multi-issue bargaining 98

8/14/2019 Multi-Issue Bargaining 98

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Games and Economic Behavior 30, 6482 (2000)

doi:10.1006/game.1999.0710, available online at http://www.idealibrary.com on

Multi-issue Bargaining with Endogenous Agenda*

Roman Inderst

Sonderforschungsbereich 504, Universitat Mannheim, 68131 Mannheim, Germany

Received August 13, 1998

The first part of this paper shows that in a noncooperative bargaining model withalternating offers and time preferences the timing of issues (the agenda) matterseven if players become arbitrarily patient. This result raises the question of whichagenda should come up endogenously when agents bargain over a set of unrelatedissues. It is found that simultaneous bargaining over packages should be a prevail-ing phenomenon, but we also point to the possibility of multiple equilibria involvingeven considerable delay. Journal of Economic Literature Classification Number: C78. 2000 Academic Press

Key Words: bargaining; agenda.

1. INTRODUCTION

Envisage a situation where two parties bargain over a set of projects (is-sues). For each project they can determine whether it will be implementedand possibly its specific realization. Assume that projects are separable and

that each subset of projects is immediately implemented after partial agree-ment on this set. Players have no access to any medium of exchange suchas monetary side payments to facilitate bargaining. This is a realistic as-sumption in many institutional settings where bribes either are illegalor conflict with (ethical) conventions. Bargaining between political partiesor between departments of an organization may satisfy this assumption.Finally, suppose that bargaining takes place in the Rubinstein (1982) ex-tensive form with alternating offers and time preferences. Define now anagenda as an ordered partition of the set of projects such that bargain-ing proceeds simultaneously over projects in a given set of the partition,

while bargaining on a specific set can only begin after agreement on all

*Financial support from the Deutsche Forschungsgemeinschaft, SFB 504, at the Universityof Mannheim, is gratefully acknowledged. The exposition of results was much improved bysuggestions from L. A. Busch and C. Pfeil.

E-mail: [email protected].

640899-8256/00 $35.00Copyright 2000 by Academic Press

All rights of reproduction in any form reserved.


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multi-issue bargaining 65

preceding sets. Hence, an agenda determines the timing of projects on thebargaining table. The first part of this paper derives equilibrium payoffsif an exogenously given agenda requires that bargaining proceed sequen-tially, simultaneously, or separately (in independent teams) over the set of

projects. The agenda can have a marked impact on payoffs, whichin con-trast to a similar result reported by Fershtman (1990)does not vanish asplayers become increasingly patient. First, bargaining simultaneously overa set of projects can improve efficiency by creating trading opportunitiesacross issues. Second, changing the agenda may have a distributive effect,and players may therefore prefer different agendas.

The impact of the agenda raises the question of which agenda should

come up endogenously. It is shown in the second part that the answer de-pends on the types of projects and on the players access to a randomizationdevice (lottery). If all projects are mutually beneficial under at least one re-alization, the game always ends immediately with simultaneous agreementover all projects. However, if the bargaining set contains strictly contro-

versial projects, multiple equilibria involving even considerable delay mayoccur if players cannot bargain over lotteries.

The role of the agenda has already been addressed in both axiomatic

and strategic models of bargaining. For the limited purpose of this paperconsideration is restricted to noncooperative contributions.1 In particular,the setting is compared to Fershtman (1990) and Busch and Horstmann(1996, 1997a).2 Fershtman considers alternating-offer bargaining over twolinear projects (linear bargaining frontiers) by two players with time pref-erences and additively separable utility functions. He analyzes sequentialagendas where the realization of utilities is postponed until both projects

are accepted. He shows that a player prefers that the first project be leastimportant to him but most important to the opponent. However, as bargain-ing frictions vanish, the impact of the agenda disappears. This is differentin the setting proposed in this paper, where utilities from agreed projectsare immediately realized.

A similar change to the model of Fershtman has been adopted by Buschand Horstman (1996, 1997a). As their work covers related issues, the maindifferences are briefly stated. Busch and Horstman (1996) consider the case

with an exogenous agenda. In contrast to our analysis in Section 2, theytreat only the two-project case with linear bargaining frontiers. Apart fromlosing generality, this restriction does not allow clear extraction of the two

1Ponsati and Watson (1996) provide an exhaustive analysis in cooperative terms.2An older unpublished paper on the role of the agenda in strategic bargaining is Herrero

(1989). We also restrict consideration to models with complete information. Issue-by-issue ne-

gotiation is used to signal bargaining power in Bac and Raff (1996) and Busch and Horstmann(1997b, c).


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66 roman inderst

basic impacts of the agenda. In another paper primarily designed to explainincomplete contracts, Busch and Horstman (1997a) partially endogenizea bargaining agenda in a simple setting with two linear projects by intro-ducing a separate bargaining round over the agenda, which precedes bar-

gaining over projects. In contrast, in Section 3 of this paper players bargainover projects without an ex ante agreed agenda. The timing of acceptedprojects becomes thus truly endogenous. Finally, confining the bargainingset to two linear projects does not allow the difference between bargaining

with mutually beneficial and strictly controversial projects to be revealed.The rest of this paper is organized as follows. Section 2 solves for equi-

libria under various exogenous agendas. Two examples illustrate how the

agenda affects the bargaining outcome. Motivated by this result, Section 3solves a model with endogenous agenda. Section 4 summarizes the results.Some proofs are relegated to an Appendix.

2. THE GAME WITH AN EXOGENOUS AGENDA

2.1. Equilibria with Various Exogenous AgendasIn what follows I consider bargaining games with two players called A

and B. Players can realize a finite set of projects denoted by Pi with i Iwhere I = 1 I is an index set of finite size I. Each project Piis characterized by a strictly decreasing, concave, and continuous functionviUi which maps Ui Ui Ui onto Vi Vi Vi.

3 Denote the inverseof this function by uiVi. Throughout the paper lowercase letters u and v

relate to functions, while capital letters U and V denote realizations. Therespective values of Ui and Vi represent the von NeumannMorgenstern(vNM) utilities of players A and B if a specific realization of project Pi ischosen. A full characterization of the utility functions is provided below.Players have no opportunity costs from implementing a specific project andthey obtain zero future utility if they break up bargaining at any pointof time.

The bargaining frontier viUi may arise in the following way. Supposethat parties bargain both over the implementation of a given project andover its specific realization with respect to the choice of a set of underlying

variables. Now viUi equals the maximum utility player B can realize ifhe or she can freely choose these parameters under the constraint that Asutility is at least Ui. Note that we admit (degenerate) projects with Ui = Uiand Vi = Vi. Define next two classes of projects.

3Note that monotonicity and concavity already imply continuity on Ui Ui Ui.


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Definition. A project Pi is called mutually beneficial if there existsa realization viUi with Ui Ui Ui such that Ui > 0 and Vi > 0. Aproject is called strictly controversial if there is no realization viUi withUi Ui Ui such that Ui 0 and Vi 0, while there exists a realization

viUi with Ui Ui Ui such that Ui > 0 or Vi > 0.

Hence, for mutually beneficial projects there exist realizations such thatboth players strictly prefer implementation to reaching no agreement onthis project. For a strictly controversial project such a realization does notexist. In what follows projects are either mutually beneficial or strictly con-troversial. Note that the definition of projects differs from the standardnotion of a convex and comprehensive bargaining set confined to nonnega-tive pairs of utilities. The bargaining frontiers (outer contours) of these setsadmit neither strictly controversial projects nor degenerate mutually bene-ficial projects. The latter projects become, however, particularly useful toisolate the impact of the agenda below. Moreover, the convexification ofthe set of feasible utilities is postponed until Section 3.2 where lotteries areintroduced.

Utility functions are additively separable, and both players have the same

constant and symmetric discount factor 0 1. If a set of projects IA Iis accepted and Pi is implemented with the choice Ui Vi at time ti whereti 0 1 , then A and B derive the respective utilities

iIA

ti Uiand

iIA

ti Vi.Attention is now restricted to exogenous agendas which prescribe how

bargaining has to proceed. We distinguish three cases.

Separate Bargaining: Time runs in discrete, equidistant periods num-

bered t = 0 1 2 Players announce in period t = 0 for each projecta pair of representatives who bargain in isolated teams according to thefollowing procedure. Players alternate in making proposals with (the rep-resentative of) player A making the first proposal at t = 0. A proposal

specifies a pair Ui Vi such that Vi = viUi with Ui

Ui Ui

. After a

proposal is made at time t, the responder can either accept or reject. If heor she accepts, the respective project is carried out immediately and util-

ities are realized. If he or she rejects, he or she becomes the proposer attime t+ 1.

Sequential Bargaining: The agenda specifies now a partition of the set Iwith I > 1 into K > 1 nonempty subsets IPk with

Kk=1 I

Pk = I. Players bar-

gain simultaneously over projects Pi with i IPk for each k 1 K

while bargaining on projects in IPk+1 has to be preceded by agreement overprojects in IPk . To be precise, players start to bargain on the set of projects

I

P

1 at t = 0. Offers are thus characterized by a family Ui ViiIP

1 withUi Ui Ui and Vi = viUi for i IP1 . After agreement in period t they


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realize the respective utilities and proceed to bargain over the set IP2 in pe-riod t+ 1. It is assumed that player A makes the first proposal on each newset of projects.4 This procedure is repeated until the last set of projects IPKis agreed and implemented. Observe that delaying an agreement for a set of

projects IPk with k < K delays also the realization of all projects belongingto sets IPk with k > k.

Simultaneous Bargaining: Now the exogenous agenda specifies thatplayers have to bargain over all projects simultaneously. Hence, this caserepresents a degenerate sequential agenda where K = 1.

I look now for subgame perfect equilibria (SPE) under the differentagendas and show that in each case equilibrium utilities are uniquely de-

termined. Consider first for a moment an (auxiliary) game where playersbargain over any nonempty set I I and break up after an agreement.(For I = I we obtain simultaneous bargaining as defined above.) To solvethis game, one has to derive the (composite) bargaining frontier for theset I. Define the boundary UI =

iI Ui and in analogy UI, VI

and VI. Define a function vU I which denotes the maximum utilityplayer B can realize under a proposal covering the set of projects I and

leaving player A with utility U. Define for player A the analogous functionuV I. It is intuitive that some characteristics of the underlying functionsviUi and uiVi carry over to both newly defined functions. In fact, uV Iis concave, continuous, and strictly decreasing, while vU I is the inverseofuV I. By applying the techniques of Rubinstein (1982) it can be shownthat the auxiliary game ends in the first period. Though equilibrium utili-ties are unique, multiple SPE may exist, as pairs of utilities may arise from

various realizations of the individual projects in I. This is particularly in-

tuitive if I contains several projects with a linear frontier of equal slope.5

Moreover, as frictions vanish with 1, equilibrium utilities converge tothose obtained by applying the (axiomatic) Nash bargaining solution to thecomposite frontier.

For I = i and I = I one can immediately apply these results for theauxiliary game to the case of separate and simultaneous bargaining. In con-trast, under a sequential agenda bargaining over a nonfinal set IPk with

k < K has to take additional account of players (continuation) utilitiesderived from subsequent bargaining over sets IPl with l k + 1 K.These discounted utilities take the form of a pair of windfall utilities

4Note that this will not represent a major restriction as the rest of this paper restrictsattention to the limit case 1 where the first-mover advantage disappears.

5Note that in Rubinsteins original proof the bargaining frontier contains for each player arealization over which he or she is completely patient. With discounting this requirement is

only satisfied if the frontier contains the choices 0 V and U 0, which may not hold in ourcase.


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U V with surely U 0 and V 0, which are realized together with the implementation of the current set IPk . The relevant bargainingfrontier is then obtained by a simple translation:

v

U I U V

= v

U U I

+ V defined on

U

UI + U UI + U

u

V I U V

= u

V V I

+ U defined on

V

VI + V VI + V

Apply now the Nash bargaining solution to the frontier vU I U V

by maximizing the Nash product U vU over the feasible set of utilities.Strict monotonicity and concavity ensure the existence of a unique choiceof utilities denoted by UNI U V and VNI U V. Finally, denote(for given ) the always unique equilibrium utilities under a separate (i.e.parallel) agenda by UP VP, under a simultaneous (i.e., composite)agenda by UC VC, and under a sequential agenda by US VS. The fol-lowing proposition is proved in the appendix.

Proposition 1. The following results hold with exogenous agendas:

(i) Separate bargaining: The game has a unique SPE and ends in thefirst period. Equilibrium utilities for 1 are uniquely characterized by UP =I

i=1 UNi 0 0 and VP =

Ii=1 V

Ni 0 0.

(ii) Simultaneous bargaining: The game ends in the first period, thereexists at least one SPE, and all SPE yield the same utilities. Equilibrium util-

ities for 1 are uniquely characterized by UC

= UN

I 0 0 and VC

=VNI 0 0.

(iii) Sequential bargaining: The set of SPE is not empty. In each SPE theset of projects IPk is accepted as soon as possible, i.e., in t = k 1. Equilibriumutilities are unique and for 1 derived by recursively applying the Nash

bargaining solution.6

2.2. Isolating the Effects of Exogenous AgendasThis section uses two simple examples to isolate two effects of the

agenda. The first effect is purely distributive and illustrated by combininga linear and a degenerate project. By combining two linear projects withdifferent slopes the second example shows how simultaneous bargaining

6Define for the last bundle IPK the utility USK = U

NIPK 0 0 and for all other bundles with

k 1 K 1 recursively USk = U

N

IPk U

Sk+1 V

Sk+1. Defining V

Sk similarly, one obtains

US = US1 and VS = VS1 .


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70 roman inderst

improves efficiency. To highlight the difference from Fershtman (1990),attention is restricted to the case where 1.

Assume first that I = 2 with a degenerate P1 represented by a singlerealization with U1 = a and U2 = b where a b > 0. P2 is linear and

characterized by V2 = g rU2 with g r > 0, U2 0 gr, and V2 0 g. As g2r

g2 maximizes the Nash product for P2, Proposition 1 yields U

P = a + g2rand VP = b + g2 for separate bargaining. For simultaneous bargaining thecomposite bargaining frontier becomes vU I = g + b rU a for U a a + g

r. Maximizing U vU I yields from Proposition 1

UC VC =

a b + g if ar > b + r

g + b + ar

2r g + b + ar

2

if b + r ar b r

a +

g

r b

if ar < b r

(1)

Utilities under separate and simultaneous bargaining are only equal forr = b

a. For r > b

aplayer B is strictly better off under simultaneous bar-

gaining, whereas for r

ba

holds, P1 can be considered to be more importantto player A than to player B if one compares the fixed payoff ratio in P1to the slope in P2. By linking P1 to P2 player As loss from delay is thusrelatively more increased than that of player B compared to separate bar-gaining over P2. This increases Bs equilibrium payoff realized from projectP2. Figure 1 illustrates an extreme example for this case.

With a sequential agenda there are two cases with either IP1 = 1 orIP1 = 2. In case bargaining proceeds first over the degenerate project P1,it is immediate that the sequential case equals that with a separate agenda.

And backward calculation shows for IP1 = 2 that payoffs are the same asunder simultaneous bargaining. This confirms the above intuition for thedifference between separate and simultaneous bargaining. As delaying P2

FIGURE 1


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also delays P1, a player gains bargaining power over the issue P2 if P1 isrelatively more important to the other party.

To illustrate the impact on bargaining efficiency, consider next the pairof linear projects

P1 v1U1 = g rU1 for U1

0

g

r

P2 v2U2 = h sU2 for U2

0

h

s

With separate bargaining the players utilities UP VP equal g2r +h

s

g

2 +h

2 . The composite frontier vU I now maximizes g + h rV1 sV2 for U 0gr

+ hs

subject to the constraints U1 + U2 U, U1 0gr

,

and U2 0hs

. For r = s, vU I is a straight line connecting the points

0 g + h and gr

+ hs

0. If player B has to make a concession to player A,he is indifferent to whether the additional utility comes from P1 or P2.This becomes different if the slopes of the two frontiers differ. With r > splayer B prefers to make a concession on P2. With r s the frontier ischaracterized by

vU I =

g + h sU if 0 U 0 and V1 > 0, whileP2 and P3 are strictly controversial with Ui > 0 and Vi < 0 for i 1 2.Player As utilities satisfy

U1 + Ui > U1 for i 2 3

U1 + U2 + U3 > U1 + Ui for i 2 3(2)

while it holds for player B that

V1 + V2 + V3 > 0

V1 > V1 + Vi for i 2 3

V1 + Vi > V1 + V2 + V3 for i 2 3

(3)

As long as V1 + V2 + V3 > 0 holds, these assumptions are surely satisfiedif is sufficiently close to 1. With only degenerate projects the strategiesof a proposer can be reduced to the proposed set It. Note that no furtheragreement is reached after time t if 1 / Irt. To avoid ambiguities assumethat the game terminates if 1 / Irt. Consider now for a given I I the

following strategies depending only on the set of remaining projects Irt: Ateven t A proposes It = I I

rt, while B accepts It if 1 It and It I.

At odd t B proposes It = I Irt, while A accepts It either if 1 / It or if

I Irt It in case 1 It.Given (2) and (3) these strategies support an equilibrium for

I

1 1 2 1 3 1 2 3

These equilibria can now be used to construct equilibria involving consid-erable delay. To give an example, suppose that until a given period t player

A proposes It = 1 2 3 while player B rejects and proposes It = 1,which is again rejected by A. If either player deviates to a different offer inperiod t t, that player is punished by implementing in t + 1 the mostunfavorable (continuation) equilibrium involving immediate acceptance ofIt = 1 if player A deviates and acceptance of It = 1 2 3 if B deviates.

At time t + 1 play switches to an equilibrium with an intermediate offerIt+1 = 1 2 or It+1 = 1 3, which is immediately accepted. Agreement


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in period t is indeed an equilibrium if t is sufficiently small and suffi-ciently high. The maximum delay which can be supported depends on theskew between the maximum and the minimum payoffs for each player inthe set of equilibria used to punish and reward players.9 The example gives

rise to the following proposition.

Proposition 3. If the set of projects contains strictly controversial projects,bargaining with endogenous agenda and without a randomization device canlead to multiple equilibria involving even considerable delay.

The existence of multiple equilibria is not surprising given the results ofvan Damme et al. (1990) on bargaining with a nonconnected frontier aris-

ing from a smallest unit of money. The set of equilibria with immediateagreement can from (2) and (3) be sustained simply due to the physical im-possibility of proposing a counteroffer which makes oneself better off whilemaking the other party at least indifferent to realizing his or her originaloffer with one period delay. Recall that constructing such counteroffers isat the heart of the proof of uniqueness in Rubinstein (1982).

To finally introduce lotteries, assume that players propose additionally foreach project in It a number p

ti 0 1 which specifies the success proba-

bility after implementation of Pi. Players realize the specified utilities if therealization is successful, while otherwise they derive zero utility from thisproject. Assume also that expected utilities from a set of accepted projectsIA where Pi with i IA is implemented in ti with realization Ui Vi andsuccess probability pi equal

iIA

piti Ui and

iIt

piti Vi. With this en-

larged strategy space it can now be proved that in any SPE of the game withan endogenous agenda all projects accepted with strictly positive probabil-

ity are indeed accepted in the first period, and that equilibrium utilities areidentical to those with an exogenous simultaneous agenda. Moreover, allprojects which are not accepted (or implemented with zero success proba-bility) are strictly controversial, and exactly one player obtains strictly neg-ative utility for any realization of any of these projects. Hence, this set doesnot allow any further mutually beneficial agreement. As the proof is virtu-ally identical to that of Proposition 2, the result is stated as a corollary.10

Corollary. In any SPE of the game with an endogenous agenda anda randomization device, all projects implemented with positive probability are

9Exact conditions are stated for a similar result in Fernandez and Glaser (1991). Note thatthe use of multiple (quasi-)stationary equilibria to construct equilibria with long delay is notnew to the literature. A summary of such approaches is provided by Avery and Zemsky (1994).

10To be precise, Proposition 2 is a special case of this assertion where only mutually ben-eficial projects are admitted. Note also that the corollary uses implicitly the (straightforward)

generalization of Proposition 1 to the case with strictly controversial projects and a random-ization device.


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76 roman inderst

accepted in the first period. The set of SPE is not empty, and equilibrium util-ities are uniquely determined. Utilities are equal to the case with an exogenous

simultaneous agenda. Moreover, the set of projects which are not accepted orwhich are accepted with zero probability does not allow any further mutually

beneficial agreement.

4. CONCLUSION

The paper presents a strategic model of multiissue bargaining with alter-nating offers and time preferences which allows the impact of the agenda to

be analyzed. It is shown that the choice of the agenda may have a distribu-tive effect and may also change the efficiency of bargaining. In contrast to aprevious result of Fershtman (1990) for a similar setting, these effects per-sist if players become increasingly patient. As players may have conflictingpreferences over various agendas, the question arises which agenda is cho-sen endogenously. It is found that players bargain simultaneously over allissues (projects) if these are mutually beneficial. If the bargaining set con-tains strictly controversial projects, an analogous result holds only if players

have access to a randomization device. With strictly controversial projectsand without lotteries there might be multiple equilibria involving even con-siderable delay.

APPENDIX

Proof of Proposition 1. Define first formally for U UI UI thefunction vU I by the program

vU I = maxUiiI

iI

viUi (4)

with the constraints iI

Ui U

Ui

Ui Ui

i I

(5)

As the objective function is surely continuous and the domain ofUiiIcompact, vU I is well defined. Define uV I analogously.

Step 1. For all nonempty I I and all U 0 V 0 it holds that:

(i) vU I U V and uV I U V are continuous, strictly de-creasing, and concave in U and V.

(ii) uvU I U V I U V = U.


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Proof. As u and v are translations ofu and v, I focus on thecase U = 0 and V = 0. The proof of assertion (i) is restricted to vU Ias it is analogous for uV I. Select for vU I a member of the (possi-bly not single-valued) set of maximizing project realizations (the argmax-

correspondence) and denote it by UiiI vU I =

iI viUi. For afeasible U < U the vector UiiI would also satisfy the constraints (5) if

As utility was U. This yields vU I vU I, and strict inequality followsfrom the strict monotonicity of all functions viUi and as the constraint (5)is surely binding. It is shown next that for any 0 1 and an arbitrarypair U U:

vU + 1 U I vU I + 1 vU I (6)Abbreviate U = U + 1 U and select again UiiI from the

argmax-correspondence for vU I, UiiI for vU I, and UiiI forvU I. Replace UiiI by the feasible linear combination Ui + 1 UiiI. Constraints (5) are still satisfied, while the concavity of vi

yields (6) from

iI

ViUi + 1 Ui iIviUi + 1 viUi

= v

U I

+ 1 v

U I

Continuity on U UI UI follows from the two previous results,while it can be derived by an -argument for U = UI. To prove assertion(ii), define V = vU I and U = uV I and claim that U > U. If U isrealized by choosing project realizations ViiI iI Vi V, iI uiVi =U then from U > U there exists a project i I where Vi < Vi. Therefore,from continuity ofui there exists a value > 0 such that

iI\i uiVi +

uiVi + U, which from

iI Vi + > V contradicts the construction ofV = vU I. The claim U < U can be similarly contradicted.

I now derive equilibrium strategies and utilities if players bargain withalternating offers over a nonempty set of mutually beneficial projectsI I and with windfall utilities U V. Call this (auxiliary) game the

I-simultaneous game. With l = I U V define Nl = UAlVAl UBl VBl by:

VAl = max

V V

I

+ V v

max

U UI + U UAl

I

VAl = vUAl I

UBl = maxU UI + U umaxV

VI + V VBl IUBl = u

VBl I

(7)


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78 roman inderst

Below the superscripts used in (7) denote whether a respective pair ofutilities is proposed by A or B in the set of SPE.

Step 2. Nl defined in (7) is uniquely determined.

Proof. Focus again on U = V = 0. As the argument is symmetric, theassertion holds if

vU I max

VI v

UI U

I

(8)

becomes zero for a unique value of U UI UI. Note first that withU = UI (8) is strictly positive, while it is nonpositive at U = UI. As

the expression is continuous from Step 1, it becomes zero for at least onechoice of U. Uniqueness follows if (8) is strictly monotone decreasing onU UI UI. For U UI U with U = minUI/UI (8) re-duces to vU I maxVI VI, which is strictly decreasing from therespective characteristic of vU I. Hence, for UI/ UI the proofis concluded. Otherwise, define V = minVI/VI and U = uV I

with uV I = v1V I from Step 1. Distinguish now between two cases.For U U (8) reduces for all U U UI to vU I vU I. ForU > U this transformation holds only for U U U, while for U U UI (8) equals vU I VI. In the latter case the monotonicityis again immediate, and it thus remains to prove that vU I vU Iis strictly monotone decreasing. This is now implied by the concavity ofvU I.

Define the set of quasi-stationary SPE as follows: In even periods A

makes a proposal which realizes utility U

A

l for A and V

A

l for B. PlayerB rejects a proposal if and only if it realizes less utility than VA I. Inodd periods B makes a proposal realizing UBl for A and VBl for B,

which A rejects if and only if he realizes less utility than UBl.

Step 3. Any SPE of an I-simultaneous game is quasi-stationary. Hence,the game ends in the first period and players realize the utilities UAlVAl. The set of SPE is moreover nonempty.

Proof. With Steps 1 and 2 the proof is a straightforward extension ofTheorem 3.4 in Osborne and Rubinstein (1990) to the case of this paper

where the relevant bargaining frontier may not contain zero-utility pointsfor both players.

Step 4. For 1 UAl and UBl converge to a limit UNI U V,while VAl and VBl converge to a limit VNI U V such that UNIU V VNI U V maximizes the product U V subject to the con-straint V = vl.


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Proof. Convergence can be easily deduced from monotonicity followingfrom (7). From Binmore et al. (1986) (BRW) the second assertion is im-mediate ifUNI U V VNI U V lies in the (adequately defined)interior of the (graph of the) frontier. Otherwise, the result follows in com-

plete analogy to BRW by considering the (from the strict monotonicity oful surely existing) derivative w.r.t. V at the respective (boundary) pairUI + U VI + V or UI + U VI + V.

The proof of assertions (i) and (ii) for a separate and simultaneousagenda follows from Steps 3 and 4 where I represents either a single projector the whole set I. Backward induction from IPK to I

P1 proves assertion (iii).

Proof of Proposition 2. Define first the set of simultaneous-proposalSPE (SPSPE): At even t with nonempty Irt A proposes I

rt U

ti V

ti iIrt

which realizes the utilities UA Irt VA Irt.

11 Player B accepts anyproposal It U

ti V

ti iIt with It I

rt which satisfies

iIt

Vti + VB Irt\It V

A Irt (9)

At odd t with nonempty Irt B makes a proposal Irt U

ti V

ti iIrt which

realizes the utilities UB Irt VB Irt. Player A accepts any proposal

It Uti V

ti iIt with It I

rt which satisfies

iIt

Uti + UA Irt\It U

B Irt (10)

The proof proceeds in four steps. The first step proves that the set ofSPSPE is nonempty. Steps 24 derive suprema and infima for both players(possibly discounted) utilities in any equilibrium and show that they areequal to the utilities defined by N I. This allows us finally to prove thatStep 1 indeed characterizes the complete set of SPE.

Step 1. The set of SPSPE is nonempty.

Proof. The proof proceeds by induction on the number of projects in Ir

tafter an arbitrary history Ht or H

t. For I

rt = 1 Proposition 1 shows that the

SPSPE strategies form indeed a continuation equilibrium (which is uniquein this case). Assume now that the assertion holds for any strict subset ofa set of remaining projects Irt with I

rt > 1. More precisely, assume that

for any t > t with Irt Irt there is a continuation equilibrium belonging to

the set of SPSPE. Consider only the strategies of player A as the argument

11With U = 0 and V = 0 in (7) the zero vector 0 0 is now surppressed in all expressions.


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80 roman inderst

is symmetric. A profitable deviating proposal for A It Uti iIt V

ti iIt

must satisfy12

iIt

Uti + UB

Irt\It

> UA

Itr

iIt

Vti + VB Irt\It V

A Itr

Uti

Ui Ui

and Vti = viUti for i It

It Irt

(11)

The first two lines use for It Itr the inductive assumption and players

supposed equilibrium strategies. It follows from the definition of N Irt,

the construction of uV I, and the loss incurred from potential delay thata deviation satisfying (11) is not feasible.

Step 2. Suppose Irt is nonempty for even t. Then UA Irt is the supre-

mum of the discounted utility player A can realize in any continuation equilib-rium. The analogous result holds for player B and odd t with the supremumVB Irt

Proof. For even t and given Irt denote the supremum of player Aspossibly discounted utility in any continuation equilibrium by UAIrt. AsUAIrt is surely bounded below by zero and bounded above by

iIrt

Ui,the supremum indeed exists. By arguing to a contradiction it is proved thatUAIrt = U

A Irt where from Step 1 UA Irt is indeed realized in a

continuation equilibrium (SPSPE). Suppose therefore UAIrt > UA Irt

holds in a specific continuation equilibrium.13 Denote Bs utility in this

equilibrium by

V

A

Ir

t and define

V

A

Ir

t = v

U

A

Ir

t Ir

t. As projects arepossibly delayed on the equilibrium path, it holds that VAIrt VAIrt as

surely VAIrt 0. Construct now a deviation for Player B from the sup-posed equilibrium path in which he or she offers in t + 1 a proposal overall remaining projects such that players realize the utilities U V with

U > UAIrt

V > VAIrt(12)

From the construction of the supremum, player A must accept. It there-fore remains to show that for any UAIrt > U

A Irt there exists a pro-posal realizing a pair U V which satisfies (12). As in the proof of Rubin-stein (1982), this is now an immediate consequence of the construction of

12This uses UB I = VB I = 0 if I is empty.13Though there might be no equilibrium in which the supremum is indeed realized, one

can choose by the nature of the supremum an equilibrium path of offers in which the utilitybecomes arbitrarily close to UAIt.


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N Irt in (7) and of the continuity of the frontier v Irt. The analogous

argument applies now to the supremum for player B.

Step 3. Suppose Irt is nonempty for even t. Then UA Irt is the infimum

of the discounted utility player A realizes in any continuation equilibrium. Theanalogous result holds for player B and odd t with the infimum VB Irt.

Proof. Take first the claim for player A and denote the infimum byUAIrt, which again surely exists. Suppose U

AIrt < UA Irt holds in

a continuation equilibrium. From Step 2 player B cannot realize a higherutility than VB Irt in any continuation equilibrium after rejection. As a

consequence, player A can strictly improve by making a proposal over allissues realizing U V with

U > UAIrt

V > V B Irt(13)

Again from (7) such a pair surely exists for all cases UAIrt < UA Irt.

Finally, from Step 1 UAIrt = UA Irt is indeed realized. The same ar-gument now applies to player B.

Step 4. Suppose Irt is nonempty for even t. Then VA Irt is the infimum

and the supremum of the discounted utility player B realizes in any continua-tion equilibrium. The analogous result holds for player A with UB Irt.

Proof. From Step 3 Bs utility in any continuation equilibrium after re-jection is bounded below by VB Irt. By construction of (7) he must there-fore realize at least VA Irt in any continuation equilibrium starting ineven t. On the other side, he cannot obtain more than this value fromStep 3 and the construction of UB Irt in (7). The same argument nowapplies to odd periods.

To finish the proof, Steps 24 show that for any period t with a nonempty

set Irt any continuation equilibrium must end the game in t with a joint pro-posal realizing the utilities specified by N Irt. To see this, note first thatfrom Steps 24 the (possibly discounted) utilities realized in continuationequilibria for even t must be exactly UA Itr for player A and V

A Irtfor player B as the respective suprema and infima coincide for each player.With costly delay, VA Irt = vU

A Irt Irt, and the strict monotonic-

ity of v Irt, it is immediate that only a simultaneous proposal is feasible.The same argument applies to odd periods. Therefore, the set of SPE iscompletely characterized by that of SPSPE in Step 1.


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82 roman inderst

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