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A Model of Voting in Boards with Heterogenous Outside Directors

Paolo Balduzzi, Clara Graziano and Annalisa Luporini

CESifo GmbH Phone: +49 (0) 89 9224-1410 Poschingerstr. 5 Fax: +49 (0) 89 9224-1409 81679 Munich E-mail: office@cesifo.de Germany Web: www.cesifo.de

A Model of Voting in Boards wth Heterogenous

Outside Directors∗

Paolo Balduzzi†

Catholic University of Milan

Clara Graziano‡

University of Udine and CESifo

Annalisa Luporini§

University of Florence and CESifo

February 24, 2010

Abstract

We analyze the voting behavior of a board of directors that has to approve or re-

ject an investment proposal that can result either in a gain or a loss. Information on

project profitability can be acquired by directors only at a cost. Insider directors, in-

dependently of their information, always vote in favor of the project. Outside directors

can be of two types: those who maximize the profit of the firm and those who care

about their own reputation. The simultaneous presence of members whose preferences

are different can be used to determine the optimal composition and size of the boards.

Our model shows that a strict majority of profit maximizing outside directors within

the board is a necessary and sufficient condition to reach a unique and optimal equi-

librium. Substituting profit-maximizing directors with reputational directors is not an

obstacle to profit maximization provided that at least one outside director is of the

profit-maximizing type. However, when there are also reputational outsiders on the

board an appropriate sequential voting protocol is required to avoid the existence of

suboptimal equilibria (in addition to the optimal one). The introduction of pre-voting

∗We thank participants from the ASSET Conference 2008, from the CEPET Workshop 2009, from the

EALE Conference 2009 and from seminars held at Stony Brook University, Catholic University of Milan,

University of Milan-Bicocca for helpful suggestions. In particular, Paolo Balduzzi thanks Patrick Leyens and

Roland Kirstein for valuable discussion. All mistakes are the authors’ own responsibility. Financial help from

IEF (Catholic University) and XXX is gratefully acknowledge.†Email: paolo.balduzzi@unicatt.it; Skype: paolo.balduzzi.‡Email: clara.graziano@uniud.it.§Email: annalisa.luporini@unifi.it.

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communication improves the outcome when there are also reputational directors on the

board, but it does not change the outcome when the board is only composed by insiders

and profit-maximizing outsiders. Finally,as information is costly, our model provides

some suggestions on the optimal size of boards.

Key words: Board of directors, Voting, Corporate Governance.

JEL clasification: G30, D71.

1 Introduction

In recent years, the study of boards of directors has become an increasingly hot topic. The

structure, composition and size of boards have been widely debated with the objective of

strengthening governance mechanisms after the Enron scandal. In this debate much of the

attention has been focused on independent directors considered as the watchdogs of insiders.

Since 2002, New York Stock Exchange (NYSE) and NASDAQ require their listed firms to

have a majority of independent directors. This requirement parallels the recommendations

of the Code of Best Practice adopted by the London Stock Exchange in 1992 according to

which boards should have at least three outside directors. Empirical literature on boards

has provided some evidence on the positive role played by outside directors. For instance,

firms with a larger number of outsiders on the board have a higher turnover performance

sensitivity for the CEO, a higher probability of selecting an outside CEO (see Weisbach [1988]

and Borokhovich, Parrino and Trapani [1996]) and a lower incidence of opportunistic timing

of stock options (Bebchuck, Grinstein and Peyer [2009]). However, overall the empirical

evidence on the relationship between board composition and firm performance is inconclusive

(see for example the survey in Bebchuck and Weisbach [2009] and Finegold, Benson and Hect

[2007]).

A peculiar element of boards is that directors represent different stakes (majority share-

holders, investors, workers, etc.). Members of the board range from the CEO of the firm

and other executives to representatives of mutual funds, bankers, CEOs of other firms, uni-

versity professors, members of the family that owns the firm and so on. In addition, even if

members are formally chosen by shareholders, it is well documented that the CEO plays an

important role in the appointment and confirmation of directors (see for example Hermalin

and Weisbach [1988, 1998]).

Diversity of members may reflect in their objectives and also outside directors may have

objectives other than profit maximization. Despite this, most of the literature on boards has

focused only on inside directors, whereas the heterogeneity of directors’ preferences and its

implications for board’s decision making hasn’t received the attention it deserves. Indeed,

generally, outside directors are assumed to perfectly represent the interest of shareholders.

Our paper studies the voting behavior of a heterogenous board that has to decide whether

to undertake a project whose returns may be positive or negative according to the realization

of the state of nature. Directors’ preferences reflect their diversity and also outside directors

may have objective different from profit maximization. In particular, we allow outside direc-

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tors to have career concern and we consider three types of directors: i) inside directors, e. g.:

the CEO and the other executive officers who are biased in favor of the project, ii) outside

directors whose objective is to maximize expected profits, i.e. directors who care about the

board taking the right decision, and iii) outside directors who care about their reputation in

the sense that they want to vote ’correctly’ but don’t care about the board’s final decision.

It is well known that the behavior of economic agents with career concern may be inefficient

and lead to distortions. We study how an optimal composition of the board may minimize

the inefficiencies resulting from the diverse objective functions of directors.

Before voting directors may incur a cost to acquire information on the profitability of the

project. Each type of director has a different incentive to acquire information. We assume

that the CEO and other insiders always favor the project because they can obtain private

benefits from its implementation. For example, the project may refer to a new plant to be

built, or a new market to enter. Insiders prefer a larger firm to a smaller one and consequently

they support the new investment. The interests of inside directors are aligned to those of

the CEO’s. This may happen either because they can extract some private benefits or just

because their career crucially depends on CEO’s decisions and, consequently, they do not

want to contrast him. Thus, inside directors independently of the state of nature, always

vote in favor of the project. As a result of this, they don’t’ have any incentive to incur the

cost to become informed.

Contrary to insiders, outside directors condition their voting strategy on the information

they have, if any. Thus, if the information cost is not too high they choose to incur it. The

difference between profit-maximizing and reputational directors relies on the fact that profit

maximizing directors want to induce the board to take the right decision while reputation-

building directors want to take the right decision but have no interest in board’s final decision

being right or wrong. They simply want to show to the market that they are right, in order

to strengthen their reputation. For example, a director can use the visibility provided by his

position on the board with the final goal to elicit (new) job offers or more directorships or

higher salary. Thus, their behavior may be different from that of profit-maximizing directors.

We assume that the market can observe individual vote, and more importantly, the realized

state of the world so that it can assess which decision was the proper one and reward directors

that vote ’correctly’. The assumption that the market can observe whether the decision is

the correct one (which is necessary for optimality of the behavior of reputation-interested

outsiders) may look unrealistic but in our opinion it is reasonable in many situations. Even

if it is the board decision that is always made public, members with reputational concerns

may communicate their votes directly to the market, for example in a press conference. In

many cases, records of the votes must acknowledge the existence of minority positions, so

the market can verify the information provided by different members.

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Our paper shows that a strict majority of outside directors within the board is a necessary

and sufficient condition to reach a unique and optimal equilibrium. This is true both when all

outsiders are profit maximizers and when there also are reputational directors. In the latter

case uniqueness of the optimal equilibrium however is only reached under a sequential voting

protocol. As information is costly, our model also provides prescriptions on the optimal size

of boards. The incentive to acquire information for profit maximizing directors negatively

depends on board size. Given the optimal board composition (which always comprises at

least one profit maximizing outsider), information costs and benefits for outside directors

then determine the optimal board size. Finally, we show that the possibility of pre-voting

communication is not a crucial feature. If the board is composed only by insiders and profit-

maximizing outsiders the introduction of pre-voting communication among directors does not

affect our previous results. Pre-voting communication instead improves the outcome when

there are also reputation concerned directors.

We focus our attention on voting mechanism and we ignore other, undoubtedly relevant,

factors that have been studied by other authors1. First, we assume that directors are informed

with a costly but exogenously given probability. Thus, we assume away any moral hazard

problem in becoming informed. Second, we rule out the possibility of abstention. This is

made for simplicity and, generally, is not completely without loss of generality. Nonetheless,

we will show that abstention is not an interesting issue in our model. Moreover, empirical

evidence shows that most boards explicitly or implicitly rule out this possibility.

Despite being built on board of directors experiences, the model can be extended to

represent different committees. For instance, some juries (e.g.: the Italian Constitutional

Court) have members appointed by different subjects. The same heterogeneity may be found

in technical committees, where politicians, bureaucrats and experts meet to provide advise.

The rest of the paper is organized as follows. Section 2 reviews the literature. Section 3

presents the main elements of the basic model. Section 4 examines the voting game in boards

with only insiders and profit-maximizing outsiders. In Section 5 we show how results change

when outside directors have different objectives. In Section 6, we introduce and formalize

the contribution of communication within the board. Finally, Section 7 concludes. All proofs

are collected in the appendix.

1It should be stressed that our committee’s objective is to aggregate private information rather than social

preferences. A similar problem is studied by Sah and Stiglitz [1988] but they assume that committee members

always vote honestly whereas in our model directors vote strategically. See Piketty [1999] for a brief review

of recent contributions about the information-aggregation role of political institutions.

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2 Related literature

One of the few papers that analyzes the decision-making process in a board where directors

have different preferences is Baranchuk and Dybvig (2008) [Consensus in Diverse Corporate

Boards, in Review of Financial Studies]. They use a new bargaining solution concept called

consensus and directors’ preferences are modelled in a spatial model where a director’s utility

depends on the distance between the decision taken and the director’s ideal decision. They

focus on grey directors and do not consider incentive for information sharing, costly effort

and delegation.

A common feature to most papers on boards is the assumption that insiders have an

informational advantage on outsiders. Then the question is how to induce insiders to share

their information with other directors (Harris and Raviv [2006], Adams and Ferreira [2006]

and Raheja [2005]). Harris and Raviv [2006] analyze the communication between insiders and

outsiders when outsiders have identical preferences and they perfectly represent shareholders.

Since they have the same objective function they behave as a single agent. The model

determines when it is optimal to have the board controlled by insiders and when by outsiders

and the extent of communication between the two groups of directors. Furthermore, the

optimal number of outside directors is determined taking into account that outside directors

have to exert effort to apply their expertise to the specific investment decision. As the number

of outsiders grows the free-riding problem becomes more severe. The optimal number of the

outsiders, balances these two effects. Adams and Ferreira [2006] study the advisory and

monitoring role of boards. They show that the two task may conflict each other and, as a

result of this, managers (the insiders) may refrain from sharing their information with outside

directors. Raheja [2005] presents a model where the board has two tasks: monitoring projects

and making the CEO succession decision. Inside directors are better informed than outside

directors on the quality of the project but their preferences are not aligned with those of

the shareholders. The optimal structure of the board depends on the characteristics of the

industry the firm belong to. When it is relatively easy to verify projects it is optimal to have

a high number of outside directors. When, on the contrary, project verification is costly or

difficult it is optimal to have a large proportion of insiders. The optimal number of outsiders

is determined by the trade-off between the cost of coordination and their voting power.

A paper close to ours is Warther [1998] who studies the voting mechanism in board

composed by the manager and two outside director. The board has to decide whether to

retain or dismiss the manager. Outside directors have all relevant information and the inside

director has none. The model focuses on how directors’ behavior is affected by the threat

of being ejected by the board. If an outside director votes to dismiss the manager and the

manager is not eventually fired, the dissenting director is ejected from the board. His model

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predicts that boards will take the dismissal/retention decision unanimously and that they

are an important source of discipline, despite the lack of debate and apparent passivity.

Unanimity in voting results also in the model of Visser and Swank [2007] who consider

a generic committee that has to make a decision on project implementation. Reputational

concerns affect members’ behavior and may distort project implementation. They show that

members want to vote unanimously since disagreement may signal lack of competence and

therefore may decrease members’ reputation. The voting rule influences the implementation

decision and they show that when information cannot be manipulated unanimity is the

optimal voting rule.

Our paper is related also to the literature on decision making and voting in committees.

In a voting perspective, the main feature of our model is that it introduces heterogeneity

since usually in this literature preferences are assumed to be homogeneous. In voting games,

heterogeneity can have two dimensions: on the one hand, it may concern the ability of the

players; on the other hand, it may regard their objective functions. The former type of

heterogeneity has been already analyzed (see, for example, Ottaviani and Sørensen [2001]).

Quite surprisingly, to our knowledge the only papers that deal with the possibility of different

preferences are Feddersen and Pesendorfer [1996, 1997] who analyze how well simultaneous

voting in large elections can aggregate private information. In the latter contribution, the

authors rule out abstention as we do in our model, and they discuss equilibrium strategies

when voters play strategically and have different preferences. Our contribution differs from

theirs along several dimensions. First, they focus on the asymptotic properties of their

models (i.e., as the number of agents goes to infinity), whereas we model small committees.

Second, in order to better discuss information transmission issues, we also consider sequential

voting games. Cite Cai, H. 2009 Costly Partecipation and heterogeneous Preferences in

Informational Committees, RAND Journal of Economics.

Our work is also related to the herding literature, which has developed in the last fifteen

years2. As in our model preferences play a relevant role, the closest references are Smith

and Sørensen [2000] and Goeree et al. [2003]: they both show how the scope for herding is

weakened when we introduce heterogenous preferences. Nonetheless, the main assumption

in this literature is that other players’ actions do not influence each player’s own payoffs. We

show that this assumption does not necessarily hold in voting games and that herding is not

a serious issue in our setting.

Finally, voting and herd behavior are both very useful in the analysis of sequential voting

games. Two main papers discuss sequential voting games3. Dekel and Piccione [2000] com-

2Seminal contributions are by Scharfstein and Stein [1990], Banerjee [1992] and Bikhchandani et al. [1992,

1998]. See also critiques and generalizations by Gale [1996], Palley [1995], and Swank and Visser [2003].3Sloth [1993] introduces and discusses sequential voting games, but only as a device to solve simultaneous

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pare simultaneous and sequential voting games where individuals observe a signal about the

true value of an option. They find that both simultaneous and sequential voting games are

effective in aggregating information. Moreover, as far as symmetric equilibria are concerned,

the sequential structure cannot do better than the simultaneous one. Our paper provides an

analysis of cases where the best equilibrium strategy profiles are asymmetric ones. Moreover,

we consider players with heterogeneous preferences. Nevertheless, even in games with homo-

geneous players, our environment is not totally symmetric: it is not necessarily the case that

every player observes a signal.

3 The model

Our model analyzes the voting behavior of a board of 2+ 1 directors composed by insiders

and outsiders. Inside directors comprise the CEO, other executives and non-executives di-

rectors with close ties to the firm. As mentioned before outside directors can be of two types:

either pure representative of shareholders’ interests (profit maximizers) or independent di-

rectors who care about their reputation. Directors have to approve (or reject) an investment

project proposed by the CEO.

Project’s value and Information

The board takes a binary decision4 by majority vote: it can accept the project (voting

“yes”) or reject it (voting “no”). If the proposal is rejected, a value of 0 is realized (status

quo). If accepted, the investment can take one of two values according to the state of nature

that can be either low () or high (): ∈ −1 1 In other words, the project can createa profit or a loss for the firm Each value has the same prior (i.e., 1

2).

Directors can obtain some information on project profitability only by incurring a non-

monetary cost, possibly different for different types of director. However, given that insiders

always favor the project, they will never incur their information cost. Consequently we focus

on the cost for outside directors and we denote it by .5 We assume that is fixed and is

the same for all outsiders. The cost represents the time and effort a director needs to gather

information on project proposal and its probability of success. By incurring , an outside

director learns the true state with probability ∈ [12 1) and with probability 1− he learns

ones.4Binary decisions are of course only a subset of the decisions actually taken by committees. Nonetheles,

we believe that many important decisions in boards or juries can be represented as binary ones, as so does

most of the literature.5Insiders’ information cost might be lower than that of the outsiders. For example inside directors might

know the true state of the world at zero cost. Such informational advantage could be a by-product of their

managing the firm. However it is inifluential in our context.

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nothing. Thus, the choice of whether to acquire information is a binary choice and directors

cannot control the accuracy of the information.6 As a consequence, the information set of

a generic member of the board is simply Ω = , with ∈ ;, when is

informed. On the contrary, the information set of an uninformed director is Ω = , ashe does not know the true state of the world. The model is a two-stage model: first directors

decide to get informed, then they participate to the board meeting and cast their vote.

Voting, communication and abstention

In the present and the following two sections we assume that there is no communication

among directors, though, in Section 5 we consider a sequential voting game. A sequential

voting mechanism can be regarded as a case where there is some information transmission

between board members (from predecessors to successors in the voting sequence). Indeed,

when directors vote sequentially, the information set also contains previous members’ voting

decisions, We discuss communication in Section 6.

Each director expresses one vote and abstention is not possible. This assumption may

seem restrictive and is worth an additional comment. Allowing abstention would bring other

issues and restrictions into the picture, which would drive attention away from the scope

of the paper. First of all, it is not clear what would happen when everyone abstains. One

could reasonably argue that the status quo wins: but is this choice less discretionary than

banning abstention? Second, it is neither clear nor straightforward what the decision rule

should be: simple majority of members or simple majority of actual votes? Third, members

in the board are actually paid to take decisions and they should be able to carry on their

duty, based on the information they observe or elicit from other members’ votes. In many

boards abstention is explicitly or implicitly ruled out (the Italian Constitutional Court and

the European Courts of Human Rights are two examples). Finally, we will show that in our

context abstention is not relevant because directors never want to abstain.

As a consequence of the no abstention assumption, the action set for each player is simply:

play “yes” to accept the project or play “no” to reject it. A strategy is a member’s voting

behavior, conditional on his information set. A mixed strategy is defined as the probability

that a member votes “yes”.

Directors’ types and utility functions

We consider three possible types: insiders, profit-maximizing outsiders and reputational

outsiders. The objectives of different types of outsiders do not necessarily clash, but they may.

Outsiders caring for shareholder objectives are interested in the final decision being correct

and consequently they choose the course of action that favors the correct final decision by

6This simplification is not made without loss of generality but nonetheless at a (quite) cheap price. See

Balduzzi [2005] for preliminary results when the signal is noisy.

9

the board as a whole. Outsiders caring about their reputation, on the contrary, want to

demonstrate that they are informed. In order to do so they want to vote correctly (i.e. vote

for approval/rejection of the project if it is profitable/unprofitable) but are not interested

in the final decision being correct. The market does not know the value of and uses the

information on directors voting behavior and on the true state of the world to update its

prior on the director’s ability to acquire (and process) information. Hence it pays to vote

correctly even if the final decision of the board is wrong. We do not model explicitly the way

in which the market forms and revises its judgements on directors’ reputation and we simply

capture this through our assumption on directors’ utility.

We assume that directors’ types are common knowledge. This assumption is stronger

than necessary since in many cases we only need the knowledge of the types of a subsets of

directors, but it greatly simplifies the analysis.

Let denote inside directors, outside directors who represents shareholders’ interest

and outside directors who care about their own reputation. Then, we call , and the

probability of voting “yes” for a member of type and respectively.

As stated before, a member of type is an “empire builder” who derives utility from the

enlargement of the firm.7 He always supports the investment project, regardless of the value

which is ex post realized. His utility is therefore a function of the final decision of the

board, , where = when the project is approved (“yes” wins) and = when the

project is rejected (“no” wins). Accordingly, () can assume the following two values:

() :

(1 =

0 = (1)

This clearly implies that always voting “yes” is a dominant strategy for the insider. For

simplicity, we abstract from additional problems, such as the member ’s reputation when

his proposal creates a loss or the reduced power when his proposal is rejected.

Both types of outside directors are risk-neutral but they have different utility functions.

The utility function of a member of type is increasing in firm’s profits and decreasing in

the cost of acquiring information If he decides to acquire information his utility is :

( ) = −

and it is ( 0) = if no effort to get informed is exerted.

7Alternatively we can interpret the behavior of this member type as the consequence of his overconfidence

in the success of the project. This however implies that the prior he attaches to = 1 is higher than 1/2.

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Given this utility function a type- member will choose the strategy that maximizes the

expected profit of the firm minus . To this end, he will condition his strategy on being

pivotal, because that is the only case where he can actually influence the outcome of the

game and therefore his own utility.

Notice that, given the values the investment can take, maximizing () is equivalent to

maximizing the probability that the board takes the correct decision (Prob). Indeed, the

latter is given by the sum of the probability that “yes” wins when the actual value of the

alternative is 1 and that “no” wins when the actual value of the alternative is −1:

Prob = (· | = 1) + [1− (· | = −1)] (2)

where the function (·) is the probability that the board as a whole votes “yes”. The expectedprofit of the firm is:

() =1

2[(| = 1) +(| = −1)] (3)

=1

21 [ (·| = 1)] + 0 [1− (·| = 1)]+ 1

2−1 [ (·| = −1)] + 0 [1− (·| = −1)]

=1

2[ (·| = 1)− (·| = −1)]

and it is straightforward to notice that expressions (2) and (3)are strategically equivalent.

A member of type is interested in his own reputation. He does not care about the final

decision of the board but only about his own. He wants to show to the market that he is

right, even when he is uninformed. In this context, “being right” means voting according to

the true state of the world, that is, voting “yes” when = 1 and voting “no” when = −1.The opposite is true for “being wrong”. As explained above we assume that the market

observes the voting behavior and the realized state of the world. Then we capture the gain

reputational outside directors can obtain from voting correctly by assuming that a -type

member has a utility function ( ) that takes one of the following two values :

( ) :

(( = | = 1) = ( = | = −1) = 1−

( = | = −1) = ( = | = 1) = 0−

where, again, is incurred only if the director decides to acquire information. Intuitively, if

a member is wrong, then the market recognizes him as surely uninformed (this will become

clearer after studying the equilibrium strategy of this member). Thus, will do anything to

take the correct decision, even if uninformed.

Optimal Strategy profile and Compensating Strategy

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Given that a board is appointed by the shareholders to run the firm in their own interests,

we measure optimality with respect to their interest only. Consequently, when defining the

optimum, we ignore the payoffs of members whose objective functions differ from that of

the shareholders (namely, insiders and reputational outsiders, ), as well as the cost of

acquiring information. We also ignore the possibility that firms’ decisions can actually affect

the welfare of the society.

Definition 1 A strategy profile is optimal (∗) when it maximizes the firm’s profits, given

all available information:

∗ = argmax( | Ω)

Before analyzing the optimal voting behavior of the board, we introduce the definition of

compensating strategy that will be useful in the following sections.

Definition 2 (Compensating strategy) Two players are compensating each other when

the following conditions are satisfied: i) they are both uninformed; ii) they play “yes” with

probabilities whose sum is equal to 1. When these probabilities take extreme values (0 or 1),

we have compensation in pure strategies.

Notice that for profit-maximizing outsiders the optimal strategy is conditional on being

pivotal. Since any strategy is optimal when the player is not pivotal, without loss of generality,

we concentrate on weakly dominant strategies.

The aim of this paper is to provide some insights on the role and importance of outside

directors within boards. To this end we proceed by steps: in the next section we only

consider profit-maximizing outsiders in addition to insiders. Then, in Section 5 we also

introduce reputational outsiders.

4 Boards with insiders and profit maximizing outsiders

Consider a board with 2 + 1 members. As a starting point consider a board uniquely

composed by inside directors. Under our assumptions such a board would always approve

the project obtaining the best choice for the shareholders half of the times (i.e., when the state

of the world is ). The expected profit for the firm would be 0. Our aim is to show under

which conditions outside directors can improve board performance. It is easy to see that the

presence of outsiders on the board would not affect the result as long as the outsiders are the

minority group. Suppose now that there are +1 insiders and outsiders. In our simplified

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setting nothing would change with respect to a board entirely composed by insiders because

the project would still be approved no matter what the outsiders do. Furthermore, since no

outsider would be pivotal, none would incur the cost to acquire information. However, the

situation changes drastically if we reverse the proportion of insiders to outsiders. In such a

case outsiders are able to reject the project if they believe it is unprofitable.

In the following we analyze the voting game in a board with 2 + 1 members where at

least + 1 directors are profit-maximizing outsiders () and the others are insiders ().

There is at least one insider, namely the CEO, who proposes the project.

We first consider the limit case in which the cost of acquiring and processing information

is zero, i.e. = 0 The following proposition and corollaries then hold.

Proposition 1 Consider an outsider-controlled board with 2+1 members where insiders al-

ways vote in favor of the project. If = 0 we can distinguish two cases.

1) if there are insiders and +1 profit-maximizing outsiders, there a unique optimal equi-

librium with expected profit () = 12[1 − (1 − )+1] Equilibrium strategies are such that

outside directors choose = 0when uninformed and they follow their information otherwise.

2) if there are − insiders ( 0) and +1+ profit-maximizing outsiders, there are

multiple equilibria that improve upon the outcome of a board entirely composed by insiders

but only one is optimal. In the optimal equilibrium, outside directors follow their information

when they are informed. When uninformed, − + 1 outsiders choose = 0 and the other

2 compensate each other in pure strategies.

Proposition 1 states that, as expected, a board dominated by profit-maximizer outsiders

improves shareholders’ welfare with respect to a board of only insiders. To reach optimality,

every uninformed outsider should not influence the outcome of the vote and must thus vote

the opposite of insiders (i.e., always “no”). This leaves informed members (if any) able to

be pivotal, which is optimal for shareholders. When no outsider is informed, the project will

be rejected and this will be the right decision only half of the times. In this case, however,

there is no way to improve the outcome. Even the decision to accept the project would

be right only with probability 12.8 Notice that in this context an outsider would never

find it profitable to abstain even if he were allowed to do so. If the number of insiders

and outsiders is almost the same ( and + 1 respectively), this equilibrium is unique.

On the contrary, with a higher number of outsiders (+ 1+ ), there also arise sub-optimal

equilibria. In the optimal equilibria any uninformed outsider still has an incentive to leave the

final decision to others, who may be informed. As the uninformed member cannot abstain, it

is optimal for him to compensate (see definition 2). This means that − outsiders will vote8As a consequence also a decision to accept with some probability would lead to the same expected profit.

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no in order to compensate insiders. The rest of the outsiders will compensate themselves.

Such ’intra.outsider’ compensation is more efficient when played in pure strategies, as it

is realized with probability one, but sub-optimal equilibria with compensation in mixed

strategies cannot be ruled out.

Corollary 1 The expected profit of the firm is not increased by increasing the proportion of

outside members above +12+1

.

Corollary 1 stresses that, for a given number of board members, increasing the proportion

of outsiders does not provide any additional benefit to shareholders, provided that outsiders

are already the majority. On the contrary, by increasing the proportion of outsiders other

equilibria may be obtained in addition to the optimal one. In particular, this means that a

board entirely composed of profit-maximizing outsiders would not be able to improve upon

the result of a board with just + 1 outsiders.9 A different way of looking at the corollary

is that, as long as a majority of outsiders is maintained, having insiders in the board is ben-

eficial for shareholders. Indeed, if the proportion of insiders reaches 49% of board member,

we have the best possible situation, because the optimal equilibrium is unique. It is worth

noting that the optimum equilibrium is reached without any communication of information

among directors. The positive role of insiders is not related to the communication of their

information to outsiders but to the fact that the presence of insiders simplifies outsiders’

strategies because, when uninformed, they know that they must compensate the partisan

vote of the insiders. This guarantees uniqueness of the optimal equilibrium. This point dif-

ferentiates our paper from previous models on board behavior that analyze the problem of

how to elicit information from insiders, as Harris and Raviv (2008) and Adams and Ferreira

(2007). Contrary to this literature where the best situation is the one that maximizes infor-

mation transmission under the existing constraints, in our setting optimality can be reached

without using insiders’ information. This is particularly interesting because insiders have a

positive role despite the fact that our model does not consider those tasks at which insiders

are better than outsiders (for example the tasks that are related to the daily management of

the firm).

Note that, the above corollary indicates that the relationship between board composition

and firm performance is highly non linear: all the benefits deriving from the presence of

profit-maximizing outsiders is captured when their fraction increases from 2+1

to n+12+1

This

may may explain why the empirical literature has not found a clear positive relation between

the number of outsiders on the board and firm performance.

Given these results, in the following, we will restrict our attention to boards with insiders

and + 1 outsiders. Corollary 1 tells us that this is the optimal proportion of outsiders but

9This result crucially depends on the assumption that directors do not communicate. See Section ...

14

says nothing as to the optimal size of the board. It turns out that if information acquisition

is not costly, expected profits are increasing in board size.

Corollary 2 When = 0 and the optimal equilibrium is played, an increase in induces

an increase in the expected profit of the firm.

Recall that the expected profit is given by

() =1

2[ (·| = 1)− (·| = −1)]

Provided that a majority of outsiders is maintained, the probability of approving the project

when the state of nature is unfavorable, (·| = −1) is not increased by an increase in because the effect of additional insiders is compensated by outsiders voting no when un-

informed. On the contrary the probability of approving the project when it is profitable,

(·| = 1) increases as is increased, because it is equal to the probability that at least

one of the +1 outsiders (who choose = 0 when uninformed) is instead informed. Hence,

adding new members is profitable for the firm. Obviously this result obtains because we

are only considering the increase in information without taking into account possible costs

of expanding board membership, as coordination costs or a smaller ability to resist CEO’s

control as suggested by Eisenberg, Sundgren and Wells (1998).

Let us now consider the general case where information acquisition is costly for outside

directors. Each outsider finds it profitable to incur information cost only if the advantage

of becoming informed is at least as large as the cost. Expanding board size therefore may

have the drawback of inducing free riding among members.10

Let [()] indicate the expected profits when outsiders decide to acquire information,

and let ∆( + 1) indicate the marginal benefit of acquiring information for the + 1

outsider:

∆(+ 1) ≡ [(+ 1)]−[()]

We can prove the following proposition.

Proposition 2 (Benchmark with c0) When 0 for the optimal equilibrium to be

played in the voting game with insiders and + 1 profit-maximizing outsiders, the cost of

acquiring information must be smaller than the benefit obtainable from the information:

∆(+ 1) =

2(1− ) ≥

so that each outside director = 1 2+ 1 decides to acquire information.

10This is pointed out also by Harris and Raviv (2006).

15

In the optimal equilibrium each outside director is more likely to be pivotal when

= −1 and therefore chooses = 0 when uninformed. It follows that information changeshis vote (and expected profits) only in the favorable state of the world. The marginal benefit

from incurring the information cost is then 2(1− ), which is decreasing in This allows

us to determine the optimal size of the board:

Corollary 3 The optimal size of the board is 2+1 where is the smallest integer such that2(1− ) ≥ and the optimal proportion of outside directors is +1

2+1

The optimal size of the board is such that all outside directors incur the information cost

because expected profits are larger when directors are informed. Contrary to what happens

when information is costless, now an increase in board size is not necessarily profitable.

Indeed, is a decreasing function of The inverse relationship between the optimal number

of directors and the information cost arises because the probability of a given director being

pivotal decreases as increases and this in turn decreases the incentive to become informed.11

An implication of corollary (3) is that in industries where it is more difficult to have the

skills necessary to evaluate the investment proposal, as in innovative industries, the optimal

dimension of the board is smaller than in more traditional industries where project evaluation

is easier (see Raheja 2005).

Corollary 4 For a given value of the incentive to acquire information is decreasing in

This follows from the fact that the higher the lower is the probability that a member

is pivotal. Indeed if one of the outside members is informed none of the others is pivotal.

On the contrary, the lower is the more important it is that each single director tries to

acquire information because he is more likely to be pivotal. Hence, for a given the free-

riding problem among members of the board becomes more severe as increases. This in

turn implies that the optimal board size decreases in .

5 Mixed Boards

We now consider the case where part of the outside directors only care about their own

reputation and we analyze the voting behavior of a board with insiders, +1− ( ≥ 0)

profit-maximizing outsiders and reputational outsiders. As in the previous section, we first

consider the limit case in which the cost of acquiring and processing information is zero, i.e.

= 0 Then the following proposition holds.

11The same endogenous determination of board size as a result of the decreased incentive to become

informed arises in Harris and Raviv (2008).

16

Proposition 3 When = 0 in a board with insiders and + 1 − ( ≥ 0) profit-

maximizing outsiders and reputational outsiders, there are multiple equilibria, only one

of which is optimal. An insider always votes “yes”; a profit maximizing outsider ( =

1 + 1− ) always votes “no” when uninformed; a reputational outsider ( = + 1 −(− 1)+ 1) always randomizes with any probability when uninformed. Independently ofhis type, an outsider always follows his signal when informed.

Again, profit-maximizing outsiders know that they can influence their own utility only

when pivotal; for this reason, they act as if they actually decided the voting outcome. Their

behavior does not change with respect to the previous section: given that insiders always

vote ’yes’, ’s best reply is compensating in pure strategies when uninformed, i.e. choosing

= 0. On the contrary, reputational outsiders do not condition their strategy on being

pivotal. Since their payoffs are independent of the final decision of the board, the best they

can do is simply to randomize whenever uninformed, i.e. to choose any ∈ [0 1]. Then,in general we should not expect a optimal equilibrium profile to be selected. Optimality

can only be obtained when there is compensation in pure strategies both from the profit

maximizing and the reputational outsiders’ side. This means that both types must play

the same strategy aimed at offsetting insiders’ strong bias. Only when every randomly

behaves as an , is the equilibrium profile optimal.

Keeping in mind such multiplicity problem, we can still find conditions for the optimal

equilibrium to be played when 0 Notice however that reputational and profit-maximizing

outsiders differ in their decision to acquire information. For a reputational outsider the benefit

from becoming informed is the increased reputation stemming from voting “correctly”. Recall

that the utility from voting correctly is equal to 1. Then when the reputational outsider is

not informed, the expected value of the benefit from voting correctly is 12. When instead

she decides to become informed, the expected benefit is +(1−)2

= +12Thus, contrary

to what happens for profit-maximizing outsiders, the gain from acquiring information for

reputational outsiders ∆ =2is independent of board size. A profit-maximizing outsider

instead continues to look at the increase in expected profit in order to make his decision.

As before, if he acquires information, expected profit changes only in the favorable state of

the world because, if uninformed, he votes against the project. Then, the next proposition

follows:

Proposition 4 When 0 for the optimal equilibrium to be played in the simultaneous

voting game with insiders and + 1 − ( ≥ 0) profit-maximizing outsiders and

reputational outsiders, it must be the case that

∆(+ 1) =

2(1− ) ≥

17

Intuitively, the marginal gain from information is higher for a reputational than for a

profit-maximizing outsider. This must be the case as reputational members fully enjoy the

possible benefit from information, whereas profit maximizing members can enjoy it only if

they are pivotal. Therefore, it is sufficient that is small enough for profit maximizing

outsiders to acquire information: if profit-maximizing outsiders find it profitable to acquire

information, a fortiori information acquisition is profitable for a reputational outsiders.

If reputational outsiders could be trusted to play the optimal equilibrium, it would be

optimal to form a board with insiders and + 1 reputational outsiders.12 Unfortunately,

as we pointed out above, with reputational outsiders multiple equilibria arise.13 A possible

solution to the multiplicity problem is to apply a protocol that imposes sequential voting

in the board. In the following subsection we analyze the sequential voting game and we

determine the sequence that must be followed in order to reach optimality.14

5.1 Sequential voting

In the sequential game, it does not matter when the insiders are voting, as their behavior is

uninformative. On the contrary, the relative position of outsiders is important. Profit maxi-

mizing outsiders should vote before reputational ones. An still compensates if uninformed,

as he is playing weakly dominant strategies and conditioning his behavior on being pivotal.

In other words, he cannot elicit any relevant information from observing the voting behavior

of an (nor that of another ). More explicitly, even if an could observe a vote revealing

for sure that the state of the world is , he would not change his compensation strategy, as

he would no longer be pivotal. On the contrary, an does not care about being pivotal: he

only cares about appearing informed. Therefore, whenever uninformed, reputational mem-

bers should follow profit-maximizing ones. Consider the first to vote. If he observes at

least one playing like an insider, then he should follow him, as that profit-maximizing out-

sider is certainly informed. But even if no profit-maximizing director deviates and plays like

the insiders, the reputational member should follow the profit-maximizing ones, as there is a

positive probability that they are informed. As for subsequent uninformed , they should

simply follow their predecessor in the voting sequence. If any is informed and his signal

is different from what previous outsiders did, then he should follow his signal. Subsequent

should follow him, as that player is certainly informed. We can state this more formally

starting again from the case where = 0.

12Notice that in this case there woud be no endogenous upper limit to the size of the board.13Notice that in equilibria other from the socially optimal one, the presence of reputational outsiders

reduces the incentive to become informed for profit maximizing outsiders. The condition in Proposition 4

becomes 2(1− )(1− ) ≥ if profit maximizers compensate in pure strategies.

14nel caso della sezione precedente sequential game non modifica

18

Proposition 5 (sequential voting with c=0) When = 0 in a board with insiders,

+ 1 − ( ≥ 0) profit-maximizing outsiders and reputational outsiders, there is a

unique equilibrium which is optimal. Insiders always vote “yes”, their vote is uninformative

and their position in the voting sequence is irrelevant. On the contrary, the order of vote

among outsiders is relevant: at least one type should vote before any type ; profit maxi-

mizing outsiders always vote “no” when uninformed; reputational outsiders always vote “no”

when uninformed, unless any previous outsider ( or ) voted “yes”. Outsiders always follow

their signal whenever informed. For different sequences of vote there are multiple equilibria.

The sequential structure works as an “implementation mechanism” for the optimal strat-

egy profile. Given the appropriate order of vote (where the first voter is an ), the previous

strategy profile is both individually and optimal. As reputational outsiders are imitating the

behavior of profit maximizing members, the replacement of an with an does not change

the voting outcome and such a board performs just like a board composed only by insiders

and profit-maximizing outsiders. The following corollary then holds.

Corollary 5 When = 0 in any optimal equilibrium profile, the proportion of type and

type is irrelevant, provided that the first voter is an .

This picture slightly changes when outsider can obtain information only at cost 0

Again, in this case we concentrate on the optimal equilibrium. With sequential voting the gain

from information for a reputational outsider depends on his position in the voting sequence

as well as on how many profit-maximizing outsiders vote before the reputational ones. Let

us consider the case with only one profit maximizing outsider who acquires information and

votes first. The reputational outsider that does not become informed can copy the behavior of

the profit-maximizing outsider voting correctly with probability +(1−)2= 1+

2= 1− (1−)

2

where(1−)2

is the probability of casting the "wrong" vote. Hence, the benefit obtained from

a reputational outsider who does not incur is:

( info) = 1− (1− )

2

where = 1 = represents the order in the reputational voting sequence (i.e. = 1

means that this is the first reputational outsiders to cast his vote). The gross benefit obtained

from a reputational outsider who pays is:

(info) = + (1− )

∙1− (1− )

2

¸= 1− (1− )+1

2

Notice that the values of () would not change if there were more than one profit maxi-

mizing outsiders who all vote before the reputational one. In that case however = 1

19

would indicate the order in the sequence of outsiders following the first one (if there are 3

s, = 3 indicates the first reputational outsider to cast his vote). We can then prove the

following Proposition.

Proposition 6 (Sequential voting with c0) When 0 for the optimal equilibrium

to be played in the sequential voting game with insiders, and + 1 − ( ≥ 0)

profit-maximizing outsiders voting before the reputational outsiders, it must be the case

that

2(1− ) ≥

From the above condition the next corollary follows immediately.

Corollary 6 The optimal size of the board is the same as in the board with only profit-

maximizing outsiders being 2+ 1 where is the smallest integer such that 2(1− ) ≥

Furthermore, it is again independent of whether voting is simultaneous or sequential.

Summarizing, mixed boards can perform efficiently provided the order vote is properly

chosen. Given an optimal order of vote, the board selects the efficient outcome as an equi-

librium. This is true even if there is no informational gain from employing insiders, which is

the usual argument for mixed boards of directors. Notice that the optimal size of the board

is the same as in the case with only profit maximizing outsiders. As a consequence the same

level of expected profits is obtained with profit maximizing outsiders or reputational ones

provided that at least one outsider is of the profit maximizing type and that reputational

directors follow the optimal voting sequence.

6 The role of communication

In this section, we examine if and how the introduction of pre-voting communication improves

the voting outcome. The main change derives from the fact that profit maximizing outsiders

have a clear incentive to communicate their information so as to induce optimal behavior

from other directors.

Communication is introduced as a pre-voting stage where members send messages about

their information set. Messages are cheap talk. Recall that the information set of a generic

member of the board is Ω = , with ∈ ;, when is informed, and

Ω = when is uninformed. Consequently, member can send a costless message

∈ ; 0 where ∈ means that is informed and ∈ 0 means that is

20

uninformed. Messages are exchanged simultaneously among directors of a given type but we

allow for selective communication in which a director sends a message to a subset of board

members. This is meant to capture the fact that profit maximizing directors may have an

incentive to coordinate on the message they send to reputational directors. When messages

are received, the information set of voting members also contains such messages. Finally, we

assume that the communication stage cannot be observed by the public.

The message strategies of different members are derived from their preferences. Insiders

cannot commit to send truthful messages because of their strong bias. Thus, they would

never be believed. This is equivalent to assuming that insiders do not send any message, i.e.

= 0

Reputational outsiders have no incentive to reveal their informed status, as their utility

depends only on their vote. So, again we assume = 015. Then, in the following we only

focus on the message strategies of profit-maximizing directors and we assume that, when

indifferent, a profit maximizing outsider always sends a truthful message.

For simplicity, we only consider the case where = 0 but results immediately generalize

to the case where 0 if we consider situations where the decision to become informed is

necessarily taken before the message stage. In other words, we refer to situations where the

process of collecting information takes time so that a director cannot strategically postpone

such decision after the message stage, trying to free ride with respect to other possibly

informed members.

6.1 Board with Insiders and Profit-maximizing outsiders

If the board is composed by insiders and + 1 profit maximizing outsiders, the (simulta-

neous) equilibrium strategies are given in the following proposition.

Proposition 7 When = 0 in a board with insiders and +1 profit-maximizing outsiders,

there are multiple equilibria, and they are all optimal. An insider does not send any message

and always votes ’yes’; an informed outsider sends a truthful message and votes according to

his information; an uninformed outsider votes according to the message(s) received if any,

and randomizes with any probability if no message is received.

When at least one outsider is informed his information is revealed, and the board surely

takes the correct decision. When no information is revealed, uninformed outsiders now

15In principle, as we do not impose that members vote consistently with their messages, reputational

members may even send wrong messages. If reputation depended on relative performances, by sending

wrong messages, directors could destroy the reputation of uninformed members of the same type. So,

messages would not be believed and, again, we could assume = 0

21

know that nobody is informed (otherwise the informed outsider would have send a message).

So they have no reason to compensate and make other outsiders pivotal. The effect of

communication on the expected profit of the firm is null. Recall that, in the absence of

communication, the wrong decision was taken with probability 1/2 only when all outsiders

were uninformed, and this is precisely what happens now that communication is allowed.

We can then conclude that in a board with insiders and + 1 profit-maximizing outsider,

communication has no effect on the performance of the board.

6.2 Mixed board

When there are both types of outside directors on the board some differences emerge because

profit maximizing outsiders have an incentive to coordinate on the message they send to

reputational directors. This modifies the outcome of the simultaneous voting game.

Proposition 8 When = 0 in a board with insiders, + 1 − ( ≥ 0) profit-

maximizing outsiders and reputational outsiders, there is a unique optimal equilibrium.

In equilibrium an insider always votes ’yes’. A profit-maximizing outsider always sends a

truthful message = to other profit maximizing outsiders. Subsequently, if informed, he

also sends a truthful message to reputational directors and votes according to his information,

but he sends a false message = to reputational directors and always votes ’no’, if

uninformed. A reputational outsider votes according to his information, if informed, and

votes according to the received messages if uninformed.

Profit maximizing outsiders now coordinate on the message they send to reputational

members to induce them to always vote ’no’ when no information is available. Reputational

directors, when observing message , are aware that the message may be false but they

follow the message because the probability that the true state is is now higher than 1/2.

Optimality is then guaranteed: as in the case without communication, it is reached because all

reputational members vote like profit-maximizing ones.16 The possibility of communication

in the simultaneous game increases the expected profit of the firm, by reducing the probability

that the board takes the wrong decision.

We know from section 5 that, in the absence of communication, mixed boards reach opti-

mality only if directors follow the optimal order of vote with at least an outsider voting before

16The same outcome can be obtained if we excluded selective communication and allowed directors to

simultaneouly send messages to all other board members. In this case an informed profit-maximizing outsider

always sends a truthful message = and votes according to his information; an uninformed proft

maximizer sends a false message = and always votes ’no’; an informed reputational outsider votes

according to his information, an uninformed reputational director votes according to his received messages if

they are consistent, and votes ’yes’ if messages report conflicting information.

22

reputational members. Such protocol serves as a means to pass information from predeces-

sors to successors in the sequence. When direct communication is possible, sequentiality is

not needed to reach the optimal outcome.

7 Conclusions

The model has analyzed the voting behavior of directors faced with the acceptance/rejection

decision on an investment project with uncertain prospects. In order to have a positive prob-

ability of becoming informed on project’s profitability, directors have to incur an information

cost. Contrary to the previous literature we have focused our attention on outside directors.

We have assumed that they may be of two different types: those who want to maximize the

profit of the firm and those who want to maximize their own reputation.

A strict majority of profit-maximizing outsiders over insiders is a necessary and sufficient

condition to maximize the profit of the firmwhen insiders are biased towards acceptance of the

project. Mixing up the identity of outsiders (i.e., substituting part of the profit-maximizing

outsiders with reputational ones) is not an obstacle to the achievement of profit maximization

provided that at least one director is of the profit maximizing type. If directors do not

communicate among themselves before voting, however, an appropriate sequential voting

protocol is required to avoid the existence of suboptimal equilibria (in addition to the optimal

one) when there are also reputational outsiders in the board. Pre-voting communication

instead does not improve the outcome when the board is only composed by insiders and

profit-maximizing outsiders.

The optimal dimension of the board is determined by comparing the advantages to the

disadvantages of expanding the membership. A large (outsider dominated) board is expected

to collect more information. However, marginal returns from information decrease in board

size for profit maximizing outsiders. Thus expanding the board may have the drawback

of inducing their free riding on costly information. Reputational members on the other

hand obtain non decreasing benefits from information acquisition. Consequently, the optimal

dimension of the board is given by the largest membership that avoids the free riding of profit

maximizers and directly depends on the amount of their information cost.

The model could be enriched along several directions. The identity of members (their

preferences) could be private information; in addition, outsiders may be of different quality

in the sense of having different probabilities of becoming informed. The model also appear to

be particularly fit for a laboratory experiment. Along these lines we will develop our future

research.

23

8 Appendix

8.1 Proof of Proposition 1

Since informed directors know the state of nature with certainty, each informed outsider votes

’yes’ when = 1 and ’no’ when = −1. Thus we focus only on the actions of uninformedoutsiders, recalling that each outsider conditions his strategy on being pivotal and that each

insider always votes ’yes’.

i) Equilibrium in the presence of insiders and + 1 outsiders

Consider a generic outsider ∈ = 1 2 +1. The probability that is

pivotal is equal to the probability that all other 6= members vote ’no’. This may happen

in both states of nature.

When = 1 is pivotal only if all other outsiders are uninformed and vote “no”, which

happens with probability:

(1− )+1Y

=1

(1− )

where ∈ [0 1) is the probability that a member 6= votes “yes”;

When = −1 is pivotal if

- all other outsiders are uninformed and vote “no”, which happens with probability

(1− )+1Y

=1

(1− ) ∈ [0 1)

- all the other outsiders are informed, which happens with probability

(1− )

- at least one of the other outsiders is informed and the others vote “no” when uninformed,

which happens with probability

X=1

−(1− )Π

where Π represents the sum of all the combination of the products of the (1− ) for the

outsiders who are uninformed .

Comparing the case where = 1 to that where = −1, it immediately follows that the

24

probability that is pivotal is higher in the bad state. Hence chooses = 0

The expected profit is given by

() =1

2[ (·| = 1)− (·| = −1)]

Considering that in this equilibrium (·| = −1) = 0 and that (·| = 1) is equal to the

probability that at least one of the +1 outsiders who choose = 0 is informed, the expected

profit is equal to

X=0

(+ 1)!

!(+ 1− )!+1−(1− ) = 1− (1− )+1 0

By direct comparison with all subsequent results, this emerges as the highest possible

expected profit. Hence, we can define it as the optimal outcome.

ii) Equilibria in the case of − insiders ( 0) and + 1 + outsiders:

Equilibrium with − outsiders choosing = 0 and 2+1 outsiders compensating

in pure strategies

We prove the existence of this equilibrium in two steps.

1. If outsiders choose = 0 and outsiders choose = 1 the best response of 6=

is to choose = 0

When = 1 is pivotal i) when all outsiders are uninformed; and ii) when at least one

of those outsiders who would choose = 1 when uninformed, is in fact informed. Then

is pivotal with probability

(1− )+1

"X

=0

!

!( − )!−(1− )

#= (1− )+1

When = −1 is pivotal i) when all outsiders are uninformed; and ii) when at least one

of those outsiders who choose = 0 when uninformed, is in fact informed. Then is

pivotal with probability

(1− )+1

"X

=0

()!

!(− )!−(1− )

#= (1− )+1

Given that (1−)+1 (1−)+1 the probability that is pivotal is higher when = −1than when = 1 Hence chooses = 0

25

2. If + 1 outsiders choose = 0 and − 1 outsiders choose = 1 the best response of

6= is to choose = 1

When = 1 is pivotal if only one of the + 1 outsiders choosing = 0 is informed

and votes yes i.e. with probability

(+ 1)(1− )+1

Given that is never pivotal when = −1 he chooses = 1

The expected profit

() =1

2[ (·| = 1)− (·| = −1)]

again has (·| = −1) = 0 and (·| = 1) equal to the probability that at least one of the+ 1 outsiders who choose = 0 is informed. () is then equal to

(·| = 1) =X

=0

(+ 1)!

!(+ 1− )!+1−(1− )

and this equilibrium coincides with the social optimum (see the last part of point i).

b) Equilibrium with − outsiders choosing = 0 and 2 + 1 outsiders com-

pensating in mixed strategies ( = 12)

Suppose 2 outsiders choosing = 12 and − outsiders choosing = 1; the best

response for the remaining outsider (denoted by ) is ∈ [0 1] Indeed, in both states of the world, is pivotal when everybody is uninformed and votes

“no”, and ii) when + 1 outsiders are informed and the others are uninformed. This

implies that is pivotal with the same probability in both states, i.e.

(1− )+1µ1

2

¶"

X=0

!

!(− )!−(1− )

µ1

2

¶#

which in turn implies that his best response is ∈ [0 1] Given that the same reasoning applies to every outsider, it immediately follows that =

12 for all outsiders sustain an equilibrium of the game. Note that if were to choose

12 the best response of a generic member 6= , would be = 0 because would

be pivotal with higher probability in = −1 than in = 1 With 2n-1 outsiders choosing

= 12 and choosing = 0 however the best response of becomes = 1 because

would be pivotal with higher probability in = 1 than in = −1

26

Symmetrically if were to choose 12 the best response of a generic member 6= ,

would be = 1 because would be pivotal with higher probability in = −1 than in = 1

With 2n-1 outsiders choosing = 12 and choosing = 1 however the best response

of becomes = 0 because would be pivotal with higher probability in = −1 than in = 1 These situations belongs to the following set of equilibria.

c) Equilibria with − outsiders choosing = 0 2 + 1 − outsiders com-

pensating in pure strategies and outsiders compensating in mixed strategies

( = 12)

If outsiders choose = 1, outsiders choose = 0 and 2(−) choose = 12the best response of is to choose ∈ [0 1] This follows immediately from the fact that

is pivotal with the same probability when = 1 and when = −1 Given that the samereasoning applies to all the outsiders except those 2 who compensate in pure strategies, it

immediately follows that = 12 for the 2( − ) outsiders who randomize, sustain an

equilibrium of the game. Moreover following the argument at the end of the previous point,

we can check that if there are (− − 1) outsiders who choose = 0 ( = 1), − who

choose = 1 ( = 0) and 2(− ) choose = 12 the best response of is to choose

= 0 ( = 1)

to be completed

8.2 Proof of Corollary 1

It follows from the proof of Proposition 1.

8.3 Proof of Corollary 2

Given that there are at least + 1 outsiders, after an increase in we still have (·| =−1) = 0 On the other hand, (·| = 1) increases as is increased, because in the equilibriumwhere + 1 outsiders choose = 0 and the other outsiders (if present) choose = 1, we

have:

(·| = 1) =X

=0

(+ 1)!

!(+ 1− )!+1−(1− ) = 1− (1− )+1

So the Corollary follows.

27

8.4 Proof of Proposition 2

The gain from acquiring information is positive only when the member is pivotal since it is

only in this case that being informed makes a difference. [(+ 1)]−[()] = 2(1− )

follows from

[(+ 1)] =1

2[ (·| = 1 + 1 info)− (·| = −1 + 1 info)]

=1

2[1− (1− )+1]

and

[()] =1

2[ (·| = 1 info)− (·| = −1 info)]

=1

2[1− (1− )]

8.5 Proof of Corollary 3

If directors get informed and the optimal equilibrium is played, expected profits are

() =1

2[ (·| = 1)− (·| = −1)]

=1

2

£1− (1− )+1

¤which is increasing in . Considering that [(+1)]−[()] =

2(1−) is decreasing in

, and that the +1 director does not acquire information if 2(1−)+1 the corollary

immediately follows.

8.6 Proof of Corollary 4

It follows immediately by differentiating with respect to the increased profit [(+1)]−[()] :

[2(1−)]

= 12(1− )−1[1− (1 + )] 0

28

8.7 Proof of Proposition ??

Given their objective function, the unique symmetric strategy of insiders is the same as in

the simultaneous voting game. As regards outsiders, in principle they could condition their

strategy on being pivotal and on the history of the game. Consider the following situation:

first, let the insiders vote, then let the remaining + 1 outsiders vote. Now, if the state

of the world is such that an informed outsiders vote exactly like an insider, then uninformed

outsiders will know the true state of the world and will play accordingly. But if such a thing

happens, then any following outsider is no longer pivotal and what he does is totally irrelevant.

Therefore the order of vote is irrelevant and each member plays as in the simultaneous game.

Note that this argument is not dependent on insiders voting first because all players know

that they always vote “yes”.

8.8 Proof of Proposition 3

The members of the board are the following:

- insiders ∈ = 1 2 ;- reputational outsiders ∈ = 1 2 ;- + 1− profit maximizing outsiders ∈ = 1 2 +1− We start from the insiders. They want to maximize the probability that the board accepts

the project and therefore always vote “yes”. For each of them, the equilibrium dominant

strategy (expressed in terms of probability of voting “yes”) is:

∗ ≡ ∗∀∈ : 1

As regards outsiders, let’s assume = 0 Then the board is composed by insiders and

profit-maximizing outsiders. We have already proved in Proposition 1 that outsiders should

compensate the positive vote by insiders by voting “no” whenever uninformed (if they are

informed, they follow the signal, as the signal is always correct). Intuitively, by compensating

and always voting “no” when uninformed, these members give the possibility to any other

− who is informed to be actually pivotal and take the correct decision. If this is true,

then a fortiori they should compensate when their number reduces (i.e., grows), as more

members are not (necessarily) playing a profit maximizing strategy. It follows that in the

heterogeneous board, when voting is simultaneous, have the following (weakly) dominant

29

strategy:

∗∀∈ :

⎧⎪⎨⎪⎩1 | Ω

= = ;0 | Ω

= = ;0 | Ω

=

⎫⎪⎬⎪⎭Finally, any , if informed (Ω

= ), votes according to his signal, as this is always

correct. If uninformed (Ω=

) then must choose the probability ∈ [0 1], whichmaximizes his expected utility. It is straightforward to work out that ’s expected utility

is:

() =1 +

2

which is independent of the probability This means that any probability ∈ [0 1] isutility maximizing for It follows that the individually optimal strategy for any is the

following:

∗∀∈ :

⎧⎪⎨⎪⎩1 | Ω =

= ;0 | Ω =

= ;∀ ∈ [0 1] Ω

=

⎫⎪⎬⎪⎭There exist multiple equilibria, only one of which is optimal (whenever every plays

= = 0 when uninformed: see Proposition 1).

8.9 Proof of Proposition 4

When 0 for the optimal equilibrium to be played in the simultaneous voting game with

insiders and +1− ( ≥ 0) profit-maximizing outsiders and reputational outsiders,

it must be the case that ∆ and ∆(+ 1) As

∆ =+ 1

2− 12=

2; and

∆(+ 1) = [(+ 1)]−[()] =

2(1− );

it immediately follows that

∆ ∆(+ 1)

8.10 Proof of Proposition 5

The members of the board are the following:

- insiders ∈ = 1 2 ;

30

- reputational outsiders ∈ = 1 2 ;- + 1− profit maximizing outsiders ∈ = 1 2 +1− For simplicity and without loss of generality, the cardinality of a player also indicates his

ordinality in the voting process among voters of his own type. That is, 1 is the first voter

among the reputational outsiders, and so on.

Finally, we call the voting decision of any generic player, that is, the vote he actually

cast and which is observed by subsequent members.

We start from the insiders. Their actions are totally uninformative, so their position

is irrelevant. Just for simplicity in the notation, their decisions are not considered in the

informational sets of other members. For each of them, the equilibrium dominant strategy

(expressed in terms of probability of voting “yes”) is:

∗ ≡ ∗∀∈ : 1

Let’s assume now that the entire set votes first. 1 has no additional information but his

own (possible) signal on which to base his action; so he behaves as in the simultaneous voting

game. His utility maximizing strategy is:

∗1 :

⎧⎪⎨⎪⎩1 | Ω1 = 1 = ;0 | Ω1 = 1 = ;

∀1 ∈ [0 1] Ω1 =

⎫⎪⎬⎪⎭

where is the probability that player votes “yes” when uninformed. On the contrary,

any subsequent ∈ \1 if uninformed, has an incentive to follow his next previous −1

From Bayes updating (see below), the probability of being right by following −1 is higher

than that one of being right by randomizing (12), for any −1. Thus should follow −1.

Notice that no herding can occur: as soon as any observes a signal, he knows that the

signal is correct and (possibly) destroys the cascade. For subsequent ∈ \1, the optimalequilibrium strategy is:

∗∈\1 :

⎧⎪⎨⎪⎩1 | Ω∈\1 = ∈\1 = ; 12−1;0 | Ω∈\1 = ∈\1 = ; 12−1;

−1 | Ω∈\1 = ; 12−1

⎫⎪⎬⎪⎭As regards − , from Proposition 3 we know that, whenever uninformed, they should

first compensate the behavior of the insiders in order to guarantee optimality. Again, by

compensating they give the possibility to any other − who is informed to be actually

31

pivotal and take the correct decision. The only difference with the simultaneous game is that

now members can actually observe if any previous − deviated and played “yes”. That

would mean that these members are informed and should be followed. Anyway, at this point

a decision would have already been taken, so it is still a (weakly) dominant strategy playing

“no” when uninformed, independently on what previous players did. As, by assumption,

0, then the number of is at most , just like the insiders. It follows that for any

∈ the optimal equilibrium strategy is:

∗∀∈ :

⎧⎪⎨⎪⎩1 | Ω

= = ; ;0 | Ω

= = ; ;0 | Ω

= ;

⎫⎪⎬⎪⎭

Proposition 3 requires for optimality that = 0∀ ∈ It follows that, if vote first,

there is a continuum of equilibria which are typically not optimal.

On the contrary, when all the vote first, although nothing changes for them, now 1

can elicit some information from their behavior. By the same reasoning as above, he has an

incentive, when uninformed, to follow the previous player (in this case, +1−). So 1 has

the following equilibrium strategy:

∗1 :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1 | Ω1 = 1 = ; ;0 | Ω1 = 1 = ; ;

if Ω1 = ; ∧

=

+1− if Ω1 = ; ∧ = ∀

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭The last two lines follow form the fact that 1 has an incentive to follow any deviating

from playing “no”, as that means he is informed. For the remaining again, nothing

changes:

∗∈\1 :

⎧⎪⎨⎪⎩1 | Ω∈\1 = ∈\1 = ; ; 12−1;0 | Ω∈\1 = ∈\1 = ; ; 12−1;

−1 | Ω∈\1 = ; ; 12−1

⎫⎪⎬⎪⎭Notice that this order of vote implies an optimal equilibrium. Every behaves like a

profit maximizing outsider, unless he is informed or follows someone who is informed for sure.

This is obviously optimal.

By using similar arguments, we can state that it is sufficient to have 1 voting before 1

to ensure optimality.

32

8.10.1 Bayes updating

− use Bayes’ rule to update their priors about the true state of the world. Suppose

2 is not informed and previously only player 1 voted (according to his equilibrium strategy

showed above). Now define the following events:

: the true state is “H”;

: 1 voted “Yes”

Whenever uninformed, 2 should follow 1 if:

Pr[ | ] 1

2(4)

This posterior will depend on the probability 1:

Pr[ | ] =12

12+ 1

2(1− )1

Given ∈ [12 1] Condition (4) is always satisfied for any 1.

This means that whenever 1 votes after, say, (who plays = 0 in equilibrium), he

still has an incentive to follow him rather than randomizing, even if is voting “no”; unless,

of course, any other voted “yes” (Pr[ | ] = 1).

8.11 Proof of Proposition 6

This proof parallels that of Proposition 4. In this case:

∆ =

2(1− ) =

∆(+ 1) = [(+ 1)]−[()] =

2(1− ) ≥

for = so that each outside director decides to acquire information. For we have

∆ ∆(+ 1) =

2(1− )

8.12 Proof of Proposition 7

First of all, note that members keep on ignoring any additional information, given their

utility function. For each of them, the equilibrium dominant strategy (expressed in terms of

33

probability of voting “yes”) is:

∗ = 1

As regards outsiders, with probability (1 − )+1 no is informed, whereas with prob-

ability 1 − (1 − )+1, at least one observes a signal (thus sends a truthful message). In

the latter case, voting strategies are straightforward: any informed sends ∈ ;every other votes accordingly. More precisely, if the revealed information is = each

outsider is pivotal. On the contrary, if the revealed information is = , then they are no

longer pivotal and can cast any vote.

When instead no is informed, any chooses the probability ∈ [0 1], which maxi-mizes his expected utility. Given the priors about the state of the world, utility is clearly

independent of so any probability ∈ [0 1] is utility maximizing for It follows that their equilibrium (weakly dominant) strategy (again in terms of probability

of voting “yes”) is:

∗ :

⎧⎪⎨⎪⎩1 | Ω = = ;

0 | Ω = = ;Ω = = ;∀ ∈ [0 1] | Ω = = ;Ω = ; 0

⎫⎪⎬⎪⎭When at least one is informed, the correct decision is taken with probability equal to

1 and the expected profit of the firm is:

() =£1− (1− )+1

¤[1

21 +

1

20] =

1

2

£1− (1− )+1

¤

When no is informed, then expected profit of the firm is:

() =1

2[ (·| = 1)− (·| = −1) = 0

as when no information is available (·| = 1) = (·| = −1)Thus, the expected profit of the firm is the same as in the case without communication.

34

8.13 Proof of Proposition 8

First of all, note that members keep on ignoring any information. For each of them, the

equilibrium dominant strategy (expressed in terms of probability of voting “yes”) is:

∗ = 1

As regards outsiders, have an incentive to coordinate on the message they send to

reputational outsiders. This implies that they will always send truthful messages to the other

profit-maximizing outsiders. With probability (1− )+1− no is informed, whereas with

probability 1− (1− )+1−, at least one is informed and sends signal ∈ . As tothe message from to , this is ∈ if at least one is informed. If all s are

uninformed instead they send the false message =

Voting strategies for are straightforward: if at least one is informed every votes

accordingly. As in Proposition 7, if the revealed information is = each outsider is

pivotal. On the contrary, if the revealed information is = , other outsiders are no longer

pivotal and can cast any vote.

Reputational outsiders follow their information when informed and follow the message

from the s when uninformed. They do so because they know that = 1 when = and

that the probability that = −1 when = is higher than 12.

It follows that the equilibrium (weakly dominant) strategy (again in terms of probability

of voting “yes”) is:

∗ :

⎧⎪⎨⎪⎩1 | Ω = = ;

0 | Ω = = ;Ω = = ;Ω = ; 0∀ ∈ [0 1] | Ω = =

⎫⎪⎬⎪⎭Finally, as regards :

∗ :

(1 | Ω = = ;Ω = = ;0 | Ω = = ;Ω = = ;

)

The correct decision is taken when at least one outsider is informed and = 1 and when

= −1 and at least one profit maximizing outsider is informed. Finally, when nobody is

informed the project is rejected and zero profits are realized. Then, expected profits are given

by:

35

() =1

2

£1− (1− )+1

¤

References

[1] Adams,R. and D. Ferreira [2007] A Theory of Friendly Boards, Journal of Finance,

62 (1), 217-250.

[2] Balduzzi, P. (2005)

[3] Banerjee, A. V. [1992]. A Simple Model of Herd Behavior, The Quarterly Journal

of Economics, 107 (3), pp. 797-817.

[4] Baranchuck, N. and P. Dybvig [2008] Consensus in Diverse Corporate Boards,

Review of Financial Studies,22 (2),716-747.

[5] Bebchuk, L., Y. Grinstein and U. Peyer [2009] Lucky CEOs and Lucky Directors

Journal of Finance, forthcoming..

[6] Bebchuk, L. and M.S. Weisbach [2009] The State of Corporate Governance re-

search, NBER Working Paper 15537, forthcoming in Review of Financial Studies

[7] Bikhchandani, S., Hirshleifer, D. and Welch, I. [1992]. A Theory of Fads, Fashion,

Custom, and Cultural Change as Informational Cascades, The Journal of Po-

litical Economy, 100 (5), pp. 992-1026.???Togliere?’

[8] Bikhchandani, S., Hirshleifer, D. and I. Welch [1998]. Learning from the behavior of

others: Conformity, Fads, and Informational Cascades, The Journal of Economic

perspectives, 12 (3), pp. 151-170.

[9] Borokhovich,K., R. Parrino and T. Trapani [1996] Outside directors and CEO

Selection, The Journal of Financial and Quantitative Analysis, 31 (3), 337-355.

[10] Dekel, E. and Piccione, M. [2000]. Sequential Voting Procedures in Symmetric

Binary Elections, The Journal of Political Economy, 108 (1), pp. 34-55.

[11] Eisenberg, Sundgren and Wells [1998] Larger Board Size and Decreasing Firm

Value in Small Firms, Journal of Financial Economics, 48, pp.35-54.

[12] Feddersen, T. and Pesendorfer, W. [1996]. The Swing Voter’s Curse, The American

Economic Review, 86 (3), pp. 408-424.

36

[13] Feddersen, T. and Pesendorfer, W. [1997]. Voting Behavior and Information Ag-

gregation in Elections with Private Information, Econometrica, 65 (5), pp. 1029-

1058.

[14] Finegold, D., G. S,Benson and D. Hect [2007]Corporate Boards and Company Per-

formance: review of research in light of recent reforms, Corporate Governance:

an international review, 15 (5), 865-878.

[15] Gale, D. [1996].What have we learned from social learning?, European Economic

Review, 40, pp. 617-628.

[16] Goeree, J. K., Palfrey, T. R. and Rogers, B. W. [2003]. Social Learning with Pri-

vate and Common Values, Working Paper 1187, California Institute of Technology,

Division of the Humanities and Social Sciences; forthcoming in Economic Theory.

[17] Harris, M. and A. Raviv [2006] A Theory of Board Control and Size, Review of

Financial Studies, forthcoming

[18] Hermalin, B. E. and M.S. Weisbach, [1988], Determinants of Board Composition,

Rand Journal of Economics, 19, 95—112.

[19] Hermalin, B. E. and M.S. Weisbach, [1998], Endogenously Chosen Boards of Direc-

tors and their Monitoring of the CEO, American Economics Review, 88, 96—118..

[20] Ottaviani, M. and Sørensen, P. [2001]. Information aggregation in debate: Who

should speak first?, The Journal of Public Economics, 81, pp. 393-421.

[21] Palley, T. I. [1995]. Safety in numbers: A model of managerial herd behavior,

The Journal of Economic Behavior and Organization, 28, pp. 443-450.

[22] Raheja, C. [2005]. Determinants of Board Size and Composition: A Theory

of Corporate Boards, Journal of Financial and Quantitative Analysis, 40 (2), pp.

283-306.

[23] Sloth, B. [1993]. The Theory of Voting and Equilibria in Noncooperative

Games, Games and Economic Behavior, 5, pp. 152-169.

[24] Swank, O. and Visser, B. [2003]. The Consequences of Endogenizing Information

for Herd Behavior, Tinberger Institute Discussion Paper N. 21.

[25] Visser, B. and O. H. Swank [2007] On Committees of Experts, Quarterly journal

of Economics, 337-372.

37

[26] Warther,V. [1998] Board Effectiveness and Board Dissent: a Model of the

Board’s Relationship to management and Shareholders, Journal of Corporate

Finance,4, pp.53-70.

[27] Weisbach, M. (1988).

38