(..q_) > qi for all gio E (qi, 1], lim^-,^

q\ and l i m ^ i

0 and l i m x _ i ip (.) < 1. An. application of Brouwer's Fixed Point Theorem now ensures the existence of q\. ' < The equilibrium path of play for the multi-round voting game is charac-terized in Figure 3.2. The first column in each interval shows the player's 67 9o 9oi 9oo 9i 9n 9io 0 | 1 1 A A 1 2 A A 1 2 A A 1 2 A Q 2 2 A Q 2 2 A Q I 1 2 2 Q 0 Figure 3.2: Equilibrium path of play in multi-round voting game choice of timing and vote if the signal received is s = 1 and the second column shows the same for s \u00E2\u0080\u0094 1. The first row shows the period at which the voter votes and the second row represents the preferred alternative. We describe the equilibrium in the multi-round game in the following proposition. P r o p o s i t i o n 3.2. The set of Perfect Bayesian equilibrium that admits a cutoff' configuration in the multi-round voting game has the following cutoff structure: Qs < Qso < Qsi A player with signal s votes for A in round 1 if her type q < qs. If q > qS7 the player waits and votes for A after observing history h n iff q < qsn. \u00E2\u0080\u00A2 We can now easily compare between the set of equilibria in the game 68 with communication and the multi-round voting game. Since the same set of equations determine the relevant cutoffs for both games, the voting cutoffs are same. Hence the set of symmetric cut-off equilibria of the multi-round game is outcome equivalent to the set of efficient responsive robust cutoff equilibria in the game with communication. This example illustrates that efficiency of flexibility of timing of votes in a voting game is not limited to a situation of aligned preferences. Our conjecture is that endogenizing the timing of votes can substitute for direct communication in aggregation of private information in a collective choice problem. Unfortunately, we don't have the general result yet. The link \u00E2\u0080\u00A2between the set of equilibria in these two types of games is not obvious. In our mechanism, to communicate her private information to others a player has to commit to a particular choice. In case of pre-play communication, a player can send a message and then deviate from her message in the voting stage. That is not feasible in the multi-round voting game. 3.4 Concluding Remarks We show that in a common interest election, full information aggregation can be achieved if flexibility in timing of votes is allowed. With symmetric and monotonic signals, this can be achieved in a symmetric Perfect Bayesian Equilibrium. Moreover, the result does not depend on the degree and na-ture of correlation between private signals. The assumption of conditionally independent signals is an almost common feature of the literature of strate-gic voting, because otherwise the analysis becomes highly complicated. Our 69 result is more general. What would happen if the preferences are diverse in a voting game with flexible timing? This will be a more difficult question to answer since even with communication the voters will have incentive to strategically withhold private information. Wi th diverse preferences, if the preferences are suffi-ciently close to induce full revelation of private information in voting with communication, the same can be achieved with the endogenous timing of votes, fn the more general case, we conjecture that there would be strong connection between the set of equilibria in the game with communication and that in the flexible timing game. In Section 3 of this chapter, we illus-trate that in the two person binary signal example. But proving it in a more general environment is an agenda for future research. How this mechanism can be implemented when the number of voters are large enough to permit anonymity? Many elections are now held online with electronic voting. Wi th advancement of Internet technology, it is now feasible to offer continuous real time updates of voting history. We conclude that if the objective of the election is information aggregation, this mechanism works well. 70 4. Committee Design with Endogenous Information 4.1 Introduction A prime task of any organization is acquisition and processing of informa-tion relevant to important decisions. Informed decisions reduce the chance of errors; guesswork increases inefficiency. Design of committees within an orga-nization for the purpose of information processing and collection is therefore very important. The Condorcet Jury Theorem of the first kind (see McLean and Hewitt [40]) states that a majority of a group is more likely than a single individual to choose the better of the two alternatives. With exogenous information, the result is trivially true if perfect information aggregation tools are available. Since a group inherently possesses more information than a single individual, whenever the problem of information aggregation can be overcome the group can do no worse. Even when information must be aggregated through (pos-sibly) imperfect mechanisms such as,voting, the Condorcet result is valid, as illustrated in the literature on strategic voting (see Feddersen and Pessendor-fer [21], Miller [43], or McLennan [41] for a sample). Chakraborty and Ghosh 71 [10] have provided a general result to this effect, fn relation to committee de-sign, these results show that with exogenous information a larger committee does better than a smaller committee under fairly general conditions. What happens if information is endogenous? Since information is a public good, with endogenous information collection there could arise a free-rider effect. When several members are entrusted with the task of information collection, each may have an incentive to save private cost by collecting less information, free riding on the information of others. However, in some cases, there is another effect that shapes the incentive for information collection at the individual level. Since, in some situations, an individual's information may not be valuable in itself, but becomes valuable only in conjunction with information provided by others, individual pieces of information are often complementary to each other. This information complementarity effect, in some cases, bolsters the incentive to collect information in a larger commit-tee and could therefore lead to better quality information collected by each member. One objective of this chapter is to illustrate the possibility of each of these effects, and show under what kind of parameter conditions (priors, cost and quality of better information, etc.) they arise. Larger committees are unambiguously better when the information com-plementarity effect applies, both because individual members collect superior information and because there are more sources of information. However, if conditions are such that larger committees are prone to'the free rider effect, the designer, in choosing committee size, may face a trade-off between the quantity and quality of information. Larger committees will base their deci-sions on several pieces of low grade information', while smaller ones act on the 72 basis of fewer pieces of high quality information. We show that the free-rider effect can be strong enough to make smaller committees informationally su-perior in some cases. The model characterizes parametric situations where it is optimal to keep committee size smaller than what is feasible. Committee design, because of its obvious importance in the process of decision-making, has attracted much attention. One strand of literature ex-tends the \"strategic information transmission\" model of Crawford and Sobel [14] to examine how a decision-maker's welfare can be influenced by varying the composition of a committee consisting of members with useful informa-tion (\"expertise\"), whose policy preferences differ from that of the decision-maker (or the organization). Gilligan and Krehbiel [26], Austeh-Smith [1, 2], Krishna and Morgan [30, 31], Battaglini [5] examine situations where all committee members have the same information but may have different pref-erences. Wolinsky [52] examines a situation where experts have the same preferences but may receive different signals. Holrnstrom [28], Dessein [17], Li and Suen [37], de Garidel-Thoron and Ottaviani [15] concentrated their focus on the effects of delegation in the same setting. Our focus in this chapter is on the incentive for collecting information by committee members23. There are a number of other studies which deal with partly similar issues. But, while we focus on the issue of committee size, most of these studies attempt to see how the incentive for information collection varies with the decision rule in a committee of fixed size. Li , Rosen and Suen [38] and Li [36] examined optimal decision rules in the context of a fixed committee size. Li [35] analyzes the organizational structure that 2 3 For an excellent survey of the existing literature on committee design with endogenous information, see Gerling, Griiner, Kiel and Schulte [25]. 73 minimizes information processing costs for a specific task. Dessein [18] also examines a model of organization with conflicting interest to study authorita-tive coordination vs. consensus in decision making. Sah and Stiglitz [49, 50] investigate similar issues of organizational design, but do not stress incen-tives to collect information. In their structure, information is exogenous. Blinder and Morgan [7] compare group versus individual decision making in an organization using an experimental study. Haleblian and Finkelstein [27], in an econometric analysis, test how managerial team size affects the orga-nizational decision-making using firm-level data and conclude that in more uncertain environments larger teams tend to do better. In the context of jury trial, Mukhopadhyay [45] identifies the free rider problem in a large jury. There are two ways in which this analysis departs from the existing lit-erature on committee design that addresses incentives to collect costly infor-mation. We consider only cheap talk mechanisms. Except for the private cost of collecting information, there is no conflict of interest among committee members or the decision maker in our model, implying that members have no incentive to withhold anything they have learned. We assume the com-mittee is advisory, and the designer cannot commit to ignore any decision relevant information ex-post, which means all the information that has been collected will be efficiently used. The only strategic choice the decision maker faces in our framework is the size of the committee. In contrast, Gerardi and Yariv [24], allow full commitment to any mechanism at the information ag-gregation stage, including ones which distort the use of available information. Persico [46] analyzes committees that make their decisions through voting, giving the designer the option to choose committee size as well as the voting 74 rule. Al l these papers assume some commitment power on the part of the designer as to how the information will be used. We think it is natural to assume in many contexts, especially those involving advisory committees, that commitment to distort the use of information once it becomes available is hard to sustain. We also allow for information of different qualities, and members can choose the precision of the information they gather. Specifically, we assume each member has access to a free informative signal, but can obtain a more precise signal if she pays a private cost. This creates a possible trade-off between quantity and quality of information from the committee designer's point of view. This aspect is absent from the other models24 of committee design that address similar issues (for example, Persico [46] and Gerardi and Yariv [24]), and has a non-trivial effect on the results. In other papers, larger committees are informationally superior to smaller ones in the weak sense, i.e. it is never the case that a smaller committee generates strictly better information than a larger one. Most papers demonstrate an upper bound to informativeness as committee size goes up. In contrast, we find situations where a smaller committee is informationally superior in the strict sense, which implies that when it comes to committee design, too many cooks may spoil the broth. One application that fits well with the model proposed in this essay is the example of a hiring committee in an academic department. Hiring com-2 4 Karoutkin and Paroush [29] has examined the quantity versus quality dilemma in an exogenous setting. In their set-up, both the quality and quantity of information are exogenous and as the committee size decreases the quality of information for each member rises automatically. In our model, this occurs as an equilibrium phenomenon (for some parametric configurations). 7 5 mittees generally consist of existing faculty members. Once the department agrees on the field of the candidate it is going to hire, the faculty as a whole wants to hire a good candidate. Hence, there is no fundamental conflict of interest among the committee members. The committee members collect in-formation about a potential recruit and then pass on their recommendations to the departmental head or the dean who then takes the final decision based on the recommendations of the committee members. The committee members get a basic idea about a candidate's qualities by attending the interview and looking at a candidate's CV. A committee mem-ber may also collect better information regarding the candidate's qualities by putting in extra effort in reading the research papers authored by the candi-date or talking to people who have better idea about the candidate from past experiences. Presumably, the extra effort induces higher personal opportu-nity cost for the committee member. Once too many members are included in a committee, each individual member's incentive to put in that extra effort would reduce. More specifically, an individual committee member may bank upon others to read the candidate's papers and make the correct recommen-dations. Since all committee members have common interest, if everybody else puts in the higher effort, this course of action saves the higher opportu-nity cost for an individual committee member without vastly compromising the efficiency of the decision. But if a large number of committee members follow the same, the efficiency of the decision would be significantly reduced. Our model analyzes a scenario almost similar to above. The organization of the chapter is as follows.- In section 2, we set up the basic model. Section 3 analyzes the model and points out the main results. 76 Section 4 concludes. 4.2 The Model An organization has to form a committee to take a decision. There are n individuals from which the committee has to be formed. The decision problem is to choose from among two options\u00E2\u0080\u0094 A (alternative) or Q (status-quo). There are two possible states of nature as well - A and Q. The common preference of each individual is such that with complete information the ^optimal decision is matched exactly with,the state. This is captured by the utility function u(d,u), where d G {A, Q} represents the ultimate decision and u is the state-of-the-world. If the decision is correct, each member of the organization gets utility equal to one. The utility from a wrong decision is 0. The preference is captured by the following utility function: f 1 iid = u . u(u,d)=\ (4.1) I 0 otherwise. The prior on the event u = A is TT . There are two signal technologies indexed by t G {h, 1} . For each technol-ogy, the signal can take one of two possible values from the set Qt = {at) qt}-A signal of type t has an accuracy pt i.e. for a t\u00E2\u0080\u0094type signal, Pr[a t|A] = Pr[g t|Q] = pt G ( | , 1 ) for t \u00E2\u0080\u0094 h,l. We also assume that the signals are conditionally independent. We assume the /-technology is costless. This means that even if an indi-vidual does not invest in acquiring information, she still gets a noisy signal 77 by default. By investing in an effort amounting to cost c, each individual can collect a signal of better quality. The better signal has a precision level ph with p h > pi. For the sake of simplicity, we will restrict ourselves to two possible values of n - 1 or 2. Increasing n to values greater than 2 does not add much insight towards understanding the main points, but complicates the analysis. We model the situation in the following way. The committee designer first chooses the committee size n (1 or 2). The member(s) of the committee then decide simultaneously whether to collect the better signal by incurring the private cost or just depend on the lower quality signal. The members report their signals to the committee designer who takes the decision after utilizing all the available information. One assumption we make here is that the committee designer cannot make any credible commitment regarding not to use any available information at the decision stage. This implies that once the signals are collected, they are utilized optimally. In other words, information is fully aggregated prior to the decision. Everybody shares a common utility function; hence there is no incentive to withhold information under this mechanism after the collection stage. The assumption regarding full information aggregation merits some dis-cussion. We do not take up the optimal mechanism design problem. Notice that if we assume the committee designer can credibly commit to any mech-anism, then the smaller committee outcome can always be mimicked with a larger committee in which the designer commits to ignore the messages of some players. But there always remains a question regarding implementabil-ity of such mechanisms. Given common interest between the decision maker 78 and the committee members it is hard to imagine that the decision maker can credibly commit to an ex post decision rule that promises to use less informa-tion than what is available to him. After all, we do not often see the heads of the departments or the deans of the faculties handing out explicit contracts to hiring committee members which say that not all recommendations would be used in decision making. We, therefore, restrict ourselves to a particular symmetric mechanism namely the ex post efficient mechanism25. However, we recognize, as has been pointed out by Gerardi and Yariv [24], for large enough n, the ex post efficient mechanism may not be the optimal mecha-nism. But that does not contradict our result. Even if we choose the optimal symmetric mechanism instead of the full information aggregation mechanism as the decision rule, a smaller committee can perform strictly better than a larger committee under certain parameter values. The intuitions behind the results remain similar. The problem we take up here is not one of finding the optimal committee size with an infinite number of potential members available. Obviously, the optimal committee size in this case is infinite. Since the default signal is somewhat informative (even if slightly), in a very large committee the prob-ability of making the correct decision is almost equal to one. One way to interpret this model is to consider it as a constrained problem. Given that only a finite number of people are available for inclusion in the committee, we discuss how committees should be formed.' For comparison across committees, we use the common expected value generated by the information collected by the committee exclusive of in-2 5 The ex post efficient mechanism is one where the committee designer uses all the available information at the decision making stage 79 formation costs. This we think is the natural welfare function when the preferences are common. If the decision affects a very large group (or an or-ganization), the individual costs are small or negligible relative to the welfare of the people affected. Henceforth, we denote this common value generated by a n-member committee by Vn (.). Before proceeding further, we need to impose some restrictions on the parameters of the model. We assume that VK~VI > c- We need this restriction to make the better signals attractive at least in some cases. For the rest of the chapter, we restrict ourselves to values of IT > | . Similar results can be obtained for IT < | because of the symmetric nature of the model. Some discussions regarding the choice of this model are worthwhile here. The standard models of committee design with endogenous information (for example, Persico [46] and Gerardi and Yariv [24]) consider a two level choice of information quality. In these models, the committee members have access to either a high quality signal or no information at all. This dampens the incentive for free riding since the alternative has no informative value. That is why these models cannot generate a strict dominance result for smaller committee size. Our model, though apparently also has a two level choice of signal quality, is actually the first approximation of a model with richer quality choice. Similar qualitative results can be obtained if we introduce a cost (which must be appropriately low) for the lower quality signal. This suggests that the result we obtain here will survive in a more general model with multi-level quality choice. 4.3 Committee Design 80 Let 9 be the vector of signals collected by all players before the decision is made. Since each individual reports their signals truthfully, 0 is completely known prior to decision-making. Then the common posterior on the state being A, defined by Bayes' Rule, is as follows: ^ ^ = m = r ^ _ ^ : ^ _ Q y (4.2) Clearly, under the no commitment mechanism, the optimal decision d*(9) is as follows: d*(9) = A 7(0) > ~. (4.3) We can write the ex-post common utility of this decision as , 7 ( 0 ) , if 7 (0) > i 1/(7(0))= , \u00E2\u0080\u00A2 ^ 1 - 7 ( 0 ) , if 7 (#)<1 Let P (0) be the probability of realization of a signal vector 9. Possible realiza-tions of 0 of course depend on the chosen signal technologies. Let T G {h, I} 2 be the vector of chosen signal technologies and QT be the set of all possible signal realizations under T. Then ex-ante common expected value from T can be written as V ( T ) = Y, P ( 0 ) U ( 7 ( 9 ) ) . (4.5) eeeT We are now in a position to discuss the equilibrium outcomes for different committee sizes. We take up that task in the next two subsections. 81 4.3.1 One member committee We first consider the single person decision problem. Without loss of general-ity, we concentrate on the range of priors TT G [|, 1]. Because of the symmetry of the structure, exactly similar results can be obtained for rr G [0, | ) . For the rest of the chapter, we assume TT > | . A i-type signal, t G {h,l} is in-formationally decision relevant if and only if TT G [ _,Pt)- In case the prior falls in this range, a signal qt pushes the posterior below | and the decision is contingent on the realization of the signal. Hence the expected utility from collecting a signal of type t without accounting for cost is given by V \ t ) = P ( q t ) ( l - 1 ( q t ) ) + P ( a t ) 1 ( a t ) for 7T G [|,Pt) a n d t = I, h. For TT > pt, the optimal decision is A independent of the signal realization. Therefore, V 1 (t) \u00E2\u0080\u0094 TT for TT > pt. Hence, the marginal benefit from collecting a /i-type signal in the single person decision problem is P h - P l if 1 < TT < pi 1 ) 1 ( T T ) = { Ph if Pl < TT < p h \u00E2\u0080\u00A2 (4.6) 0 otherwise Notice that b 1 (TT) is a continuous function of TT. The individual collects h if and only if b 1 (TT) > c. This condition induces an interval of priors over which the /i-type signal is collected by the individual. We summarize the finding in the following lemma. L e m m a 4.1. For any c < Ph \u00E2\u0080\u0094 pi, in a one member committee, a h-type signal will be collected if and only if it G [_,.Ph ~ c]. 82 Proof. Since b 1 (IT) is strictly decreasing in IT for all n G [puPh], and reaches 0 at ph, for any c > 0, 61 (IT) > c induces a unique interval for any c < Ph~Pi = The social value function from this individual decision (ignoring the cost) is then given by 2 6 Since Ph \u00E2\u0080\u0094 c > pi, the /-type signal has no decision relevance in the range where the to-type signal is not being collected and hence cannot affect the social value. One observation may be worth mentioning here. Notice that as pi falls, Ph \u00E2\u0080\u0094 Pi rises. As we mentioned earlier, ph \u00E2\u0080\u0094 Pi indicates the upper bound on the range of cost parameter for which an individual can be induced to collect a high quality signal. This shows that the presence of a free informative signal may sometimes lower the incentive for providing effort to find a better signal. In other words, free information may be expensive from a social point of view. 4.3.2 Two member committee Next we move on to our analysis of a two member committee. Let ti'E {to, 1} denote player i's choice of signal technology and V2 (tj, tj) represent the pay-off to players when players i and j choose f and tj respectively. For different 2 6 We are slightly abusing our notation here. We write the value function assuming that optimal decision d* is taken for all ranges of prior, given the incentive constraint for the individual. We do this throughout the chapter for the sake of comparison across the committees. max b 1 (n) . I ( 4 . 7 ) 83 parameter values, three types of pure strategy equilibria may exist in a two member committee: (i) (h, h), (ii) (h, I) or (I, h) and (iii) (I, I). We next characterize the necessary and sufficient conditions for existence of these pure strategy equilibria in terms of the value function. 1. V 2 (h, h) \u00E2\u0080\u0094 V 2 (h, I) > c is necessary and sufficient for (h, h) to be an equilibrium. 2. V 2 (h, h) \u00E2\u0080\u0094 V 2 (h, I) < c and V 2 (h, I) \u00E2\u0080\u0094 V 2 (I, I) > c are necessary and sufficient for (h, I) and (I, h) to be equilibria. 3. V 2 (h, I) \u00E2\u0080\u0094 V 2 (1,1) < c is necessary and sufficient for (1,1) to be an equilibrium. These conditions follow directly from the definition of Bayesian Nash Equilibrium which is the equilibrium concept we use throughout this essay. Since we are in an environment with common preferences where information is a public good, players' ex-ante utilities (net of cost) depend only on the types of signal technologies chosen by them, but not on the exact combination of choices. Hence, the value functions mentioned above are sufficient statistics for characterizing the equilibria in this environment. An immediate corollary of the above conditions is that at least one pure strategy equilibrium exists for all parameter values. The necessary and sufficient conditions for existence of different pure strategy equilibria also show that neither (h, h) nor (/, I) can coexist with (h, I) or (/, h) as equilibria. Moreover, for parameter values such that (h, I) is an equilibrium, (l,h) is an equilibrium as well. Our next lemma proves 84 that when (1,1) or (h,h) is the unique pure strategy equilibrium, no mixed strategy equilibrium can exist. L e m m a 4.2. For parameter values such that (1,1) or (h,h) is the unique pure strategy equilibrium, then it is the unique equilibrium of the two person game. Proof. First consider that (I, I) is the unique pure strategy equilibrium. Then, V 2 (h, h) - V2 (h, I) < c and V2 (h, I) - V 2 (1,1) < c. Now consider any mixed strategy o for player j where o is the probability of playing h. Generically, for a to be part of a mixed strategy equilibrium, we must have the following a V 2 (h, h) + (l- a) V 2 (h, l ) - c = aV2 (I, h) + (1 - a) V2 (I, I) or, equivalently a [V2 (h, h) - V 2 (I, h)] +(l-a) [V2 (h, I) - V 2 (I, I)] = c. From the two strict inequalities described above, and the fact that V 2 (h, I) = V2 (l,h), it follows that, for any o __ 0. and g (ph) < 0. An application of Intermediate Value The-orem then ensures that $ is non-empty. The boundedness obviously follows from definition of To see that $ is closed, assume the contrary. Then there exists a sequence {7r n} such that g (Trn) = 0 for all n, but lim n_oo g (TTn) ^ 0. But this contradicts continuity of g (.). | 88 Since $ C 9t, the last Lemma ensures that both max, and min, $ exist and are unique. Let us denote these two by n lm a x and n lm i n respectively. Now define c\ = b 1 (n lmax) and ci = b 1 {ir lmiD) \u00E2\u0080\u00A2 Since 6 1 (.) is strictly decreasing in this range, C\ < c-i- An inspection of b 1 (.) and b 2 (.; I) reveals that we essen-tially confront two possible scenarios. If II (a;, q^) < II q{), then TTLMAX = ^Lin e {^-(ai,qh) ,U(qhqi)). On the other hand, if U ( a h q h ) > U(qhqi), then n lmin = II (qu q{) and -n lmax = II (a,, g h ) . Finally, for any c, define ff = max {n\ (c) , ff/ (c)}, where 7r^ ( C ) and 7f; (c) are as defined in Lemma 4.3 and Lemma 4.4 respectively. Let T* denote the vector of equilibrium signal technologies chosen by a committee of size n. Now, we can state our first proposition. Proposition 4.1. 1. c < C\ and ix \u00C2\u00A3 (ph \u00E2\u0080\u0094 c, ft] are sufficient for T{* = /, and existence of a T2* \u00E2\u0082\u00AC {(h, h), (h, I), (I, h)}. 2. c > C2 and TT \u00C2\u00A3 (ffj (c), \u00E2\u0080\u0094 c] are necessary and sufficient for Tj* = / i andT; = (1,1) . Proof. See Appendix A.2. | Proposition 4.1 describes the effect of committee size on incentives. It identifies conditions under which the larger committee is subject to either the free rider effect or the information complementarity effect, and proves neither of these are empty. The first part provides a sufficient condition for information complementarity effect to come into play. Under these condi-tions, the equilibrium signal quality is I in a one member committee, but in a two member committee, there always exists an equilibrium with at least one h-type signal. The second part of the proposition provides necessary and 89 sufficient conditions for the free-rider effect to apply. It identifies parameter zones where a one member committee collects a /i-type signal in equilibrium, while in a two member committee the only equilibrium is (I, I). The result is quite intuitive. The incentive to free-ride is more intense .when the cost of quality information is high. \u00E2\u0080\u00A2 On the other hand, signals become complementary towards more extreme priors, since the cumulative signals must be sufficiently strong to have any relevance to the decision. Hence, for the complementarity effect, we need a sufficiently low value of c along with relatively large prior values. Exactly the opposite is true for the free-rider effect. We need a sufficiently high c along with not so extreme priors for this to happen. Some discussion regarding the set of equilibria for different parameter values is worthwhile here. In the following two figures, we illustrate these in the case of a particular set of parameter values, namely when n (qt, qi) < n (a;, qh) . The other cases can be dealt with similarly. We illustrate the above proposition with the help of a numerical example. For parameter values Vh = 0.8 and Vi = 0.6, the values of ci and c2 in the above proposition can be easily computed to be 0.0727 and 0.1077 respec-tively. Figures 4.1 and 4.2 are drawn with parameter values Vh = 0.8 and Pi = 0.6 to show the pure strategy equilibria for different values of prior. In Figure 4.1, we take c = 0.02 < c\. The panel at the top right hand corner of Figure 4.1 identifies the pure strategy equilibria when the prior falls in different zones. For vr close to \ (TT \u00E2\u0082\u00AC A = [0.5000,0.5625)), the bigger committee admits (h,l) and (l,h) equilibria. For extreme values of TT (TT G E = (0.9118,1.0000]), neither type of committee puts in an ef-90 Note: The above figure is drawn with parameter values pi = 0.6, = 0.8 and c = 0.02. Figure 4.1: Pure Strategy Equilibria for Different Prior Values: c < Ci. fort to collect a h\u00E2\u0080\u0094type signal. For other values of TT , the bigger commit-tee admits a (h,h) equilibrium. When rr falls in C = (0.7800,0.8214] or D = (0.8214,0.9118], one (h,h) equilibrium exists in a two member com-mittee while a single member committee does not put any effort at all into gathering the better signal2 8. For values of TT G B = [0.5625,0.7800], the bigger committee collects two h\u00E2\u0080\u0094type signals while the single member com-2 8 This is just for the purpose of illustration. We ca.n have other parameter configurations satisfying the conditions mentioned in Proposition 4.1 such that values of prior exist where a one member committee collects I in equilibrium while a two member committee admits (h,l) or (l,h) as equilibria. 91 Note: The above figure is drawn with parameter values pi = 0.6, p^ = 0.8 and c \u00E2\u0080\u0094 0.15. Figure 4.2: Pure Strategy Equilibria for Different Prior Values: c > c2. mittee collects only one in equilibrium. B, C and D are the zones where a clear domination of the complementarity effect over the free-rider effect can be seen as identified in Proposition 1. Figure 4.2 illustrates the set of equilibria for c = 0.15 > c2. Given that c > c2, for priors very close to \ (TT G A = [0.5000,0.6042]), 7\* - h and T 2 = (h, I) or (/, h). Both these equilibria generate the same social value29 as the single member committee for this particular situation. Then, we have a 2 9 0 f course, there exist mixed strategy equilibria for the bigger committee in this range, which may do strictly better than the smaller committee outcome. 92 range of priors (TT \u00E2\u0082\u00AC B = (0.6042,0.6500]) where T* = h and T2* = (I, I). This is the parameter zone we identified in part 2 of Proposition 4.1. Towards the extreme prior values belonging to the interval C \u00E2\u0080\u0094 (0.6500,1.0000], Tj* = I and T2* = (1,1). Notice that in both cases illustrated above, for values of prior close enough to \ , Tj* = h, and T2* = (h,l) or (l,h), that is the smaller and the bigger committees are informationally equivalent. In other words, both committees generate the same level of decision relevant information and hence social value. This is what Persico [46] and Gerardi and Yariv [24] identified. Their results show that given the parameters, there exists an upper bound on the social value that can be generated by increasing committee size. In their models, in the absence of an exogenous cost of designing a bigger com-mittee, a smaller committee cannot do strictly better. In our next propo-sition, we provide conditions for the strict dominance result of the smaller committee. In part 2 of Proposition 4.1 we have identified the conditions under which a one member committee collects h in equilibrium, while the unique equi-librium in the two member committee is (1,1). This is necessary for welfare dominance of the smaller committee, but may not be sufficient. In our next proposition, we show-that for all values of and pi, we can find parame-ter zones where Tj* \u00E2\u0080\u0094 h and T2* = (1,1) become sufficient for strict welfare dominance of the smaller committee. P r o p o s i t i o n 4.2. For allpi,ph \u00E2\u0082\u00AC (\, 1) such that P h > Pi, there exist TT and c such that the smaller committee dominates the larger committee welfarewise. Proof. See Appendix A.2. | 93 We illustrate again with numerical examples. We show in the proof of 2 Proposition 4.2 that if ph > p2+^_piy2 > = h a n d F 2 = (M) a r e sufficient for strict welfare dominance of the smaller committee. For parameter values ph = 0.8 and p ; = 0.6, ph > p2+^_pi)2- For these values of ph and ft, c2 =;0.1077 and .Tr^ = 0.6923 can be easily computed. Now, at Tt = 0.6923, V 2(l,l) \u00E2\u0080\u0094 0.6923.' For any c > c2, the values of TT for which T2* = h and T2* = (1,1) are strictly less than 0.6923. Since V 2(l,l) is increasing in TT, hence V 2(l, I) < V l(h) \u00E2\u0080\u0094 0.8 in the relevant zone. Next we consider parameter values ph \u00E2\u0080\u0094 0.8 and pi = 0.7. Now, ph is 2 strictly less than p 2 + ^ _ p [ ^ \u00E2\u0080\u00A2 We choose c = 0.05 such that (h,h) is not an equilibrium. In the range of prior given by (0.6190,0.7500), T\u00C2\u00A3 = h and T2* = (1,1). To show that this no longer suffices for strict welfare dominance of the smaller committee, we choose TT = 0.74 where V 2(l,l) = 0.8008, but V l(h) = 0.8. But for lower values of TT, in particular for values of TT in the interval (0.6190,0.7381), V 1 (h) > V 2 (1,1) still holds. Notice that so far we have chosen the common value of each individual for welfare comparisons. Suppose instead we choose a utilitarian welfare function that sums up each individual's net welfare to obtain social welfare. As before a necessary condition for welfare dominance of the smaller committee is that the smaller committee chooses to collect a h-type signal while the larger committee collects only Z-type signals. As shown in Proposition 4.1 this can happen only if TT \u00E2\u0082\u00AC (^i(c) ,ph~ c] and c > c2. Notice that social welfare if information is collected by a smaller committee is now 2V 1 (h) \u00E2\u0080\u0094 c, while in case of the larger committee social welfare is 2V 2(l, I). Suppose we again take Ph = 0.8 and pi \u00E2\u0080\u0094 0.6. As we have already shown, for these values of 94 Ph and pi, C2 \u00E2\u0080\u0094 0.1077 and 7ilmin = 0.6923. Suppose we choose c = 0.15. Then the highest value of n for which Tx* = h and T2* = (I, I) is 0.65. At TT = 0.65,2V 2(U) = 1.344. Since V2{l,l) is increasing in n, 2V2(l,l) < 1.344 in the relevant zone, while 2Vl(h) \u00E2\u0080\u0094 c = 1.45. Hence, for these parameter values the smaller committee dominates the bigger committee welfarewise even for utilitarian welfare function. This example illustrates the robustness of our result with respect to the specification of the welfare function. For high enough c, the incentive to free ride dominates the positive in-centive towards good quality signals arising out of complementarity between the signals in a larger committee, at least for some values of the prior. In a smaller committee, the lower free riding incentive may induce the individ-uals to collect a larger number of high quality signals, making the smaller committee better from a social point of view. This indicates the quality -quantity trade-off that the committee designer faces in an environment with multiple information qualities. Notice that the last proposition is a strict violation of the Condorcet Jury Theorem of the first kind. Even under the assumption of full information aggregation, a smaller committee strictly dominates a bigger committee. In a two level signal quality choice model with the alternative to the informative signal being a completely uninformative one, this result cannot be obtained. It can be easily seen in this model'by verifying that as pi \u00E2\u0080\u0094> \,-b1 (TT) \u00E2\u0080\u0094> b2 ( T T ; I) for all values of n. 4.4 Concluding Remarks 95 This essay doesn't attempt to give a general characterization of the solution to optimal committee design problems. Previous analyzes have provided a weak dominance result for smaller committees in that there is an upper bound on the value of information generated by increasing the committee size. We, on the other hand, argue that not allowing committee members choice over the quality of information they collect misses an important aspect of the problem. 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[53] Young, Peyton (1986): \"Optimal Voting Rules.\" Journal of Economic Perspectives (Winter) 9: 51-64- \u00E2\u0080\u00A2 - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ' -[54] Young, Peyton (1988): \"Condorcet's Theory of Voting.\" American Po-litical Science Review 82: 1231-44. 103 A. Appendices A . l Appendix to Chapter 2 A.1.1 Proof of Lemma 2.3 Proof. We first express V (m + 2; q D ) \u00E2\u0080\u0094 V (m; q D ) expl ic i t ly in terms of the parameters as the following: V(m. + 2;qD)-V(m;qD)= ~ { x J P * ^ ~ * ( 1 \" TO) . . : _ : 2 \ m + 2 _ x ( 1 _ p ) : r ( 1 _ y r ) f / / j x=fi . , (m)+l ^ x=0 \ X ' m / \ x=Rs{m) 104 To prove this lemma we first show that for a generic 1 < R < m, E (\u00E2\u0084\u00A2y (i - v)m-x - E {m+x v (i - pr+2-x x=0 ^ ' ' x=0 ^ ' ' m + l ^ { l - p ) m + l - R p R { - ^ - { l - p ) \ (A.2) 7V \" r \m+l and E r V ^ i - p f - E \ \ ) p m + 2 - x a - P)X x=R ^ ' x-R+X ^ ' We use the method of induction to prove the above results. For notational convenience we write * (R) = E (TV (! - p ) m _ x - E (m+x \ x (! - p ) m + 2 _ x and TO / \ TO+2 \u00E2\u0080\u00A2 I 9 \ V'(i2) = E ( m ) p m \" s ( 1 - p ) a - E l p m + 2 - x a-p) x -X = H W '. . . X=R+I \ x y For R = 1, the first equality holds. That both and of equa-tion [A.2] are equal to p (1 \u00E2\u0080\u0094 p) m \u00E2\u0080\u0094 (m + 1) p(l \u00E2\u0080\u0094 p ) m + 1 can be verified after some algebraic manipulation. Now suppose that the equality holds for some arbitrary R. We show that then the equality holds for R + 1. We need to 105 show that ' ( f l + [ ) - l's+0 ( 1 - \" R ~ V + 1 {\u00C2\u00A3TT - <> - < A - 4 > But, m-R+l m+l\ , m + l - R R { R by induction hypothesis. After cancelling (1 \u00E2\u0080\u0094 p)m R p R from equation [A.4] and some simplification we are left to prove the following: m \ , fm + l\ ,H 2 (m\ fm + 2 s - i ; ' 1 - \" - I \u00C2\u00AB J ( 1 - p ) + U r U + i J p ( 1 - p ) H ) P _ ( ' R + 1 1 ) P ( 1 \" P ) which can be shown to be true using the fact that (n^1) \u00E2\u0080\u0094 (\") = {x\u00E2\u0084\u00A2i)-Next we consider the second equality. That equation [A.3] holds for R = m can be seen from verifying that both LHS and RHS of [A.3] are equal to p 2 (1 \u00E2\u0080\u0094 p)m \u00E2\u0080\u0094 mp (1 \u00E2\u0080\u0094 p ) m + 1 . Suppose equation [A.3] holds for some arbitrary R + l. We will show that then it will hold for R. The induction hypothesis 106 tells us that V (R + 1) = (Rtl)P m~ R (1 - {P - S i ) \u00E2\u0080\u00A2 N o w > 4>(R) = ^ ( i ? + i ) + ( ^ ) p m - f i ( i - p ) i i - ( ^ 1 2 ) p m - ^ 1 ( i - p ) i ? + 1 R+ijr v r / r \u00E2\u0084\u00A2+i + ( fljp-\u00C2\u00AB (1 _ p ) \u00C2\u00AB _ + i ] ^ (1 - p) We need to show that {^R) = O^ 1^ (1\" p)* {P \" ^ T i ) \u00E2\u0080\u00A2 (A'5) After cancelling p m _ - R (1 \u00E2\u0080\u0094 p)^ from both sides of equation [A.5] and some simplification we are left to prove fl+i>(i-p) - UJ (1-p)+U)^U+i> (1\"p) m + l \ 2 / m which can be shown to be true again using the relation (n^1) \u00E2\u0080\u0094 (\") = (fix)-Since equations [A.2] and [A.3] hold for a generic R,, we-can now write V (rri + 2; qjj) \u00E2\u0080\u0094 V (m; QD) in the following manner: V(m + 2;qD) - V { m ; q D ) = TT (1 - qD) 0 (Rs (m)) + (1 - TT) qD^ (Rs (m)) = (* ^ V - pr+i-Mm) P ^ ( M ) { - (i - P) } - (i -\Rs(m)J [ m + l J nm+l-Rs(m) /-\u00E2\u0080\u00A2 _ ^ ^ ( m ) / \u00E2\u0080\u009E Rs (m) + p^-KW (1 - p)H^m> p - (1 - T T ) q D \ (A.6) m + l 107 Suppose that V (m; qo) \u00E2\u0080\u0094 V (m \u00E2\u0080\u0094 2; qo) > 0. We first consider the case when ttl < I < p . If ^ g l > l - p , then V (m + 2; to) - V (m; qD) > 0. Suppose < 1 - p. Then V (m + 2; - V (m; qD) > 0 if and only if (1 _ p ) m + i - \u00C2\u00AB . M p * s ( m ) 7 r ( i _ ^ ) p _ ^ g l ( 1 _ p r - H s ( m ) ^ s ( m ) _ l 7 r ( 1 _ t o ) p - ^ g i But, /?s (m) < 1 /?., (m) - 1 < Rs (m) < 1 m + l ~ 2 ' m - 1 _ m + l ~ 2 Hence, using R s (m \u00E2\u0080\u0094 2) = Z?s (m) \u00E2\u0080\u0094 1 from Lemma 2.2 in equation [A.6] and writing m \u00E2\u0080\u0094 2 in place of m we find V(m;qD) - V ( m - 2 ; q D ) ' U M - 1 ) K l - P r ~ M m ) ^ { ^ - ( 1 - P ) } ^ ( 1 - q D ) > o p m - i i . M ( 1 _ p ) \u00C2\u00AB . ( m ) - l ( 1 _ ^ 1 _ p _ ^Mzl { 1 _ p ) m - R s ( m ) p M m ) _ l n { 1 _ q D ) p - M ^ l ' . S i n C e 1 - p - 5 ^ 1 > when *-<\">-1 < ^ ^ m - 1 F m + l 1/ (m; to) - V {m - 2;qD) > 0 V {m + 2;qD) - V [m; qD) > 0. Now consider the other possibility that > ^ > 1 - p. If ^ f f i < P, 108 then V (m + 2; qD)-V (m; to) > 0. Suppose Rf^ > p. Then V (m + 2; to)-V (m; to) > 0 if and only if p m+l -\u00C2\u00AB s (m) ( 1 _ p)R.(m) ( 1 _ ^ ^ ^ ) _ ( 1 _ p ) {I - p ) m + 1 - R s { m ) p ^ h (I - qD) ^ 1 P p m - ^ W ( 1 _ p ) ^ M - l ( 1 _ ^ M _ ( 1 _ p ) ( l _ p r - \u00C2\u00AB . ( - ) p f l . ( m ) - l 7 r ( 1 _ g | ? ) ^ - p But, ^ > i > ^tr > I Hence, ' m+l 2 m\u00E2\u0080\u00941 m+l 2 ' V (m; qD) \u00E2\u0080\u0094 V (m \u00E2\u0080\u0094 2; qD) >0 r W (1 - p ) * ^ ) - 1 (1 - vr)^ ^ - (1 - p ) ^ (1 - P ) \u00E2\u0080\u0094 ^ M p ^ M - % (1 - to) < - p Since ^ - ( 1 - p ) > S^-t-* w h e n ^ W - 1 > M a i oi cc flj(m) ;> K s ( m ) - i nen m _j . m + 1 , m + l r 7 7 1 - 1 ^ V (m; 0 =^ K (m + 2; to) - V' (m; to) > 0 in this case as well. This completes the proof of the lemma. | A.1.2 Proof of. Lemma 2.4 x Proof. We have to deal with two cases separately: Case 1: pj > 1; The jury puts relatively higher weight on acquitting the guilty. Case 2: pj < 1. The jury puts relatively higher weight on convicting the innocent. 109 \u00E2\u0080\u00A2 Case 1: pj > 1. The uninformative decision of the jury in this case is C. Hence, V (0; qp) \u00E2\u0080\u0094 -(1 - ir)qD. Now suppose n 0 is the smallest positive integer such that ajury of size n 0 votes informatively. Then R s (no) = R s (no + 1) = 1. For any n < no, the jury votes uninformatively for C. Hence, R s (no) = 1. Now R s (no + 1) \u00E2\u0080\u0094 1 or 2 by Lemma 2.2. For any n 0 > 1, if R s (n 0 + 1) = 2, then again by Lemma 2.2, R s (no \u00E2\u0080\u0094 1) = 1 contradicting the hypothesis that no is the smallest integer such that the jury votes informatively. For no = 1, Rs (2) = 1 follows directly from the fact that pj > 1. We will argue that V (n0; qD) > V (0; qD) V (n 0 + 2; qD) > V (n0; qD) and V (n0 + 1; qD) > V (0; qD) => V (n 0 + 3; qD) > V (n 0 + 1; q D ) . Then an application of Lemma 2.3 will be sufficient for the proof in this case. Suppose V(no]qo) > V(0;qD)- Then, using Rs(no) \u00E2\u0080\u0094 1, we can show that (1 \u00E2\u0080\u0094 TT) qoP n\u00C2\u00B0 > TT (1 \u00E2\u0080\u0094 qjj) (1 \u00E2\u0080\u0094 p) n\u00C2\u00B0 follows. As shown in the proof of Lemma 2.3, we can write V (n 0 + 2; q^) \u00E2\u0080\u0094 V (n0; qo) as n 0 + 1 1 > 1 ' ^ + 1 M i - 9 * ) u - p r V (nQ + 2; qD) - V (n0; qD) ^ ( l - q D ) ( l - p ) ^ p [ ^ - ( l - p ) + ( l - T T ) q D p n \u00C2\u00B0 ( l - p ) ( p - ^ j ( l - p ) (p > 0 D + l 1 n 0 + 1 + p n 0 + 1 ( l - p ) where we made use of the fact (1 \u00E2\u0080\u0094 IT) qoP 710 > vr (1 \u00E2\u0080\u0094 qo) (1 \u00E2\u0080\u0094 p) n\u00C2\u00B0 and 110 The subcase for V (no + 1; qo) > V (0; qo) can be treated similarly. Case 2: pj < 1. The uninformative decision of the jury in this case is A. Hence, V (0; qo) = - T T ( 1 - qD) . Again, as in Case 1, suppose no is the smallest positive integer such that a jury of size n 0 votes informatively. Then Rs (no) \u00E2\u0080\u0094 no and Rs (no + 1) = no + 1. For any n < no, the jury votes uninformatively for A. Hence, Rs (no) = no. Now Rs (no + 1) = n 0 + 1 or n 0 by Lemma 2.2. For any n 0 > 1, if Rs (no + 1) = \"-Oi then again by Lemma 2.2, Rs (no \u00E2\u0080\u0094 1) = no \u00E2\u0080\u0094 1 contradict-ing the hypothesis that no is the smallest integer such that the jury votes informatively. For n-o \u00E2\u0080\u0094 1, Rs (2) = 2 follows directly from the fact that PJ < 1. Now we can proceed similarly as in Case 1 to show that V (no; qo) > V(0;qD) => V (n 0 + 2; qD) >,V(n0;qD) and V(n0 + l;qD) > V(0;qD) V(nQ + l;qD)>V{n0 + 3;qD)- I A.2 Appendix to Chapter 4 A.2.1 Proof of Lemma 4.3 Proof. In writing b2 (TT; I) explicitly in terms of the parameters, we have to consider two separate cases. Case I: II (ahqh) > U (quqi) I l l Using equations [4.10] and [4.9], we can write b 2(n;l) = < P h - p 2 - 2 p i { l - p i ) n if TT __