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5 – 6 March 2010 CESifo Conference Centre, Munich
A joint initiative of Ludwig-Maximilians University’s Center for Economic Studies and the Ifo Institute for Economic Research
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: [email protected] 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: [email protected]; Skype: paolo.balduzzi.‡Email: [email protected].§Email: [email protected].
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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-
3
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.
5
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
6
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
7
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.
8
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.
10
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.
13
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
¤
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