A Rationale for Unanimity in Committees∗

Yves Breitmoser† Justin Valasek‡

May 16, 2019

Abstract

Existing theoretical and experimental studies have established that unanimity is a poordecision rule for promoting information aggregation. Despite this, unanimity rules are fre-quently used in committees making decisions on behalf of society. This paper shows thatrequiring unanimous decisions can facilitate truthful communication and optimal informa-tion aggregation when voters face consequences not only for the outcome of the collectivedecision, but also for how they personally voted. Theoretically, we show that majority rulesuffers from a free-rider problem in this setting, since agents’ votes are not always pivotal.Unanimity mitigates free-riding since all agents are equally responsible for the commit-tee’s decision. We test our predictions in a controlled laboratory experiment. As predicted,if unanimity is required, subjects are more truthful, respond more to others’ messages, andare ultimately more likely to make the optimal decision. Our work therefore provides arationale for a unanimity rule in settings where committee members are held accountable,formally or informally, for their individual voting decisions.

Keywords: committees, incomplete information, decision rules, cheap talk, informationaggregation, laboratory experiment

JEL Classification Codes: D71, D72, C90.

∗Thanks to Marina Agranov, Nageeb Ali, Inacio Bo, Agustin Casas, Chang-Koo Chi, Guillaume Frechette,Dan Friedman, Steffen Huck, Navin Kartik, Aniol Llorente-Saguer, Andrea Mattozzi, Krisztina Molnar, RebeccaMorton, Salvatore Nunnari, Tom Palfrey, Karl Schlag, Andy Schotter, Sebastian Schweighofer-Kodritsch, PeterNorman-Sorensen, and Arturas Rozenas as well as seminar participants at the Helsinki School of Economics,University of Bristol, Norwegian School of Economics, University of Vienna, Southern Denmark University,University of Mannheim, NYU CESS Conference, CESifo PSE Conference and Rotterdam PE Workshop for theirhelpful comments and suggestions. A special thanks to Rune Midjord and Tomas Rodrıguez Barraquer for thediscussions and insight that helped shape the motivation for this paper. A version of this paper was previouslycirculated under the title “The Value of Consensus: Information Aggregation in Committees with Vote-ContingentPayoffs.” Financial support of the WZB Berlin and the DFG (project BR 4648/1 and CRC TRR 190) is greatlyappreciated. Corresponding author: Justin Valasek.

†Bielefeld University. Contact e-mail: yves.breitmoser@uni-bielefeld.de‡Norwegian School of Economics (NHH), WZB Berlin and CESifo. Contact e-mail: justin.valasek@nhh.no

1 Introduction

Committees are a ubiquitous institution for making social decisions in the presence of uncer-tainty. Ideally, committees aggregate the private information of their members and thus makemore informed decisions than could be made by any one individual in isolation. This intuitionwas first formalized by de Condorcet (1785), who showed that if all individuals hold privateinformation that is more likely to be “right” than “wrong,” and if all individuals vote accordingto their private information, then a sufficiently large committee that votes via a majority rulewill choose the “right” option with arbitrary precision. However, in many real-world settings,committee members may face consequences not only for the outcome of the collective deci-sion, but also for how they personally voted. For example, members of criminal-trial juriesmay be exposed to ex-post regret for voting to convict a defendant who is later shown to beinnocent, politicians may wish to signal their ideological position to the electorate, and FDAexperts may seek to avoid blame for approving a drug that proves to have severe side-effects. Inthese cases, the assumption that individuals will vote sincerely may fail since, relative to deci-sions made by a single agent, decision-making by majority dilutes individual responsibility forthe committee’s decision, making committee members more likely to vote according to theirvote-contingent biases.

In this paper, we explore the effect of vote-contingent payoffs on communication and informa-tion aggregation in committees. Additionally, we consider the impact of the decision rule onbehavior. In practice, committees use a wide range of rules to reach decisions: For example,FDA committees use a majority rule when deciding whether to approve a new pharmaceuti-cal drug for general use, while many criminal trials require the jury to unanimously vote toeither convict or acquit. Based on the findings of the existing theoretical and experimental lit-erature, however, the use of a unanimity rule to aggregate information is puzzling—existingstudies have shown that unanimity is a uniquely poor decision rule for promoting informationaggregation (see Feddersen and Pesendorfer, 1998, Guarnaschelli et al., 2000, Persico, 2004,Austen-Smith and Feddersen, 2006, and Bouton et al., 2017a), and at best produces the sameresults as majority (Coughlan, 2000, Guarnaschelli et al., 2000 and Goeree and Yariv, 2011).1

In contrast, our analysis suggests that uniformly enforcing responsibility for the committee’sdecision may mitigate a voting bias that occurs in many real-world settings where committeemembers can be held accountable, formally or informally, for their individual voting decisions.Specifically, we show theoretically and experimentally that when committee members receivevote-contingent payoffs and deliberate prior to voting, a unanimity rule can facilitate truthfulcommunication of private information and optimal voting behavior.

Our analysis builds on two largely independent strands of literature. On one hand, the lack ofrobustness of information aggregation when committee members face vote-contingent payoffshas been well-documented by a number of theoretical studies, and payoffs linked to a commit-

1Two important exceptions are Bond and Eraslan (2010) and Chan et al. (2017). Bond and Eraslan (2010) showthat in a model where a proposer chooses which options will be voted over, a unanimity rule can be ex anteoptimal in cases where the preferences of the proposer and the committee differ. Chan et al. (2017) consider theimpact of the decision rule on the length of the committee’s deliberation process. They show that a unanimityrule can promote information aggregation by committees, since it gives more patient members control over whento terminate deliberation. In contrast, the mechanism we present here operates via the committee members’communication and voting behavior, rather than endogenous proposals or the length of deliberation.

1

tee member’s vote are relevant across a range of settings: committee members may face regretor blame for supporting an incorrect decision (for an application concerning FDA committeessee Friedman and Friedman, 1990, p. 208, and Midjord et al., 2017), career concerns (Visserand Swank, 2007 and Levy, 2007), and expressive or moral biases (Brennan and Buchanan,1984, Tyran, 2004, Huck and Konrad, 2005, Feddersen et al., 2009, and Morgan and Vardy,2012). Additionally, vote-contingent payoffs are highly relevant for committee members thatface electoral pressures: e.g., individual voting records affect the reelection chances of legisla-tors (Canes-Wrone et al., 2002) and reelection concerns affect the decisions of elected judgeson state supreme courts (Hall, 1992).

On the other hand, a similarly comprehensive literature analyzes information aggregation anddecision rule design when voting is preceded by open discussion. Deliberation is observed inmany committee settings prior to voting, such as juries, expert panels and political decision-making bodies, and deliberation can significantly improve a committee’s ability to aggregateinformation in equilibrium under any decision rule (for a theoretical argument, see Coughlan,2000, and Gerardi and Yariv, 2007; for experimental evidence, see Guarnaschelli et al., 2000,and Goeree and Yariv, 2011). Pre-vote deliberation has also been widely studied in the casewhere committee members have conflicting preferences over the committee decision (that is,when agents disagree about the optimal committee decision for some information set). Forexample, Li (2001) and Austen-Smith and Feddersen (2006) show that, theoretically, heteroge-nous preferences lead to the non-existence of truthful communication as part of an equilibriumstrategy. Experimentally, however, Goeree and Yariv (2011) show that in committees withheterogeneous payoffs, deliberation results in a high degree of truthful communication andinformation aggregation despite theoretical predictions to the contrary.2

Our paper merges these two strands of the literature and shows that the impact of vote-contingentpayoffs on the effectiveness of pre-vote deliberation and information aggregation is fairly dras-tic, implying a strong rationale for the widespread occurrence of unanimity rules. That is, whilethe standard Condorcet model assumes that agents have preferences over the committee deci-sion and the revealed state, we generalize this model to include the possibility that agents alsohave preferences over their individual vote. We show that under majority, such vote-contingentpayoffs can introduce a free-rider problem that eliminates equilibria where committee mem-bers truthfully communicate their private signals and vote informatively. In particular, if thereexists an information set at which agents prefer to individually vote for one option and have thecommittee select the other option (as we discuss in more detail following the introduction ofthe model, the examples of vote-contingent payoffs we provide here can result in this conditionbeing satisfied), then the “free-riding” condition is satisfied and under majority there are noequilibria with truthful communication and committee-optimal decisions.

In contrast, under a unanimity rule there exists a symmetric equilibrium with truthful com-munication and committee-optimal decisions for any profile of vote-contingent payoffs. Inparticular, we consider the use of a decision rule that requires unanimous agreement: that is,the committee reaches a decision only when all members of the committee vote for the sameoption. This form of unanimity rule is consistent with the legal definition of unanimity where,

2In contrast, the impact of vote-contingent payoffs on the effectiveness of pre-vote deliberation is an understudiedtopic—to the best of our knowledge, communication and vote-contingent payoffs in committees has only beenaddressed within the context of career concerns, and has focused on the optimal level of transparency. We reviewthis literature below.

2

e.g., for a decision to be reached either all jury members must vote “guilty” or all membersmust vote “not guilty.”3 Accordingly, we model unanimity as a repeated series of informalstraw polls, where a final decision is reached only when the straw poll is unanimous. Thiscontrasts with a status-quo unanimity rule, common in international organizations, where oneoption is designated as a status-quo and unanimous agreement is required for any change fromthe status quo.4 As we will show below, a status-quo unanimity rule is partially robust to vote-contingent payoffs, while a unanimity rule that requires unanimous agreement for any decisioneliminates the free-rider problem for any vote-contingent payoff.

To clarify the intuition, we consider a (very) simple numerical example: Assume that the stateof the world is binary, R(ed) or B(lue), with the committee deciding by majority, and beingasked to match (or, “predict”) the state of the world. There are three committee members, andeach receive an independent signal about the true state of the world. Each signal is correctwith probability 0.6, and each committee member gets a payoff of 1 if the committee decisionmatches the state of the world. Additionally, each committee member receives a payoff of 0.1 ifthey vote for R; we denote this “expressive” payoff by K. (In our analysis of the full model, wewill consider vote-contingent payoffs of a much more general form to capture more complexvote-contingent payoffs; e.g. FDA committee members who may receive a negative payoff onlyif they vote for approving a drug, the drug is approved by the committee, and it results in severeside effects.)

If two committee members receive B signals and all truthfully communicate their signal, theyknow that decision B is correct with probability 0.6 and decision R is correct with probability0.4. Hence, if their vote is pivotal, then voting B is individually optimal since 0.6− 0.4 =0.2 > K. However, it is not individually optimal for all committee members to vote for B,since then no member’s vote is pivotal under majority. Instead, it is a mutual best responsefor committee members to vote for B with a probability of 50 percent each, which yields a 50percent probability that the committee selects option R given common knowledge of two signalsfor B. This illustrates how vote-contingent payoffs introduce a collective-action problem to thevoting stage that prevents the committee from reaching the optimal decision.5

This voting bias towards option R distorts information aggregation in the voting stage and,additionally, gives committee members an incentive to misrepresent their private signal: As-sume there exists an equilibrium with truthful communication and that the other two committeemembers have signals of B. A committee member with a private signal of R knows that if theytruthfully communicate, and the other two committee members have signals of B, there is onlya 50 percent probability that B is chosen. If they deviate by misreporting a B signal (and indi-vidually vote for R), however, the other two committee members will vote as if there are threesignals for B and this yields an 81 percent probability that B is chosen. Therefore, despite the

3See American Bar Association (2013) for detailed description of jury deliberations. In federal jury trials in the US,if a unanimous decision cannot be reached after extensive jury deliberation, a “hung jury” results and a mistrialis declared. A mistrial, however, is not equivalent to an acquittal. As described by the American Bar Association(2013), a hung jury implies that: “The case is not decided, and it may be tried again at a later date before a newjury.”

4In the economic literature, unanimity has been most commonly modeled as a status-quo unanimity rule (seeFeddersen and Pesendorfer, 1998 and Maggi and Morelli, 2006). For a notable exception, see Coughlan (2000).

5To illustrate the result as simply as possible, we focus on voting strategies that are symmetric and do not conditionon the individual signal or message here. However, in the analysis we will show that our main results do notrequire this assumption.

3

relatively low expressive payoff of K = 0.1, there is no such equilibrium in which committeemembers vote for B with positive probability and truthfully communicate their private signals.6

As a result, communication is strategic rather than truthful, information aggregation is com-promised, and social efficiency declines. In contrast, voting by unanimity ensures that truthfulcommunication and committee-optimal decisions occur in equilibrium for all values of K.

From a more general perspective, we show that homogeneity of preferences over the committeedecision is not a sufficient condition for the existence of an equilibrium with truthful commu-nication and committee-optimal information aggregation when agents also have heterogeneouspreferences over the profile of votes. Accordingly, the novel insight of our theoretical analy-sis is that, given homogeneous preferences over the committee decision and vote-contingentpayoffs, unanimity outperforms majority because unanimity ensures homogeneous payoffs forevery admissible voting profile. In turn, the heterogeneity of payoffs under majority resultsin the non-existence of an equilibrium with truthful communication and committee-optimalinformation aggregation whenever the free-riding condition is satisfied.

While the theory is suggestive, the prediction that unanimity dominates majority is not conclu-sive. In general, there are multiple equilibria under both voting rules; for example, there existso-called babbling equilibria where communication is random. Additionally, behavioral biasesdue to lying aversion, incorrect Bayesian updating, or level-k reasoning may also interfere withthe theoretical prediction; e.g. earlier experimental work has shown that subjects overreporttheir true signals relative to theoretical benchmarks (Goeree and Yariv, 2011). In such cases,a controlled laboratory experiment can be a useful tool to generate insights about observedbehavior in an analogous choice environment.

In the second part of our paper, we present the results of a laboratory experiment that comparescommittee behavior under majority and unanimity in the presence of expressive payoffs. In a2×2 design, we compare the two decision rules for two different levels of expressive payoffs.Across all four treatments, the effects are highly consistent with the theory: Under majority,subjects systematically misreport their messages in a pre-vote round of binary cheap talk and,indeed, this effect is mitigated under unanimity. As predicted, unanimity also outperformsmajority in terms of information aggregation—to the best of our knowledge, this finding con-stitutes the first experimental evidence of a setting where a unanimity rule is strictly preferableto majority (previous experiments in settings with pre-vote deliberation, such as Guarnaschelliet al., 2000, and Goeree and Yariv, 2011, find either no significant difference between unanim-ity and majority, or less information aggregation under unanimity). Thus, we conclude thatvote-contingent payoffs substantially enhance the comparative efficiency of unanimity votingand may help explain its widespread use.

Finally, we analyze the behavioral mechanism underlying this result. First, we find that theprivate signal and co-players’ messages have a similar impact on a subject’s vote under una-nimity, but that subjects are relatively less likely to respond to co-players’ messages undermajority. Second, we classify agents into distinct strategy types using a finite mixture model-ing approach and find that under majority rule 20–26 percent of subjects pursue a “free-rider”strategy that is biased towards falsely reporting the non-expressive option and personally vot-

6Note that reporting message of B given a signal of R also results in a 25 percent probability that B is chosen whenthere is only one signal for B. For K = 0.1, however, the expected loss is (individually) outweighed by the gain ofselecting B with a higher probability given two signals for B.

4

ing for the expressive option. In contrast, the proportion of subjects classified as free-ridersunder unanimity is not significantly different from zero (with point estimates at 0). In all, thesefindings provide strong evidence supporting the hypothesis that the higher degree of truthfulcommunication and lower degree of “strategic” behavior under unanimity has a positive causaleffect on information aggregation.

Next, we review the related literature. Section 2 introduces the general model and presents ourmain results, and Section 3 characterizes the existence of equilibria with truthful communica-tion and analyzes several extensions of the model. Section 4 describes our experiment testingthe predictions of the model. Section 5 presents the analysis of the experimental results andSection 6 concludes. The formal proofs, additional theoretical analyses, the experimental in-structions and a number of robustness checks of the econometric results are provided in theAppendix.

1.1 Literature

The issue of optimal voting rules in committees was first addressed in Austen-Smith and Banks(1996), who show that for committees of finite size, information aggregation is only achievedunder the decision rule that features sincere voting as an equilibrium strategy.7 Next, Feddersenand Pesendorfer (1998) went on to show that as the size of the committee approaches infinity,all q-rules other than status-quo unanimity aggregate information. The reason for the failureof information aggregation under status-quo unanimity is that an agent’s individual vote isonly pivotal when all other agents have voted for the non-veto option, implying that it is notoptimal for an agent to veto based on their private information only. This reluctance to “veto”under a status-quo unanimity rule has also been observed experimentally by Guarnaschelliet al. (2000), who find that majority rule outperforms status-quo unanimity in a setting withoutcommunication or vote-contingent payoffs. More recently, Bouton et al. (2017a,b) demonstratethat a voting mechanism that allows agents to vote for either option, or to veto, can outperformboth majority and a status-quo unanimity rule.

The results referenced above, however, pertain only to settings where committees do not de-liberate prior to voting. Coughlan (2000) finds that if voting is preceded by a round of cheap-talk communication, then both majority and status-quo unanimity admit equilibria with perfectinformation aggregation (also see Gerardi and Yariv, 2007, for a comprehensive analysis ofcommunication and voting). Experimentally, Guarnaschelli et al. (2000) and Goeree and Yariv(2011) find that allowing subjects to either conduct a “straw poll” or communicate via chat priorto voting largely eliminates the discrepancy found between majority and status-quo unanimityabsent communication.

One strand of this literature has focused on the efficacy of communication when agents haveconflicting preferences over the committee decision (see Li, 2001, and Austen-Smith and Fed-dersen, 2006). In this case, agents have an incentive to deviate from truthful communication inan attempt to bias the committee decision towards their preferred option. In contrast, the mech-anism we highlight here does not rely on heterogenous preferences over the committee deci-

7For comprehensive overviews of the literature on information aggregation in voting, see Li and Suen (2009),Gerling et al. (2005), and Palfrey (2016).

5

sion. Instead, due to heterogenous preferences over the vote profile, agents have an incentiveto misreport their signal to persuade other agents to vote for the non-expressive option, whilepersonally free-riding and voting to obtain the expressive payoff. We show that this mechanisminduces some subjects to adopt a free-riding strategy, which may explain why we find evidencefor strategic communication with vote-contingent payoffs, while Goeree and Yariv (2011) findthat the rate of truthful communication is not significantly affected by heterogeneous prefer-ences.

Lastly, our paper is related to the literature on reputation payoffs in committees, which con-siders the Holmstrom (1999) model of career concerns applied to a committee setting. Here,reputation payoffs depend on the agent’s vote relative to the aggregate voting profile, as agentsseek to maximize the principal’s ex-post belief that the agent is of high ability. This literatureprimarily considers the question of the optimal level of transparency (see Fehrler and Hughes,2018, for a review). Visser and Swank (2007), however, consider the optimal decision rule ina setting with communication and reputation payoffs (additionally, Levy, 2007, analyzes theoptimal decision rule in a setting without communication). They find that, similar to the set-ting analyzed in the literature on conflicting preferences, an agent’s incentive to misreport theirprivate information stems from heterogeneity in preferences over the committee decision.8

In contrast, we demonstrate that with vote-contingent payoffs, committee heterogeneity can bea function of the underlying decision rule: With majority, the committee can always achieve theoptimal decision even when some committee members vote for a different option, which allowsfor heterogeneous payoffs and can result in a free-rider problem. Unanimity rule, however,holds all members equally responsible for the committee decision, which facilitates committee-optimal behavior by enforcing homogeneous payoffs.

2 The Model and General Results

2.1 Framework

We consider a standard Condorcet setting with pre-vote communication (cheap talk): An odd-numbered committee of N agents, i ∈ 1,2, ...,N with N ≥ 3, chooses between two optionsR(ed),B(lue). The committee decision, denoted by X ∈ R,B,D, is made via a vote, whereeach committee member submits a vote, vi ∈ R,B, simultaneously with no abstentions. Here,decision “D” denotes “disagreement” in case unanimity is required but not achieved.

The underlying state of the world is denoted by ω ∈ Ω = R,B. Agents do not observe thestate of the world, and all agents have a common prior over the state of the world, denotedPR = Pr(ω = R), that specifies that each state is equally likely (PR = 1/2). Additionally, eachagent receives a private signal regarding the state, si, drawn i.i.d. from a finite signal spaceS according to the probability distributions pω ∈ ∆(S). Each signal is partially informative,pR(s) 6= pB(s), but no signal perfectly reveals the state of the world in the sense that pω(s)> 0

8Fehrler and Hughes (2018) also provide a theoretical and experimental analysis of communication and reputationpayoffs; in their setting, committee members misreport the precision of their signal when they have a low-precisionsignal and communication is observed by the principal.

6

for all s∈ S and ω∈Ω. We define S= SN and let s∈ S indicate a specific profile of signals. Priorto voting, agents engage in communication: agents simultaneously send messages, mi ∈ S, thatare publicly observable (agents are free to send any message, regardless of their private signal).9

After the voting stage, a decision rule dv ∈ Majority,Unanimity, defined below, maps votes(vi) into the committee decision X .

Consistent with our motivation of studying information aggregation with vote-contingent pay-offs, the committee members’ payoffs are a function of the committee decision, the underlyingstate of the world, and the agent’s vote vi. Formally, agents’ preferences are represented by autility function, u(X ,ω,vi) : R,B,D×R,B2 → R. For conceptual reasons, let us split upthe utility function into

u(X ,ω,vi) = u1(X ,ω)+u2(X ,ω,vi),

where u1(X ,ω) represents the common-value payoff for matching the committee decision tothe state of the world, and u2(X ,ω,vi) represents the vote-contingent payoffs. This formulationis fairly general, allowing for vote-contingent payoffs that can vary based on the committeedecision and the state of the world; for example, an FDA committee member may receive anegative reputation payoff only if a drug that is approved by the committee, is later shownto result in severe side-effects, and the committee member individually voted to approve thedrug. We impose the restrictions that u1(ω,ω) > u1(X ,ω) and u(ω,ω,vi) > u(X ,ω,v′i) for allvi,v′i,ω and X 6= ω, which implies that agents’ payoffs from the committee choosing the optionthat matches the state is greater than any vote-contingent payoff—these restrictions are notstrictly necessary, but imply a payoff structure analogous to the standard models of informationaggregation, where agents’ primary motivation is to match the decision to the state.

We compare the decision rules of majority and unanimity (we also consider a status-quo una-nimity rule in Section 3.2). We use the standard definition of a majority decision rule:

Definition 1 (Majority). Each player submits a vote vi ∈ R,B. The final decision X is theoption that receives a larger number of votes.

For a unanimity decision rule, we also model the process by which the committee coordinateson a unanimous decision as a sequence of (internal) straw polls:

Definition 2 (Unanimity). The straw poll proceeds in rounds, and each round each playersubmits a vote vt

i ∈ R,B, t ∈ 0,1, . . .. If the straw poll is unanimous then voting stops,each agent’s final vote vi is equal to their vote in the straw poll, and the committee decision isequal to the unanimous decision. Otherwise, a new round of the straw poll begins. In case ofperpetual disagreement, X = D.

For simplicity, we focus on stationary strategies in the voting stage. While we do not includea discount term, including discounting between intermediate rounds of voting does not qualita-tively impact our results. As we show later, an equilibrium of the game exists in which agentsplay stationary strategies and reach a decision in the first round of voting, so theoretically thestraw poll is not necessary for agents to exchange information. Also, in our baseline model

9Deliberation is observed in many committee settings prior to voting, including the examples we introduce below.Following earlier papers such as Coughlan (2000); Guarnaschelli et al. (2000), we model deliberation as a stagewhere agents can publicly communicate their signals prior to voting.

7

vote-contingent payoffs are contingent on the final vote (vi) only; we consider the case of ex-pressive payoffs in the communication and coordination (straw-poll voting) stages in Section3.2. Lastly, in case of perpetual disagreement, X = D, all payoffs are zero (u(D,ω,vi) = 0).

The procedure for the unanimity rule specified above is just one of many ways to model theprocess by which the committee coordinate on a unanimous decision. We illustrate our resultsusing this particular procedure since it follows the general process often used in committees:agents first publicly share their private signals, then “deliberate” using a sequence of straw polls,and reach a final decision when the outcome of a straw poll is unanimous (see Guarnaschelliet al., 2000; Goeree and Yariv, 2011). Note that the feature of unanimity rule that is mostimportant for our analysis is that for a given decision to be chosen, all agents must unanimouslyvote for that decision.

We make one additional assumption regarding payoffs. Let Eω[u(X ′,ω,vi = X)|s] denote theexpected utility, taken over the states of the world ω, when the committee decision is X ′, playeri votes vi = X (X may equal X ′), and signal profile is s:

Assumption 1. For all s, there exists an decision X ′ such that Eω[u(X ′,ω,vi =X ′)|s]> u(D,ω,vi)=0.

This assumption entails that, given knowledge of the signal profile s, there exists a possible de-cision X ′ such that all committee members are better off if the committee reaches this decisionrather than deliberating in perpetuity.

The timing of the game is as follows:

1. Nature draws state ω ∈ R,B and sends private signals s = (si).2. Committee members observe si and simultaneously send messages mi ∈ S.3. Committee members observe m and simultaneously submit votes ∈ R,B.4. (Unanimity only) If the vote is non-unanimous, then the committee proceeds to a new

round of voting (stage 3 is repeated).5. The committee decision is taken, the state is revealed and payoffs accrue.

The equilibrium concept we consider is symmetric Perfect Bayesian Nash (a formal definition isprovided in the Appendix). By symmetry, we require that agents with the same signal who alsosent the same message vote R with the same probability. Our theoretical results establish (non-)existence and properties of equilibria sustaining truthful communication. That is, all messagesare sent with positive probably and all information sets relevant for defining the (stationary)strategies of our agents are on the path of play. Beliefs are therefore uniquely specified byBayes’ Rule and need not be specified explicitly. Given the beliefs consistent with Bayes’Rule, expected utilities are also well-defined, and equilibria are defined straightforwardly.

We denote the game by Γ = 〈PR,S, pω,N,dv,u〉. Agents’ strategies are pairs (σi,τi), where:

• σi(mi|si) is the probability of sending message mi ∈ S after receiving signal si ∈ S,• τi(si,mi,m) is the probability of vote R after signal si, own message mi, and the aggregate

message profile m ∈ S.10

10Since we consider stationary strategies and focus on the existence of an equilibrium with truthful communicationunder Unanimity, we omit a t subscript on τi without loss of generality.

8

Lastly, we introduce a definition of what we refer to as the Free-Riding property. We call itthe Free-Riding property since one of our main results (presented in Section 2.3) shows thatwhen vote-contingent payoffs result in this property being satisfied, they cause an incentive tofree-ride under Majority.

Definition 3 (Free-Riding). A game Γ = 〈PR,S, pω,N,dv,u〉 satisfies Free-Riding if there existsa vector of signals s′ such that the following two conditions are satisfied:

1. Given a committee decision for X∗ such that

Eω[u(X∗,ω,vi = X∗)|s′]> Eω[u(X ′,ω,vi = X ′)|s′],

with X ′ 6= X∗, agents individually prefer to vote for X ′,

Eω[u(X∗,ω,vi = X ′)|s′]> Eω[u(X∗,ω,vi = X∗)|s′].

2. Take Ns to be the set of agents with signal s given s′; there is no signal set S⊆ S such that| ∪s∈S Ns| = (N + 1)/2, i.e. such that the agents with signals in S constitute a minimalwinning coalition.

The first condition of the Free-Riding property requires that vote-contingent payoffs are non-trivial in the sense that they may induce a conflict between collective and individual interestfor at least one signal profile. Absent vote-contingent payoffs (u2 = 0), Γ does not satisfy thiscondition since agents’ utilities are independent of their individual votes: u(X ,ω,vi 6= X) =u(X ,ω,vi = X) for all ω. In contrast, a simple expressive payoff for voting for R satisfiescondition 1 straightforwardly since, holding the committee decision constant, agents receive ahigher payoff from individually voting for R regardless of s.

The second condition of the Free-Riding property implies that agents cannot always coordinateon who may free-ride based solely on their signals s′. As a result, there can be no symmetricpure strategy equilibrium where the winning option always receives exactly (N+1)/2 votes un-der Majority. For example, a simple expressive payoff for voting for R satisfies both conditionsof the Free-Riding property since for any homogenous signal vector (s′ such that s′i = s′j for alli, j) where option B maximizes the expected value of u1, each agent prefers that the committeeselects B, but individually prefers to vote R.

2.2 Examples and Applications

Next, we illustrate how the main applications listed in the Introduction fit our framework. Aswe will show, all of these examples imply vote-contingent payoffs that satisfy the first conditionof Free-Riding property and therefore may cause distortions in information aggregation underMajority. The Free-Riding property, however, also requires a restriction on the signal set. Wewill assume that the signal set satisfies the second free-riding condition when discussing theexamples below (this assumption holds true if, for example, signals are binary).

Application 1: Jury Trials In criminal trials, jurors seek to minimize their expected regretfrom taking an incorrect decision—either convicting a defendant that is shown to be innocent,

9

or acquitting a defendant who is shown to be guilty (see Kaplan, 1968, and Ellsworth, 1994,pp. 55–56). In this environment, jurors who vote in opposition to the jury decision may lessentheir individual regret from an incorrect conviction or acquittal. That is, given an incorrect jurydecision, convicting an innocent defendant or acquitting a guilty defendant, a juror would bebetter off if they had individually voted to respectively acquit or convict. Formally, this exampletranslates into vote-contingent payoffs such that

u2(Convict, Innocent, vi = Convict)< u2(Convict, Innocent, vi = Acquit)

and

u2(Acquit, Guilty, vi = Acquit)< u2(Acquit, Guilty, vi = Convict),

which are captured by our model.

Moreover, Kaplan (1968) specifically suggests that the level of regret for an incorrect convic-tion, and hence the associated vote-contingent payoff, may be greater than the level of regretfrom an incorrect acquittal. If this asymmetry is high enough, then the Free-Riding propertywill be satisfied and, as we show later, the imbalance in the vote-contingent payoffs from mit-igating type I and type II errors can distort the jury decision under Majority. In contrast, Una-nimity rule, such as is practiced in criminal jury trials in the U.S., avoids the collective-actionproblem that arises under Majority.

Application 2: Party-line Voting11 Members of legislative bodies are sometimes askedto pass a policy measure that could be socially optimal, but that is unpopular with votersex ante. Under Majority, this can give rise to a collective-action problem: while societymay be better off in expectation if the reform is passed, if the reform ends up being sociallyinefficient legislators’ chances of reelection will be larger if they individually vote againstthe reform. Formally, after normalizing the payoff of voting to reject the policy to zero,u2(X ,socially inefficient,vi = Accept Policy) < 0 and the Free-Riding property is satisfied bythe subset of signals where passing the reform is socially optimal in expectation.

Our results suggest that this collective action can distort voting outcomes under Majority, andmay cause individual legislators to have an incentive to withhold negative information about thereform in an attempt to manipulate colleagues into voting for the reform. Unanimity, however,solves the collective-action problem and sustains truthful communication in equilibrium. Thismay justify intervention at the level of the political party to require that all party membersvote for the reform (see Levitt, 1996; Snyder and Groseclose, 2000, for empirical analyses ofthe influence of the party in legislative voting)—if party leaders are able to enforce a strictparty-line vote (de facto unanimity among the governing coalition), then the collective-actionproblem will be mitigated.

Application 3: FDA Committees Posicor, a drug approved by the FDA to treat high bloodpressure, was withdrawn after being linked with over 100 deaths. David Willman of the LATimes later won a Pulitzer Prize in part for his article analyzing the FDA committee’s decision

11We would like to thank Referees 1 and 3 for both independently suggesting this application.

10

to approve the drug by a vote of 5-3 (Willman, 2000). His article explicitly names two ofthe committee members who voted against the drug, and cites a third committee member whocould not participate in the vote due to financial conflict as stating “You do wonder how theworld would perceive it. I’m glad I didn’t vote. . . ”

This anecdote highlights that committees of experts may be exposed to vote-contingent pay-offs in cases where the committee makes a high-profile error (see Midjord et al., 2017, for ananalysis of vote-contingent payoffs for type I and type II individual errors in a setting withoutcommunication). For example, if an FDA committee votes to approve a drug, the drug resultsin severe side-effects, and the committee member individually votes to approve the drug,

u2(X = Approve, ω = Not Approve, vi = Approve)< 0.

While other outcomes may also result in vote-contingent payoffs, the FDA incorrectly approv-ing a drug is likely more salient than an incorrect drug rejection or a correct approval/rejection.Therefore, similar to Case 1, the vote-contingent payoff in the case of an incorrect approvalmay be relatively large, which can result in the Free-Riding property being satisfied.

Application 4: EU Institutions Other voting bodies that may be subject to vote-contingentpayoffs include the European council and other EU institutions composed of politicians electedat the domestic level. For example, take the European Stability Mechanism (ESM), which takesdecisions regarding the management and disbursement of bail-out funds to individual Eurozonecountries. Decisions in the ESM are currently taken by a status-quo unanimity rule among thefinance ministers of Eurozone countries. However, recent proposals to replace the ESM witha European Monetary Fund (EMF) have discussed reforming the decision rule (see EuropeanCommission, 2017; Sapir and Schoenmaker, 2017).

Bailouts of other EU countries, however, are not always popular with politicians’ domesticaudiences. Since position-taking (individual votes) at the international level may influence do-mestic approval (Putnam, 1988; Hagemann et al., 2017a,b), politicians will consider the impactof their EU-level vote on their chances of reelection. If domestic politicians face an audiencecost for voting to approve a bailout, independent of whether a bailout is necessary to securethe financial stability of the Eurozone, then u2(X ,ω,vi = Bailout)< u2(X ,ω,vi = No Bailout)and the Free-Riding property is satisfied by the subset of signals where a bailout is optimal.In such cases, where voting against the status quo of “No Bailout” yields a negative (relative)vote-contingent payoff, a status-quo unanimity rule is “robust,” while adopting a majority ruleresults in a collective-action problem that will make it more difficult for the EMF to approvenecessary bailouts.

Application 5: Moral Bias When voting for one option is morally or ethically appealing,committee members may be subject to an “expressive” payoff based on their individual vote(see Tyran, 2004; Feddersen et al., 2009). For example, all members of a tenure committeemay prefer to only grant tenure to high quality candidates. However, all else equal, committeemembers may also prefer to not vote against giving tenure to one of their close colleagues.Formally, a simple moral or expressive bias can be represented by a separable utility functionthat consists of a payoff for matching the committee decision to the state of the world and a

11

positive vote-contingent payoff for one of two options that is independent of the committeedecision or the realized state; e.g.

u1(X ,ω)+u2(X ,ω,vi) = IX=ω +K ∗ Ivi=R,

where K ∈ (0,1) represents the moral bias of voting for option R, and Icond takes a value of 1if the condition is satisfied and 0 otherwise. This example satisfies the Free-Riding propertystraightforwardly, and is analogous to the example that we utilize for our laboratory experiment.

2.3 General Results

In this subsection, we present our two main results comparing Unanimity and Majority withinthe general framework. In the following section, we present a more detailed analysis for the spe-cific case of a simple expressive (moral) payoff. Our aim here is not to be exhaustive—gameswith communication generally admit a multitude of equilibria—rather we focus on characteriz-ing the conditions under which the model supports equilibria with truthful communication andoptimal committee decisions (we provide a more complete characterization of equilibria for theparameters we use in the experiment in Appendix B).

Definition 4 (Optimal Information Aggregation). We define (Committee-) Optimal InformationAggregation as replicating the decision taken by a single decision-maker (DM) with preferencesu(X ,ω,vDM) = u(X ,ω,vi), who has access to the complete profile of signals, s, and whose votedetermines the committee decision, X = vDM.12

We refer to the decision that optimally aggregates information as the committee-optimal deci-sion. We also distinguish between optimal information aggregation and information aggrega-tion: information aggregation implies that the committee selects the option that is most likelyto coincide with the state of the world given the set of private signals (s); (committee-) optimalinformation aggregation, however, takes into account the vote-contingent payoffs, which im-plies that, say, option R may be optimal even when ω = B is more likely. We acknowledge thatin cases where a committee takes a decision on behalf of society, and the payoffs from match-ing the decision to the state of the world are symmetric, social welfare may be maximized byinformation aggregation since the committee members’ vote-contingent payoffs may be smallrelative to the social impact of choosing the more likely option. Therefore, while we focus onoptimal information aggregation in the theoretical analysis, we will consider both benchmarkswhen interpreting our experimental results.

Our first main result illustrates the free-riding problem that can occur under Majority whencommittee members are subject to vote-contingent payoffs. It establishes that the Free-Ridingproperty is indeed a sufficient condition for the non-existence of a symmetric equilibrium withtruthful communication and committee-optimal decisions.

12This concept of optimality is distinct from a DM who has access to s and sets the vote profile (vi) to maximizeaggregate utility since, for some signal profiles, the DM would be better off choosing a heterogenous profile ofvotes. For example, with a simple expressive payoff for R the DM would be better off with a minimal majority forB, since this would allow a minority of voters to receive the expressive payoff of voting for R. However, the abovedefinition facilitates the comparison of the two decision rules, since the committee-optimal decision is equivalentfor both Unanimity and Majority, and is equivalent to the decision a committee member would select when signalsare truthfully communicated and their vote is pivotal. We discuss this distinction below when relevant.

12

Proposition 1 (Non-existence under Majority). For any game Γ with dv = Majority satisfy-ing Free-Riding, there does not exist a symmetric equilibrium with truthful communication(σ(si|si) = 1) and committee-optimal decisions.

In a game that satisfies Free-Riding, vote-contingent payoffs introduce a coordination problemdue to the fact that, under majority rule, it is possible for the committee to select a given optioneven when a minority of agents vote for the other option. Therefore, under Majority it willnot be a best response for all agents to vote for the committee-optimal decision given a signalprofile of s′, since a single agent can deviate to voting for the other option and receive a higherexpected utility. This implies that, given truthful communication, it is not a best-response foragents to play a symmetric voting strategy that selects the committee-optimal decision for allsignal profiles.

Note that Proposition 1 does not fully preclude the existence of an equilibrium with informationaggregation when the Free-Riding condition holds: in the special case where the signal spaceis binary and both signals are equally informative, the committee can aggregate informationwithout truthful communication as long as all agents vote for the option that is most likelygiven their private signal.13 However, as we establish below in the context of the model witha simple expressive payoff, aggregating information absent informative communication is onlyan equilibrium for “small” vote-contingent payoffs—if expressive payoffs are sufficiently large,then there are no equilibria under Majority that result in information aggregation.

Proposition 1 starkly contrasts with our main result for Unanimity.

Proposition 2 (Existence under Unanimity). For any game Γ with dv = Unanimity, there existsa symmetric equilibrium with truthful communication and committee-optimal decisions.

Under Unanimity, all committee members must vote for an option for it to be selected. There-fore, even for games satisfying Free-Riding, given truthful communication it is a best-responsefor all agents to vote for the committee-optimal decision, since no agent can deviate to a dif-ferent voting strategy without preventing the committee from selecting the committee-optimaldecision. Likewise, any deviation from truthful communication will either result in the samedecision, or move the committee decision away from the optimal decision.

In order to illustrate the result more formally, consider the following sets:

SR =

s ∈ S |Eω(u(X = R,ω,vi = R)|s)≥ Eω(u(X = B,ω,vi = B)|s),

SB =

s ∈ S |Eω(u(X = R,ω,vi = R)|s)< Eω(u(X = B,ω,vi = B)|s).

That is, SR and SB constitute a partition of the set of signal profiles, S; SR is the set of signalprofiles under which i weakly prefers decision R given vi = R, and SB is the set of signal profilesunder which i strictly prefers decision B given vi = B. By symmetry, this implies that SR is theset of signal profiles under which R is committee optimal, and SB is the set of signal profilesunder which B is committee optimal.

13If the signals are not equally informative, however, then truthful communication may be necessary to aggregateinformation even under Majority.

13

Now consider voting behavior τ∗ under Unanimity such that vti = R if m ∈ SR and vt

i = Bif m ∈ SB. Given truthful communication, τ∗ is a best response to itself as it implementsthe committee-optimal decision and deviations yield either delay or perpetual disagreement.Given τ∗ in turn, no agent has an incentive to deviate from truthful communication: truthfulcommunication results in the committee-optimal decision, and any deviation will result in eitherthe same outcome, or X = R (B) for some s ∈ SB (SR). This implies that under Unanimity anequilibrium exist with σi(si|si) = 1 and τi(X ,ω,vi) = τ∗(X ,ω,vi).

Independently, both Proposition 1 and Proposition 2 relate to results in the existing literature oninformation aggregation in committees.14 Taken together, however, the novel insight providedby the combination of Propositions 1 and 2 is that the committee’s decision rule is relevant forefficiency, even in settings with communication and homogeneous preferences over the com-mittee decision (u1(X ,ω)), when agents have heterogenous preferences over the voting profile(u2(X ,ω,vi)). Proposition 1 shows that heterogenous preferences over the voting profile resultin a free-rider problem under Majority in cases where agents’ preferences diverge between thecommittee decision and their individual vote. In contrast, Proposition 2 shows that Unanim-ity avoids this free-rider problem by requiring a homogenous vote profile for any committeedecision.

While Proposition 1 shows that vote-contingent payoffs can distort voting, vote-contingent pay-offs can also impact the committee’s ability to truthfully share private information. In thefollowing section, we provide more details on the distortionary impact that vote-contingentpayoffs have on both voting and communication by analyzing the (non)existence of equilibriawith truthful communication under Majority for a simplified version of the general model.

3 The (im)possibility of truthful communication

To explore the impact of vote-contingent payoffs on communication under Majority, and to es-tablish testable predictions for the experiment, we explicitly characterize equilibria with truthfulcommunication for a relevant yet tractable class of models that consider an expressive (moral)payoff for voting for one of the options. This model contains the above examples on EU insti-tutions and moral bias as special cases. The other three examples, party-line voting, jury trialsand FDA committees, require a distinction of vote-contingent payoffs for type I and type II er-rors; however, given sufficient asymmetry between the vote-contingent payoffs for the differenterrors, a corresponding example can straightforwardly be developed along similar lines.

A simple model of expressive payoffs with binary signals Each committee member re-ceives a private signal from a binary signal space (S is binary), si ∈ R,B, with Pr(si = x|ω= x)equal to α ∈ (1/2,1) for any x ∈ R,B. For the common-value component of payoffs, eachagent receives a payoff of 1 if the committee chooses the option that matches the underlyingstate of the world, and a payoff of 0 otherwise; for the vote-contingent component of payoffs,

14As discussed in the introduction, it is well-known that vote-contingent payoffs can cause information aggregationto fail. Additionally, Proposition 2 follows from the same reasoning behind Theorem 1 in Austen-Smith andFeddersen (2006), who analyze communication in a setting with heterogenous preferences over the committeedecision (also see Gerardi and Yariv, 2007).

14

each agent has a simple expressive voting bias and receives a payoff of K ∈ (0,1) conditionalon voting for option R. That is:

u1(X ,ω)+u2(X ,ω,vi) = IX=ω +K ∗ Ivi=R.

This voting bias is independent of the state of the world and the decision of the committee. Wedenote this simplified game by Γ = 〈K,PR,α,N,dv〉, with PR, N and dv as defined above.

Given the symmetric structure of the model, a positive payoff of K for voting for R is equivalentto a negative payoff of −K for voting for B. Therefore, the model captures both positive andnegative expressive payoffs. Also, since the signal space is binary, s and m can be characterizedby the number of signals and messages, respectively, of B. We denote the aggregate number ofsignals of B by S# (S# = ∑i Isi=B where Isi=B takes a value of 1 if si = B and 0 otherwise), andthe aggregate number of messages of B by M# (M# = ∑i Imi=B).

3.1 Characterization of Truthful Equilibria

Benchmark results As a benchmark, we first clarify that the existence of an equilibrium withoptimal information aggregation under Unanimity obtains in any game Γ.

Corollary 1 (Truthful Messaging under Unanimity). In any game Γ with dv = Unanimity, anequilibrium exists with truthful messaging (σ(si|si) = 1) and committee-optimal decisions.

This result follows directly from Proposition 2: given that agents optimally aggregate informa-tion in the voting stage, they have no incentive to deviate from truthful communication.

In contrast, we explicitly demonstrate when and in which sense truthful communication andresponsive voting fails to be sustainable in equilibrium under majority voting. We begin bycharacterizing a well-known set of equilibria, existent in most games of information aggre-gation through voting, that do feature truthful communication in the communication stage:regardless of messages, all agents voting R is a mutual best response under Majority. Clearly,any message strategy can be sustained in these “non-responsive equilibria,” including truthfulcommunication.

Proposition 3 (Non-Responsive Equilibrium). In any game Γ with dv = Majority, there isa symmetric equilibrium where all agents truthfully communicate (σ(si|si) = 1) and vote foroption R for any signal, message and profile of messages (τ(si,mi,M#) = 1).

The rationale for Proposition 3 is that when τ(si,mi,M#) = 1 for all i 6= j, then j’s vote cannotbe pivotal, which implies that v j = R is a best response. Given K > 0 there is no non-responsiveequilibrium with τ(si,mi,M#) = 0 under Majority, since if i’s vote is not pivotal, then i has a bestresponse of voting for R and receiving the expressive payoff. In the remainder of this section,we analyze equilibria with Responsive Voting, which we define as equilibria where τ(si,mi,M#)is not constant.

Responsive voting Next, we consider the existence of equilibria with truthful communicationand responsive voting under Majority. In the analysis, three qualitatively similar but technically

15

distinct cases need to be considered. We consider all cases in detail in the appendix, but for easeof exposition here, let us focus on voting behavior that constitutes a mutual best response giventruthful communication under the restrictions that (i) voting behavior conditions only on M#

and (ii) among the set of mutual best responses it maximizes the probability that the committeechooses the option with the majority of private signals. We denote such voting strategies asτM. In the other cases, agents may condition their voting behavior on their individual message,they might simply vote all R for some M#, or they might vote B with low probability, whichyields results that are qualitatively similar for all classes of equilibria. Given our focus here, wesimplify the notation of τ(si,mi,M#) to τM(M#) for the remainder of this subsection.

The following lemma characterizes voting strategies in potential equilibria sustaining truthfulcommunication and responsive voting satisfying this restriction.

Lemma 1 (Voting Stage: Majority). In any game Γ with dv = Majority, for any symmetricequilibrium that exhibits truthful communication, σ(si|si) = 1, and responsive voting strategiesτM = τ∗(M#), there exists S′ ≥ (N +1)/2 such that

• τ∗(M#) = 1 if M# < S′,• τ∗(M#) ∈ (0,1) if M# ≥ S′.

We characterize S′ and τ∗(M#) in the proof of Lemma 1 in the Appendix. Intuitively, a neces-sary condition for agents to vote for B with positive probability is that M# is high enough suchthat the expected common-value payoff of selecting B is larger than the expressive payoff, K.However, this is not a sufficient condition; τ∗(M#)< 1 can only be a symmetric best response ifthe expected common-value payoff of selecting B times the probability of being pivotal, giventhat other agents play τ∗(M#), is equal to the expressive payoff. Thus, S′ is the minimum valueof M# such that this condition is satisfied for some τM(M#) ∈ (0,1).

More generally, Lemma 1 states that when R is committee optimal, all committee members votefor R with probability one in any truthful, responsive equilibrium. However, there is no truthful,responsive equilibrium in which all agents vote for B when B is committee optimal. Due to thefree-rider problem highlighted in Proposition 1—each committee member would prefer that amajority vote for B, but to individually belong to the minority of agents that vote for R—allpotential responsive symmetric equilibria involve mixing in the voting stage. This mixing overvi introduces an aggregate bias toward option R, since there is a positive probability that thecommittee will select R even when B is committee optimal.

Lemma 1 characterizes the potential form of responsive equilibria with truthful communicationunder Majority. However, the result does not imply that such an equilibrium always exists.Based on Lemma 1, we next show that the existence of an equilibrium with truthful commu-nication and responsive voting can be ruled out if the expressive payoff K is above a certainthreshold.

Lemma 2 (Truthful Messaging under Majority). In any game Γ with dv = Majority, there existsK′> 0 such that equilibria exhibiting truthful communication and responsive voting do not existif K > K′.

We explicitly characterize K′ in the proof of Lemma 2 in the Appendix, but essentially, at K′ itis no longer optimal to vote B if there are just (N +1)/2 signals for B. As a result, K′ is fairlylow even in small committees, as will become clear with our experimental design.

16

Lemma 2 shows that the free-rider problem in the voting stage has a knock-on effect on themessaging stage, eliminating equilibria with truthful communication and responsive voting forsufficiently high values of K. The intuition for this result is as follows: Independent of theirown vote, each agent would prefer that the committee select option B when there are moresignals for B (information aggregation). However, as detailed in Lemma 1, given truthful com-munication the expressive payoff biases the committee’s decision toward R, since agents playmixed strategies at the voting game for M# ≥ S′. Deviating from truthful communication andmessaging B following a signal for R allows agents to reduce this bias since the probability thateach agent votes for B is increasing in M#.

Because of this, agents face a tradeoff when considering a deviation to messaging B followinga signal for R: on one hand, this decreases the committee’s voting bias towards R; on theother hand, it may imply that the committee selects B when R has a higher number of signals.This tradeoff implies that truthful messaging and responsive voting can be an equilibrium forlow levels of K, despite the committee’s voting bias towards R. For large enough K, however,deviating to messaging B (and voting R) following a signal for R is individually optimal, therebyeliminating equilibria with truthful communication and responsive voting. We shall refer to thisstrategy, falsely reporting B while personally voting R, as the “free-riding” strategy.

Information aggregation in non-truthful equilibria Finally, to conclude the analysis ofMajority’s ability to aggregate information, we characterize the equilibria that aggregate infor-mation without truthful messaging. Under Unanimity, truthful messaging is a necessary condi-tion for optimal information aggregation since, to reach the optimal decision, each agent needsto condition their vote on the aggregate signal profile. Under Majority, however, informationaggregation, in the sense of selecting the option that is most likely given the set of signals, canbe achieved without communication as long as each agent votes their signal (vi = si; note thatthis voting behavior only aggregates information when all signals are equally informative). Ournext result establishes that an equilibrium with uninformative messaging and “sincere voting”(vi = si) exists under Majority only for comparably low values of K.

Lemma 3 (Sincere Voting). In any game Γ with dv = Majority, there exists an equilibrium withbabbling at the message stage, e.g. σ(R|si) = 1 for all i, and sincere voting, vi = si, at the votingstage if and only if K ≤ K

′′. Moreover, K

′′ ≤ K′, where K′ is the threshold value specified inLemma 2.

Similar to Lemma 2, Lemma 3 shows that sincere voting is only an equilibrium when K issmall, and in particular smaller than K′. Also, note that there are no equilibria with partiallyinformative messaging and sincere voting (vi = si) for K > K

′′.15

Importantly, Lemma 3 also provides insight regarding the necessity of communication for in-formation aggregation through voting with vote-contingent payoffs: given K > K′′, if agentsbabble in the communication stage there is no symmetric τ < 1 that can be sustained in equilib-rium (in any information set) if voting is by Majority. Unanimity is trivially robust also in thisrespect: if agents babble in the communication stage, they can still use the multiple rounds of

15If messaging is partially informative, then there exists some M# such that Pr(ω=B|si =B,M#)≤ Pr(ω=B|si =B),in which case it is not an equilibrium for i with si = B to vote B for all M# since given K > K

′′voting for B is not a

best response when the probability that the state of the world is equal to B is less than or equal to Pr(ω = B|si = B).

17

voting to “communicate” their private information. That is, there exists a non-stationary equi-librium in which agents babble in the communication stage, vote their signal in the first roundof voting, and, if the first round of voting is non-unanimous, in the second round of votingunanimously vote for the option that is revealed to be committee-optimal.

Summary Given that K′ is greater than or equal to K′′, we can summarize the results ofLemma 2 and 3 in the following proposition.

Proposition 4 (Truthful Communication and Information Aggregation under Majority). In anygame Γ with dv = Majority, for K > K′ there are no equilibria with truthful communicationand responsive voting, and no equilibria with information aggregation or optimal informationaggregation.

Proposition 4 allows us to make a more precise comparison between Majority and Unanim-ity rules: relative to Unanimity, where an equilibrium exists with truthful communication andcommittee-optimal information aggregation for any level of the expressive payoff, if the expres-sive payoff is high all responsive equilibria under Majority lack both truthful communicationand information aggregation.16 Our experimental design will exploit this particular observa-tion, but before we go further in this respect, let us address two important robustness concernsand an extension of the baseline model.

3.2 Extensions and Robustness

In order to conveniently discuss robustness and extensions, we continue to restrict our attentionto the simple model with a binary signal space and an expressive payoff, K, for option R. Incontrast to the analysis above, some of our results will be sensitive to the sign of K. Therefore,to facilitate the exposition of the results, we will explicitly consider both negative and positivevote-contingent payoffs; i.e. K ∈ (−1,0)∪ (0,1).

Status-Quo Unanimity rule In our baseline analysis, we consider a unanimity rule that re-quires that all committee members vote unanimously for an option to be chosen. This differsfrom a status-quo unanimity rule, where one option is designated as the status-quo ex ante, andis selected if one or more committee members vote for the status quo.

Definition 5 (Status-quo unanimity). Each player submits a vote vi. The final decision X is Bif all players vote B, and is R otherwise.

In our analysis of the status-quo unanimity rule, we find that it is “robust” to positive (relative)payoffs for voting for the status quo, but faces similar problems as Majority when committeemembers face negative (relative) vote-contingent payoffs.

16Note that Proposition 4 does not detail the properties of equilibria under Majority with informative, but not truthful,communication. We do not characterize all equilibria of the model with partially truthful communication, but inAppendix B, we detail the equilibria of the model under Majority for the parameters we use in the experiment andshow that if K is sufficiently high then the non-responsive equilibrium is the only equilibrium (i.e. X = R) and thatthis condition is satisfied by our “high expressive payoff” treatment.

18

Proposition 5 (Status-Quo Unanimity).

1. In any game Γ with K > 0 and dv = Status-Quo Unanimity, there exists an equilibriumwith truthful messaging and committee-optimal decisions.

2. In any game Γ with K < 0 and dv = Status-Quo Unanimity, there is no symmetric equi-librium with truthful messaging and committee-optimal decisions.

When committee members face a positive payoff for voting for the status quo, then an equilib-rium with truthful communication and committee-optimal voting exists under both non-status-quo and status-quo unanimity rules. For example, if committee members receive a positiveexpressive payoff for voting for R and option B is committee optimal, then all committee mem-bers must forgo the expressive payoff to choose the committee-optimal option. Thus, in the caseof K > 0, a status-quo rule eliminates the collective-action problem that occurs under Majoritywhen voting for B is committee optimal.

However, if committee members face a negative payoff for voting for the status quo, then astatus-quo unanimity suffers from the same free-riding problem as majority. For example,if committee members receive a negative expressive payoff for voting for R and option R iscommittee optimal, then only one committee member must pay the negative expressive payoffunder a status-quo unanimity rule to choose the committee-optimal option—that is, a status-quo unanimity rule functions as sub-majority rule for option R, requiring only a single vote forR to be selected. Therefore, analogous to a majority rule, given truthful communication thereis no symmetric equilibrium in which the committee selects R with probability one whenever Ris committee optimal.

Proposition 5 suggests that a status-quo unanimity rule is only sufficient for institutions wherecommittee members face positive (relative) vote-contingent payoffs for a single option. Forexample, if FDA committee members only receive vote-contingent payoffs when they vote toapprove a bad drug, then a status-quo unanimity rule with a status quo of reject would admit anequilibrium with truthful communication and committee-optimal information aggregation. Insituations where committee member may face vote-contingent payoffs that vary on a case-by-case basis, however, a unanimity rule that requires a unanimous decision is necessary for fullrobustness to vote-contingent payoffs.

Expressive payoffs in the communication stage In certain situations agents may also riskbeing exposed to expressive payoffs based on their communications to the committee. Forexample, if agents receive a negative reputation-based payoff for voting for B, then they mightreceive a similar payoff if it is revealed that they supported B in the communication stage. Inthis extension, we detail when communication payoffs prevent truthful communication underUnanimity.

To fix ideas, consider the following example: in addition to vote-contingent payoffs, assumeagents receive a negative payoff of −K if they vote for B, or if it is publicly revealed that theycommunicated support for B. The probability that the committee’s communication is publiclyrevealed is equal to δ. As in the baseline model, we normalize the expressive payoffs as apositive value for voting/communicating R. The vote-contingent payoffs in the extended model

19

are as follows:

u2(X ,ω,vi,mi) =

δK, if vi = B,mi = RK, if vi = R,0, otherwise.

The following proposition shows that Unanimity is robust to communication-based payoffs ifthe risk of communication becoming public is of positive but comparably small relevance nextto one’s final vote.

Proposition 6 (Truthful Messaging with Communication Payoffs). In any game Γ with K > 0,dv = Unanimity, and B is committee-optimal given S# = N, there exists a δ′ such that if δ < δ′,an equilibrium of the game exists with truthful messaging and committee-optimal decisions.

Note that even under Unanimity, agents’ messages are not always pivotal for the committeedecision. Therefore, agents have an incentive to report R given a signal of B, analogous tothe incentive to vote R under Majority. In contrast to the voting stage, however, the probabil-ity of a message being pivotal is strictly positive even when other agents play pure (truthful)strategies. Therefore, truthful messaging can be supported as an equilibrium strategy even withcommunication-based payoffs as long as δ is not large.

Expressive payoffs in the coordination stage Finally, we consider the related case whereagents are exposed to expressive payoffs in the coordination stage, when they coordinate on thefinal voting decision (straw poll), and detail the robustness of Unanimity to this extension ofthe model. To make the contrast as stark as possible, we consider the case where agents receivea payoff of K for any round in which vt

i = R. That is, given a unanimous decision is reached inround T , agents receive the following vote-contingent payoffs:

u2(X ,ω,vti) = K×

T

∑t=1

Ivti=R.

The following proposition details the partial robustness of Unanimity rule to expressive payoffsfor voting in the straw poll.

Proposition 7 (Truthful Messaging with Coordination Payoffs).

1. In any game Γ with K < 0 and dv = Unanimity, an equilibrium with truthful messagingand committee-optimal decisions exists where the committee reaches a decision in thefirst round of voting in the straw poll.

2. In any game Γ with K > 0 and dv = Unanimity, there is no stationary equilibriumwith truthful messaging where the committee selects B with probability one when B iscommittee-optimal.

Intuitively, when expressive payoffs are negative, then agents have an incentive to reach a de-cision in the first round of voting to avoid additional expressive payoffs that may accrue in anyadditional rounds of voting. When expressive payoffs are positive, however, then agents havean incentive to vote for as many rounds as possible, accruing positive expressive payoffs eachtime they vote for R. Therefore, while vt

i = R is a best response given M# < (N +1)/2 (deviat-ing to vi = B is not individually optimal) when expressive payoffs are positive, if all agents vote

20

for B with certainty for some M#, then agent i has a best response to deviate to vti = R, since

voting in the straw poll will proceed indefinitely and i will receive K in each round.

4 Experiment

While the theoretical insights provided by Propositions 1, 2 and 4 are suggestive, they do notprovide a fully satisfactory answer as to which decision rule is optimal in a setting with vote-contingent payoffs. For example, both decision rules admit multiple equilibria, even underthe assumption of symmetric strategies, leading to ambiguity regarding the theoretical predic-tions. Moreover, there are well-documented behavioral factors that may impact the theoreticalpredictions: a single agent voting naively is enough to destroy the efficiency of informationaggregation under unanimity rules, and a preference for truth-telling (lying aversion) may mit-igate the negative predictions for Majority. Additionally, there are more subtle ambiguities ifwe allow for asymmetric strategies: if (N−1)/2 agents play a fixed strategy τ(M#) = 1 for anyM#, then the remaining (N +1)/2 agents face an effective decision rule of unanimity, allowingfor optimal information aggregation.

This indeterminacy of the theoretical predictions leaves scope for further exploration. In sucha setting, where the theoretical predictions are suggestive but not definitive, laboratory exper-iments can be a useful tool to generate insights on observed behavior in an analogous choiceenvironment. In this section, we explain how our experimental design may help us to detailthe differences in communication and voting under Majority and Unanimity, and to explorethe mechanisms behind the observed differences in aggregate behavior under the two decisionrules.

4.1 Experimental Design

The experiment closely implements our model of voting with expressive payoffs and binary sig-nals, using a 2×2 treatment design with “High” and “Low” expressive payoffs under Majorityand Unanimity. The experimental implementation closely follows the related experiments ofGuarnaschelli et al. (2000) and Goeree and Yariv (2011). In particular, we use neutral lan-guage, communicate probabilities and signals to subjects using balls drawn from urns, andprovide feedback about the actual state of world and composition of payoffs after each round.A detailed description follows and a translation of the instructions and a screenshot are providedas supplementary material. The experiments were conducted at the WZB/TU experimental lab-oratory in Berlin in May, June and November of 2016. Subjects were recruited using ORSEE(Greiner, 2015) and the experiment was programmed in Z-Tree (Fischbacher, 2007).

The four treatments are summarized in Table 1. The sum of the expressive payoff K, whichcommittee members get after voting R, and common-value payoff Pc, which committee mem-bers get after a committee decision that agrees with the state of the world, is always equal to to50 points. In the treatments with “low” expressive payoffs, we set K = 10 and Pc = 40, and inthe treatments with “high” expressive payoffs, we set K = 15 and Pc = 35. The payoffs werecalibrated such that K > K′ in both the Low and High treatments, and we discuss the theoret-

21

ical predictions in detail in the following subsection. In both cases, we conduct sessions withboth unanimity and majority voting. The precision of each subject’s signal was constant acrosstreatments and equal to α = 0.6. Subjects were paid according to the sum of points accumu-lated across all 50 games, and one point corresponded to one Euro cent in all treatments.17 Theexperiment lasted between 75 and 105 minutes and subjects earned between 19 and 22 Euroson average across sessions.

Table 1: Overview of experimental treatments

Label Decision rule Pc K #Subjects #Sessions #Games

Majority-Low Majority 40 10 48 4 50Majority-High Majority 35 15 45 4 50Unanimity-Low Unanimity 40 10 45 4 50Unanimity-High Unanimity 35 15 48 4 50

For each treatment, we ran four sessions with either 9 (two sessions) or 12 (fourteen sessions)participants. In all cases, two sessions were run simultaneously to increase anonymity. Uponarrival at the laboratory, subjects were seated randomly. An experimental assistant then handedout printed versions of the instructions and read the instructions out loud. Subsequently, sub-jects filled in a computerized control questionnaire verifying their understanding of the instruc-tions, and the experiment did not start until all subjects had answered all questions correctly.The subjects then played 50 voting games in committees of size three (N = 3), with randomrematching after each game (see Figure 3 in Appendix C for a composite screenshot). Aftereach game, the subjects received feedback on the state, payoffs, aggregate behavior and theaggregate signal profile. Under Majority, the timing of each round was as follows.

Majority Observe private signal si ∈ R,B. Send a public message to their group mi ∈ R,B.Observe message profile. Submit vote vi ∈ R,B. Observe state, votes, outcome, andpayoffs for this game.

Under Unanimity, the timing of each round was identical to Majority aside from the votingstage. We did not allow for an infinite number of rounds of voting as in the theoretical model.Instead, subjects were given three chances to reach a unanimous decision, after which all sub-jects were assigned a default vote of R. This decision rule replicates the most important featureof Unanimity—that the final vote profile is uniform for any committee decision—and insuresthat a decision is reached in a reasonable time frame. This “default option” is the conservativechoice in relation to our experimental hypotheses, as it ensures that the frequency of commit-tee decisions equal to the non-expressive option under Unanimity is not driven by the defaultoption. Additionally, analogous to the Unanimity rule, this game structure also admits an equi-librium in which all agents truthfully communicate their private messages and uniformly vote

17This approach is known as the “pay all” method. Charness et al. (2016) review the evidence for and against “payall” in contrast to the approach to “pay one” round and conclude that “in general, . . . both are useful methods”,suggesting that practical considerations such as potential bankruptcy (as in auctions, which would favor “pay one”)should be considered when designing experiments. The practical considerations that let us to choose to “pay all”are the comparability with earlier experiments (such as Goeree and Yariv, 2011) and the potential introduction ofbackground risk by “pay one”, which would be a potential confound in our risky environment.

22

for the committee-optimal option.18

Unanimity The only difference to Majority is in the voting stage: Submit vote v1i ∈ R,B. If

vote is unanimous proceed to Outcome Stage. Otherwise, again submit a vote v2i ∈R,B.

If vote is unanimous proceed to Outcome Stage. Otherwise, submit a final vote v3i R,B.

If vote is not unanimous, all subjects are assigned the vote vi = R. Proceed to Outcomestage.

The format of allowing for multiple straw polls is similar to the procedure used in many jury de-liberations. It is possible, however, that subjects use the multiple rounds of voting as additionalcommunication and that this richer strategic space explains the higher degree of informationaggregation we observe under Unanimity. We address this concern in a robustness sectionin the Appendix, and show that the higher degree of information aggregation observed underUnanimity is almost entirely driven by groups that reach a decision in the first round of voting,see Figure 7, which suggests that additional rounds of voting under Unanimity is not drivingour results. Upon completion of the experiment, subjects left the laboratory and were paidindividually in a separate room by an experimental assistant.

4.2 Theoretical Predictions and Experimental Research Questions

Lemma 2 shows that for a wide range of expressive payoffs, truthful communication is ruledout in responsive equilibria under Majority voting.19 The expressive payoffs induced in theexperiment, 10/40 and 15/35, are comparably small in the sense that they do not dominate thecommon-value payoffs from matching the committee decision to the state, but are large enoughto satisfy K ≥ K′ as defined in Lemma 2. That is, in both Majority treatments, expressivepayoffs are large enough to suppress truthful communication in equilibrium, which yields thefollowing hypothesis.

Hypothesis 1. Messages are more likely to be truthful under Unanimity than under Majorityafter R signals but not after B signals.

Lemma 2 shows that Majority may induce asymmetric misreporting since agents best-respondto truthful reporting by sending truthful messages given a signal of B, and misreport given asignal of R to induce their co-players to vote for B. Additionally, Lemma 1 shows that agentswho receive a signal of R and misreport will vote R, given that their co-players vote for B with

18This result follows as a corollary of Proposition 2—since the final profile of votes is homogenous in all decisions,any deviation from truthful communication and committee-optimal voting leaves agents weakly worse off. Ad-ditionally, since B is optimal only after three signals B for both high- and low-expressive payoffs, this decisionrule admits an equilibrium where agents optimally aggregate information by babbling in the message stage andvoting sincerely. As we show below, however, a relatively high number of subjects communicate truthfully underunanimity and respond to messages of other subjects in their voting behavior, and 66 percent of groups reach aunanimous decision in the first round of voting.

19To reiterate, in responsive equilibria, committee members do not ignore the information contained in the messages.There always exist non-responsive equilibria with truthful communication under Majority, namely equilibria wheresubjects always vote R regardless of messages, where unilateral attempts to respond to the messages are payoffirrelevant, see Proposition 3.

23

a higher probability. We refer to this as the free-rider problem of the Majority rule. The aboveanalysis formally established that free-riding is optimal in response to players who are truthfuland expect others to be truthful, which clarifies the general intuition underlying free-riding.

While free-riding is optimal in response to players who are truthful and expect others to betruthful, it is not necessarily sustained in an equilibrium of the game. Since evidence fromgames of information aggregation in committees shows that many subjects communicate truth-fully even when they have a clear incentive to misreport their signal (see Goeree and Yariv,2011, and Abeler et al., 2018, for an analysis of lying aversion in sender-receiver games moregenerally), we provide two sets of equilibrium predictions. In Appendix B, first we explore theequilibria of the games used in the experiment assuming all agents are “rational,” and secondlywe consider the optimal behavior of rational agents if a fraction γ∈ [0,1] of all agents are truth-ful and expect others to be truthful.20 This “behavioral model” is equal to the rational modelwhen γ = 0, while all agents play truthful strategies when γ = 1.

When γ = 0, we show analytically that for the high-expressive payoff, among the plausiblecandidates, no rational equilibria with responsive voting exist (moreover, a numerical search didnot reveal any equilibria with responsive voting). For the low-expressive payoff, we identify aresponsive equilibrium where, contrary to the prediction made above, only agents who messageB vote for B with positive probability and agents with a signal of B mix between messaging Rand B. However, when we allow for values of γ sufficiently high—for our experimental games,the threshold condition is γ ≥ 0.13—then the rational agents are predicted to play free-ridingstrategies in equilibrium in both treatments. We hypothesize that this condition on γ will besatisfied given the evidence from the earlier study mentioned above. Indeed, we estimate thatroughly 45 percent of subjects play truthful strategies in the Majority treatments, underliningthe relevance of the predictions from the behavioral model.

As a result of the free-rider problem, Proposition 4 predicts that Unanimity will outperformMajority in terms of optimal information aggregation. Additionally, based on the behavioralmodel we predict that information aggregation will disproportionately fail when the aggregateprofile of signals indicates that the committee should select B: If a majority of subjects receiveB signals, then the free-rider problem implies that R will be selected with positive probability.If a majority of subjects receive R signals, however, then the committee will select R with highprobability since subjects playing a free-rider strategy will vote R regardless of the messages.Therefore, our model suggests the following hypothesis.

Hypothesis 2. Information is aggregated more efficiently under Unanimity when the committee-optimal decision is B but not when the committee-optimal decision is R.

Conditional on resolving Hypotheses 1 and 2 as predicted, theoretically, the channel for theincreased level of information aggregation under Unanimity could be twofold. On the onehand, committee members may anticipate and account for the more truthful messages underUnanimity. On the other hand, even if subjects hold the belief that their co-players’ messagesare truthful, Lemma 1 shows that Majority induces a coordination problem in the voting stagethat may still prevent optimal information aggregation.

20Naturally, the own type is private information in this model. Loosely, our behavioral model thus resembles earliermodels; for example Kreps et al. (1982) who analyze a behavioral model of cooperation in finitely repeated games.

24

Therefore, we continue our analysis by identifying the mechanisms behind any relative decreasein information aggregation under Majority. First, we consider the question of whether subjectsaccount for the average level of truthful reporting in their voting decisions.

Hypothesis 3. Subjects anticipate and respond to more truthful messages under Unanimity.

Second, Lemmas 1 and 2 predict that agents will best-respond to truthful reporting by free-riding, i.e. by misreporting R signals to induce B votes of their co-players while personallyvoting for R. This prediction, confirmed also by the the aforementioned equilibrium analysisin Appendix B.3, provides us with an opportunity to verify the causal mechanism predicted byour model.

Hypothesis 4. A significant share of subjects strategically misreport under Majority to freeride.

If the findings of the experiment negate this prediction, then subjects may be falsely reportingtheir signals for reasons other than what we predict, and hence our model would be falsified.

5 Analysis of the experiment

We address the four experimental hypotheses successively. Table 2 provides a first overview ofthe experimental results, delineating the truthfulness of messages and the level of informationaggregation by decision rule, expressive payoffs and signal.

Table 2: Average treatment effects in relation to Questions 1 and 2

Expressive payoff Low HighDecision Rule Majority Unanimity Majority Unanimity

Truthful message if R signal 90% 96%∗∗∗ 79% 94%∗∗∗

Truthful message if B signal 84% 86% 87% 86%

Committee-optimal decision R 80% 79% 88% 86%Committee-optimal decision B 61% 83%∗∗∗ 60% 83%∗∗∗

Mann-Whitney-Wilcoxon test (Unanimity = Majority); ∗ p < 0.1, ∗∗ p < 0.05, ∗∗∗ p < 0.01

5.1 Are messages more truthful under Unanimity?

The upper two lines of Table 2, and similarly Figure 1, provide information on the truthfulnessof communication, disaggregated by treatment and signal as required to evaluate Hypothesis 1.The asymmetric prediction is satisfied for both signal R and signal B: Given a signal of R, inboth the Low and High expressive payoff conditions, messages are significantly more truthfulunder Unanimity than under Majority. That is, with a signal of R, misreporting increases from4% to 10% given Low expressive payoffs, and from 6% to 21% given High expressive payoffs.

25

.8.8

5.9

.95

1A

vera

ge M

essa

ge T

ruth

ful

Red BlueSignal

Majority Unanimity

Truthful Messaging, Low Expressive Payoff

.75

.8.8

5.9

.95

Ave

rage

Mes

sage

Tru

thfu

l

Red BlueSignal

Majority Unanimity

Truthful Messaging, High Expressive Payoff

Figure 1: Average levels of truthful messaging (mi = si) by signal and treatment.

In contrast, truthful reporting given a signal of B is very stable across all conditions: Thedifferences between Unanimity and Majority are very small (at most two percentage points) andfar from being significant, confirming the second part of Hypothesis 1. Across all treatments,the average rate of truthful reporting, roughly 85 percent, is lower than might be expected giventhe results of Guarnaschelli et al. (2000) (their experiment considers homogeneous payoffs,but is otherwise comparable). Focusing only on Unanimity, however, the relative frequency oftruthful messages is comparable with Guarnaschelli et al. (2000).

Result 1. As predicted (Hypothesis 1), messages are more truthful in Unanimity after R signalsand equally truthful after B signals.

A related point apparent in Figure 1 is worthy of mention. Under Majority, the level of misre-porting given a signal of R more than doubles for the High vote-contingent payoff treatment,from 10% to 21%. This finding is consistent with the quantitative increase in the incentive tostrategically misreport given a signal of R. In contrast, the average level of truthful reporting bysignal is stable across the Low and High treatments under Unanimity. This finding corroboratesResult 1 and suggests that, under Majority, the level of truthful communication is sensitive tothe size of the incentive to engage in strategic misreporting. Section C in the Appendix furthershows that the results are highly robust to experience.

5.2 Is information aggregated more efficiently under Unanimity?

While truthful communication is a necessary condition for the committee to behave optimally,the most pertinent comparison between Majority and Unanimity is the ability of the committeeto efficiently aggregate the private information of its members. Figure 2 shows the committeedecision as a function of the aggregate profile of signals, and the lower two lines in Table 2provide the relative frequencies of committee-optimal decisions, disaggregated by whether Ror B is optimal—across all conditions examined here, the committee-optimal decision is B ifand only if all subjects have received B signals.21 As detailed in Hypothesis 2, our prediction

21In the low-expressive payoff treatment, aggregate utility is maximized if the committee selects B given two signalsof B and only two committee members vote for B.

26

0.2

.4.6

.81

Mea

n O

utco

me

Blu

e

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, Low Expressive Payoff

0.2

.4.6

.81

Mea

n O

utco

me

Blu

e

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, High Expressive Payoff

Figure 2: Average levels of outcome B as a function of the aggregate profile of signals (notmessages) by treatment.

is that there is no difference between Unanimity and Majority if the expressive option, R, iscommittee optimal, but that under Unanimity the committee will be more likely to select thenon-expressive option, B, when it is optimal.

The experimental results are very sharp. If R is committee optimal, there is virtually no dif-ference in the probability that the committee selects R between Unanimity and Majority: 80%versus 79% for Low, and 88% versus 86% for High expressive payoffs. If B is committee op-timal, the difference is large and highly significant: 22 percentage points (61% versus 83%)for Low expressive payoffs, and 23 percentage points (60% versus 83%) for High expressivepayoffs. In all, the probability that the committee correctly aggregates information given threesignals for B increases by over a third under Unanimity.22

Result 2. As predicted (Hypothesis 2), the committee is significantly more likely to select theoptimal option under Unanimity if the non-expressive option, B, is committee optimal, and thedifference between the two decision rules is negligible if R is optimal.

5.3 Do subjects anticipate more truthful messages under Unanimity?

Having established that Unanimity results in more truthful messaging and increased informa-tion aggregation (Hypotheses 1 and 2) we now turn to the more subtle questions regardingmechanism (Questions 3 and 4). For a first pass at exploring the effect of messages under thedifferent decision rules, we estimate a discrete choice model that considers each individual’svoting decision as a function of the voting rule, and the information known by the subjects atthe time they take their voting decision. Table 3 summarizes the estimation.

The regression results indicate that subjects respond to both an own signal of B and to co-player’s messages for B by voting for B with a higher probability. Moreover, under Unanimity,

22The difference between the two decision rules is also slightly larger in the 2nd half of the experiment (see SectionC in the supplementary appendix). Additionally, note that subjects are more likely to select B given two signals ofB under Unanimity, although this difference is attenuated in the second half of the experiment.

27

the impact of a message for B from a co-player has a similar impact on voting behavior as anown signal for B. Under Majority, however, subjects are significantly less likely to respondto co-players’ messages of B, relative to their own signal (see the negative coefficient on theinteraction of “Majority” and “Other’s messages”). This strongly suggests that the higher pro-portion of misreporting observed under Majority negatively impacts the committee’s ability toefficiently aggregate the information of its members.

Table 3: Probit estimations to explain individual votes/opinions for B

Vote/Opinion B High

Own signal B 0.282(0.0174)

∗∗∗

Number of others’ messages B 0.298(0.0112)

∗∗∗

Majority 0.0624(0.0252)

∗

Majority × own signal B 0.00908(0.0361)

Majority × others’ messages B −0.109(0.0218)

∗∗∗

Constant −0.171(0.00893)

∗∗∗

N 4650Subject-level clustered standard errors in parentheses.∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

This direct comparison between the decision rules is suggestive of the predicted Hypothesis3. However, we can make the analysis more precise by controlling for differences in expectedpayoffs from voting B under the two decision rules. That is, while we see an increased re-sponsiveness to co-players’ messages under Unanimity, the choice environment in the votingstage is not directly comparable to Majority. Therefore, we supplement the finding of the dis-crete choice model by investigating the impact of co-player’s messages on the subjects’ impliedbeliefs, taking into account the actual differences in expected payoffs.

In line with Hypothesis 3, our hypothesis is that subjects’ beliefs are more responsive to co-players’ messages under Unanimity. We test this hypothesis by estimating a simple structuralequation model of the decision making process.23 The equation system directly implements thestrategic game played by the subjects. First, a subject’s belief about the true state ω ∈ R,B isa function of the own signal s ∈ R,B and the opponents’ messages m2,m3 ∈ R,B,

Pr(ω = R|s,m2,m3) =1

1+ expα1(Is=B−0.5)+α2(Im2=B + Im3=B−1)(1)

with belief parameters α1,α2. Here we use the indicators Is=B, Im2=B, Im3=B to indicate whetherthe own signal (s) or the opponents’ messages (m2,m3) are equal to B (value 1) or not (value 0).

23An arguably simpler approach would be to plainly ask subjects about their beliefs, but in the context of beliefsunderlying strategic decisions, the elicited beliefs have been found to be incompatible with the chosen actions(Costa-Gomes and Weizsacker, 2008). Even without such obstacles, robustly incentive-compatible elicitation ofbeliefs, prior and after revelation of messages, is not simple either and may distract or appear obtrusive to subjects.

28

Second, a subject’s belief about the voting outcome X ∈ R,B, conditional on the own votev ∈ R,B and the number of B messages, is24

Pr(X = R|vi) =1

1+ expβ1 · Iv=B +β2|m1+m2+m3= 1−Pr(X = B|vi). (2)

Here, β1 captures the weight of the own vote, and β2|· captures the expectation about the op-ponents’ votes as a function of the message profile. Specifically, β2|· is a vector of four values,where β2|0 denotes the expectation in the case where there are zero B messages, and β2|1, β2|2,β2|3 concern the cases of one, two, or three B messages. In conjunction, these two beliefs definethe probability that the voting outcome X is equal to the true state ω,

Pr(X = ω|vi) = Pr(ω = R) ·Pr(X = R |v)+Pr(ω = B) ·Pr(X = B |v).

Finally, using Pc and K to denote the payoff from the voting outcome being correct (X = ω)and the expressive payoff from voting R, respectively, voting for R has probability

Pr(v = R) =1

1+ exp−λ ·Pc · (Pr(X = ω|vi = R)−Pr(X = ω|vi = B))−λ ·K,

allowing for logistic errors (with precision parameter λ ≥ 0). Note that this model is fairlygeneral. Depending on how the belief parameters (α1,α2,β1,β2|·) relate to their empiricalcounterparts, the model is compatible with (ir)rational expectations, overshooting in Bayesianupdating, cursed beliefs and level-k beliefs. The empirical counterparts (α1, α2, β1, β2|·) can beestimated simply by logit regressions. The rational signal and message weights α1 and α2 areestimated by regressing the true state of the world ω on the signal and messages, using Eq. (1),and the rational outcome weights β1 and β2|· are estimated by regressing the probability thatthe correct decision is made (X = ω) on the own vote and the message profile, using Eq. (2).

We control for expected payoffs as observed in the experiment, while still allowing for twoforms of behavioral biases, by requiring (1) the belief weights α1,α2 to be jointly proportionalto their rational expectation counterparts, which allows for overshooting (base rate fallacy) orconservative belief formation as observed in many experiments, and (2) the vote weights β2|· tobe jointly proportional to their empirical counterparts, which allows for over- or underestimat-ing the predictability of the co-players’ votes. In the extreme case, β2|· = 0, subjects believetheir co-players are perfectly unpredictable (level-1, Stahl and Wilson, 1995). If 0 < β2|· < β2|·,subjects underestimate the predictability of others as observed by Weizsacker (2003), Goereeand Holt (2004) and Eyster and Rabin (2005), and if β2|· = β2|·, subjects hold rational expec-tations of the mapping from their co-players’ messages to votes. We report on two robustnesschecks invoking rational expectations in either dimension below, but the results are very robustin general.

We find that subjects overshoot in belief formation, given signals and messages, and thereforehold rather strong beliefs. To see this effect as clearly as possible, it is best to look at caseswhere the co-players’ messages contradict the own signal. That is, we look at beliefs aboutthe true state ω after a private B signal and two R messages of co-players, and after a privateR signal and two B messages of co-players. Given the above notation, our Hypothesis 3 that

24Slightly abusing notation, we use m1 +m2 +m3 as shortcut for Im1=B + Im2=B + Im3=B.

29

Table 4: Structural equation analysis of beliefs (ω = Red) as function of signals and messages

Empirical Baseline Rational 1 Rational 2

First halves of sessionsBeliefs after R signal and two B messages from othersPrMaj(ω = R|s = R,m2 = m3 = B) 0.476

(0.033)0.407(0.011)

0.478(0)

0.394(0.012)

PrUna(ω = R|s = R,m2 = m3 = B) 0.452(0.031)

0.328−−(0.018)

0.461−−(0)

0.361−(0.011)

Beliefs after B signal and zero B messages from othersPrMaj(ω = R|s = B,m2 = m3 = R) 0.505

(0.03)0.593(0.011)

0.522(0)

0.606(0.012)

PrUna(ω = R|s = B,m2 = m3 = R) 0.559(0.029)

0.672++

(0.018)0.539++

(0)0.639+(0.011)

Log-Likelihood −2021.5 −2280.2 −2312.4

Robustness checksRational state beliefsRational voting beliefs

Second halves of sessionsBeliefs after R signal and two B messages from othersPrMaj(ω = R|s = R,m2 = m3 = B) 0.435

(0.029)0.397(0.012)

0.465(0)

0.378(0.013)

PrUna(ω = R|s = R,m2 = m3 = B) 0.37(0.035)

0.321−−(0.016)

0.422−−(0)

0.336−−(0.009)

Beliefs after B signal and zero B messages from othersPrMaj(ω = R|s = B,m2 = m3 = R) 0.527

(0.031)0.603(0.012)

0.535(0)

0.622(0.013)

PrUna(ω = R|s = B,m2 = m3 = R) 0.551(0.027)

0.679++

(0.016)0.578++

(0)0.664++

(0.009)

Log-Likelihood −1850.2 −2019.6 −2075.5

Robustness checksRational state beliefsRational voting beliefs

Note: In the baseline model we allow for mistakes in Bayesian updating when forming state beliefs and votingbeliefs. That is, the respective belief parameters (α1,α2) and β2|· are allowed to be arbitrarily scaled vectors oftheir rational expectation counterparts (α1, α2) and β2|· as estimated from logistic regressions. The robustnesschecks enforce rational expectations by equating the respective belief parameters with their rational expectationcounterparts. The respective likelihoods are significantly worse than that of the baseline model (showing thatsubjects do actually not hold rational expectations), but the main result that implicit beliefs differ between majorityand unanimity treatments is robust nonetheless. As before, significance of differences between majority andunanimity estimates is indicated by plus and minus signs (++/−− at p < .05 and +/− at p < .1 using bootstrappedp-values), next to the unanimity treatment estimates. All standard errors are bootstrapped.

30

beliefs react more strongly to the co-players’ messages under Unanimity, since messages aremore truthful in this case, amounts to

PrMaj(ω = R|s = B,m2 = m3 = R)< PrUna(ω = R|s = B,m2 = m3 = R),PrMaj(ω = R|s = R,m2 = m3 = B)> PrUna(ω = R|s = R,m2 = m3 = B).

Based on the estimates of the structural equation models, namely α1,α2 in conjunction with Eq.(1), these beliefs can be computed straightforwardly. Since the beliefs are based on estimatesof α1,α2, we bootstrap their distributions to test our hypothesis (resampling at the subjectlevel to account for the panel character of the data, stratifying to acknowledge the treatmentstructure). Parameters are estimated by maximum likelihood25 and both the standard errors ofthe parameters as well as the p-values of the null hypotheses are also bootstrapped.

The results are reported in Table 4. First, looking at the empirically true probabilities, wecan see that in all cases, subjects should tend to follow the opponents’ messages when bothare the same despite contradicting the own signal. For example, in the second halves of thesessions, after an R signal and two B messages from the opponents, the empirical probabilitiesthat the state is R are 43.5% and 37% under Majority and Unanimity, respectively. This showsthat messages should be given weight—and more so under Unanimity treatments than underMajority, as the empirical probabilities deviate relatively more from 50-50 under Unanimity.The remaining columns of Table 4 show that, in all cases, subjects’ beliefs indeed deviatemore from 50-50, in the direction of the messages, in unanimity treatments than in majoritytreatments. The differences are significant, obtain robustly in both the first and the secondhalves of the sessions, and the robustness checks assuming rational expectations forming eitherstate or outcome beliefs confirm the result. Based on this, we conclude that subjects anticipateand account for the higher probability of truthful messages under Unanimity.

Result 3. Subjects’ beliefs react more strongly to co-players’ messages under Unanimity, show-ing that they respond to the increased truthfulness of messages.

5.4 Do subjects strategically misreport to free-ride?

Lastly, we explore the theoretical prediction that subjects will best-respond to truth-telling bymisreporting and free-riding. For suggestive evidence regarding free-riding, it is instructive toconsider the average voting strategy of subjects the case of three messages for B (M# = 3). Inthis case, for the Majority/High expressive payoff treatment where misreporting is the mostcommon, subjects who misreport their signal vote for B just 16 percent of the time, relative to69 percent for subjects who sent a truthful message of B (16 percent is also much lower thanthe analogous rate under Unanimity/High, which is 54 percent; see Table 5 in the appendix).This low rate of voting for B is consistent with the free-riding strategy of misreporting B andthen voting for R.

To identify whether Majority causes subjects to play the “free-riding” strategy, however, weneed to link messages and voting at the individual level. Establishing this link is critical, asfree-riding is not the only conceivable reason to misreport signals. Another reason is to attempt

25To maximize the likelihood, we first use the gradient-free NEWUOA approach (Powell, 2006), which is compara-bly robust (Rios and Sahinidis, 2013), and secondly a Newton-Raphson algorithm to ensure convergence.

31

persuasion. For example, while our model assumes that agents are risk neutral, if subjectshave heterogenous risk preferences, they may hold different preferences regarding the optimaloption conditional on a given set of signals. Therefore, subjects who are, say, less risk aversethan average may choose to misreport R-signals to increase the probability that the committeechooses B given two signals for B. The difference to free-riding is that persuasive misreportingis followed up by also voting for the misreported option (in case opponents sent two messagesfor B), while free-riders will vote R regardless of their opponents’ messages.

Overall, it is easy to think of at least five classes of individual strategies, summarized in Ta-ble 5a. In addition to the three misreporting strategies outlined above, “strategic Red/Blue”as persuasive misreporting and “free-riding” as strategic misreporting, we also allow for sub-jects who are honest in the messaging stage and believe that other agents message honestly(“honest/naive”), which is our equilibrium prediction for Unanimity, and subjects with noisybehavior as a residual family to collect the players that do not fit into either of the other fourclasses (“noisy”). Such apparently noisy behavior may result from either misunderstanding thegame or, more likely, from playing inconsistently over the course of the session. The inclusionof the noisy type thus serves as a robustness check of both the statistical classification and ofthe adequacy of the experimental set-up more generally. Our objective will be to evaluate ourHypothesis 4 that subjects more frequently use free-riding strategies in majority treatments andin turn honest/naive strategies in unanimity treatments.

The voting strategies we assign to the honest/naive type follow from the theoretical predictionsof Lemma 1 and in all cases, we allow for noise. Specifically, the honest type will vote for Rgiven two or more messages/signals for R. With two messages/signals for B, the honest typewill vote for B with an intermediate probability, and with three messages/signals for B they willvote for B with a high probability. The messaging and voting strategies of the strategic typesare then assigned relative to the honest type: The strategic Red type is more likely to messageand vote R, while the strategic Blue type is more likely to message and vote B. The free-ridertype, on the other hand, is more likely to message B and vote R.26 The detailed definitionsare provided in Table 5a. We allow for πLie,πLow,πMedium,πHigh to be free parameters in theestimation to avoid an inadequate specification.

A statistically efficient approach to determine the latent weights of the strategy classes is finitemixture modeling (MacLachlan and Peel, 2000). Such mixture models have been commonlyused in behavioral analyses since Stahl and Wilson (1994, 1995), and have more recently beenused, for example, to study individual heterogeneity in dictator games (Cappelen et al., 2007)and individual behavior in repeated games (Dal Bo and Frechette, 2011; Fudenberg et al., 2012;Breitmoser, 2015). Finite mixture modeling is a general approach allowing for probabilistic as-signment of subjects to strategy classes, which resolves a number of concerns with deterministicassignments,27 but is otherwise comparable to cluster analyses.28

26While the relative comparisons follow from the theory, the exact division into the strategy classes was calibratedusing the aggregate voting strategies reported in Table 6 in the Appendix.

27Deterministic approaches that assign each subject to a strategy class based on some distance measure are sensitiveto the distant measure chosen, are ambivalent near the boundaries of each class, and do not reflect the degree of(un)certainty about a subject’s classification. Furthermore, deterministic classification requires the distances to bereliably measured. In our case, however, they would be based on only few observations per information set.

28In most behavioral cluster analyses, each data point (subject) is represented by vectors with few elements. In suchcases, we can plot the individual estimates and “mark” the cluster areas. This approach is inadequate here, aseach subject is characterized by choices in many different information sets (namely, fourteen) with comparably

32

Table 5: Do subjects play more honest strategies in unanimity voting? Results of the mixture analysis

(a) Definition of the classes of strategies

Messages Votingµ(A) µ(B) π(A,A,0) π(B,A,0) π(A,A,1) π(A,B,1) π(B,B,1) π(B,A,1) π(A,A,2) π(A,B,2) π(B,B,2) π(B,A,2) π(A,B,3) π(B,B,3)

Noise .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5 .5Honest 1 0 1 1 1 1 1 1 πMed 1 πMed πLow πMed πLowStratRed 1 πLie 1 1 1 1 1 1 1 1 1 πMed 1 πMedStratBlue 1−πLie 0 1 1 1 1 1 πMed πMed 1 πMed πLow πMed πLowFreeride 1−πLie 0 1 1 1 1 1 πHigh πHigh 1 πHigh πMed πHigh πMed

Note: µ(s) is the probability of sending message R given the signal s ∈ R,B. π(s,m,M) is the probability of voting R as a function of one’s signal s, message m, and the number M# of B messages overall (i.e. inaggregate over all players). The parameters (πLie,πLow,πMed,πHigh) allow adaptation to subjects’ behavior, with the theoretical ex-ante hypothesis πLow < πMed < πHigh.

(b) Strategy weights and parameters across treatments (bootstrapped standard errors in parentheses)

Strategy weights in population Strategy parametersNoise Honest StratRed StratBlue FreeRide ε πLie πHigh πMed πLow ICL-BIC

All games per sessionMajority 35 15 0.12

(0.05)0.44(0.11)

0.19(0.07)

0(0.01)

0.26(0.09)

0.04(0)

0.42(0.11)

0.73(0.06)

0.35(0.04)

0.08(0.03)

6368.79

Majority 40 10 0.07(0.04)

0.45(0.12)

0.25(0.08)

0.03(0.03)

0.2(0.07)

Unanimity 35 15 0.07(0.04)

0.78(0.06)

++ 0.11(0.05)

0.04(0.03)

0(0)

−−

Unanimity 40 10 0.04(0.03)

0.75(0.07)

++ 0.18(0.06)

0.02(0.03)

0(0.02)

−−

Majority 0.09(0.03)

0.44(0.1)

0.22(0.06)

0.01(0.02)

0.23(0.06)

0.04(0)

0.41(0.11)

0.72(0.05)

0.35(0.04)

0.08(0.03)

6345.24

Unanimity 0.06(0.03)

0.77(0.05)

++ 0.15(0.04)

0.03(0.02)

0(0.01)

−−

Robustness check 1: 1st halves per sessionMajority 0.13

(0.04)0.59(0.11)

0.12(0.06)

0.05(0.04)

0.12(0.08)

0.04(0)

0.61(0.13)

0.79(0.13)

0.36(0.04)

0.14(0.05)

3304.43

Unanimity 0.1(0.03)

0.78(0.04)

++ 0.12(0.03)

0(0.01)

0(0)

−−

Robustness check 2: 2nd halves per sessionMajority 0.1

(0.03)0.48(0.15)

0.19(0.06)

0.01(0.02)

0.22(0.11)

0.03(0.01)

0.41(0.2)

0.86(0.06)

0.38(0.09)

0.06(0.05)

3009.04

Unanimity 0.07(0.03)

0.71(0.08)

++ 0.17(0.06)

0.04(0.02)

0(0.02)

−−

Note: This table provides the statistical support for our observation that subjects use more honest/naive strategies in unanimity treatments and more free-riding strategies in majority treatments. The table reportsthe weights of the five predicted strategies in the population, the estimated strategy parameters (πLie,πLow,πMed,πHigh), the bootstrapped standard errors, and the goodness-of-fit measures ICL-BIC. The upper panelprovides the estimates for the entire sessions, the lower panel provides robustness checks focusing on either first halves and second halves of the sessions. In the upper panel, we report estimates distinguishing eitherall treatments or only majority and unanimity treatments. Plus and minus signs indicate significant differences (++ at p < .05 and + at p < .1 using bootstrapped p-values) of the strategy weights in the unanimitytreatments compared to weights in the respective majority treatments. The ICL-BICs show that the latter more parsimonious approach is statistically more adequate, but the main results are robust in either case. Theyalso hold robustly if we focus on either the first or the second halves of the sessions, as shown in the lower panel.

33

We assume that ex-ante, a subject plays strategy k ∈K with probability ρk, and that each subjectsticks with the chosen strategy throughout the analyzed interactions. The statistical model isfully described by the ex-ante strategy weights ρ = (ρ1,ρ2, . . .) and the strategy parametersπ = (πLie,πLow,πMed,πHigh) discussed above. Formally, given a subject pool S# and our set ofobservations O= oss∈S, let P(os|π,k) denote the probability of the choices os made by subjects ∈ S assuming s plays strategy of type k with parameters π. Then, the likelihood function

LL(ρ,π |O) = ∑s∈S

log ∑k∈K

ρk ·P(os|π,k)

is maximized over (ρ,π) to estimate the ex-ante strategy weights ρ we are interested in. Thestrategy parameters π are not of direct interest for our research hypothesis, but allow us to testwhether the estimates align with our ex-ante predictions, which also serves as a robustnesscheck. Given the observed choices O, we can determine the posterior class assignment of eachsubject s ∈ S simply by applying Bayes Rule.

This approach does not require us to commit to distance functions and expresses the degree of(un)certainty as a function of behavior by implying probabilistic posterior beliefs. As usual,we maximize the likelihood by the expectation-maximization (EM) algorithm (see e.g. Ar-cidiacono and Jones, 2003), and in the maximization step we again first use the gradient-freeNEWUOA approach and secondly a Newton-Raphson algorithm to ensure convergence. Modeladequacy is measured using ICL-BIC (Biernacki et al., 2000), which penalizes both superflu-ous model components and excessive parameterization. ICL-BIC has been shown to enablereliable estimation of the number of components (in our case, strategy classes) in the popula-tion (Fonseca and Cardoso, 2007). Finally, standard errors are bootstrapped by replacementat the subject level to account for the panel character of the data, using stratified resamplingacknowledging the treatment structure.

Table 5b presents the estimated strategy weights and strategy parameters, alongside the boot-strapped standard errors and statistical tests of our hypothesis. The main results are that 44 and45% of the subjects use honest/naive strategies in the Majority treatments, compared to 78 and75% in the Unanimity treatments, and that 26 and 20% of the subjects free-ride in the Majoritytreatments but no subjects free-ride in the Unanimity treatments.

The respective differences between Majority and Unanimity treatments are all highly signifi-cant and as hypothesized. The result is robust to pooling the Majority treatments and Unanim-ity treatments, respectively, and robust to focusing on either the first halves or second halves ofeach session, as shown in the lower panel of Table 5b. Further, the strategic parameters satisfyπLow < πMed < πHigh, showing that the subjects use the strategies as predicted, and the share ofunclassified (“noisy”) players is around or below 10%, showing that subjects use their respec-tive strategies consistently throughout the session. Finally, in our robustness checks reported inTable 8 (see Appendix C), we find that none of the strategy classes are superfluous, althoughsome weights are small, in the sense that eliminating either class increases the ICL-BIC mea-sure of model adequacy. With these robustness checks in mind, we conclude as follows.

Result 4. Subjects use honest/naive strategies more frequently in Unanimity and strategicallymisreport to free-ride in Majority.

few observations in each case. Loosely speaking, the finite mixture analysis determines a common denominatoracross information sets with regards to strategies.

34

Our experimental results show that expressive payoffs lead to strategic communication and in-efficient information aggregation when committees take decisions via Majority rule. Addition-ally, relative to Unanimity, we demonstrate that subjects are less responsive to other subjects’messages under Majority, both in terms of the voting decision and their implicit ex-post beliefsregarding the state of the world. We find evidence that this decrease in the effectiveness ofcommunication is due to the fact that, under Majority, a subset of subjects adopt a “free-riding”strategy, falsely reporting the non-expressive option to encourage other subject to vote for thisoption, while personally voting for the expressive option.

6 Conclusion

In this paper, we generalize the standard model of information aggregation through voting toaccount for the possibility that committee members receive vote-contingent payoffs. Using atheoretical model, we show that when the committee aggregates votes via a majority rule, thereare no equilibria with truthful communication and responsive voting despite the fact that com-mittee members have homogeneous preferences over the committee decision. In contrast, anequilibrium with truthful communication and committee-optimal decisions exists under una-nimity rule for any vote-contingent payoffs, as long as committee members have homogeneouspreferences over which committee decision to take given the aggregate profile of signals. Thisfinding contrasts with previous results in the literature, which have suggested that unanimity isa uniquely bad decision rule for aggregating information, and suggests a novel rationale for theuse of a unanimity rule: in settings with vote-contingent payoffs, efficiency can only be assuredunder a decision rule that uniformly enforces responsibility for the committee decision acrossall committee members.

We test the predictions of the model using laboratory experiments. Our experimental resultsbroadly support for the theoretical predictions. We find that, relative to a unanimity rule, sub-jects are more likely to falsely report their signal and committee decisions are less likely toaggregate private information under majority rule. Moreover, we identify that this decreasein information aggregation can be attributed to subjects adopting a “free-rider” strategy undermajority, which leads to less effective communication and sub-optimal committee decisions.

ReferencesAbeler, J., Nosenzo, D., and Raymond, C. (2018). Preferences for truth-telling. Econometrica,

Forthcoming.

American Bar Association (December 02, 2013). How Courts Work. (Retrieved fromhttps://www.americanbar.org/groups/public education/resources/law related educationnetwork/how courts work/jurydeliberate/).

Arcidiacono, P. and Jones, J. B. (2003). Finite mixture distributions, sequential likelihood andthe em algorithm. Econometrica, 71(3):933–946.

35

Austen-Smith, D. and Banks, J. (1996). Information Aggregation, Rationality, and the Con-dorcet Jury Theorem. American Political Science Review, 90(1):34–45.

Austen-Smith, D. and Feddersen, T. J. (2006). Deliberation, preference uncertainty, and votingrules. American political science review, 100(02):209–217.

Biernacki, C., Celeux, G., and Govaert, G. (2000). Assessing a mixture model for clusteringwith the integrated completed likelihood. IEEE transactions on pattern analysis and machineintelligence, 22(7):719–725.

Bond, P. and Eraslan, H. (2010). Strategic voting over strategic proposals. The Review ofEconomic Studies, 77(2):459–490.

Bouton, L., Llorente-Saguer, A., and Malherbe, F. (2017a). Get Rid of Unanimity Rule: TheSuperiority of Majority Rules with Veto Power. Journal of Political Economy, Forthcoming,Forthcoming.

Bouton, L., Llorente-Saguer, A., and Malherbe, F. (2017b). Unanimous rules in the laboratory.Games and Economic Behavior, 102:179–198.

Breitmoser, Y. (2015). Cooperation, but no reciprocity: Individual strategies in the repeatedprisoner’s dilemma. The American Economic Review, 105(9):2882–2910.

Brennan, G. and Buchanan, J. (1984). Voter choice: Evaluating political alternatives. AmericanBehavioral Scientist, 28(2):185–201.

Canes-Wrone, B., Brady, D., and Cogan, J. (2002). Out of step, out of office: Electoral account-ability and House members’ voting. American Political Science Review, 96(1):127–140.

Cappelen, A. W., Hole, A. D., Sørensen, E. Ø., and Tungodden, B. (2007). The pluralism offairness ideals: An experimental approach. The American Economic Review, 97(3):818–827.

Chan, J., Lizzeri, A., Suen, W., and Yariv, L. (2017). Deliberating Collective Decisions. Reviewof Economic Studies, Forthcoming.

Charness, G., Gneezy, U., and Halladay, B. (2016). Experimental methods: Pay one or pay all.Journal of Economic Behavior & Organization, 131:141–150.

Costa-Gomes, M. A. and Weizsacker, G. (2008). Stated beliefs and play in normal-form games.The Review of Economic Studies, 75(3):729–762.

Coughlan, P. (2000). In Defense of Unanimous Jury Verdicts: Mistrials, Communication, andStrategic Voting. American Political Science Review, 94(2):375–393.

Dal Bo, P. and Frechette, G. R. (2011). The evolution of cooperation in infinitely repeatedgames: Experimental evidence. The American Economic Review, 101(1):411–429.

de Condorcet, M. (1785). Essai sur l’application de l’analyse a la probabilite des decisionsrendues a la pluralite des voix’. L’imprimerie royale.

Ellsworth, P. (1994). Some steps between attitudes and verdicts. In Hastie, R., editor, Inside theJuror: The Psychology of Juror Decision Making. Cambridge University Press, Cambridge,UK.

36

European Commission (2017). Proposal for a Council Regulation on the Establishment of theEuropean Monetary Fund. Brussels.

Eyster, E. and Rabin, M. (2005). Cursed equilibrium. Econometrica, 73(5):1623–1672.

Feddersen, T., Gailmard, S., and Sandroni, A. (2009). Moral Bias in Large Elections: Theoryand Experimental Evidence. American Political Science Review, 103(2):175–192.

Feddersen, T. and Pesendorfer, W. (1998). Convicting the Innocent: The Inferiority ofUnanimous Jury Verdicts under Strategic Voting. The American Political Science Review,92(1):23–35.

Fehrler, S. and Hughes, N. (2018). How Transparency Kills Information Aggregation: Theoryand Experiment. American Economic Journal: Microeconomics, 10(1):181–209.

Fischbacher, U. (2007). z-tree: Zurich toolbox for ready-made economic experiments. Experi-mental economics, 10(2):171–178.

Fonseca, J. R. and Cardoso, M. G. (2007). Mixture-model cluster analysis using informationtheoretical criteria. Intelligent Data Analysis, 11(2):155–173.

Friedman, R. and Friedman, M. (1990). Free to Choose. A Personal Statement. HoughtonMifflin Harcourt.

Fudenberg, D., Rand, D. G., and Dreber, A. (2012). Slow to anger and fast to forgive: Cooper-ation in an uncertain world. The American Economic Review, 102(2):720–749.

Gerardi, D. and Yariv, L. (2007). Deliberative voting. Journal of Economic Theory, 134:317–338.

Gerling, K., Gruner, H. P., Kiel, A., and Schulte, E. (2005). Information acquisition and deci-sion making in committees: A survey. European Journal of Political Economy, 21(3):563–597.

Goeree, J. K. and Holt, C. A. (2004). A model of noisy introspection. Games and EconomicBehavior, 46(2):365–382.

Goeree, J. K. and Yariv, L. (2011). An experimental study of collective deliberation. Econo-metrica, 79(3):893–921.

Greiner, B. (2015). Subject pool recruitment procedures: organizing experiments with orsee.Journal of the Economic Science Association, 1(1):114–125.

Guarnaschelli, S., McKelvey, R. D., and Palfrey, T. R. (2000). An experimental study of jurydecision rules. American Political Science Review, 94(02):407–423.

Hagemann, S., Bailer, S., and Herzog, A. (2017a). Signals to Their Parliaments? Governments’Strategic use of Votes and Policy Statements in the EU Council. Mimeo.

Hagemann, S., Hobolt, S., and Wratil, C. (2017b). Government responsiveness in the europeanunion: Evidence from council voting. Comparative Political Studies, 50(6):850–876.

Hall, M. (1992). Electoral Politics and Strategic Voting in State Supreme Courts. The Journalof Politics, 54(2):427–446.

37

Holmstrom, B. (1999). Managerial Incentive Problems: A Dynamic Perspective. The Reviewof Economic Studies, 66(1):169–182.

Huck, S. and Konrad, K. (2005). Moral Cost, Commitment, and Committee Size. Journal ofInstitutional and Theoretical Economics, 161:575–588.

Kaplan, J. (1968). Decision Theory and the Factfinding Process. Stanford Law Review,20:1065–1092.

Kreps, D. M., Milgrom, P., Roberts, J., and Wilson, R. (1982). Rational cooperation in thefinitely repeated prisoners’ dilemma. Journal of Economic theory, 27(2):245–252.

Levitt, S. (1996). How do senators vote? disentangling the role of voter preferences, partyaffiliation, and senator ideology. The American Economic Review, 86(3):425–441.

Levy, G. (2007). Decision Making in Committees: Transparency, Reputation, and Voting Rules.The American Economic Review, 97(1):150–168.

Li, H. (2001). A Theory of Conservatism. Journal of Political Economy, 109(3):617–636.

Li, H. and Suen, W. (2009). Viewpoint: Decision-making in committees. Canadian Journal ofEconomics/Revue canadienne d’economique, 42(2):359–392.

MacLachlan, G. and Peel, D. (2000). Finite mixture models. Wiley series in probability andstatistics. Wiley.

Maggi, G. and Morelli, M. (2006). Self-enforcing voting in international organizations. Amer-ican Economic Review, 96(4):1137–1157.

Midjord, R., Rodrıguez Barraquer, T., and Valasek, J. (2017). Voting in large committees withdisesteem payoffs: A ‘state of the art’ model. Games and Economic Behavior, 104:430–443.

Morgan, J. and Vardy, F. (2012). Mixed Motives and the Optimal Size of Voting Bodies.Journal of Political Economy, 120(5):986–1026.

Palfrey, T. (2016). Experiments in Political Economy. In Roth, A. and Kagel, J., editors, TheHandbook of Experimental Economics, pages 347–434. Princeton University Press.

Persico, N. (2004). Committee Design with Endogenous Information. The Review of EconomicStudies, 71(1):165–191.

Powell, M. J. (2006). The NEWUOA software for unconstrained optimization without deriva-tives. In Large-scale nonlinear optimization, pages 255–297. Springer.

Putnam, R. (1988). Diplomacy and Domestic Politics: The Logic of Two-Level Games. Inter-national Organization, 42(3):427–460.

Rios, L. M. and Sahinidis, N. V. (2013). Derivative-free optimization: a review of algorithmsand comparison of software implementations. Journal of Global Optimization, 56(3):1247–1293.

Sapir, A. and Schoenmaker, D. (2017). The Time is Right for a European Monetary Fund.Bruegel Policy Brief, 4:1–7.

38

Snyder, J. and Groseclose, T. (2000). Estimating party influence in congressional roll-callvoting. American Journal of Political Science, 44(2):193–211.

Stahl, D. O. and Wilson, P. W. (1994). Experimental evidence on players’ models of otherplayers. Journal of economic behavior & organization, 25(3):309–327.

Stahl, D. O. and Wilson, P. W. (1995). On players’ models of other players: Theory andexperimental evidence. Games and Economic Behavior, 10(1):218–254.

Tyran, J.-R. (2004). Voting when money and morals conflict: an experimental test of expressivevoting. Journal of Public Economics, 88(7):1645–1664.

Visser, B. and Swank, O. H. (2007). On Committees of Experts. The Quarterly Journal ofEconomics, 122(1):337–372.

Weizsacker, G. (2003). Ignoring the rationality of others: evidence from experimental normal-form games. Games and Economic Behavior, 44(1):145–171.

Willman, D. (December 19, 2000). With study results kept secret, nation got another blood-pressure drug. Los Angeles Times, (Retrieved from http://www.pulitzer.org/winners/david-willman).

39

A Proofs for Sections 2 and 3

A.1 Perfect Bayesian Equilibrium

Recall that all equilibrium candidates that we consider formally are truthful, implying that bothmessages are sent with positive probability and that all information sets relevant for definingthe (stationary) strategies of our agents are therefore on the path of play. As a result, beliefs areuniquely specified by Bayes’ Rule, and the resulting expected utility of player i under strategyprofile (σ,τ), given information set I, is denoted as Ui(σ,τ|I). Given this definition, strategyprofile (σ, τ) is part of a PBE if (i) the messaging strategy σ is optimal, i.e.

Ui(σ, τ |I= si)≥Ui((σ′i, σ−i), τ |I= si) for all σ

′i

in both messaging information sets si ∈ S, and (ii) the voting strategy τ is optimal, i.e.

Ui(σ, τ |I= (si,mi,m))≥Ui(σ,(τ′i, τ−i) |I= (si,mi,m)) for all τ

′i

in all voting information sets (si,mi,m).

A.2 Proofs for Section 2.3

Proof of Proposition 1: We prove the result by contradiction. Assume an equilibrium, (σ∗,τ∗)exist with truthful communication and committee-optimal outcomes. Given truthful commu-nication, m = s. Therefore, for each m, the associate set of τ∗(si,mi,m) must imply that thecommittee selects option R (B) with probability one if R (B) is committee-optimal given m.

Take s = s′, and take R to be the committee optimal outcome given s′ (the same argument holdswhen B is committee optimal). In this case, Pr(X = R|τ∗(si,mi,s′)) = 1, which implies that amajority of agents must vote vi = R with probability one.

Given condition (2) of Free-Riding, however, there is no partition of s′, s1,s2, such that allagents with the same signal are in the same subset of s′, and ∑

i Is′i∈s1= (N +1)/2. Therefore,

since (σ∗,τ∗) is a symmetric pure strategy that selects X = R with probability one when s = s′,it must be the case that τ∗(si,mi,s′) specifies that more than (N + 1)/2 agents play vi = R.However, given X = R, it is individually optimal to vote for B for s = s′. Therefore, it is a bestreply for agents with τ(si,mi,s′) = 1 to deviate to vi = B, given that no individual agent’s voteis pivotal for the committee decision. This implies that (σ∗,τ∗) cannot be an equilibrium.

Proof of Proposition 2: First, note that Unanimity limits the outcome set to points with vi = X ,or perpetual delay. Therefore, given utility functions of the form u : X ,ω,vi→R, agents havehomogenous payoffs at all terminal nodes—this implies that there exists an equilibrium withtruthful communication and coordination on the optimal outcome.

For clarity, however, we prove the result by contradiction. Assume there exists a deviationfrom the symmetric strategy σ(si|si) = 1, and τ(si,mi,m) = 1 for m ∈ SR and τ(si,mi,m) = 0for m ∈ SB that results in a strictly positive increase in i’s expected payoffs, where SR and SB

40

are defined as in the main text as the subset of signal profiles where agents prefer X = R andX = B respectively given vi = X .

Given that Unanimity limits the outcome set to points such that vi = X (or perpetual delay),any deviation that changes i’s expected payoffs requires that the expected committee outcomechanges as a function of the underlying vector of signals for at least one profile of signals.This implies that any profitable deviation results in either (1) X = B with positive probabilityfor some s ∈ SR, (2) X = R with positive probability for some s ∈ SB, (3) that no decision isreached for some s (X = D), or some combination of all three. However, by the definition ofSR, SB and Assumption 1, all payoff-relevant changes (1), (2) and (3) imply that i’s expectedpayoffs decrease or remain constant, which is a contradiction.

A.3 Proofs for Section 3.1

Before proceeding with the proofs for Section 3.1, we introduce some additional notation.First, let piv(x,M) denote the probability that i’s vote is pivotal under Majority when M−1 ∈(N−1)/2,(N +1)/2, ...,N−1 agents vote R with probability x, and all other agents vote Rwith probability one:

piv(x,M) =(M−1)!(N−1

2

)!(M− N+1

2

)!x(N−1)/2(1− x)M−(N+1)/2,

Next, let S denote the minimum number of total signals for B (S#) required for B to be committeeoptimal. That is:

S =

N +1, if Pr(ω = B|S#)−Pr(ω = R|S#)< K ∀ S#,minS# | Pr(ω = B|S#)−Pr(ω = R|S#)≥ K, else.

Committee-optimal information aggregation prescribes that the committee play strategies suchthat they select option R when the actual S# is smaller than S, and option B when S# is greateror equal to S.

Proof of Proposition 3: First, given that τ j(, , ,)= 1 for j 6= i, the committee outcome is indepen-dent of vi, which implies that Pr(X = ω|vi = R,M#,mi,si)+K > Pr(X = ω|vi = B,M#,mi,si).Therefore, τi(, , ,) = 1 is a best response.

Second, given τi(, , ,) = 1, the committee outcome is independent of mi, which implies that forany messaging behavior, a corresponding equilibrium exists with τi(, , ,) = 1.

Proof of Lemma 1: Define S′ as follows:

S′ =

N +1, if piv(0.5,N)[Pr(ω = B|S#)−Pr(ω = R|S#)]< K ∀ S#,minS# | piv(0.5,N)[Pr(ω = B|S#)−Pr(ω = R|S#)]≥ K, else.

For M# < S′, the following expression holds for all x ∈ [0,1] since piv(x,N) is maximized at

41

x = 0.5:

piv(x,N)[Pr(ω = B|M#)−Pr(ω = R|M#)]≤ piv(0.5,N)[Pr(ω = B|M#)−Pr(ω = R|M#)]< K.

This implies that for M# < S′, piv(τM(M#),N)Pr(X =ω|vi =R,M#,mi,si)+K > piv(τM(M#),N)Pr(X =ω|vi =B,M#,mi,si) for any τM(M#) and that there is no truthful equilibrium in which τM(M#)<1 is a best response for M# < S′.

For M# ≥ S′, note that piv(x,N) is equal to zero for x = 0,1, and piv(x,N) is strictly increasingover the domain [0,0.5) and strictly decreasing over (0.5,1]. Therefore, there exists a uniqueτM(M#) ∈ (0,0.5] such that the best-response condition holds, and this τM(M#), denoted asτ∗(M#), is implicitly characterized by the following equation:

piv(τ∗(M#),N) =K

Pr(ω = B|M#)−Pr(ω = R|M#). (3)

Additionally, by the symmetry of piv(x,N) about 0.5, piv(1− τ∗(M#),N)[Pr(ω = B|M#)−Pr(ω = R|M#)] = K, and 1− τ∗(M#) is the unique equilibrium in [0.5,1).

Lemma 1 characterizes the form of equilibria with truthful communication and responsive vot-ing given the assumption that voting behavior does not conditional on the individual signal andmessage; i.e. τ(si,mi,M#) = τM(M#). Below, we extend Lemma 1 to include equilibria withtruthful communication and message-contingent voting. First, we define S

′′(M#) as follows:

S′′(M#) = piv

(N−1

2(M−1),M#

)[Pr(ω = B|S# = M#)−Pr(ω = R|S# = M#)

]−K.

Lemma 1’ (Voting Stage: Majority, Message-Contingent). Given dv = Majority, equilibriawith truthful communication, σ(si|si) = 1, and responsive voting either take the form outlinedin Lemma 1 (uniform voting), or the following form (message-contingent voting):

• τ∗(R,R,M#) = 1.• τ∗(B,B,(N +1)/2) = 0 if S = (N +1)/2.• τ∗(B,B,M#) ∈ (0,1) if M# ≥ S

′′(M#).

• τ∗(B,B,M#) = 1 else.

Proof of Lemma 1’: We begin by showing that τ∗(B,B,M#) constitutes a best response givenτ(R,R,M#) = 1. First, if S = (N + 1)/2 and M# = (N + 1)/2, then it is a best response for ito set vi = B if i is pivotal with probability one. Therefore, τ∗(B,B,(N + 1)/2) = 0 is a bestresponse since τ(R,R,M#) = 1, ensuring that i with si = B is pivotal when M# = (N +1)/2.

Next, note that piv(x,M) is strictly increasing over the domain [0,(N − 1)/(2(M− 1))) andstrictly decreasing over ((N− 1)/(2(M− 1)),1]. Therefore, given M# ≥ S

′′(M#) there exists

τ∗(B,B,M#) ∈ (0,(N− 1)/(2(M− 1))) that is a best response by the same argument as in theproof of Lemma 1.

Likewise, if M# < S′′(M#) and M# 6= (N +1)/2, then τ(B,B,M#) = 1 is a best response to any

τ∗(B,B,M#) ∈ [0,1].

Next, we show that τ(R,R,M#) = 1 is a best response given τ∗(B,B,M#). If M# < S′′(M#) or

42

M# = (N+1)/2, then i with si = R is never pivotal, which implies that vi = R is a best response.If M# ≥ S

′′(M#), then τ(R,R,M#) = 1 is a best response if:

piv((1− τ

∗(B,B,M#)),M# +1)< piv

((1− τ

∗(B,B,M#)),M#)Simplifying this expression gives:

N−12

< M#(1− τ∗(B,B,M#)),

which holds for τ∗(B,B,M#) ∈ (0,(N−1)/(2(M−1))) and M# ≥ (N +3)/2.

Additionally, note that for τ(R,R,M#) ∈ (0,1) and τ(B,B,M#) ∈ (0,1) to constitute a best re-sponse for a given M#, the probability of being pivotal for i with si = R must equal the probabil-ity of being pivotal for i with si = B. This implies that any equilibrium with truthful communi-cation and responsive voting for both si = R and si = B take the form τ(R,R,M#) = τ(B,B,M#)(i.e. τ(si,mi,M#) = τM(M#)).

Lastly, τ(R,R,M#)≤ 1 and τ(B,B,M#) = 1 and τ(R,R,M#)< 1 for some M# cannot constitutea best response, since given τ(R,R,M#) < 1 and τ(B,B,M#) = 1, the probability that i’s voteis pivotal is only greater than zero for M# < (N + 1)/2. However, when M# < (N + 1)/2,then τ(R,R,M#) = 1 maximizes payoffs independent of other’s voting behavior. Therefore,all equilibria with truthful communication and responsive voting take the form τ(R,R,M#) =τ(B,B,M#) (Lemma 1), or τ(R,R,M#) = 1 and τ(B,B,M#)≤ 1 (Lemma 1’).

Proof of Lemma 2: We provide a general proof of Lemma 2 by analyzing three separate cases.

CASE 1: τ(si,mi,M#) = τM(M#): This case corresponds to the form of equilibria we focus onin the main body of the text.

We prove the result by contradiction. Assume an equilibrium exists with truthful messaging inthe deliberation stage, and that all agents play strategy τM(M#) = τ∗(M#) in the voting stage.Note that the formulation of the Lemma implies that given K, τ∗(M#)> 0 for M# = N (that is,S′ ≤ N, otherwise the only equilibrium is the non-responsive equilibrium).

Let K denote the difference in the posterior probabilities of both states conditional on (N+1)/2signals for B.

K = Pr(

ω = B|S# = N+12

)−Pr

(ω = R|S# = N+1

2

)That is, K is the relative expected utility that i receives when the committee selects option B,given (N +1)/2 signals for B. Additionally, take K′ = piv(0.5,N)K.

The proof stems from the following two observations: (1) since K > K′ implies that S′ >(N + 1)/2, when M# = (N + 1)/2, the unique best response is for all agents to vote for R(τ∗(M#) = 1); (2) by Equation 3 in the proof of Lemma 1, τ∗(M#) is decreasing in M#.

Now, consider the expected payoff of an agent, i′, who has a signal of R, but who deviates tomi′ = B. Also, assume that i′ plays strategy τ∗(M−1)—that is, conditional on S#, i′ continuesto play the same strategy as under truthful communication. This implies that i′’s expected

43

expressive payoff is unchanged conditional on S#, and i′’s expected payoffs change only dueto the change in the probability that X = B given S#. First, note that by (1), τ∗(M#) = 1 forM# ≤ (N + 1)/2, and therefore Pr(X = B|S# ≤ (N − 1)/2,mi′ = B) = 0, which implies thatdespite the deviation, B will never be chosen when R receives more signals. Second, by (2),Pr(X = B|S# ≥ (N + 1)/2,mi′ = B) > Pr(X = B|S# ≥ (N + 1)/2,mi′ = R) for S# > S′− 1; i.e.the probability that B is chosen when B receives the majority of signals is higher given i′’sdeviation.

Therefore, since i′’s expected utility is strictly higher given an increase in Pr(X = B|S# > N/2),setting mi′ = B is a best response. (Note, however, that the strategy (σ(R) = σ(B) = 0,τ∗(M#−1)) is not a best reply—given that other agents play the mixed strategy τ∗(S#−1), i′ has a bestreply of voting R for all M# (σ(R) = σ(B) = 0,τ(M#) = 1)).

This shows that for K >K′= piv(0.5,N)K, there is no equilibrium with truthful communicationwhere agents play τ∗(M#) in the voting stage.

CASE 2: τ(si,mi,M#) = τM(M#),(1− τM(M#))), τ(M#) = 1: Note that it is possible that equi-libria with truthful communication feature voting behavior that, e.g., alternates between τ∗(M#),(1− τ∗(M#)) and τ(M#) = 1 (non-responsive) for M# ∈ (N +3)/2,(N +5)/2, ...,N. In thiscase, Pr(X = B|M#,τ(si,mi,M#)) is not necessarily weakly increasing in M#, which impliesthat the proof provided for Case 1 does not apply, since i′ with si′ = R deviating to mi′ = Bcould lower the probability that B is selected for some M#.

However, for large enough K, it will still be a best response for i with a signal of R to deviateto mi = B. Take S′ defined as above, i.e.:

S′ =

N +1, if piv(0.5,N)[Pr(ω = B|S#)−Pr(ω = R|S#)]< K ∀ S#,minS# | piv(0.5,N)[Pr(ω = B|S#)−Pr(ω = R|S#)]≥ K, else.

To see that a lower bound on K exists such that there are no equilibria with truthful communica-tion and responsive voting for K larger than this bound, take K large enough so that S′ = N. ByLemma 1, the only possible responsive equilibrium that features truthful communication anduniform voting has τ∗(M#)< 1 iff M# = N. Therefore, given truthful communication, there is apositive probability that the committee selects B only if S# = N. However, in this case, it is nota best response for i′ with si′ = R to set mi′ = R, since a unilateral deviation to mi′ = B resultsin a positive probability that the committee selects B when S# = N− 1. Since this argumentholds regardless of whether agents play τ∗(M#) or (1−τ∗(M#)) (given S′=N, agents must voteresponsively for M# = N for the equilibrium to be responsive) in the voting stage, this showsthat given K large enough so that S′ = N, no equilibria exist with truthful communication andresponsive voting.

CASE 3: MESSAGE-CONTINGENT VOTING (τ∗(R,R,M#) = 1,τ∗(B,B,M#)): Next we con-sider the case of message-contingent voting outlined in Lemma 1’. First, note that similarto Case 2, there may exist two values of τ that constitute a best response for M# ≥ S

′′(M#),

τ′(B,B,M#) ∈ (0,(N− 1)/(2(M− 1))) and τ′′(B,B,M#) ∈ ((N− 1)/(2(M− 1)),1). Also, bythe same argument as in Case 1, if K > [Pr(ω = B|S#)− Pr(ω = R|S#)] for S# = (N + 1)/2,then there are no equilibria with message-contingent voting and τ∗(B,B,M#) = τ′(B,B,M#) forM# ≥ S

′′(M#), since agents with si = R would profit from a deviation to mi = B.

44

Additionally, equilibria where agents play τ′(B,B,M#) for some M#≥ S′′(M#), and τ′′(B,B,M#)

for other M# ≥ S′′(M#) are ruled out for K high enough by the proof of Case 2, since strategies

in the two cases are equivalent when S′ = N.

NOTE: N = 3: Lastly, we note that when N = 3 (the case we consider in our experiment), ifK > Pr(ω = B|S#)− Pr(ω = R|S#) for S# = 2, then by Cases 1-3, no symmetric equilibriumexists with truthful communication and responsive voting.

Proof of Lemma 3: First, we define K′′

as follows:

K′′= Pr(S# = (N +1)/2|si = B)[Pr(ω = B|S# = (N +1)/2)−Pr(ω = R|S# = (N +1)/2)].

Assume i receives a blue signal, si = B. Since messaging is uninformative and since, given astrategy of vi = si, i’s vote is pivotal only when S# = (N +1)/2, i’s expected relative payoff ofvoting B is equal to:

Pr(S# = (N +1)/2|si = B)[Pr(ω = B|S# = (N +1)/2)−Pr(ω = R|S# = (N +1)/2)]−K,

which is positive iff K < K′′.

Second, to see that K′′ ≤ K′, note that K′ = [piv(0.5,N)/Pr(S# = (N+1)/2|si = B)]K

′′. There-

fore, K′′ ≤K′ if Pr(S# = (N+1)/2|si =B)≤ piv(0.5,N). Expanding Pr(S# = (N+1)/2|si =B)

gives:

Pr(

S# =N +1

2

∣∣∣∣si = B)= Pr(ω = B|si = B)

[Pr(

S# =N +1

2

∣∣∣∣ω = B,si = B)]

+Pr(ω = R|si = B)[

Pr(

S# =N +1

2

∣∣∣∣ω = R,si = B)]

.

Next, note that Pr(S# = (N+1)/2|ω=B,si =B) = piv(α,N) (since α is equal to the probabilitythat si = ω) and Pr(S# = (N +1)/2|ω = R,si = B) = piv(1−α,N) = piv(α,N), which impliesthat Pr(S# = (N +1)/2|si = B) = piv(α,N)≤ piv(0.5,N).

A.4 Proofs for Section 3.2

For the following proofs, we define σ∗ ≡ σ(si|si) = 1 and τ∗(si,mi,M#) as follows:

τ∗(si,mi,M#) =

1, if M# < S,0, if M# ≥ S.

where S, defined in the previous subsection, is the minimum number of signals for B requiredfor X = B to be committee-optimal.

Proof of Proposition 5: For (1), by contradiction, assume there exists a profitable deviation from(σ∗,τ∗(si,mi,M#)), (σ′,τ′(si,mi,M#)). For i to be better off, (σ′,τ′(si,mi,M#)) must introducea positive probability that either (i) vi = R for some S# ≥ S, or (ii) X = B for some S# < S

45

(or both)—any other changes in payoff-relevant outcomes would decrease i’s expected utility.However, under status-quo unanimity, if vi = R then X = R, and if X = B then vi = B, whichimplies that both (i) and (ii) decrease i’s expected utility.

(2) follows as a corollary to Proposition 1: Note that given K < 0, the free-riding condition issatisfied for any S# < (N + 1)/2 other than S# = 1. Therefore, given truthful communication,there is no symmetric equilibrium in which Pr(X = R|S#) = 1 for S# ∈ 0, [2,(N− 1)/2] bythe same argument as in the proof of Proposition 1.

Proof of Proposition 6: We prove the result by construction. By Corollary 1, τ∗(si,mi,M#) is abest response to truthful communication.

Next, we show that σ∗ is a best response to truthful communication given τ∗(si,mi,M#) and δ

small. If si = R, then mi = R is a best response by Corollary 1, since deviating to mi = B resultsin an additional payoff loss of δK. If si = B, then deviating to mi = R results in a payoff gainof δK. However, given τ∗(si,mi,M#), deviating to mi = R results in X = R when i’s message ispivotal; i.e. S# = S. Therefore, the relative expected payoff of deviating to mi = R is equal to:

δK−Pr(S# = S|si)[Pr(ω = B|S)−Pr(ω = R|S)−K].

Since this expression is nonpositive for δ ≤ δ′ = (Pr(S# = S|si)/K)[Pr(ω = B|S)− Pr(ω =R|S)−K], truthful messaging is a best response to truthful communication and τ∗(si,mi,M#)for δ≤ δ′.

Lastly, we show that there is no joint deviation in mi,vi that is a best response to (σ∗,τ∗(si,mi,M#)).Again, if si = R, then this follows from Corollary 1. If si = B and mi = R, then τ∗(si,mi,M#)is a best response for M# 6= S−1. Moreover, the voting outcomes are equivalent to (σ(si|si) =1,τ∗(si,mi,M#)) for all S# 6= S−1.

Therefore, the only possible deviation from (σ(si|si) = 1,τ∗(si,mi,M#)) that could be a bestresponse is mi = R, τ∗(si,mi,M#)) for M# 6= S− 1 and τ(B,R, S− 1)) = 0. In this case, thecommittee would not reach an agreement when M# 6= S−1, and the relative expected payoff ofdeviating is:

δK−Pr(S# = S|si)[Pr(ω = B|S−K],

which is strictly lower than zero for δ≤ δ′.

Proof of Proposition 7: (1) If K ∈ (−1,0), then existence of an equilibrium with truthful com-munication and τ = τ∗(si,mi,M#) follows from the same logic as in the proof of Proposition2: Any deviation that changes i’s payoffs results in either (1) X = B and vi = B with positiveprobability for some S# < S, (2) X = R with positive probability for some S# ≥ S, or (3) that thecommittee decision is delayed past the first round of voting for some M#, or some combinationof all three. Since (1), (2), and (3) all imply a weak decease in i’s expected payoffs, no deviationfrom (σ∗,τ∗(si,mi,M#)) is profitable.

(2) For K ∈ (0,1), take M#′ ≥ S. If agents play τ∗(si,mi,M#), then all agents vote B withprobability one when M# = M#′ . Therefore, if i deviates to τ(si,mi,M#′) = 0, no agreement isreached in any voting round when M# = M#′ . Moreover, i receives a payoff of K in each roundof voting, which is a profitable deviation.

46

B Extension of the theoretical analysis: Non-truthful, respon-sive equilibria and population with behavioral agents

In this section, we extend the theoretical analysis in the main text in two ways to formally estab-lish predictions for the experiment. First, we establish an existence condition for and the shapeof non-truthful, responsive equilibria of games with large K, i.e. with K large enough such thattruthful, responsive equilibria do not exist. Second, to account for the behavioral regularity thatsome subjects in experimental games of information aggregation with communication reporttruthfully even when lying is a best response (see Goeree and Yariv, 2011 and Abeler et al.,2018 for an analysis of lying aversion in sender-receiver games more generally), we character-ize the equilibrium behavior of rational agents when the population contains behavioral agentswho are truthful and expect everybody to be truthful. Since the main purpose of this section isto inform our experimental analysis, we restrict our attention to the game with N = 3.

Regarding the existence of non-truthful, responsive equilibria of games with large K—i.e.K > Pr(ω = B|M# = (N + 1)/2)− Pr(ω = R|M# = (N + 1)/2) = 0.2, which is satisfied inboth treatments in our experiment—we find that such an equilibrium exists if K ≤ (α2− (1−α)2)/(α2 +(1−α)2) = 0.385 given α = 0.6 (see Subsection B.1). This condition is satisfiedin the treatment with K = 10/40, but not in the treatment with K = 15/35. Hence, responsivevoting is compatible with payoff maximization (in equilibrium) in exactly one of the two treat-ments. Subsection B.2 explicitly characterizes the non-truthful, responsive equilibrium in caseK = 10/40.

Regarding the predictions of the game with behavioral agents, in Subsection B.3, we character-ize equilibria in case a fraction γ of all agents are truthful and expect everybody to be truthful,while the remaining fraction 1−γ are rational agents seeking to maximize expected payoffs—with the own type being private information of the agents. We find that given a relatively smallproportion of behavioral agents (around 13% for our experimental parameters; we estimatethat around 50% of our subjects play such “honest-naive” strategies in our experiment) then anequilibrium exists where the rational agents free-ride: i.e. they misreport a message of B givena signal of R (σi(R|R)< 1) and vote for R without an incentive to deviate unilaterally.

B.1 On the existence of Non-Truthful, Responsive Equilibria

If K > Pr(ω=B|M# = 2)−Pr(ω=R|M# = 2), where truthful responsive equilibria do not exist,in any responsive equilibrium agents must misreport at least one of their signals with positiveprobability. We derive an existence condition for equilibria (of rational agents) where the agentsmay misreport exactly one of their signals. As far as the existence condition itself is concerned,this comes without loss of generality, as misreporting both signals with positive probabilitysimply diminishes their informational content and thus weakens the incentives to subsequentlyvote B. Misreporting just one signal with positive probability may actually strengthen messagequality, however: misreporting R signals (by sending R messages after B signals) with positiveprobability implies that if agents see two B messages, there is a positive probability that thereare three B signals instead of just two, thus increasing the posterior probability that the stateof the world is B after two B messages and hence raising the incentive to vote B after two B

47

messages. Agents with B messages are then best off voting B even for relatively large K.

Without loss of generality, as far as constructing a threshold for K is concerned, we considerexistence of responsive equilibria where agents vote B with positive probability in exactly oneinformation set, while voting R regardless of messages in all other information sets. For anygiven K, it may additionally be sustainable in equilibrium that agents vote B with positiveprobability also in some or all of the other information sets, but at the threshold for K, theydo so in exactly one information set: Given any informative message strategy, the more Bmessages there are, the higher is the probability that B is the state of the world, but the moreB messages there are, the more difficult it is also coordinate on who votes B. With many Bmessages, any symmetric equilibrium sustaining B-voting is mixed and any given player is lesslikely to be pivotal—hence, they have less of an incentive to vote B. At the threshold for K,B-voting can be sustained in the generically unique information set that balances these twoeffects best. As a side note, it is not generally the information set with (N + 1)/2 messagesfor B (which minimizes coordination costs of B-voting) nor the one with N messages for B(which maximizes the posterior)—depending on α and N, it might also be an intermediateinformation set. In our case with N = 3, however, there are no intermediate information sets,which simplifies the analysis.

For notational simplicity, we focus on the class of equilibria where agents vote B only if theyobtained a B signal and sent a B message. This is in general the smallest group of agentsthat holds a majority in case there are at least (N + 1)/2 messages for B, and thus the groupof agents with the highest probability of being pivotal when voting B. Formally, that is, weconsider equilibria in strategies (σ,τ) where s′ ∈ R,B exists such that the probabilities ofmessaging R and voting R are, respectively,

σ(mi|si) =

σ < 1, for si = s′,1, for si 6= s′,

τ(si,mi,M#) =

τ(M#)≤ 1, for si = B,mi = B,M#,1, else,

with τ(M#)< 1 for some M#. As indicated, in such equilibria agents play strategies with mixedmessaging given signal s′ (rendering messages non-truthful), and only agents with si = mi = Bvote for B with positive probability. The following analysis distinguishes the two cases s′ = Rand s′ = B.

Case 1: s′ = R By analyzing this case, we implicitly verify whether “free-riding” strategiescan be sustained in equilibrium given rational agents. Given s′ = R, the above general familyof strategies refines to the following set of strategies:

σ1(mi|si) =

σR < 1, for si = R,1, for si = B.

τ1(si,mi,M#) =

τ < 1, for si = B,mi = B,M# = 3,1, else.

The following result shows that while the free-riding strategy is a best response to the strategies

48

with truthful communication and responsive voting as outlined in the main text, they are notpart of an equilibrium if all agents are rational.

Result 5. Given K > Pr(ω=B|M# = 2)−Pr(ω=R|M# = 2), there are no responsive equilibriawith strategies of the form (σ1(mi|si),τ

1(si,mi,M#)).

The result follows as a corollary to Lemma 2 and Lemma 3. In short, subjects have a strictincentive to deviate to messaging B with probability 1 after R signals, rendering messagesuninformative.

Proof of Result 5: Given τ < 1, Pr(X = B|S#)> 0 iff S# = 2,3. Therefore, σ(R|R) = 0 is a strictbest response for any τ ≤ 1, and the only possible (σR,τ) that may constitute an equilibriumis (0,τ). However, given σR = 0, M# = 3 for any S#, which implies that τ < 1 cannot be anequilibrium by Lemma 3.

Case 2: s′=B This is the theoretically interesting case when agents are rational. As discussedabove, we consider the existence of equilibria where (σ,τ) take the following form:

σ2(mi|si) =

1, for si = R,σB < 1, for si = B.

τ2(si,mi,M#) =

0 for si = B,mi = B,M# = 2,τ≤ 1, for si = B,mi = B,M# = 3,1, else.

Let us begin by outlining in more detail why these strategies are a theoretically interesting can-didate for being part of an equilibrium. If σ(B|B)< 1, then the expected number of signals forB given two messages for B is higher than two. Hence, Pr(ω = B|M# = 2)> Pr(ω = B|S# = 2),which improves the incentive to vote B after 2 B messages, i.e. to play τi(B,B,2) = 0. Sec-ond, misreporting R after B signals may indeed be rational: if all agents have a signal of Band both other agents send a message of B, then agent i is better off sending a message ofR, since the opponents may then vote B without coordination costs. This yields outcome Bwith higher probability, Pr(X = B|M# = 2) = 1 > Pr(X = B|M# = 3). As a downside, mis-reporting R after B signals increases the bias towards outcome R, but this is unavoidable, asthe first best where agents report truthfully is not attainable in equilibrium if K is sufficientlylarge.Therefore, (σ2(mi|si),τ

2(si,mi,M#)) may be part of an equilibrium for some (σB,τ) evenwhen K > Pr(ω = B|M# = 2)−Pr(ω = R|M# = 2).

To begin with, consider the information set (B,B,3), i.e. the agent in question holds a B signal,sent a B message, and overall there are three B messages. Given the assumed strategy profile,all agents must have received B signals and now they are about to play a mixed strategy equi-librium where they vote R with probability τ each. It is straightforward to verify that such anequilibrium exists if

K ≤ 12· α

3− (1−α)3

α3 +(1−α)3 .

Alternatively, in the information set I = (B,B,2), where the agent in question holds a B signal,

49

sent a B message, and overall there are two B messages, the posterior probability that ω = Bgiven the above strategy profile is

Pr(ω = B|I) = α3 ·3σ2(1−σ)+3α2(1−α) ·σ2

α3 ·3σ2(1−σ)+3α2(1−α) ·σ2 +(1−α)3 ·3σ2(1−σ)+3(1−α)2α ·σ2 ,

and given this, an agent is willing to vote B in information set (B,B,2) if and only if

Pr(ω = B|I)≥ Pr(ω = A|I)+K ⇔ K ≤ 2∗Pr(ω = B|I)−1.

Thus we obtain the upper bound for K by maximizing the posterior over σ,

ddσ

α3 ·3σ2(1−σ)+3α2(1−α) ·σ2

α3 ·3σ2(1−σ)+3α2(1−α) ·σ2 +(1−α)3 ·3σ2(1−σ)+3(1−α)2α ·σ2 = 0.

This yields the first-order condition

α3(2−3σ)+α2(1−α)2α3σ(1−σ)+α2(1−α)σ

=α3(2−3σ)+α2(1−α)2+(1−α)3(2−3σ)+(1−α)2α2α3σ(1−σ)+α2(1−α)σ+(1−α)3σ(1−σ)+(1−α)2ασ

,

which has a unique zero at σ = 0. Intuitively, as σ tends to zero, the posterior tends into thedirection of the posterior for three B signals. Formally, using L’Hospital,

limσ→0

Pr(ω = B|I) = limσ→0

α3σ(2−3σ)+α2(1−α)2σ

α3σ(2−3σ)+α2(1−α)2σ+(1−α)3σ(2−3σ)+(1−α)2α2σ,

= limσ→0

α3(2−6σ)+α2(1−α)2α3(2−6σ)+α2(1−α)2+(1−α)3(2−6σ)+(1−α)2α2

=α3 +α2(1−α)

α3 +α2(1−α)+(1−α)3 +(1−α)2α=

α2

α2 +(1−α)2 ,

the maximal posterior converges to the posterior that obtained if two out of two signals were B,which is better than two out of three (which is attainable with truthful communication), but notquite as good as three out of three as in the information set (B,B,3). In this latter case, agentsrandomize, however, which implies that they are comparably unlikely to be pivotal. Given theoptimal posterior in case (B,B,2), the threshold for K follows as

K∗ = 2 · α2

α2 +(1−α)2 −1 =α2− (1−α)2

α2 +(1−α)2 .

Since

K(3|3,α)≤ K∗(2|3,α) ⇔ 12· α

3− (1−α)3

α3 +(1−α)3 ≤α2− (1−α)2

α2 +(1−α)2 ,

which holds true for all α ≥ 0.5, the relevant threshold is therefore K∗. Near the threshold,agents vote B only in the information set (B,B,2).

Result 6. If K ≥ K∗ = α2−(1−α)2

α2+(1−α)2 , there are no responsive equilibria with strategies of the form

(σ2(mi|si),τ2(si,mi,M#)).

50

Given α = 0.6, we obtain K∗ = 0.385 for our experiment. Since 15/35 ≥ K∗, responsiveequilibria sustaining messaging and voting strategies as outlined above do not exist if K =15/35—implying that there are only babbling equilibria in our treatment with K = 15/35. Anon-truthful, responsive equilibrium for the treatment with K = 10/40 is characterized next.Note that since K = 10/40 is below both of the thresholds characterized above, voting B can besustained in both information sets (B,B,2) and (B,B,3).

B.2 A responsive equilibrium for K = 10/40

In our treatment with K = 10/40, there exist several equilibria with similar form where agentsvote B with positive probability in at least one information set. We shall characterize the equi-librium that is most responsive in the sense that B may result in the largest number of informa-tion sets and in each case with highest-possible probability. This equilibrium has the followingform,

σ(mi = R|si) =

1, if si = Rσ ∈ [0,1], if si = B (4)

τ(vi = R|I) =

0, if I = (B,B,2)τ ∈ (0,1], if I = (B,B,3)1, else.

(5)

where σ,τ are in general functions of K and α. Given our values of K = 10/40 and α = 0.6,we will be able to give a specific numeric solution.

Result 7. Consider the game with Majority voting, N = 3, α = 0.6 and K = 10/40. There existtwo equilibria that are compatible with Eqs. (4)-(5), and they are characterized as

σ =1330

1763±10√

114τ =

38±√

11476

.

In the following analysis, we derive the equilibrium values of σ and τ, and hence prove the re-sult. Individual optimality of the behavior in the remaining voting information sets is straight-forward, as voting R is strictly optimal if all opponents vote R and voting B in information set(B,B,2) is strictly optimal for the agents with B signals because they are pivotal (given that Kis below the threshold K∗ characterized in Result 6). Messaging R after signal R is establishedsimilarly straightforwardly, noting that messaging B after signal R affects the outcome only ifthere are two more messages for B under the above voting strategy, and in this case, it willdiminish the probability that B actually results—rendering this deviation unprofitable.

B.2.1 Messaging after signal B

Throughout the following analysis, we use τ = 1− τ instead of τ itself, which is slightly moreconvenient.

51

Sending message m1 = B after signal s1 = B yields outcome B iff one of the following threecases applies:

• both opponents received B signals, both sent B messages, and at least two of three voteB,

• both opponents received B signals, exactly one sent a B message, and both this agent andI vote B,

• exactly one opponent received a B signal, sent a B message, and both this agent and Ivote B.

Outcome R obtains in all other cases. Hence, the expected payoff of sending message B is

Π(m1 = B|s1 = B) = Pr(ω = R|s1 = B) ·Pr(x = B,v1 = B|ω = R,m1 = s1 = B) ·0+Pr(ω = R|s1 = B) ·Pr(x = B,v1 = R|ω = R,m1 = s1 = B) ·K+Pr(ω = R|s1 = B) ·Pr(x = R,v1 = B|ω = R,m1 = s1 = B) ·1+Pr(ω = R|s1 = B) ·Pr(x = R,v1 = R|ω = R,m1 = s1 = B) · (1+K)

+Pr(ω = B|s1 = B) ·Pr(x = B,v1 = B|ω = B,m1 = s1 = B) ·1+Pr(ω = B|s1 = B) ·Pr(x = B,v1 = R|ω = B,m1 = s1 = B) · (1+K)

+Pr(ω = B|s1 = B) ·Pr(x = R,v1 = B|ω = B,m1 = s1 = B) ·0+Pr(ω = B|s1 = B) ·Pr(x = R,v1 = R|ω = B,m1 = s1 = B) ·K,

and the expected payoff of sending message R is

Π(m1 = R|s1 = B) = Pr(ω = R|s1 = B) ·Pr(x = B,v1 = R|ω = R,m1 = R,s1 = B) ·K+Pr(ω = R|s1 = B) ·Pr(x = R,v1 = R|ω = R,m1 = R,s1 = B) · (1+K)

+Pr(ω = B|s1 = B) ·Pr(x = B,v1 = R|ω = B,m1 = R,s1 = B) · (1+K)

+Pr(ω = B|s1 = B) ·Pr(x = R,v1 = R|ω = B,m1 = R,s1 = B) ·K.

Here and in the following, a “comma” is used to indicate a logical “and”. The probabilities of

52

the various outcomes can be written as

Pr(X = B,v1 = B|ω = R,m1 = s1 = B)= 2Pr(s2 = B,s3 = R|ω = R) ·Pr(m2 = B|s2 = B)

+Pr(s2 = B,s3 = B|ω = R) ·[2Pr(m2 = B,m2 = R|s2 = s3 = B)+ . . .

]= 2α(1−α) ·σ+(1−α)2 ·2σ(1−σ)+(1−α)2 ·σ2 · (τ3 +2τ

2(1− τ))

Pr(X = R,v1 = B|ω = R,m1 = s1 = B)= Pr(s2 = B,s3 = B|ω = R) ·Pr(m2 = B,m2 = B|s2 = s3 = B) ·Pr(X = R| · · ·)= (1−α)2

σ2(1− τ)2

τ

Pr(X = B,v1 = R|ω = R,m1 = s1 = B)= Pr(s2 = B,s3 = B|ω = R) ·Pr(m2 = B,m2 = B|s2 = s3 = B) ·Pr(X = B| · · ·)= (1−α)2 ·σ2 · τ2(1− τ)

Pr(X = R,v1 = R|ω = R,m1 = s1 = B)

= α2 +2α(1−α)(1−σ)+(1−α)2(1−σ)2 +(1−α)2

σ2 · ((1− τ)3 +2(1− τ)2

τ)

Pr(X = B,v1 = B|ω = B,m1 = s1 = B)

= 2α(1−α) ·σ+α2 ·2σ(1−σ)+α

2 ·σ2 · (τ3 +2τ2(1− τ))

Pr(X = R,v1 = B|ω = B,m1 = s1 = B)

= α2σ

2 · (1− τ)2τ

Pr(X = B,v1 = R|ω = B,m1 = s1 = B) = α2 ·σ2 · τ2(1− τ)

Pr(X = R,v1 = R|ω = B,m1 = s1 = B)

= (1−α)2 +2α(1−α)(1−σ)+α2(1−σ)2 +α

2σ

2 · ((1− τ)3 +2(1− τ)2τ)

Pr(X = B,v1 = R|ω = R,m1 = R,s1 = B) = (1−α)2 ·σ2 · τ2

Pr(X = R,v1 = R|ω = R,m1 = R,s1 = B) = 1− (1−α)2 ·σ2 · τ2

Pr(X = B,v1 = R|ω = B,m1 = R,s1 = B) = α2 ·σ2 · τ2

Pr(X = R,v1 = R|ω = B,m1 = R,s1 = B) = 1−α2 ·σ2 · τ2

53

Thus, we can write the expected payoffs as

Π(m1 = B|s1 = B) =

0+(1−α) · (1−α)2 ·σ2 · τ2(1− τ) ·K+(1−α) · (1−α)2

σ2(1− τ)2

τ

+(1−α) · (α2 +2α(1−α)(1−σ)+(1−α)2(1−σ)2 +(1−α)2σ

2 · ((1− τ)3 +2(1− τ)2τ)) · (1+K)

+α · (2α(1−α) ·σ+α2 ·2σ(1−σ)+α

2 ·σ2 · (τ3 +2τ2(1− τ)))

+α ·α2 ·σ2 · τ2(1− τ) · (1+K)

+0+α · ((1−α)2 +2α(1−α)(1−σ)+α2(1−σ)2 +α

2σ

2 · ((1− τ)3 +2(1− τ)2τ)) ·K

Π(m1 = R|s1 = B) =

(1−α) · (1−α)2 ·σ2 · τ2 ·K+(1−α) · (1− (1−α)2 ·σ2 · τ2) · (1+K)

+α ·α2 ·σ2 · τ2 · (1+K)

+α · (1−α2 ·σ2 · τ2) ·K

and thus we obtain

Π(m1 = B|s1 = B) =

(1−α)3 ·σ2 · τ2(1− τ) ·K +(1−α)3σ

2(1− τ)2τ

+(1−α) · (α2 +2α(1−α)(1−σ)+(1−α)2(1−σ)2 +(1−α)2σ

2 · ((1− τ)3 +2(1− τ)2τ)) · (1+K)

+α · (2α(1−α) ·σ+α2 ·2σ(1−σ)+α

2 ·σ2 · (τ3 +2τ2(1− τ)))

+α3 ·σ2 · τ2(1− τ) · (1+K)

+α · ((1−α)2 +2α(1−α)(1−σ)+α2(1−σ)2 +α

2σ

2 · ((1− τ)3 +2(1− τ)2τ)) ·K

Π(m1 = R|s1 = B)

= (1−α)(1+K)+(1−α)3 ·σ2 · τ2 ·K +α3 ·σ2 · τ2 +α ·K

= 1+K−α+(1−α)3 ·σ2 · τ2 ·K +α3 ·σ2 · τ2

Solving the indifference condition Π(m1 = B|s1 = B) = Π(m1 = R|s1 = B) for σ at α = 0.6will yield σ ∈ (0,1) given the τ ∈ [0,1] as characterized next. Note that it is straightforward toverify that indifference may obtain if K = 10/40 but not if K = 15/35, indicating that such anequilibrium may exist only if K = 10/40 (as already shown with Result 6).

B.2.2 Voting B in information set (B,B,3)

As defined above, τ is the probability of voting B, i.e. Pr(v1 = B|M# = 3,s1 = B) = τ. Here,the condition M# = 3 indicates that all three agents sent B signals. Given the assumed strategyprofile, we also know

Pr(v2 = B|M# = 3,s1 = B) = Pr(v3 = B|M# = 3,s1 = B) = τ,

54

assuming σ > 0, as B messages are sent only after B signals—hence if we have three B mes-sages, then each one got a B signal and votes B with probability τ. Thus, in information set(B,B,3), the expected payoffs of voting R and B can be written as

Π(v1 = R|M# = 3,s1 = B) = K +Pr(ω = B|·) · τ2 +(1−Pr(ω = B|·)) · (1− τ2)

Π(v1 = B|M# = 3,s1 = B) = Pr(ω = B|·) · (1− (1− τ)2)+(1−Pr(ω = B|·)) · (1− τ)2

where Pr(ω = B|·) denotes the posterior probability of being in state B conditional on (M# =3,s1 = B). Again noting that under the assumed strategy profile, having three B messages isidentical to having three B signals, this posterior is

Pr(ω = B|M# = 3,s1 = B) =α3

α3 +(1−α)3 .

Refining expected payoffs, we obtain

Π(v1 = R|M# = 3,s1 = B) = K +α3 · τ2

α3 +(1−α)3 +(1−α)3 · (1− τ2)

α3 +(1−α)3

Π(v1 = B|M# = 3,s1 = B) =α3 · (1− (1− τ)2)

α3 +(1−α)3 +(1−α)3 · (1− τ)2

α3 +(1−α)3

and equating to solve for the mixed equilibrium yields

K +α3 · τ2 +(1−α)3 · (1− τ2)

α3 +(1−α)3 =α3 · (1− (1− τ)2)+(1−α)3 · (1− τ)2

α3 +(1−α)3

K =α3 · (1− (1− τ)2− τ2)+(1−α)3 · ((1− τ)2− (1− τ2))

α3 +(1−α)3

K =α3 · (2τ−2τ2)+(1−α)3 · (−2τ+2τ2)

α3 +(1−α)3

K =α3− (1−α)3

α3 +(1−α)3 ·2(τ− τ2)

τ− τ2 =

K2· α

3 +(1−α)3

α3− (1−α)3 .

This equation has two real solutions if K = 10/40, given α= 0.6, namely τ= 0.36 and τ= 0.64,or to be precise,

τ =38±

√114

76.

With τ = 0.36, we get σ = 0.80 from the above indifference condition, and with τ = 0.64, weget σ = 0.71. Specifically, using the first solution (τ ≈ 0.36), the indifference condition in themessage stage refines to(

10√

114−1763)

σ2 +1330σ

9500= 0 ⇒ σ =

13301763−10

√114

55

and with the second solution, we obtain(10√

114−1763)

σ2−1330σ

9500= 0 ⇒ σ =

13301763+10

√114

.

B.3 Equilibrium strategies in a population with behavioral agents

Next, we characterize equilibrium strategies of rational agents (share 1−γ> 0) in a populationwith behavioral agents (share γ > 0 in the population), who generally message truthfully andbelieves all others do so as well. Types are private information of agents, and the behavioraltype was called honest-naive in the experimental analyses. In this section, we establish that formoderate values of γ an equilibrium exists where rational agents play a “free-riding” strategy.For orientation, in the experimental analysis, we estimate γ ≈ 0.5, and as a corollary of thetheoretical analysis (which implicitly covers the case γ = 1), it follows that free-riding can besustained for γ sufficiently close to 1.

Given their beliefs, behavioral agents naively vote for the committee-optimal option assumingall messages are honest, and as indicated, they message truthfully themselves. Hence, theirstrategy is represented formally as follows.

σ(mi = B|si = R) = 0 σ(mi = B|si = B) = 1

τ(vi = R|I = ω) =

1, if ω 6= (B,B,3)0, if ω = (B,B,3)

Note that it is committee optimal to vote B only if there are three B messages assuming allmessages are honest. As for the free-riding strategy of the rational agents, we consider thefollowing class of strategies.

σ(mi = B|si = R) = σ σ(mi = B|si = B) = 1 (6)

τ(vi = R|I = ω) =

1, if ω 6= (B,B,3)τ, if ω = (B,B,3) (7)

That is, as defined above, the free-riding rational agents honestly message B signals but strate-gically message B even after R signals with positive probability, with the incentive to subse-quently vote R (free riding), and they vote B themselves only if they got a B signal, sent a Bmessage and both other players also sent B messages. Within this framework, we establish thefollowing result:

Proposition 8. Fix K ∈ 10/40,15/35. If γ ≥ 0.13, there exists an equilibrium in whichrational agents play according to Eqs. (6), (7) with

σ = 1 τ ∈

1, if K = 15/35,1, if K = 10/40 and γ≥ 5

√2495+183

808 ≈ 0.535,0, if K = 10/40 and γ≤ 5

√685+158108 ≈ 0.251,

(0,1), if K = 10/40, else.

56

B.4 Proof of Proposition 8

In all of the following, we again use τ = 1− τ to abbreviate the expressions.

Trivially, it is individually optimal to vote R if all others do so in any state. Hence, to establishthat the claimed strategy profile is part of an equilibrium for rational agents, we only need toevaluate voting in informatinon set (B,B,3) and messages in either information set (i.e. aftereither signal). Lemma 4 shows that truthfully messaging B signals is part of an equilibriumstrategy given the voting strategies in Eq. (7). Lemma 5 then shows that messages after Rsignals take the claimed form, and finally, Lemma 6 shows that voting in state (B,B,3) takesthe claimed form.

Lemma 4 (Messages after B signals). Rational agents have no incentive to deviate unilaterallyfrom σ(mi = B|si = B) = 1 if γ≥ 0.13 (for all τ ∈ [0,1]).

Proof. We analyze optimal behavior of agent 1, that of the other agents follows by symmetry.Agent 1 receives signal s1, sends message m1, votes v1, and as above, the outcome is denotedas X , the information set is denoted as I, and the state of the world is ω.

Case 1: If both opponents are behavioral (probability γ2), then by construction the own mes-sage is relevant only if both of them send B messages (following B signals). Otherwise, atleast one of them has an R signal, sends an R message and and both vote R. In case both op-ponents send B messages, the own message determines the voting outcome given the assumedstrategies.

• m1 = R yields outcome R

• m1 = B yields outcome B

Case 2: If exactly one opponent is behavioral and the other one rational (probability 2(1−γ)γ), then the own message again matters only if both opponents send B messages. Given thestrategic messaging and voting of the rational opponent, the outcome in this case, as a functionof the own message m1, is:

• m1 = R yields outcome R,

• m1 = B and vote v1 = R yields outcome B with probability Pr(s2 = B|m2 = B,s1 = B,s3 =B) · τ, otherwise R.

• m1 = B and vote v1 = B yields outcome B

Case 3: If both opponents are of the rational type, any R message triggers R voting of all, andthus again, the own message matters only if both opponents send B messages. In this case,

• m1 = R yields outcome R,

57

• m1 = B and vote v1 = R yields outcome B with probability Pr(s2 = s3 = B|m2 = m3 =B,s1 = B) · τ2, otherwise R.

• m1 = B and vote v1 = B yields outcome B with probability Pr(s2 = s3 = B|m2 = m3 =B,s1 = B) · (τ2 +2τ(1− τ))+2 ·Pr(s2 = B,s3 = R|m2 = m3 = B,s1 = B) · τ, otherwise R.

Expected payoffs: Hence, we can focus on the implications of the own message (after signalR) in case both opponents send B messages. Throughout the following analysis, we will haveto distinguish the three cases conditioning on the number of behavioral agents amongst the op-ponents. We distinguish these cases by means of a superscript to probabilities Pr indicating thenumber of behavioral opponents. That is, Pr2 refers to the case of two behavioral opponents,Pr1 refers to the case of one behavioral opponent, and Pr0 refers to the case of zero behavioralopponents. Thus, the expected payoff of an R message conditional on then reaching an infor-mation set I = (s1 = B,m1 ∈ R,B,m2 = m3 = B) is, noting that following the R message allagents vote R,

Π(m1 = R|I) = K + γ2 ·Pr2(ω = R|I)+2(1− γ)γ ·Pr1(ω = R|I)+(1− γ)2 ·Pr0(ω = R|I)

The expected payoff of B message conditional on then reaching the information set I = (s1 =B,m1 ∈ R,B,m2 = m3 = B) is, noting that agent 1 will vote R with probability 1− τ,

Π(m1 = B|I) = (1− τ) ·Π(m1 = B|I,v1 = R)+ τ ·Π(m1 = B|I,v1 = B)

where the expected payoffs are

Π(m1 = B|I,v1 = R) = K + γ2 ·Pr2(ω = B|I)

+2(1− γ)γ ·[Pr1(ω = B|I) ·Pr1(s2 = B|I,ω = B) · τ+Pr1(ω = R|I) ·

(1−Pr1(s2 = B|I,ω = R) · τ

)]+(1− γ)2 ·

[Pr0(ω = B|I) ·Pr0(s2 = s3 = B|I,ω = B) · τ2 +Pr0(ω = R|I) ·

(1−Pr0(s2 = s3 = B|I,ω = R) · τ2)]

Π(m1 = B|I,v1 = B) = γ2 ·Pr2(ω = B|I)+2(1− γ)γ ·Pr1(ω = B|I)

+(1− γ)2 ·[Pr0(ω = B|I) ·Pr0(X = B|I,ω = B,v1 = B)+Pr0(ω = R|I) ·

(1−Pr0(X = B|I,ω = B,v1 = B)

)]with

Pr0(X = B|I,ω = B,v1 = B) = Pr0(s2 = s3 = B|I,ω = B) · (τ2 +2τ(1− τ))

+2 ·Pr0(s2 = B,s3 = R|I,ω = B) · τ.

The underlying probabilities are straightforwardly verified to be as follows, given that we knowinformation set m2 = m3 = B and s1 = B.

58

Pr1(s2 = B|I,ω = B) =α3

α3 +(1−α)α2 ·σ=

α

α+(1−α) ·σ

Pr0(s2 = s3 = B|I,ω = B) =α3

α3 +2(1−α)α2σ+(1−α)2α ·σ2 =α2

α2 +2(1−α)ασ+(1−α)2σ2

Pr0(s2 = B,s3 = R|I,ω = B) =2(1−α)α2σ

α3 +2(1−α)α2σ+(1−α)2α ·σ2 =2(1−α)ασ

α2 +2(1−α)ασ+(1−α)2 ·σ2

Pr1(s2 = B|I,ω = R) =(1−α)3

(1−α)3 +α(1−α)2σ=

1−α

1−α+ασ

Pr0(s2 = s3 = B|I,ω = R) =(1−α)3

(1−α)3 +2(1−α)2ασ+α2(1−α)σ2 =(1−α)2

(1−α)2 +2(1−α)ασ+α2σ2

Pr2(ω = B|I) = α3

α3 +(1−α)3

Pr1(ω = B|I) = α3 +(1−α)α2 ·σα3 +(1−α)α2 ·σ+(1−α)3 +α(1−α)2σ

Pr0(ω = B|I) = α3 +2(1−α)α2σ+(1−α)2α ·σ2

α3 +2(1−α)α2σ+(1−α)2α ·σ2 +(1−α)3 +2(1−α)2ασ+α2(1−α)σ2

Case 1: K = 10/40. Given the above characterization, the difference in expected payoffs atK = 10/40, and given α = 0.6 as well as σ = 1, is

Π(m1 = R|I,σ = 1)−Π(m1 = B|I,σ = 1) =

10192 (γ−1)2τ3−28 (γ−1) (1743γ−793) τ2 +175

(352γ2−352γ+65

)τ−24700γ2

45500

Noting that expected payoffs are cubic in τ, there are two local extrema in τ,

τ =−√

635649γ2−361998γ+185224+1743γ−7931092γ−1092

,

the smaller one of which is the local maximum in [0,1] relevant to our analysis. Evaluated atthis maximum, the difference in expected payoffs is straightforwardly verified to be negative if(approximately) γ≥ 0.13, in which case messaging B is thus optimal.

Case 2: K = 15/35. Note that we may focus on τ = 0 in this case (as claimed in the proposi-tion), which in conjunction with α = 0.6 and K = 15/35 yields

Π(m1 = R|I,σ = 1, τ = 0)−Π(m1 = B|I,σ = 1, τ = 0) =−19γ

35,

which is obviously non-positive for all γ≥ 0, implying that messaging B is again optimal.

Lemma 5 (Messages after R signals). Rational agents have no incentive to deviate unilaterallyfrom σ(mi = B|si = R) = 1 for all γ, τ ∈ [0,1].

59

Proof. Similarly to the message after B signal (Lemma 4), the own message matters only ifboth opponents’ messages are B. Given this, and again using the notation used in the proof ofLemma 4, the expected payoff of an R message conditional on then reaching the informationset I = (s1 = R,m1 = R,m2 = m3 = B) is, noting that following the R message all agents voteR,

Π(m1 = R|I) = K + γ2 ·Pr2(ω = R|I)+2(1− γ)γ ·Pr1(ω = R|I)+(1− γ)2 ·Pr0(ω = R|I)

The expected payoff of B message conditional on then reaching the information set I = (s1 =R,m1 = B,m2 = m3 = B) is, noting that agent 1 will vote R in any case now (contrasting tovoting behavior after B signals),

Π(m1 = B|I) = K + γ2 ·Pr2(ω = B|I)

+2(1− γ)γ ·[Pr1(ω = B|I) ·Pr1(s2 = B|I,ω = B) · τ+Pr1(ω = R|I) ·

(1−Pr1(s2 = B|I,ω = R) · τ

)]+(1− γ)2 ·

[Pr0(ω = B|I) ·Pr0(s2 = s3 = B|I,ω = B) · τ2 +Pr0(ω = R|I) ·

(1−Pr0(s2 = s3 = B|I,ω = R) · τ2)]

The underlying probabilities are as follows given (m2 = m3 = B,s1 = R) and the assumedstrategy profile

Pr1(s2 = B|I,ω = B) =α3

α3 +(1−α)α2 ·σ=

α

α+(1−α) ·σ

Pr0(s2 = s3 = B|I,ω = B) =α3

α3 +2(1−α)α2σ+(1−α)2α ·σ2 =α2

α2 +2(1−α)ασ+(1−α)2σ2

Pr1(s2 = B|I,ω = R) =(1−α)3

(1−α)3 +α(1−α)2σ=

1−α

1−α+ασ

Pr0(s2 = s3 = B|I,ω = R) =(1−α)3

(1−α)3 +2(1−α)2ασ+α2(1−α)σ2 =(1−α)2

(1−α)2 +2(1−α)ασ+α2σ2

Pr2(ω = B|I) = α2(1−α)

α2(1−α)+(1−α)2α= α

Pr1(ω = B|I) = α2(1−α)+(1−α)2α ·σα2(1−α)+(1−α)2α ·σ+(1−α)2α+α2(1−α)σ

=α+(1−α) ·σ

α+(1−α) ·σ+(1−α)+ασ=

α+(1−α) ·σ1+σ

Pr0(ω = B|I) = α2(1−α)+2(1−α)2ασ+(1−α)3 ·σ2

α2(1−α)+2(1−α)2ασ+(1−α)3 ·σ2 +(1−α)2α+2(1−α)α2σ+α3σ2

Case 1: K = 10/40. Given the above characterization, the difference in expected payoffs atK = 10/40 (given α = 0.6) in response to σ = 1 is

Π(m1 = R|I,σ = 1)−Π(m1 = B|I,σ = 1) =−(2 (1− γ) τ+5γ) (3 (1− γ) τ+5γ)

125,

which is non-positive for all γ.

60

Case 2: K = 15/35. The payoff difference is independent of K following the R signal, sincethe agent will eventually vote R regardless of the message. Hence, the arguments from the caseK = 10/40 apply equally here.

Lemma 6 (Voting in (B,B,3)). Rational agents have no incentive to deviate unilaterally fromvoting behavior as claimed in Prop. 8.

Proof. Recall that we may focus on characterizing τ, the probability of voting B in the informa-tion set (B,B,3). In this information set, agent 1 got a B signal, sent a B message, and in totalthere are three B messages (i.e. both opponents sent B messages). We begin by listing the im-plications of voting R, distinguishing three cases again depending on the number of behavioralagents amongst the opponents.

• Two behavioral agents: both will vote B, hence B results and the own vote is irrelevant

• One behavioral agent: behavioral agent votes B, sophisticated one votes B with probabil-ity Pr(s2 = B|I) · τ, and only if both do, B results

• No behavioral agent: sophisticated agents both vote B with prob Pr(s2 = s3 = B|I) · τ2

Given this, and using the notation introduced for Lemma 4 again, the expected payoff of votingR is

Π(v1 = R|I) = K + γ2 ·Pr2(ω = B|I)

+2(1− γ)γ ·[Pr1(ω = B|I) ·Pr1(s2 = B|I,ω = B) · τ+Pr1(ω = R|I) ·

(1−Pr1(s2 = B|I,ω = R) · τ

)]+(1− γ)2 ·

[Pr0(ω = B|I) ·Pr0(s2 = s3 = B|I,ω = B) · τ2 +Pr0(ω = R|I) ·

(1−Pr0(s2 = s3 = B|I,ω = R) · τ2)]

If the agent is voting B, the three cases amount to

• Two behavioral agents: both will vote B, hence B results and the own vote is irrelevant

• One behavioral agent: behavioral agent votes B, hence B results

• No behavioral agent: sophisticated agents both vote B with prob Pr(s2 = s3 = B|I) ·τ2, exactly one votes B with the probability defined below, and in either case B results,otherwise, if both vote R, R results

Thus, the expected payoff of voting B in the information set I = (B,B,3), is

Π(v1 = B|I) = γ2 ·Pr2(ω = B|I)+2(1− γ)γ ·Pr1(ω = B|I)+(1− γ)2 ·

[Pr0(ω = B|I) ·Pr0(X = B|I,ω = B,v1 = B)

+Pr0(ω = R|I) ·Pr0(X = R|I,ω = R,v1 = B)]

where, in addition to the terms defined above,

61

Pr0(s2 = B,s3 = R|I,ω = R) =2(1−α)2ασ

(1−α)3 +2(1−α)2ασ+α2(1−α)σ2 =2(1−α)ασ

(1−α)2 +2(1−α)ασ+α2σ2

Pr0(s2 = s3 = R|I,ω = R) =α2(1−α)σ2

(1−α)3 +2(1−α)2ασ+α2(1−α)σ2 =α2σ2

(1−α)2 +2(1−α)ασ+α2σ2

Pr0(X = B|I,ω = B,v1 = B) = Pr0(s2 = s3 = B|I,ω = B) · (1− (1− τ)2)+2Pr0(s2 = R,s3 = B|I,ω = B) · τPr0(X = R|I,ω = R,v1 = B) = Pr0(s2 = s3 = B|I,ω = R) · (1− τ)2 +2Pr0(s2 = R,s3 = B|I,ω = R) · (1− τ)

+Pr0(s2 = s3 = R|I,ω = R)

Given α = 0.6 and σ(mi = B|si) = 1 for both si ∈ R,B, the difference in expected payoffs is

Π(v1 = R|I)−Π(v1 = B|I)

=364 (γ−1)2

τ2−2 (γ−1) (748γ−273) τ+808γ2−366γ+1625K−4421625

.

First, fix K = 10/40. The difference is convex and quadratic in τ, implying that it can becharacterized straightforwardly. The difference is positive in response to τ = 0 if

γ≥ 5√

2495+183808

≈ 0.535,

implying that voting R is a best response to itself in that case. Note that this case comprises theextreme case τ≈ 1, where rational agents obviously vote R in this information set (as implicitlyestablished in the main text). The difference is negative in response to τ = 1 if

γ≤ 5√

685+158108

≈ 0.251,

implying that that voting B is a best response to itself in that case. Otherwise, the equilibriumaction is mixed in the considered state, with

τ =−√

2√

132696γ2−137592γ+43771−748γ+273364γ−364

.

Second, in case K = 15/35, the payoff difference is easily verified to be positive for all γ,implying that the rational agents are generally best off voting R.

C Additional Material and Robustness Analyses

Additional graphs and tables Figure 3 provides a composite screen-shot that displays allqueries and all pieces of information that were available to subjects at some point during theexperiment. Table 6 describes the relative frequency and number of votes for B across allinformation sets in all four treatments.

62

Learning In this subsection we show that the qualitative results presented in above are robustto excluding the first 25 rounds of the experiment (learning). Figures 5 and 6 replicate theFigures referred to when discussing Questions 1 and 2 for the restricted data set comprisingonly the second halves of all sessions. The patterns are virtually indistinguishable. Table 7replicates the probit regression referred to in the discussion of Question 3, including a dummy“Late” distinguishing whether subjects are in the first or second halves of their sessions. Withonly two exceptions, “Late” again is insignificant. The remaining statistical results relied uponin the discussion of Questions 3 and 4 in the main text distinguish first and second halves ofsessions explicitly, thus establishing robustness to learning explicitly.

Multiple voting rounds under Unanimity As we discuss in the main text, there is an asym-metry in our operationalization of Majority and Unanimity. In particular, to replicate a unanim-ity decision rule that requires a unanimous decision and allows multiple rounds of voting, weallowed for up to three rounds of voting in the Unanimity treatment. Here we explore whetherthe multiple rounds of voting had a direct impact on the level of information aggregation—inparticular, whether multiple rounds of voting contribute to the higher degree of informationaggregation observed under Unanimity. To address this question, we compare the level of in-formation aggregation between groups that reached a unanimous decision in the first round ofvoting and groups that required multiple rounds of voting to reach a unanimous decision or whodid not reach a unanimous decision after three rounds of voting.

If the multiple rounds of voting were instrumental in driving the higher degree of informationaggregation under Unanimity, then we would expect to see that groups that voted for multiplerounds did a better job at aggregating information relative to groups that reached a unanimousdecision in the first round. However, as seen in Figure 7, this is not the case: the higher degreeof information aggregation observed under Unanimity is entirely driven by groups that reacheda unanimous decision in the first round of voting.

Finite mixture model: robustness checks Table 8 shows that eliminating components (strat-egy classes) from the analysis leads to worse values of the information criterion ICL-BIC, sug-gesting that no component should be eliminated.

63

Figure 3: Original screenshot indicating placement (a translated version with detailed explana-tion follows on the next page)

Note: This screenshot simultaneously displays all queries and all pieces of information that were available atsome point during the experiment. All items are in the positions they had been displayed, but they were not allsimultaneously visible.

64

Figure 4: Translated screenshot

Note: This screenshot simultaneously displays all queries and all pieces of information that were available at somepoint during the experiment. All items are in the positions they had been displayed, and they were displayed inthe following order.

1. Show urns and drawn ball (displayed for the entire game)Shows the two jars (“Blue Urn” and “Red Urn”) and the ball drawn (“Your ball”). These items remain on the screen for the entiregame.

2. After five seconds, query for message (no time limit)Now the box “Your Message” appears with the two balls underneath to choose from. Subjects submit the message by clicking “OK”,there is no time limit. Once the message is submitted, the box disappears.

3. When all messages are submitted, they are displayed (displayed for the remainder of the game)Now the box “Messages” on the left appears, displaying the messages of all three subjects. These items remain on the screen for therest of the game.

4. After five seconds, query for vote (no time limit)Now the box “Your Vote” appears with the two options to choose from. Subjects submit their vote by clicking “OK”, there is no timelimit. Once the vote is submitted, the box disappears.

5. When all votes are submitted, they are displayed (displayed for the remainder of the game)Now the box “Votes” on the left appears, displaying the votes of all three subjects. These items remain on the screen for the rest of thegame (in Majority or in Unanimity if decision unanimous or the third vote was taken) or disappear (in Unanimity otherwise, wherethe voting stage is restarted).

6. After five seconds, the decision taken by the committee (“Majority Decision”), the urn originally chosen by Nature (“Actual Urn”)and the payoff information is displayed (“Points”). The majority decision and numbers displayed here are entirely artificial. Theinformation remains on the screen for 10 seconds, after which a new game starts.

65

Table 6: Proportion and number of votes for B

Treatment High Low———————— ————————

M si/mi Majority Unanimity Majority Unanimity0 red/red 0.02 (198) 0.00 (399) 0.02 (304) 0.00 (404)0 blue/red 0.08 (24) 0.00 (63) 0.23 (53) 0.00 (43)

1 red/red 0.05 (464) 0.01 (487) 0.07 (523) 0.02 (511)1 red/blue 0.09 (46) 0.00 (17) 0.14 (36) 0.12 (17)1 blue/blue 0.08 (218) 0.02 (258) 0.08 (276) 0.01 (276)1 blue/red 0.22 (64) 0.06 (63) 0.18 (101) 0.03 (75)

2 red/red 0.25 (228) 0.35 (240) 0.36 (237) 0.62 (192)2 red/blue 0.06 (110) 0.19 (42) 0.32 (57) 0.47 (19)2 blue/blue 0.40 (456) 0.41 (516) 0.56 (507) 0.63 (435)2 blue/red 0.31 (55) 0.62 (39) 0.49 (45) 0.66 (35)

3 red/blue 0.16 (85) 0.54 (13) 0.45 (20) 0.83 (12)3 blue/blue 0.69 (302) 0.93 (263) 0.69 (241) 0.96 (231)

Proportion of votes for B as a function of the aggregate message profile (M) andthe individual signal/message (number of observations are reported in parenthe-ses). We use first-round votes for the unanimity treatments.

.8.8

5.9

.95

1A

vera

ge M

essa

ge T

ruth

ful

Red BlueSignal

Majority Unanimity

Truthful Messaging, Low Expressive Payoff, 2nd Half

.75

.8.8

5.9

.95

Ave

rage

Mes

sage

Tru

thfu

l

Red BlueSignal

Majority Unanimity

Truthful Messaging, High Expressive Payoff, 2nd Half

Figure 5: Truthful reporting for rounds 26−50.

66

0.2

.4.6

.81

Mea

n O

utco

me

Blu

e

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, Low Expressive Payoff, 2nd Half

0.2

.4.6

.81

Mea

n O

utco

me

Blu

e

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, High Expressive Payoff, 2nd Half

Figure 6: Information aggregation for rounds 26−50.

0.2

.4.6

.81

Mea

n O

utco

me

Blu

e

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, Agree 1st Round

0.2

.4.6

.8M

ean

Out

com

e B

lue

0 1 2 3Total Signals Blue

Majority Unanimity

Information Aggregation, Disagree 1st Round

Figure 7: Information aggregation by signal profile, decomposed by whether the group reacheda decision in the first round (left figure), or voted for multiple rounds (right graph).

67

Table 7: Probit estimations to explore subject learning

(1) (2) (3)Vote Blue Low High JointOwn signal B 1.477∗∗∗ 1.627∗∗∗ 1.544∗∗∗

(0.168) (0.158) (0.111)

Number of other’s messages B 1.692∗∗∗ 1.622∗∗∗ 1.656∗∗∗

(0.136) (0.129) (0.0928)

Majority 1.372∗∗∗ 1.357∗∗∗ 1.329∗∗∗

(0.284) (0.330) (0.209)

Majority*own signal -0.510∗∗ -0.513∗ -0.493∗∗∗

(0.198) (0.216) (0.140)

Majority*other’s messages -0.902∗∗∗ -0.905∗∗∗ -0.890∗∗∗

(0.162) (0.161) (0.112)

Late -0.250 -0.611∗∗ -0.341∗

(0.252) (0.228) (0.166)

Late*own signal 0.180 0.0851 0.105(0.132) (0.141) (0.0939)

Late*other’s messages 0.215 0.168 0.128(0.173) (0.144) (0.110)

Late*majority -0.110 0.0482 -0.0738(0.249) (0.266) (0.175)

Late*majority*other’s messages -0.147 -0.00879 -0.0322(0.191) (0.179) (0.127)

High -0.322∗∗∗

(0.0833)

Constant -3.264∗∗∗ -3.447∗∗∗ -3.186∗∗∗

(0.240) (0.258) (0.168)N 4650 4650 9300Subject-level clustered standard errors in parentheses, “late” indicates round > 25.∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001

68

Table 8: Robustness check on estimated strategy weights, testing whether all strategy classeshave significant weight. The test based on ICL-BIC (less is better), and we find that no strategyclass may be eliminated without increasing ICL-BIC. Format is equal to Table 5b

Strategy weights in population Strategy parametersNoise Honest StratRed StratBlue FreeRide ε πLie πHigh πMed πLow ICL-BIC

All games per sessionMajority 35 15 0.12 0.44 0.19 0 0.26 0.04 0.42 0.73 0.35 0.08 6368.79Majority 40 10 0.07 0.45 0.25 0.03 0.2Unanimity 35 15 0.07 0.78 0.11 0.04 0Unanimity 40 10 0.04 0.75 0.18 0.02 0

Majority 35 15 0.12 0.2 0.18 0.51 0.05 0.14 0.62 0.23 0.04 6746.9Majority 40 10 0.11 0.25 0.22 0.42Unanimity 35 15 0.1 0.13 0.7 0.06Unanimity 40 10 0.06 0.21 0.53 0.19

Majority 35 15 0.16 0.21 0.05 0.58 0.05 0.13 0.75 0.25 0.05 6770.39Majority 40 10 0.18 0.25 0.05 0.53Unanimity 35 15 0.13 0.8 0 0.07Unanimity 40 10 0.09 0.52 0.11 0.27

Majority 35 15 0.12 0.44 0.19 0.26 0.04 0.42 0.72 0.35 0.08 6368.55Majority 40 10 0.07 0.48 0.25 0.21Unanimity 35 15 0.11 0.78 0.11 0Unanimity 40 10 0.04 0.78 0.18 0

Majority 35 15 0.13 0.57 0.13 0.16 0.04 0.54 0.5 0.44 0.17 6468.46Majority 40 10 0.07 0.62 0.11 0.2Unanimity 35 15 0.07 0.81 0.08 0.04Unanimity 40 10 0.04 0.79 0.15 0.02

Majority 35 15 0.13 0.22 0.64 0.05 0.13 0.35 0.19 0.25 6948.35Majority 40 10 0.11 0.32 0.57Unanimity 35 15 0.1 0.13 0.76Unanimity 40 10 0.06 0.27 0.67

Majority 35 15 0.16 0.25 0.59 0.05 0.13 0.74 0.25 0.05 6766.47Majority 40 10 0.18 0.3 0.53Unanimity 35 15 0.13 0.8 0.07Unanimity 40 10 0.09 0.64 0.27

Majority 35 15 0.23 0.66 0.11 0.05 0.67 0.68 0.43 0.15 6677.39Majority 40 10 0.16 0.75 0.09Unanimity 35 15 0.11 0.82 0.07Unanimity 40 10 0.04 0.82 0.14

Majority 0.09 0.44 0.22 0.01 0.23 0.04 0.41 0.72 0.35 0.08 6345.24Unanimity 0.06 0.77 0.15 0.03 0

Majority 0.11 0.23 0.19 0.47 0.05 0.14 0.62 0.22 0.03 6728.42Unanimity 0.08 0.17 0.61 0.14

Majority 0.17 0.23 0.05 0.55 0.05 0.13 0.75 0.25 0.05 6758.34Unanimity 0.11 0.66 0.06 0.17

Majority 0.09 0.46 0.22 0.23 0.04 0.42 0.72 0.35 0.08 6349.6Unanimity 0.08 0.78 0.15 0

Majority 0.1 0.6 0.12 0.18 0.04 0.54 0.5 0.44 0.17 6448.95Unanimity 0.06 0.8 0.12 0.03

Majority 0.12 0.27 0.61 0.05 0.13 0.35 0.19 0.25 6934.63Unanimity 0.08 0.2 0.72

Majority 0.17 0.27 0.56 0.05 0.13 0.75 0.25 0.05 6754.68Unanimity 0.11 0.72 0.17

Majority 0.19 0.71 0.1 0.05 0.67 0.67 0.43 0.15 6663.68Unanimity 0.08 0.82 0.1

69