Sequential aggregation of verifiable information

Journal of Public Economics 95 (2011) 1447–1454

Contents lists available at ScienceDirect

Journal of Public Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j p u b e

Sequential aggregation of verifiable information ☆ Volker Hahn CER-ETH, Center of Economic Research at ETH Zurich, ZUE D13, 8092 Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 23 September 2010 Received in revised form 10 June 2011 Accepted 12 June 2011 Available online 22 June 2011 JEL classification: D71 D82

a b s t r a c t We introduce the notion of verifiable information into a model of sequential debate among experts who are motivated by career concerns. We show that self-censorship may hamper the efficiency of information aggregation, as experts withhold evidence contradicting the conventional wisdom. In this case, silence is telling and undermines the prevailing view over time if this view is incorrect. As a result, withholding arguments about the correct state of the world is only a temporary phenomenon, and the probability of a correct decision always converges to one as the group of experts becomes large. For small groups, a simple mechanism the principal can use to improve decision-making is to appoint a devil's advocate. © 2011 Elsevier B.V. All rights reserved.

Keywords: Experts Committees Career concerns Verifiable information Information aggregation

1. Introduction According to the well-known Jury Theorem established by Condorcet (1785), groups are more likely to reach correct decisions than individuals because idiosyncratic errors of individuals wash out when votes are aggregated. By contrast, in a highly influential book Janis (1972) presents case-study evidence on foreign-policy committees to bolster his case that concurrence-seeking in groups may induce self-censorship in the sense that arguments contradicting the prevailing opinion are withheld. As a consequence, wrong decisions may be taken, even if group members have strong objections privately. He has coined the term “groupthink” for this psychological drive for consensus. In this paper we present a model of sequential debate among experts that reconciles these seemingly opposing views on the efficiency of information aggregation in groups. Our framework is based on Ottaviani and Sørensen (2001) (henceforth abbreviated as OS). Building on the foundational work of Scharfstein and Stein (1990), they propose a model of sequential debate among experts who are motivated by career concerns. OS show that herding phenomena and informational cascades create a bound to the amount of information that can be aggregated if experts are uninformed about their own ability.

☆ I would like to thank Hans Gersbach, Florian Morath, Wendelin Schnedler, workshop participants in Bonn, two anonymous referees and Kai Konrad, the editor, for many valuable comments and suggestions. E-mail address: [email protected]. 0047-2727/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2011.06.004

While OS consider cheap talk,1 we introduce the assumption of verifiable information into a model of sequential debate. The assumption of verifiable information is particularly plausible with regard to experts. Experts may be able to present data or hard facts to bolster their case. Moreover, they may invest in making information hard by providing a detailed explanation for their views.2 The essence of our model is that, even if information is verifiable, it may be possible for experts to withhold it. This will be in an expert's interest if the evidence is detrimental to his reputation. More specifically, we present a model populated by a principal who has to choose between two actions with uncertain outcomes and a number of experts who engage in a sequential debate about which decision is correct. Experts are privately endowed with pieces of verifiable information about the state of the world. Like in OS, they are interested in creating the impression of expertise. This can be justified by the observation that a public perception of high competence may enable an expert to achieve more prestigious positions in the future. A favorable assessment of his ability may also enable him to earn a higher wage when moving on to another position. Our model provides us with three main insights. First, for a large set of parameters we show that experts practice self-censorship. In order to preserve their reputation, they may withhold arguments that do not concur with conventional wisdom and present only evidence in favor of it, where we say that an argument concurs with “conventional wisdom” 1

The canonical model of cheap talk is due to Crawford and Sobel (1982). See Dewatripont and Tirole (2005) for a model in which senders and receivers can invest in making soft information hard. 2

1448

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

(or the “prevailing view”) if the corresponding state appears more likely to be correct than the alternative state, taking into account the equilibrium interpretation of all actions that experts have chosen so far. Second, withholding arguments about the correct state of the world is only a temporary phenomenon in large groups of experts. This is a consequence of the observation that experts’ silence is telling in our model. Although initially experts will present only evidence supporting an incorrect view about the state of the world if the prior beliefs are sufficiently biased towards this view, the amount of evidence in favor of the incorrect view will be comparably meager. The scarcity of evidence in favor of conventional wisdom will induce all players to revise their assessments of the correct state of the world over time. At some point, these assessments will have shifted so much that experts will dare to present arguments in favor of the correct state of the world. Third and consequently, we establish that the probability of the sequential debate of experts leading to a correct decision of the principal will converge to one if the number of experts becomes large. This contrasts with the finding in OS that herding problems pose a limit to the amount of information that can be aggregated (see their Lemma 1). It is well-known that in models of cheap talk typically a large number of equilibria exist. As a consequence, attention is frequently restricted to a particular equilibrium like the most informative one, although it may be unclear why this equilibrium may be chosen. By contrast, all of our results do not depend on which perfect Bayesian Nash equilibrium is selected. Moreover, we derive a range of parameter values under which equilibrium is unique and entails an efficient aggregation of information. Intuitively, this obtains if the mere fact that experts possess evidence is conducive to their reputation and thus induces them to present their arguments. In this case, the accuracy of the information released by experts is circumstantial for their reputation. We also study several model variants. We use the special case of a single expert to illustrate the severity of self-censorship in our model by demonstrating that a single expert may withhold information that would affect the decision of the principal. By contrast, in a model where communication is cheap talk, a single expert would suppress his private information only in cases where this information would be immaterial to the principal's decision. A simple mechanism the principal could use to improve decision-making is to appoint an advocatus diaboli. More specifically, we show that the quality of decision-making may be improved if an expert is only allowed to challenge the consensus view but not to present evidence in its favor. In addition, we prove that our findings about information aggregation in large groups of experts extend to (i) a model where experts propose their arguments simultaneously, (ii) the case where the state of the world cannot be observed directly after the principal has made her decision, and (iii) a framework with an asymmetric information structure. Finally, we examine the optimal order of speech. In contrast to OS's model, the anti-seniority rule, whereby experts with the lowest probability of high competence move first, may be inferior even in a two-member committee. Our paper is organized as follows. In Section 2, we review additional papers related to ours. Section 3 outlines the model. The special case of a single expert and the general model with an arbitrary number of experts are analyzed in Sections 4 and 5. We consider extensions to our model in Section 6. Section 7 concludes. Most of the proofs are relegated to an online appendix (as Appendices D-K). 2. Related literature We have already mentioned that our model is related to the literature on the aggregation of private information by voting, which goes back to Condorcet (1785). More recent treatments of the subject pursue a game-theoretic approach that takes into account the fact that

it may not be in the agents’ interests to vote in line with their private information (see Austen-Smith and Banks (1996), Feddersen and Pesendorfer (1997), Wit (1998)). 3 Dekel and Piccione (2000) analyze sequential voting in a gametheoretic framework. They find that equilibria of a simultaneous voting game are also equilibria under sequential voting. Consequently, they argue that the results found in the literature on herd behavior and informational cascades do not immediately extend to models of voting because, in the latter models, voters condition their action on being pivotal. 4 Their model has been extended to include the possibility of abstention (see Battaglini (2005)) and the desire to vote for the winning candidate (see Callander (2007)). Glazer and Rubinstein (1998) examine different mechanisms to elicit private information from experts who are interested in the outcome of decision-making or wish their own recommendation be accepted. In contrast to these models, we assume that agents care about their reputation for being highly competent. Thus our framework belongs to the literature on experts with career concerns (see Ottaviani and Sørensen (2006), Visser and Swank (2007), Gersbach and Hahn (2008), and Hahn (2008)). Visser and Swank (2007) show that in such a model experts may vote for the a priori unconventional decision. In addition, there are incentives to show a united front. As mentioned before, our paper is also related to the work of OS, who study the optimal order of speech for a group of experts. In our paper, we introduce verifiable information into a model of experts with career concerns. There is a thriving literature dealing with the problem of the disclosure of verifiable (or certifiable) information (see Bolton and Dewatripont (2005), Ch. 5, for an overview). In these models, the informed party cannot claim to be of a type different from her true type, but she may withhold some information. This literature has identified conditions under which full disclosure occurs in equilibrium (see Grossman and Hart (1980), Milgrom (1981), Grossman (1981), Okuno-Fujiwara et al. (1990), and Seidmann and Winter (1997)). 5 In contrast with these papers, information is partially verifiable in our framework, as uninformed experts cannot prove that they possess no private information about the state. Partially verifiable information has been considered by Shin (1994), Lipman and Seppi (1995), and Forges and Koessler (2005). However, none of these papers have studied information aggregation in committees. To our knowledge, there are three papers on verifiable or partially verifiable information and voting. Bade and Rice (2009) examine how the possibility of disseminating information prior to an election affects extremists’ incentives to acquire information. Mathis (2006) and Schulte (2010) study committees in which members can engage in a simultaneous exchange of verifiable information before they vote. Schulte (2010) identifies conditions on agents’ preferences that guarantee full disclosure of information, while Mathis (2006) analyzes the impact of the voting rule. In these papers, agents have heterogeneous preferences and are interested in the outcome of the decision. Communication serves as a means of influencing the vote. In the present paper, experts are heterogeneous with respect to their

3 For a lucid review of the literature on information aggregation and communication in committees see Austen-Smith and Feddersen (2009). 4 For the literature on herd behavior and informational cascades, see Scharfstein and Stein (1990), Banerjee (1992), Bikhchandani et al. (1992), and Ali and Kartik (2010). In these models, decision makers with similar preferences may suppress their private information if the actions of other agents suggest that the opposite action is correct. This precludes efficient information aggregation. 5 Giovannoni and Seidmann (2007) compare the amount of information conveyed in games of cheap talk to the respective amount in a framework with verifiable information. Wolinsky (2003) proposes a model of verifiable messages in which the receiver is uncertain about the sender's preferences. Gersbach and Keil (2005) examine the reallocation of budgets and tasks in a public organization as a means of eliciting verifiable information about productivity improvements from agents.

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

abilities and are motivated by career concerns. They use communication to improve their reputation as competent individuals. 6 In our paper, disclosure may be only partial if one of the states appears significantly more likely to be correct than the other state. In this case, the failure to provide an argument that is in line with the prevailing view creates the suspicion that the expert withholds information about the other state being correct. The effect that the failure to release information favorable to the sender may be interpreted as indicative of the sender withholding unfavorable information occurs also in Example 3 of Okuno-Fujiwara et al. (1990), for instance. We show that this effect is sufficiently strong for a correct decision to be taken with certainty in large committees. 3. Model We consider a model inhabited by a principal and N ≥ 1 experts, indexed by i = 1,…, N. Experts engage in a sequential debate about the correct state of the world. Subsequently, the principal makes a decision, based on the information revealed in the debate. After the decision has been taken, all players observe the correct state of the world, and the ability of experts is assessed. In the following, we give a detailed account of the informational structure and the motivations of players. Two different states of the world are possible, which we label s=− 1 and s=1. The prior probabilities of the states are πs ∈]0, 1[(π1 +π− 1 =1). There are two types of experts, highly competent (H) and less competent ones (L). The ability of an individual expert is unknown both to the expert himself and to all other players. All players assign the common probability κi ∈]0, 1[ to the event of expert i being highly competent.7 An expert of ability type τ∈{H, L} receives a correct argument (Ai =s) with probability ρτ, an incorrect argument (Ai =−s) with probability μ τ, and no argument (Ai = 0) with probability 1 − ρτ − μ τ (ρτ, μ τ N 0, ρτ + μ τ ≤ 1). Conditional on the state s, all Ai's are independent. We make the following assumptions: ρH N ρL (the probability of receiving a correct argument is higher for an H-type than for an L-type), ρH + μ H N ρL + μ L (the probability of receiving some argument is higher for H than for L), ρH N μ H, and ρL N μ L (for both types, receiving correct arguments is more likely than incorrect ones). 8 Together ρH N ρL and ρH + μ H N ρL + μ L imply ρH(1 − ρL − μ L) N ρL(1 − ρH − μ H) and thus ρH ð1−μ L Þ N ρL ð1−μ H Þ:

ð1Þ

The latter inequality has the intuitive interpretation that the probability of an expert being highly competent is higher if he has observed a correct argument than in the case where he has not observed the wrong argument.9 It will be convenient to define the following variables: ρi : = κi ρH + ð1−κi ÞρL

ð2Þ

μ i : = κi μ H + ð1−κi Þμ L

ð3Þ

We note that ρi (μ i ) is the probability of an expert of unknown level of expertise receiving a correct (incorrect) argument. Our previous assumptions directly imply 0 b μ i b ρi b 1. 6 The present paper also differs from these contributions in that we focus on sequential debate and the probability of a correct decision for large numbers of experts. 7 An analysis of the case with known own abilities is available upon request. 8 The assumptions ρH N μH and ρL N μL ensure that an argument Ai ∈ {− 1, + 1} indeed supports the view that the state is s = Ai in the sense that, upon observing Ai, a player attributes a higher probability to the state being s = Ai than before. Suppose, for example, ρL b μL held. Then an argument in favor of a particular state would make this state appear less likely to be correct for low values of κi. 9 Notice that the probability of an expert being highly competent, conditional on his having observed the correct argument, is (κiρH)/(κiρH + (1 − κi)ρL). Given that an expert has not observed a wrong argument, his probability of being of type H amounts to (κi(1 − μH)))/(κi(1 − μH) + (1 − κi)(1 − μL)). Comparing both expressions yields (1).

1449

There is a sequential debate among experts, where the order of speech is exogenously given and is assumed to be i = 1, 2,…, N, without loss of generality. We use → a ∈ f−1; 0; 1gN to denote the complete vector of arguments raised by all experts. In particular, the argument revealed by expert i is denoted by ai ∈ {− 1, 0, 1}, where ai = 0 means that expert i announces no argument. Upon observing Ai ∈ {− 1, 1}, expert i can choose between truthful revelation (ai = Ai) and withholding the information (ai = 0). If expert i has not observed an argument (Ai = 0), he faces a singleton choice set (ai ∈ {0}). Thus we assume that experts cannot prove that they possess no argument. Having observed all arguments → a raised in the debate, the principal chooses an action σ ∈ {− 1, 1}. This action delivers utility 1 to the principal if it corresponds to the correct state of the world s. The principal's utility is 0 otherwise. 10 After the principal has made her decision, the correct state of the world can be observed by all players. In line with OS, experts’ utility is equal to the probability outside observers (the market) assign to the event of the expert being highly competent, after observing the complete debate → a and the correct state of the world s. This can be motivated in several ways. For example, highly competent experts may earn higher wages in the future. 11 Moreover, a favorable perception of their competency may enable experts to reach more prestigious positions. Finally, they may draw direct utility from being perceived as highly competent. In the following, we characterize perfect Bayesian Nash equilibria of this game of sequential debate. In equilibrium, all experts behave optimally in the debate, given the strategies of the other experts and the way the market forms its assessment of experts’ abilities. The principal makes an optimal decision, based on the information revealed in the debate. Moreover, the market forms Bayesian updates of the probabilities of experts being highly competent. 4. Single expert We begin our analysis with the case of a single expert (N = 1). Later we will show how these results can be extended to an arbitrary number of experts. For simplicity, we omit the index for the expert (i = 1) in this section. If the expert has observed no argument (A = 0), his decision problem is trivial because his only possible choice is a = 0. Hence we focus on A ∈ {− 1, 1} in the following. In order to derive the expert's optimal behavior, we have to compute the competence the market assigns to him in all possible contingencies. We use κ(a, s) to denote the market's assessment of the expert's competence if he has chosen a ∈ {− 1, 0, 1} and the correct state of the world is s. Next we introduce λ(A) = Pr(a = A|A), which gives the probability of the expert proposing his argument if he has obtained A ∈ {− 1, 1}. Consequently, if the expert has received argument A, he remains silent with probability 1 − λ(A). Now we are in a position to give expressions for κ(.,.), which pinpoint the expert's utility: κð1; 1Þ = κð−1; −1Þ =

ρH κ ρ

ð4Þ

κð1; −1Þ = κð−1; 1Þ =

μH κ μ

ð5Þ

κð0; 1Þ =

1−ρH λð1Þ−μ H λð−1Þ κ 1−ρλð1Þ−μλð−1Þ

κð0; −1Þ =

1−ρH λð−1Þ−μ H λð1Þ κ 1−ρλð−1Þ−μλð1Þ

ð6Þ

ð7Þ

10 Note that the values 0 and 1 only represent convenient normalizations. These normalizations do not affect our results. 11 The incentives to signal a high level of ability in order to achieve higher wages in the future were first modeled explicitly by Holmström (1999).

1450

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

If information is verifiable (a = − 1 or a = + 1), beliefs about the expert's behavior do not affect the assessment of his competence. By contrast, the case a = 0 leaves room for interpretation because this type cannot be certified. Accordingly, Eqs. (6) and (7) depend on λ(1) and λ(− 1), while Eqs. (4) and (5) do not. After these preparations, we can state the following lemma, which is proved in Appendix A: Lemma 1. For all π1, π− 1 ∈]0, 1[ with π1 + π− 1 = 1 and all possible beliefs about his behavior, the expert strictly prefers to reveal at least one argument. Moreover, if μ L ð1−ρH Þ ≤ μ H ð1−ρL Þ;

ð8Þ

the expert will reveal both arguments in equilibrium. If (8) does not hold, argument A ∈ {− 1, + 1} will definitely be withheld in equilibrium for sufficiently small πA. This lemma guarantees that there is always one type of argument that is made public with certainty. Moreover, the expert will reveal both arguments if Eq. (8) holds. In order to give an intuition for Eq. (8), we compare two formal expressions. First, we state the probability of the expert being of type H if he has not received a correct argument, which implies that he has either received a wrong argument or no argument. This probability is given by (κ(1− ρH))/(κ(1− ρH) + (1− κ)(1 − ρL)). Second, we consider the probability of the expert being of high competency, given that he has received a wrong argument. This probability is (κμ H)/(κμ H + (1− κ)μ L). It is easy to see that Eq. (8) is equivalent to the statement that the first probability is weakly lower than the second. As a consequence, Eq. (8) has the interpretation that the information that the expert has not observed the correct argument would be less favorable to this expert's reputation than the information that the expert has observed the wrong signal. This finding can be seen in light of the unraveling argument, which implies full disclosure of verifiable information under fairly general conditions (see Grossman and Hart (1980), Milgrom (1981), and Grossman (1981)). According to this argument, an agent with the most favorable information always has an incentive to release this information. As a result, the agent with the second-most favorable information prefers disclosure, as he would be pooled with worse types otherwise. Thus, in turn, the agent of the third-most favorable type benefits from disclosure. This line of reasoning is then repeated to show that full disclosure obtains. In our framework, the unraveling argument can be applied if Eq. (8) holds. An argument in line with the prevailing view, i.e. an argument A ∈ {− 1, + 1} for which πA N 1/2, can be interpreted as the most favorable signal. Therefore an expert will find it beneficial to disclose it. From our previous discussions, we know that Eq. (8) holds in situations where an expert who is known to have received a wrong argument has a higher probability of being highly competent than an expert who has not observed the correct signal. Hence an expert with an argument contradicting the prevailing view does not want to be pooled with the non-certifiable type A = 0 and makes his argument public. As a consequence, full disclosure obtains. If Eq. (8) fails to hold, experts may practice self-censorship and withhold one type of argument. According to Lemma 1, this will happen with certainty if the argument is sufficiently unlikely to be correct. 5. Arbitrary number of experts Now we turn to the more general case with N ≥ 1 and consider the decision of an arbitrary expert i. The market's assessment of his ability will be based only on the state of the world and the information publicly available at the time when i made his decision, → a ðiÞ := ða1 ; …; ai−1 Þ. → a ðiÞ can be used to compute updated versions of π1 and π− 1, which will

→  a ðiÞ and π a ðiÞ . It is crucial to note that, for a be denoted by π → 1

−1

given → a ðiÞ∈f−1; 0; 1gi−1 , expert i's behavior can be characterized in

the same way as the behavior of a single expert,
provided that
π1 and   π− 1 are replaced by the updated probabilities π1 → a ðiÞ and π−1 → a ðiÞ . We give a detailed account of the updating process for probabil
 ities πs → a ðiÞ in Appendix E (the results are reiterated in Appendix B at the end of this paper). There we demonstrate that the updating process conveniently formalized in terms of log-likelihoods

is most
→  ln πs → a ðiÞ =π−s a ðiÞ . We show that the release of arguments + 1 and -1 always makes the respective states appear more likely. Because arguments + 1 and −1 correspond to the verifiable types, the magnitude of these shifts in beliefs is independent of the strategies chosen by the expert in equilibrium. On the contrary, the impact of


 ai = 0 on ln πs → a ði + 1Þ =π−s → a ði + 1Þ does depend on expert i's equilibrium strategy. For example, if the expert makes both types of argument public in equilibrium, ai = 0 will reveal no information
→ 
→  about the stateof the s a ði + 1
Þ = π s a ðiÞ or

→
→world and  hence
π
→  ln πs a ði + 1Þ =π−s a ði + 1Þ = ln πs a ðiÞ =π−s → a ðiÞ equivalently. If the expert releases only + 1 but not -1 in equilibrium, then ai = 0 will make state -1 appear more probable. This effect has the intuitive explanation that an expert who does not present an argument may withhold evidence in favor of s = − 1 if λ(− 1) = 0 and λ(1) = 1. In this sense, silence is telling in our model. With the help of these definitions, we are able to give a precise meaning to the term “prevailing view”. The prevailing view (or conventional wisdom) supports s∈{−1, + 1} at expert i's turn if


 ln πs → a ðiÞ =π−s → a ðiÞ N 0. The prevailing view after the complete debate → a determines the principal's optimal choice of σ. More specifically,

→
→ the principal
will
choose σ=1 N 0. She will choose 
→if ln π1 a =π−1 a → σ=−1 if ln π1 a =π−1 a b 0. In line with Lemma 1, a unique equilibrium obtains if (8). In this equilibrium, all experts present their arguments, irrespective of the arguments’ types. If (8) is violated, experts may withhold arguments that contradict the prevailing view. For example, sufficiently large


 values of ln π1 → a ðiÞ =π−1 → a ðiÞ entail that expert i always withholds arguments indicating that the state of the world is s = − 1. Hence one might conjecture that, even for a large number of experts, the principal will never make a correct decision if there is a sufficiently strong but incorrect prior belief about the state of the world. This is, however, untrue. Suppose, for example, that the correct state of the world is −1, but that ln(π1/π− 1) is large. Although, at the beginning of the debate, experts will present only arguments supporting s = 1, they are not likely to find many arguments in line with this view. As a consequence, experts will remain silent frequently, which will shift the common assessment of the state of


 the world towards −1. At some point, ln π1 → a ðiÞ =π−1 → a ðiÞ will be sufficiently low such that some expert i will find it worthwhile to present an argument supporting s = − 1. Therefore large groups of experts will always enable the principal to make correct decisions. To make these arguments more precise, we have to specify a procedure how new members are added to an existing group of experts. We start from a particular equilibrium of the game with N experts and a vector of competencies (κi)iN= 1. Then we introduce an additional expert N + 1 (with a level of competence κN + 1 ∈]0, 1[). Importantly, the strategies of experts i = 1,…, N in the equilibrium of the original game correspond to an equilibrium in the game extended in this way. Thus we can consider the same strategies of experts i = 1,…, N as before. In Appendix F we show Proposition 1. Adding an individual expert to the committee leads to a weakly higher probability of the principal choosing the correct option. If N → ∞, this probability converges to 1. The proposition is a variant of the Condorcet Jury Theorem. For very large groups of experts, the correct decision is taken despite the fact that arguments contradicting the prevailing view may be withheld. By contrast, for non-verifiable information, a similar proposition does not hold. As shown by OS in their Lemma 1, experts would always pool for sufficiently informative priors and no

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

additional information would be revealed. As a result, a wrong decision may be taken for non-verifiable information. The convergence result in Proposition 1 rests on (i) our finding that for each expert i there is at least partial disclosure of arguments (see Lemma 1), (ii) the effect that under partial disclosure an expert's silence is informative about the state, and (iii) the finding that, loosely speaking, there is a minimum expected gain in information about the state of the world that is revealed in all steps of the debate. 12 In our model, each contribution to the debate ai is a signal about the state, the precision of which is affected by the heterogeneous levels of κi. The principal makes an optimal decision, based on the vector of signals (ai)iN= 1. This is reminiscent of studies that analyze optimal decision-making rules for a number of votes with different degrees of precision (see Ben-Yashar and Nitzan (1997)). In our model, at least three additional complications arise. First, there are three possible realizations of each signal (ai ∈ {− 1, 0, + 1}) rather than two, as assumed in Ben-Yashar and Nitzan (1997). Second, the signals are correlated in an endogenously determined way because each expert i's decision to release his argument Ai depends on → a ðiÞ. 13 Third, the information content of a particular signal depends on the realization of the other signals. 14 Proposition 1 provides the new result that, if the three aforementioned complications arise simultaneously, the probability of a correct decision under an optimal decision rule converges to one when the number of signals goes to infinity. Additionally, we obtain a corollary to Lemma 1 and Proposition 1: Corollary 1. Independent of π1 and π− 1, the probability of expert N suppressing correct arguments converges to zero as N → ∞. In particular, the corollary holds if experts always withhold information about the correct state of the world at the beginning of the game because the priors π− 1 and π1 are sufficiently biased. Therefore withholding correct arguments is always a temporary phenomenon. This distinguishes our model from standard models of herding in which informational cascades for the incorrect state of the world continue indefinitely (see Scharfstein and Stein (1990), Banerjee (1992), and Bikhchandani et al. (1992)).

6. Model variants In this section, we consider several variants of our basic model. We demonstrate that in general first-best decision-making cannot be attained, even if there is only a single expert. A simple mechanism is proposed and shown to improve decision-making: the appointment of a devil's advocate. Finally, we prove that our findings about information aggregation continue to hold (i) if all experts announce their arguments simultaneously, (ii) if the state of the world cannot be observed directly, and (iii) if the information structure is asymmetric. In contrast to OS, we find that for two-member committees the antiseniority rule may be harmful to the principal. 12 The third finding is shown in Lemma 3 in Appendix G. Our set of assumptions on the ρ's and μ's can be separated into two parts. While ρH N ρL and ρH + μH N ρL + μL are sufficient to ensure Lemma 1, the additional assumptions ρH N μH and ρL N μL guarantee reasonable properties of information aggregation (see footnote 3) and the convergence result in Proposition 1. 13 Boland (1989) presents several variants of the Condorcet Jury Theorem for dependent votes and varying voter competencies. However, the correlation pattern across votes considered there differs from the one across ai's in our setup. Ladha (1992) derives a sufficient condition on the average correlation of dichotomous votes that guarantees a version of the Condorcet Jury Theorem to hold. His condition cannot be applied to our model due to the first and third complication mentioned in the main text. 14 The information revealed by ai = 0 about the state of the world depends on → a ðiÞ in general. Depending on whether disclosure by expert i is full (λi(1) = 1 and λi(− 1)= 1) or partial (λi(1) = 1 and λi(− 1)= 0 or λi(1) = 0 and λi(− 1) = 1), ai = 0 may have no effect on the beliefs about the state or may shift them towards -1 or + 1.

1451

6.1. Quality of decision for a single expert In their analysis of sequential debate with non-verifiable information, OS show in their Lemma 2 that the first-best is implemented when the last expert is the only one to be herding with positive probability. The reason for their finding is that the last expert herds exactly in situations where his signal would not affect the principal's decision anyway. Now we examine whether this result extends to our framework of verifiable information. Equivalently, we consider whether a single expert entails the first-best, i.e. whether the decision taken by the principal for N = 1 always corresponds to the decision she would take if she obtained the expert's private information. In Appendix H, we draw on the definitions C1 :=

1−μ i 1−ρi

C2 :=

μ L ð1−ρH Þ−μ H ð1−ρL Þ ρH ð1−μ L Þ−ρL ð1−μ H Þ

ð9Þ

ð10Þ

to show Proposition 2. Suppose N = 1, (8) does not hold, and ρ1=μ 1 N πα=π−α N maxfC1 ; 1 = C2 g for some α ∈ {− 1, 1}. Then it is impossible to reach the first-best. Proposition 2 highlights that experts’ desire to appear competent impedes truthful information revelation. An expert may withhold verifiable information that contradicts conventional wisdom even if, given this information, the expert believes that the conventional wisdom is more likely to be wrong than correct. Notably, the expert keeps information secret precisely in situations where this information would be valuable to the principal because it would affect her decision. Conversely, the expert releases information that strengthens an already existing bias. This information is worthless to the principal because it does not influence her choice. In this sense, problems of withholding information are more severe in our model than in OS. 6.2. Mechanisms to extract more information The finding of the previous subsection raises the question how the principal could improve decision-making. One conceivable approach would be to design a complex scheme that stipulates payments to each individual expert as a function of the state of the world, the action chosen by himself and, in the case of more than one expert, the behavior of his colleagues. Especially in the public sector, the set of available incentive schemes is limited (see Wilson (1989)). Thus we restrict our attention to a simple mechanism here: the appointment of an advocatus diaboli.15 In the case of a single expert, this means that the expert is only allowed to raise the argument that corresponds to the a priori less likely view but must not raise evidence in line with conventional wisdom. While we disregard complex payment schemes here, a simple specification of sufficiently high sanctions on non-complying experts can be used to prevent them from presenting evidence in line with the consensus view. Proposition 3. Suppose in addition to N = 1, μ L(1− ρH) N μ H(1− ρL), and ρ1=μ 1 N πα=π−α N maxfC1 ; 1 = C2 g for some α ∈ {− 1, 1} (i.e. the conditions stated in Proposition 2), C1(ρH − ρL)/(μ L − μ H) N πα/π− α holds. Then the advocatus-diaboli mechanism implements the first-best.

15 The appointment of a devil's advocate goes back to the processes of canonization and beatification by the Roman Catholic Church and was introduced in the sixteenth century (see Catholic Encyclopedia. New York: Robert Appleton Company. 1913). According to Janis (1972), it was used successfully in the formulation of the Marshall Plan and the handling of the Cuban missile crisis.

1452

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

While, in general, sophisticated payment schemes are plausible to yield superior outcomes, the simple advocatus diaboli mechanism ensures the first-best in the scenario considered in Proposition 3. Why does assigning the role of an advocatus diaboli to an expert have the potential to improve decision-making? Intuitively, if the expert must not announce arguments that are in line with the a priori more likely view, the only way he can signal his competence is by challenging this view. 16 These arguments are particularly valuable to the principal because they may affect her decision, whereas arguments that strengthen the prevailing opinion about the state of the world have no effect on her choice. An additional condition appears in Proposition 3 on top of the ones required for Proposition 2. If the conditions required for Proposition 2 hold but not the additional condition in Proposition 3, the advocatusdiaboli mechanism cannot induce the expert to raise counter-arguments, although this would be valuable to the principal. Accordingly, the appointment of a devil's advocate does not affect the probability of the principal taking a correct decision. But under the more stringent set of conditions in Proposition 3 it is possible for the advocatus diaboli mechanism to lead to a strict increase in the performance of decisionmaking.

6.3. Simultaneous debate Lemma 1 can also be used to characterize experts’ behavior in a debate where arguments are submitted simultaneously. It is straightforward to adjust the proof of Proposition 1 to show that the principal makes a correct decision with certainty for very large committees.17 This finding holds even if prior beliefs about the state of the world are so strongly tilted towards the incorrect state of the world that all experts withhold evidence in support of the correct state of the world. Intuitively, information that is in line with the incorrect state of the world is released by all experts in this case. However, if only few experts find this kind of evidence, the principal will learn that the consensus view is wrong.

6.4. State of the world unobservable We have assumed in this paper that the state of the world is perfectly revealed after the principal has made her decision. This has the convenient implication that each expert does not have to consider the consequence of his action for the future course of the debate. If we made the assumption that the state of the world is not revealed by the principal's decision, this feature would break down and the model would be intractable in general (the complications arising in this case are described in more detail in Appendix K). However, for very large committees, the strategy profiles representing equilibria in our model would also correspond to equilibria in this model variant. Loosely speaking, sufficiently precise information about the state of the world will be revealed, even if a single expert deviates from his equilibrium strategy. Thus the strategy profiles remain individually optimal. To sum up, for very large committees, there are always equilibria in which the probability of a correct decision converges to one. This conclusion holds for both sequential and simultaneous debate.

1−ρH λð−1Þ−μ H λð1Þ κ. As can be verified easily, this 1−ρλð−1Þ−μλð1Þ expression is increasing in λ(1) if (8) does not hold. As a consequence, conditional on s = − 1, withholding A = − 1 is less harmful to the expert if λ(1) = 1 than in the case where λ(1) = 0. This explains why an expert is more inclined to raise an argument challenging the consensus view if he must not announce arguments in line with this view. 17 Because the same patterns of experts’ behavior occur (releasing both arguments or releasing one type of argument), Lemma 3, which is crucial for Proposition 1, holds also for simultaneous debate. 16

According to (7), κð0; −1Þ =

6.5. Asymmetric information structure The information structure of our basic model is symmetric in the sense that probabilities ρH, ρL, μH, and μL are identical across both states of the world. In the following, we specify how far this assumption can be relaxed without impinging on our results. We introduce ρτs and μ τs as the probabilities of an expert i receiving Ai = s or Ai = − s, conditional on the state of the world s ∈ {− 1, + 1} and his ability τ ∈ {H, L}. With the generalized set of assumptions s

s

ρH N ρL ;

∀s ∈ f−1; + 1g;

ð11Þ

ρH + μ H N ρL + μ L ; ∀s ∈ f−1; + 1g;

ð12Þ

s

s

s

−s

s

ρτ N μ τ ;

s

∀s ∈ f−1; + 1g; τ ∈ fH; Lg;

ð13Þ

straightforward extensions of Lemma 1 and Proposition 1 hold, as shown in Appendix L. 6.6. Optimal order of speech OS address the question whether it is optimal to adopt the antiseniority rule whereby committee members speak in order of increasing expertise (κ1 b κ2 b … b κN). They prove that, while the anti-seniority is weakly superior for two-member committees (see p. 404 in their paper), it is not necessarily superior to alternative orders for larger committees (see their Proposition 2). Our model is even less supportive of the anti-seniority rule: Proposition 4. In two-member committees, the anti-seniority may be inferior to the alternative order of speech. The proof is given in Appendix J. 18 There we construct an example of a two-member committee with κ1 ≠ κ2, whose crucial features are described in the following. The prior is biased towards the state being 1. The other parameters are chosen in a way such that the principal will not find it optimal to choose −1 if she merely knows that both experts have not observed an argument in favor of 1. However, if the principal receives the additional information that the less able expert has observed a signal supporting −1, she will prefer to choose −1. Suppose, for instance, that both experts obtained the argument −1, which entails that it would be optimal for the principal to choose −1. Irrespective of which expert moves first, he will never present the argument −1 in our example because it appears to be too unlikely to be correct. However, if the more able expert moves first, the observation that he is silent will be sufficiently informative for the less able expert to raise the argument −1. As a result, the principal opts for −1. On the other hand, under the anti-seniority rule, both experts will not raise their arguments because the less able expert's silence is not enough to encourage the more able expert to raise argument −1. As a consequence, there is not sufficient evidence for the principal to choose −1. In short, letting the more able expert move first is detrimental in OS because his statements induce the weaker expert to herd. In this paper, the new effect occurs that giving the more able expert the opportunity to speak first can encourage the weaker agent to speak when the more able expert remains silent. 7. Conclusions In this paper, we have proposed a model of sequential debate among experts who have private verifiable information about the state of world and are motivated by career concerns. We have 18 It is also possible to construct examples where the anti-seniority rule corresponds to the optimal order of two experts.

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

demonstrated that self-censorship may hamper the aggregation of information and may lead to wrong decisions by small groups, even if experts have objections privately. These findings are in line with the pessimistic view of group decision-making held by Janis (1972). An expert may withhold evidence that conflicts with the prevailing view, even if, based on this information, he believes the prevailing opinion to be wrong and even if the evidence would induce the principal to revise her decision. In this case, decision-making can be improved by assigning the role of a devil's advocate to an expert. Despite the problems created by self-censorship, the principal will never choose the wrong option if the group of experts becomes large. Intuitively, it may be the case that experts present only evidence in line with common wisdom initially. However, if common wisdom is wrong, only few experts will be able to provide evidence in its support. As more and more experts remain silent, players revise their beliefs about the state of the world. In this sense, experts’ silence is telling and will encourage other experts to challenge the prevailing view after some time. In the end, this will lead to a correct choice by the principal, which lends support to the Condorcetian perspective that decision-making guarantees an optimal decision for large groups. Appendix A. Proof of Lemma 1 A.1. Derivation of a condition guaranteeing that it is profitable for the expert to announce A Suppose the expert has observed A ∈ {− 1, 1}. Then the probability of A corresponding to the correct state of the world is ðπA ρÞ=ðπA ρ + π−A μ Þ, which takes into account the facts that the probability of the expert observing a correct argument is ρ and that the probability of his observing a wrong argument is μ. In a similar vein, the expert estimates the probability of − A being the correct state of the world to be ðπ−A μ Þ=ðπA ρ + π−A μ Þ. After these preparations we can state a condition guaranteeing that it is advantageous for the expert to announce his argument A: π−A μ πA ρ κð A; −AÞ κð A; AÞ + πA ρ + π−A μ πA ρ + π−A μ

ð14Þ

πA ρ π−A μ κð0; AÞ + κð0; −AÞ ≥ πA ρ + π−A μ πA ρ + π−A μ If the condition is violated, it is profitable for the expert to withhold A. With the help of Eqs. (4)–(7), (14) can be equivalently stated as    1−ρH λð−AÞ−μ H λð AÞ μ H ρ 1−ρH λð AÞ−μ H λð−AÞ ≥ π−A μ − : πA ρ H − ρ 1−ρλð AÞ−μλð−AÞ 1−ρλð−AÞ−μλð AÞ μ 

positive for A=−1 and thus μ L −μ H +λ(+1)(ρL μ H −ρH μ L)N 0. Then Eq. (16) can be rewritten for A=1 and A=−1 as π1 μ −μ H + λð−1ÞðρL μ H −ρH μ L Þ 1−ρλð + 1Þ−μλð−1Þ ; ≥ L ρH −ρL + λð−1ÞðρL μ H −ρH μ L Þ ⋅ 1−ρλð−1Þ−μλð + 1Þ π−1

πA

ρH −ρL + λð−AÞðρL μ H −ρH μ L Þ μ −μ H + λð−AÞðρL μ H −ρH μ L Þ ≥ π−A L : 1−ρλð AÞ−μλð−AÞ 1−ρλð−AÞ−μλð AÞ

ð17Þ

π1 ρ −ρL + λð + 1ÞðρL μ H −ρH μ L Þ 1−ρλð + 1Þ−μλð−1Þ : ð18Þ ≤ H μ L −μ H + λð + 1ÞðρL μ H −ρH μ L Þ ⋅ 1−ρλð−1Þ−μλð + 1Þ π−1 Because

μ L −μ H + λð−1ÞðρL μ H −ρH μ L Þ ρH −ρL + λð−1ÞðρL μ H −ρH μ L Þ b1

and

ρH −ρL + λð + 1ÞðρL μ H −ρH μ L Þ μ L −μ H + λð + 1ÞðρL μ H −ρH μ L Þ N

1,

which follows from ρH +μ H N ρL +μ L, the right-hand side of Eq. (17) is strictly smaller than the right-hand side of Eq. (18). Hence, at least one of the inequalities Eqs. (17) and (18) holds. A.3. We show that in any equilibrium both arguments are released by the expert if μ L(1 − ρH) ≤ μ H(1 − ρL) According to the previous step, one argument is always released in equilibrium. Without loss of generality, we assume that this argument is A = − 1. Consequently, in equilibrium λ(− 1)= 1. Inserting λ(− 1)= 1 into (17) reveals that the right-hand side of Eq. (17) will be negative if μ L(1− ρH) ≤ μ H(1− ρL). This implies that the expert will also present A = + 1 in equilibrium. □ Appendix B. Updating process In this Appendix, we summarize our findings from Appendix E
 about the updating process for probabilities πs → a ðiÞ :  1 0 
!   i Pr aj js; → a ð jÞ πs → a ði + 1Þ πs
 = ln A; ð19Þ + ∑ ln@  ln π−s π−s → a ði + 1Þ j=1 a ð jÞ Pr aj j−s; →
 where Pr aj js; → a ð jÞ is the probability of expert j choosing aj, conditional on → a
ð jÞ and s being the correct state of the world. The ! Pr ai js; → a ðiÞ
 for all three possible actions ai ∈ {− 1, values of ln a ðiÞ Pr ai j−s; → 19 0, 1} are
 !   Pr sjs; → a ðiÞ ρi
 ln N0 = ln μi Pr sj−s; → a ðiÞ

for ai = s;

ð20Þ

for ai = −s;

ð21Þ

ln


 !   Pr −sjs; → a ðiÞ μ
 = ln i b 0 → ρi Pr −sj−s; a ðiÞ

ln


 !   Pr 0js; → a ðiÞ 1−ρi λi ðsÞ−μ i λi ð−sÞ
 : = ln 1−ρi λi ð−sÞ−μ i λi ðsÞ Pr 0j−s; → a ðiÞ

ð15Þ Applying the definitions of μ and ρ (see Eqs. (2) and (3)) and rearranging yields

1453

ð22Þ

□ Appendix C. Supplementary data

ð16Þ Supplementary data to this article can be found online at doi:10. 1016/j.jpubeco.2011.06.004. A.2. We show that Eq. (16) holds for at least one argument, ∀ λ(+ 1), λ(− 1) ∈ [0, 1] Due to ρH N ρL and Eq. (1), the left-hand side of Eq. (16) is strictly positive. If the right-hand side of Eq. (16) was weakly negative for A=−1, then Eq. (16) would be fulfilled for A=−1 and the claim would hold. Hence we assume in the following that the right-hand side is strictly

References Ali, S. Nageeb, Kartik, Navin, 2010. Observational Learning with Collective Preferences. mimeo, August. 19 λi(A) will in general depend on the arguments raised by experts j = 1,…, i − 1. However, to simplify notation, we abstain from making this dependency explicit.

1454

V. Hahn / Journal of Public Economics 95 (2011) 1447–1454

Austen-Smith, David, Banks, Jeffrey S., 1996. Information aggregation, rationality, and the Condorcet jury theorem. The American Political Science Review 90 (1), 34–45 March. Austen-Smith, David, Feddersen, Timothy J., 2009. Information aggregation and communication in committees. Philosophical Transactions of the Royal Society B: Biological Sciences 364 (1518), 763–769. Bade, Sophie, Rice, Andrew, 2009. Political Advocacy with Collective Decision Making. mimeo. Banerjee, Abhijit V., 1992. A simple model of herd behavior. Quarterly Journal of Economics 107 (3), 797–817 August. Battaglini, Marco, 2005. Sequential voting with abstention. Games and Economic Behavior 51 (2), 445–463 May. Ben-Yashar, Ruth, Nitzan, Shmuel, 1997. The optimal decision rule for fixed-size committees in dichotomous choice situations: the general result. International Economic Review 38 (1), 175–186 February. Bikhchandani, Sushil, Hirshleifer, David, Welch, Ivo, 1992. A theory of fads, fashion, customs, and cultural change as informational cascades. Journal of Political Economy 100 (5), 992–1026. Boland, Philip J., 1989. Majority systems and the Condorcet jury theorem. The Statistician 38 (3), 181–189. Bolton, Patrick, Dewatripont, Mathias, 2005. Contract Theory. MIT Press, Cambridge, MA. Callander, Steven, 2007. Bandwagons and momentum in sequential voting. The Review of Economic Studies 74 (3), 653–684 July. Crawford, Vincent P., Sobel, Joel, 1982. Strategic information transmission. Econometrica 50 (6), 1431–1451 November. Condorcet, Marquis de, 1785. An Essay on the Application of Analysis to the Probability of Decisions Rendered by a Plurality of Votes. In Iain McLean and Arnold B. Urken, editors, Classics of Social Choice. Ann Arbor: University of Michigan Press, 1995. Dekel, Eddie, Piccione, Michele, 2000. Sequential voting procedures in symmetric binary elections. Journal of Political Economy 108 (1), 34–55 February. Dewatripont, Mathias, Tirole, Jean, 2005. Modes of communication. Journal of Political Economy 113 (6), 1217–1238. Feddersen, T., Pesendorfer, W., 1997. Voting behavior and information aggregation in elections with private information. Econometrica 65, 1029–1058 September. Forges, Françoise, Koessler, Frédéric, 2005. Communication equilibria with partially verifiable types. Journal of Mathematical Economics 41 (7), 793–811. Gersbach, Hans, Hahn, Volker, 2008. Should the individual voting records of CentralBank Council members be published? Social Choice and Welfare 30 (4), 655–683 May. Gersbach, Hans, Keil, Marten, 2005. Productivity improvements in public organisations. The Economic Journal 115 (505), 671–688 July.

Giovannoni, Francesco, Seidmann, Daniel J., 2007. Secrecy, two-sided bias and the value of evidence. Games and Economic Behavior 59 (2), 296–315. Glazer, Jacob, Rubinstein, Ariel, 1998. Motives and implementation: on the design of mechanisms to elicit opinions. Journal of Economic Theory 79 (2), 157–173. Grossman, Sanford J., 1981. The informational role of warranties and private disclosure about product quality. Journal of Law and Economics 24 (3), 461–483 December. Grossman, S.J., Hart, O.D., 1980. Disclosure laws and takeover bids. The Journal of Finance 35 (2), 323–334 May. Hahn, Volker, 2008. Committees, sequential voting and transparency. Mathematical Social Sciences 63 (3), 366–385 November. Holmström, Bengt, 1999. Managerial incentive problems: a dynamic perspective. The Review of Economic Studies 66 (1), 169–182 January. Janis, Irving L., 1972. Victims of groupthink: a psychological study of foreign-policy decisions and fiascoes. Houghton Mifflin, Oxford, England. Ladha, Krishna K., 1992. The Condorcet jury theorem, free speech, and correlated votes. American Journal of Political Science 36, 617–634. Lipman, Barton L., Seppi, Duane J., 1995. Robust inference in communication games with partial provability. Journal of Economic Theory 66 (2), 370–405. Mathis, Jérôme, 2006. Deliberation with Partially Verifiable Information. mimeo, March. Milgrom, Paul R., 1981. Good news and bad news: representation theorems and applications. The Bell Journal of Economics 12 (2), 380–391. Okuno-Fujiwara, Masahiro, Postlewaite, Andrew, Suzumura, Kotaro, 1990. Strategic information revelation. The Review of Economic Studies 57 (1), 25–47 January. Ottaviani, Marco, Sørensen, Peter Norman, 2001. Information aggregation in debate: who should speak first? Journal of Public Economics 81 (3), 393–421. Ottaviani, Marco, Sørensen, Peter Norman, 2006. Reputational cheap talk. The RAND Journal of Economics 37 (1), 155–175. Scharfstein, David, Stein, Jeremy, 1990. Herd behavior and investment. The American Economic Review 80 (3), 465–479 June. Schulte, Elisabeth, 2010. Information aggregation and preference heterogeneity in committees. Theory and Decision 69, 97–118. Seidmann, Daniel J., Winter, Eyal, 1997. Strategic information transmission with verifiable messages. Econometrica 65 (1), 163–169 January. Shin, Hyun Song, 1994. News management and the value of firms. The RAND Journal of Economics 25 (1), 58–71 Spring. Visser, Bauke, Swank, Otto, 2007. On committees of experts. Quarterly Journal of Economics 122 (1), 337–372 February. Wilson, James Q., 1989. Bureaucracy: What Government Agencies Do and Why They Do It. Basic Books. Wit, Jörgen, 1998. Rational choice and the Condorcet jury theorem. Games and Economic Behavior 22 (2), 364–376. Wolinsky, Asher, 2003. Information transmission when the sender's preferences are uncertain. Games and Economic Behavior 42 (2), 319–326.