The Condorcet Jury Theorem with information acquisition
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Chen, Jun Article The Condorcet Jury Theorem with information acquisition Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Chen, Jun (2021) : The Condorcet Jury Theorem with information acquisition, Games, ISSN 2073-4336, MDPI, Basel, Vol. 12, Iss. 4, pp. 1-33, https://doi.org/10.3390/g12040079 This Version is available at: https://hdl.handle.net/10419/257561 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
games Article The Condorcet Jury Theorem with Information Acquisition Jun Chen Citation: Chen, J. The Condorcet Jury Theorem with Information Acquisition. Games 2021,12, 79. https://doi.org/10.3390/g12040079 Academic Editors: Hans-Theo Normann and Ulrich Berger Received: 27 August 2021 Accepted: 19 October 2021 Published: 25 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Economics and Management School, Wuhan University, Wuhan 430072, China; [email protected] Abstract: We analyze a committee decision in which individuals with common preferences are uncertain which of two alternatives is better for them. Members can acquire costly information. Private signals and information choice are both continuous. As is consistent with Down’s rational ignorance hypothesis, each member acquires less information in a larger committee and tends to acquire zero information when the committee size goes to infinity. However, with more members, a larger committee can gather more aggregate information in equilibrium. The aggregate information is infinite with the size going to infinity if and only if marginal cost at “zero information acquisition” is zero. When the marginal cost at “zero information acquisition” is positive, the probability of making an appropriate decision tends to be less than one. Keywords: information acquisition; the Condorcet Jury Theorem; jury size; committee decision 1. Introduction The classical Condorcet Jury Theorem (CJT) argues that (i) increasing the size of one committee raises the probability that an appropriate (right) decision is made, and (ii) the probability of making the appropriate decision goes to one with the size of one committee going to infinity. The theorem developed out of a study by de Condorcet [1] of decision-making process in societies when group members have private information. Recent literature on committee decisions has pointed out that if information acquisition is costly, the CJT may fail to hold. The reasoning is that each member has little incentive to acquire private information because he has a negligible probability of affecting the outcome in a large committee and thus he can free ride on the information of other members (see Gerling et al. [2] for a survey). Existing literature on the CJT with information acquisition employs one of two modeling methods. In the first, members of a committee can only decide whether or not to acquire the private information; the quality of the information and the information cost is given. In these models, the proportion of members acquiring information is non-monotone with respect to the committee size, and there is an optimal size maximizing the aggregate information (see Mukhopadhaya [3] , Koriyama and Szentes [4] , Gershkov and Szentes [5] , Gerardi and Yariv [6] and Persico [7] ). The second one makes use of binary signals and allows members to decide the quality of signals. Martinelli [8 , 9] has shown that although each individual acquires less information in a larger committee, the probability of making an appropriate decision can be either increasing or decreasing in the committee size, and it does not necessarily go to one as the size tends to infinity. We think that existing research has contributed substantially towards understanding the group decision processes with information acquisition. However, we believe that in many environments, both the signals and the quality of information choice are continuous. Arguably some results regarding the CJT in the model with continuous signals need to be revisited.1 In this paper we focus on a group decision problem in which members have common preferences, but they do not know which of two alternatives is better for them. 2 Members have no free information, but can decide how much private information they acquire. Games 2021,12, 79. https://doi.org/10.3390/g12040079 https://www.mdpi.com/journal/games
Games 2021,12, 79 2 of 33 Society decides the committee size and the decision rule that defines how each member’s report contributes to the final decision. Proposition 3characterizes the linear equilibria where each individual’s report is linear in his/her signal. We show that the decision threshold will not affect the final decision because each individual’s report will adjust according to the threshold. Therefore, for a given set of information, the committee’s decision is the same as the first-best decision, which is shown in Proposition 1. Therefore what is concerned is how committee members acquire the private information. We can show that each member’s information acquisition is different from the first-best information choice, which is shown in Proposition 2. Proposition 4shows us that members have less incentives to acquire information in a larger committee. This is consistent with Down’s rational ignorance hypothesis and it is reasonable since information is a public good in equilibrium, and therefore committee members are more likely to free ride on the information of others in a large committee. However, a larger committee tends to gather more aggregate information, which is confirmed in Proposition 5. Therefore, to make the appropriate decision, the optimal choice for the society is to maximize the committee size when there is no participation cost for committee members. Proposition 6shows the asymptotic probability of aggregate information acquired by a committee. If C0( 0 ) = 0 the limit of aggregate information tends to infinity; if C0( 0 ) is positive the limit of aggregate information is finite and the society cannot obtain the appropriate decision with probability 1. Moreover we show that the limit of aggregate information is a continuous and monotonically decreasing function of C0( 0 ) , with its limit being infinite when C0( 0 ) tends to zero. Combining Corollary 2and Proposition 6 we see that the information acquisition is asymptotically efficient, and universal or near universal participation, given that the society is very large and there is no participation cost, is preferable. Next we relax the assumption that individuals are indifferent between the two choices prior to observations. Proposition 8shows that the rational ignorance hypothesis still holds; and Proposition 9shows that the aggregate information gathered by a committee is non-decreasing in the committee size. Furthermore, the limit of the aggregate information is a function of C0( 0 ) . Proposition 9also shows that this function is discontinuous, but it is continuous and monotonically decreasing when the marginal cost at zero information acquisition is small enough. It tends to infinity when the marginal cost at zero information acquisition tends to zero. Taken together, our results show that the rational ignorance hypothesis is generally satisfied in the committee decision with information acquisition, but a larger committee serves the society better than what the rational ignorance hypothesis indicates at first glance. Furthermore, the probability of making the appropriate decision might not be able to tend to 1. Its limit is 1 if and only if C0(0)is zero. Furthermore, even if the committee members can only report 0 and 1, Propositions 10 and 11 show that the limit of the probability of the appropriate decision goes to 1 if and only if the marginal cost at zero information acquisition is zero and the limit is strictly less than 1 if and only if the marginal cost at zero information acquisition is positive, although the rational ignorance hypothesis still holds irrespective of the information cost function. This conclusion differs from Martinelli [8] : in a strategic voting model with binary signals, he shows that the limit of the appropriate decision goes to 1 if and only if both the marginal cost and the second-order derivative at zero information acquisition are zero; the reason is that the information is coarser than ours so that there needs to be stricter conditions for the CJT to hold. Following the work of Triossi [11] , we extend our analysis into the model where the committee members have heterogeneous information cost functions. We show that a larger committee will acquire more information in Proposition 12. However, the aggregate information goes to infinity if and only if the probability is positive for skill parameters whose marginal cost at zero information acquisition is zero.
Games 2021,12, 79 3 of 33 We then extend the analysis into more general continuous distributions. Proposition 13 shows that if the member can report a real number, the probability of the appropriate decision tends to 1 if and only if the marginal cost at zero information acquisition is 0. If members can only report 0 and 1, then Proposition 14 shows that when the conditional distributions satisfy the monotone likelihood ration property (MLRP), the limit of the probability of the appropriate decision is 1 if and only if the marginal cost at zero information acquisition is 0. The paper proceeds as follows. Section 2introduces the model. Section 3derives the first-best solution. Section 4derives the equilibrium, and Section 5investigates the effects of committee size on social welfare and information acquisition in equilibrium. Section 6 extends our analysis into the model where individuals in the society are biased towards one alternative prior to observations. Section 7does some extensions and shows that the conclusions are still valid in other settings. Section 8concludes the paper. 2. The Model There is a society consisting of N(∈N) ex-ante identical individuals. The underlying state of the world, ω∈Ω , can take one of the two values, 0 and 1, with the common prior Pr(ω= 1 ) = 1 −Pr(ω= 0 ) = γ∈( 0, 1 ) . The society has to make a binary decision d∈ { 0, 1 } . There is no interest conflict among individuals. Each individual has a benefit u(d,ω)if decision dis made when the underlying state is ω. In particular, u(d,ω) = 0, if d=ω −α, if d=0 and ω=1 −β, if d=1 and ω=0 where α,β>0 represents the severity of type-I error and type-II error, respectively. The society randomly chooses n individuals to form a committee and determines the decision rule. 3 Each member needs to pay some cost to gather the private information. As in Li [12] , Duggan and Martinelli [13] and Li and Suen [14] we assume the signals are continuous. The private signal of member iis si=ω+εiwith εi∼ N(0, 1/qi) when (s)he pays the cost C(qi) , where the cost function satisfies C0≥ 0, C00 ≥ 0 and C( 0 ) = 0. When C(qi) = cqi with c> 0 the cost function is linear; otherwise it is nonlinear. Furthermore, cov(εi,εj) = 0 for all i6=j. For notation convenience we adopt the method of Ganuza and Penalva [15] and Amir and Lazzati [16] : the information choice is for member i to choose from a family of joint cumulative distributions {F(si,ω;qi)} indexed by the precision qi . Suppose the probability density function (PDF) is f(si,ω;qi). Let s,(s1 , s2 , ··· , sn) be the signal profile and q,(q1 , q2 , ··· , qn) be the information profile.Then each information profile qinduces a distribution F(s,ω;q). After receiving the private signal, member i does some reports according to his private signal and the signal precision to the society ϕi:R×R+→R and the final decision is made according to the reports of all members. Let ri=ϕi(si , qi) be the report value when the precision is qi and the signal is si , and the decision rule is d=ψ(r1,r2,··· ,rn). We want to analyze the effects of the decision rule ψ and the committee size n on members’ behavior in equilibrium. Therefore we will try to solve the equilibrium of the game composed by nindividuals in the committee. Formally the game is Γψ=hI,Σ,Hi
Games 2021,12, 79 4 of 33 where I={ 1, 2, ··· , n} is the set of players, Σ=×n i=1Σi is the nonempty set of purestrategy profile with Σi⊂R+×R being each player’s pure strategy set and H:Σ−→ Rn is the combined payoff function, where hi(σ)∈R is i ’s payoff under pure-strategy profile σ . A pure strategy for player i in Γψ consists of a pair (qi , ϕi) , where qi∈R+ and ϕi: R×R+→R is a Borel measurable function from the signal set and information acquisition set into reports. The payoff for iis hi(σ) = u(d,ω)−C(qi). Given the strategy profile σ, the expected payoff for player iis E[hi(σ)] = ZRn×Ωhi(σ)dF(s,ω;q) = ZRn×Ωu(ψ(ϕ(s,q)),ω)dF(s,ω;q)−C(qi) Furthermore the social welfare is measured by the average payoff per capita: W,u(d,ω)− n ∑ i=1 C(qi)/N 3. The First-Best Solution As a benchmark, we derive the first-best solution when there is no information asymmetries, and the society chooses the decision d and the information profile q to maximize the ex-ante social welfare. Since there are no information asymmetries, the decision rule is a function of the signal profile s and the information profile q ; so the firstbest decision is d=κ(s , q) , where the decision rule κ:Rn×Rn +→R is a Borel measurable function. Given the distribution F(s , ω ; q) and its PDF f(s , ω ; q) , the unconditional PDF of the profile sis f(s;q) = (1−γ) n ∏ i=1 f(si|ω=0; qi) + γ n ∏ i=1 f(si|ω=1; qi) The problem for society can be expressed as max q∈Rn +(− n ∑ i=1 C(qi)/N+ZRn×Ωmax d∈{0,1} E[u(d,ω)|s,q]dF(s,ω;q)) Backward induction implies that we can at first solve the optimal decision rule in the second stage and then solve the optimal information profile in the first stage. In the second stage, the expected payoff from the decision d= 0 is −αPr[ω= 1 |s , q] and the expected payoff from the decision d= 1 is −βPr[ω= 0 |s , q] . So, it is optimal to choose d=1 if and only if4 −βPr[ω=0|s,q]≥ −αPr[ω=1|s,q]⇐⇒ Pr[ω=1|s,q] Pr[ω=0|s,q]≥β α Or equivalently it is optimal to choose d=1 if and only if ∏n i=1f(si|ω=1; qi) ∏n i=1f(si|ω=0; qi)≥β(1−γ) αγ Given that f(si|ω= 1; qi) and f(si|ω= 0; qi) are PDFs of normal distributions, with mean 0 and 1, respectively, we have the optimal decision rule.
Games 2021,12, 79 5 of 33 Proposition 1. A first-best decision rule for the society is κ(s,q) = (0; if ∑n i=1qisi Q<s∗ 1; if ∑n i=1qisi Q≥s∗(1) where Q =∑n i=1qiis the committee’s aggregate information and s∗=1 2+ln Λ Q(2) with Λ=β(1−γ) αγ (3) From Proposition 1we know that when the weighted sum of signals is large enough the best choice for the society is to choose d= 1. The parameter Λ defined by Equation (3) is the cost of type-II error relative to type-I error when there is no private information and all members make decisions based on the common prior. Li [12] and Laslier and Weibull [17] , in committee decision models with different information structures, show that the cost in Equation (3) is critical in determining each committee’s decision behavior. In the special case Λ= 1, the threshold is s∗= 1 / 2, independent of the precision Q . Given the common prior members are indifferent between the two choices. When Λ> 1, we have β( 1 −γ)>αγ ; given the common prior members prefer d= 0. In this case s∗> 1 / 2 and larger precision Q decreases the threshold. On the contrary, when Λ< 1 we have β( 1 −γ)<αγ ; given the common prior members prefer d= 1. In this case s∗< 1 / 2 and larger precision Q increases the threshold. Furthermore in both cases the threshold s∗tends to 1/2 when Qgoes to infinity. For convenience we call the model with Λ= 1a priori balance and contrarily the model with Λ6=1 is called a priori imbalance. Given the optimal decision rule, the expected aggregate benefit is ZRn×Ωmax d∈{0,1} E[u(d,ω)|s,q]dF(s,ω;q) = −αγG1(s∗)−β(1−γ)(1−G0(s∗)) (4) where G1(·) and G0(·) are conditional distributions of ∑qisi/Q , when ω= 1 and ω= 0, respectively. Denote the densities by g1(·) and g0(·) , respectively. Then we can see that given s∗there is αγg1(s∗) = β(1−γ)g0(s∗)(5) Given the optimal decision in Equation (1), the society then chooses the information profile to maximize the social welfare: max q∈Rn +(−αγG1(s∗)−β(1−γ)(1−G0(s∗)) − n ∑ i=1 C(qi)/N) Applying the envelope theorem and Equation (5), and taking partial derivative of Equation (4) with respect to qi, we can have the social marginal value of information: ν(Q),β(1−γ)φ(s∗√Q) 2√Q(6) where φ(·)is the PDF of the standard normal distribution. Taking derivative of ν(Q)with respect to Qwe have dν(Q) dQ =β(1−γ)φ(s∗√Q) 16Q5/2 h−Q2−4Q+4(ln Λ)2i
Games 2021,12, 79 6 of 33 We can see that sgn(dν(Q)/dQ) = sgn(−Q2− 4 Q+ 4 (ln Λ)2) . Note that −Q2− 4 Q+ 4 (ln Λ)2 is negative for all Q∈R+ when Λ= 1. It is monotonically decreasing in Q∈R+ , and is positive at Q= 0 when Λ6= 1; furthermore, the unique positive real root of the equation dν(Q)/dQ =0 given Λ6=1 is e Q=2q1+ (ln Λ)2−2>0 (7) Therefore, when Q>e Q the marginal value of information in the ( a priori) imbalance model is decreasing in Q , while when Q<e Q the marginal value of information in the imbalance model is increasing. Lemma 1. (i) If Λ=1,ν(Q)is monotonically decreasing in Q with lim Q→0ν(Q) = +∞and lim Q→+∞ν(Q) = 0. (ii) If Λ6=1, then dν(Q) dQ S0if and only if Q Te Q and lim Q→0ν(Q) = lim Q→+∞ν(Q) = 0 When Λ= 1, the information value is a concave function of qi ; in this case, there will be a unique first-order information acquisition. When Λ6= 1, the function ν(Q) is plotted in Figure 1; it is firstly increasing from 0 to its peak and then decreasing. This implies that the value of the information is non-concave, it is very similar to the classic result of Radner and Stiglitz [18] and Chade and Schlee [19] . In a principal-agent model, Lindbeck and Weibull [20] show that the information value for the agent is similar to Figure 1. In their model the information acquisition choice is determined by the agent ability while in our model the information acquisition is determined by the cost defined in Equation (3). 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 =0.5 =1 =5 Figure 1. Marginal Value. For clarification, we will at first discuss the a priori balance case, i.e., Λ= 1 and later we will show that the results can be extended to the a priori imbalance case.
Games 2021,12, 79 7 of 33 Given the assumption that Λ= 1, the optimal threshold of the choice is s∗= 1 / 2 and the first-order condition gives the first-best information choice; the properties of social marginal value of information guarantees the existence and uniqueness of the first-best information choice. Proposition 2. Suppose Λ= 1. Then the first-best information gathering choice ˆ q=q is uniquely determined by β(1−γ)φ(s∗√Q) 2√Q=C0(q). N Since the social marginal value of information is determined by the aggregate information, each member has the same first-best information acquisition. 4. Equilibrium In this section we want to solve the equilibrium of the game Γψ given the decision rule ψ . Note that the society cannot observe each individual’s information choice, the decision rule based on the signal quality is not applicable anymore. However, we assume that the society follows the average decision rule, i.e., the society takes decision d= 1 if and only if the average of all reports is large enough. More precisely, we assume that given the report profile (r1,r2,··· ,rn), the decision rule is ψ(r) = (0, if r1+r2+···+rn n<R 1, if r1+r2+···+rn n≥R(8) then the formal definition of the equilibrium is now given by Definition 1. A pure strategy profile (q∗,ϕ∗)∈Σis a Nash equilibrium of Γψif, for each i ∈ I, (q∗ i,ϕ∗ i)∈arg max qi∈R+,ϕi∈RZRn×Ωu(ψ(ϕi(si,qi),ϕ∗ −i(s−i,q∗ −i)),ω)dF(s,ω;qi,q∗ −i)−C(qi) Although the game we are studying is a one-shot game, we can still distinguish between the information acquisition stage and the report stage. Following Hauk and Hurkens [21] and Amir and Lazzati [16] , we can firstly solve the report game assuming an exogenous profile of information choice q , then the equilibrium information choice is given by the condition that there is no incentive for any player to unilaterally deviate from q∗ ; given that member i ’s deviation can only affect his own report since the deviation is unobservable. We will solve the equilibrium with reports linear in private signals, which we call linear equilibrium. The following proposition shows that there are infinitely many such equilibria. Proposition 3. Suppose Λ= 1. Then there are infinite linear equilibria in the game Γψ . In each equilibrium the committee member i ∈ I reports r∗ i=ϕ∗ i(si,qi) = aisi+bi(9) where ai=λ·qiwith λ∈R++ (10) and bisatisfies n ∑ i=1 bi=nR −λ 2Q−λln Λ(11)
Games 2021,12, 79 8 of 33 and acquires the private information q∗=q>0, which is uniquely determined by ν(Q) = C0(q)(12) where Q =nq. Note that given equilibrium report shown in Equations (9)–(11) we have n ∑ i=1 r∗ i|ω∼ NλQω,λ2Q So given the information profile q and the report strategy in Equations (9)–(11), the expected utility is E[u(ψ(r∗),ω)] −C(qi) =−αγΦh(s∗−1)pQi−β(1−γ)n1−Φhs∗pQio−C(qi) =E[u(κ(s,q),ω)] −C(qi) where Φ(·) is the cumulative distribution function of the standard normal distribution, s∗ is defined in Equations (2) and (3), and κ(s , q) is the first-best decision rule. Therefore the reports in equilibrium should be that the final decision according to the decision rule ψ in Equation (8) is the same as that all signals and information choices are observed directly and the decision rule is κ in Equation (1). The threshold R cannot affect the final decision and the information acquisition. From Equation (12) we know that the marginal value of information is a function of the aggregate information; this implies the information is fully shared. Therefore the marginal value of information is the same for all members and all members would acquire the same information in equilibrium. Furthermore, as shown in Lemma 1, ν(Q) is monotonically decreasing and it tends to infinity when Q goes to 0 and tends to 0 when Q goes to infinity, Equation (12) has a unique positive solution for any information cost function. Furthermore, since each member’s information choice does not take into consideration other individuals’ benefit from the information acquisition, committee members cannot acquire efficiently sufficient private information. Formally, Corollary 1. Suppose Λ= 1. Thenfor eachcommittee with size n∈N , q∗<ˆ q and d(ˆ q/q∗)/dN >0 . 5. Committee Size and Social Welfare In this section we want to discuss the effects of the committee size on the social welfare and the information choice. We denote by q∗(n) = q∗(n , N) and Q∗(n) = Q∗(n , N) since each member’s information acquisition is independent of the society size.5 5.1. Rational Ignorance This subsection focuses on the effects of a finite committee size. Note that the marginal benefit is monotonically decreasing in the committee size, each individual has less incentives to acquire information in a larger committee. Proposition 4. Suppose Λ=1. Then in any linear equilibrium dq∗(n) dn <0and lim n→+∞q∗(n) = 0 This proposition is consistent with Down’s rational ignorance hypothesis (see [ 22 , 23 ]): each individual has less incentives to invest in political information acquisition in a larger committee, and each individual tends to acquire no private information as the size of the committee goes to infinity.
Games 2021,12, 79 15 of 33 0 10 20 30 40 50 0 1 2 3 4 5 6 7 8 Figure 6. Aggregate Information in Equilibrium When Λ6=1. 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 25 30 35 40 45 50 Figure 7. Asymptotic Aggregate Information When Λ6=1. 7. Extensions In this section I want to extend the model from three aspects: first of all, I will show that the limit of probability of the appropriate decision goes to 1 if and only if the marginal cost at zero information acquisition is zero when the committee members can only report 0 or 1; then I will show that the conclusions in the above sections holds when members have heterogeneous cost functions; finally I will check if the conclusions are still applicable for more general continuous distributions.
Games 2021,12, 79 16 of 33 7.1. Strategic Voting In this subsection I assume that each member can only report 0 or 1, and that the final decision follows the τ-rule: ψs(r) = (0, if r1+r2+···+rn n<τ 1, if r1+r2+···+rn n≥τ(15) where τ∈(0, 1)and nτis an integer.14 There is strategic voting [ 27 – 29 ]. According to Li et al. [30] and Duggan and Martinelli [13] there exists a cutoff equilibrium such that given qi, member ireports according to ri=1si≥t∗ i. I want to solve for the symmetric equilibrium such that all members acquire the same private information q∗ and follow the same report strategy characterized by the threshold t∗. Now suppose all members except for i follow the strategy (q∗ , t∗) , then the payoff of player iis Pr[piv|ω=0]u(d=1, ω=0)Φ[−ti√qi]Pr[ω=0] +Pr[piv|ω=1]u(d=0, ω=1)Φ[(ti−1)√qi]Pr[ω=1]−C(qi)(16) plus a constant that is independent of player i’s strategy. In the above expression, Pr[piv|ω] = n−1 nτ−1Φ[−(t∗−ω)pq∗]nτ−1Φ[(t∗−ω)pq∗]n−nτ is member i ’s conditional probability of being pivotal given the underlying state ω∈ { 0, 1 } . Taking derivative of Equation (16) w.r.t. ti , we can see that a necessary condition for an optimal threshold for member iis given by Jτ(n,ti) = 0 (17) where Jτ(n,ti),Φ[−(t∗−1)√q∗ Φ[−t∗√q∗]]nτ−1Φ[(t∗−1)√q∗] Φ[t∗√q∗]n−nτφ[(ti−1)√qi] φ[ti√qi]−Λ(18) Furthermore, Equation (16) shows that the marginal value of information of member iis: Vn(qi),−Pr[piv|ω=0]u(d=1, ω=0)φ[ti√qi]Pr[ω=0]ti 2√qi +Pr[piv|ω=1]u(d=0, ω=1)φ[(ti−1)√qi]Pr[ω=1]ti−1 2√qi Then we can show: Lemma 2. lim n→+∞Pr[piv|ω=0] = lim n→+∞Pr[piv|ω=1] = 0. That is to say, the probability of being pivotal tends to zero as the committee size goes to infinity. This implies that limn→+∞Vn(qi) = 0. Therefore, if C0( 0 )> 0, and if the committee size is large enough, the marginal value of information is strictly smaller than the marginal cost. Therefore, Proposition 10. Suppose C0( 0 )> 0and the reporting space is { 0, 1 } . There exists an n such that for all n ≥n, q∗(n) = 0, and lim n→+∞Pr[d=1|ω=0]>0and lim n→+∞Pr[d=0|ω=1]>0
Games 2021,12, 79 17 of 33 When C0( 0 )> 0, members have no incentive to acquire any information when the size is large enough. In this case, each member would report 0 when Λ> 1. Therefore the limit of probability of the appropriate decision is strictly less than 1 as long as the marginal cost at zero information acquisition is positive. Now suppose C0(0) = 0, then we have Lemma 3. Suppose C0( 0 ) = 0and the reporting space is { 0, 1 } . There exists an ˆ n such that for all n≥ˆ n each committee member reports following ∀i , ri=1si≥t∗ , where the threshold t∗ is implicitly defined as: t∗=1 2+ln Λ+ (nτ−1)lnhΦ[−t∗√q∗] Φ[−(t∗−1)√q∗]i+ (n−nτ)lnhΦ[t∗√q∗] Φ[(t∗−1)√q∗]i q∗(19) and the information choice q∗is implicitly defined as: Vn(q∗) = n−1 nτ−1β(1−γ)Φ[−t∗√q∗]nτ−1Φ[t∗√q∗]n−nτφ[t∗√q∗] 2√q∗=C0(q∗)(20) The threshold in Equation (19) is a solution of Equation (17): the existence of the threshold has been proved by Duggan and Martinelli [13] . Equation (20) solves the equilibrium information acquisition by equating the marginal value to the marginal cost. Note that as the committee size goes to infinity, the marginal value tends to zero; this implies that the equilibrium information acquisition tends to zero as the committee size goes to infinity. This is consistent with the rational ignorance, which is shown in Proposition 11. The existence of the solution in Equation (20) is guaranteed by the conclusion that lim q→0Vn(q) = +∞and lim q→+∞Vn(q) = 0 One more condition for the positive information acquisition is that the payoff with information acquisition is greater than max{−αγ , −β( 1 −γ)} , this is guaranteed by the conclusion in Proposition 11: when the committee is large enough, the probability of the appropriate decision is very close to 1 and the cost paid by each member tends to zero, and therefore it is beneficial for each member to acquire some information when the committee is large enough. Before the next proposition, we have the following lemma. Lemma 4. e H=lim sup n→+∞ φ[(t∗−1)√q∗] φ[t∗√q∗] Φ[−(t∗−1)√q∗] Φ[−t∗√q∗] is finite. The next proposition shows that the rational ignorance applies and the CJT is valid as long as the marginal cost at zero information acquisition is zero. Proposition 11. Suppose C0(0) = 0and the reporting space is {0, 1}. Then (i) lim n→+∞q∗(n) = 0; (ii) lim n→+∞Pr[d=1|ω=0] = lim n→+∞Pr[d=0|ω=1] = 0. The first conclusion shows that the rational ignorance theorem is still valid. This is because the marginal value of information tends to zero as the committee size goes to infinity.
Games 2021,12, 79 18 of 33 The second conclusion shows that the CJT is valid as long as the marginal cost at zero information acquisition is zero. The proof of the second part follows the idea of Duggan and Martinelli [13] . In the proof we show that Equations (17) and (18) and Lemma 4imply lim n→+∞Φ[−(t∗−1)√q∗] Φ[−t∗√q∗]τΦ[(t∗−1)√q∗] Φ[t∗√q∗]1−τ =1 and the above equation implies lim n→+∞Φ[−t∗pq∗]<τ<lim n→+∞Φ[−(t∗−1)pq∗] Therefore, when ω= 0, the probability of each member reporting 1 is less than τ and by the strong law of large numbers, the ratio of members reporting 1 is less than τ . Similar logic applies when ω=1. 7.2. Heterogeneous Information Acquisition In this subsection I want to extend the results into the balance model with heterogeneous information cost functions. Formally, I suppose that each individual’s cost function is from the set {C(q,k)} , which is indexed by the parameter κ∈K,[k , ¯ k] . k represents the information acquisition skill. The cost function satisfies the condition ∂2C(q , k)/∂q∂k≥ 0, which implies that increasing k increases the marginal cost of information, and ∂C(q , k)/∂k≥ 0. The distribution of the parameter is H:K→[ 0, 1 ] . 15 Denote by kn the skill profile, and kn+1= (kn , kn+1) is the skill profile with n+ 1 members, and the first nmembers’ skill is kn. Note that the cost function will not affect the reporting strategy in equilibrium in Proposition 3. Therefore the marginal value of information is still given by ν(Q) and the equilibrium information is determined by equating the marginal benefit to the marginal cost. Formally, Lemma 5. Suppose Λ= 1and the reporting space is R . Then there is a threshold k∗(n , kn) such that in any linear equilibrium q∗ i(n , kn) = 0if ki>k∗(n , kn) ; and if ki≤k∗(n , kn) , q∗ i(n,kn) = qi, where qiis uniquely determined by ν(qi+Q∗ −i(n,kn)) = ∂C(qi,ki) ∂qi where Q∗ −i(n,kn) = ∑j6=iq∗ j(n,kn). The intuition is that individual i acquires positive information if and only if ν(Q∗ −i(n , kn)) >∂C(qi , ki)/∂qi|qi=0 . We know that ∂C(qi , ki)/∂qi|qi=0 is non-decreasing in ki . Then when ki is too large, the marginal cost at zero information acquisition is too high for member i to acquire any information. This process is shown in Figure 8: if member i ’s skill is k1 , Cq( 0, k1)>ν(Q∗ −i) and (s)he has no incentive to acquire any information; if member i ’s skill is k2 , Cq( 0, k2)<ν(Q∗ −i) and there is one unique intersection between the marginal cost and marginal value, member iacquires positive information. Denote by16 K,k:∂C(q,k) ∂q|q=0=∂C(q,k) ∂q|q=0 the set of skill parameters whose marginal cost at zero information acquisition equals to that of k. We can see that it is nonempty since k∈ K. Given the information acquisition in Lemma 5, the ex-ante aggregate information is E[Q∗(n)] = ZKnQ∗(n,kn)dH(kn)(21)
Games 2021,12, 79 19 of 33 Figure 8. Information Acquisition with Heterogeneous Information Costs. We have the following conclusions: Proposition 12. Suppose Λ= 1and individuals in the society have heterogeneous information cost functions. Then (i) q∗ i(n+1, kn+1)≤q∗ i(n,kn)for all i ∈ I and n ∈N, and if Pr[k∈ K]>0, there is lim n→+∞q∗ i(n,kn) = 0for all i ∈ I (ii) Q∗(n,kn)≤Q∗(n+1, kn+1)for all i ∈ I and n ∈N, and therefore dE[Q∗(n)] dn >0 (iii) if Pr[k∈ K]>0and ∂C(q,k) ∂q|q=0=0, then lim n→+∞ E[Q∗(n)] = +∞ (iv) if Pr[k∈ K]>0and ∂C(q,k) ∂q|q=0=c>0, then lim n→+∞ E[Q∗(n)] = ν−1(c) The first part of Proposition 12 shows that the Down’s rational ignorance still holds when the cost functions are heterogeneous. Furthermore, as shown in the Appendix A, suppose there is one committee with skill profile kn and now one more member with skill kn+1 participates the committee. When kn+1≥k∗(n , kn) , the participation of member n+ 1 would not change the others’ information choice. If kn+1<k∗(n , kn) , the participation of member n+ 1 will move k∗ downwards, and therefore decrease each member’s information acquisition. Furthermore, from the conclusions in part (iii) and (iv) we can see that when Pr[k∈ K]>0, the limit of each member’s acquisition is 0 as the size tends to infinity. The second part of the proposition shows that the ex-ante aggregate information is larger in larger committee. Intuitively, when kn+1≥k∗(n , kn) , the participation of member n+ 1 does not change the aggregate information. However, if kn+1<k∗(n , kn) , then either
Games 2021,12, 79 20 of 33 members with less marginal cost or more members acquire positive private information, the aggregate information increases. Since one more member into the committee will either not change or increase the aggregate information, the ex-ante aggregate information is monotonically increasing in the committee size. Part (iii) and (iv) study the asymptotic properties. We can see that the property is determined by the distribution of skills and the marginal cost at zero information acquisition with lowest skill. If ∂C(q , k)/∂q|q=0= 0 and Pr[k∈ K]> 0, then limn→+∞Q∗(n , kn) = +∞ since there are infinite members with skills whose marginal cost at zero information acquisition is zero in the profile kn : if the limit is finite, than all members whose skill is in the set K acquires positive information. Since every skill profile leads to the infinite aggregate information when the size goes to infinity, the limit of the ex-ante aggregate information acquisition is infinite. Similarly, when ∂C(q , k)/∂q|q=0=c> 0 and Pr[k∈ K]> 0, there is limn→+∞Q∗(n , kn) =ν−1(c) since otherwise the members whose skill is in the set K acquire positive information. From Equation (21) we know that the ex-ante aggregate information approaches ν−1(c)when the committee size goes to infinity. 7.3. General Continuous Distributions In the above analysis we assume the normal distribution. In this subsection I want to extend the analysis into other continuous distributions. Formally I assume the conditional PDFs f(si|ω= 0; qi) and f(si|ω= 1; qi) are both continuous in si and qi ; they have the same support (S , S) where S , S∈[−∞ , +∞] . I assume that the conditional distributions have mean ω and precision qi . I want to see if the conclusions about the CJT are still valid when the reporting space is R and the society follows the average decision rule. Then I want to check if the conclusions are still valid in the strategic voting model and the society follows the τ-rule. First of all, suppose the society follows the average decision rule and the reporting space is R . Then note that we are trying to solve the symmetric linear equilibrium, in which each agent’s report function is linear in its own signal and all members acquire the same private information. The distribution of the average reports is determined by the average of all signals. According to Lindeberg-Lévy Central Limit Theorem,17 √n 1 n n ∑ i=1 si−ω!|ωd −→ N0, 1 q This implies that in equilibrium lim n→+∞ E[u(ψ(r∗),ω)] = −αγΦ[(s∗−1)pQ]−β(1−γ)Φ[−s∗pQ] Therefore, when the committee size is large enough, the marginal value of information is very close to ν(Q). According to this we have the following conclusions: Proposition 13. Suppose the continuous conditional PDFs are f(si|ω= 0; qi) and f(si|ω= 1; qi)and the reporting space is R. (i) lim n→+∞q∗(n) = 0; (ii) If C0(0) = 0, then lim n→+∞Q∗(n) = +∞and lim n→+∞Pr[d=0|ω=0] = lim n→+∞Pr[d=1|ω=1] = 1; (iii) If C0(0) = c>0, then lim n→+∞Q∗(n) = ν−1(c)and lim n→+∞Pr[d=ω|ω]<1.
Games 2021,12, 79 21 of 33 Since in the limit the marginal value of information is close to ν(Q) , each member tends to acquire no information when the size goes to infinity. The second and third points in Proposition 13 follow the same intuition as in Proposition 6. Now I want to test the CJT if each member can only report 0 and 1, and the society follows τ -rule in Equation (15). I assume the conditional PDFs satisfy the monotone likelihood ratio property (MLRP): Assumption 1. The likelihood ratio, f(s|ω= 1; q)/f(s|ω= 0; q) , is weakly increasing on s for all s ∈(S,S). Note that the payoff of member iis Pr[piv|ω=0]u(d=1,ω=0)[1−F(ti|ω=0; qi)] Pr[ω=0] +Pr[piv|ω=1]u(d=0, ω=1)F(ti|ω=1; qi)Pr[ω=1]−C(qi) plus a constant independent of i ’s strategy. The conditional probability of being pivotal is Pr[piv|ω] = n−1 nτ−1[1−F(t∗|ω,q)]nτ−1F(t∗|ω,q)n−nτ The equation for the threshold now is Jτ,f(n,t∗) = 0 where Jτ,f(n),1−F(t∗|ω=1; q) 1−F(t∗|ω=0; q)nτ−1F(t∗|ω=1; q) F(t∗|ω=0; q)n−nτf(t∗|ω=1; q) f(t∗|ω=0; q)−Λ Duggan and Martinelli [13] have proved the existence of the threshold for given precision qand Assumption 1. The marginal value of information is: Vn(q) = −Pr[piv|ω=0]u(d=1, ω=0)Pr[ω=0]∂F(t∗|ω=0; q) ∂q +Pr[piv|ω=1]u(d=0, ω=1)Pr[ω=1]∂F(t∗|ω=1; q) ∂q Note that with continuous PDFs, limn→+∞Pr[piv|ω= 1 ] = limn→+∞Pr[piv|ω= 0 ] = 0, which implies limn→+∞Vn(q) = 0. Therefore, when C0( 0 )> 0, and the committee is large enough, there is no symmetric equilibrium with positive information acquisition. Furthermore, the limit of marginal value of information being zero implies that each member tend to acquire no information even when C0( 0 ) = 0. Therefore, we have the following conclusions: Proposition 14. Suppose the continuous conditional PDFs are f(si|ω= 0; qi) and f(si|ω= 1; qi)and the reporting space is {0, 1}. (i) lim n→+∞q∗(n) = 0; (ii) If C0(0) = c>0, then lim n→+∞Pr[d=0|ω=0]<1and lim n→+∞Pr[d=1|ω=1]<1; (iii) If C0(0) = 0, then lim n→+∞Pr[d=0|ω=0] = lim n→+∞Pr[d=1|ω=1] = 1.
Games 2021,12, 79 22 of 33 The above proposition shows that the conclusions in Proposition 11 are still valid for more general continuous distributions satisfying MLRP. When C0( 0 ) = c> 0 and the committee is large enough, members have no incentive to acquire any information and therefore the limit of the probability of making an appropriate decision is strictly less than 1. When C0(0) = 0, we can show that lim n→+∞1−F(t∗|ω=0; q∗)<τ<lim n→+∞1−F(t∗|ω=1; q∗) Therefore the strong law of large numbers implies that the limit probability of the right decision tend to be 1 when C0( 0 ) = 0. Furthermore since the probability of the right decision tend to be 1, and each member’s information acquisition tends to zero, the equilibrium information is determined by equating the marginal cost to the marginal value when the committee is large enough. 8. Conclusions In a model where there is no interest conflict among individuals but the information is costly, we show that committee members have less incentive to acquire information in a larger committee if the committee size is large enough and each member tends to acquire zero information when the committee size goes to infinity. However, the aggregate information is increasing in the size; the CJT is partly verified. Furthermore, whether the probability of making the appropriate decision tends to one depends on the information cost function. We show that aggregate information tends to infinity if and only if the marginal cost at zero information acquisition is zero. If the marginal cost at zero information acquisition is positive, the aggregate information is bounded from above and there is some probability of making a wrong decision even when there are infinite members; in this case the CJT is not valid. The basic model is very parsimonious. In real life individuals may have interest conflicts and the information structure may be more complicated. Hence it would be interesting to investigate the CJT following two avenues. Firstly, it is very common that there are interest conflicts among individuals. Li et al. [30] has shown us that members have incentives to manipulate information when there are preference conflicts, and therefore partition equilibrium is the only monotone equilibria. Depending on the disagreement zone, the data partition may be different. It would be very interesting to introduce the preference conflicts into the model. However, from Section 7 we see that in the equilibrium with two partitions, the marginal value of information tends to zero as the committee size goes to infinity and there is an equation for the threshold; therefore we can conjecture that the limit of the probability of the right decision tends to one if and only if the marginal cost at zero information acquisition is 0 and the limit is strictly less than 1 if and only if the marginal cost at zero information acquisition is positive. Secondly, when there are participation costs, members may choose to abstain. McMurray [31] has shown that the quality of the signals will affect whether or not to participate or abstain in a voting game. It would be very interesting to extend the analysis into our model. We can predict that the participation cost and the choice of abstain will affect the information acquisition in equilibrium, and therefore the CJT. Funding: Financial support from Knut and Alice Wallenberg Research Foundation and the Fundamental Research Funds for the Central Universities is gratefully acknowledged. Institutional Review Board Statement: No applicable. Informed Consent Statement: No applicable. Data Availability Statement: No applicable. Acknowledgments: I am indebted to Jörgen Weibull, Mark Voorneveld, Erik Lindqvist, Tore Ellingsen, Xavier Vives, Paul Segerstrom, Jens Josephson, Chloé Le Coq, Topi Miettinen, Pauli
Games 2021,12, 79 23 of 33 Murto, Rune Stenbacka, Paula Mäkelä and Juuso Välimäki for their guidance and comments. I thank the two anonymous reviews for their insightful suggestions and comments. All errors are my own. Conflicts of Interest: The author declares no conflict of interest. Appendix A. The Proofs Appendix A.1. Proof of Proposition 1 The conclusion directly follows the calculations before the proposition. Hence it is ommitted here. Appendix A.2. Proof of Lemma 1 The monotonicity of ν(Q) follows the calculation before the lemma. We now prove the limits. When Λ=1, the marginal value is ν(Q) = β(1−γ)φ√Q 2 2√Q When Q goes to 0, the numerator goes to β( 1 −γ)/ 2 while the denominator goes to 0, hence the marginal value goes to infinity. On the other hand, when Q goes to infinity, the numerator goes to 0 and the denominator goes to infinity; the maginal value goes to 0. When Λ6=1, the marginal value is ν(Q) = β(1−γ)φh√Q 2+ln Λ √Qi 2√Q It is obvious that when Qgoes to infinity, ν(Q)goes to 0. When Qgoes to 0, there is lim Q→+∞ν(Q) = lim Q→+∞ 1 2√2β(1−γ)1 √Q exp"√Q 2+ln Λ √Q2 2# =lim Q→+∞ −1 2√2β(1−γ) exp"√Q 2+ln Λ √Q2 2#ln Λ+Q 2 =0 Appendix A.3. Proof of Proposition 2 The conclusion directly follows the first order condition that social marginal value of information equals to the social marginal cost. The uniqueness of the solution follows the properties of ν(Q)and C0(q). Appendix A.4. Proof of Proposition 3 Given the information acquisition profile q , suppose committee member j with j6=i reports rj=ϕj(sj,qj) = ajsj+bj, then given the conditional distribution of sj, there is rj|ω∼ N ajω+bj,a2 j qj! Therefore, ∑j6=irj n∼ N ∑j6=iaj nω+∑j6=ibj n,1 n2∑ j6=i a2 j qj!
Games 2021,12, 79 24 of 33 Furthermore, given signal si and the report strategy profile ϕ(s , q) = r , the benefit for committee member iis E[u(d,ω)|si;q] = ZRn−1×Ωu(ψ(ϕ(s,q)),ω)dF(s−i,ω|si;qi,q−i) =ZRn−1γf(s|ω=1; q) f(si;qi)u(ψ(r−i,ri), 1) + (1−γ)f(s|ω=0; q) f(si;qi)u(ψ(r−i,ri), 0)ds−i =−αγ f(si|ω=1; qi) f(si,qi)Φ n q∑j6=ia2 j/qjR−ri n−∑j6=iaj n−∑j6=ibj n −β(1−γ)f(si|ω=0; qi) f(si,qi) 1−Φ n q∑j6=ia2 j/qjR−ri n−∑j6=ibj n where f(si ; qi) = Pr[ω= 1 ]f(si|ω= 1; qi) + Pr[ω= 0 ]f(si|ω= 0; qi) is the unconditional probability density function of signal si. The optimal report for member igiven other members’ reports is ˆ ϕi∈arg max ri∈RE[u(ψ(ϕ−i(s−i,q−i),ri),ω)|si,q] The FOC implies αγ f(si,|ω=1; qi) f(si;qi)φ n q∑j6=ia2 j/qjR−ri n−∑j6=iaj n−∑j6=ibj n = β(1−γ)f(si|ω=0; qi) f(si;qi)φ n q∑j6=ia2 j/qjR−ri n−∑j6=ibj n Or equivalently, ri=qi∑j6=ia2 j/qj ∑j6=iaj sj+nR −∑ j6=i bj−1 2∑ j6=i aj−∑j6=ia2 j/qj ∑j6=iajln Λ+qi 2 Therefore given the report of all members except for i ’s are linear in their signals, the best response of member i ’s report should also be linear in his signal si . Suppose ri=ϕi(si,qi) = aisi+bi, then there is ai=qi∑j6=ia2 j/qj ∑j6=iaj (A1) bi=nR −∑ j6=i bj−1 2∑ j6=i aj−∑j6=ia2 j/qj ∑j6=iajln Λ+qi 2(A2) From our assumption we know that the above equation system is valid for all i∈ {1, 2, ··· ,n}. From (A1) we know the solution for aiis ai=λ·qiwith λ∈R++ (A3) Plugging (A3) into (A2) we have n ∑ i=1 bi=nR −λ 2Q−λln Λ
Games 2021,12, 79 31 of 33 If when Cq(0, k) = 0 and Q∗(∞,k∞)<+∞, then for all ki∈ K, there is Cq(0, ki)<ν(qi+Q∗ −i) and therefore q∗ i(∞,k∞)>0. This implies lim n→+∞Q∗(n,kn)≥lim n→+∞n·min{qi(n,kn):ki∈ K}= +∞ This contradicts with the assumption that Q∗(∞,k∞)<+∞. Therefore, we have lim n→+∞q∗ i(n,kn) = 0 since Cq(0, ki)>ν(0+Q∗ −i(∞,k∞)) for all ki/∈ K and Cq(0, ki) = ν(0+Q∗ −i(∞,k∞)) for all ki∈ K Furthermore, according to Lebesgue’s monotone convergence theorem, there is lim n→+∞ E[Q∗(n)] = ZKnlim n→+∞Q∗(n,kn)dH(kn) = +∞ Similarly, if Cq( 0, k) = c> 0, and Q∗(∞ , k∞)6=ν−1(c) . Then if Q∗(∞ , k∞)>ν−1(c) , then for all i∈ I , there is Cq( 0, ki)>ν( 0 +Q∗ −i(∞ , k∞)) , and therefore q∗ i(∞ , k∞) = 0, which implies Q∗(∞ , k∞) = 0, which contradicts our assumption. On the contrary, if Q∗(∞ , k∞)<ν−1(c) , then for all i∈ K , there is Cq( 0, ki)<ν( 0 +Q∗ −i(∞ , k∞)) and therefore q∗ i(∞ , k∞)> 0; therefore Q∗(∞ , k∞) = ∞ , which contradicts with our assumption. Therefore, we have limn→+∞q∗ i(n,kn) = 0 since Cq(0, ki)>ν(0+Q∗ −i(∞,k∞)) for all ki/∈ K and Cq(0, ki) = ν(0+Q∗ −i(∞,k∞)) for all ki∈ K Furthermore, according to Lebesgue’s monotone convergence theorem, there is lim n→+∞ E[Q∗(n)] = ZKnlim n→+∞Q∗(n,kn)dH(kn) = ν−1(c) Appendix A.20. Proof of Proposition 13 According to the explanation before Proposition 13, we can prove this by applying the same method as the proof of Propositions 6and 9. Appendix A.21. Proof of Proposition 14 According to the explanation before Proposition 14, we can prove this by applying the same method as the proof of Propositions 10 and 11. Notes 1 Cai [10] has studied another group-decision environment where group behavior, signals and information choice are all continuous. 2 In practice many group decisions have these characteristics. For example, juries need to decide if one person is a criminal or not; a recruiting committee needs to determine if one person is suitable for one position or not. 3When n=N, all individuals in the society need to join in the committee. 4We assume that when the two choices are indifferent, the society prefers d=1. 5Note that since the information acquisition is independent of the society size, there is lim n→+∞q∗(n) = lim n→+∞lim N→+∞q∗(n,N) = lim N→+∞q∗(g(N),N)
Games 2021,12, 79 32 of 33 and lim n→+∞Q∗(n) = lim n→+∞lim N→+∞Q∗(n,N) = lim N→+∞Q∗(g(N),N) where g:N→Nis a nondecreasing function satisfying g(N)≤Nand limN→+∞g(N) = +∞. 6 For more discussion on the participation cost on the effects of CJT and the voting behavior, see Krishna and Morgan [24] and the references therein. 7 To understand the form of C0( 0 ) , we have one toy example: imagine that each member starts out with the same prior belief about the underlying state and has an access to an infinitely long text with relevant but scattered information about the state. The information flow follows the Brownian motion with a constant drift: dΓt=θdt+ζde Wt , where e Wt is a Brownian motion and ζ> 0 measures how noisy the signal is. Each member acquires information by determining the time T , which results in the signal sT=T−1ΓT∼ N(θ,ζ/T) . The information acquisition cost function is C(q) = C(T/ζ) and it is the time cost. If each member needs to pay cfor each unit of time, we have C0(0) = c>0. However, if the time cost is C(T) = T2, we have C0(0) = 0. 8 The conclusions in Proposition 6are very similar to the ones in Burguet & Vives [25] : in one social learning model with information acquisition, Burguet & Vives [25] show that the aggregate information gathered over time tends to infinity if and only if the marginal cost at zero information acquisition is zero; in our model, the aggregate information is changing in the committee size but not the time. 9To see the convexity of the marginal value, note that there is d2ν(Q) dQ2=β(1−γ)φs∗√Q 128Q9/2 h−Q2−4Q+4(ln Λ)2−Q2−20Q+4(ln Λ)2−16Q3+2Q2i Therefore, ν(Q) is convex when Q is quite close to 0 or Q is sufficiently large. When Q is close to e Q , ν(Q) is concave. This means that it is possible that there are more than two intersections between the marginal value and the marginal cost less than e Q/n ; however, there are at most one intersection larger than e Q/n. 10 We are focusing on the pure-strategy equilibrium. Since the value of information is convex when q is less than e Q/n in equilibrium, mixed strategies in information acquisition might lead to larger payoff. However, as is shown below, when the committee size increases, the optimal information acquisition must be larger than e Q/n , where the value of information is concave. Therefore, taking mixed strategies into consideration would not change our conclusions about the CJT. 11 Suppose Λ6= 1. Note that the social marginal value of information equals to the marginal benefit. This implies that the first-best information acquisition q∗in the imbalance model is: (i) if C0(0)/N>ν(e Q),ˆ q=0; (ii) if C0( 0 )/N≤V(e Q) , there exists an n∗ such that for all committee size n≥n∗ , the first-best information acquisition ˆ q is uniquely determined by ˆ q=sup{q:ν(Q) = C0(q)/N}. 12 There is one equilibrium in which nobody acquires information no matter what the committee size is. However, I have shown in the proof of Proposition 7that when the committee size is large enough, committee members can get a higher payoff if (s)he can acquire a bit information. 13 When C0(Q)≥ν(Q) for all Q≥ 0, then for a committee with only one member there is no information acquisition. If this condition is violated, it is possible that each member acquires some information no matter how large the committee is; however, it is also possible that each member would not acquire any private information until the committee size is large enough. 14 Here I ignore the unanimity rule. Duggan and Martinelli [13] have shown that the solution of unanimity rule is different from other rules. Furthermore, the assumption nτ being an integer is for notation convenience, and if it is not, replace nτ by dnτe and all other calculations are the same. 15 ¯ kcan be either finite or infinite and the distribution Hcan be either discrete or continuous. 16 The conclusion in this section can be extended into the model where the cost function is parameterized by multiple parameters. For example, if the cost function is C(q , k1 , k2 , ··· , km) satisfying ∂2C(q , k1 , ··· , km)/∂q∂kj≥ 0 for kj∈[kj , ¯ kj] , then the set can be defined as K,((k1,··· ,km):∂C(q,k1,··· ,km) ∂q|q=0=∂C(q,k1,··· ,km) ∂q|q=0) And then the conclusions in Proposition 12 can be extended in a similar way. 17 From the proof of Proposition 3we know that the average of signals being normally distributed is sufficient for the existence of linear equilibrium. Since the average of signals converges to a normal distribution, when the committee size is very large, there exists symmetric linear equilibria.
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