A resurrection of the Condorcet Jury Theorem

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1 Theoretcal Economcs 4 (2009), / A resurrecton of the Condorcet Jury Theorem YUKIO KORIYAMA Département d Économe, École Polytechnque, Palaseau BALÁZS SZENTES Department of Economcs, Unversty College London Ths paper analyzes the optmal sze of a delberatng commttee where () there s no conflct of nterest among ndvduals and () nformaton acquston s costly. The commttee members smultaneously decde whether to acqure nformaton, and then make the ex-post effcent decson. The optmal commttee sze, k, s shown to be bounded. The man result of ths paper s that any arbtrarly large commttee aggregates the decentralzed nformaton more effcently than the commttee of sze k 2. Ths result mples that overszed commttees generate only small neffcences. KEYWORDS. Votng, nformaton aggregaton, costly nformaton. JEL CLASSIFICATION. D72, D INTRODUCTION The classcal Condorcet Jury Theorem (CJT) states that large commttees can aggregate decentralzed nformaton more effcently than small ones. Its orgn can be traced to the dawn of the French Revoluton, when Mare-Jean-Antone-Ncolas de Cartat, marqus de Condorcet (1785) nvestgated the decson-makng processes n socetes. 1 Recent lterature on commttee desgn has ponted out that f nformaton acquston s costly, the CJT fals to hold. The reasonng s that f the sze of a commttee s large, a commttee member realzes that the probablty that she can nfluence the fnal decson s small compared to the cost of nformaton acquston. As a result, she mght prefer to avod ths cost and free-rde on the nformaton of others. Therefore, large commttees mght generate lower socal welfare than small ones. These results suggest that n the presence of costly nformaton acquston, optmally choosng the sze of a commttee s both an mportant and delcate ssue. In ths paper, we characterze the Yuko Koryama: yuko.koryama@polytechnque.edu Balázs Szentes: b.szentes@ucl.ac.uk The authors are grateful to semnar partcpants at the Unversty of Chcago, Berkeley, Unversty of Rochester, École Polytechnque, Arzona State Unversty, Maastrcht Unversty, Unversty of Hawa, and Unversty College London for helpful comments. Fnancal support from the NSF s gratefully acknowledged. 1 Summares of the hstory of the CJT can be found n, for example, Grofman and Owen (1986), Mller (1986), and Gerlng et al. (2005). Copyrght c 2009 Yuko Koryama and Balázs Szentes. Lcensed under the Creatve Commons Attrbuton- NonCommercal Lcense 3.0. Avalable at

2 228 Koryama and Szentes Theoretcal Economcs 4 (2009) welfare loss assocated wth overszed commttees and show that ths loss s surprsngly small n certan envronments. Therefore, as long as the commttee sze s large enough, the careful desgn of a commttee mght not be as mportant as t was orgnally thought to be. In fact, f ether the nformaton structure s ambguous, or the commttee has to make decsons n varous nformatonal envronments, t mght be optmal to desgn the commttee to be as large as possble. Commttee desgn receves consderable attenton from economsts because, n many stuatons, groups rather than ndvduals make decsons. Informaton about the desrablty of the possble decsons s often decentralzed: ndvdual group members must separately acqure costly nformaton about the alternatves. A classcal example s a jury tral where a jury has to decde whether a defendant s gulty or nnocent. Each juror ndvdually obtans some nformaton about the defendant, at some cost of effort (payng attenton to the tral, nvestgatng evdence, etc.). Lkewse, when a frm s facng the decson whether to mplement a project, each member of the executve commttee can collect nformaton about the proftablty of the project (by spendng tme and exertng effort). Yet another example s the decson of an academc department to hre a new member. Each member of the recrutng commttee must revew the applcatons ndvdually before makng a collectve decson. What these examples have n common s the fact that nformaton acquston s costly and often unobservable. The exact setup analyzed n ths paper s descrbed as follows. A group of ndvduals has to make a bnary decson. There s no conflct of nterest among the group members, but they have mperfect nformaton about whch decson s best. Frst, k ndvduals are asked to serve on a commttee. Then the commttee members smultaneously decde whether to nvest n an nformatve sgnal. Fnally, the commttee makes the optmal decson gven the acqured nformaton. We do not explctly model how the commttee members communcate and aggregate nformaton. Instead, we smply assume that they end up makng the ex post effcent decson. 2 The only strategc choce an ndvdual must make n our model s whether to acqure a sgnal upon beng selected to serve on the commttee. The central queston of our paper s the followng: how does the commttee sze, k, affect socal welfare? Frst, for each k, we fully characterze the set of equlbra (ncludng asymmetrc and mxed-strategy equlbra). We show that there exsts k P ( ) such that whenever k k P, there s a unque equlbrum n whch each commttee member nvests n nformaton wth probablty one. Furthermore, the socal welfare generated by these equlbra s an ncreasng functon of k. If k > k P, then there are multple equlbra, many of whch nvolve randomzaton by the members. We show also that the socal welfare generated by the worst equlbrum n the game, where the commttee sze s k, s decreasng n k f k > k P. The optmal commttee sze k s defned as the smallest commttee sze such that there s an equlbrum n the commttee wth k members that maxmzes socal welfare among all equlbra n any commttee. We 2 Snce there s no conflct of nterest among the ndvduals, t s easy to desgn a mechansm that s ncentve compatble and effcently aggregates the sgnals. Alternatvely, one can assume that the collected nformaton s hard.

3 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 229 prove that the optmal commttee sze s ether k P or k P + 1. Ths mples that the CJT fals to hold: large commttees can generate smaller socal welfare than small commttees. We show, nevertheless, that f the sze of the commttee s larger than k, even the worst equlbrum generates hgher socal welfare than the unque equlbrum n the commttee of sze k 2. That s, the welfare loss due to an overszed commttee s qute small. For our results to hold, we need an assumpton on the dstrbuton of the sgnals. Ths assumpton, stated formally n the next secton, requres the margnal beneft from a sgnal to decrease fast. Lterature Revew Although the Condorcet Jury Theorem provdes mportant support for the bass of democratc decson makng, many of the premses of the theorem have been crtczed. Perhaps most mportantly, Condorcet assumes sncere votng. That s, each ndvdual votes as f she were the only voter n the socety. Ths means that an ndvdual votes for the alternatve that s best, condtonal on her sgnal. Austen-Smth and Banks (1996) show that n general, sncere votng s not consstent wth equlbrum behavor. Ths s because a ratonal ndvdual votes condtonal not only on her sgnal, but also on her beng pvotal. Feddersen and Pesendorfer (1998) show that as the jury sze ncreases, the probablty of convctng an nnocent can actually ncrease under unanmty rule. A varety of papers show, however, that even f the voters are strategc, n certan envronments the outcome of votng converges to the effcent outcome as the number of voters goes to nfnty. Feddersen and Pesendorfer (1997) nvestgate a model n whch preferences are heterogeneous and each voter has a prvate sgnal concernng whch alternatve s best. They construct an equlbrum for each populaton sze, such that the equlbrum outcome converges to the full nformaton outcome as the number of voters goes to nfnty. The full nformaton outcome s defned as the result of a votng game where all nformaton s publc. Myerson (1998) shows that asymptotc effcency can be acheved even f there s populaton uncertanty; that s, f a voter does not know how many other voters there are. In contrast, the Condorcet Jury Theorem mght fal to hold f the nformaton acquston s costly. Mukhopadhaya (2003) consders a model smlar to ours where voters have dentcal preferences, but nformaton acquston s costly. He shows by numercal examples that mxed-strategy equlbra n large commttees may generate lower expected welfare than pure-strategy equlbra n small commttees. 3 Martnell (2006) also ntroduces a cost of nformaton acquston nto a model where sgnals are bnary. He allows the precson of the sgnals to depend contnuously on the amount of nvestment and proves that f the cost and the margnal cost of the precson are zero at the zero level of precson, then the decson s asymptotcally effcent. More precsely, f the sze of the commttee converges to nfnty, then there s 3 The results are qute dfferent f the votng, rather than the nformaton acquston, s costly: see e.g. Börgers (2004).

4 230 Koryama and Szentes Theoretcal Economcs 4 (2009) a sequence of symmetrc equlbra n whch each member nvests only a lttle, and the probablty of a correct decson converges to one. 4 We thnk that Martnell (2006) contrbutes substantally towards the understandng of the effcency propertes of group decson makng when there s no fxed cost assocated wth nformaton acquston. However, we beleve that the exstence of a fxed cost s an essental feature of many envronments. Indeed, one has to pay the prce of a newspaper, even f t wll be thrown away later. The management of a company has to pay for a consultant, even f the consultant s work wll be completely gnored. Smlarly, a juror has to st through a tral, even f he decdes not to pay any attenton. What these examples have n common s that a postve cost must be nvested n the sgnals even f the precson s zero. These examples also suggest that exertng more effort mght lead to more accurate nformaton, though ths s not modeled n our paper. Hence, we thnk that the result n our paper s an mportant complement to Martnell (2006). If nformaton acquston has fxed as well as varable costs, then an asymptotc effcency result, lke the one n Martnell (2006), does not hold. However, we conjecture that our result s stll vald n some form. That s, the effcency loss due to large commttees s small. 5 Numerous papers analyze the optmal decson rules n the presence of costly nformaton. Persco (2004) dscusses the relatonshp between the optmal decson rules and the accuracy of the sgnals. He shows that a votng rule that requres a strong consensus n order to upset the status quo s optmal only f the sgnals are suffcently accurate. The ntuton for the extreme case, where the decson rule s unanmty rule, s the followng. Under unanmty rule, the probablty of beng pvotal s small. However, ths probablty ncreases as the sgnals become more accurate. Therefore, n order to provde a voter wth an ncentve to nvest n nformaton, the sgnals must be suffcently accurate. L (2001), Gerard and Yarv (2008), and Gershkov and Szentes (forthcomng) show that the optmal votng mechansm sometmes nvolves ex post neffcent decsons. That s, the optmal mechansm mght specfy neffcent decsons for certan sgnal profles. We beleve that n many stuatons such a commtment devce s not avalable, whch s why we smply restrct attenton to ex post effcent decson rules. We beleve that ths s the approprate assumpton n the context of a delberatng commttee n whch there s no conflct of nterest among ndvduals. Secton 2 descrbes the model. The man theorems are stated and proved n Secton 3. Secton 4 concludes. Some of the proofs are relegated to the Appendx. 2. THE MODEL There s a populaton consstng of N (> 1) ndvduals. The state of the world, ω, can take one of two values, 1 and 1, wth Pr[ω = 1] = π (0, 1). The socety must make a 4 In hs accompanyng paper, Martnell (2007) analyzes a model n whch nformaton has a fxed cost, voters are heterogeneous n ther costs, and abstenton s not allowed. He shows that f, on the one hand, the support of the cost dstrbuton s not bounded away from zero, asymptotc effcency can be acheved. If, on the other hand, the cost s bounded away from zero, and the number of voters s large, nobody acqures nformaton n any equlbrum. 5 The dffculty of solvng these models s that t s partcularly hard to characterze the set of all equlbra.

5 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 231 decson, d, whch s ether 1 or 1. There s no conflct of nterest among ndvduals. Each ndvdual has a beneft of u (d,ω) f decson d s made when the state of the world s ω. In partcular, 0 f d = ω u (d,ω) = q f d = 1 and ω = 1 (1 q) f d = 1 and ω = 1, where q ( (0, 1)), ndcates the severty of a type-i error. 6 Each ndvdual can purchase a sgnal at a cost c (> 0) at most once. Sgnals are d across ndvduals, condtonal on the realzaton of the state of the world. The ex post payoff of an ndvdual who nvests n nformaton s u c. Each ndvdual maxmzes her expected payoff. There are two stages of the decson-makng process. At Stage 1, k ( N ) members of the socety are desgnated at random to serve on a commttee. At Stage 2, the commttee members decde smultaneously and ndependently whether or not to nvest n nformaton. Then, the effcent decson s made gven the sgnals collected by the members. We do not model explctly how commttee members delberate at Stage 2. Snce there s no conflct of nterest among the members, t s easy to desgn a communcaton protocol that effcently aggregates nformaton. Alternatvely, one can assume that the acqured nformaton s hard. Hence, no communcaton s necessary for makng the ex post effcent decson. We focus merely on the commttee members ncentves to acqure nformaton. Next, we turn to the defnton of socal welfare. Frst, let µ denote the ex post effcent decson rule. That s, µ s a mappng from sets of sgnals nto possble decsons. If the sgnal profle s (s 1,..., s n ), where n s the number of sgnals acqured, then µ(s 1,..., s n ) = 1 ω [u (1,ω) s 1,..., s n ] ω [u ( 1,ω) s 1,..., s n ]. Socal welfare s measured by the expected sum of the payoffs of the ndvduals, s1,...,s n,ω[n u (µ(s 1,..., s n ),ω) c n], where the expectaton also takes nto account possble randomzaton by the ndvduals. That s, n can be a random varable. If the commttee s large, then a member mght prefer to save the cost of nformaton acquston and choose to rely on the opnons of others. By contrast, f k s too small, there s too lttle nformaton to aggregate, and thus the fnal decson s lkely to be neffcent. The questons are: What value of k maxmzes ex-ante socal welfare? How bg s the welfare loss n sub-optmal commttees? Next, we defne the optmal commttee sze formally. 6 In the jury context, where ω = 1 corresponds to the nnocence of the suspect, q ndcates the severty of the error of convctng an nnocent.

6 232 Koryama and Szentes Theoretcal Economcs 4 (2009) DEFINITION. The optmal sze of the commttee s the smallest commttee sze k ( ) such that there s an equlbrum n the commttee wth k members that maxmzes socal welfare among all equlbra n any commttee. In our model, the optmal commttee sze s k f () there s an equlbrum n the commttee wth k members that maxmzes socal welfare among all equlbra n any commttee, and () each member acqures nformaton wth postve probablty n ths equlbrum. Ths s because a member who does not nvest n nformaton can be elmnated from the commttee wthout changng the ncentves of the other members. We compute socal welfare n all equlbra n suboptmal commttees and compare the welfare loss n these commttees. Snce the sgnals are d condtonal on the state of the world, the expected beneft of an ndvdual from the ex post effcent decson s a functon of the number of sgnals acqured. We defne ths functon as follows: η(n) = s1,...,s n,ω[u (µ(s 1,..., s n ),ω)]. We assume that the sgnals are nformatve about the state of the world, but only mperfectly so. That s, as the number of sgnals goes to nfnty, the probablty of makng the correct decson s strctly ncreasng and converges to one. Formally, the functon η s strctly ncreasng and lm n η(n) = 0. An ndvdual s margnal beneft from collectng an addtonal sgnal, when n sgnals are already obtaned, s g (n) = η(n + 1) η(n). Note that lm n g (n) = 0. For our man theorem to hold, we need the followng assumpton. ASSUMPTION 1. The functon g s log-convex. 7 That s, g (n + 1)/g (n) s ncreasng n n ( ). Whether or not ths assumpton s satsfed depends only on the prmtves of the model that s, on the dstrbuton of the sgnals and on the parameters q and π. An mmedate consequence of the assumpton s the followng. REMARK 1. The functon g s decreasng. PROOF. Suppose to the contrary that there exsts an nteger n 0 such that g (n 0 +1) > g (n 0 ). Snce g (n + 1)/g (n) s ncreasng n n, t follows that g (n + 1) > g (n) whenever n n 0. Hence g (n) > g (n 0 ) > 0 whenever n > n 0. Ths mples that lm n g (n) 0, whch s a contradcton. Next, we explan that Assumpton 1 essentally means that the margnal value of a sgnal decreases rapdly. Notce that the functon g beng decreasng means that the 7 The standard defnton of convex functons requres the functons to have convex domans. Ths defnton, however, can be naturally extended to functons wth non-convex domans by requrng the convexty nequalty to hold only on the domans (see Peters and Wakker 1987).

7 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 233 margnal socal value of an addtonal sgnal s decreasng. We thnk that ths assumpton s satsfed n most economc and poltcal applcatons. How much more does Assumpton 1 requre? Snce g s decreasng and lm n g (n) = 0, there always exsts an ncreasng sequence {n m } m=1 such that g (n m ) g (n m + 1) s decreasng n m. Hence, t s stll natural to restrct attenton to nformaton structures where the second dfference n the socal value of a sgnal, g (n) g (n + 1), s decreasng. Recall that Assumpton 1 s equvalent to (g (n) g (n +1))/g (n) beng decreasng. That s, Assumpton 1 requres that the second dfference n the value of a sgnal not only decreases, but does so at an ncreasng rate. In general, t s hard to check whether ths assumpton holds because t s often dffcult (or mpossble) to express g (n) analytcally. The next secton provdes examples where the assumpton s satsfed. 2.1 Examples for the log-convexty assumpton Frst, suppose that the sgnals are normally dstrbuted around the true state of the world. The log-convexty assumpton s satsfed for the model where π + q = 1. That s, the socety would be ndfferent between the two possble decsons f nformaton acquston were mpossble. The assumpton s also satsfed even f π + q 1, f the sgnals are suffcently precse. Formally, we have the followng result, whch, lke the other results n ths secton, s proved n the Appendx. PROPOSITION 1. Suppose that s N (ω,σ). () If q + π = 1 then Assumpton 1 s satsfed. () For all q, π, there exsts ɛ q,π > 0 such that Assumpton 1 s satsfed f ɛ q,π > σ. In our next example the sgnal s ternary that s, ts possble values are { 1, 0, 1}. In addton, Pr(s = ω ω) = p r, Pr(s = 0 ω) = 1 r, and Pr(s = ω ω) = (1 p)r. Notce that r ( (0, 1)) s the probablty that the realzaton of the sgnal s nformatve, and p s the precson of the sgnal condtonal on beng nformatve. PROPOSITION 2. Suppose that the sgnal s ternary. Then for any r ( (0, 1)) there exsts a threshold p(r ) (0, 1) such that f p > p(r ), Assumpton 1 s satsfed. Next, we provde an example where the log-convexty assumpton s not satsfed. Suppose that the sgnal s bnary that s, s { 1, 1} and Pr(s = ω ω) = p, Pr(s = ω ω) = 1 p.

8 234 Koryama and Szentes Theoretcal Economcs 4 (2009) PROPOSITION 3. If the sgnal s bnary then Assumpton 1 s not satsfed. Snce bnary sgnals are commonly assumed n the lterature on commttee desgn, we further explan the negatve result n Proposton 3. For smplcty, assume that the pror s symmetrc. In ths case, the ex post optmal decson after recevng a set of sgnals s 1 f and only f there are (weakly) more sgnals 1 n the set than sgnals 1. The margnal socal value of a sgnal can be postve only f the ex post optmal decson s dfferent from the one wthout ths sgnal wth postve probablty. Ths can happen only f there are as many sgnals 1 as sgnals 1 pror to recevng the addtonal sgnal, and n partcular, f the number of sgnals s even. Ths mples that the margnal socal values of the second, fourth, sxth, etc. sgnals are all zero. Hence, the functon g s not even decreasng. Notce that n ths example, even f the cost of nformaton s zero there always exsts an equlbrum n whch only one ndvdual obtans a sgnal. Indeed, f an ndvdual knows that there s exactly one sgnal collected by the others, he has no ncentve to nvest n nformaton because he cannot have any effect on the optmalty of the fnal decson. Ths argument s due to the fact that each sgnal realzaton has the same strength. Were an ndvdual to know that hs sgnal would be potentally very nformatve, he would nvest f the cost s low enough, because, at least wth small probablty, hs nformaton nduces a more nformatve decson. Most of the lterature assumes that commttee members aggregate nformaton by a bnary votng procedure where members vote ether yes or no (or abstan). Assumng bnary sgnals n ths case s perhaps less problematc because even f a member has a strong sgnal he cannot communcate t va hs vote. We beleve, however, that allowng varaton n the strength of the sgnals s an mportant and realstc feature of the world. Allowng, for example, that a juror leans slghtly towards a gulty verdct and at the same tme another juror s sure that the defendant s nnocent mght have mportant consequences. We beleve also that even f there s bnary votng at the end of the delberaton, commttee members are able to communcate ther nformaton to the others, at least partally. Focusng on the delberaton mght be more mportant than analyzng the votng game. 3. RESULTS Ths secton s devoted to our man theorems. We frst characterze the set of equlbra for all k ( ). The next subsecton shows that f k s small, the equlbrum s unque, and each member ncurs the cost of nformaton (Proposton 4). Secton 3.2 descrbes the set of mxed-strategy equlbra for suffcently large k (Proposton 5). Fnally, Secton 3.3 states and proves the man results (Theorems 1 and 2). 3.1 Pure-strategy equlbrum Suppose that the sze of the commttee s k. If the frst k 1 members acqure nformaton, the expected gan from collectng nformaton for the k th member s g (k 1). She

9 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 235 s wllng to nvest f ths gan exceeds the cost of the sgnal; that s, f c < g (k 1). (1) Ths nequalty s the ncentve compatblty constrant that guarantees that a commttee member s wllng to nvest n nformaton f the sze of the commttee s k. 8 PROPOSITION 4. Let k denote the sze of the commttee. There exsts k P such that there exsts a unque equlbrum n whch each member nvests n a sgnal wth probablty one f and only f k mn{k P, N }. Furthermore, the socal welfare generated by these equlbra s monotoncally ncreasng n k ( mn{k P, N }). PROOF. Recall from Remark 1 that g s decreasng and lm k g (k ) = 0. Therefore, for any postve cost 9 c < g (0), there exsts a unque k P such that g (k P ) < c < g (k P 1). (2) Frst, we show that f k k P then there s a unque equlbrum n whch each commttee member nvests n nformaton. Suppose that n an equlbrum, the frst k 1 members randomze accordng to the profle (r 1,..., r k 1 ), where r [0, 1] denotes the probablty that the th member nvests. Let I denote the number of sgnals collected by the frst k 1 members. Snce the members randomze, I s a random varable. Notce that I k 1 and r1,...,r k 1 [g (I )] g (k 1) because g s decreasng. Notce also that from k k P and (2), t follows that g (k 1) > c. Combnng the prevous two nequaltes, we get r1,...,r k 1 [g (I )] > c. Ths nequalty mples that no matter what the strateges of the frst k 1 members are, the k th member strctly prefers to nvest n nformaton. From ths observaton, the exstence and unqueness of the pure-strategy equlbrum follow. It remans to show that f k > k P, such a pure-strategy equlbrum does not exst. If k > k P, then g (k 1) < c. Therefore, the ncentve compatblty constrant, (1), s volated, and there s no equlbrum where each member ncurs the cost of the sgnal. Fnally, we must show that the socal welfare generated by these pure-strategy equlbra s ncreasng n k ( mn{k P, N }). Notce that snce N > 1, c < g (k 1) = η(k ) η(k 1) < N (η(k ) η(k 1)). 8 In what follows, we gnore the case where there exsts k such that c = g (k ). Ths equalty does not hold genercally, and would have no effect on our results. 9 If c > η(1) η(0), then nobody has an ncentve to collect nformaton, hence k P = 0.

10 236 Koryama and Szentes Theoretcal Economcs 4 (2009) g (k ) k c FIGURE 1. Expected gan g (k ) and the cost c. After addng N η(k 1) c k, we get N η(k 1) c(k 1) < N η(k ) c k. The left-hand sde s the socal welfare generated by the equlbrum n a commttee of sze k 1, whle the rght-hand sde s the socal welfare nduced by a commttee of sze k. Fgure 1 s the graph of g (k ) and c when s N (ω, 1), π =.3, q =.7, and c = The expected gan s decreasng and log-convex. In ths example, k P = 11. The amount of nformaton purchased n any equlbrum s neffcently small. Ths s because when a commttee member decdes whether or not to nvest, she consders her prvate beneft rather than the socety s beneft. Snce nformaton s a publc good, ts socal beneft s bgger than ts ndvdual beneft. Hence, the total number of sgnals acqured n an equlbrum s smaller than the socally optmal number. Ths s why the socal welfare s monotoncally ncreasng n the commttee sze k as long as k k P. Mukhopadhaya (2003) has proved a result that corresponds to the statement of Proposton 4 for the case where the sgnals are bnary. He has also shown by numercal examples that mxed-strategy equlbra can yeld lower expected welfare n large commttees than n small commttees. Our analyss goes further by analytcally comparng the expected welfare of all mxed-strategy equlbra.

11 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem Mxed-strategy equlbrum Suppose now that the sze of the commttee s larger than k P. We consder strategy profles n whch the commttee members can randomze when makng a decson about ncurrng the cost of nformaton acquston. The followng proposton characterzes the set of mxed-strategy equlbra (ncludng asymmetrc ones). We show that each equlbrum s characterzed by a par of ntegers (a,b). In the commttee, a members nvest n a sgnal wth probablty one and b members acqure nformaton wth a postve probablty that s less than one. The rest of the members, k (a + b) n number, do not ncur the cost. We call such an equlbrum a type-(a,b) equlbrum. PROPOSITION 5. Let the commttee sze be k (> k P ). There exsts an equlbrum where a members nvest for sure, b members nvest wth probablty r (0, 1), and k (a + b) members do not nvest, f and only f where the frst two nequaltes are strct whenever b > 0. a k P a + b k, (3) PROOF. Frst, we explan that f, n an equlbrum n whch one member nvests wth probablty r 1 (0, 1) and another nvests wth probablty r 2 (0, 1), then r 1 = r 2. Snce the margnal beneft from an addtonal sgnal s decreasng, our games exhbt strategc substtuton. That s, the more nformaton the others acqure, the less ncentve a member has to nvest. Hence, f r 1 < r 2, then the ndvdual who nvests wth probablty r 1 faces more nformaton n expectaton and has less ncentve to nvest than the ndvdual who nvests wth probablty r 2. On the other hand, snce r 1, r 2 (0, 1), both ndvduals must be exactly ndfferent between nvestng and not nvestng, a contradcton. Now, we formalze ths argument. Let r [0, 1] ( = 1,..., k ) be the probablty that the th member collects nformaton n an equlbrum. Suppose that r 1, r 2 (0, 1) and r 1 > r 2. Let I 1 and I 2 denote the number of sgnals collected by members 2, 3,..., k and by members 1, 3,..., k, respectvely. Notce that snce r 1 > r 2 and g s decreasng, r2,r 3,...,r k [g (I 1 )] > r1,r 3,...,r k [g (I 2 )]. (4) On the other hand, a member who strctly randomzes must be ndfferent between nvestng and not nvestng. Hence, for j = 1, 2, rj,r 3,...,r k [g (I j )] = c. Ths equalty mples that (4) should hold wth equalty, whch s a contradcton. Therefore, each equlbrum can be characterzed by a par (a,b) where a members collect nformaton for sure and b members randomze but collect nformaton wth the same probablty. It remans to show that there exsts a type-(a,b) equlbrum f and only f (a,b) satsfes (3). Frst, notce that whenever k > k P, n all pure-strategy equlbra k P members

12 238 Koryama and Szentes Theoretcal Economcs 4 (2009) nvest wth probablty one and the remanng members never nvest. In addton, the par (k P, 0) satsfes (3). Therefore, we have to show only that there exsts an equlbrum of type-(a,b) where b > 0 f and only f (a,b) satsfes a < k P < a + b k. (5) Suppose that a members nvest n nformaton for sure and b 1 nvest wth probablty r. Let G (r ; a,b) denote the expected gan from acqurng nformaton for the (a + b)th member. That s, b 1 b 1 G (r ; a,b) = r (1 r ) b 1 g (a + ). =0 We clam that there exsts a type-(a,b) equlbrum f and only f there exsts r (0, 1) such that G (r ; a,b) = c. Suppose frst that such an r exsts. We frst argue that there exsts a type-(a,b) equlbrum n whch b members nvest wth probablty r. Ths means that the b members, who are randomzng, are ndfferent between nvestng and not nvestng. The a members who nvest for sure strctly prefer to nvest because the margnal gan from an addtonal sgnal exceeds G (r ; a,b). Smlarly, those members who do not nvest, who number k (a +b), are strctly better off not nvestng because ther margnal gans are strctly smaller than G (r ; a,b). Next, we argue that f G (r ; a,b) = c does not have a soluton n (0, 1), then there exsts no type-(a,b) equlbrum. Ths mmedately follows from the observaton that f b members are strctly randomzng, they must be ndfferent between nvestng and not nvestng, and hence G (r ; a,b) = c. Therefore, t s suffcent to show that G (r ; a,b) = c has a soluton n (0, 1) f and only f (5) holds. Notce that G (r ; a,b) s strctly decreasng n r because g s strctly decreasng. Also observe that G (0; a,b) = g (a ) and G (1; a,b) = g (a + b 1). By the Intermedate Value Theorem, G (r ; a,b) = c has a soluton n (0, 1) f and only f G (1; a,b) < c < G (0; a,b), whch s equvalent to g (a + b 1) < c < g (a ). (6) Recall that k P satsfes g (k P ) < c < g (k P 1). Snce g s decreasng, (6) holds f and only f a < k P and a + b > k P. That s, the two strct nequaltes n (5) are satsfed. The last nequalty n (5) must hold because a + b cannot exceed the sze of the commttee, k. Fgure 2 graphcally represents the set of pars (a,b) that satsfy (3). Accordng to the prevous proposton, there are several equlbra n whch more than k P members acqure nformaton wth postve probablty. A natural queston to ask s: can these mxed-strategy equlbra be compared from the pont of vew of socal welfare? The next proposton partally answers ths queston. We show that f one fxes the number of members who acqure nformaton for sure, then as the number of randomzng members ncreases, the socal welfare generated by the equlbrum decreases. Ths proposton plays an mportant role n determnng the optmal sze of the commttee.

13 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 239 k b k P + 1 k P 1 k P k P + 1 k a FIGURE 2. The set of mxed-strategy equlbra. PROPOSITION 6. Suppose that k s such that there are both type-(a,b) and type- (a, b + 1) equlbra. Then the type-(a, b) equlbrum generates strctly hgher socal welfare than the type-(a, b + 1) equlbrum. In order to prove ths proposton we need the followng result, whch s proved n the Appendx. LEMMA 1. () G (r ; a,b) > G (r ; a,b + 1) for all r (0, 1]. () r a,b > r a,b+1, where r a,b and r a,b+1 are the solutons for r of G (r ; a,b) = c and G (r ; a,b + 1) = c respectvely. PROOF OF PROPOSITION 6. Suppose that a members collect nformaton wth probablty one and b members do so wth probablty r. Let f (r ; a,b) denote the beneft of an ndvdual; that s, b b f (r ; a,b) = r (1 r ) b η(a + ). Clearly =0 f (r ; a,b) b b 1 b b = r 1 (1 r ) b η(a + ) r (b )(1 r ) b 1 η(a + ). r Notce that =1 b b 1 = b 1 and =0 b b 1 (b ) = b.

14 240 Koryama and Szentes Theoretcal Economcs 4 (2009) Therefore the rght-hand sde of the prevous equalty can be rewrtten as b b 1 b 1 b 1 b r 1 (1 r ) b η(a + ) b r (1 r ) b 1 η(a + ). 1 =1 After changng the notaton n the frst summaton, ths can be further rewrtten as b 1 b 1 b 1 b 1 b r (1 r ) b 1 η(a + + 1) b r (1 r ) b 1 η(a + ) =0 Ths last expresson s just bg (r ; a,b), and hence we have Next, we show that Snce f (0; a,b) = f (0; a,b + 1) = η(a ), f (r a,b ; a,b) f (r a,b+1 ;a,b + 1) =0 =0 b 1 b 1 = b r (1 r ) b 1 η(a + + 1) η(a + ). =0 f (r ; a,b) = bg (r ; a,b). r f (r a,b ; a,b) f (r a,b+1 ; a,b + 1) > b(r a,b r a,b+1 )c. (7) = f (r a,b ; a,b) f (0; a,b) f (r a,b+1 ; a,b + 1) f (0; a,b + 1) = b ra,b 0 G (r ; a,b)d r b ra,b+1 By part () of Lemma 1, ths last dfference s larger than b ra,b 0 G (r ; a,b)d r b ra,b G (r ; a,b)d r = b G (r ; a,b + 1) d r. ra,b r a,b+1 G (r ; a,b)d r. By part () of the lemma, we know that r a,b+1 < r a,b. In addton, snce G s decreasng n r, ths last expresson s larger than b(r a,b r a,b+1 )G (r a,b ; a,b). Recall that r a,b s defned such that G (r a,b ; a,b) = c and hence we can conclude (7). Let S(a, b) denote the socal welfare n the type-(a, b) equlbrum; that s, Then S(a,b) = N f (r a,b ; a,b) c(a + b r a,b ). S(a,b) S(a,b + 1) = N f (r a,b ; a,b) c(a + b r a,b ) [N f (r a,b+1 ; a,b + 1) c(a + b r a,b+1 )] > N b(r a,b r a,b+1 )c cb(r a,b r a,b+1 ) = (N 1)cb(r a,b r a,b+1 ) > 0, where the frst nequalty follows from (7), and the last one follows from part () of Lemma 1.

15 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem The man theorems Frst, we show that the optmal commttee sze s ether k P or k P + 1. Second, we prove that f k > k, then even the worst possble equlbrum yelds hgher socal welfare than the unque equlbrum n the commttee of sze k 2. THEOREM 1. The optmal commttee sze, k, s ether k P or k P + 1. We emphasze that for a certan set of parameter values, the optmal sze s k = k P, and for another set, k = k P + 1. PROOF. Suppose that k s the optmal sze of the commttee and the equlbrum that maxmzes socal welfare s of type-(a,b). By the defnton of optmal sze, a + b = k. If b = 0, then all of the commttee members nvest n nformaton n ths equlbrum. From Proposton 4, k k P follows. In addton, Proposton 4 states that the socal welfare s ncreasng n k as long as k k P. Therefore, k = k P follows. Suppose now that b > 0. If there exsts an equlbrum of type-(a,b 1), then, by Proposton 6, k s not the optmal commttee sze. Hence f the sze of the commttee s k, there does not exst an equlbrum of type-(a,b 1). By Proposton 5, ths mples that the par (a,b 1) volates the nequalty chan (3) wth k = k. Snce the frst and last nequaltes n (3) hold because there s a type-(a,b) equlbrum, t must be the case that the second nequalty s volated. That s, k P a + b 1 = k 1. Ths mples that k k P + 1. Agan, from Proposton 4, t follows that k = k P or k P + 1. Next, we turn our attenton to the potental welfare loss due to overszed commttees. THEOREM 2. In any commttee of sze k (> k ), all equlbra nduce hgher socal welfare than the unque equlbrum n the commttee of sze k 2. The followng lemma plays an mportant role n the proof. We pont out that the proof of ths lemma s the only place where we use the log-convexty assumpton (Assumpton 1). For our prevous results, we need the functon g only to be decreasng, whch s a consequence of Assumpton 1. LEMMA 2. For all k 1 and, g (k 1){g ( ) g (k )} {g (k ) g (k 1)}{η( ) η(k )}, (8) wth equalty f and only f = k or k 1. PROOF OF THEOREM 2. Recall that S(a, b) denotes the expected socal welfare generated by an equlbrum of type-(a,b). Usng ths notaton, we have to prove that S(k 2, 0) < S(a,b). From Theorem 1, we know that k = k P or k P +1. By Proposton 4, S(k P 2, 0) < S(k P 1, 0). Therefore, n order to establsh S(k 2, 0) < S(a,b), t s enough to show that for all pars of (a,b) that satsfy (3). S(k P 1, 0) < S(a,b) (9)

16 242 Koryama and Szentes Theoretcal Economcs 4 (2009) Notce that f a + members nvest n nformaton, whch happens wth probablty b r a,b (1 r a,b ) b n a type-(a,b) equlbrum, the socal welfare s N η(a + ) c(a + ). Therefore, b b S(a,b) = r a,b (1 r a,b ) b N η(a + ) c(a + ) =0 b b = r a,b (1 r a,b ) b N η(a + ) c c a =0 b b = N r a,b (1 r a,b ) b η(a + ) c(a + b r a,b ). =0 In the last equaton, we use the dentty b b =0 ra,b (1 r a,b ) b = b r a,b. Therefore, (9) can be rewrtten as b b N η(k P 1) c(k P 1) < N r a,b (1 r a,b ) b η(a + ) c(a + b r a,b ). =0 Snce a k P 1 by (3) and b N, the rght-hand sde of ths nequalty s larger than b b N r a,b (1 r a,b ) b η(a + ) c(k P 1 + N r a,b ). =0 Hence t suffces to show that b b N η(k P 1) c(k P 1) < N r a,b (1 r a,b ) b η(a + ) c(k P 1 + N r a,b ). =0 After addng c(k P 1) to both sdes and dvdng through by N, we have b b η(k P 1) < r a,b (1 r a,b ) b η(a + ) c r a,b. (10) =0 The left-hand sde s the payoff of an ndvdual f k P 1 sgnals are acqured by others, whle the rght-hand sde s the payoff of an ndvdual who s randomzng n a type-(a,b) equlbrum wth probablty r a,b. Snce ths ndvdual s ndfferent between randomzng and not collectng nformaton, the rght-hand sde of (10) can be rewrtten as b 1 b 1 r a,b (1 r a,b ) b 1 η(a + ). =0 Hence (10) s equvalent to b 1 b 1 η(k P 1) < r a,b (1 r a,b ) b 1 η(a + ). (11) =0

17 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 243 By Lemma 2, b 1 b 1 g (k P 1) r a,b (1 r a,b ) b 1 g (a + ) g (k P ) =0 g (k P ) g (k P 1) b 1 b 1 r a,b (1 r a,b ) b 1 η(a + ) η(k P ). =0 (12) Notce that b 1 b 1 r a,b (1 r a,b ) b 1 g (a + ) = c < g (k P 1), (13) =0 where the equalty guarantees that a member who s randomzng s ndfferent between nvestng and not nvestng, and the nequalty holds by (2). Hence from (12) and (13), g (k P 1){g (k P 1) g (k P )} > g (k P ) g (k P 1) b 1 b 1 r a,b (1 r a,b ) b 1 η(a + ) η(k P ). Snce g (k P 1) g (k P ) > 0, ths nequalty s equvalent to =0 b 1 b 1 g (k P 1) > η(k P ) r a,b (1 r a,b ) b 1 η(a + ). =0 Fnally, snce η(k P ) g (k P 1) = η(k P 1), ths nequalty s just (11). The two graphs of Fgure 3 show the socal welfare n the worst equlbrum as a functon of the commttee sze for an example. In the example, the pror s symmetrc, and the parameters are N = 100, s N (ω, 1), π =.3, p =.7, and c = In ths case we have k P = 11, and k = 12. The two graphs are the same except that the scalngs of the vertcal axes dffer. One can see that the welfare loss due to overszed commttees s qute small. 4. CONCLUSION In ths paper, we dscuss the optmal commttee sze and the potental welfare losses assocated wth overszed commttees. We focus on envronments n whch there s no conflct of nterest among ndvduals, but nformaton acquston s costly. Frst, we confrm that the optmal commttee sze s bounded. In other words, the Condorcet Jury Theorem fals to hold: larger commttees mght nduce smaller socal welfare. However, we show also that the welfare loss due to overszed commttees s surprsngly small. In an arbtrarly large commttee, even the worst equlbrum generates welfare hgher than does an equlbrum n a commttee wth two less members than the optmal commttee. Our results suggest that carefully desgnng commttees mght be not as mportant as has been thought.

18 244 Koryama and Szentes Theoretcal Economcs 4 (2009) Socal welfare Commttee sze (k ) Commttee sze (k ) Socal welfare FIGURE 3. Socal welfare as a functon of the commttee sze k for N = 100, π = 0.3, q = 0.7, σ = 1, c = k P = 11, k = 12. APPENDIX LEMMA 3. Suppose that η ( C 1 ( + )) s absolutely contnuous and strctly ncreasng, and η (k + 1)/η (k ) s strctly ncreasng for k > ɛ, where ɛ 0. Let g (k ) = η(k + 1) η(k ) for all k 0. Then g (k + 1)/g (k ) < g (k + 2)/g (k + 1) for k ɛ. PROOF. Fx k ( ɛ). Notce that η (k + 2)/η (k + 1) < η (t + 2)/η (t + 1) s equvalent to η (k + 2)η (t + 1) < η (k + 1)η (t + 2). Therefore η (k + 2) k +1 k η (t + 1) d t < η (k + 1) k +1 k η (t + 2) d t η (k + 2)[η(k + 2) η(k + 1)] < η (k + 1)[η(k + 3) η(k + 2)].

19 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 245 It follows that η (k + 2) η (k + 1) η(k + 3) η(k + 2) g (k + 2) < = η(k + 2) η(k + 1) g (k + 1). (14) Smlarly, for all t (k, k + 1), η (k + 2)/η (k + 1) > η (t + 1)/η (t ) s equvalent to η (k + 2)η (t ) > η (k + 1)η (t + 1). Therefore η (k + 2) k +1 k It follows that η (t ) d t > η (k + 1) k +1 k η (t + 1) d t η (k + 2)[η(k + 1) η(k )] > η (k + 1)[η(k + 2) η(k + 1)]. η (k + 2) η (k + 1) From (14) and (15) t follows that > η(k + 2) η(k + 1) η(k + 1) η(k ) g (k + 1) g (k + 2) < g (k ) g (k + 1) g (k + 1) =. (15) g (k ) for all k ɛ. PROOF OF PROPOSITION 1. The sum of normally dstrbuted sgnals s also normal: k =1 s N (ωk,σ k ). The densty functon of k =1 s condtonal on ω s k 1 σ k φ =1 s ωk σ k where φ(x) = (2π) 1/2 exp( x 2 /2). The ex post effcent decson rule s gven by 1 f s s k θ µ(s 1,..., s k ) = 1 f s s k < θ, where θ = (σ 2 /2) log[(1 q)(1 π)/qπ] s the cut-off value. Hence, for k \ {0}, η(k ) = qπpr µ(s 1,..., s k ) = 1 ω = 1 (1 q)(1 π) Pr µ(s 1,..., s k ) = 1 ω = 1 θ k = qπφ σ (1 q)(1 π)φ θ + k k σ (16) k where Φ s the cdf of standard normal dstrbuton. If k = 0, η(0) = max{ qπ, (1 q)(1 π)}. (17) Notce that the rght-hand sde of (16) converges to that of (17) as k goes to zero. Part(). If q + π = 1, then qπ = (1 q)(1 π) and θ = 0. Hence k η(k ) = 2qπΦ σ and η (k ) = qπ σ 1 k φ for k > 0. k σ

20 246 Koryama and Szentes Theoretcal Economcs 4 (2009) Therefore η (k + 1) η = (k ) k k + 1 exp 1 2σ 2 s ncreasng n k (> 0). From Lemma 3, settng ɛ to be zero, t follows that g (k + 1)/g (k ) s ncreasng n k. Part(). By the defnton of θ, θ k qπφ σ = (1 q)(1 π)φ θ + k k σ. k Applyng ths to take the dervatve of (16), we get θ k θ k η (k ) = qπφ σ k k σ k θ k 1 = qπφ σ k σ k. + θ + k k σ k Next, we argue that for any ɛ (> 0), η (k + 1)/η (k ) s ncreasng for all k > ɛ f σ s suffcently small. For k > 0, η (k + 1) k φ θ (k +1) σ k +1 η = (k ) k + 1 φ k θ k = σ k + 1 exp 1 k k 1 θ 2 = k + 1 exp 2σ 2 k (k + 1) 1 k L 2 = k + 1 exp σ 2 1 exp 8k (k + 1) 2σ 2 exp 1 θ 2 2θ + k + 1 2σ 2 k +1 θ 2 2σ 2 k 2θ + k where L = log{(1 q)(1 π)/(qπ)}. Now suppose that k > ɛ. The last term n (18) has no nfluence on whether η (k + 1)/η (k ) s ncreasng. The second term converges to 1 as σ goes to 0. Obvously, the frst term s strctly ncreasng n k. Hence, η (k + 1)/η (k ) s ncreasng n k (> ɛ) f σ s suffcently small. By settng ɛ (0, 1) and usng Lemma 3, we have shown that g (k + 1)/g (k ) < g (k + 2)/g (k + 1) for all k 1. It remans to show that g (1)/g (0) < g (2)/g (1). From the argument n the proof of Lemma 3, t follows that η (2)/η (1) < η (t + 1)/η (t ) for all t (1, 2) mples η (2)/η (1) < g (2)/g (1). Hence t s enough to show that g (1)/g (0) < η (2)/η (1) for suffcently small σ. Snce lm σ 0 g (0) = η(0) > 0, t s enough to show that lm σ 0 [g (1)/{η (2)/η (1)}] = 0. In order to establsh ths equalty, t s obvously enough to show that (18) η(k ) lm σ 0 η (2)/η = 0 for k {1, 2}. (19) (1) Remember L = log{(1 q)(1 π)/(qπ)}. By (16), for k > 0, Lσ k η(k ) = qπ Φ 2 k + exp(l)φ Lσ k σ 2 k, σ

21 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 247 whch mples η(k ) O E (Φ( k /σ)) as σ Usng (18), η (2)/η (1) O E (exp( 1/2σ 2 )) = O E (φ(1/σ)) as σ 0. By l Hôptal s Rule, for k {1, 2}, Φ k /σ φ k /σ k /σ 2 lm = lm σ 0 φ(1/σ) σ 0 φ(1/σ)(1/σ 3 = 0, ) whch mples (19). PROOF OF PROPOSITION 2. Frst, we clam that the ex post effcent decson rule µ : { 1, 0, 1} k { 1, 1} s the cut-off rule 1 f k =1 µ(s 1,..., s k ) = s θ, 1 f k =1 s < θ (20), where θ = log[(1 q)(1 π)/qπ]/ log[p/(1 p)]. (s 1,..., s k ) s a permutaton of Suppose that the sgnal sequence Then µ(s 1,..., s k ) = 1 f 1,..., 1, 0,..., 0, 1,..., 1. (21) }{{}}{{}}{{} a k a b b ω [u (ω, 1) s 1,..., s k ] = (1 q)(1 π) Pr[s 1,..., s k ω = 1] Pr[s 1,..., s k ] In addton, Hence, µ(s 1,..., s k ) = 1 f or equvalently, > ω [u (ω, 1) s 1,..., s k ] = qπ Pr[s 1,..., s k ω = 1]. Pr[s 1,..., s k ] Pr[s 1,..., s k ω = 1] = (p r ) a (1 r ) k a b (r (1 p)) b Pr[s 1,..., s k ω = 1] = (p r ) b (1 r ) k a b (r (1 p)) a. (1 q)(1 π)p b (1 p) a > qπp a (1 p) b, log (1 q)(1 π) qπ a b > log p = θ. 1 p Snce k =1 s = a b, (20) follows. Now we consder the case where p converges to 1. Let ɛ denote 1 p and let Pr[a,b] denote the probablty of a sgnal sequence that s a permutaton of (21). Then Pr[a,b ω = 1] = C k (a,b)(1 ɛ) a ɛ b Pr[a,b ω = 1] = C k (a,b)(1 ɛ) b ɛ a, 10 O E s a verson of Landau s O, whch descrbes the exact order of the expresson. Formally, f (x) O E (g (x )) as x a f and only f there exsts M > 0 such that lm x a f (x)/g (x) = M.

22 248 Koryama and Szentes Theoretcal Economcs 4 (2009) where C k (a,b) = [k!/(a!b!(k a b)!)]r a +b (1 r ) k a b. 11 Notce that C k (a,b) s ndependent of ɛ and symmetrc wth respect to a and b. We have 12 Pr[a b 1 ω = 1] = C k (0, 1)ɛ +O(ɛ 2 ) Pr[a b 0 ω = 1] = C k (0, 0) + {C k (1, 0) + C k (1, 1)}ɛ +O(ɛ 2 ). Observe that θ < 1 f p s close enough to one. Wthout loss of generalty, assume that q + π 1. Then 1 < θ 0. Hence η(k ) = qπpr[a b 1 ω = 1] (1 q)(1 π) Pr[a b 0 ω = 1] = qπc k (0, 1)ɛ (1 q)(1 π)[c k (0, 0) + {C k (1, 0) + C k (1, 1)}ɛ] +O(ɛ 2 ). Then g (k ) = η(k + 1) η(k ) = A(k ) + B(k )ɛ +O(ɛ 2 ), where A(k ) = (1 q)(1 π)d k (0, 0) B(k ) = qπd k (0, 1) (1 q)(1 π)[d k (1, 0) + D k (1, 1)] and D k (a,b) = C k +1 (a,b) C k (a,b). Usng ths notaton, g (k + 1) = A(k + 1) + B(k + 1)ɛ +O(ɛ2 ) A(k + 1) B(k +1) 1 + A(k +1) ɛ +O(ɛ2 ) g (k ) A(k ) + B(k )ɛ +O(ɛ 2 = ) A(k ) 1 + B(k ) A(k ) ɛ +O(ɛ2 ) A(k + 1) B(k + 1) = 1 + A(k ) A(k + 1) B(k ) ɛ +O(ɛ 2 ). A(k ) We want to show that g (k + 1)/g (k ) s ncreasng n k f ɛ s suffcently small. Snce A(k + 1)/A(k ) = 1 r, t s suffcent to show that B(k )/A(k ) s convex n k. It s straghtforward to see that D k (0, 1) D k (0, 0) = D k (1, 0) D k (0, 0) = (k + 1)r (1 r )k k r (1 r ) k 1 (k + 1)r (1 r ) k r (1 r ) k +1 (1 r ) k = (1 r ) 2 (1 r ) s a polynomal of k wth degree 1, hence t has no nfluence on the convexty of B(k )/A(k ). On the other hand, D k (1, 1) D k (0, 0) = (k + 1)k r 2 (1 r ) k 1 k (k 1)r 2 (1 r ) k 2 (1 r ) k +1 (1 r ) k = (k + 1)k r 2 (1 r ) k (k 1)r 2 (1 r ) 3 (1 r ) 2 has a postve coeffcent of k 2. Hence we conclude that B(k )/A(k ) s convex n k. 11 Defne C k (a,b) = 0 f k < a + b. 12 f (x ) O(g (x )) as x 0 f and only f there exsts δ > 0, M > 0 such that x < δ mples f (x)/g (x) < M.

23 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 249 PROOF OF PROPOSITION 3. rule s defned by As n the proof of Proposton 2, the ex post effcent decson µ(s ) = 1 f s θ 1 otherwse, where θ = log[(1 q)(1 π)/qπ]/ log[p/(1 p)]. By symmetry, we can assume θ 0 wthout loss of generalty. Frst, suppose θ > 1. Then η(k ) = qπ for k < θ, and the margnal beneft from an addtonal sgnal s zero for k < θ 1. Therefore g (k +1)/g (k ) s not well-defned. Second, suppose 0 θ < 1. We consder two dfferent cases dependng on whether k s even or odd. Case 1: Suppose k = 2m, where m. Then the (2m + 1)-st sgnal makes a dfference f and only f m of the frst 2m sgnals are postve and m are negatve, and the (2m +1)-st sgnal s postve (denote ths stuaton as pv e ). In such a case, the socal decson changes from 1 to 1. Hence the gan s q f ω = 1 and the loss s (1 q) f ω = 1. Therefore, the expected margnal beneft s g (2m) = q Pr[ω = 1, pv e ] (1 q)pr[ω = 1, pv e ] 2m 2m = q π p m (1 p) m p (1 q) (1 π) p m (1 p) m (1 p) m m 2m = {pqπ (1 p)(1 q)(1 π)} p m (1 p) m. m Case 2: Suppose k = 2m + 1, where m. Then the (2m + 2)-nd sgnal makes a dfference f and only f the frst 2m + 1 sgnals contan m + 1 postve and m negatve sgnals and the (2m + 2)-nd sgnal s negatve (denote ths stuaton as pv o ). In such a case, the socal decson changes from 1 to 1. Hence the loss s q f ω = 1, and the gan s 1 q f ω = 1. Therefore the expected margnal beneft s g (2m + 1) = q Pr[ω = 1, pv o ] + (1 q)pr[ω = 1, pv o ] 2m + 1 = q π p m +1 (1 p) m +1 m 2m (1 q) (1 π) p m +1 (1 p) m +1 m 2m + 1 = { qπ + (1 q)(1 π)} p m +1 (1 p) m +1. m Recall that 0 θ < 1, whch s equvalent to pqπ (1 p)(1 q)(1 π) > 0 and (1 q)(1 π) qπ 0. If θ > 0, g (2m + 2) g (2m + 1) pqπ (1 p)(1 q)(1 π) =, (1 q)(1 π) qπ whch s a constant functon of m, so that Assumpton 1 does not hold. If θ = 0, then g (2m + 1) = 0 and g (2m + 2)/g (2m + 1) s not well-defned.

24 250 Koryama and Szentes Theoretcal Economcs 4 (2009) PROOF OF LEMMA 1. (). Notce that b 1 b 1 G (r ; a,b) = r (1 r ) b 1 g (a + ). =0 Snce r (1 r ) b 1 = r (1 r ) b + r +1 (1 r ) b 1, Snce g s decreasng, b 1 b 1 r G (r ; a,b) = (1 r ) b + r +1 (1 r ) b 1 g (a + ). =0 b 1 b 1 b 1 b 1 G (r ; a,b) > r (1 r ) b g (a + ) + r +1 (1 r ) b 1 g (a + + 1) =0 b 1 b 1 b b 1 = r (1 r ) b g (a + ) + r (1 r ) b g (a + ) 1 = =0 =0 =1 b b 1 b 1 + r (1 r ) b g (a + ), 1 =0 where the frst equalty holds because we have just redefned the notaton n the second summaton, and the second equalty holds because, by conventon, n n 1 = n+1 = 0 for all n. Fnally, usng b 1 b 1 b + 1 =, we have G (r ; a,b) > b b r (1 r ) b g (a + ) = G (r ; a,b + 1). =0 (). By the defntons of r a,b and r a,b+1, we have and by part () of ths lemma, c = G (r a,b ; a,b) = G (r a,b+1 ; a,b + 1), G (r a,b+1 ; a,b + 1) < G (r a,b+1 ; a,b). Therefore G (r a,b ; a,b) < G (r a,b+1 ; a,b). Snce G (r ; a,b) s strctly decreasng n r, r a,b > r a,b+1 follows. PROOF OF LEMMA 2. The statement of the lemma s obvous f {k 1, k }. It remans to show that (8) holds wth strct nequalty whenever / {k 1, k }. Frst, notce that for any postve sequence {a j } 0, f a j +1/a j < a j +2 /a j +1 for all j, then k a k j = +1 > a j a k 1 for all k > 1 and for all {0,..., k 2}. k 1 j = a j

25 Theoretcal Economcs 4 (2009) A resurrecton of the Condorcet Jury Theorem 251 Assumpton 1 allows us to apply ths result to the sequence a j = g (j ), and hence, for all k 1, k g (k ) g (k 1) > j = +1 g (j ) η(k + 1) η( + 1) k 1 = for all {0,..., k 2}. j = g (j ) η(k ) η( ) Snce η(a ) > η(b) f a > b, ths mples that for all {0,..., k 2}, g (k )[η( ) η(k )] < g (k 1)[η( + 1) η(k + 1)]. (22) Smlarly, for a postve sequence {a j } 0, f a j +1/a j < a j +2 /a j +1 for all j, then a k a k 1 < j =k +1 a j 1 j =k a j for all k 1 and for all k + 1. Agan, by Assumpton 1, we can apply ths result to the sequence a j = g (j ) and get g (k ) g (k 1) < j =k +1 g (j ) η( + 1) η(k + 1) 1 j =k g (j ) = for all > k. η( ) η(k ) Multplyng through by g (k 1)(η( ) η(k )), we get (22). That s, (22) holds whenever / {k 1, k }. After subtractng g (k 1)(η( ) η(k )) from both sdes of (22), we get (8). REFERENCES Austen-Smth, Davd and Jeffrey S. Banks (1996), Informaton aggregaton, ratonalty, and the Condorcet jury theorem. Amercan Poltcal Scence Revew, 90, [229] Börgers, Tlman (2004), Costly votng. Amercan Economc Revew, 94, [229] Condorcet, marqus de (Mare-Jean-Antone-Ncolas de Cartat) (1785), Essa sur l applcaton de l analyse à la probablté des décsons rendues à la pluralté des vox. Imprmere Royale, Pars. [227] Feddersen, Tmothy and Wolfgang Pesendorfer (1997), Votng behavor and nformaton aggregaton n electons wth prvate nformaton. Econometrca, 65, [229] Feddersen, Tmothy and Wolfgang Pesendorfer (1998), Convctng the nnocent: The nferorty of unanmous jury verdcts under strategc votng. Amercan Poltcal Scence Revew, 92, [229] Gerard, Dno and Leeat Yarv (2008), Informaton acquston n commttees. Games and Economc Behavor, 62, [230] Gerlng, Kerstn, Hans Peter Grüner, Alexandra Kel, and Elsabeth Schulte (2005), Informaton acquston and decson makng n commttees: A survey. European Journal of Poltcal Economy, 21, [227]