Journal of Economics

, Volume 111, Issue 1, pp 69–95

Voting in small committees

Authors

    • Catholic University of Milan, IEF and CIFREL
  • Clara Graziano
    • University of Udine and CESifo
  • Annalisa Luporini
    • University of Florence and CESifo
Article

DOI: 10.1007/s00712-012-0321-5

Cite this article as:
Balduzzi, P., Graziano, C. & Luporini, A. J Econ (2014) 111: 69. doi:10.1007/s00712-012-0321-5

Abstract

We analyze the voting behavior of a small committee that has to approve or reject a project proposal whose return is uncertain. Members have diverse preferences: some of them want to maximize the expected value, while others have a bias towards project approval and ignore their information on the project value. We focus on the most efficient use of scarce information when members cannot communicate prior to voting, and we provide insights on the optimal composition of the committee. Our main result is that the presence of biased members can improve the voting outcome, by simplifying the strategies of unbiased members. Thus, committees with diverse members perform as well as homogeneous committees, and even better in some cases. In particular, when value-maximizing members outnumber biased members by one vote, the optimal equilibrium becomes unique.

Keywords

VotingSmall committeesCommittees composition

JEL Classification

D71D72

1 Introduction

In many committees, members are nominated by (and thus represent the interests of) different institutions and this reflects in different voting behaviors. Consider, for instance, boards of directors whose objective, in principle, is to maximize firm value. Directors represent different stakeholders, (majority and minority shareholders, investors, workers, etc.) whose objectives may not be aligned. Another example1 is provided by monetary policy committees, where some members are chosen within the staff of the central bank while other members are appointed by external bodies, such as the Government (the Bank of England Monetary Policy Committee is a typical example). In this case, internal members are usually more concerned about inflation while external members are more concerned about unemployment. In general, empirical studies show that members belonging to different groups have significant differences in their voting behaviors, and that these differences can be explained by factors such as political pressure or the channel of appointment, especially when committee members face retention decisions (see, for example, Sheperd 2009; Harris et al. 2011).

The present paper analyzes the effect of member heterogeneity by studying the voting behavior of a small committee that has to approve or reject a project. We consider two types of players: expected value maximizers and biased members who always vote in favor of the project, even disregarding their private information. Then, the following question arises: why should biased members be allowed on this committee? Our model shows that their presence is beneficial in the absence of pre-voting communication because it ensures uniqueness and optimality of the equilibrium strategy profile. The intuition is that the bias provides certainty about some members’ strategies thus simplifying the responses of the others, and therefore reducing the number of (otherwise) multiple equilibria. In particular, we explore the behavior of uninformed value-maximizing members. Given that they want to maximize the probability that the committee makes the correct decision, they face the question of how to avoid influencing the decision and let informed members determine it. The equilibrium voting strategies prescribe that uninformed unbiased members systematically contrast the vote of biased members. Indeed, in many small committees dissent voting is commonly observed (Spencer 2006). On the basis of the equilibrium voting strategies, we determine the optimal composition of the committee, consisting in letting unbiased members outnumber biased members by just one vote. Our result is consistent with the actual composition of some committees such as the Bank of England Monetary Policy Committee or the Italian Constitutional Court.

The paper is organized as follows. Section 2 reviews the main literature. Section 3 presents the basic model. Section 4 examines, as a benchmark, the voting game in a committee composed only of value-maximizing members. Section 5 introduces biased members and analyzes if and how results change when members have different objectives. Then, in Sect. 6 we discuss some assumptions of the model. Finally, Sect. 7 concludes. All proofs are collected in the Appendix.

2 Related literature

Since Condorcet’s seminal contribution, namely his Jury Theorem, the literature about voting has been constantly growing. A lot of papers have generalized the Jury Theorem,2 and many others have extended voting games to include both naive and strategic voting.3

Traditionally, in this literature the aim of voters has been to aggregate information, with the assumption that taking the right decision (that is, guessing the correct state of the world) was the common objective of all the players. In fact, we believe this is not the case, as heterogeneity of preferences is well documented both in large and in small elections.4 Feddersen and Pesendorfer (1996, 1997) have focused on heterogeneity in large elections, showing that full information aggregation is still possible. In particular, they show that the probability of electing the “wrong” candidate asymptotically goes to zero.

Since the seminal contribution by Austen-Smith (1990), information sharing and voting in small committees has been increasingly analyzed along with the possibility of communication.5 Nonetheless, in small committees, the presence of preference heterogeneity and (possible) resulting conflicts of interests appear to be a relevant problem, as information sharing and aggregation may be severely limited by strategic behavior (see also Gerling et al. 2005).

Thus, one possibility is to look for optimal voting rules to minimize information losses. In a standard Condorcet Jury Theorem framework, Chwe (1999) suggests to provide minority members with optimal incentives to participate in voting, in order not to waste their information. Things become more complicated when the relevant issue is not to find optimal voting rules but, rather, an optimal way to aggregate “relevant” information.6 Li et al. (2001) examine a two-person committee where each member receives a private signal and reports his information. Since members have conflicting interests, strategic considerations induce information misreporting and there is no truth-telling equilibrium. Conflict of interest prevents full information aggregation also in larger committees, as shown by Maug and Yilmaz (2002). These authors suggest to group voters into two separate classes because such a voting mechanism may alleviate the incentive to withhold information when voters have strong conflicts of interests and individual information differs. Thus, voting decisions become more informative. Examining a committee of experts, Wolinsky (2002) suggests to solve the problem in a similar way, by partitioning members in different groups.7

We take a different approach to the problem of information sharing among committee members by examining how the voting outcome can be optimized when members do not communicate. Without imposing any explicit revelation mechanism and without looking for optimal voting rules or protocols, we suggest an optimal composition rule as a sufficient device to provide the highest possible level of information aggregation, even when biased members do not use their private information. The positive role of the latter in our model relies on the fact that their presence on the committee eliminates multiple (and possibly suboptimal) equilibria.

3 The model

A committee is composed of \(2n+1,n\ge 1,\) members who have to decide by majority vote whether to approve a project (voting “yes”) or reject it (voting “no”). If the proposal is rejected, a value of \(0\) is realized. When accepted, the project yields value \(v=-1\) if the state of the world is low (\(L\)), and \(v=1\) if the state of the world is high (\(H\)). Thus, \(v:\left\{ -1,1\right\} \).

Each state, and thus each value, has the same prior (i.e., \(\frac{1}{2}\)). This implies that when members have no information on the state of the world, there is no one choice that dominates the other. Given these probabilities, the highest expected value that can be achieved by voting correctly (rejecting the project in \(L\) and approving it in \(H\)) is \(\frac{1}{2}\). Note that a single uninformed decision maker would always obtain an expected value of \(0\).

We consider a simple information structure where, before voting, any member of the committee learns the true state with probability \(\alpha \in \left[\frac{1}{2},1\right)\) and learns nothing with probability \(1-\alpha .\)8 As a consequence, the information set of a generic member \(i\) is \(\Omega _{i}=\{\omega _{i}\}\), with \(\omega _{i}\in \left\{ H;L\right\} \), when \(i\) is informed, while it is \(\Omega _{i}=\{H,L\} \), when \(i\) does not know the true state of the world. The probability of being informed is identically and independently distributed across members. Each member does not know who else is informed. Furthermore, we consider the case in which committee members can become informed at no cost. As we point out at the end of Sect. 4, introducing a fixed cost for acquiring information may set an upper bound to the optimal size of the committee.

We assume that members cannot communicate prior to voting, and as usual in the literature on committee voting, we do not consider abstention.9 Both assumptions are discussed in Sect. 6. Given that abstention is not allowed, the action set of each player has only two elements: vote “yes” to accept the project, and “no” to reject it. A strategy \(s_{i}\) is a member \(i\)’s voting behavior, conditional on his information set. A mixed strategy is defined as the probability that a member votes “yes”.

The committee is composed of unbiased members who want to maximize the expected value of the project, \(E(v),\) and of biased members who want to approve the project independently of the state of the world. We assume that all members are risk neutral and that their types are common knowledge. Let \( M\) denote value-maximizing members, and \(B\) members with a bias. Then, we call \(m\) and \(b\) the probabilities of voting “yes” for an uninformed member of type \(M\) and \(B \) respectively.

The utility function of an \(M\) type can be directly expressed in terms of the expected value of the project: \(u_{M}(E(v))=E(v).\) Then, an \(M\) member will choose the strategy that maximizes \(E(v)\). Notice that, given the values the project can take, maximizing \(E(v)\) is equivalent to maximizing the probability that the committee makes the correct decision. Indeed, the latter is equal to the sum of the probabilities that “yes” wins when the actual value of the alternative is \(1\) and that “no” wins when the actual value of the alternative is \(-1\):
$$\begin{aligned} \frac{1}{2}\left\{ Y(\cdot \mid v=1)+[1-Y(\cdot \mid v=-1)]\right\} , \end{aligned}$$
where the function \(Y(\cdot |\cdot )\) is the conditional probability that the board as a whole votes “yes”. The expected value of the project is:
$$\begin{aligned} E(v)=\frac{1}{2}[1Y(\cdot |v=1)-1Y(\cdot |v=-1)] \end{aligned}$$
and it is straightforward to notice that the two expressions are strategically equivalent.

Given that a member can influence the outcome (and consequently his own utility) only when he is pivotal, conditioning the vote on being pivotal is a weakly dominant strategy for an \(M\) member who wants to maximize \(E(v).\) We concentrate only on equilibria where the \(M\) members choose such strategies.10 The solution concept we use is Bayesian Nash Equilibrium.

The utility function of a \(B\) member positively depends on the approval of the project.11 A \(B\) member always supports the project, regardless of the value which is ex post realized. His utility \(u_{B}\), therefore, depends on the final decision of the committee, and can take the following two values: \(u_{B}=1\) if the project is approved, and \(u_{B}=0\) if the project is rejected. This clearly implies that always voting “yes” is a dominant strategy for a \(B\) member. For simplicity, we abstract from additional problems, such as a \(B\) member’s potential loss of reputation when the approval of the project creates a loss.

Finally, before analyzing the voting behavior of the committee, we introduce the definition of compensating strategy that will be useful in the following sections.

Definition 1

(Compensating strategy) Two members are playing compensating strategies when the following conditions are jointly 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 and 1), compensation is in pure strategies.

Note that this definition describes strategies but does not require members to know who is informed and who is not. Compensation is an ex ante concept. It may well happen that one of the two individuals who compensate when uninformed is in fact informed, so that compensation does not necessarily obtain ex post.

4 The benchmark

We define our benchmark as a committee only composed of members who want to maximize \(E(v)\), i.e. members of type \(M.\) Clearly, whenever an \(M\) member is informed, he votes according to his information. The issue is to define what an uninformed \(M\) member should do. Intuitively, any uninformed member has an incentive to leave the final decision to the others, who may be informed. It can then be shown that there are four types of equilibria differing as to the behavior of uninformed members.

The first consists of an (asymmetric) equilibrium, where all but one member compensate for each other in pure strategies when uninformed, while the remaining member plays any strategy (when uninformed). Given that one of the members can actually play “yes” with any probability, and that the identity of those members who vote “yes” and those who vote “no” (as well as that of the member who possibly uses a mixed strategy) is interchangeable, there exists in fact a multiplicity of such equilibria all of which yield the same expected value. Considering that members condition their strategies on being pivotal, no symmetric equilibrium in pure strategies is possible.12 The second consists of a (symmetric) unique equilibrium where all the members compensate for each other in mixed strategies, voting “yes” with probability \(1/2.\) The third consists of an equilibrium where an even number of uninformed members compensate for each other in pure strategies, while the remaining members compensate in symmetric mixed strategies, voting “yes” with probability \(1/2\). Finally, the last consists of an equilibrium where there are more members voting “yes” (“no”) than members voting “no” (“yes”), while the others choose the same mixed strategy, voting “yes” with probability lower (greater) than \(1/2.\) Again there is a multiplicity of equilibria of the third and fourth type because the identity of the players is interchangeable and because there can be different numbers of players choosing mixed strategies.

In order to better understand the nature of the first type of equilibria, consider a simple example with \(\alpha = {\frac{1}{2}}\) and \(n=2\), so that there are five members \(M_{i}, i=1,2,3,4,5.\) Suppose that four members vote “yes” when uninformed. The remaining member knows that he is pivotal only if two members vote “no”. But, given the above strategies, this happens only if the two members who vote “no” are in fact informed. Then, the remaining member should vote “no”. This tells us that (as members condition their strategies on being pivotal) there cannot exist a symmetric equilibrium in pure strategies where everybody votes “yes” (neither, by the same argument, an equilibrium where everybody votes “no”). Moreover, it immediately appears that a situation where four members vote “yes” (or “no”), when uninformed, cannot be an equilibrium: voting “yes” (“no”) is not the best response for an uninformed individual when there are already three members following the “yes” (“no”) strategy. His best response is to vote the opposite of another uninformed member, thus giving rise to the asymmetric equilibrium. Once there are three members voting “yes” and two members voting “no”, nobody has an incentive to change his strategy. Consider one of those member voting “yes”. The probability that he is pivotal is the same in both states, so he is indifferent between voting “yes” or “no”.

The expected value of these equilibria can be easily computed for \(\alpha =1/2\). In the case where two members vote “no” when uninformed, e.g. \(m_{1}=m_{2}=0,\) and \(m_{3}=m_{4}=m_{5}=1\), the expected value is
$$\begin{aligned} E(v)^{PS}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=\frac{1}{2}\left[ 1- \frac{1}{8}\right] =\frac{7}{16}<\frac{1}{2} \end{aligned}$$
which is clearly equal to the value that obtains if three members vote “yes” when uninformed, e.g. \( m_{1}=m_{2}=m_{3}=0,\) and \(m_{4}=m_{5}=1\)
$$\begin{aligned} E(v)^{PS}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=\frac{1}{2}\left[ \frac{7 }{8}-0\right] =\frac{7}{16}<\frac{1}{2}. \end{aligned}$$
Moreover it can be easily checked that \(E(v)\) does not change even if there is one member choosing a mixed strategy. For example when \(m_{1}=m_{2}=0, m_{3}=m_{4}=1\) and \(m_{5}=1/2\)
$$\begin{aligned} E(v)^{PS}&=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)] \\&=\frac{1}{2}[1-[1-Y(\cdot |v=1)]-Y(\cdot |v=-1)] \\&=\frac{1}{2}\left[ 1-2\frac{1}{16}\right] =\frac{7}{16}<\frac{1}{2}. \end{aligned}$$
Notice that, in the spirit of Condorcet, the expected value is bigger than the value obtained by a single decision maker (\(0\)). Still, this committee cannot provide full information (yielding \(E(v)^{FI}=\frac{1}{2}\)) as it may collectively be uninformed. In other words, the committee does not always make the correct decision. When all the members are uninformed, the decision (whatever it is) is correct with probability \(1/2.\) Moreover, the decision is wrong with probability \(1/2\) if the only informed members are those voting according to the actual state even if uninformed (those choosing \(m_{i}=1\) if uninformed when the actual state is \(v=1\), or those choosing \(m_{i}=0\) if uninformed when the actual state is \(v=0\)).
Consider now the second type of equilibrium where all the uninformed members compensate for each other in (symmetric) mixed strategies. Intuitively, in this equilibrium any member randomizes as long as he is pivotal in both states of the world with the same probability, given the other members’ strategies. But this is simultaneously true for any single member, only if all the members are pivotal in each state of the world with the same probability. The only profile which is compatible with this logic is then the one where all the members compensate for each other in mixed strategies, voting “yes” with probability \(1/2.\) This argument rules out any other possible equilibrium in mixed strategies: whenever a member is not indifferent between voting “ yes” or “no” (because the probability that he is pivotal in a state is higher than the probability that he is pivotal in another state), he plays a pure strategy. But then, other members will have an incentive to deviate from any mixed strategy to compensate for his pure strategy. Again, we can easily compute the expected value for the five member case with \(m_{1}=m_{2}=m_{3}=m_{4}=m_{5}={\frac{1}{2}}\) and \(\alpha =\frac{1}{2},\) obtaining
$$\begin{aligned} E(v)^{MS}&= \frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)] \\&= \frac{1}{2}\left[ 1-\left[ Y(\cdot |v=-1)+(1-Y(\cdot |v=1)\right]\right] \\&= \frac{1}{2}\left[ 1-\frac{53}{256}\right]=\frac{203}{512}<\frac{7}{16} \end{aligned}$$
where \(\left[ Y(\cdot |v=-1)+(1-Y(\cdot |v=1)\right] \) is the probability of making the wrong decision. Thus, the expected value falls short of that obtainable in the pure strategy equilibrium.
The third type of equilibrium looks like a combination of the previous two. In terms of expected value, these equilibria perform better than the mixed strategy ones but worse than equilibria with compensation in pure strategies. The expected value for the five member case with \(\,m_{1}=1,\,m_{2}=0,\,m_{3}=m_{4}=m_{5}={\frac{1}{2}}\) and  \(\alpha =\frac{1}{2},\) is
$$\begin{aligned} E(v)^{MPS}&= \frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)] \\&= \frac{1}{2}\left[ 1-\left[ Y(\cdot |v=-1)+(1-Y(\cdot |v=1)\right]\right]\\&= \frac{1}{2}\left[ 1-.171874\right] =.414063;\quad \frac{203}{512}<.414063<\frac{7}{16} \end{aligned}$$
These mixed equilibria also dominate equilibria of the fourth type where there are more members voting “yes” (“no”) than members voting “no” (“yes”), while the remaining members choose the same mixed strategy, voting “yes” with probability lower (greater) than \(1/2.\) An example of the fourth type of equilibrium is found in the proof of the following Proposition, which generalizes our findings.

Proposition 1

(Benchmark) In a voting game with \(2n+1\) members of type \(M,\) informed members always play according to their information. There exist four types of equilibria differing as to the behavior of uninformed members: \((i)\) multiple equilibria where \(n\) members always vote “no”, \(n\) members always vote “yes”, and one member chooses \(m\in \left[ 0,1\right] ; (ii)\) a unique equilibrium in mixed strategies where \(2n+1\) members randomize with probability \({\frac{1}{2}}; (iii)\) multiple equilibria where \(2n-2k,\)\(k=1,2, \ldots , n-1,\) members compensate in pure strategies and \(2k+1\) members play \(m_{k}=1/2\); \((iv)\) multiple equilibria where \((n-k_{1})\) members choose \(m_{j}=0\ (m_{z}=1)\) and \((n-k_{2})\) members choose \(m_{z}=1\) (\(m_{j}=0\)) with \(k_{1}<k_{2}\le n\) while all the others, \(k\ne j,z\), choose the same mixed strategy \(m_{k}> \frac{1}{2}\)\((m_{k}<\frac{1}{2}).\) Equilibria of type (i) yield expected value \(E(v)^{*}=\frac{1}{2}[1-(1-\alpha )^{n+1}].\) All other equilibria yield a lower expected value.

The intuition for this result is the same as in the five-member case: uninformed members do not want to influence the outcome of the voting process, so they compensate for each other, and leave the final decision to possibly informed members. This is also what happens in the second kind of equilibria where uninformed players compensate for each other in mixed strategies. Compensation is more effective when played in pure strategies, as it is realized with probability one: \(E(v)\) is higher in the equilibria where \(2n\) agents compensate in pure strategies than in the symmetric equilibrium in mixed strategies or in the equilibria where only part of the members choose pure strategies while the others choose mixed strategies. Then, in what follows we take the level of \(E(v)^{*}\) as our benchmark.

Definition 2

Any equilibrium that yields \(E(v)^{*}=\frac{1}{2}[1-(1-\alpha )^{n+1}]\) is defined optimal.

Notice that \((1-\alpha )^{n+1}\) is the probability that the decision is wrong, given by the probability that all the members are uninformed, plus the probability that the only informed members are those voting according to the actual state even if uninformed (those choosing \(m_{i}=1\) if uninformed when the actual state is \(v=1\), or those choosing \(m_{i}=0\) if uninformed when the actual state is \(v=-1\)).13

Just for illustration, we draw in Figs. 1 and 2 the relationship between \( E(v)\) and the probability of having informed members (\(\alpha \)), in committees with five and nine members. In both graphs, we compare the optimal equilibrium outcome (thin line) with the symmetric mixed strategy equilibrium outcome (thick line).
https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0321-5/MediaObjects/712_2012_321_Fig1_HTML.gif
Fig. 1

E(v) and \(\alpha \) in the five-member committee

https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0321-5/MediaObjects/712_2012_321_Fig2_HTML.gif
Fig. 2

E(v) and \(\alpha \) in the nine-member committee

The graphs show a positive relationship between \(E(v)\) and \(\alpha \) in both equilibria. They also show that the mixed strategy equilibrium never yields a higher expected value than the one with compensation in pure strategies. When all the members are informed (\(\alpha \)\(=1\)), there is no difference between the two equilibria and \(E(v)\) is the same. It is also clear that \(E(v)\) is growing in \(n.\) These relations are formalized by Corollary 1 which immediately follows from \(E(v)^{*}=\frac{1}{2}[1-(1-\alpha )^{n+1}].\)

Corollary 1

When an optimal equilibrium is played, \(E(v)^{*}\) is increasing both in \(n\) and in \(\alpha \) at a decreasing rate.

The positive relation between \(E(v)^{*}\) and \(n\) recalls the central idea of the Jury Theorem. The expected value grows with \(n\) because there is information aggregation (although imperfect, as we have seen in the five member case). In the present paper we do not address the issue of the optimal committee size. We take the size as exogenous, as it is likely to be determined on the ground of other criteria than the optimality of the voting behavior of the committee.14 Moreover, the optimal size of the committee can be bounded by information costs (e.g., see Persico 2004). When information acquisition is costly, each member has an incentive to acquire information only insofar as his benefit, which depends on the probability of being pivotal, is not lower than his cost. As in our equilibrium the probability of being pivotal decreases in size, this may set a limit to the optimal size of the committee.15

5 Heterogeneous preferences

Having determined the optimal equilibria, we compare this benchmark to the outcome of a committee composed of members with heterogeneous preferences. Consider again a committee with five members (\(n=2\)), in the case where \(\alpha ={\frac{1}{2}} ,\) but let now members be either of type \(M\) or \(B.\) If \(B\) members hold the majority, the case is trivial because the committee always approves the project and the \(M\) members are never pivotal. Then, we concentrate on the remaining two interesting cases in which the committee is composed of: (i) four value-maximizing members \(M_{i},\)\(i=1,2,3,4,\) and one biased member \(B_{5};\) and (ii) three value-maximizing members \(M_{i},\)\(i=1,2,3,\) and two biased members \(B_{j}, j=4,5\).

We start from the latter case and we show that there exists a unique equilibrium. Given that \(B\) members always vote “yes”, independently of their information, \(M\) members vote according to their information whenever informed, and vote “no” when they are uninformed. Such rejection is optimal, because the probability that an uninformed member is pivotal is higher when the state of the world is \(L\). In this case, we can easily compute:
$$\begin{aligned} E(v)_{M=B+1}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=\frac{1}{2}\left[ \frac{7}{8}-0\right] =\frac{7}{16} \end{aligned}$$
Quite surprisingly, the performance of this committee is the same as the optimal performance of the committee composed of unbiased members only. In addition, this equilibrium is unique. Thus, we can say that the heterogeneous committee ensures the optimal outcome, provided that the \(M\) members outnumber \(B\) members by just one vote.

Consider now the case where the committee is composed of \(M_{i},\)\(i=1,2,3,4, \) and \(B_{5}.\) Recall that the dominant strategy of an informed \(M\) member is to vote according to his information, and that of the unique \(B\) member is to always approve the project. Then, three types of equilibria emerge, differing as to the behavior of uninformed \(M\) members. In the equilibria of the first type, two of the \(M\) members compensate for each other in pure strategies when uninformed, and the other two \(M\) members vote “no”. In the equilibrium of the second type, all of the four \(M\) members play the same mixed strategy, when uninformed. In the equilibria of the third type, one of the \(M\) members votes “no”, while the other three \(M\) members play the same mixed strategy, when uninformed.

While the equilibrium of the first type is optimal, yielding expected value
$$\begin{aligned} E(v)_{M>B+1}^{PS}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=\frac{1}{2}\left[ \frac{7}{8}-0\right] =\frac{7}{16} \end{aligned}$$
the equilibrium of the second type is suboptimal. We find in fact the following solution: \(m_{1}=m_{2}=m_{3}=m_{4}=\frac{5-\sqrt{13}}{6},\) and
$$\begin{aligned} E(v)_{M>B+1}^{MS}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=.384973<\frac{7}{16} \end{aligned}$$
Intuitively, when the \(M\) members outnumber the \(B\) members by more than one, the former have different choices. It may be the case that one single member offsets the bias of \(B_{5}\) in pure strategies and the remaining members compensate for each other in pure strategies, or that all of the \(M\) members play the same mixed strategy with the aim to collectively contrast the bias of \(B_{5}\). This is the reason why the symmetric mixed strategy of the \(M\) members in the second type of equilibrium is now biased towards rejection, \(m_{i}<1/2,\)\(i=1,2,3,4\).
Moreover, another type of equilibrium is possible where the \(M\) members collectively contrast the bias of \(B_{5}\): one biased member always votes “yes”, one uninformed unbiased member plays a pure strategy (\(m_{1}=0\)), and the remaining members, whenever uninformed, play a symmetric mixed strategy (\(m_{2}=m_{3}=m_{4}=\frac{1}{3}\)). Such an equilibrium yields:
$$\begin{aligned} E(v)_{M>B+1}^{PMS}=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]=.388889. \end{aligned}$$
Hence:
$$\begin{aligned} E(v)_{M>B+1}^{MS}<E(v)_{M>B+1}^{PMS}<E(v)_{M>B+1}^{PS}. \end{aligned}$$
With more than \(5\) members, there may arise additional equilibria, analogous to the equilibria of the fourth type in Proposition 1. Such equilibria however are shown to be suboptimal in Proposition 2, which generalizes our results.

Proposition 2

Consider a committee with \(2n+1\) members where members of type \(B\) always approve the project and informed members of type \(M\) always vote according to their information. We can distinguish two cases:
  1. (i)

    if there are \(n\) members of type \(B\) and \(n+1\) members of type \(M\) the game has a unique equilibrium in which all the members of type \(M\) always vote “no” when uninformed. This unique equilibrium is optimal;

     
  2. (ii)

    if there are \(n-k\) members of type \(B\) (\(n>k>0\)) and \(n+1+k\) members of type \(M\), the voting game may have multiple equilibria. There always exists optimal equilibria where \(2k\) members of type \(M\) compensate for each other in pure strategies and the remaining \(n-k+1\) members of type \(M\) vote “no” when uninformed; other equilibria are suboptimal.

     
 

From the above proposition, Corollary 2 immediately follows.

Corollary 2

The expected value \(E(v)^{*}\) is not increased by increasing the proportion of value-maximizing members above \(\tfrac{n+1}{2n+1}\).

For a given size of the committee, increasing the proportion of the \(M\) members is not profitable, provided they already hold the majority. By increasing the proportion of the \(M\) members, optimal equilibria can still be obtained but there may also exist other equilibria. Thus, if value-maximizing members outnumber biased members by only one vote, the situation is greatly simplified with respect to our benchmark case because the optimal equilibrium is unique. Hence, we refer to this composition as optimal and to such a committee as the optimal heterogeneous committee.

We know from Corollary 1 that increasing the size of the committee (increasing \(n\)), increases \(E(v)^{*}\), when the optimal equilibrium is played. Since an increase in \(n\) may now result in an increase in the number of the \(B\) members, this point deserves some attention. Consider the case with \(n\) members of type \(B\) and \(n+1\) members of type \(M\) and recall that
$$\begin{aligned} E(v)=\frac{1}{2}[Y(\cdot |v=1)-Y(\cdot |v=-1)]. \end{aligned}$$
The probability of approving the project when the state of nature is unfavorable, \(Y(\cdot |v=-1),\) is still equal to zero after an increase in \(n,\) because the effect of additional biased members is compensated for by the additional \(M\) members voting “no” when uninformed. On the contrary, the probability of approving the project when it is profitable, \(Y(\cdot |v=1),\) increases with \(n\), because it is equal to the probability that at least one of the \(n+1\) members of type \(M\) is informed. Hence adding new members (including biased ones) is profitable.16

6 Discussion

We have assumed that the members of the committee cannot abstain and cannot communicate. We now briefly comment on these two assumptions.

Abstention

What happens if we relax our no-abstention assumption? First of all, note that allowing abstention would bring some new issues and restrictions into the picture. For instance, it should be arbitrarily stated what happens when everyone abstains, and an ad hoc rule (i.e., a probability to implement the project) should be applied. Moreover, it is unclear whether to look at simple majority of members or simple majority of actual votes.

In general, however, abstention would improve the performance of a committee composed only of unbiased members (our benchmark). With simple majority of actual votes, there clearly is an equilibrium in which uninformed members abstain. Notice that any ad hoc rule for the case in which all members abstain can induce such a behavior. In fact, if at least one member is informed, the correct decision is made with certainty. In other words, the wrong decision is made with probability \(\frac{1}{2}\) only when nobody is informed. If the committee is composed of \(2n+1\) unbiased members, in this equilibrium the expected value is \(E(v)=\frac{1}{2} [1-(1-\alpha )^{2n+1}]>E(v)^{*}.\)

Things are more complicated if there are \(k\) biased members, with \(1\le k<n. \) In this case, \(k\) unbiased members should not abstain when uninformed but should vote “no” in order to contrast the biased members’ votes.17 Moreover, as it may occur that \(k\) unbiased members vote “no”, while the other \(M\) members abstain, the tie-breaking rule should prescribe not to implement the project so as to take care of the case where the only informed \(M\) members are among those who (when uninformed) contrast the \(B\) members.

Notice however that the no-abstension assumption implies no loss of generality in the case of our optimal heterogeneous committee (\(n+1\) unbiased members and \(n\) biased members). Indeed, suppose abstention is allowed: biased members always vote “yes” and any unbiased member still finds it optimal to condition his strategy on being pivotal. The reason is the following. Suppose in a symmetric equilibrium all the other unbiased members abstain when uninformed. If the remaining one is pivotal, this means that the other unbiased members voted “no”, hence voting “no” is optimal. This is true for all unbiased members, hence “abstention” is not an equilibrium strategy. This is true even in all other asymmetric (putative) equilibria where some unbiased members vote “no” when uninformed and some others abstain. The most extreme case obtains when all the other unbiased members vote “no” when uninformed. Still, the probability that at least one of the other members voted “no” because he is informed and the true state is \(v=-1\) is higher than the probability that they all voted “no” because they are uninformed (the only case where the remaining member would be indifferent among voting “yes”, “no” or abstaining). Hence, voting “no” instead of abstaining is optimal (formula and calculation are the same as the ones in the Proof of Proposition 2).

Communication

If we allow members to communicate prior to voting, unbiased members will certainly communicate any information they possess because information aggregation among them may improve the outcome. Indeed, it can be easily verified that perfect information aggregation among \(M\) members is feasible. It suffices that a single \(M\) member is informed to reach the correct decision. Coherently with the literature (see, e.g. Gerling et al. 2005), communication can be introduced as a pre-voting stage where members of type \(M\) simultaneously send costless messages about their information sets. Recall that the information set of a generic member \(i\) is \(\Omega _{i}=\{\omega _{i}\}\), with \(\omega _{i}\in \left\{ H;L\right\} \), when \(i\) is informed, and \(\Omega _{i}=\{H,L\}\) when \(i\) is uninformed. Consider the case where member \(i\) can send a message \(\sigma _{i}\in \left\{ H,L,0\right\} ,\) where \(\sigma _{i}=0\) means that \(i\) sends no information. Messages update \(M\) members’ information sets.18 Moreover, assume that an informed member, whenever indifferent, sends a truthful message. Then, equilibrium strategies prescribe that each informed \(M\) member sends a truthful message and votes according to his information. Furthermore, each uninformed \(M\) member receiving at least one \(\sigma _{i}=L\) votes according to the received message(s), and indifferently votes either “yes” or “no” otherwise. This implies that in equilibrium full aggregation of information among \(M\) members is possible. As a result, it is optimal to increase the number of unbiased members as much as possible, because the outcome now depends on their (absolute) number and not on their proportion to biased members as in the optimal committee of the no-communication case. If the committee is composed of \(2n+1\) unbiased members, the expected value becomes \(E(v)=\frac{ 1}{2}[1-(1-\alpha )^{2n+1}]\) because the wrong decision is made with probability \(1/2\) only when nobody is informed, and this occurs with probability \((1-\alpha )^{2n+1}.\)

However, if we consider the optimal committee of the no communication case with \(n+1\) members of type \(M\) and \(n\) members of type \(B,\) the introduction of communication cannot improve on the outcome \(E(v)^{*}.\) In such a committee, the right decision is made with probability \(1\) when at least one out of the \(n+1\)\(M\) members is informed, and with probability \(\frac{1}{2}\) when no \(M\) member is informed. But this is precisely what happens in the case without communication, and consequently the expected value of the project reaches the same level \(E(v)^{*}=\frac{ 1}{2}[1-(1-\alpha )^{n+1}]\). Indeed, the voting strategies of the \(M\) members in the case with no communication (contrasting biased members and leaving the decision to possibly informed members) minimize the information required to reach the best possible outcome, \(E(v)^{*}.\)19 We can then conclude that, for a committee composition with \(n+1\) members of type \(M\) and \(n\) members of type \(B\), communication cannot improve the voting outcome.

7 Conclusions

We have analyzed the voting behavior of a small committee that has to approve or reject a project whose return is uncertain. Members have heterogenous preferences: some members want to maximize the expected value while others have a bias towards project approval and disregard their private information. More precisely, we have shown that, in the absence of communication among members, heterogeneous committees can function at least as well as committees with homogeneous value-maximizing members. In particular, when value maximizers outnumber biased members by just one vote, the presence of biased members can improve the voting outcome by simplifying the strategies of the value maximizers: the equilibrium becomes unique and yields the optimal outcome. For a given committee size, increasing the number of value-maximizing members above the minimum that ensures majority does not increase the expected value and gives rise to additional suboptimal equilibria.

Despite being quite simple, we believe our framework can be easily applied to explain voting behaviors in a number of different small decisional bodies such as monetary policy committees, juries, boards of directors, and so on. In all of these committees, it is not uncommon to observe dissent voting. We explain such dissent as the result of optimal voting strategies, given an optimal composition rule of the committee itself. Furthermore, our result shows that the composition actually used in some small committees (for instance, in the Bank of England Monetary Policy Committee or the Italian Constitutional Court) is optimal if members are diverse and communication is limited.

Footnotes
1

Additional examples are special juries as Supreme or Constitutional Courts, and technical committees, where politicians, bureaucrats and experts meet to provide advice.

 
2

See, for instance, Duggan and Martinelli (2001), Myerson (1998) and McLennan (1998). See also Piketty (1999) for a brief review of recent contributions about the information-aggregation role of political institutions.

 
3

See Austen-Smith and Banks (1996) and an experiment on the use of strategic voting by Eckel and Holt (1989).

 
4

See for instance Blinder (2006), Jung (2011), Riboni and Ruge-Murcia (2007) for the case on Monetary Policy Committees.

 
5

This strand of literature has been reviewed by Austen-Smith and Feddersen (2009). See also Adams and Ferreira (2007), Harris and Raviv (2008), and Raheja (2005) for communication in boards of directors.

 
6

Berk and Bierut (2004) suggest that often in small committees it is technologically or politically unfeasible to implement optimal voting rules, thus binding voting to be based on simple majority.

 
7

Committee members have heterogeneous preferences also in the model of Cai (2009). The focus of Cai, however, is the optimal size of the committee rather than its composition.

 
8

Alternatively, we can assume that every member observes a signal that is perfectly informative with probability \(\alpha \) and is totally noisy with probability \(1-\alpha \). Referring to this interpretation, it can be shown that our results would not qualitatively change if the signal were only partially noisy (Balduzzi 2005).

 
9

Of course we acknowledge some exceptions, such as Morton and Tyran (2008). Note also that in many committees abstention is explicitly or implicitly ruled out (juries, the European Courts of Human Rights and the Italian Constitutional Court are some examples).

 
10

This rules out “uninteresting” equilibria where each member is never pivotal. For instance when every member votes “yes” independently of his information. However it can be easily verified that all these equilibria yield \(E(v)=0.\)

 
11

Obviously, nothing substantial in our results would change if a biased member always supported rejection.

 
12

On this point, see footnote 10.

 
13

That \((1-\alpha )^{n+1}\) represents the probability of making the wrong decision is shown in the Proof of Proposition 1, point (v).

 
14

For example, the need to represent different stakeolders or to balance different powers or just plain political criteria.

 
15

We do not develop this aspect in the present paper because it would distort the attention away from our main focus. Costly information acquisition can be studied along the lines set in Persico (2004) who analyzes the optimal voting rule (in terms of both incentives to acquire information and information aggregation) in a similar setting. See also Harris and Raviv (2008) on the effects of costly information acquisition on the optimal size of boards of directors.

 
16

Clearly our comment on the optimal size of the committee following Corollary 1 still applies.

 
17

For simplicity, we only consider pure strategies for the \(M\) members.

 
18

Alternatively we can assume that messages are exchanged among all the members and enter everybody’s information set. Notice however that biased members cannot commit to send truthful messages because of their strong bias. Thus, they would never be believed. This is equivalent to assuming that they do not send any message, i.e. \(\sigma _{B}=0.\) On the other hand members of type \(B,\) given their preferences, would not change their strategies even if they received a message revealing that the state of nature is \(L\). For these reasons we focus on the message strategies of the \(M \) members.

 
19

The introduction of communication results in the expansion of the set of equilibria. When no information is revealed, \(M\) members now know that nobody is actually informed and thus have no strategic reason to contrast biased members and make other unbiased members pivotal. They can indifferently cast any vote and thus multiple equilibria arise: there now also exist equilibria where some unbiased members vote “no” after observing \(\sigma _{i}=H\). These additional equilibria may entail an unconvincing behavior on the part of the \(M\) members, nonetheless they all yield \(E(v)^{*}=\frac{1}{2}[1-(1-\alpha )^{n+1}]\). Notice however that, contrary to what happens in the type (ii) equilibria of Proposition 1, this multiplicity does not entail any coordination problem: whatever the choice of the \(M\) members receiving message \(\sigma _{i}=H\), an equilibrium with expected value \(E(v)^{*}\) is reached.

 

Acknowledgments

We thank two anonymous referees, Ferdinando Colombo, Luigi Vannucci, and Giulio Pianigiani for very helpful suggestions. We also thank participants at the ASSET Conference 2008, the CEPET Workshop 2009, the EALE Conference 2009, the CESifo Area Conference in applied Micro 2010 and at seminars held at Stony Brook University, Catholic University of Milan, University of Milan-Bicocca for comments on previous versions of the paper. All mistakes are the authors’ own responsibility. Financial help from IEF (Catholic University) and PRIN 2008 is gratefully acknowledged.

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© Springer-Verlag Wien 2012