−vAv2HvL − 2vA

√vHv5/2

L − 2vAvHv2L − v5/2

H v3/2L

−2v3/2H v5/2

L − 2v2Hv2

L

)

For vA <√

vHvL, A clearly prefers to acquire information, while for vA =√vHvL it is indifferent. For vA slightly larger than

√vHvL it prefers ignorance

while for vA approaching vH it might prefer to acquire information again. This

implies we have to be careful about staying in an interior solution, in other

words we need vL ≥ (1−q)2v2AvH

((1−q)vA+vH)2 or vA ≤ vH√

vL

(1−q)(√

vH−√vL)

.

Let q = 12 . Then the difference in utility for group A between complete-

information and asymmetric information is equal to

∆πA |q= 12

=v3

A (√

vH −√vL)

2 (vA −√vH

√vL)

2(vA + vH)2(vA + vL)2(vAvH + vAvL + 2vHvL)2

×((vH + vL)

(v3

A − 3v2A

√vH

√vL − 3vAvHvL − 3v3/2

H v3/2L

)

−2vA√

vH√

vL

(v2

H + 4vHvL + v2L

)− 4v2

Hv2L

)

We can show that this is unambiguously positive for vA <√

vLvH and negative

for vA >√

vLvH given that we are in an interior solution. For vH > 9vL the

condition for an interior solution is binding. So for vH < 9vL vA can be as

high as vH . Let us plug this into the expression in brackets: v4H − 5v7/2

H

√vL −

14v5/2H v3/2

L − 5v3/2H v5/2

L − 2v3HvL − 7v2

Hv2L. This is clearly strictly negative for all

vH < 9vL. For vH > 9vL we insert the highest possible vA into the expression in

APPENDIX 113

brackets carries the sign of:

−(

4v3/2H − 7vH

√vL + v3/2

L

)

which is always negative for vH > 9vL.

Disclosing Information. To see whether group B prefers to disclose or not it is

sufficient to look at group A’s effort difference between full and asymmetric

information. Since less investment of the opponent is strictly preferred given a

fixed investment, it is even more so, if B can in addition optimally react. If A

invests more under complete information against B, B will clearly prefer asym-

metric information. Define ∆xi := xCIi − xAI

i , i = AH, AL. Then the difference

in A’s effort is equal to

∆xAH =v2

AvH

(vA + vH)2− vLvHv2

A ((1 − q)√

vL + q√

vH)2


=qv2

Av3/2H (

√vH −√

vL) (vA −√vH

√vL)

(vA + vH)2 (vHvL + vA ((1 − q)vL + qvH))2

× (qvA√

vH√

vL + qvAvH + 2(1 − q)vAvL

+ qv3/2H

√vL + (2 − q)vHvL

)

∆xAL =v2

AvL

(vA + vL)2− vLvHv2

A ((1 − q)√

vL + q√

vH)2


= −(1 − q)v2Av3/2

L (√

vH −√vL) (vA −√

vH√

vL)

(vA + vL)2 (vHvL + vA ((1 − q)vL + qvH))2

×((1 − q)

(vA

√vH

√vL + vAvL +

√vHv3/2

L

)

+ 2qvAvH + qvHvL + vHvL) .

At vA =√

vLvH A’s effort is identical, while for vA >√

vLvH A underinvests

against a high-value opponent and overinvests against a low-value one under

asymmetric information. The opposite holds true for vA <√

vLvH . Hence it

follows that for vA >√

vLvH a high-value B prefers not to disclose, while a

low-value one prefers disclosure and vice versa for vA <√

vLvH . Now let us

consider the ex-ante expected utility of group B when it has not yet learned its

value. Define ∆πi := πCIi − πAI

i , i = H, L, B. Then


E[∆πB] = q∆πL + (1 − q)∆πH

=−(1 − q)qvA (

√vH −√

vL)2 (vA −√

vH√

vL)

(vA + vH)2(vA + vL)2(qvA (vH − vL) + vAvL + vHvL)2

×[(

v2H − v2

L

)qv2

A

(v2

A + vA√

vHvL + 4vHvL

)

+ qvA

(2v2

AvHvL + 2vAv3/2H v3/2

L + 2v2Hv2

L

)(vH − vL)

+ v4Av2

L + v3A

(2v3/2

H v3/2L + 2v2

HvL +√

vHv5/2L + 4vHv2

L + 2v3L

)

+ vA

(4v3

Hv2L + 6v2

Hv3L + 3v5/2

H v5/2L

)

+ v2A

(2v5/2

H v3/2L + 4v3/2

H v5/2L + 2v3

HvL + 7v2Hv2

L + 6vHv3L

)

+ 2v3Hv3

L + 2qv3A

(v3

H − v3L

)].

Hence, for vA =√

vLvH group B is also indifferent in expectation whether to

disclose or not, while for vA >√

vLvH it prefers not to disclose and for vA <√vLvH disclosure is optimal.

Signaling of Valuation. Now lobbying group B has the possibility to expend

money before the contest in order to signal its valuation. To show whether and

when a separating equilibrium exists, consider the following set up. Each group

L and H can send a costly signal to A before the contest, which we denote

by si. The signal is completely unproductive and only serves the signaling

objective. We assume signaling costs are c(s) = s for both groups. The game

has a separating equilibrium when it is possible for L to send a signal which H

does not want to mimic and vice versa.

We first look at vA >√

vLvH . In this situation we know from the above

discussion that L prefers complete information, while H is better off under

asymmetric information. Hence, L would like to signal its type and H would

like to hinder it by mimicking its behavior by setting sH = sL. A’s beliefs are

the following: that any signal sB ≥ sL indicates B has identity L, otherwise

B has identity H. Because the signal is costly individual rationality implies

sB ∈ {0, sL}. In a separating equilibrium we must have that sL = sL and sH = 0.

That is, each group’s incentive compatibility (IC) constraint has to hold and no

group has an incentive to mimic the behavior of the other. The respective IC

APPENDIX 115

constraints are

ICL :v3

L

(vL + vA)2− sL ≥ vL −

2vA√

vLvH

vA + vH+

v2AvH

(vA + vH)2,

ICH :v3

H

(vA + vH)2≥ vH − 2vA

√vHvL

vA + vL+

v2AvL

(vA + vL)2− sL.

It is easily shown that it is not possible to find sL > 0 fulfilling both inequalities

simultaneously. Hence, there does not exist a separating equilibrium when

vA >√

vLvH , and as a result no information is transferred and both groups

engage in an incomplete information contest.

Now turn to vA ≤ √vLvH . In this case, it is H who wants so signal its

identity to overcome incomplete information, while L wants to hinder it. A

believes it is facing H in the contest whenever the signal is sB ≥ sH . Otherwise

it believes it is facing L. Individual rationality implies now sB ∈ {0, sH}. In a

separating equilibrium we must have sL = 0 and sH = sH . The respective IC

constraints are now

ICL :v3

L

(vL + vA)2≥ vL −

2vA√

vLvH

vA + vH+

v2AvH

(vA + vH)2− sH , (3.4)

ICH :v3

H

(vA + vH)2− sH ≥ vH − 2vA

√vHvL

vA + vL+

v2AvL

(vA + vL)2. (3.5)

It is now easily verified that there exists a range of signals sH for which both

inequalities hold simultaneously. From the intuitive criterion (Cho and Kreps,

1987) it follows that the equilibrium value of sH makes L exactly indifferent

between mimicking H or not, so that (3.4) holds with equality. Then we have

s+H = vL +v2

AvH

(vA + vH)2− v3

L

(vL + vA)2− 2vA

√vLvH

vA + vH> 0.

Also, if sH = s+H the beliefs of group A are correct and therefore there exists a

separating equilibrium. Note, however, that all values sH > s+H also support a

separating equilibrium as long as (3.5) still holds. Therefore, we proved that

a separating equilibrium with endogenous information transmission exists if

and only if vA ≤√vLvH , which proves the proposition.


To see this, first note that (i) reaction functions are hump-shaped and (ii) reach a

maximum where xA = xB, i.e. where the reaction function crosses the 45 degree


line (for a proof see Yildirim (2005)). Moreover, we find an equilibrium on this

line exactly when vA = vB, i.e. when the game is symmetric. Let us denote

complete-information symmetric efforts for vA = vL by xL and for vA = vH by

xH . Keeping the valuation of the opponent fixed, a group’s effort is strictly

increasing in its own valuation. So let vA increase from vL to vH . Then the

effort of the L-value type is strictly decreasing (strategic substitute) and the

effort of the H-value type is strictly increasing (strategic complement). If the

opponent is of the L-value type, xA increases from xL to some xHL > xL. To

the contrary, if the opponent is of the H type xA increases from some xLH < xL

to xH . Note that xH > xHL > xL > xLH , i.e. if the opponent is of the H-value

type A’s effort is at the beginning lower and at the end higher compared to the

L-value type. Accordingly, by continuity there has to be some vA ∈ (vL,vH) for

which efforts against both types of the other group are identical and equal to

xA.

If vA = vA group A will spend the same lobbying effort in the complete

information games and in the asymmetric information game in equilibrium.

Accordingly, both types of group B will choose the same effort independent of

the informational environment, implying A’s costs and winning probabilities

are identical and thus A is indifferent between both information regimes. �


Proof. We showed in Lemma 3.6.1 that at vA = vA both groups are indifferent

between complete information and asymmetric information. We now prove

also Proposition 3.6.1. To do this we need to analyze the derivative of both

groups’ difference in utilities between complete and asymmetric information.

We derive some preliminary results concerning effort comparative statics at

vA = vA under both informational arrangements with respect to changes in vA.

We then use these results to prove first the information acquisition part of the

proposition and then also information disclosure.

Preliminaries. Here we derive some comparative statics results we need later

on. Because we do not have closed form solutions for equilibrium efforts we

totally differentiate the systems of first-order conditions and use Cramer’s rule.

Under complete information the system of first-order conditions is:

APPENDIX 117

∂πCIAL

∂xCIAL

|vA=vA=

∂pL(xCIAL, xCI

L )

∂xCIAL

vA − 1!= 0

∂πCIAH

∂xCIAH

|vA=vA=

∂pH(xCIAH , xCI

H )

∂xCIAH

vA − 1!= 0

∂πCIL

∂xCIL

|vA=vA= −∂pL(xCI

AL, xCIL )

∂xCIAL

vL − 1!= 0

∂πCIH

∂xCIH

|vA=vA= −∂pH(xCI

AH , xCIH )

∂xCIAH

vH − 1!= 0

Letting i = L, H, totally differentiating these first order conditions yields the

following matrix system:

∂2 pi

∂(xCIAi )

2 vA∂2 pi

∂xCIAi∂xCI

i

vA

− ∂2 pi

∂xCIi ∂xCI

Ai

vi − ∂2 pi

∂(xCIi )2 vi

︸︷︷︸=Ai

dxCIAi

dvAdxCI

idvA

=

(− ∂pi

∂xCIAi

0

)

Define

Ai1 =

− ∂pi

∂xCIAi

∂2 pi

∂xCIAi ∂xCI

i

vA

0 − ∂2 pi

∂(xCIi )2 vi

, Ai2 =

∂2 pi

∂(xCIAi)

2 vA − ∂pi

∂xCIAi

− ∂2 pi

∂xCIi ∂xCI

Ai

vi 0

.

From Cramer’s rule it follows that∂xCI

Ai∂vA

= |Ai1||Ai| as well as

∂xCIi

∂vA= |Ai2|

|Ai| in equilib-

rium. Hence,

∂xCIAL

∂vA|vA=vA

=−∂2 pL

∂x2L(

∂2 pL

∂x2A

∂2 pL

∂x2L

−(

∂2 pL∂xA∂xL

)2)

v2A

> 0, (3.6)

∂xCIAH

∂vA|vA=vA

=−∂2 pH

∂x2H(

∂2 pH

∂x2A

∂2 pH

∂x2H

−(

∂2 pH∂xA∂xH

)2)

v2A

> 0, (3.7)

∂xCIL

∂vA|vA=vA

=

∂2 pL∂xA∂xL(

∂2 pL

∂x2A

∂2 pL

∂x2L

−(

∂2 pL∂xA∂xL

)2)

v2A

< 0, (3.8)


∂xCIH

∂vA|vA=vA

=

∂2 pH∂xA∂xH(

∂2 pH

∂x2A

∂2 pH

∂x2H

−(

∂2 pH∂xA∂xH

)2)

v2A

> 0. (3.9)

These comparative statics show how equilibrium efforts at vA = vA react to

changes in vA if there is complete information.

Under asymmetric information the system of first-order conditions is:

∂πAIA

∂xAIA

|vA=vA=

(q

∂pL(xAIA , xAI

L )

∂xAIA

+ (1 − q)∂pH(xAI

A , xAIH )

∂xAIA

)vA − 1

!= 0

∂πAIL

∂xAIL

|vA=vA= −∂pL(xAI

A , xAIL )

∂xAIL

vL − 1!= 0

∂πAIH

∂xAIH

|vA=vA= −∂pH(xAI

H , xAIH )

∂xAIH

vH − 1!= 0

Totally differentiating yields the following matrix system:

(q

∂2 pL

∂(xAIA )2 + (1 − q) ∂2 pH

∂(xAIA )2

)vA q

∂2 pL

∂xAIA ∂xAI

L

vA (1 − q) ∂2 pH

∂xAIA ∂xAI

H

vA

− ∂2 pL

∂xAIL ∂xAI

A

vL − ∂2 pL

∂(xAIL )2 vL 0

− ∂2 pH

∂xAIH ∂xAI

A

vH 0 − ∂2 pH

∂(xAIH )2 vH

︸︷︷︸=B

dxAIA

dvAdxAI

LdvAdxAI

HdvA

=

−q∂pL

∂xAIA

− (1 − q) ∂pH

∂xAIA

0

0

Define

B1 =

−q∂pL

∂xAIA

− (1 − q) ∂pH

∂xAIA

q∂2 pL

∂xAIA ∂xAI

L

vA (1 − q) ∂2 pH

∂xAIA ∂xAI

H

vA

0 − ∂2 pL

∂(xAIL )2 vL 0

0 0 − ∂2 pH

∂(xAIH )2 vH

B2 =

(q

∂2 pL

∂(xAIA )2 + (1 − q) ∂2 pH

∂(xAIA )2

)vA −q

∂pL

∂xAIA

− (1 − q) ∂pH

∂xAIA

(1 − q) ∂2 pH

∂xAIA ∂xAI

H

vA

− ∂2 pL

∂xAIL ∂xAI

A

vL 0 0

− ∂2 pH

∂xAIH ∂xAI

A

vH 0 − ∂2 pH

∂(xAIH )2 vH

APPENDIX 119

B3 =

(q

∂2 pL

∂(xAIA )2 + (1 − q) ∂2 pH

∂(xAIA )2

)vA q

∂2 pL

∂xAIA ∂xAI

L

vA −q∂pL

∂xAIA

− (1 − q) ∂pH

∂xAIA

− ∂2 pL

∂xAIL ∂xAI

A

vL − ∂2 pL

∂(xAIL )2 vL 0

− ∂2 pH

∂xAIH ∂xAI

A

vH 0 0

It follows again from Cramer’s rule that

∂xAIA

∂vA=

|B1||B| ,

∂xAIL

∂vA=

|B2||B| ,

∂xAIH

∂vA=

|B3||B| ,

and hence

∂xAIA

∂vA|vA=vA

= (3.10)

− ∂2 pH

∂x2H

∂2 pL

∂x2L(

∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2))

v2A

> 0

∂xAIL

∂vA|vA=vA

= (3.11)

∂2 pL

∂xA∂xL

∂2 pH

∂x2H(

∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2))

v2A

< 0

∂xAIH

∂vA|vA=vA

= (3.12)

∂2 pH

∂xA∂xH

∂2 pL

∂x2L(

∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2))

v2A

> 0

Those comparative statics are the marginal change of equilibrium efforts under

asymmetric information if vA changes at vA.

Information Acquisition. We showed in Lemma 3.6.1 that if vA = vA group

A is indifferent between ignorance and complete information. To prove the

proposition we show that the derivative of the difference of utilities of A with

respect to vA is non-zero at vA = vA. Using

pi =f (xA)

f (xA) + f (xi)


and xi = xiB, i = H, L to shorten the exposition, the derivative of ∆πA at vA is

equal to

∂∆πA

∂vA|vA=vA

=

[(1 − q)

(∂pH

∂xA

(∂xCI

AH

∂vA− ∂xAI

A

∂vA

)+

∂pH

∂xH

(∂xCI

H

∂vA− ∂xAI

H

∂vA

))

+ q

(∂pL

∂xA

(∂xCI

AL

∂vA− ∂xAI

A

∂vA

)+

∂pL

∂xL

(∂xCI

L

∂vA− ∂xAI

L

∂vA

))]vA

−((1 − q)

∂xCIAH

∂vA+ q

∂xCIAL

∂vA

)+

∂xAIA

∂vA.

We know that vA > 0, 0 < q < 1.∂pH∂xA

= ∂pL∂xA

= 1vA

and∂pL∂xL

=− 1vL

<∂pH∂xH

=− 1vH

<

0 follow from the first order conditions of the two groups. The derivative

simplifies to

∂∆πA

∂vA|vA=vA

= −((1 − q)

vH

(∂xCI

H

∂vA− ∂xAI

H

∂vA

)+

q

vL

(∂xCI

L

∂vA− ∂xAI

L

∂vA

))vA.

This derivative will only be zero if a change in vA induces the same effect on B’s

complete-information effort as on its asymmetric information effort, or if they

just offset each other for the two types weighted by the probability q and their

valuation. The relevant comparative statics were derived in equations (3.6),

(3.7), (3.11), and (3.12).∂2 pL

∂x2A

< 0,∂2 pH

∂x2A

< 0,∂2 pH

∂x2H

> 0 and∂2 pL

∂x2L

> 0 follow from the

shape of the CSF.∂2 pL

∂xAxL> 0 and

∂2 pH∂xAxH

< 0 come from the fact that at vA = vA

A is an underdog against an opponent with valuation vH but a favorite against

an opponent with valuation vL. Using this, the derivative of the difference in

utilities equals

∂∆πA

∂vA|vA=vA

= −

(∂2 pL

∂x2L

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

((∂2 pL

∂xA∂xL

)2− ∂2 pL

∂x2L

∂2 pL

∂x2A

))

(∂2 pL

∂x2A

∂2 pL

∂x2L−(

∂2 pL

∂xA∂xL

)2)(

∂2 pH

∂x2A

∂2 pH

∂x2H−(

∂2 pH

∂xA∂xH

)2)

vA vH vL

×

∂2 pL

∂xA∂xLvH

(∂2 pH

∂x2A

∂2 pH

∂x2H−(

∂2 pH

∂xA∂xH

)2)

(∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2))

+

∂2 pH

∂xA∂xHvL

((∂2 pL

∂xA∂xL

)2− ∂2 pL

∂x2A

∂2 pL

∂x2L

)q (1 − q)

(∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A−(

∂2 pL

∂xA∂xL

)2))

APPENDIX 121

which has the sign of

−(

∂2 pL

∂x2L

(∂2pH

∂x2H

∂2pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)− ∂2 pH

∂x2H

(∂2 pL

∂x2L

∂2pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2))

.

This term relates∂xCI

AH∂vA

|vA=vAto

∂xCIAL

∂vA|vA=vA

. For∂xCI

AH∂vA

|vA=vA>

∂xCIAL

∂vA|vA=vA

it will

be negative and for∂xCI

AH∂vA

|vA=vA<

∂xCIAL

∂vA|vA=vA

it will be positive. For our CSF

given in equation (3.2) it will always be negative. This means that starting at

xLA = xH

A a slight increase in vA will lead to a relatively higher increase in effort

on the part of group A against the high-type opponent.9 Hence, we find that

at vA = vA the derivative of ∆πA is strictly negative. Thus, there exist some

valuations vA > vA where ignorance is bliss.

Information Disclosure. At vA = vA group B is exactly indifferent whether it

discloses its information or not, ex-ante as well as ex-interim, as group A always

chooses the same lobbying effort. Let us now vary vA marginally from there.

The derivative of the difference in the expected utility of player B between

complete information and asymmetric information with respect to vA at vA

can be written as

∂∆πB

∂vA|vA=vA

= (1 − q)

[vH

(−∂pH

∂xA

(∂xCI

AH

∂vA− ∂xAI

A

∂vA

)− ∂pH

∂xH

(∂xCI

H

∂vA− ∂xAI

H

∂vA

))

−(

∂xCIH

∂vA− ∂xAI

H

∂vA

)]

+ q

[vL

(− ∂pL

∂xA

(∂xCI

AL

∂vA− ∂xAI

A

∂vA

)− ∂pL

∂xL

(∂xCI

L

∂vA− ∂xAI

L

∂vA

))

−(

∂xCIL

∂vA− ∂xAI

L

∂vA

)]

=

((1 − q)vH

(∂xAI

A

∂vA− ∂xCI

AH

∂vA

)+ q vL

(∂xAI

A

∂vA− ∂xCI

AL

∂vA

))1

vA,

where we use pi =f (xA)

f (xA)+ f (xi)and xi = xi

B, i = H, L to shorten the exposition.

We know that vA > 0, 0 < q < 1.∂pH∂xA

= ∂pL∂xA

= 1vA

and∂pL∂xL

=− 1vL

<∂pH∂xH

=− 1vH

< 0

9Note that for more general CSF the opposite case can arise and A increases its effort more againstthe low-type opponent. Then there will be a value of ignorance for vA < vA.


follow from the first-order conditions of the two groups. The relevant equilib-rium comparative statics of efforts were derived in equations (3.6), (3.7), and(3.10). Using these in the derivative yields

(∂2 pH

∂x2H

((∂2 pL

∂xA∂xL

)2− ∂2 pL

∂x2L

∂2 pL

∂x2A

)+ ∂2 pL

∂x2L

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2))

(∂2 pL

∂x2A

∂2 pL

∂x2L−(

∂2 pL

∂xA∂xL

)2)(

∂2 pH

∂x2A

∂2 pH

∂x2H−(

∂2 pH

∂xA∂xH

)2)

v3A

(3.13)

×

(∂2 pH

∂x2H

vH

((∂2 pL

∂xA∂xL

)2− ∂2 pL

∂x2A

∂2 pL

∂x2L

)+ ∂2 pL

∂x2L

vL

(∂2 pH

∂x2A

∂2 pH

∂x2H−(

∂2 pH

∂xA∂xH

)2))

q (1 − q)(

∂2 pL

∂x2L(1 − q)

(∂2 pH

∂x2H

∂2 pH

∂x2A

−(

∂2 pH

∂xA∂xH

)2)+ ∂2 pH

∂x2H

q

(∂2 pL

∂x2L

∂2 pL

∂x2A

−(

∂2 pL

∂xA∂xL

)2)) < 0,

where we use∂2 pL

∂x2A

< 0,∂2 pH

∂x2A

< 0,∂2 pH

∂x2H

> 0 and∂2 pL

∂x2L

> 0, which follow from the

shape of the CSF.∂2 pL

∂xAxL> 0 and

∂2 pH∂xAxH

< 0 come from the fact that at vA = vA

A is an underdog against an opponent with valuation vH but a favorite against

an opponent with valuation vL and

∂2pH

∂x2H

((∂2 pL

∂xA∂xL

)2

− ∂2pL

∂x2L

∂2 pL

∂x2A

)+

∂2pL

∂x2L

(∂2pH

∂x2H

∂2 pH

∂x2A

−(

∂2pH

∂xA∂xH

)2)> 0.

This term relates∂xCI

AH∂vA

|vA=vAto

∂xCIAL

∂vA|vA=vA

. For∂xCI

AH∂vA

|vA=vA>

∂xCIAL

∂vA|vA=vA

it will

be positive and for∂xCI

AH∂vA

|vA=vA<

∂xCIAL

∂vA|vA=vA

it will be negative. For our CSF

given in equation (3.2) it will always be positive. This means that starting at

xLA = xH

A a slight increase in vA will lead to a relatively higher increase in effort

on the part of group A against the high-type opponent. Hence, we find that at

vA = vA the derivative in (3.13) is strictly negative.

Putting together the information disclosure and information acquisition

part, the proof of the proposition follows from the proof of Corollary 3.5.1.

Proof of Propositions 3.7.1 and 3.7.2

Proof. Expected aggregate effort with contest success function pi =xi

xi+xjunder

complete information is equal to

E

∑

i={A,B}xCI

i

=

vA (((1 − q)vH + qvL)vA + vLvH)

(vA + vH) (vA + vL),

APPENDIX 123

while expected aggregate effort under one-sided asymmetric information is

equal to

E

∑

i={A,B}xAI

i

= ((1 − q)

√vH + q

√vL)

((1 − q) 1√

vH+ q 1√

vL

)

(1

vA+((1−q)

vH+ q

vL

)) .

Their difference is equal to

E[∑∆x

]=

vA (((1 − q)vH + qvL)vA + vLvH)

(vA + vH) (vA + vL)

− ((1 − q)√

vH + q√

vL)

((1 − q) 1√

vH+ q 1√

vL

)

(1

vA+((1−q)

vH+ q

vL

))

=(1 − q) qvA (

√vH −√

vL)2 (vA −√

vHvL) (vA (√

vHvL + vH + vL) + vHvL)

(vA + vH)(vA + vL)(qvA(vH − vL) + vL(vA + vH)).

It is easily observed that this is positive for vA >√

vHvL and negative otherwise,

hence, proving Proposition 3.7.1.

Efficiency implies that the informational regime should be chosen to maxi-

mize q xAxA+xL

+ (1 − q) xHxA+xH

as we assume vL ≤ vA ≤ vH . We get

∆

(q

xA

xA + xL+ (1 − q)

xH

xA + xH

)

= − (1 − q)qvA(vH − vL)(v2

A − vHvL

)

(vA + vH)(vA + vL)(qvA(vH − vL) + vL(vA + vH)),

which is positive for vA <√

vHvL and negative else.


Proof. Full information strategies for a match with valuations vi > vj are given

by the bidding distribution functions

Fj(x;vj,vi) =vi − vj

vi+

x

vi

Fi(x;vi ,vj) =x

vj,

for x ∈ [0,vj], see Hillman and Riley (1989) or Baye et al. (1996). In the fol-

lowing let Fi(x;vj) indicate the bidding distribution of group i facing another

group j and denote the corresponding density function by fi(x;vj). The ex-ante


expected complete information payoffs are

πCIH = vH − vA

πCIL = 0

πCIA = q (vA − vL) .

Those results are standard and the proofs can be found for example in Hillman

and Riley (1989) or Baye et al. (1996). Using the equilibrium strategies it is

easily verified that expected aggregate effort is equal to

XCI = q∫ vL

0( fA(x;vL) + fL(x;vA)) x dx

+ (1 − q)∫ vA

0( fA(x;vH) + fH(x;vA)) x dx

= q∫ vL

0

(x

vL+

x

vA

)dx + (1 − q)

∫ vA

0

(x

vA+

x

vH

)dx

=q

2

(v2

L

vA+ vL

)+

(1 − q)

2

(vA +

v2A

vH

)

and that expected allocative efficiency (the ex-ante probability that the player

with higher valuation wins) equals

EFCI = q∫ vL

0FCI

L (x;vA) f CIA (x;vL)dx + (1 − q)

∫ vA

0FCI

A (x;vH) f CIH (x;vA)dx

= q∫ vL

0

(vA − vL

vA+

x

vA

)1

vLdx + (1 − q)

∫ vA

0

(vH − vA

vH+

x

vH

)1

vAdx

= (1 − q)

(1 − vA

2vH

)+ q

(1 − vL

2vA

).

Under one-sided asymmetric information consider first the case where vA is

relatively small, vA ≤ vA ≡ vL

q+vLvH

(1−q). We then find that A’s bidding/effort dis-

tribution function has a mass point at zero. The groups’ equilibrium strategies

are given by the distribution functions

FAIA (x;vL,vH) =

{vH−(1−q)vA

vH− qvA

vL+ x

vLfor x ∈ [0,qvA]

vH−vAvH

+ xvH

for x ∈ [qvA,vA]

FAIL (x;vA) =

x

qvAfor x ∈ [0,qvA]

FAIH (x;vA) =

x − qvA

(1 − q)vAfor x ∈ [qvA,vA].

APPENDIX 125

That those distribution functions indeed characterize an equilibrium is easily

verified and we leave this to the reader (a proof is available upon request).

Equilibrium payoffs in this case are

πAIA = 0 < πCI

A = q (vA − vL)

πAIH = vH − vA = πCI

H

πAIL = vL

vH − (1 − q)vA

vH− qvA > πCI

L = 0.

A prefers complete information while B weakly prefers asymmetric informa-

tion – the L-type is better off while the H-type is indifferent.

Expected aggregate effort is equal to

XAIvA≤vA

= q∫ qvA

0

(f AIA (x;vL,vH) + f AI

L

)x dx

+ (1 − q)∫ vA

qvA

(f AIA (x;vL,vH) + f AI

H

)x dx

=∫ qvA

0

(x

vA+

x

vL

)dx +

∫ vA

qvA

(x

vA+

x

vH

)dx

=vA

(q2vA(vH − vL) + vL(vA + vH)

)

2vHvL,

and expected allocative efficiency equals

EFAIvA≤vA

= q∫ qvA

0FAI

L (x;vA) fA(x;vL)dx + (1 − q)∫ vA

qvA

FAIA (x;vH) fH(x;vA)dx

= q∫ q vA

0

x

q vA

1

vLdx

+ (1 − q)∫ vA

qvA

(vH − (1 − q)vA

vH− qvA

vL+

x

vL

)1

(1 − q)vAdx

=q2vAvH − (q − 1)vL[(q − 1)vA + 2vH]

2vHvL.

Now consider vA > vA = vL

q+vLvH

(1−q). Here only L’s effort distribution has a mass

point, which is at zero.

FAIA (x;vL,vL) =

{x

vLfor x ∈ [0, x]

xvH

+(

1 − (1−q)vAvH

)(1 − vL

vH

)for x ∈ [x, x]

FAIL (x;vA) =

x

qvA+ 1 − vL

qvA+

vL (1 − q)

qvHfor x ∈ [0, x]

FAIH (x;vA) =

x

(1 − q)vA+

vL

vH− vL

(1 − q)vAfor x ∈ [x, x],


where x = vL − (1 − q)vAvLvH

and x = vL + (1 − q)vA

(1 − vL

vH

). The correspond-

ing expected equilibrium payoffs are

πAIA = qvA − vL +

(1 − q)vAvL

vH< πCI

A = q (vA − vL)

πAIH = vH − vL − vA (1 − q)

(1 − vL

vH

)> vH − vA = πCI

H

πAIL = 0 = πCI

L .

B prefers asymmetric information, since the H-type is better off while the L-

type is indifferent, whereas A prefers full information. Ex-ante expected aggre-

gate effort is equal to

XAIvA>vA

=∫ x

0

(f AIA (x;vL) + f AI

L (x;vA))

x dx

+∫ x

x

(f AIA (x;vL) + f AI

L (x;vA))

x dx

=vL(vA + vL)((q − 1)vA + vH)

2

2v2HvA

+(q − 1)(vA + vH)((q − 1)vA(vH − 2vL)− 2vHvL)

2v2H

and expected allocative efficiency equals

EFAIvA>vA

= q∫ x

0FAI

L (x;vA) fA(x;vL)dx + (1 − q)∫ x

xFAI

A (x;vL) fH(x;vA)dx

=vAvH

((q2 − 1

)vA + 2vH

)− vL((q − 1)vA + vH)

2

2vAvH2

.

To complete the proof note that when A and B disagree on information

transmission, we assumed their payoffs to be smaller than max{

πCIi ,πAI

i

}, the

exact value depending on how exactly information transmission works. There-

fore, the disclosure of information is weakly dominated for B and staying igno-

rant is weakly dominated for A.

Continuous uniform distribution

In this section we discuss the model when the type space is not binary but a

continuous uniform distribution. This gives us an idea as to how general our

results are.

APPENDIX 127

Let us assume that B’s value is distributed uniformly on [v,v], with vA ∈[v,v]. In case both lobbying groups know their respective valuations, equilib-

rium efforts are equal to

xCIi

(vi,vj

)=

v2i vj

(vi + vj

)2.

A’s expected utility under complete information is

πCIA =

∫ v

v

v3A

(vA + vB)2

dF(vB) =1

v − v

(v3

A

vA + v− v3

A

vA + v

),

while B gets in this situation

πCIB =

v3B

(vB + vA)2

,

and in expectation this equals

E[πCIB ] =

v3A

vA+v + 3v2A ln[vA + v]− 2vAv + v2

2 −(

v3A


2

)

v − v.

The expected utility of lobbying group A if it does not know the value of group

B

πAIA =

1

v − v

(∫ v

v

xA

xA + xB(vB)dvB

)vA − xA.

Taking the derivative and setting it equal to zero

∂πAIA

∂xA=

1

v − v

(∫ v

v

xB(vB)

(xA + xB(vB))2

dvB

)vA − 1

!= 0

we get A’s first order condition. Plugging this into group B’s reaction func-

tion xB(xA) = max{√

xAvB − xA,0}

we can solve for the equilibrium efforts.

Focussing on interior solutions we get the following equilibrium efforts:

xAIA =

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

xAIB =

√√√√vB

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)−

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2


A and B’s equilibrium utility under one-sided asymmetric information is equalto

πAIA =

2vA(√

v−√v)

vA(ln[v]−ln[v])+(v−v)

v − v

(∫ v

v

1√vB

dvB

)vA −

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

=

2vA(√

v−√v)


v − v2(√

v −√v)

vA −

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

πAIB =

√vBxA − xA√

vBxAvB −√

vBxA + xA = vB − 2√

xAvB + xA

= vB − 2

√√√√ 2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)vB +

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

and B’s expected utility before it learns its type

E[πAIB ] =

v − v

2− 4

3

(v

32 − v

32

)√√√√ 2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

+

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

.

Now we consider the incentives to disclose or acquire information. The

difference in utilities for A and B is equal to

∆πA =1

v − v

(v3

A

vA + v− v3

A

vA + v

)−

(2vA(√

v−√v))

2


v − v

+

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

,

∆πB =v3

B

(vB + vA)2− vB − 2

√√√√ 2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)vB

+

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

.

Ex-ante, before B knows its valuation the difference in expected utility is equal

to

APPENDIX 129

1 2 3 4 5

1

2

3

4

5

v

vA

1 2 3 4 5

1

2

3

4

5

v

vA

1 2 3 4 5

1

2

3

4

5

v

vA

panel a) panel b) panel c)

Figure 3.4: Difference in expected utility for lobbying group A (panel a)) and B (panelb)) as well as zone of agreement (panel c).

∆E[πB] =

v3A


2 −(

v3A


2

)

v − v

− v − v

2+

4

3

(v

32 − v

32

)√√√√ 2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

−

2vA

(√v −√

v)

vA (ln[v]− ln[v]) + (v − v)

2

.

These expressions are quite unwieldy and Hence, we illustrate the equivalents

of Propositions 3.3.1 to 3.7.2 only graphically. Normalizing the lowest valua-

tion to one, v = 1, we plot the differences in utility for A as well as B (from an

ex-ante) between full and asymmetric information in Figure 3.4. v is plotted on

the abscissa while vA is on the ordinate. We plot only valuation pairs for which

an interior solution exists. In the lightgray regions the lobbying groups prefer

ignorance/non-disclosure, while in the darkgray region the lobbying groups

prefer to acquire/disclose information. If A is relatively weak, information dis-

closure is favorable for both players while if A is relatively strong both players

prefer asymmetric information exactly as in our baseline set-up in Section 3.2.

We find that players generally agree whether to disclose B’s valuation. In-

terestingly, only in a small region where A has an about average valuation, in

other words vA is close to E[vB], the players’ preferences diverge. In these cases

B prefers disclosure while A prefers to stay ignorant about B’s value. This can

be seen in panel c) of Figure 3.4.

To illustrate Propositions 3.7.1 and 3.7.2 we plot the difference in expected


1 2 3 4 5

1

2

3

4

5

vv

A1 2 3 4 5

1

2

3

4

5

v

vA

panel a) panel b)

Figure 3.5: Difference in aggregate effort (panel a)) and expected allocative efficiency(panel b)).

aggregate effort and expected efficiency under complete and asymmetric in-

formation. Figure 3.5 illustrates these differences. In the darkgray region dis-

closure leads to lower expected aggregate effort or higher expected allocative

efficiency while in the lightgray region non-disclosure is preferable.

Overall we find that our results under a continuous uniform distribution

are remarkably similar to the ones under only two types of player B, vH and

vL.

Chapter 4

Imperfect Property Rights: The Role ofHeterogeneity and Uncertainty

Philipp Denter and Dana Sisak†

†We would like to thank Dirk Burghardt, Lukas Inderbitzin, Martin Kolmar, John Morgan, HendrikRommeswinkel, Philip Schuster, Stergios Skaperdas, Rudi Stracke, Darjusch Tafreschi and seminarparticipants at University of St.Gallen and University of California, Irvine for their insightful commentsand suggestions. The kind hospitality of UC Berkeley and UC Irvine is gratefully acknowledged. Bothauthors gratefully acknowledge financial support of the Swiss National Science Foundation.

132 CHAPTER 4. IMPERFECT PROPERTY RIGHTS

4.1 Introduction

Among all rules and laws those that are designed to assign and enforce prop-

erty rights are probably the most important from an economic point of view.

Property rights, if properly enforced, imply the control of economic goods and

resources and are as such fundamental for all market transactions. Maybe

most importantly, property rights allow the holder to obtain income from a

good or resource. As an example, a person’s incentives to work depend on the

expected share of earned income that is obtained. This is not very surprising

and there is a lot of empirical evidence for this claim, see for example Field

(2007). Similarly, the incentives to invest hinge to a great deal on the expected

share of returns obtained, as has been shown for example in a recent study by

Hornbeck (2010). This has of course also important implications for economic

development. Mehlum et al. (2005) show in a theoretical analysis how the lack

of well enforced property rights dampens incentives to invest and so might

lead to poverty traps. Skaperdas and Syropoulos (2002) and Garfinkel et al.

(2008) show that trade may decreases welfare if property rights are insecure.

Similarly, property rights enforcement is important for the internalization of

externalities, as has been noted already in the classical works of Coase (1960)

and Demsetz (1964, 1967). Overall, property rights are important to shape in-

centives in various settings efficiently. Nevertheless, property rights are usually

only imperfectly enforced.

In this paper we identify uncertainty as sufficient for imperfect enforce-

ment. We interpret the process of enforcement as a contest game between an

appropriator and an individual trying to defend his property, the defender. If

the defender deters the appropriator from appropriation, property rights are

perfectly enforced. Our point of departure is a situation absent incomplete

information, where deterrence may be possible if the defender is able to com-

mit to enforcement effort. As soon as we introduce uncertainty about relevant

characteristics of the appropriator, perfect enforcement breaks down and en-

forcement is imperfect. The actual magnitude of uncertainty does not play a

role, and neither does the defender’s attitude towards risk or his strength in the

contest. Introducing a state as an enforcer does not alter imperfect enforcement,

either.

The creation or enforcement of property and other rights has been the sub-

ject of study for many researchers in the social sciences during the past decades.

The early literature started with the seminal works of Coase (1960), Demsetz

(1964) and Becker (1968). While the first two articles are mainly concerned with


the effect of imperfect property rights enforcement and when individuals may

have an incentive to create property rights, Becker tries to answer from a nor-

mative point of view what is the optimal extent of rule enforcement if there

was a social planner. About a decade later another literature developed with

the aim to explain why and how property rights may actually emerge absent

any enforcing institution as the state. The point of departure is usually a state

of anarchy.1 Because in anarchy there do not exist rules or institutions by def-

inition, it is possible to study the prerequisites for rules and institutions, like

property rights, to emerge. Those results are important for our understanding

of modern societies. After all, enforcement of property rights is, even in the

presence of a state, to a great deal the responsibility of individuals or firms.

To quote Friedman (1994), “legal rules are in large part a superstructure erected

upon an underlying structure of self-enforcing rights”. For example, households

buy lockable doors, bike locks, or erect walls and fences around their houses,

department stores hire security personnel, buy video surveillance systems or

burglar alarms. Empirically, this is reflected by the fact that individual spend-

ing for property rights enforcement exceeds public spending (see for example

Shavell (1991) and the references therein). For a self-interested individual there

is no reason to respect others’ possession unless the costs of disrespecting prop-

erty are too high. If somebody can further his goals by stealing from others he

will probably do exactly that. As a consequence, property rights are inherently

challenged and a contest for property emerges. Early contributions to this lit-

erature are due to Umbeck (1977, 1981), who analyzed how claims to property

evolved during the California gold rush and showed that violence was a ma-

jor impediment for the creation of secure property rights. Other important

contributions include the articles of Grossman and Kim (1995) and Grossman

(2001). There it is shown that in a contest for possession the possessor might

be able to secure his possession perfectly if he is sufficiently strong and able

to commit to some level of defensive effort, for example by moving first and

building a wall or fence. The assumption that the defender has the opportunity

to move first has been adopted frequently and reflects incumbency. Another

interesting paper is due to Muthoo (2004), in which it is shown that secure

property rights may emerge over time. This finding is similar to the results

in Hafer (2006). More recent contributions are Gintis (2007), Kolmar (2008), or

Hoffmann (2010). Hoffmann (2010) studies how property rights may emerge

1To our knowledge, the first paper studying the implications of anarchy is Bush and Mayer (1974).Influential articles analyzing economic behavior under anarchy are for example Skaperdas (1992) orHirshleifer (1995). For an overview consult Skaperdas (2006) or Garfinkel and Skaperdas (2007).


in a barter economy. Gintis (2007) discusses how the endowment effect can ex-

plain the emergence of perfectly secure property rights. Finally, Kolmar (2008)

shows that even if secure property rights emerge – that is property remains

unchallenged – production incentives are nevertheless distorted. The reason

is that securing property rights implies deterrence, and a rational defender of

property only deter appropriation at the margin.

A common theme in many of these papers is that perfect enforcement is

possible if the defender is sufficiently strong, because then he can deter the

appropriator. As a result, power and incumbency determine whether or not

property rights are secure. However, in all those papers uncertainty does not

play a role. In this paper we stress the importance of uncertainty, for it is likely

to prevent deterrence and hence impedes the creation of secure property.

We analyze a game of conflict in which a defender has the opportunity to

publicly commit resources to fend off an appropriator. Because there are appro-

priators of different types (abilities), appropriators are heterogeneous, and the

defender does not know which appropriator is trying to challenge his property.

This seems quite realistic, since, loosely speaking, people usually do not know

in advance who is going to try to break into their houses, or whether somebody

will attempt to break in at all. We show that this simple structure is sufficient to

destroy the possibility of deterrence of all types and thus property rights can-

not be secure. The reason for this finding is, metaphorically speaking, that all

possible appropriators will stand in front of the same wall. In order to defend

property at lowest possible costs, the defender builds the wall only slightly

higher than the appropriator can climb. When there is uncertainty about the

appropriator’s ability it is not clear what is the optimal height and the defender

builds a wall that is optimal in expectation, and thus deters only some average

appropriator. We show that it will never be in the interest of the defender to

build the wall high enough to deter all types of appropriators. In a first step we

analyze the conflict game using the canonical lottery contest success function

(CSF) to model the conflict. We provide exact conditions for different degrees of

security of property rights to emerge and derive a formula which yields the ex

ante probability of successfully defending the good or resource. We interpret

this probability as the degree of enforcement of property rights. We proceed

and show in a more genera model that the result is robust.

While the primary objective of this paper is to highlight the importance of

uncertainty for property rights enforcement, we also contribute to the literature

on the theory of contests. First, we provide a characterization of the equilibrium

of a ratio-form contest with sequential moves and arbitrary continuous type

4.2. THE MODEL 135

uncertainty. We thereby generalize the analysis of Linster (1993) along many

lines. We show that in our setting of type uncertainty the probability to win

the contest may be negatively correlated with the expected relative strength of

the defender, which seems counterintuitive at first glance. Other papers deal-

ing with informational asymmetries in contests and conflict are for example

Hurley and Shogren (1998), Wärneryd (2003), and Slantchev (2010). Hurley

and Shogren (1998) look at rent-seeking games with one-sided asymmetric in-

formation and show how different beliefs about the strength of the opponent

alter equilibrium behavior. Wärneryd (2003) analyzes contest games between

two players that are ignorant about their own type in the setting of a common-

value conflict (that is, both contestants value the prize equally). Slantchev (2010)

analyzes how choosing a position in a bargaining game prior to a possible con-

flict functions as a signal about a player’s actual strength. He finds that it is

often optimal to appeal to Sun Tzu’s principle of “feigning weakness”. Another

recent related article is due to Chassang and i Miquel (2010). Here the authors

highlight the importance of uncertainty for the stability of peace, respectively

for the outbreak of overt conflict or war. They show that uncertainty generates

incentives to start preemptive conflict, and thereby uncertainty makes deter-

rence less likely. This is in line with our findings in this paper.

The chapter is organized as follows: In the next section we describe the

basic model and define different degrees of security of property. Section 4.3 an-

alyzes a simple contest game of self-enforcement of property rights to develop

an intuition for our main result and discusses comparative static results. Sec-

tion 4.4 studies a generalized model and shows that the developed intuition is

quite general. Section 4.5 shows that also in the presence of a state our results

do not change, even though the state may generate economies of scale in the

process of enforcement. In Section 4.6 we discusses our results in relation to

other literatures in economics.

4.2 The Model

4.2.1 Basic Set-up

Consider two players i ∈ {D, A}, where the letters are mnemonics for defender

and appropriator respectively. D is in possession of a good, which A wants

to appropriate. Enforcement of property rights is in the responsibility of D,

which may for example be due to a weak state. In the context of property

rights this assumption is quite realistic. Clearance rates of property crimes are


consistently low in most countries. For example, the average clearance rate of

burglaries in the U.S. in 2002 was some 14 percent, if victims reported the theft.

However, it is estimated that only every third victim actually reports burglary

(U.S. Department of Justice, 2002). There is also evidence that burglars do

not think about the consequences they face in the event of being caught. The

deterrence effect of punishment is insignificant (U.S. Department of Justice,

2002). That being said for the U.S., public enforcement of property rights is in

most other countries even worse.2 Hence, the major impediment for burglary

and related crimes is usually private spending for self-enforcement.

D and A may be heterogeneous. For specificity, we assume they are hetero-

geneous in their valuation of the good vi. D’s valuation for the good is common

knowledge and is given by vD, whereas A has private information about his

valuation vA. vA is drawn from G(vA), which has the compact and strictly

positive support [vA,vA]⊂ R+ and is differentiable. All this is common knowl-

edge. Throughout the paper we contrast the incomplete information scenario

to one in which there is complete information. In the latter case deterrence

and therefore secure property rights are the outcome if vD is sufficiently high.

As will become apparent later on this does not hold true under incomplete

information.

Note that although we model heterogeneity of players by emphasizing their

different utilities derived from the good, our results are identical if heterogene-

ity in other dimensions is added. In fact, it is uncertainty about A’s willingness

to put in effective effort that matters, which is also influenced by other factors

like marginal costs and effectiveness of effort.

Each player’s probability pi to succeed in the conflict is a function of both

players’ efforts. Specifically, we consider

pD(xD, xA) =

{1 if xA = 0,

xDxD+xA

else.(4.1)

and pA = 1 − pD. In Section 4.4 we show that our results hold more generally.

pA(xD,0) = 1 reflects that the good is in D’s possession in the first place, and A

needs to spend at least some resources to steal it. Each player i chooses effort xi,

and costs of effort are Ci(xi) = xi. We look at sequential moves where D moves

2The Heritage Foundation and the Wall Street Journal rank countries according to the degree ofpublicly enforced property rights on a 0 to 100 scale. For example, a score of 40 suggests “[t]hecourt system is highly inefficient, and delays are so long that they deter the use of the court system”and “expropriation is possible”. Countries with a score equal to or lower than 40 include Argentina,Paraguay, and Bolivia in South America or Croatia, Serbia, and Russia in Europe. See http://www.

globalpropertyguide.com/.

4.2. THE MODEL 137

first, respectively is able to make a commitment. This is a natural assumption

in our context. Fortification or target hardening is usually a durable good and

we usually see somebody erecting a wall or a fence around his house before

somebody else tries to break in.

Given the above assumptions D’s objective is

maxxD∈R+

UD(xD, xA) = vD

∫ vA

vA

p (xD, xA(vA; xD)) dG(vA)− xD. (4.2)

For A, who holds full information after observing D’s effort, maximizing utility

corresponds to

maxxA∈R+

UT(xD, xA) = vA (1 − p(xD, xA))− xA. (4.3)

Because D does not know the type of A we have an incomplete information

game and the equilibrium concept we employ is Bayesian Nash equilibrium.

4.2.2 Concerning Property Rights

Before going into the details of the model a word on our interpretation of secure

property rights is in order. We define a property right analogous to for example

Grossman (2001), Gintis (2007), and Kolmar (2008) as the right of incumbency

or initial claims.3

The appropriator spends effort xA in the contest. If xA(vA; xD) > 0 type vA

is active and challenges D’s property. If xA(vA; xD) = 0 type vA stays passive.

Define π to be the equilibrium probability of observing xA(vA; xD)> 0. We can

use this convention to define three different regimes of security of property:

Definition 4.2.1. Property rights are

1. secure, if the equilibrium probability of appropriative activity is zero, that is the

good is definitively unchallenged and π = 0 in equilibrium. Property rights are

perfectly enforced.

2. imperfect, if the equilibrium probability of appropriative activity is strictly be-

tween zero and one, and 0 < π < 1 in equilibrium.

3This is for example also in the tradition of Locke, who wrote in §27 of his Second Treatise: “Thoughthe Earth, and all inferior Creatures be common to all Men, yet every Man has a Property in his own Person.This no Body has any Right to but himself. The Labour of his Body, and the Work of his Hands, we may say,are properly his. Whatsoever then he removes out of the State that Nature hath provided, and left it in, he hathmixed his Labour with, and joyned to it something that is his own, and thereby makes it his Property. It beingby him removed from the common state Nature placed it in, hath by this labour something annexed to it, thatexcludes the common right of other Men”. See http://press-pubs.uchicago.edu/founders/documents/

v1ch16s3.html.


3. insecure, if the equilibrium probability of appropriative activity is equal to one,

π = 1.

In defining secure property rights we exactly follow papers such as Grossman

(2001), Gintis (2007), or Kolmar (2008). Secure property rights in our defini-

tion are equivalent to the de facto recognition of incumbency, as extensively

discussed in Gintis’ paper. Many authors including the just mentioned have

shown that property rights can be secure in the sense just defined if incum-

bency enables to commit resources to defense and if the incumbent or defender

is sufficiently strong. If property rights are not secure, they do not have to be

insecure, because we define property rights security in a probabilistic way. In

this we depart from the aforementioned papers. Insecure property rights in

our definition resemble the Hobessian jungle in which there is no recognition

of property whatsoever and property is always challenged. Imperfect property

rights as defined above is a form of security in between the two polar cases.

4.3 Equilibrium

4.3.1 Complete Information: A Benchmark

To establish a benchmark we let G(vA) converge to one point, such that vA =

vA = vA. In this case there is no uncertainty and hence there is complete infor-

mation. The equilibrium of this game is our benchmark.

Proposition 4.3.1. The defender enforces his property with probability

p∗D(x∗D, x∗A) = min

{vD

2vA,1

}.

Proof. See for example Linster (1993).

From this it follows that A does not spend any effort in equilibrium if and

only if vD ≥ 2vA. Therefore, if D is sufficiently strong there is deterrence and

property rights are secure. Otherwise, property rights are insecure. Imperfect

property rights cannot emerge. We now turn to the incomplete information

game.

4.3.2 Incomplete Information

In this section we look at how uncertainty affects the level of security of prop-

erty in equilibrium. If D moves first or is able to make a commitment he

4.3. EQUILIBRIUM 139

considers A’s optimal reaction while optimizing. A’s reaction function is easily

found:

xA(vA; xD) = max{0,√

xD vA − xD}. (4.4)

It follows immediately that the appropriator tries to appropriate only when

xD < vA. For all higher efforts xD there is deterrence. For property rights to be

secure this has to hold for all types vA. Let θ(v) ≡∫ vA

v v−1/2A dG(vA), where v

is the marginal type of A participating actively in the contest, v = max{vA, xD}.

Note that θ(v) = θ(xD), since for all xD < vA the density of vA is zero. To

determine D’s optimization problem this formulation is useful. Plugging (4.4)

in (4.2) and using θ(xD) it is then easily verified that D’s optimization problem

corresponds to:

maxxD∈R+

vD [G(xD) +√

xD θ(xD)]− xD.

A simply reacts according to his reaction function (4.4). Proposition 4.3.2 shows

the equilibrium.

Proposition 4.3.2. D’s unique equilibrium effort in the sequential game is (implicitly)

determined by

x+D =

(vD θ (v)

2

)2

> 0, (4.5)

where v = max{vD, x+D}. Player A’s equilibrium effort as the follower (F) follows

immediately and is

x+A = max

{√vA x+D − x+D,0

}.


D’s effort in equilibrium is monotonically increasing in vD and always

strictly positive. However, this is not true for A, who might stay passive. If

all types of A stay passive we would have secure property in equilibrium. If

to the contrary all types of A are active property is insecure. We next look for

a condition that tells us when this is the case. It is obvious from (4.4) that the

appropriator with the lowest valuation stays passive in the conflict whenever

x+D ≥ vA. It is this type who is deterred first and therefore this type’s equilib-

rium effort determines whether property is insecure or not.

Corollary 4.3.1. All types of appropriators are active in the equilibrium of the conflict


game if and only if

vD ≤ vD ≡ 2√

vA

θ(vA).

Accordingly, if this condition is met property rights are insecure and π = 1−G(vA) =

1. If this condition is not met some types of A stay passive and property rights are

imperfect or secure.

Proof. Follows immediately from Proposition 4.3.2 and (4.4).

The corollary tells us that when D gets too strong relative to A he spends

so much effort in equilibrium that some types of A prefer to stay inactive in

equilibrium, that is partial deterrence occurs. In order to have secure property,

however, full deterrence is necessary. To answer whether property can be secure

in an equilibrium we need to look more closely at x+D. In particular, property is

secure if and only if x+D ≥ vA.

Corollary 4.3.2. In equilibrium D always chooses effort

x+D < vA, (4.6)

implying π = 1 − G(x+D) > 0 and hence property rights are never secure.

Proof. If we let x+D → vA on the RHS of (4.5), θ(v) gets zero in the limit, and so

does the RHS. This yields a contradiction since then 0 = x+D = vA > 0.

The corollary shows that uncertainty is sufficient to explain imperfectly en-

forced property rights. Note that the result does not depend on the possible

range of vA, since any interval [vA,vA] with positive measure suffices. Only in

the limit, when vA = vA = vA, secure property can be sustained in an equilib-

rium.

What is the intuition for this result? Assume vD ≥ 2vA. Without uncer-

tainty D would spend exactly xD = vA ∈ [vA,vA], hence deter A and enforce

property rights perfectly. But when there is uncertainty he prefers not to do

that and enforces only imperfectly. To see the intuition behind the result it is

illustrative to look at a simple example. Assume vD = 8 and vA ∈ {2,4} with

equal probabilities. Without uncertainty D would deter both types of A and

secure property rights perfectly. In the presence of asymmetric information D

faces uncertainty in his decision making. By spending xD = 4 he would deter

both types of A, however, he would also spend too much with a probability of

50 percent. Effort higher than 2 has, with a 50 percent probability, no benefit

at all. By increasing effort the expected marginal benefit from effort decreases

4.3. EQUILIBRIUM 141

1 2 3 4 5

-1

1

2

3

1 2 3 4 5

-1

1

2

3

xDxD

∂UD

∂xD

∂UD

∂xD

Figure 4.1: Derivative of D’s utility function with respect to xD against both playerswith vD = 8, vA = 4, and vA = 2. In the right panel we see the derivative ofD’s objective function in case vA = 4 is zero at xD = 4. In the left panel wesee that at this effort the derivative in case of vA = 2 is negative, implyingthe overall derivative has to be negative, too, and therefore the first ordercondition for an optimum cannot hold.

faster in the presence of uncertainty, because at the same time the marginal ben-

efit of effort and the probability that effort has a benefit at all decrease. Because

by increasing effort the fraction of deterred types of A increases, the marginal

benefit of effort decreases not only due to diminishing marginal products but

also as a consequence of a decreasing probability that effort has a positive ben-

efit at all. However, efforts have always to be paid in full, whether or not effort

is effective. In our example the marginal benefit of effort at xD = 4 is with equal

probability zero and one, while marginal costs are equal to one for sure. Hence,

the first order condition for an optimum does not hold but effort has to be lower

in equilibrium. This intuition is also graphically illustrated in Figure 4.1. In the

case of a continuous distribution as described in the model, the probability of

deterrence is continuously increasing in xD in the interval [vA,vA]. Therefore,

as xD → vA the probability of deterrence approaches one, and therefore the

expected marginal benefit of effort approaches zero.

Having shown that property rights are not secure in equilibrium, it is of

interest to look more closely at the ex ante probability that D actually keeps

the good. If the probability to lose possession is economically not significant

the implications of uncertainty for an individual are rather negligible and our

relatively strict definition of security drives the results, rather than economic

significance. There are again two different sources of security of property: first,

some types of A might be deterred, and this increases the probability of keep-

ing the good. Second, those types that are not deterred do not succeed for

sure but get the good only with some probability, which is determined by the


CSF. In other words, because of deterrence, there is some probability not to

encounter an appropriator at all, and there is also some probability to keep the

good conditional on having a conflict. We need to consider both probabilities,

what simply corresponds to integrating over the CSF, given the equilibrium

strategies:

Corollary 4.3.3. The ex ante probability that D keeps the good is

φ =∫ vA

vA

x+Dx+D + x+A(·, ·)

dG(vA)

= G(x+D) +θ(x+D)

2vD

2,

where x+D and xSA are defined as in Proposition 4.3.2.

Proof. This follows immediately from the definition of the CSF in (4.2) and

Proposition 4.3.2.

To get an idea of the magnitude of the impact of uncertainty it is instructive

to proceed with an example distribution. For specificity, assume the appropri-

ator’s valuation is uniformly distributed on [(1 − σ)v,v], σ ∈ [0,1], and assume

vD = 2v. This ensures that the defender would perfectly secure property rights

against all realizations of the appropriator’s valuation if he had perfect informa-

tion. σ is a spread parameter and may vary between zero and one, the former

being the limit when uncertainty vanishes. When σ increases uncertainty gets

larger. The expected valuation of the appropriator, however, decreases. The fol-

lowing formula establishes the connection between the spread and the ex ante

probability to keep possession in our example:

φu(σ) =σ(σ + 3) + 4

(σ + 2)2. (4.7)

A nice property is that φu is independent of v, what makes it quite easy to

analyze. If there was no uncertainty, σ = 0, we get φu(0) = 1, as we should

have expected. As uncertainty increases φu decreases. In Figure 4.2 we plot φu

against σ. An interesting question is now how uncertainty translates into the

degree of enforcement of property rights. For example, if we are interested in

finding the value of σ necessary to generate an ex ante chance of 1% or 5 % to

lose possession, we find σ1% = 0.04168 or σ5% = 0.25403 respectively. Therefore,

if uncertainty is about 25 % relative to v, which is less than 13 % relative to vD,

the probability to lose is some 5 %. However, φu cannot fall below 8/9 ≈ 88.9%,

reflecting that D is still the stronger player. Actually, as σ increases the expected

4.4. GENERALIZING THE MODEL 143

0.0 0.2 0.4 0.6 0.8 1.0

0.85

0.90

0.95

1.00

0.0 0.2 0.4 0.6 0.8 1.0

2.0

2.5

3.0

3.5

4.0

σσ

φu κ

Figure 4.2: Left panel: The ex ante probability φu to keep property as a functionof σ, if vA ∈ [(1 − σ)v,v] and vD = 2v. Right panel: Expected relativevaluation κ ≡ E[vD]/E[vA ] =

42−σ as a function of σ. While φu is strictly

monotonically decreasing in σ the opposite is true for κ. Therefore, in thepresence of uncertainty κ may be a poor measure of strength in a contest,contrary to complete information contests.

valuation of A decreases, and as a consequence D becomes relatively stronger

in expectation.

Note that in the example the probability to win the contest is actually

negatively correlated to the ratio of expected valuation, κ ≡ E[vD]/E[vA ] =

vD/E[vA], something that never happens absent imperfect information. The in-

tuition is that higher uncertainty benefits D because with a given level of effort

the fraction of deterred appropriators is increasing. This makes it attractive to

lower effort to save costs, but this necessarily brings some formerly deterred

types back into the game. As a consequence, D is better off but loses more

often. A consequence of this finding is that standard measures of strength in

contests, as for example the valuation, marginal costs, or effort productivity,

are not necessarily meaningful in the presence of uncertainty.

4.4 Generalizing the Model

In this section we now show that our main result in Corollary 4.3.2 does not

depend on the particular CSF we employed above to model the conflict, nor

on the assumption of uncertainty neutrality. We proceed by first proving that

deterrence is generally possible, and then show that under uncertainty full

deterrence cannot be an equilibrium outcome.

Each player i ∈ {D, A} values the contested good by vi. Both derive utility


from wealth, which is given by

wi =

{vi − xi if i wins in the contest,

−xi else.

A player’s utility is defined over wealth and given by ui = ui(wi) where u′i is

strictly positive and finite and u′′i ≤ 0. So we allow for risk aversion. Both

players put effort xi ≥ 0. D’s probability of winning the contest is

pD(xD, xA) =

{f (xD)

f (xD)+ f (xA)if f (xD) + f (xA) > 0,

1 else,(4.8)

where f (0) = 0, f ′ > 0 and f ′′ ≤ 0.4 A wins with a probability pA = 1 − pD.

(4.8) is a standard CSF, see for example Konrad (2009). If we let f (x) = x this is

the CSF we employed before.

Player i’s von Neumann–Morgenstern utility function is

Ui(xi, xj) = pi(xi, xj)ui(vi − xi) + (1 − pi(xi, xj))ui(−xi). (4.9)

As before D is able to publicly commit effort, for example by moving first.

The following lemma is now important to prove our main result later on.

Lemma 4.4.1. Assume each player maximizes a von Neumann–Morgenstern expected

utility function as defined in (4.9) and probabilities are determined by (4.8). Denote

A’s best response by xA(vA; xD). Then there exists µ(vA) such that

1. if xD ≥ µ(vA) the optimal response is xA(vA; xD) = 0,

2. if xD < µ(vA) the optimal response is xA(vA; xD) > 0,

3. and µ(vA) is increasing in vA, µ′(vA) > 0.


The lemma states that in the above described framework for each type of A

there exists a unique finite level of effort xD that is necessary to deter A and so

to secure property rights. This threshold level is increasing in the valuation of

A. The lemma is important because it states that deterrence is technologically

possible, so we do not assume the result of the following proposition by making

deterrence infeasible:

4In an earlier version of this paper we show that our results also hold for a more general class ofCSFs that allow for deterrence under full information.

4.5. WHEN THERE IS A STATE 145

Proposition 4.4.1. Let the distribution of types of A be G(vA) with density g(vA) and

support [vA,vA] ⊂ R+, where vA < vA and denote D’s optimal effort by x+D. Then

in equilibrium it always holds that x+D < µ(vA) and as a consequence thereof there is

never full deterrence and property rights are never perfectly enforced.


The result that a very small amount of incomplete information suffices to

guarantee imperfect enforcement is quite general. Apart from property rights

enforcement this is also interesting in other circumstances. For example, in

international relations piling stocks of defensive weapons is often used as a

deterrent for other countries to declare overt conflict or war. However, in the

presence of uncertainty deterrence is possible only probabilistically, as we have

shown. This result is similar to a recent result of Chassang and i Miquel (2010),

who show that a little amount of incomplete information can impede deterrence

and hence lead to overt conflict, because uncertainty may create incentives for

preemptive attacks.

4.5 When There is a State

The above reasoning immediately carries over to the case of state enforced prop-

erty rights, or property rights which are partly state and partly self-enforced, as

for example in Konrad and Skaperdas (2010). Even if the state is able to gener-

ate economies of scale in the process of enforcement, this does not change our

results as long as marginal costs of enforcement are positive. To see this assume

the following simple extension of the model in Section 4.2. As before, D may

spend xD for self-enforcement of property rights. However, now total effective

enforcement effort is e(γ1,γ2, xD) ≡ γ1xD + γ2, where the γ1 and γ2 are en-

forcement instruments under the control of the state. γ2 may be interpreted as

a guaranteed minimum level of enforcement effort, whereas γ1 influences D’s

marginal costs of effective enforcement effort, for example because the state

subsidizes lockable doors and burglar alarms (γ2 > 1). As a consequence, the

probability of enforcement is

∫ vA

vA

e(γ1,γ2, xD)

e(γ1,γ2, xD) + xA(vA; e(γ1,γ2, xD))dG(vA).

Let the state’s cost functions be C1(γ1) and C2(γ2) with strictly positive marginal

costs. The sequence of events is the following:

1. The state determines γ1 and γ2.


2. D determines xD.

3. A determines xA.

We assume the state moves first because usually states are longer lived than

individuals and an individual takes usually as given the level of state enforce-

ment. That D moves second follows from the same reasoning as above. The

state then maximizes

maxγ1,γ2

∫ vA

vA

e(γ1,γ2, xD)

e(γ1,γ2, xD) + xA(vA; e(γ1,γ2, xD))dG(vA)vS − C1(γ1)− C2(γ2),

taking into account the optimal reactions of both D and A. It is straightforward

to show that for A and D this situation is now strategically equivalent to the

one in Section 4.3, except that D’ marginal costs of effort are now 1/γ1 and

he is constrained to choose effort e ≥ γ2. Hence, he will always choose xD <

(vA − γ2)/γ1 unless γ1 = ∞ or γ2 = vA. In those cases, however, the state

is the de facto single enforcer of property rights. So it suffices to focus on

the state’s behavior without taking into account D’s optimal reaction to study

whether there may be full deterrence. The state is in the identical position as

D in our baseline model above with the only difference being the cost function.

But since marginal costs are strictly positive, it follows immediately from the

earlier discussions that the state does not fully deter A, either. The derivative

of the state’s objective function at γ2 = vA is −C2(vA) < 0, and at γ1 = ∞ is

−C1(∞) < 0. Therefore, the state will not fully deter A, either, even though it

may be able to generate economies of scale in the process of enforcement.

4.6 Discussion

Property rights and other rules and laws will not be perfectly enforced in an

equilibrium. But what is the deeper reason for that? In our framework infra-

marginal losses against all already deterred types drive the result. However, on

a more fundamental level this is due to the inability of the defender to make

type contingent defensive efforts. Uncertainty about the appropriator’s type

forces the defender to built the same defense for all possible contingencies, and

it is then not optimal to deter all appropriative activities. This enables some

types of appropriators to make positive expected rents, which would not nec-

essarily be the case without uncertainty. This parallels many findings from the

economics of information. For example, it is also the impossibility to write type

contingent contracts that drive the results in the seminal work of Akerlof (1970).

4.6. DISCUSSION 147

Because sellers have to charge the market price for all types of cars, leading to

a race to the bottom in quality, only bad quality cars survive. In our example

it is not possible to built up type or state contingent fortification. Note that

this holds true for any number of policy instruments, as long as contingent

instruments are not feasible.

Moreover, the appropriator gains from uncertainty, and thus receives in-

formation rents. That uncertainty benefits some players and allows them to

receive information rents is something we find quite often in the contract the-

ory and principal-agent literature. For example, in a Baron and Myerson (1982)

regulation problem, where the authority does not know the cost function of the

monopolist, we usually find the low cost types to benefit from the possibility of

high cost types. Another example for such information rents is second degree

price discrimination.

Finally, on a more technical note, for deterrence to be technically possible

appropriators’ and violators’ reaction functions need to be downward sloping

and must cross the abscissa at some point. Equivalently, efforts need to be

strategic substitutes over a given range of the reaction function. This is what we

find in standard oligopoly models with quantity competition. In this respect

Cournot oligopoly models and standard (imperfectly discriminating) contest

models are similar, as has been noted by Mehlum and Moene (2002) and others.

However, while in Cournot competition we find strategic substitutes over the

whole range of the strategy space, this is not true in contest models, in which

we usually find hump-shaped best responses. For low values of the opponent’s

effort we find strategic complements and strategic substitutes only later (Dixit,

1987). However, if we want to analyze deterrence this is exactly the region

we are interested in. So, because we find property rights are never secure in

a contest with uncertainty, in Cournot oligopoly models with uncertainty a

Stackelberg leader will never deter all possible types of competing firms, either.

This is in line with Gal-Or (1987), who showed that in an environment with

uncertainty “the preemptive capabilities of a Stackelberg leader are reduced”.


Appendix


Proof. D maximizes

vD

∫ vA

vA

[xD

xD + xA(vA; xD)

]dG(vA)− xD

s.t. xA = max{√vA xD − xD,0} ,

or equivalently

G(v)vD + vD√

xDθ(v)− xD,

where v = max{x+D ,vA} and θ(v) ≡∫ vA

v v−1/2A dG(vA) as defined above in Sec-

tion 4.3. To prove existence of a maximum it is sufficient to note that this

function is continuous (since it is differentiable) on the compact and convex

set [0,vD], and therefore it follows from the extreme value theorem that a maxi-

mum exists. Uniqueness and the applicability of the first order condition follow

from its strict concavity, the second order condition is

−vDθ(xD)

4x3/2D

− vDG′(xD)

2xD< 0.

The first order condition reads

θ(v)vD

2√

x+D

+ vD G′(v)− vD

√1

xD

√xD G′(v)− 1

!= 0.

Simple manipulations reveal that

x+D =

(θ(v)vD

2

)2

.


Proof. Given the above assumptions A maximizes

EuA = pA(xA, xD)uA(vA − xA) + (1 − pA(xA, xD))uA(−xA) (4.10)

=f (xA)

f (xA) + f (xD)uA(vA − xA) +

f (xD)

f (xA) + f (xD)uA(−xA)

APPENDIX 149

It is easy to see that his utility from staying passive in the contest, that is

from spending effort xA = 0, is u0 ≡ uA(0). Moreover, all efforts xA > vA are

strictly dominated. Denote the utility level from putting xA = vA by uvA≡

pA(xA, xD)uA(0) + (1 − pA(xA, xD))uA(−vA) ≤ u0.

The first derivative of (4.10) is

∆ =∂EuA

∂xA

=f ′(xA) f (xD)

( f (xA) + f (xD))2uA(vA − xA)−

f (xA)

f (xA) + f (xD)u′(vA − xA)

︸︷︷︸≡I

− f ′(xA) f (xD)

( f (xA) + f (xD))2uA(−xA)−

f (xD)

f (xA) + f (xD)u′(−xA)

︸︷︷︸≡I I

.

Except for the first term all of those terms are negative in sign. The first two

terms are the impact of increasing effort marginally on winning, while the

third and fourth term describe the effect on losing. Increasing effort has two

opposing effects on the winning utility. First, winning becomes more likely,

what is positive. Second, if he wins he gets less, what is negative. The third

and fourth term are both negative, reflecting that increasing effort decreases

the probability to lose and at the same time decreases the utility in the event

of losing. It is straightforward to verify that I is strictly decreasing in xA. That

directly implies f (xA)/( f (xA) + f (xD))uA(vA − xA) is strictly concave and is

hence either a monotonic function in xA, increasing or decreasing, or its graph

is inverse U-shaped. Because I I is negative EuA must also be either monotonic

in xA, increasing or decreasing, or it is inverse U-shaped. However, we know

that u0 > uvA, implying that it cannot be monotonically increasing and must be

decreasing on some interval. Of course, if EuA is inverse U-shaped ∆ must be

zero at some point, at which utility is strictly higher than uA. Then the best

response must be positive, xA(vA; xD)> 0. If the derivative is strictly decreasing

A’s optimal choice is xA(vA; xD) = 0. To find out which of those two possible

solutions is the correct one we have to evaluate ∆ at xA = 0:

∆|xA=0 =f ′(0)f (xD)

[uA(vA)− uA(0)]− u′A(0).

It is now easily verified by inspection of ∆|xA=0 that this is a monotonic function


in xD and that there exists a threshold value xD = µA(vA), such that

Sign[∆|xA=0] =

+ if xD < µA(vA),

0 if xD = µA(vA),

− if xD > µA(vA).

If xD ≥ µA(vA) it is true that ∆|xA=0 is either zero or negative. Otherwise ∆|xA=0

is positive. From the above discussion it follows immediately that xA(vA; xD)>

0 if and only if xD < µ(vA). The threshold is given by

µ(vA) = f−1

(f ′(0)

u′A(0)

[uA(vA)− uA(0)]

),

from which it immediately follows that µ(vA) is continuously differentiable

and µ′(vA) > 0.


Proof. In Lemma 4.4.1 we found that there exists a threshold level of effort xD

denoted by µ(v), which is strictly increasing and differentiable. Therefore, its

inverse vA = µ−1(xD) exists and is also continuously differentiable. We then

define

ω(xD) ≡ µ−1(xD).

ω(xD) is the marginal type vA that is deterred given effort xD. If ω(xD) < vA

all thieves are active. ω(xD) ≥ vA implies all thieves are deterred.

We are now able to define D’s expected utility function:

EuD =∫ vA

vA

p(xD, xA(vA; xD))uD(vD − xD)

+ (1 − p(xD, xA(vA; xD))) uD(−xD)dG(vA)

=

(G(ω(xD)) +

∫ vA

ω(xD)p(xD, xA(vA; xD))dG(vA)

)uD(vD − xD)

+

(1 − G(ω(xD))−

∫ vA

ω(xD)p(xD, xA(vA; xD))dG(vA)

)uD(−xD)

The derivative with respect to xD is the following:

APPENDIX 151

Γ = uD(vD − xD)

×[∫ vA

ω(xD)

(∂pD(·, ·)

∂xA

∂xA(·, ·)∂xD

+∂pD(·, ·)

∂xD

)dG(vA)

− ω′(·)g(ω(·))pD(·, ·) + ω′(·)g(ω(·))]− uD(−xD)

×[∫ vA

ω(xD)

(∂pD(·, ·)

∂xA

∂xA(·, ·)∂xD

+∂pD(·, ·)

∂xD

)dG(vA)

− ω′(·)g(ω(·))pD(·, ·) + ω′(·)g(ω(·))]

− u′D(vD − xD)

(G(ω(xD)) +

∫ vA

ω(xD)pD(·, ·)dG(vA)

)

− u′D(−xD)

(1 − G(ω(xD))−

∫ vA

ω(xD)pD(·, ·)dG(vA)

)

Fortunately, we are able to boil down this expression significantly. To show that

xD cannot optimally be equal to or larger than µ(vA), it is sufficient to show that

Γ|xD=µ(vA)is negative. Therefore, using pD(xD, xT) = f (xD)/( f (xD) + f (xT)),

let xD = µ(vA) and note that ω(µ(vA)) = vA and G(vA) = 1 by definition. This

simplifies the derivative already significantly:

Γ|xD=µ(vA)= uD(vD − µ(vA))

×(

g(vA)ω′(µ(vA))−

f (µ(vA))g(vA)ω′(µ(vA))

f (xA(vA,µ(vA))) + f (µ(vA))

)

+ u(−µ(vA))

×(

f (µ(vA))g(vA)ω′(µ(vA))

f (xA(vA,µ(vA))) + f (µ(vA))− g(vA)ω

′(µ(vA))

)

− u′D(vD − µ(vA)).

Now we can use that xA(vA;µ(vA)) = 0 by definition of µ(vA) as well as f (0) =

0 to see that the only remaining term is

Γ|xD=µ(vA)= −u′

D(vD − µ(vA)) < 0.

This is the marginal cost of effort weighted by the probability to encounter a

thief with a valuation lower than vA. The derivative is negative, and therefore

the necessary condition for a maximum is not fulfilled. Hence, it cannot be

optimal to deter type vA. Moreover, continuity of Γ implies this is true not only

for this type, who has measure zero, but for some interval of types [v′A,vA] with

strictly positive measure. Therefore, property rights must be imperfect.

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Curriculum Vitae

Education

2007 - 2012 PhD in economics and finance,

University of St. Gallen,

2001 - 2006 Diploma in economics (Diplom-Volkswirt),

Johannes Gutenberg University, Mainz,

Dongbei University of Finance and Economics, Dalian.

Professional Experience

2007 - 2012 Research and teaching assistant,

University of St. Gallen,

2011 Visiting researcher,

University of California, Irvine,

2007 - 2008 Student assistant,

Johannes Gutenberg University, Mainz.

July 2012 Philipp Denter

essays on political economy contests - university of st ... · marco, and dirk. a very special...

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