A Resurrection of the Condorcet Jury Theorem

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1 A Resurrecton of the Condorcet Jury Theorem Yuo Koryama and Balázs Szentes yz Septemer 19, 2008 Astract hal , verson 1-3 Jun 2009 Ths paper analyzes the optmal sze of a deleratng commttee where, () there s no con ct of nterest among ndvduals, and () nformaton acquston s costly. The commttee memers smultaneously decde whether or not to acqure nformaton, and then, they mae the ex-post e cent decson. The optmal commttee sze,, s shown to e ounded. The man result of ths paper s that any artrarly large commttee aggregates the decentralzed nformaton more e cently than the commttee of sze 2. Ths result mples that overszed commttees generate only small ne cences. 1 Introducton The classcal Condorcet Jury Theorem (CJT) states that large commttees can aggregate decentralzed nformaton more e cently than small ones. Its orgn can e traced to the dawn of the French Revoluton when Mare Jean Antone Ncolas Cartat le Marqus de Condorcet (1785, translaton 1994) nvestgated the decson-mang processes n socetes. 1 A recent lterature on commttee desgn has ponted out that f the nformaton acquston s costly, the CJT fals to hold. The reasonng s that f the sze of a commttee s large, a commttee memer realzes that the proalty that she can n uence the nal decson s too 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, larger commttees mght generate lower socal welfare than smaller ones. These results suggest that n the presence of costly nformaton acquston, optmally choosng the sze of a commttee s oth an mportant and delcate ssue. In ths paper, we characterze the welfare loss assocated to overszed commttees, and we 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 Département d Econome, Ecole Polytechnque, Palaseau Cedex 91128, France. y Department of Economcs, Unversty College London, Gower Street, London WC1E 6BT, Unted Kngdom. z The authors are grateful to semnar partcpants at the Unversty of Chcago, Unversty of Rochester, UCL, Ecole Polytechnque, Arzona State Unversty, Maastrcht Unversty, and Unversty of Hawa for helpful comments. 1 Summares of the hstory of the CJT can e found n, for example, Grofman and Owen (1986), Mller (1986), and Gerlng, Grüner, Kel, and Schulte (2003). 1

2 hal , verson 1-3 Jun 2009 commttee mght not e as mportant as t was orgnally thought to e. In fact, f ether the nformaton structure s amguous, or the commttee has to mae decsons n varous nformatonal envronments, t mght e optmal to desgn the commttee to e as large as possle. Commttee desgn receves a consderale attenton y economsts ecause, n many stuatons, groups rather than ndvduals mae decsons. Informaton aout the desralty of the possle decsons s often decentralzed: ndvdual group memers must separately acqure costly nformaton aout 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 otan some nformaton aout the defendant at some e ort cost (payng attenton to the tral, nvestgatng evdences, etc.). Lewse, when a rm s facng the decson whether or not to mplement a project each memer of the executve commttee can collect nformaton aout the pro talty of the project (y spendng tme and exertng e ort). Yet another example s the hrng decsons of academc departments. Each memer of the recrutng commttee must revew the applcatons ndvdually efore mang a collectve decson. What these examples have n common s the fact that nformaton acquston s costly and often unoservale. The exact setup analyzed n ths paper s descred as follows. A group of ndvduals has to mae a nary decson. There s no con ct of nterest among the group memers, ut they have mperfect nformaton aout whch decson s the est. Frst, ndvduals are ased to serve n a commttee. Then, the commttee memers smultaneously decde whether or not to nvest n an nformatve sgnal. Fnally, the commttee maes the optmal decson gven the acqured nformaton. We do not explctly model how the commttee memers communcate and aggregate nformaton. Instead, we smply assume that they end up mang the ex-post e cent decson. 2 The only strategc choce an ndvdual must mae n our model s the choce whether or not to acqure a sgnal upon eng selected to serve n the commttee. The central queston of our paper s the followng: how does the commttee sze,, a ect socal welfare? Frst, for each, we fully characterze the set of equlra (ncludng asymmetrc and mxed-strategy equlra). We show that there exsts a P (2 N) such that whenever P, there s a unque equlrum n whch each commttee memer nvests n nformaton wth proalty one. Furthermore, the socal welfare generated y these equlra s an ncreasng functon of. If > P, then there are multple equlra and many of them nvolve randomzatons y the memers. We also show that the socal welfare generated y the worst equlrum n the game, where the commttee sze s, s decreasng n f > P. The optmal commttee sze,, s de ned such that () f the commttee sze s, then there exsts an equlrum that maxmzes socal welfare, and () n ths equlrum, each memer nvests n nformaton wth postve proalty. We prove that the optmal commttee sze,, s ether P or P + 1. Ths 2 Snce there s no con ct of nterest among the ndvduals, t s easy to desgn a mechansm whch s ncentve compatle and e cently aggregates the sgnals. Alternatvely, one can assume that the collected nformaton s hard. 2

3 mples that the CJT fals to hold: large commttees can generate smaller socal welfare than smaller commttees. Nevertheless, we show that f the sze of the commttee s larger than, even the worst equlrum generates hgher socal welfare than the unque equlrum n the commttee of sze 2. That s, the welfare loss due to an overszed commttee s qute small. hal , verson 1-3 Jun 2009 Lterature Revew Although the Condorcet Jury Theorem provdes mportant support for the ass of democratc decson mang, many of the premses of the theorem have een 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 est, condtonal on her sgnal. Austen-Smth and Bans (1996) showed that n general, sncere votng s not consstent wth equlrum ehavor. Ths s ecause a ratonal ndvdual votes not only condtonal on her sgnal, ut also on her eng pvotal. Feddersen and Pesendorfer (1998) have shown that as the jury sze ncreases, the proalty of convctng an nnocent can actually ncrease under the unanmty rule. A varety of papers have shown, however, that even f the voters are strategc, n certan envronments the outcome of a votng converges to the e cent outcome as the numer of voters goes to n nty. Feddersen and Pesendorfer (1997) nvestgate a model n whch preferences are heterogeneous and each voter has a prvate sgnal concernng whch alternatve s est. They construct an equlrum for each populaton sze, such that the equlrum outcome converges to the full nformaton outcome as the numer of voters goes to n nty. The full nformaton outcome s de ned as the result of a votng game, where all nformaton s pulc. Myerson (1998) has shown that asymptotc e cency can e acheved even f there s populaton uncertanty; that s, a voter does not now how many other voters there are. In contrast, the Condorcet Jury Theorem mght fal to hold f the nformaton acquston s costly. Muhopadhaya (2003) has consdered a model, smlar to ours, where voters have dentcal preferences ut nformaton acquston s costly. He has shown y numercal examples that mxedstrategy equlra n large commttees may generate lower expected welfare than pure-strategy equlra n small commttees. 3 Martnell (2006) also ntroduced cost of nformaton acquston and, he allows the precson of the sgnals to depend contnuously on the amount of nvestment. Martnell (2006) 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 e cent. More precsely, f the sze of the commttee converges to n nty, then there s a sequence of symmetrc equlra n whch each memer nvests only a lttle, and the proalty of a correct decson converges to one. 4 3 The results are qute d erent f the votng, rather than the nformaton acquston, s costly, see e.g. Borgers (2004). 4 In hs accompanyng paper, Martnell (2007) analyzes a model n whch nformaton has a xed cost, voters are heterogeneous n ther costs, and astenton s not allowed. On the one hand, the author shows that f the 3

4 hal , verson 1-3 Jun 2009 We thn that Martnell (2006) contruted sustantally towards the understandng of the e cency propertes of group decson mang f there s no xed cost assocated to nformaton acquston. However, we eleve that ths xed cost aspect s an essental feature of many envronments. Indeed, one has to pay the prce of a newspaper even f he decdes to throw t away later. The management of a company has to pay for a consultant even f the wor of the consultant wll e completely gnored. Smlarly, a juror has to st through the tral even f he decdes not to pay any attenton. Hence, we thn that our paper s an mportant complementary result to Martnell (2006). Numerous papers have analyzed the optmal decson rules n the presence of costly nformaton. Persco (2004) dscusses the relatonshp etween 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 only optmal f the sgnals are su cently accurate. The ntuton for the extreme case, where the decson rule s the unanmty rule, s the followng: under the unanmty rule, the proalty of eng pvotal s small. However, ths proalty ncreases as the sgnals ecome more accurate. Therefore, n order to provde a voter wth an ncentve to nvest n nformaton, the sgnals must e su cently accurate. L (2001), Gerard and Yarv (2006), and Gershov and Szentes (2004) have shown that the optmal votng mechansm sometmes nvolves ex-post ne cent decsons. That s, the optmal mechansm mght specfy ne cent decsons for certan sgnal pro les. We eleve that there are many stuatons where such a commtment devce s not avalale. That s why we smply restrct attenton to ex post e cent decson rules. We eleve that ths s the approprate assumpton n the context of a deleratng commttee n whch there s no con ct of nterest among ndvduals. Secton 2 descres 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,!, can tae one of two values: 1 and 1. Furthermore, Pr [! = 1] = 2 (0; 1). The socety must mae a decson, d, whch s ether 1 or 1. There s no con ct of nterest among ndvduals. Each ndvdual has a ene t of u (d;!) f decson d s made when the state of the world s!. In partcular, 8 >< 0 f d =!; u (d;!) = q f d = 1 and! = 1; >: (1 q) f d = 1 and! = 1; support of the cost dstruton s not ounded away from zero, asymptotc e cency can e acheved. On the other hand, f the cost s ounded away form zero and the numer of voters s large, noody acqures nformaton n any equlrum. 4

5 hal , verson 1-3 Jun 2009 where q 2 (0; 1), ndcates the severty of type-i error 5. 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 payo of an ndvdual who nvests n nformaton s u c. Each ndvdual maxmzes her expected payo. There are two stages of the decson-mang process. At Stage 1, ( N) memers of the socety are desgnated to serve n the commttee at random. At Stage 2, the commttee memers decde smultaneously and ndependently whether or not to nvest n nformaton. Then, the e cent decson s made gven the sgnals collected y the memers. We do not model explctly how commttee memers delerate at Stage 2. Snce there s no con ct of nterest among the memers, t s easy to desgn a communcaton protocol that e cently aggregates nformaton. Alternatvely, one can assume that the acqured nformaton s hard. Hence, no communcaton s necessary for mang the ex-post e cent decson. We focus merely on the commttee memers ncentves to acqure nformaton. Next, we turn our attenton to the de nton of socal welfare. Frst, let denote the ex-post e cent decson rule. That s, s a mappng from sets of sgnals nto possle decsons. If the sgnal pro le s (s 1 ; :::; s n ), where n s the numer of acqured sgnals, then (s 1 ; :::; s n ) = 1, E! [u (1;!)j s 1 ; :::; s n ] E! [u ( 1;!)j s 1 ; :::; s n ]. The socal welfare s measured as the expected sum of the payo s of the ndvduals, that s, E s1; ;s n;! [Nu ( (s 1 ; :::; s n ) ;!) cn] ; (1) where the expectaton also taes nto account the possle randomzaton of the ndvduals. That s, n can e a random varale. If the commttee s large, then a memer mght prefer to save the cost of nformaton acquston and choose to rely on the opnons of others. On the other hand, f s too small, there s too lttle nformaton to aggregate, and thus the nal decson s lely to e ne cent. The queston s: What s the optmal that maxmzes ex-ante socal welfare? To e more spec c, the optmal sze of the commttee s, f () the most e cent equlrum, n the commttee wth memers, maxmzes the socal surplus among all equlra n any commttee, and () each memer acqures nformaton wth postve proalty n ths equlrum. Snce the sgnals are d condtonal on the state of the world, the expected ene t of an ndvdual from the ex post e cent decson s a functon of the numer of sgnals acqured. We de ne ths functon as follows: (n) = E s1; ;s n;! [u ( (s 1 ; ; s n ) ;!)]. We assume that the sgnals are nformatve aout the state of the world, ut only mperfectly. That s, as the numer of sgnals goes to n nty, the proalty of mang the correct decson 5 In the jury context, where! = 1 corresponds to the nnocence of the suspect, q ndcates how severe error t s to convct an nnocent. 5

6 s strctly ncreasng and converges to one. Formally, the functon s strctly ncreasng and lm n!1 (n) = 0. An ndvdual s margnal ene t from collectng an addtonal sgnal, when n sgnals are already otaned, s g (n) = (n + 1) (n) : Note that lm n!1 g (n) = 0. For our man theorem to hold, we need the followng assumpton. Assumpton 1 The functon g s log-convex. Ths assumpton s equvalent to g (n + 1) =g (n) eng ncreasng n n (2 N) : Whether or not ths assumpton s sats ed depends only on the prmtves of the model, that s, on the dstruton of the sgnals and on the parameters q and. An mmedate consequence of ths assumpton s the followng. hal , verson 1-3 Jun 2009 Remar 1 The functon g s decreasng. Proof. Suppose y contradcton that there exsts an nteger, n 0 2 N, 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!1 g (n) 6= 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 eng decreasng means that the margnal socal value of an addtonal sgnal s decreasng. We thn that ths assumpton s sats ed n most economc and poltcal applcatons. How much more does Assumpton 1 requre? Snce g s decreasng and lm n!1 g (n) = 0, there always exsts an ncreasng sequence fn m g 1 m=1 N, 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 d erence 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) eng decreasng. That s, Assumpton 1 requres that the second d erence n the value of a sgnal does not only decrease, ut decreases at an ncreasng rate. In general, t s hard to chec whether ths assumpton holds ecause t s often d cult (or mpossle) to express g (n) analytcally. The next secton provdes examples where Assumpton 1 s sats ed. 2.1 Examples for Assumpton 1 Frst, suppose that the sgnals are normally dstruted around the true state of the world. The log-convexty assumpton s sats ed for the model where + q = 1. That s, the socety would e nd erent etween the two possle decsons f nformaton acquston were mpossle. The assumpton s also sats ed even f + q 6= 1 f the sgnals are su cently precse. Formally: 6

7 Proposton 1 Suppose that s N (!; ). () If q + = 1 then Assumpton 1 s sats ed. () For all q;, there exsts an " q; > 0, such that Assumpton 1 s sats ed f " q; >. Proof. See the Appendx. In our next example the sgnal s ternary, that s, ts possle values are f 1; 0; 1g. In addton, Pr (s =!j!) = pr; Pr (s = 0j!) = 1 r; and Pr (s =!j!) = (1 p) r: Notce that r (2 (0; 1)) s the proalty that the realzaton of the sgnal s nformatve, and p s the precson of the sgnal condtonal on eng nformatve. Proposton 2 Suppose that the sgnal s ternary. Then, there exsts a p (r) 2 (0; 1) such that, f p > p (r), Assumpton 1 s sats ed. hal , verson 1-3 Jun 2009 Proof. See the Appendx. Next, we provde an example where the log-convexty assumpton s not sats ed. Suppose that the sgnal s nary, that s, s 2 f 1; 1g and Pr (s =!j!) = p; Pr (s =!j!) = 1 p: Proposton 3 If the sgnal s nary then Assumpton 1 s not sats ed. Proof. See the Appendx. 3 Results Ths secton s devoted to the proofs of our man theorems. To that end, we rst characterze the set of equlra for all (2 N). The next susecton shows that f s small, the equlrum s unque and each memer ncurs the cost of nformaton (Proposton 4). Secton 3.2 descres the set of mxed-strategy equlra for large enough (Propostons 5). Fnally, Secton 3.3 proves the man theorems (Theorems 1 and 2). 3.1 Pure-strategy equlrum Suppose that the sze of the commttee s. If the rst 1 memers acqure nformaton, the expected gan from collectng nformaton for the th memer s g( 1). She s wllng to nvest f ths gan exceeds the cost of the sgnal, that s, f c < g ( 1) : (2) Ths nequalty s the ncentve compatlty constrant guaranteeng that a commttee memer s wllng to nvest n nformaton f the sze of the commttee s. 6 6 In what follows, we gnore the case where there exsts a 2 N such that c = g (). Ths equalty does not hold genercally, and would have no e ect on our results. 7

8 Proposton 4 Let denote the sze of the commttee. There exsts a P 2 N; such that, there exsts a unque equlrum n whch each memer nvests n a sgnal wth proalty one f and only f mn P ; N. Furthermore, the socal welfare generated y these equlra s monotoncally ncreasng n mn P ; N. Proof. Recall from Remar 1 that g s decreasng and lm!1 g () = 0: Therefore, for any postve cost c < g (0) 7, there exsts a unque P 2 N such that g P < c < g P 1 : (3) hal , verson 1-3 Jun 2009 Frst, we show that f < P then there s a unque equlrum n whch each commttee memer nvests n nformaton. Suppose that n an equlrum, the rst 1 memers randomze accordng to the pro le (r 1 ; :::; r 1 ) ; where r 2 [0; 1] denotes the proalty that the th memer nvests. Let I denote the numer of sgnals collected y the rst ( 1) memers. Snce the memers randomze, I s a random varale. Notce that I 1, and E r1;:::;r 1 [g (I)] g ( 1) ecause g s decreasng. Also notce that from P and (3), t follows that g ( 1) > c. Comnng the prevous two nequaltes, we get E r1;:::;r 1 [g (I)] > c. Ths nequalty mples that no matter what the strateges of the rst ( 1) memers are, the th memer strctly prefers to nvest n nformaton. From ths oservaton, the exstence and unqueness of the pure-strategy equlrum follow. It remans to show that f > P, such a purestrategy equlrum does not exst. But f > P, then g ( 1) < c: Therefore, the ncentve compatlty constrant, (2), s volated and there s no equlrum where each memer ncurs the cost of the sgnal. Fnally, we must show that the socal welfare generated y these pure-strategy equlra s ncreasng n mn P ; N. Notce that snce N > 1, c < g ( 1) = () ( 1) < N (() ( 1)). After addng N( 1) c, we get N( 1) c ( 1) < N() c. The left-hand sde s the socal welfare generated y the equlrum n commttee of sze 1, whle the rght-hand sde s the socal welfare nduced y the commttee of sze. 8

9 hal , verson 1-3 Jun 2009 Fgure 1: Expected gan g () and the cost c Fgure 1 s the graph of g () and c, where s N (!; 1), = :3, q = :7, and c = The expected gan s decreasng and log-convex. In ths example, P = 11. The statement n Proposton 4 s what Muhopadhaya (2003) has shown n the case where the sgnal s nary. He has also shown y numercal examples that mxed-strategy equlra n large commttees can yeld lower expected welfare than small commttees. Our analyss goes further y analytcally comparng the expected welfare of all mxed-strategy equlra. The amount of purchased nformaton n equlrum s ne cently small. Ths s ecause when a commttee memer decdes whether or not to nvest, she consders her prvate ene t rather than the socety s ene t. Snce nformaton s a pulc good, ts socal ene t s gger than the ndvdual ene t. Hence, the total numer of sgnals acqured n an equlrum s smaller than the socally optmal amount. Ths s why the socal welfare s monotoncally ncreasng n the commttee sze, as long as P. 7 If c > (1) (0); then noody has ncentve to collect nformaton, hence P = 0: 9

10 3.2 Mxed-strategy equlrum Suppose now that the sze of the commttee s larger than P. We consder strategy pro les n whch the commttee memers can randomze when mang a decson aout ncurrng the cost of nformaton acquston. The followng proposton characterzes the set of mxed-strategy equlra (ncludng asymmetrc ones). We show that each equlrum s characterzed y a par of ntegers (a; ). In the commttee, a memers nvest n a sgnal wth proalty one, and memers acqure nformaton wth postve ut less than one proalty. The rest of the memers, a (a + ) numer of them, do not ncur the cost. We call such an equlrum a type-(a; ) equlrum. hal , verson 1-3 Jun 2009 Proposton 5 Let the commttee sze e > P. Then, for all equlra there s a par (a; ), such that a memers nvest for sure, memers nvests wth proalty r 2 (0; 1), and (a + ) memers do not nvest. In addton a P a + ; (4) where the rst two nequaltes are strct whenever > 0. Proof. Frst, we explan that f, n an equlrum n whch one memer nvests wth proalty r 1 2 (0; 1) and another nvests wth proalty r 2 2 (0; 1), then r 1 = r 2. Snce the margnal ene t from an addtonal sgnal s decreasng, our games exht strategc susttuton. That s, the more nformaton the others acqure, the less ncentve a memer has to nvest. Hence, f r 1 < r 2 then the ndvdual who nvests wth proalty r 1 faces more nformaton n expectaton and has less ncentve to nvest than the ndvdual who nvests wth proalty r 2. On the other hand, snce r 1 ; r 2 2 (0; 1) oth ndvduals must e exactly nd erent etween nvestng and not nvestng, a contradcton. Now, we formalze ths argument. Let r 2 [0; 1] ( = 1; ; ) e the proalty that the th memer collects nformaton n equlrum. Suppose that r 1, r 2 2 (0; 1), and r 1 > r 2. Let I 1 and I 2 denote the numer of sgnals collected y memers 2; 3; :::; and y memers 1; 3; :::;, respectvely. Notce that snce r 1 > r 2 and g s decreasng, E r2;r 3;:::;r [g (I 1 )] > E r1;r 3;:::;r [g (I 2 )]. (5) On the other hand, a memer who strctly randomzes must e nd erent etween nvestng and not nvestng. Hence, for j = 1; 2 E rj;r 3;:::;r [g (I j )] = c. (6) Ths equalty mples that (5) should hold wth equalty, whch s a contradcton. Therefore, each equlrum can e characterzed y a par (a; ) where a memers collect nformaton for sure, and memers randomze ut collect nformaton wth the same proalty. It remans to show that there exsts a type-(a; ) equlrum f and only f (a; ) sats es (4). Frst, notce that whenever > P, n all pure-strategy equlra P memers nvest wth 10

11 proalty one, and the rest of the memers never nvests. In addton, the par P ; 0 sats es (4). Therefore, we only have to show that there exsts an equlrum of type-(a; ) equlrum where > 0 f and only f (a; ) sats es a < P < a + : (7) hal , verson 1-3 Jun 2009 Suppose that n a commttee, a memers nvest n nformaton for sure and 1 nvests wth proalty r. Let G(r; a; ) denote the expected gan from acqurng nformaton for the (a + )th memer. That s, X 1 1 G(r; a; ) = r (1 r) 1 g (a + ). =0 We clam that there exsts a type-(a; ) equlrum f and only f there exsts an r 2 (0; 1) such that G (r; a; ) = c. Suppose rst that such an r exsts. We rst argue that there exsts a type- (a; ) equlrum n whch memers nvest wth proalty r. Ths means that the memers, those that are randomzng, are nd erent etween nvestng and not nvestng. The a memers, who nvest for sure, strctly prefer to nvest ecause the margnal gan from an addtonal sgnal exceeds G (r; a; ). Smlarly, those memers who don t nvest, a (a + ) numer of them, are strctly etter o not nvestng ecause ther margnal gans are strctly smaller than G (r; a; ). Next, we argue that f G (r; a; ) = c does not have a soluton n (0; 1) then there exsts no type- (a; ) equlrum. But ths mmedately follows from the oservaton that f memers are strctly randomzng, they must e nd erent etween nvestng and not nvestng and hence G (r; a; ) = c. Therefore, t s su cent to show that G (r; a; ) = c has a soluton n (0; 1) f and only f (7) holds. Notce that G(r; a; ) s strctly decreasng n r ecause g s strctly decreasng. Also oserve that G(0; a; ) = g (a) and G(1; a; ) = g (a + 1). By the Intermedate Value Theorem, G(r; a; ) = c has a soluton n (0; 1) f and only f G(1; a; ) < c < G(0; a; ), whch s equvalent to g (a + 1) < c < g (a) : (8) Recall that P sats es g P < c < g P 1 : Snce g s decreasng, (8) holds f and only f a < P and a + > P. That s, the two strct nequaltes n (7) are sats ed. The last nequalty n (7), must hold ecause a + cannot exceed the sze of the commttee,. Fgure 2 graphcally represents the set of pars (a; ) whch satsfy (4). Accordng to the prevous proposton, there are several equlra n whch more than P memers acqure nformaton wth postve proalty. A natural queston to as s: can these mxed-strategy equlra e compared from the pont of vew of socal welfare? The next proposton partally answers ths queston. We show that f one xes the numer of memers who acqure nformaton for sure, then the larger the numer of memers who randomze s, the smaller 11

12 hal , verson 1-3 Jun 2009 Fgure 2: The set of mxed-strategy equlra the socal welfare generated y the equlrum s. Ths proposton plays an mportant role n determnng the optmal sze of the commttee. Proposton 6 Suppose that 2 N, such that there are oth type-(a; ) and type-(a; + 1) equlra. Then, the type-(a; ) equlrum generates strctly hgher socal welfare than the type-(a; + 1) equlrum. In order to prove ths proposton we need the followng results. Lemma 1 () G (r; a; ) > G (r; a; + 1) for all r 2 (0; 1], and () r a; > r a;+1, where r a; and r a;+1 are the solutons for G (r; a; ) = c and G (r; a; + 1) = c n r, respectvely. Proof. See the Appendx. Proof of Proposton 6. Suppose that a memers collect nformaton wth proalty one, and memers nvest wth proalty r. Let f (r; a; ) denote the ene t of an ndvdual, that s, f(r; a; ) = X =0 r (1 r) (a + ): 12

13 (r; a; = X =1 r 1 (1 r) (a + ) X 1 r ( ) (1 r) 1 (a + ). 1 1 Notce that = and ( ) =. Therefore, the rght-hand sde of the 1 prevous equalty can e rewrtten as X 1 r 1 (1 r) (a + ) 1 =1 =0 =0 X 1 1 r (1 r) 1 (a + ). hal , verson 1-3 Jun 2009 After changng the notaton n the rst summaton, ths can e further rewrtten: X 1 1 r (1 r) 1 (a + + 1) =0 X 1 1 = =0 X 1 1 r (1 r) 1 (a + ) =0 r (1 r) 1 (a + + 1) (a + ). Ths last expresson s just G (r; a; ), and hence, we have Next, we show that Snce f(0; a; ) = f(0; a; + 1) = (r; a; = G (r; a; ). f (r a; ; a; ) f (r a;+1 ; a; + 1) > (r a; r a;+1 ) c. (9) f (r a; ; a; ) f (r a;+1 ; a; + 1) = f (r a; ; a; ) f (0; a; ) f (r a;+1 ; a; + 1) f (0; a; + 1) = Z ra; 0 G (r; a; ) dr Z ra;+1 0 G (r; a; + 1) dr. By part () of Lemma 1, ths last d erence s larger than Z ra; G (r; a; ) dr 0 Z ra;+1 Z ra; 0 G (r; a; ) dr = G (r; a; ) dr. r a;+1 By part () of Lemma, we now that r a;+1 < r a;. In addton, snce G s decreasng n r, ths last expresson s larger than (r a; r a;+1 ) G (r a; ; a; ). Recall that r a; s de ned such that G (r a; ; a; ) = c and hence we can conclude (9). Let S(a; ) denote the socal welfare n the type-(a; ) equlrum, that s: S(a; ) = Nf (r a; ; a; ) c (a + r a; ). 13

14 Then, S (a; ) S (a; + 1) = Nf (r a; ; a; ) c (a + r a; ) [Nf (r a;+1 ; a; + 1) c (a + r a;+1 )] > N (r a; r a;+1 ) c c (r a; r a;+1 ) = (N 1) c (r a; r a;+1 ) > 0; where the rst nequalty follows from (9), and the last one follows from part () of Lemma The Proofs of the Theorems Frst, we show that the optmal commttee sze s ether P or P + 1. Second, we prove that f > then even the worst possle equlrum yelds hgher socal welfare than the unque equlrum n the commttee of sze 2: hal , verson 1-3 Jun 2009 Theorem 1 The optmal commttee sze,, s ether P or P + 1: We emphasze that for a certan set of parameter values, the optmal sze s = P, and for another set, = P + 1. Proof. Suppose that s the optmal sze of the commttee and the equlrum that maxmzes socal welfare s of type-(a; ). By the de nton of optmal sze, a + =. If = 0, then all of the commttee memers nvest n nformaton n ths equlrum. From Proposton 4, P follows. In addton, Proposton 4 also states that the socal welfare s ncreasng n as long as P. Therefore, = P follows. Suppose now that > 0. If there exsts an equlrum of type-(a; 1), then, y Proposton 6, s not the optmal commttee sze. Hence, f the sze of the commttee s, there does not exst an equlrum of type-(a; 1). By Proposton 5, ths mples that the par (a; 1) volates the nequalty chan (4) wth =. Snce the rst and last nequaltes n (4) hold ecause there s a type-(a; ) equlrum, t must e the case that the second nequalty s volated. That s, P a + 1 = 1. Ths mples that P + 1. Agan, from Proposton 4, t follows that = P or P + 1. Next, we turn our attenton to the potental welfare loss due to overszed commttees. Theorem 2 In any commttee of sze (> ), all equlra nduce hgher socal welfare than the unque equlrum n the commttee of sze 2. The followng lemma plays an mportant role n the proof. We pont out that ths s the only step of our proof that uses Assumpton 1. Lemma 2 For all 1 and 2 N, g ( 1) fg () g ()g fg () g ( 1)g f () ()g ; (10) and t holds wth equalty f and only f = or 1. 14

15 Proof of Theorem 2. Recall that S (a; ) denotes the expected socal welfare generated y an equlrum of type-(a:). Usng ths notaton, we have to prove that S ( 2; 0) < S (a; ). From Theorem 1, we now that = P or P + 1. By Proposton 4, S P 2; 0 < S P 1; 0. Therefore, n order to estalsh S ( 2; 0) < S (a; ), t s enough to show that S P 1; 0 < S (a; ) ; (11) hal , verson 1-3 Jun 2009 for all pars of (a; ) whch satsfy (4). Notce that f a+ memers nvests n nformaton, whch happens wth proalty ra; (1 r a; ) n a type-(a; ) equlrum, the socal welfare s N(a + ) c(a + ). Therefore, X h S (a; ) = r a; (1 r a; ) N (a + ) c (a + ) =0 ( X h = r a; (1 r a; ) N (a + ) c ) ca =0 ( X ) = N r a; (1 r a; ) (a + ) c (a + r a; ) : =0 In the last equaton, we used the dentty P =0 e rewrtten as N P 1 c P 1 < N Snce a P ( X =0 ra; (1 r a; ) = r a;. Therefore, (11) can ) ra; (1 r a; ) (a + ) c (a + r a; ). 1 y (4) and N, the rght hand sde of the prevous nequalty s larger than ( X ) N ra; (1 r a; ) (a + ) c P 1 + Nr a; =0 Hence t su ces to show that N P 1 c P 1 < N After addng c P ( X =0 ) ra; (1 r a; ) (a + ) c P 1 + Nr a; : 1 to oth sdes and dvdng through y N, we have ( X ) ( P 1) < r a;(1 r a; ) (a + ) cr a; : (12) =0 The left-hand sde s a payo of an ndvdual f P 1 sgnals are acqured y others, whle the rght-hand sde s the payo of an ndvdual who s randomzng n a type-(a; ) equlrum wth proalty r a;. Snce ths ndvdual s nd erent etween randomzng and not collectng nformaton, the rght-hand sde of (12) can e rewrtten as X 1 1 r a;(1 r a; ) 1 (a + ): =0 15

16 Hence (12) s equvalent to By Lemma 2 X 1 1 ( P 1) < r a;(1 r a; ) 1 (a + ): (13) =0 g P 1 ( X 1 1 r a;(1 r a; ) 1 g(a + ) g P ) (14) =0 > g P g P 1 ( X 1 1 r a;(1 r a; ) 1 (a + ) P ) : =0 hal , verson 1-3 Jun 2009 Notce that X 1 1 r a;(1 r a; ) 1 g(a + ) = c < g P 1 ; (15) =0 where the equalty guarantees that a memer who s randomzng s nd erent etween nvestng and not nvestng, and the nequalty holds y (3). Hence, from (14) and (15), g P 1 g P 1 g P > g P g P 1 ( X 1 1 r a;(1 r a; ) 1 (a + ) P ) : Snce g P 1 g P > 0; the prevous nequalty s equvalent to =0 g P 1 > P X 1 1 r a;(1 r a; ) 1 (a + ). Fnally, snce P g P 1 = P 1, ths nequalty s just (13). =0 The two graphs of Fgure 3 show the socal welfare n the worst equlrum as a functon of the commttee sze. In ths numercal example, the pror s symmetrc and the parameters are chosen such that N = 100; s N (!; 1), = :3; p = :7; and c = 10 4 : In addton, P = 11 and = 12: The two graphs are ndeed the same graph except that the scalngs of the vertcal axes are d erent. One can see that the welfare loss due to overszed commttees s qute small. 4 Concluson In ths paper, we have dscussed the optmal commttee sze and the potental welfare losses assocated wth overszed commttees. We have focused on envronments n whch there s no con ct of nterest among ndvduals ut nformaton acquston s costly. Frst, we have con rmed that the optmal commttee sze s ounded. In other words, the Condorcet Jury Theorem fals to hold, that s, larger commttees mght nduce smaller socal welfare. However, we have also showed that the welfare loss due to overszed commttees s surprsngly small. In an artrarly large commttee, 16

17 hal , verson 1-3 Jun 2009 Fgure 3: Socal Welfare, as a functon of the commttee sze 17

18 even the worst equlrum generates a hgher welfare than an equlrum n a commttee n whch there are two less memers than n the optmal commttee. Our results suggest that carefully desgnng commttees mght e not as mportant as t was thought to e. 5 Appendx Lemma 3 Suppose that 2 C 1 (R + ) s asolutely contnuous, strctly ncreasng and 0 ( + 1) = 0 () s strctly ncreasng for > "; where " 0. Let g () = ( + 1) () for all 0: Then g ( + 1) =g () < g ( + 2) =g ( + 1) for ": Proof. Fx a ( ") : Notce that 0 ( + 2) = 0 ( + 1) < 0 (t + 2) = 0 (t + 1) s equvalent to 0 ( + 2) 0 (t + 1) < 0 ( + 1) 0 (t + 2). Therefore, hal , verson 1-3 Jun 2009 It follows that 0 ( + 2) Z +1 0 (t + 1) dt < 0 ( + 1) Z +1 0 (t + 2) dt, 0 ( + 2) [ ( + 2) ( + 1)] < 0 ( + 1) [ ( + 3) ( + 2)]. 0 ( + 2) ( + 3) ( + 2) g ( + 2) 0 < = ( + 1) ( + 2) ( + 1) g ( + 1) : (16) Smlarly, for all t 2 (; + 1), 0 ( + 2) = 0 ( + 1) > 0 (t + 1) = 0 (t) s equvalent to 0 ( + 2) 0 (t) > 0 ( + 1) 0 (t + 1). Therefore, It follows that 0 ( + 2) Z +1 0 (t) dt > 0 ( + 1) Z +1 0 (t + 1) dt, 0 ( + 2) [ ( + 1) ()] > 0 ( + 1) [ ( + 2) ( + 1)]. 0 ( + 2) 0 ( + 1) From (16) and (17) t follows that for all ": > ( + 2) ( + 1) ( + 1) () g ( + 1) g () < g ( + 2) g ( + 1) = g ( + 1) : (17) g () Proof of Proposton 1. The sum of normally dstruted sgnals are also normal; P =1 s N!; p. The densty functon of P =1 s condtonal on! s 0 P 1 p s 1! p A where (x) = (2) 1=2 exp x 2 =2 : The ex post e cent decson rule s gven y (s 1 ; ; s ) = 1 f s s ; and (s 1 ; ; s ) = 1 f s s < ; 18

19 where = 2 =2 log [(1 q) (1 ) =q] s the cut-o value. Hence, for 2 Nn f0g, () = q Pr [ (s 1 ; ; s ) = 1j! = 1] (1 q) (1 ) Pr [ (s 1 ; ; s ) = 1j! = 1] + = q p (1 q) (1 ) p where s the cdf of standard normal dstruton. If = 0, (18) (0) = max f q; (1 q) (1 )g : (19) Notce that the rght hand sde of (18) converges to that of (19) as goes to zero. Part() If q + = 1; then q = (1 q) (1 ) and = 0. Hence p! () = 2q and 0 () = q p! 1 p for > 0: hal , verson 1-3 Jun 2009 Therefore, 0 ( + 1) 0 () = r exp 2 2 s ncreasng n (> 0) : From Lemma 3, settng " to e zero, t follows that g ( + 1) =g () s ncreasng n 2 N: Part() Frst, we argue that for any " (> 0) ; 0 ( + 1) = 0 () s ncreasng for all > " f s su cently small. For > 0; 0 ( + 1) 0 () = = = r + 1 (+1) p +1 r + 1 = p r exp ( + 1) r L exp 2 exp 8 ( + 1) exp exp where L = log f(1 q) (1 ) = (q)g : Now suppose that > ". The last term n (20) has no n uence on whether 0 ( + 1) = 0 () s ncreasng or not. The second term converges to 1 as goes to 0. Ovously, the rst term s strctly ncreasng n. Hence, 0 ( + 1) = 0 () s ncreasng n (> ") ; f s su cently small. By settng " 2 (0; 1) and usng Lemma 3, we have shown that g ( + 1) =g () < g ( + 2) =g ( + 1) for all 1: It remans to e shown that g (1) =g (0) < g (2) =g (1) : From the argument n the proof of Lemma 3, t follows that 0 (2) = 0 (1) < 0 (t + 1) = 0 (t) for all t 2 (1; 2) mples 0 (2) = 0 (1) < g (2) =g (1) : Hence, t s enough to show that g (1) =g (0) < 0 (2) = 0 (1) for su cently small : Snce lm!0 g (0) = (20) (0) > 0; t s enough to show that lm!0 [g (1) = f 0 (2) = 0 (1)g] = 0. In order to estalsh ths equalty, t s ovously enough to show that lm!0 () 0 (2) = 0 = 0 for 2 f1; 2g : (21) (1) 19

20 hal , verson 1-3 Jun 2009 Rememer L = log f(1 q) (1 ) = (q)g : By (18), for > 0, ( () = q L p! 2 p + exp (L) whch mples () 2 O E L p!) 2 p p= as! 0: 8 Usng (20), 0 (2) = 0 (1) 2 O E exp 1=2 2 = O E ( (1=)) as! 0. By l Hôptal s Rule, for 2 f1; 2g ; p= p= p= 2 lm = lm!0 (1=)!0 (1=) (1= 3 = 0; ) whch mples (21). Proof of Proposton 2. Frst, we clam that the ex post e cent decson rule : f 1; 0; 1g! f 1; 1g s the followng cut-o rule: 8 >< 1 f (s 1 ; ; s ) = >: 1 f X s ; =1 X s < ; where = log [(1 q) (1 ) =q] = log [p= (1 p)] : Suppose that the sgnal sequence (s 1 ; ; s ) s a permutaton of 8 9 >< >= >: 1; {z ; 1 ; 0; ; 0; 1; ; 1 } {z } {z } >; : (23) a a Then (s 1 ; ; s ) = 1 f In addton, E! [u (!; 1) js 1 ; ; s ] = (1 q) (1 ) Pr [s 1; ; s j! = 1] Pr [s 1 ; ; s ] > E! [u (!; 1) js 1 ; ; s ] = q Pr [s 1; ; s j! = 1] : Pr [s 1 ; ; s ] =1 (22) Pr [s 1 ; ; s j! = 1] = (pr) a (1 r) a (r (1 p)) and Pr [s 1 ; ; s j! = 1] = (pr) (1 r) a (r (1 p)) a : Hence, (s 1 ; ; s ) = 1 f (1 q) (1 ) p (1 p) a > qp a (1 p) ; or equvalently, log a > log (1 q)(1 ) q p 1 p = : 8 O E s a verson of Landau s O, whch descres the exact order of the expresson. Formally, f (x) 2 O E (g (x)) as x! a 9M > 0 s.t. lm x!a jf (x) =g (x)j = M: 20

21 Snce P =1 s = a, (22) follows. Now, we shall consder the case where p converges to 1. Let " denote 1 p and let Pr [a; ] denote the proalty of a sgnal sequence whch s a permutaton of (23). Then, Pr [a; j! = 1] = C (a; ) (1 ") a " ; Pr [a; j! = 1] = C (a; ) (1 ") " a ; where C (a; ) = [!=(a!! ( a )!)] r a+ (1 r) a. 9 Notce that C (a; ) s ndependent of " and symmetrc wth respect to a and. We have 10 Pr [a 1j! = 1] = C (0; 1) " + O " 2 ; and hal , verson 1-3 Jun 2009 Pr [a 0j! = 1] = C (0; 0) + fc (1; 0) + C (1; 1)g " + O " 2 : Oserve that < 1 f p s close enough to one. Wthout loss of generalty, assume that q + 1: Then Then where 1 < 0: Hence, () = q Pr [a 1j! = 1] (1 q) (1 ) Pr [a 0j! = 1] = qc (0; 1) " (1 q) (1 ) [C (0; 0) + fc (1; 0) + C (1; 1)g "] + O " 2 : and D (a; ) = C +1 (a; ) g ( + 1) g () g () = ( + 1) () = A () + B () " + O " 2 ; A () = (1 q) (1 ) D (0; 0) ; (24) B () = qd (0; 1) (1 q) (1 ) [D (1; 0) + D (1; 1)] ; = = C (a; ) : Usng these notatons, A ( + 1) 1 + B(+1) A(+1) " + O "2 = A () 1 + B() A() " + O ("2 ) B ( + 1) B () 1 + " + O " 2 : (25) A ( + 1) A () A ( + 1) + B ( + 1) " + O "2 A () + B () " + O (" 2 ) A ( + 1) A () We want to show that g ( + 1) =g () s ncreasng n f " s su cently small. Snce A ( + 1) =A () = 1 r; t s su cent to show that B () =A () s convex n : It s straghtforward to see that D (0; 1) D (0; 0) = D (1; 0) D (0; 0) = ( + 1) r (1 r) r (1 r) 1 ( + 1) r (1 r) r (1 r) +1 (1 r) = (1 r) 2 (1 r) s a polynomal of wth degree 1, hence t has no n uence on the convexty of B () =A () : On the other hand, D (1; 1) D (0; 0) = ( + 1) r2 (1 r) 1 ( 1) r 2 (1 r) 2 (1 r) +1 (1 r) = ( + 1) r2 (1 r) ( 1) r 2 (1 r) 3 (1 r) 2 9 De ne C (a; ) = 0 f < a f (x) 2 O (g (x)) as x! 0 f and only f 9 > 0; M > 0 s.t. jxj < mples jf (x) =g (x)j < M: 21

22 hal , verson 1-3 Jun 2009 has a postve coe cent of 2 : Hence we conclude B () =A () s convex n. Proof of Proposton 3. de ned y As n the proof of Proposton 2, the ex post e cent decson rule s (s) = ( 1 f P s ; 1 otherwse, where = log [(1 q) (1 ) =q] = log [p= (1 p)] : By symmetry, we can assume 0 wthout loss of generalty. Frst, suppose > 1: Then () = an addtonal sgnal s zero for < q for < ; and the margnal ene t from 1. Therefore, g ( + 1) =g () s not well-de ned. Second, suppose 0 < 1: We shall consder two d erent cases dependng on whether s even or odd. Case 1: Suppose = 2m; where m 2 N. Then, the (2m + 1)-st sgnal maes a d erence f and only f the rst 2m sgnals have a te etween postve and negatve sgnals 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! = 1; and the loss s (1 q) f! = 1: Therefore, expected margnal ene t s g (2m) = q Pr [! = 1; pv e ] (1 q) Pr [! = 1; pv e ] 2m = q p m (1 p) m p (1 q) (1 ) m 2m = fpq (1 p) (1 q) (1 )g m p m (1 p) m : 2m p m (1 p) m (1 p) m Case 2: Suppose = 2m + 1; where m 2 N. Then, the (2m + 2)-nd sgnal maes a d erence f and only f the rst (2m + 1) sgnals contans (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! = 1; and the gan s (1 q) f! = 1: Therefore, expected margnal ene t s g (2m + 1) = q Pr [! = 1; pv o ] + (1 q) Pr [! = 1; pv o ] 2m + 1 = q p m+1 (1 p) m+1 + (1 q) m 2m + 1 = f q + (1 q) (1 )g m (1 ) p m+1 (1 p) m+1 : 2m + 1 p m+1 (1 p) m+1 Recall that 0 < 1; whch s equvalent to pq (1 p) (1 q) (1 ) > 0 and (1 q) (1 ) q 0: If > 0; g (2m + 2) g (2m + 1) pq (1 p) (1 q) (1 ) = ; (1 q) (1 ) q whch s a constant functon of m, and hence, Assumpton 1 does not hold. g (2m + 1) = 0 and g (2m + 2) =g (2m + 1) s not well-de ned. Proof of Lemma 1. Part (). Notce that X 1 1 G (r; a; ) = r (1 r) 1 g (a + ) : =0 m If = 0; then 22

23 Snce r (1 r) 1 = r (1 r) + r +1 (1 r) 1, hal , verson 1-3 Jun 2009 Snce g s decreasng G (r; a; ) > X 1 1 r G (r; a; ) = (1 r) + r +1 (1 r) 1 g (a + ). = = =0 =0 X 1 1 X 1 r (1 r) 1 g (a + ) + r +1 (1 r) 1 g (a + + 1) =0 =0 X 1 1 X r (1 r) 1 g (a + ) + r (1 r) g (a + ) 1 X 1 =0 + =1 1 r (1 r) g (a + ), 1 where the rst equalty holds ecause we have just rede ned thenotaton n the second summaton, n n and the second equalty holds ecause, y conventon, = = 0 for all n 2 N. Fnally, 1 n usng + =, we have 1 G (r; a; ) > X =0 r (1 r) g (a + ) = G (r; a; + 1). Part (). By the de ntons of r a; and r a;+1, we have and y part () of ths lemma, Therefore c = G (r a; ; a; ) = G (r a;+1 ; a; + 1) ; G (r a;+1 ; a; + 1) < G (r a;+1 ; a; ) : G (r a; ; a; ) < G (r a;+1 ; a; ) Snce G (r; a; ) s strctly decreasng n r; r a; > r a;+1 follows. Proof of Lemma 2. The statement of the lemma s ovous f 2 f 1; g. It remans to show that (10) hold wth strct nequalty whenever =2 f 1; g. Frst, notce that for any postve sequence, fa j g 1 0, f a j+1=a j < a j+2 =a j+1 for all j 2 N, then P a j=+1 > a j P a 1 1 j= a j for all > 1 and for all 2 f0; :::; 2g : Assumpton 1 allows us to apply ths result for the sequence a j = g (j), and hence, for all 1 P g () g ( 1) > j=+1 P g (j) 1 j= g (j) = ( + 1) ( + 1) () () for all 2 f0; :::; 2g : 23

24 Snce (a) > () f a >, ths mples that for all 2 f0; :::; 2g g () [ () ()] < g ( 1) [ ( + 1) ( + 1)]. (26) Smlarly, for a postve sequence fa j g 1 0 f a j+1=a j < a j+2 =a j+1 for all j 2 N, then P a j=+1 < a j P a 1 1 j= a j for all 1 and for all + 1: Agan, y Assumpton 1, we can apply ths result to the sequence a j = g (j) and get P g () g ( 1) < j=+1 P g (j) 1 j= g (j) = ( + 1) ( + 1) () () for all >. hal , verson 1-3 Jun 2009 Multplyng through y g ( 1) ( () ()), we get (26). That s, (26) holds whenever =2 f 1; g. After sutractng g ( 1) ( () ()) from oth sdes of (26) we get (10). References [1] D. Austen-Smth, J.S. Bans Informaton Aggregaton, Ratonalty, and the Condorcet Jury Theorem Amercan Poltcal Scence Revew 90(1) (1996) [2] R. Ben-Yashar, S. Ntzan The nvaldty of the Condorcet Jury Theorem under endogenous decsonal slls Economcs of Governance 2(3) (2001) [3] T. Börgers, Costly Votng Amercan Economc Revew 94 (2004) [4] Le Marqus de Condorcet Essa sur l applcaton de l analyse à la proalté des décsons rendues à la pluralté des vox Les Archves de la Revoluton Françase, Pergamon Press (1785) [5] P. Coughlan In Defense of Unanmous Jury Verdcts: Mstrals, Communcaton, and Strategc Votng Amercan Poltcal Scence Revew 94(2) (2000) [6] T. Feddersen, W. Pesendorfer Swng Voter s Curse Amercan Economc Revew 86 (1996) [7] T. Feddersen, W. Pesendorfer Votng Behavor and Informaton Aggregaton n Electons wth Prvate Informaton Econometrca 65(5) (1997) [8] T. Feddersen, W. Pesendorfer Convctng the Innocent: The Inferorty of Unanmous Jury Verdcts under Strategc Votng Amercan Poltcal Scence Revew 92(1) (1998) [9] D. Gerard, L. Yarv, Informaton Acquston n Commttees mmeo (2006) 24

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