Costly Participation and Heterogeneous Preferences in

Informational Committees∗

Hongbin Cai†

Abstract

Informational committees are groups of people who are designated to gather information.

This paper develops a simple model of committee size based on costly participation and prefer-

ence heterogeneity. Committee members exert non-verifiable efforts in gathering information

and report their findings to a principal, who then chooses policy. They may have policy

preferences that differ from the principal’s and hence may distort their reports. In a setting

in which the information structure and policy preferences are both represented by normal

random variables, I characterize an equilibrium under the mean decision rule and derive the

optimal committee size. I show that when effort costs are sufficiently high, preference hetero-

geneity can provide members additional incentives to gather information, thus mitigating the

moral hazard problem in information gathering. Consequently, the optimal committee size

and the principal’s expected payoff can increase in the heterogeneity of committee members’

policy preferences.

∗An early version of the paper was circulated under the title: “Optimal Committee Design With Heterogeneous

Preferences.” I would like to thank the editor (Jacques Cremer), two anonymous referees, Dirk Bergemann, Larry

Blume, Yeon-Koo Che, Estelle Cantillon, Hanming Fang, John Kennan, David Levine, Hao Li, Stephen Morris,

Giuseppe Moscarini, Joe Ostroy, David Pearce, James Peck, Nicola Persico, Uday Rajan, John Riley, Mike Ryall,

Sergei Severinov, Leeat Yariv, Lixin Ye, and seminar participants at a number of universities for helpful comments.

I thank the Department of Economics and the Cowles Foundation at Yale University for hospitality and support.

†Department of Economics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095-1477. Tel: 310-794-6495. Fax:

310-825-9528. e-mail: cai@econ.ucla.edu

1 Introduction

Consider a firm investing in a single project where the efficient project size is unknown and

dependent on many uncertain factors such as future input prices, future labor market conditions,

or technology trends. The more the actual project size deviates from the efficient level, the

more (and increasingly more) costly it will be to the firm.1 Lacking good information about the

efficient project size, the firm selects a certain number of agents from a large pool of candidates

who have expertise in assessing projects to gather information (e.g., independent board directors,

consultants). If selected, each agent exerts effort to collect information (e.g., doing research,

consulting specialists, reading files, etc.) and reports his finding to the firm. After receiving

reports from all selected agents, the firm chooses the project size. I call such a group of agents

designated to gather information an “information committee.”2

To a certain degree, members of the information committee share the firm’s objective to

have a project size as close to the efficient level as possible, either because they care about their

reputation and thus care about the value of the firm they serve or because they have stakes in the

firm. However, there are two problems that prevent members from having incentives perfectly

aligned with that of the firm. First, information gathering is both costly to members and not

easily observed by the firm. When such costs are sufficiently high, members will shirk in gathering

information and then make non-informative reports to the firm. Second, committee members

may have preferences over project size different from the firm’s. For example, a member may

prefer larger projects because he gets additional direct benefits (e.g., more consulting businesses)

or indirect benefits (e.g., more prestige), or may prefer smaller projects because he is concerned

about environmental impacts. Members with preference biases have incentives to distort their

information in order to manipulate the firm’s decision in their favor. How should the firm

1For example, suppose the firm’s cost function is C = K + aθ2− 2aθx + ax

2 = K + a(x − θ)2, where K is

the fixed cost, x is the actual project size the firm chooses, a is a positive cost parameter, and θ is the unknown

efficient project size.

2Other examples include expert panels, outside advisors (e.g., reviewers on academic recruiting and promotion),

sports judges, etc.

1

structure such an information committee and what determines the optimal committee size?

In this paper I develop a simple model of informational committees in which information

gathering is costly and committee members have heterogeneous preferences. I argue that when

information gathering is quite costly to committee members, under certain conditions, hetero-

geneous preferences provide members incentives to gather information to offset their shirking

tendency caused by the participation costs. The basic idea is as follows. I assume that com-

mittee members discover their exact preferences over the firm’s decision during the information

gathering process. This captures situations in which the decision problem is complex and agents’

potential preference biases are caused by non-obvious factors whose impacts on agents’ welfare

can only be learned through careful study of the underlying decision problem. In such cases, only

by participating and gathering information can committee members influence the firm’s decision

so that their own interests can be better protected. This incentive effect is stronger when mem-

bers’ preferences are more heterogeneous ex ante. I show that when the moral hazard problem

in information gathering is sufficiently severe, under certain conditions, the optimal committee

size and the firm’s expected payoff increase in the heterogeneity of committee policy preferences.

Specifically, in our model a principal, the firm in the example above, makes a policy decision

that is represented as a point on the real line. The outcome of a policy decision depends on the

state of the world, which also lies on the real line but is unknown when the decision is made. The

principal selects N agents to form a committee to gather information. Each member receives one

noisy signal about the state of the world from exerting non-verifiable efforts. Member recommen-

dations are then reported to the principal, who aggregates them and makes a policy decision.

Before joining the committee, agents are identical and unbiased, but each member learns his

own policy preference during the process of gathering information. In a specific setting in which

the information structure and policy preferences are both represented by normal random vari-

ables, I characterize an equilibrium that is Pareto optimal from the viewpoint of the committee

members, under a mean decision rule such that the policy implemented is simply the mean of

the member recommendations. In this equilibrium which is called the “simple reporting equilib-

rium,” members with no policy biases report truthfully, and those with policy biases exaggerate.

2

When the principal cannot commit to a policy rule, the simple reporting equilibrium remains

an equilibrium with an uninformative prior. In this equilibrium, members’ information about

the state of the world is transmitted and incorporated in policy decisions, though the quality

of the final decision is reduced by the noisy reports of the preference-biased agents. The two

assumptions, the mean decision rule in the commitment case and the uninformative prior in the

non-commitment case, are reasonable in our context, but certainly restrictive and ad hoc. The

problems of finding an optimal decision rule in the commitment case and finding an informative

equilibrium for any prior in the non-commitment case are still open.

Given the simple reporting equilibrium, the committee members’ incentives to gather infor-

mation and then the optimal committee size are studied. The optimal committee size is always

smaller than the first best level, because a member’s marginal information contribution is de-

creasing in policy bias. In equilibrium, decisions will be more volatile than the first best because

of noisy reports from biased members and because of the smaller number of signals generated

by a suboptimally small committee. I derive comparative statics of the optimal committee size

and the principal’s expected payoff with respect to the parameters of the model. Attendance

and information gathering costs, quality of prior knowledge about the state of the world, and

signal quality are found to affect the optimal committee size and the principal’s expected payoff

in economically intuitive ways. Most interestingly, the optimal committee size and the principal’s

expected payoff can increase in the heterogeneity of members’ policy preferences. This happens

when the moral hazard problem in information gathering severely limits the feasible committee

size. Because members with policy biases can manipulate the policy in their favor, heterogeneous

policy preferences provide additional incentives to gather information, thus mitigating the moral

hazard problem in information gathering. Of course, the heterogeneity of policy preferences also

has a direct negative effect because it increases the noisiness of committee members’ reports and

hence the principal’s decision. When the positive participation effect dominates the negative

noisiness effect, the heterogeneity of policy preferences can increase the optimal committee size

and the principal’s expected payoff.

In different settings with costly participation and heterogeneous agents, Campbell (1999)

3

and Osborne, Rosenthal and Turner (2000) show that agents with smaller participation costs

or more extreme preferences are more likely to participate in collective decisions. Consequently,

collective decisions will tend to be decided by such “active” and “extreme” agents instead of by the

median voter’s preference, and will vacillate between extreme policies. Thus, the combination of

costly participation and heterogeneous preferences prevents collective decisions from efficiently

aggregating society preferences. While the incentive effect of preference heterogeneity in my

model is similar in some sense to that in these papers, the mechanism and the implications

are quite different. In particular, since I focus on information aggregation instead of preference

aggregation, preference heterogeneity can be welfare improving.

The paper is organized as follows. The next section presents the model; Section 3 then

presents the first best solution. In Section 4 I characterize the simple reporting equilibrium

under the mean decision rule, and derive its efficiency properties. Section 5 studies the committee

members’ incentives to gather information and derives the optimal committee size. Based on the

comparative statics, I then present the implications of the model. Extensions of the model are

discussed in Section 6, related literature is discussed in Section 7, and concluding remarks are

made in Section 8.

2 The Model

A decision maker (the principal) must choose a policy x ∈ �, which yields an outcome y = x−θ,

where the state of the world θ ∈ � is unknown when the decision is taken. The principal and all

agents have a common prior about θ represented by the distribution G(·). The principal’s utility

function is V (y) = −y2 = −(x− θ)2, hence her ideal policy is x = θ.

The principal selects N agents from a large pool to form an information committee. All

agents are identical ex ante. There is a fixed attendance cost of c per person to be a committee

member, for which the principal must compensate each member. After the committee is formed,

individual members decide simultaneously and non-cooperatively whether to gather information

(This will be called the “information gathering stage”). A member i can get at most one signal

4

of θ, denoted θi = θ+ εi, where εi is independently drawn from an identical distribution function

F (·) with mean zero and variance σ2ε for all i. In the process of gathering information, a member

also discovers his own policy preference, which is represented by an ideal policy outcome ti. Ex

ante, ti are independent and drawn from the same mean-zero distribution T (·). All {εi} and

{ti} are independent from each other and across i. I assume that member i’s utility function is

ui(ti, θ) = −(y− ti)2 = −(x−θ− ti)

2. Hence, the most preferred policy for a member with policy

preference ti is x = θ + ti.

The assumption that committee members discover their preferences during the information

gathering process but not before applies to situations in which the policy decision is complex and

committee members’ preference biases are caused by non-superficial factors that are discovered

through careful study of the policy issues. For example, a board director of company X may find

that an investment project of company X could affect his own company Y in some subtle way in

the long run. For other cases when committee members’ preference biases are caused by tastes

over obvious characteristics (i.e., ideology), they may know the direction, but perhaps not the

strength, of their preference biases before gathering information. In such cases, one expects that

the principal will try to reduce obvious preference biases when selecting committee members.

Information gathering is costly (e.g., reading files, doing research): each member has to

pay an information cost of k to gather information. I assume that information gathering is

non-verifiable by the court, so the principal cannot induce information gathering by making

contingent promises to compensate committee members for information costs. For simplicity, I

also assume that information gathering is observable. At the end of the information gathering

stage, it becomes common knowledge who has gathered information and who has not. In Section

6.1, I show that the main results continue to hold when information gathering is not observable.

Both ti and θi are member i’s private information. Once committee members have gath-

ered information, they report their recommendations simultaneously and non-cooperatively (the

“information aggregation stage”). I suppose that information is soft so that each informed mem-

ber can report anything in the support of θi, i.e., the real line. Denote a report profile by

θN = {θi}Ni=1

. Note that it is without loss of generality to restrict a report to be a single number.

5

If asked to report ti in addition to θi, a committee member i can always claim ti = 0.

I will consider two kinds of decision-making environments. In the first case, the principal

commits to a simple decision rule based on the aggregate reports of the committee members. In

the second case, without commitment she acts on the reports of the committees.

Any equilibrium in which all committee members collect information must satisfy the follow-

ing two incentive compatibility conditions.3

(i) Incentive Compatible Signal Reporting: Suppose the decision rule is x(θN ). In

the information aggregation stage, given his private signal and the set of members who

collected information and their reporting strategies, each informed member chooses a report

to maximize his expected utility. Formally, for any member i ∈ N with a signal θi and a

policy preference ti, denote his reporting strategy by θi = si(θi, ti), and other members’

reporting strategies by θ−i = s−i(θ−i, t−i) = {sj(θj, tj)}j �=i, where θ−i = {θj}j �=i and

t−i = {tj}j �=i. Given other members’ reporting strategies, member i’s expected utility is

ui(θi, ti)(si, s−i) = −E{θ,θ−i,t−i}[(x(θ

N )− θ− ti)2|ti, θi] = −E{θ,θ

−i,t−i}[(x(θN )− θ− ti)

2|θi]

where the last equality is due to the independence of ti with t−i and θ−i. The IC condition

for the information aggregation stage requires that for every (θi, ti), given s∗−i(θ−i, t−i),

−E{θ,θ−i,t−i}[(x(s

∗i , s

∗−i)− θ − ti)

2|θi] ≥ −E{θ,θ−i,t−i}[(x(si, s

∗−i)− θ − ti)

2|θi] (1)

Let s∗ = (s∗i , s∗−i) be an equilibrium in the reporting game. Denote ui(θi, ti)(s

∗) as member

i’s expected equilibrium payoff when his signal is θi and his policy preference is ti.

(ii) Incentive Compatible Information Production: In the information production stage,

given that other members will gather information, each committee member must be willing

to gather information as well. Formally, suppose the following: (i) members N\i will

3If some members do not collect information in equilibrium, they will not be chosen in the optimal committee

design.

6

collect information; (ii) if member i also collects information, the subsequent reporting

game with all informed N members is played according to an equilibrium s∗ = (s∗i , s∗−i);

(iii) if member i does not collect information, the subsequent reporting game with the N\i

informed members is played according to an equilibrium s∗−i under a decision rule x(θ−i).

The IC condition for the information production stage requires that for each member i,

E{θi,ti}[ui(θi, ti)(s∗)]− k ≥ E{ti}[ui(ti, x(s

∗−i))] (2)

where ui(ti, s∗−i) is i’s expected utility with policy preference ti in the equilibrium s∗−i.

In the commitment case, I assume that if member i does not collect information, the

principal uses the mean decision rule on the N − 1 reports of the members who have

collected information.4 In the non-commitment case, I assume that the principal and the

N − 1 informed members ignore member i and play along the subgame perfect equilibrium

path with a committee of N − 1 members. As will be clear, this also results in the mean

decision rule on the N − 1 signals under our assumption of uninformative prior.5

The principal’s payoff is the expected information efficiency of committee decisions EV , minus

the total attendance costs Nc, subject to the incentive constraints of the members, i.e.,

S = EV (x(θN )− θ)−Nc = −E[x(θN )− θ)]2 −Nc (3)

This can be understood as follows. The decision-maker’s expected utility is EV minus imple-

mentation costs (i.e., total monetary payments). She maximizes the expected utility subjects

to committee members’ participation and incentives constraints. Since information gathering is

not verifiable, the decision-maker cannot motivate committee members to gather information by

4In Section 6.1, I argue that the results of the paper are still valid if the principal uses the mean decision rule

on all N reports, where the uninformed member i’s report is taken to be zero.

5In the non-commitment case, a harsher punishment for shirking is that the principal and the N − 1 informed

members play the “babbling” equilibrium if member i does not collect information. The equilibrium I use in the

punishment phase strengthens the IC condition for information production.

7

paying them k. Instead, their incentives to gather information must come from the expected

benefits they get by becoming informed and hence being able to influence the policy decisions.

When the IC constraints are binding, i.e., the expected benefits from gathering information equal

to k, then the principal just needs to pay committee members the attendance costs to satisfy

their participation constraints. Thus, the total monetary payments equal Nc; and I can reduce

the decision-maker’s constrained optimization problem by plugging in the binding participation

constraints, giving us the objective function as in Equation (3), subject to the incentive con-

straints only. When the IC constraints are not binding, i.e., the expected benefits from gathering

information exceed k, then committee members receive rents from participating in the decision

process. Presumably in this case the principal could save money by reducing the attendance fees

or even asking the agents to pay a lump sum to join the committee. However, this is not likely

to be feasible, and I rule out this possibility. For instance, if the agents are not completely sure

that the project is legitimate or not, then they would refuse to pay the attendance costs (or any

lump sum payment to the principal) or refuse to join the committee.6

In the commitment case when the principal commits to a fixed decision rule x(θN ), the two IC

conditions discussed above define an equilibrium under x(θN ). Anticipating such an equilibrium,

the principal will choose an optimal N to maximize Equation (3). In the non-commitment

case when the principal acts on the committee’s information, her optimal policy will depend on

her updated belief about θ following the information production and reporting stages. Given a

history H of the game, let θ be the principal’s Bayesian updated belief of θ, and µ and σ2 be θ’s

expectation and variance. Then, the principal chooses a policy x that satisfies the following,

maxx

Eθ[V |H]−Nc = −Eθ[(x− θ)2|H]−Nc = −(x− µ)2 − σ2 −Nc

Clearly, given the principal’s posterior belief θ, she will simply choose x∗ = µ. This and the IC

conditions constitute a Perfect Bayesian Nash Equilibrium (PBNE) of the game with a committee

size N . Anticipating this equilibrium at the beginning of the game, the principal will choose an

6In such a case, the principal would not want to “fake” a project to induce agents to join the committee and

to gather information, since a “fake” project by definition does not bring any benefit to the principal.

8

optimal N to maximize Equation (3).

3 The First Best Solution

As a benchmark, I derive the first best solution when there are no incentive problems. The

principal only needs to determine how many signals to get and what decision rule to use.

Given a signal profile θN , let θ be the posterior belief of θ, and µ and σ2N

be θ’s expectation

and variance. The subscript N of the posterior variance indicates that the posterior variance

depends on the number of signals N . With the signal profile θN , E[V (x− θ)|θN ] equals −(x−

µ)2−E[(µ−θ)2|θN ] = −(x−µ)2−σ2N. Clearly the principal’s optimal policy is x∗ = µ. Under this

policy, her expected utility is simply the negative of the posterior variance, which is independent

of the specific signal profile. Hence, for any N , the principal’s expected payoff is

S = EV (x∗ − θ)−Nc = −σ2N −Nc (4)

Since σ2N

is decreasing in N , there exists an N∗ maximizing Equation (4) that satisfies

σ2N−1 − σ2N ≥ c ≥ σ2N − σ2N+1 (5)

Thus Proposition 1 follows.

Proposition 1 In the first best solution, the committee size is N∗, the integer that maximizes

Equation (4). The optimal policy is x∗ = µ, the posterior mean of θ given a signal profile θN∗

.

Proposition 1 is easy to understand. Given any signal profile, the principal maximizes her

expected payoff by choosing the policy equal to her best estimate of θ, the posterior mean.

Consequently, her expected utility simply becomes the negative of the posterior variance. Then

the optimal number of signals trades off the value of information (in terms of reduction in

posterior variance) with attendance costs.

When information gathering efforts cannot be monitored and information is private and soft,

the first best solution is generally not attainable. The next proposition shows two special cases

when the first best solution is attainable.

9

Proposition 2 When c ≥ k, the first best solution is attainable if either:

(i) Information is hard, θi = θi: informed members must report their signals truthfully; or

(ii) committee members have no policy biases: for every i, ti = 0 with probability one.

Proof: See the Appendix.

Proposition 2 (i) shows that even though individual members incur information costs and

efforts cannot be monitored, the first best solution can be achieved if information is hard and

information cost is not greater than attendance cost. The reason is as follows. Each committee

member’s expected utility as a function of the number of signals can be expressed as some constant

plus the information value of the signals, provided the policy choice is always the posterior mean

for any signal profile. Consequently, the marginal value of one additional signal to a committee

member is the same as that to the principal. So when the committee size is set at the first best

level N∗ and the information cost is no greater than the attendance cost, each member has the

right incentives to collect information if other members collect information. Therefore, in my

setup, there is no free rider problem in exerting efforts among committee members. Of course, if

c < k, then it is not an equilibrium such that all members in an N∗ member committee gather

information, and hence the first best is not achievable.

Part (ii) of Proposition 2 indicates that if committee members have homogeneous prefer-

ences, incentives to distort information do not arise and hence the first best is implementable

(despite moral hazard problems in information gathering). In contrast, the recent literature on

the Condorcet Jury Theorem using unanimous voting rules, e.g., Austen-Smith and Banks (1996),

Feddersen and Pesendorfer (1997, 1998), and Persico (2004), shows that even with homogeneous

policy preferences information may not be fully revealed under unanimity voting.7 The reason

is that unanimity is not the optimal decision rule with homogeneous jurors.8 In contrast, my

7The “swing voter’s curse” phenomenon, termed by Feddersen and Pesendorfer, arises under unanimous voting

because one member’s vote is decisive precisely when all other members have different signals. However, Martinelli

(2002) shows that unanimous voting can be optimal when members do not have homogeneous preferences.

8However, the first best can be implemented with unanimous voting preceded by a communication stage (Cough-

10

model does not employ unanimous voting rules which allows homogeneous preferences to lead to

truthful reports as an optimal solution.

For tractability, in the rest of the paper I focus on the normal distribution case where the

prior belief about θ, signal noises εi, and policy biases ti are all normally distributed. Specifically,

I suppose that the prior distribution of the true state G(θ) is normal with mean µ, variance σ2θ,

and precision τθ = 1/σ2θ. Without loss of generality, µ can be normalized to zero. Suppose

θi = θ + εi, where εi is iid normal with mean zero, variance σ2ε , and precision τε = 1/σ2ε . By the

standard Bayesian updating rule, the posterior of θ given a signal profile θN , θ, has a normal

distribution with mean µ = Nτεζ/(τθ +Nτε) and variance (τθ +Nτε)−1, where ζ =

∑i∈N θi/N

is the sample mean. Then Equation (4) becomes

EV (x∗ − θ)−Nc = −(τθ +Nτε)−1−Nc

This function is concave in N , and the first order condition gives the optimal number of signals

N∗ = (τεc)−0.5

− τθ/τε (N∗ = 0 when c ≥ τε/τ

2

θ). I will ignore the integer problem.

Proposition 3 In the normal distribution case, the first best committee size is N∗ = (τεc)−0.5−

τθ/τε, and the first best decision rule is x∗ = N∗τεζ/(τθ +N∗τε).

Proposition 3 is very intuitive. Expectedly, the optimal committee size N∗ decreases in

attendance cost. It also decreases in the precision of the prior τθ, because the better the prior

information is about the state of the world, the less need there is for new signals. Since ∂N∗/∂τε =

τ−2ε [τθ − 0.5(τε/c)0.5], N∗ is not globally monotonic in the signal precision τε. When signal

precision is relatively small (τε ≤ 4cτ2θ), N∗ increases as signals are more informative. But as

signal precision increases beyond a certain level (τε > 4cτ2θ), the marginal value of signal begins

to decline and hence N∗ becomes decreasing in τε. Note that when signal precision is sufficiently

low (τε < cτ2θ), the marginal value of signal is less than the marginal cost, so no signal should be

collected. This is an example of the nonconcavity first identified by Radner and Stiglitz (1984).

The corollary below follows immediately from Propositions 2 and 3 and will be useful later.

lan, 2000). This is not true when players have different preferences (Doraszelski, Gerardi and Squintani, 2003).

11

Corollary 1 When either information is hard or committee members have no policy biases, the

optimal committee size is N∗ = (τεc)−0.5

− τθ/τε if c ≥ k, and is (τεk)−0.5 − τθ/τε if c < k.

It is assumed that committee members’ policy biases {ti} are independently and identically

distributed with mean zero and variance σ2t. In this model, σ2

tcan be interpreted as indicating the

(ex ante) heterogeneity of members’ policy preferences, because a large σ2t means that committee

members’ ideal policies are more likely to be far from that of the principal’s and to be far from

each other’s. Since σ2tis known but the realizations of {ti} are not known before the formation

of the committee, this ex ante notion of heterogeneity is what matters for committee size.

4 Simple Reporting Equilibrium

Assuming that all N committee members have gathered information in the information produc-

tion stage, in this section I focus on the information aggregation and decision-making stages. I

first consider the commitment case and then the non-commitment case.

4.1 Commitment Case: Mean Decision Rule

In the first case to be considered, I assume that the principal commits to a fixed decision rule

equal to the mean of the reports by the N committee members. In the context of continuous

signals and decisions, such a “mean decision rule” is simple and natural. In an environment

similar to this with uncertainty about the state of the world and normally distributed signals,

Martinelli (2001) presents an equilibrium in which voters’ expectations about the state of the

world–which determine how they vote–depend on the average of two competing parties’ pol-

icy platforms.9 In many situations, the decision maker implicitly commits through reputation

mechanisms (e.g. reputation of rubber-stamping) to some decision rule that depends only on the

aggregate information of the committee, in which case the mean decision rule can be thought as

a first approximation.

9The mean decision rule is also analyzed in other contexts, e.g., Gibbons (1988) and De Sinopoli and Iannantuoni

(2000).

12

Define M = N + τθ/τε as the “adjusted” sample size. Intuitively, M is the total amount of

information contained in the sample and in the prior knowledge, where the prior knowledge counts

as τθ/τε signals. Then the posterior mean of θ given a true signal profile θN can be conveniently

written as∑

i θi/M , or the “adjusted” sample mean. The proposition below presents a reporting

equilibrium under the mean decision rule.

Proposition 4 Under the mean decision rule x =∑

i∈N θi/N , it is an equilibrium for member

i with a policy preference ti to report s∗i(θi, ti) = Nθi/M +Nti.

Proof: See the Appendix.

In the equilibrium identified in Proposition 4, those members without policy biases report

truthfully after making the adjustment from N to M , and those with policy biases exaggerate

by N times their policy bias. Under the mean decision rule and given a true signal profile θN ,

this equilibrium yields a final decision of x =∑

i∈N θi/N =∑

i∈N θi/M +∑

i∈N ti. The intuition

of Proposition 4 is shown for the case of member i. If all other committee members follow their

equilibrium strategies, then in expectation the sum of reports by those N\i members will be

unbiased. So when i does not have any policy bias (ti = 0), reporting truthfully will keep the

sum of all reports unbiased and hence will result in an unbiased policy, which maximizes agent i’s

utility. When agent i has a policy preference ti, he would like to distort the policy choice by ti.

To do so, he needs to exaggerate by Nti under the mean decision rule, so reporting Nθi/M +Nti

is optimal.

For ease of exposition, denote this equilibrium as the “simple reporting equilibrium”. Note

that in this equilibrium, committee members’ information about the state of the world θ is trans-

mitted and incorporated in the principal’s decision, while their policy biases enter the principal’s

decision as additional noise. In the equilibrium decision x =∑

i θi/M +∑

i ti, the first term is

precisely the posterior mean of θ given the true signal profile θN and the second term is the sum

of committee members’ policy biases. It follows that

E[V ] = −E[(x− θ)2] = −Nσ2t− σ2

ε/M (6)

13

Thus, the expected loss of the equilibrium decision consists of two terms: Nσ2t reflects the

uncertainty about committee members’ preferences, and σ2ε/M reflects the residual uncertainty

about the true state of the world.

For member i with signal θi and policy preference ti, expected utility is

ui(θi, ti) = −(N − 1)σ2t− σ2

ε/M (7)

Note that this is independent of member i’s signal and preference bias, so it is also his uncondi-

tional expected utility before he knows his policy preference and receives his signal. The reason

is the following. Because a committee member with a policy bias ti exaggerates by Nti, his

expectation is that the policy will be biased from the true posterior mean by ti, which is ideal

from his point of view. Hence, given any θi and ti, member i’s expected utility depends on only

two terms, both of which are independent of θi and ti: (N − 1)σ2treflects his uncertainty about

other N − 1 members’ preferences and σ2ε /M is the posterior variance based on true signals.

It is easy to see that the simple reporting equilibrium is not the only reporting equilibrium.10

However, Proposition 5 below shows that the simple reporting equilibrium (weakly) Pareto dom-

inates all other possible equilibria.

Proposition 5 Under the mean decision rule, no other reporting equilibrium has greater expected

payoffs for the principal or committee members than those in the simple reporting equilibrium.

Proof: See the Appendix.

The idea behind Proposition 5 is the following. In the simple reporting equilibrium every

member reveals his true signal but not his policy preference. In any other reporting equilibrium,

if one member believes that other members will deviate from truthfully revealing their signals,

then he will try to “neutralize” other members’ expected deviations conditional on his own

information. The net effect of such deviations is to add mean zero noise in their reports. So the

final decision cannot be more informative than in the simple reporting equilibrium.

10For instance, for any constants {bi}N such that

∑i∈N

bi = 0, the strategy profile si(θi, ti) = Nθi/M+Nti+bi

is an equilibrium that yields the identical expected payoffs for all the players as in the simple reporting equilibrium.

14

4.2 Non-Commitment Case

Now consider the case in which the principal acts on the committee’s information without com-

mitment. I assume τθ = 0, that is, the prior is uninformative. It reflects the situation where

there is no prior knowledge about the state of the world except that a prior θ does not take some

values more likely than others, hence all information comes from the sample.

Proposition 6 With an uninformative prior, the following strategy profile is an equilibrium: (i)

member i with a policy preference ti reports s∗i(θi, ti) = θi +Nti; (2) upon receiving reports θN ,

the principal chooses a policy of x =∑

i∈N θi/N .

Proof: See the Appendix.

Proposition 6 says that with an uninformative prior, committee members’ reporting strategies

in the simple reporting equilibrium studied above is a part of an equilibrium, wherein the principal

simply uses the mean report as her policy. Given the principal’s mean decision rule, the preceding

subsection shows that the committee members’ reporting strategies are their best responses. The

additional equilibrium requirement in the non-commitment case is that the mean decision rule

must be the principal’s best response to the committee members’ reporting strategies. From

the principal’s point of view, the committee members simply report their true signals with some

additional noises (Nti). Hence a report profile can be thought of a different “true” signal profile

with smaller precision. With a non-informative prior, the posterior mean of θ conditional on a

signal profile is simply the sample mean. Therefore, the principal’s best response is indeed to

choose a policy equal to the mean report.

The assumption of an uninformative prior is strong, but is quite commonly used in Bayesian

updating theory. In economics, the assumption is made in many papers on global games (see

the survey by Morris and Shin, 2003), and papers on strategic information transmission (e.g.,

Battaglini, 2003). Uninformative priors can be a reasonable approximation for complex decision-

making situations in which there is no good prior knowledge about the state of the world.11

11When τθ > 0, the mean decision rule will not be the principal’s best response to the committee members’

reporting strategies. It can be checked that when the committee members report si(θi, ti) = Nθi

M+ Nti where

15

5 Optimal Committee Size and Implications

Having characterized the simple reporting equilibrium and established its efficiency property, I

now analyze the information production stage. By Equation (7), if all N\i committee members

gather information, member i’s expected utility from gathering information is given by

E{θi,ti}ui(θi, ti)− k = −(N − 1)σ2t − σ2ε /M − k

If member i does not gather information, the decision rule will be x∗−i =∑

j �=i θi/(N − 1) and

the reporting strategies of other committee members will be sj(θj , tj) = (N − 1)θj/(M − 1) +

(N − 1)tj for all j �= i. In this case member i’s expected payoff is

−E{θ−i,t−i,ti}{Eθ[(x

∗−i − θ − ti)

2|θ−i, t−i, ti]} = −σ2t − σ2ε /(M − 1)− (N − 1)σ2t (8)

Therefore, if all other members gather information, member i will also gather information if

σ2ε /(M − 1)− σ2ε /M + σ2t ≥ k (9)

The right hand side of Equation (9) is simply the cost of becoming informed. The left hand

side is the benefit to member i from becoming informed given that all other members gather

information. If he gathers information, his signal makes the policy more informative, yielding a

benefit of σ2ε/(M−1)−σ2ε/M to himself. Moreover, once he becomes informed, he can manipulate

the policy in his own favor. In a sense, by becoming informed, every member is insured against

the uncertainty of his own policy preference, yielding an additional benefit of σ2t .

Equation (9) has some interesting implications for participation in committees. When σ2t is

large relative to k, then every member has an incentive to gather information, despite the fact

that information gathering is personally costly. This will tend to be true when the policy has

M = N + τθ

τε, the principal’s best response is to choose a policy of

x =N + τθ/τε

N + τθ/τε +M2τθ/τt

∑i∈N

θi

N

When τθ/τt is very small, this will be sufficiently close to the mean decision rule, so the strategy profile of the

simple reporting equilibrium in Proposition 4 is an “epsilon equilibrium” in the sense of Radner (1981).

16

a larger impact on members and ex ante preferences are not easily discernible. In such cases,

if committee memberships are not regulated, there will be “too many” volunteers even with

uncompensated information costs, leading to over-sized committees.

Now the optimal committee size can be found. From Equation (9), the optimal committee

size cannot exceed σε/(k − σ2t )0.5− τθ/τε when k > σ2t . From Equation (6), the principal’s net

expected utility with a committee of N members is

S = EV −Nc = −σ2ε /M −Nσ2t −Nc (10)

The solution that maximizes this function is σε/(c+ σ2t )0.5− τθ/τε. Thus, Proposition 7 follows,

Proposition 7 When c ≥ k − 2σ2t , the optimal committee size N is given by [τε(c+ σ2t )]−0.5 −

τθ/τε. When c < k − 2σ2t , the optimal committee size N is given by [τε(k − σ2t )]

−0.5 − τθ/τε.

Below I present the implications of the model in several corollaries.

Corollary 2 The optimal committee size N is smaller than the first best solution N∗.

Corollary 2 follows immediately from Propositions 3 and 7. The reason is that committee

members do not always report truthfully and hence their reports are not as informative as what

they would be if there were no incentives to distort information. As is apparent from comparing

N∗ and N , the added noise in the reports has the same effect as having a fixed cost of σ2t in

addition to attendance cost c (when c ≥ k − 2σ2t ).

Corollary 3 Policy decisions are more volatile than in the first best solution.

There are two reasons for Corollary 3. First, for a fixed committee size, the variance of

equilibrium policy is greater than that of the first best policy since committee members with

policy biases distort their information. Comparing the equilibrium policy x =∑

i θi/M +∑

i ti

with the first best policy x∗ =∑

i θi/M , we see that while both are unbiased estimates of θ, their

posterior variances are different: Var(x) = σ2ε /M +Nσ2t and Var(x∗) = σ2ε/M . Secondly, since

the optimal committee size is smaller than in the first best solution, equilibrium policy is less

17

informed than that in the first best solution. With the optimal committee size, the posterior

variance of the first best equilibrium policy is σ2ε /M + Nσ2t where M = N + τθ/τε, while the

posterior variance of the equilibrium policy is σ2ε /M∗ where M∗ = N∗ + τθ/τε > M .

Corollary 4 The optimal committee size and the principal’s expected payoff decrease (at least

weakly) in attendance and information costs (c and k). High precision of the prior τθ decreases

the optimal committee size but increases the principal’s expected payoff.

Corollary 4 is apparent by inspection. The principal’s expected payoff, S = −σ2ε/M−N(σ2t +

c), increases in τθ because M is independent of τθ and N is decreasing in τθ.

Corollary 5 The optimal committee size is increasing in the precision of signals for low preci-

sion, but decreases when precision is above a certain level. The principal’s expected payoff always

increases in the precision of signals.

Proof: Let γ = max{c+σ2t , k−σ2t }. Since ∂N/∂τε = τ−2ε [τθ−0.5(τε/γ)0.5], we can see that when

τε ≤ 4γτ2θ, N is increasing in τε; when τε > 4γτ2

θ, N becomes decreasing in τε. Since the principal’s

expected payoff can be written as S = −(γ/τε)0.5−Nγ, we have γ−1∂S/∂τε = τ−2ε [(τε/γ)0.5−τθ].

It is non-negative when τε > γτ2θ, which must hold for N ≥ 0. Q.E.D.

As in the first best solution, the optimal committee size is not globally monotonic in the

signal precision. On the other hand, the principal always gains from more precise signals. For

a fixed committee size, more precise signals improve the quality of policy decisions. This effect

dominates the committee size effect when signal precision decreases the optimal committee size.

Recalling that σ2t represents the heterogeneity of committee preferences, the following corol-

lary relates optimal committee size to heterogeneity.

Corollary 6 When c+ 2σ2t≥ k (c+ 2σ2

t< k), the optimal committee size decreases (increases)

in the heterogeneity of committee members’ policy preferences.

18

When c+2σ2t≥ k, added noise in committee members’ reports decreases the value of having

more signals, so an increase of σ2tdecreases the optimal committee size. When k > c + 2σ2

t,

motivating members to gather information is a dominant concern, so an increase of σ2tincreases

the optimal committee size. Because in the simple reporting equilibrium committee members with

policy biases can effectively manipulate the policy, becoming informed gives them full insurance

against uncertainty about their own policy preferences, which provides additional incentives to

gather information. When information cost k is significantly large so that the committee size is

constrained by committee members’ tendency to shirk in gathering information, greater or more

uncertain stakes in the policy decision can help mitigate the shirking problem in information

production and make a larger committee more informative. Compared to the case in which

either information is hard or committee members have no policy biases (Corollary 1), the optimal

committee size is thereby greater when information is soft and committee members have strong

policy biases.

Corollary 7 When c + 2σ2t≥ k (c + 2σ2

t< k), the principal’s expected payoff is decreasing

(increasing) in the heterogeneity of committee members’ policy preferences.

Proof: Since S = −σ2ε/M − Nγ, where γ = max{c + σ2

t, k − σ2

t}, by the Envelope Theorem,

d(S)/dγ = −N < 0. So the principal’s expected payoff is always decreasing in γ. The corollary

then follows by the definition of γ. Q.E.D.

Corollary 7 shows that committee members’ policy biases and distortion of information may

improve the principal’s expected payoff. This can happen when the moral hazard problems in

information production are so severe that the IC condition for information production is binding.

It will tend to occur when information gathering is hard to monitor and is relatively costly for

committee members.

19

6 Extensions

6.1 Unobservable Information Gathering

In this subsection I show that the main results still hold when information gathering is not ob-

servable. Specifically, I argue that Proposition 4 continues to hold with unobservable information

gathering. To see this, suppose all committee members except member 1 gather information and

follow the reporting strategy s∗i(θi, ti) = Nθi/M+Nti for i �= 1. From Proposition 4, we know that

if member 1 gathers information, then his best reporting strategy is s∗1(θ1, t1) = Nθ1/M +Nt1.

If he does not gather information and sends a message s1, it can be verified that his expected

payoff is

u1(s1, s∗

−1) = −

s21N2

−(N − 1)σ2

θ

M2(M − 1)−

σ2ε

M − 1−Nσ2

t (11)

Therefore, member 1 should send a report of s1 = 0 if he does not gather information. The

intuition is simple. Since s∗i(θi, ti) = Nθi/M+Nti for i �= 1, the average report excluding member

1’s report will still be unbiased. Without knowing his own policy bias, member 1’s optimal report

is zero so that the overall average report remains unbiased.

The remaining question is whether member 1 would want to choose not to gather information

and report zero. But clearly this cannot be the case. Compared with member 1’s deviation

payoff in Equation (8) when information gathering is observable, member 1 gets an even smaller

deviation payoff when information is unobservable by the amount of(N−1)σ2

θ

M2(M−1) in Equation (11).

This is due to the fact that the mean decision does not give an unbiased estimator of θ with only

N − 1 real signals, which reduces member 1’s expected payoff. Since member 1 has the incentive

to gather information when information gathering is observable, i.e., Equation (9) holds, then he

will also have the incentive to do so when information gathering is unobservable. Therefore, the

main results extend to the case with unobservable information gathering.

6.2 Ex Ante Heterogeneous Committee Members

I assume that agents are ex ante identical, so the only issue for committee design is the committee

size. From the analysis of the model, it is not difficult to obtain some implications for the optimal

20

committee composition when agents are not ex ante identical. Due to space limitations, only

a brief informal discussion is provided here. Consider first the case in which agents differ in

terms of information gathering abilities so that agent i can get a signal θi = θ + εi, where εi

is an independent normal noise with precision τ iε . For simplicity, suppose committee members

i ∈ N1 are more adept at gathering information than members j ∈ N2: τiε = τ1ε > τ jε = τ2ε, where

N1∪N2 = N . In this case, the policy decision rule should be modified to x =∑

i∈N1θiτ1ε/(N1τ1ε+

N2τ2ε)+∑

j∈N2θjτ2ε/(N1τ1ε+N2τ2ε). Hence the information of those members i ∈ N1 with better

quality signals will be given greater weights in the policy decisions. All the current results should

follow without substantive modifications. In forming the committee, the principal prefers the

more productive agents and will select the less productive agents only after the available pool of

the more productive agents is exhausted.

Now consider the case in which agents differ in their ex ante policy preferences. If some

members are likely to have larger biases than others (perhaps due to their having large stakes in

the policy decisions), then they will add more noise in their reports. All the analysis and results

of the current model should carry through with minor modifications. However, by Corollary

7, the optimal committee composition is not trivial. When c ≥ k, the principal would like to

select agents with the smallest biases. However, when moral hazard problems in information

production are severe (c < k), then the principal needs to select agents with biases large enough

such that 2σ2t ≥ k − c even though there are other people with smaller biases.

Osborne et al. (2000) offer a different but related explanation of why committees may be com-

prised of members with greater stakes. In the Osborne model, members with greater stakes can

pull committee decisions toward their preferred positions by participating more than members

with smaller stakes. In the present model, members with greater stakes are more motivated to

gather information, because they can be better insured against policy uncertainty by manipulat-

ing their reports. Osborne et al. focuses on preference aggregation, which implies that committees

occupied by members with greater stakes are not optimal (low participation by moderates and

high randomness of decisions). In contrast, a focus on information aggregation allows solutions

where members with greater stakes serve on information committees in order to overcome the

21

moral hazard problem of information gathering.

7 Related Literature

This paper is related to several established lines of research in the existing literature. One is the

strategic information transmission (“cheap talk”) literature originated from the seminal work of

Crawford and Sobel (1982), see, for example, Gilligan and Krehbiel (1987, 1989), Austen-Smith

(1993), Wolinsky (1999), Krishna and Morgan (2001a, b), Morris (2001), Battaglini (2002),

and Morgan and Stocken (2003). In particular, De Garidel-Thoron and Ottaviani (2000) show

that when the expert’s bias is unknown there exists an equilibrium whose outcome achieves the

optimum (i.e., constrained delegation outcome).12 Strategic information transmission is also

a core element of my model, but my setting (heterogeneous preferences and information with

normal prior and normal noise) and focus (optimal committee design) are different from the

above works.

This paper is also related to the literature on jury decisions. Strategic voting in jury-type

committee decisions has attracted much attention recently (See, Austen-Smith and Banks (1996),

Feddersen and Pesendorfer (1997, 1998), Chwe (1999), Li, Rosen and Suen (2001), Persico (2004),

Gerardi and Yariv (2002), and others). The main differences between the current paper and the

jury decisions literature is that I consider situations with a continuous policy space and that I

am not concerned with voting mechanisms.

This paper is most closely related to Mukhopadhaya (2003) and Persico (2004), both of which

study optimal committee size focusing on information production and aggregation in a jury-like

environment. Assuming majority voting, Mukhopadhaya (2003) shows that a smaller jury may

make more informed decisions than a larger one because the free rider problem in information

production can be less severe in a smaller jury. I assume away the free rider problem in my

model. The optimal committee size is smaller than the first best in my model because of the

12Like Holmstrom (1984) and Dessein (2002), the main focus of de Garidel-Thoron and Ottaviani (2000) is to

study delegation as an alternative to strategic information transmission.

22

incentive problems in reporting information truthfully and the moral hazard problem in gathering

information. Persico (2004) derives the optimal voting rule in a jury of any size under which

each juror votes sincerely, then characterizes the optimal jury size that trades off the marginal

value of an additional signal under the optimal voting rule with its production cost. My model

differs from Persico (2004) in several aspects. First, instead of jury decisions, I consider policy

decisions that are on the real line and committee members who gather and report information to

a principal to make informed decisions. Second, unlike Persico (2004), I suppose that committee

members have heterogeneous preferences. In equilibrium, committee members add noise to their

reports in order to manipulate the principal’s decision. Third, I do not consider the effects of

various voting mechanisms.

8 Conclusion

In this paper I develop a simple model of committee size based on costly participation and

heterogeneous preferences. Under certain conditions, I show that when effort costs are sufficiently

high, preference heterogeneity can provide members additional incentives to gather information,

thus mitigating the moral hazard problem in information gathering. Consequently, the optimal

committee size and the principal’s expected payoff can increase in the heterogeneity of committee

members’ policy preferences.

This paper can be extended in many directions. The analysis can be extended to distributions

that are not normal, especially those conjugate distribution families whose posterior distributions

can be easily computed. Like most existing literature, I have assumed that all players have

quadratic utility functions. It is desirable to investigate how robust the results of my model are

to more general specification of utility functions. More importantly, it would be interesting to

solve for the optimal mechanism in the commitment case and the most informative equilibrium

in the non-commitment case with any given priors.

23

9 Appendix

Proof of Proposition 2: The first best solution requires a committee size of N∗ and the

optimal decision rule x∗ = µ: the posterior mean. For case (i), the question is whether committee

members have the right incentives to gather information. For any member i ∈ N∗, given all other

members will collect information, his expected utility if he also collects information will be

Eui − k = −E{θN∗

,ti}{Eθ[(µ− θ − ti)

2|θN∗

, ti]} − k = −σ2t − σ2N∗ − k

If i does not collect information, the remaining N∗ − 1 members report their signals. Since

the decision rule will be the posterior mean of the N∗−1 signals, in this case member i’s expected

utility is −σ2N∗−1 − σ2t , where σ2N∗−1 is the posterior variance with N∗ − 1 signals. Hence, given

that all other members gather information, member i also collects information if

σ2N∗−1 − σ2N∗ ≥ k

By the definition of N∗ (Equation 5), the above inequality holds when c ≥ k. Therefore, it is an

equilibrium for every member in N∗ to exert effort in collecting information.

For case (ii), it is easy to see that reporting truthfully is an equilibrium given the optimal

decision rule. Since for all i, ti = 0, committee member i’s expected utility coincides with that

of the principal. So, if all other committee members report truthfully, committee member i will

also report truthfully under the optimal decision rule. The argument of case (i) implies that the

incentive compatibility condition for information production is satisfied. Q.E.D.

Proof of Proposition 4: It is necessary to show that under the mean decision rule, committee

members’ reporting strategies s∗i (θi, ti) = Nθi/M+Nti are best responses to each other. Consider

informed member i’s best response to any reporting strategies by other members s−i(θ−i, t−i). Let

x be the true “adjusted” sample mean: x = (∑

i∈N θi)/M , i.e., the posterior mean condition on

the true signal profile θN : x = Eθ[θ|θN ]. When ti = 0, i’s reporting strategy s∗i (θi, 0) maximizes

ui(θi, 0)(si, s−i) = −E{θ,θ−i,t−i}[(

∑j �=i θj

N+

si(θi, 0)

N− θ)2|θi]

24

= −E{θ−i,t−i}{Eθ[(

∑j �=i θj

N+

si(θi, 0)

N− x+ x− θ)2|θi, θ−i, t−i]}

= −E{θ−i,t−i}[(

∑j �=i θj

N+

si(θi, 0)

N− x)2|θi]−E{θ

−i}{Eθ[(x− θ)2|θi, θ−i]}

= −E{θ−i,t−i}[(

∑j �=i θj

N+

si(θi, 0)

N− x)2|θi]−

σ2εM

(12)

where the last equality follows from the fact that the posterior variance is (τθ + Nτε)−1 =

(τεM)−1 = σ2ε/M . Similarly, for ti �= 0, s∗i (θi, ti) maximizes

ui(θi, ti)(si, s−i) = −E{θ−i,t−i}[(

∑j �=i θj

N+

si(θi, ti)

N− x− ti)

2|θi, ti]−σ2εM

Then it must be the case that s∗i (θi, ti) = s∗i (θi, 0)+Nti is i’s best reporting strategy for ti �= 0. If

not, then si(θi, 0) = s∗i (θi, ti)−Nti would yield a higher value for ui(θi, 0)(si, s∗−i) than s∗i (θi, 0).

Since in equilibrium s∗i (θi, ti) = s∗i (θi, 0) +Nti for every i and ti, Equation (12) becomes

ui(θi, 0)(si, s−i) = −E{θ−i,t−i}[(

∑j �=i(s

∗j(θj , 0) +Ntj)

N+

si(θi, 0)

N− x)2|θi]−

σ2εM

= −E{θ−i,t−i}[(

∑

j �=i

tj +∑

j �=i

sj(θj , 0)−Nθj/M

N+

si(θi, 0)−Nθi/M

N)2|θi]−

σ2εM

= −Et−i(∑

j �=i

tj)2 −Eθ

−i[(∑

j �=i

sj(θj, 0)−Nθj/M

N+

si(θi, 0)−Nθi/M

N)2|θi]−

σ2εM

The last equality follows from the fact that (i) t−i are independent with θi and θ−i, and (ii)

Et−i

∑j �=i tj = 0.

Since ti’s are independent, Et−i(∑

j �=i tj)2 =∑

j �=iE(t2j ) = (N−1)σ2t . Let ∆(θi) = Eθ−i[∑

j �=i(sj(θj, 0)−

Nθj/M)|θi] =∑

j �=iEθj [(sj(θj , 0)−Nθj/M)|θi]. Then we have

ui(θi, 0)(si, s−i) = −1

N2Eθ

−i[(∑

j �=i

(sj(θj, 0)−Nθj/M)−∆(θi))2|θi]

−1

N2[si(θi, 0)−Nθi/M +∆(θi)]

2 − (N − 1)σ2t − σ2ε /M (13)

Since only the second term on the right hand side of Equation (13) is affected by si(θi, 0),

member i’s best response function when ti = 0 is

25

si(θi, 0) = Nθi/M −∆(θi) = Nθi/M −∑

j �=i

Eθj [(sj(θj , 0)−Nθj/M )|θi] (14)

Any {s∗i (θi, 0)}i∈N that satisfies Equation (14) gives rise to a reporting equilibrium with

s∗i (θi, ti) = s∗(θi, 0) + Nti for ti �= 0. Clearly s∗i(θi, 0) = Nθi/M for all i ∈ N constitutes a

solution to (14). Therefore, s∗i(θi, ti) = Nθi/M + Nti for all i and ti are an equilibrium under

the mean decision rule x =∑

i∈N θi/N .

In this equilibrium, x =∑

i∈N θi/N =∑

i∈N si(θi, 0)/N +∑

i∈N ti =∑

i∈N θi/M +∑

i∈N ti.

So conditional on a true signal profile θN , the principal’s expected payoff is

Eθ[V |θN ] = −Eθ[(x− θ)2|θN ] = −Nσ2t − σ2ε/M

Since this is independent of the signal profile θN , it is also the ex ante expected payoff for the

principal before the realization of signals. Since s∗i(θi, ti) = s∗(θi, 0) + Nti, it must be that

ui(θi, ti)(s∗

i, s∗−i) = ui(θi, 0)(s

∗

i, s∗−i) for all i and all ti. From Equation (13), we have

ui(θi, ti) = −(N − 1)σ2t − σ2ε /M

This proves the proposition. Q.E.D.

Proof of Proposition 5: Let {si(θi, ti)} be any reporting equilibrium. Let δi(θi) = si(θi, 0)−

Nθi/M . Since Equation (14) must hold, then δi(θi) = −∆(θi) = −∑

j �=iEθj [(sj(θj, 0)−Nθj/M)|θi] =

−∑

j �=iEθj [δj(θj)|θi]. Therefore, for any θi,∑

j∈N Eθj [δj(θj)|θi] = 0.

Under the mean decision rule, we have x =

∑i∈N

θi

N=

∑i∈N

si(θi,0)

N+∑

i∈N ti =

∑i∈N

θi

M+

∑i∈N

δi(θi)

N+∑

i∈N ti. Conditional on a true signal profile θN ,

E[V |θN ] = −Eθ[(x− θ)2|θN ]

= −Eθ[(

∑i∈N θiM

+

∑i∈N δi(θi)

N− θ)2|θN ]−Nσ2t

= −(∑

i∈N δi(θi))2

N2−Nσ2t − σ2ε /M

26

Since the first term is non-negative and will remain non-negative after taking the expectation

over θN , this is less than E[V ] in the simple reporting equilibrium.

For the committee members’ expected payoffs, Equation (13) now becomes

ui(θi, ti) = −1

N2Eθ

−i[(∑

j �=i

δj(θj)−∑

j �=i

E[δj(θj)|θi])2|θi]− (N − 1)σ2t − σ2ε /M

Since the first term is non-negative, this is less than ui(θi, ti) in the simple reporting equilibrium.

Q.E.D.

Proof of Proposition 6: By Proposition 4, the reporting strategies of the committee members

are clearly optimal under the principal’s decision rule. It only needs to be shown that given

committee members’ reporting strategies, the principal’s optimal policy is indeed x =∑

i∈N θi/N .

Given committee members’ reporting strategies and a report profile θN , the principal gets a

sample of N signals about θ: θi = θi +Nti = θ+ εi +Nti. Let λi = εi +Nti, then {λi} are i.i.d

normal random variables with mean zero and variance σ2λ= σ2ε +N2σ2t , or precision τλ = τετt

N2τε+τt.

By the standard Bayesian updating rule, the posterior mean of θ given θN is

E[θ|θN ] =Nτλ

τθ +Nτλ

∑i θiN

Clearly the principal’s best response to the committee members’ reporting strategies is to

choose a decision equal to this posterior mean of θ. When τθ = 0, this reduces to∑

i θi/N .

Therefore, the proposed strategy profiles indeed constitute an equilibrium. Q.E.D.

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