A Resurrection of the Condorcet Jury Theorem

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1 A Resurrecton of the Condorcet Jury Theorem Yuo Koryama Balázs Szentes Department of Economcs, Unversty of Chcago June 13, 2007 Abstract 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 or not to acqure nformaton, and then, they mae the ex-post effcent decson. The optmal commttee sze,, s bounded. The man result of ths paper s that any arbtrarly large commttee can aggregate the decentralzed nformaton more effcently than the commttee of sze 2. Ths result suggests that havng larger commttees generates relatvely small neffcences. 1 Introducton The classcal Condorcet Jury Theorem (CJT) states that large commttees can aggregate decentralzed nformaton more effcently than small ones. Its orgn ascends to the dawn of the French Revoluton when Mare Jean Antone Ncolas Cartat le Marqus de Condorcet [1785, translaton 1994] nvestgated decson-mang processes n socetes. 1 Recently, a growng lterature on commttee desgn ponted out that f the nformaton acquston s costly, the CJT fals to hold. The reason 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 too small compared to the cost of nformaton acquston. As a result, she mght prefer to save ths cost and free-rde on the nformaton of others. Therefore, too large 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 an mportant and delcate ssue. On the one hand, we also dentfy a welfare loss assocated to overszed commttees. On the other hand, we show that ths loss s surprsngly small n certan envronments E. 59th Sreet, Chcago, IL The authors are grateful to the semnar partcpants at worshop at Unversty of Chcago for helpful comments. 1 Summares of the hstory of the CJT can be found n, for example, Grofman and Owen [1986], Mller [1986], or Gerlng, Grüner, Kel, and Schulte [2003]. 1

2 The reason why commttee desgn receves a consderable attenton by economsts s that, n many stuatons, groups rather than ndvduals mae decsons. Informaton about the desrablty of the possble decsons s often decentralzed: Indvdual group members must separately acqure costly nformaton about the alternatves. A classcal example s a tral jury where a jury has to decde whether a defendant s gulty or nnocent. Each juror may ndvdually obtan some nformaton about the defendant at some effort cost (payng attenton to the tral, nvestgatng evdences etc.). Another example s a frm facng a 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 hrng decsons of academc departments. Each member of the recrutng commttee s supposed to revew the applcatons before mang a collectve decson. What s common n these examples s that nformaton acquston s costly and often unobservable. The specfc setup analyzed n ths paper s the followng. A group of ndvduals has to mae a bnary decson. There s no conflct of nterest among the group members, but they have mperfect nformaton about whch decson s the best. Frst, ndvduals are ased to serve n a commttee. Then, the commttee members smultaneously decde whether or not to nvest nto an nformatve sgnal. Fnally, the optmal decson s made gven the acqured nformaton. We do not explctly model how the commttee members communcate and aggregate nformaton. Instead, we just assume that they end up mang the ex-post effcent decson. 2 The only strategc choce an ndvdual has to mae n our model s whether or not to acqure a sgnal upon beng selected to serve n the commttee. The central queston of our paper s: How does the commttee sze,, affect socal welfare? Frst, for each, we fully characterze the set of equlbra (ncludng asymmetrc and mxedstrategy ones). We show that there exsts a p such that whenever p, there s a unque equlbrum n whch each commttee member nvests nto nformaton wth probablty one. In addton, the socal welfare generated by these equlbra s an ncreasng functon of. If> p, then there are multple equlbra, and many of them nvolve randomzatons by the members. We also show that the socal welfare generated by the worst equlbrum n the game where the commttee sze s s decreasng n. The optmal commttee sze,,sdefned such that () f thecommtteeszes, then there exsts an equlbrum that maxmzes socal welfare, and () n ths equlbrum, each member nvests nto nformaton wth postve probablty. It turns out that the optmal commttee sze,,sether p or p +1. Ths 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 equlbrum generates hgher socal welfare than the unque equlbrum n the commttee of sze 2. That s, the 2 Snce there s no conflct of nterest among the ndvduals, t s easy to desgn a mechansm whch s ncentve compatble and effcently aggregates the sgnals. Alternatvely, one can assume that the collected nformaton s hard. 2

3 welfare loss due to an overszed commttee s qute small. Lterature Revew The Condorcet s Jury Theorem provdes an mportant support for the bass of democracy. In hs orgnal wor, Condorcet assumed nave (sncere) votng, that s, each voter behaves as f he or she was the only decson maer n the socety. Ths s not compatble wth ratonal behavor n general, because nave votng mght be dfferent from the best response gven nave votng by the others. Austen-Smth and Bans [1996] showed that ratonal ndvduals do not necessarly tae the same acton n the collectve decson mang process as n the ndvdual one. Ths s because a voter s ratonal choce s based on the condton n whch her vote s pvotal. Feddersen and Pesendorfer [1998] showed that t mght be even wrong to follow the ntuton accompaned wth the nave votng. As the jury sze ncreases, the probablty of convctng an nnocent ncreases when the voters tae ratonal actons under the unanmty rule, because beng pvotal s a strong sgn of the suspect to be gulty. A strong mplcaton from these papers s that an approprate assumpton should be a strategc votng rather than nave one. Under the assumpton of strategc votng, a varety of papers addressed the queston of asymptotc effcency of votng as an nformaton aggregaton devce. Feddersen and Pesendorfer [1997] showed that prvate nformaton s effcently aggregated under strategc votng as the commttee sze goes to nfnty. Intuton s that the belef over states converges to the dstrbuton where the result of votes s almost always ted condtonal on beng pvotal, thus the wnner s elected by those who tae nformatve acton. Hence the each voter has an ncentve to vote nformatvely. Myerson [1998] showed that the asymptotc effcency holds n Posson Games where there s populaton uncertanty. However, Condorcet Jury Theorem mght fal f the nformaton acquston s costly 3. Muhopadhaya [2003] showed that a larger jury may mae a worse decson, because of the free rder problem. Ths result s farly ntutve. When the commttee sze becomes large, each commttee member becomes less responsble n the decson mang process, hence becomes less wllng to pay the postve cost to acqure nformaton. Therefore CJT fals n the Muhopadhaya s settng. He showed by numercal examples that mxed-strategy equlbra n large commttees may yeld a lower expected welfare than the optmal small commttee. The paper most related to ours n sprt s probably Martnell [2006]. The author also ntroduced cost of nformaton acquston and, n addton, he allows the precson of the sgnals contnuously depend on the amount of nvestment. Martnell [2006] proves that f the cost, and the margnal cost of the precson s zero at zero level of precson, then the decson s asymptotcally effcent. That s, f the sze of the commttee converges to nfnty, there s a sequence of symmetrc equlbra n whch each member nvests only a lttle, and the probablty of correct 3 The results becomes qute dfferent f we assume that the cost occurs when the voters go to vote, nstead of when they collect nformaton. For costly votng, see e.g. Borgers [2004] 3

4 decson converges to one. Our paper emphaszes the fxed cost aspect of nformaton acquston and should be vewed as a complementary result to Martnell [2006]. In hs accompanyng paper Martnell [2007], he showed that there s no equlbrum wth nformaton acquston when the cost s heterogeneous and prvate nformaton. However n hs model, abstenton s not allowed 4. Persco [2004] dscusses the relatonshp between decson rules and the accuracy of the sgnal. A votng rule whch requres a strong consensus to upset the status quo (n the extreme, unanmty) can be optmal only f the sgnal s suffcently accurate. The ntuton s as follows. Under the unanmty rule, the probablty of beng pvotal s small and t becomes even smaller f the sgnal s not accurate. Hence the sgnal needs to be suffcently accurate to gve an ncentve for a voter to nvest n the costly nformaton. L [2001] showed that the optmal decson rule s ex post suboptmal, when the pror belef s based toward one of the two alternatves. Gerard, Yarv [2006] also found that an ex ante optmal devce s ex post neffcent n general. Ths s because there s a trade-off between lettng agents to acqure nformaton and extractng the maxmal amount of nformaton from them. They also found that the optmal sze s bounded 5, but does not necessarly concde wth the maxmum number of agents who can be nduced to purchase nformaton n equlbrum. Gershov, Szentes [2004] showed that the optmal mechansm s sequental and characterzed the ex ante optmal votng scheme among the ex post effcent ones. The socal planner follows the cut-off decson rule. Secton 2 descrbes the model. The man result of the paper s 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 tae one of two values: 1 and 1. Furthermore,Pr[ω =1]=π (0, 1). The socety must mae a decson, d, whch s ether 1 or 1. There s no conflct of nterest among ndvduals. Each of them has a beneft ofu (d, ω) f decson d s made when the state of the world s ω. Inpartcular, 0 f d = ω, u (d, ω) = q f d = 1 and ω =1, (1 q) f d =1and ω = 1, where q (0, 1) s the parameter ndcatng the severty of type-i error 6. Each ndvdual can purchase a sgnal at a cost c (> 0) at most once. Sgnals are d across ndvduals condtonal 4 If abstenton s allowed, there may be mxed-strategy equlbra n large commttee wth nformaton acquston. 5 They have ths result even though the cost s not ncluded n the defnton of socal surplus functon. 6 In the jury context, where ω =1corresponds to the nnocence of the suspect, q ndcates how severe error t s to covct an nnocent. 4

5 on the realzaton of the state of the world. The ex-post payoff of an ndvdual who nvests nto nformaton s u c. Each ndvdual maxmzes her expected payoff. There are two stages of the decson-mang process. At Stage 1, ( N) members of the socety s desgnated to serve n the commttee at random. At Stage 2, the members decde smultaneously and ndependently whether or not to nvest nto nformaton. Then, the effcent decson s made gven the sgnals collected by the members. We do not model explctly how do commttee members delberate at Stage 2. Snce there s no conflct of nterest among the members, t s easy to desgn a communcaton protocol whch effcently aggregates nformaton. Alternatvely, one can assume that the acqured nformaton s hard and hence, no communcaton s necessary for mang the ex-post effcent decson. We focus merely on the ncentves of the commttee members to acqure nformaton. Next, we turn our attenton 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 and f the sgnal profle s (s 1,..., s ) then µ (s 1,..., s )=1 E ω [u (1,ω):s 1,..., s ] E ω [u (1,ω):s 1,..., s ]. The socal welfare s measured as the expected sum of the payoffs of the ndvduals, that s, NE s1,,,.s,ω [u (µ (s 1,..., s ),ω) c], (1) where the expectaton also taes nto account the possble randomzaton of the ndvduals, that s, 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. On the other hand, f s too small, there s too lttle nformaton to aggregate, and hence the fnal decson s lely to be neffcent. The queston s: What s the optmal that maxmzes ex-ante socal welfare? To be more specfc, the optmal sze of the commttee s, f() themosteffcent equlbrum, n the commttee wth members, maxmzes the socal surplus among all equlbra n any commttee, and () each member acqures nformaton wth postve probablty n ths equlbrum. Snce the sgnals are d condtonal on the state of the world, the expected beneft ofan ndvdual from the ex post effcent decson s a functon of the number of sgnals acqured. Defne ths functon as follows: η () =E s1,,s,ω [u (µ (s 1,,s ),ω)]. We assume that the sgnals are nformatve, but only mperfectly, about the state of the world. That s, as the number of sgnals goes to nfnty, the probablty of mang the correct decson s strctly ncreasng and converges to one. Formally, the functon η s strctly ncreasng and lm η () =0. An ndvdual s margnal beneft from collectng an addtonal sgnal, when sgnals are already obtaned, s g () =η ( +1) η (). 5

6 Note that lm g () =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 ( +1)/g () beng ncreasng n ( N). Whether or not ths assumpton s satsfed depends on the dstrbuton of the sgnals and also on the parameters q and π. An mmedate consequence of ths assumpton s the followng. Remar 1 The functon g s decreasng. Proof. Suppose by contradcton that there exsts an nteger, n 0 N, suchthat,g (n 0 +1) > g (n 0 ). Snce g ( +1)/g () s ncreasng n, t follows that g (n +1)>g(n) whenever n n 0. Hence, g (n) >g(n 0 ) > 0 whenever n>n 0. Thsmplesthatlm g () 6= 0, a contradcton. In general, t s hard to chec whether ths assumpton holds because t s often dffcult (or mpossble) to express g () analytcally. The next secton provdes examples where Assumpton 1 s satsfed. 2.1 Examples for Assumpton 1 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 was mpossble. The assumpton s also satsfed even f π + q 6= 1f the sgnals are suffcently precse. Formally: Proposton 1 Suppose s N (ω,σ). () If q + π =1then Assumpton 1 s satsfed. () If σ s small enough then Assumpton 1 s satsfed. Proof. See the Appendx. In our next example the sgnal s trnary, that s, ts possble values are { 1, 0, 1}. In addton Pr (s = ω ω) = pr, Pr (s =0 ω) = 1 r, 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. Proposton 2 Suppose that the sgnal s trnary. Then, there exsts a p (r) (0, 1) such that, f p > p (r), Assumpton 1 s satsfed. 6

7 Proof. See the Appendx. Next, we provde an example where the logconvexty 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. Proposton 3 If the sgnal s bnary then Assumpton 1 s not satsfed. Proof. See the Appendx. 3 Results Ths secton s devoted to the proofs of our man theorems. To that end, we frst characterze the set of equlbra for all. The next subsecton shows that f 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 large enough (Propostons 5). Fnally, Secton 3.3 proves the man theorems (Theorems 1 and 2). 3.1 Pure-strategy equlbrum Suppose that the sze of the commttee s. Ifthefrst 1 members acqure nformaton, the expected gan from collectng nformaton for the th member 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 compatblty constrant guaranteeng that a commttee member s wllng to nvest nto nformaton f the sze of the commttee s. Proposton 4 Let denote the sze of the commttee. There exsts a P N, such that, there exsts a unque equlbrum n whch each member nvests nto a sgnal wth probablty one f and only f mn { p,n}. Furthermore, the socal welfare generated by these equlbra s monotoncally ncreasng n mn P,N ª. Proof. Recall from Remar 1 that g s decreasng and lm g () =0. Therefore, for any postve cost c<g(0) 7, there exsts a unque P N such that g P <c<g P 1. (3) Frst we show that f < P then there s a unque equlbrum n whch each commttee member nvests nto nformaton. Suppose that n an equlbrum the frst 1 members randomze 7 If c>η(1) η(0), then nobody has ncentve to collect nformaton, hence P =0. 7

8 accordngtotheprofle (r 1,...,r 1 ), where r [0, 1] denotes the probablty that the th member nvests. Let I denote the number of sgnals collected by the frst ( 1) members. Snce the members randomze I s a random varable. Notce that I 1, and E r1,...,r 1 [g (I)] g ( 1) because g s decreasng. Also notce that from < p and (3) t follows that g ( 1) >c. Combnng the prevous two nequaltes, we get E r1,...,r 1 [g (I)] >c. Ths nequalty mples that no matter what the strateges of the frst ( 1) members are, the th member strctly prefers to nvest nto nformaton. From ths observaton, the exstence and unqueness of the pure-strategy equlbrum follow. It remans to show that f > P such a pure-strategy equlbrum does not exsts. But f p then g () <c.therefore the ncentve compatblty constrant (2) s volated, and there s no equlbrum where each member ncurs the cost of the sgnal. Fnally, we have to show that the socal welfare generated by these pure-strategy equlbra s ncreasng n ( mn { p,n}). Notce that snce N>1, c<g( 1) = η() η( 1) <N(η() η( 1)). After addng Nη( 1) c we get Nη( 1) c ( 1) <Nη() c. The left-hand sde s the socal welfare generated by the equlbrum n commttee of sze 1 whle the rght-hand sde s the socal welfare nduced by the commttee of sze. Ths s what Muhopadhaya [2003] showed n the case where sgnal s bnary. He also showed by numercal examples that mxed-strategy equlbra n large commttees would yeld lower expected welfare than small commttees. Our analyss goes further by analytcally comparng the expected welfare of all mxed-strategy equlbra ncludng asymmetrc ones. Fgure 1 s the graph of g () and c. The amount of purchased nformaton n equlbrum s neffcently small. The reason s that although nformaton s a publc good, when a commttee member decdes whether or not to nvest, she only consders her prvate beneft nstead of the socetes. Hence, the total number of sgnals acqured s smaller than the socally optmal amount, and the socal welfare s monotoncally ncreasng n the commttee sze, aslongas s smaller than the frst best level. 8

9 Fgure 1: Expected gan g () and the cost c 3.2 Mxed-strategy equlbrum Supposenowthattheszeofthecommtteeslargerthan p. We consder strategy profles n whch the commttee members can randomze when mang 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 nto a sgnal wth probablty one, and b members acqure nformaton wth postve but less than one probablty. The rest of the members, a (a + b) of them, do not ncur the cost. We call such an equlbrum a type-(a, b) equlbrum. Proposton 5 Letthecommtteeszebe> p. Then, for all equlbra there s a par (a, b) such that, a members nvests for sure, b members nvests wth probablty r (0, 1), and (a + b) members do not nvest. In addton a P a + b, (4) where the frst two nequaltes are strct whenever b>0. Proof. Frst, we explan that f, n an equlbrum a member nvests wth probablty r 1 (0, 1) and another one wth probablty r 2 (0, 1), then r 1 = r 2. Snce the margnal beneft from 9

10 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,,) be the probablty that the member collects nformaton n equlbrum. Suppose that r 1, r 2 (0, 1), andr 1 >r 2. Let I 1 and I 2 denote the number of sgnals collected by members 2, 3,..., and by members 1, 3,...,, respectvely. Notce, that snce r 1 >r 2,andg s decreasng E r2,r 3,...,r (g (I 1 )) >E r1,r 3,...,r (g (I 2 )). (5) On the other hand, a member who strctly randomzes must be ndfferent between nvestng and not nvestng, and hence, for j =1, 2 E rj,r 3,...,r (g (I j )) = c. (6) Ths equalty mples that (5) should hold wth equalty, 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 (4). Suppose that n a commttee a members nvest nto nformaton for sure and b 1 nvests wth probablty r. LetG(r; a, b) denote the expected gan from acqurng nformaton for the (a + b)th member. That s, µ 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 an r (0, 1) such that G (r; a, b) =c. Suppose frstthatsuchanr exsts. We argue frst, 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 collectng nformaton and not collectng t. The a members, who nvest for sure, strctly prefer to nvest, because there s margnal gan from an addtonal sgnal exceeds G (r; a, b). Smlarly, those members who don t nvest, a (a + b) of them, 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. But 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 must hold. Therefore, t s suffcent to show that G (r; a, b) =c has a soluton n (0, 1) f and only f (4) holds. Notce that G(r; a, b) and strctly decreasng n r, becauseg 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, 10

11 Fgure 2: The set of mxed-strategy equlbra 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). (7) Recall that P satsfes g P <c<g P 1. Snce g s decreasng, (1) holds f and only f a P 1 and a + b 1 P, that s, a +1< P +1 a + b. The last nequalty n (4), a + b, must hold because a + b cannot exceed the sze of the commttee,. Fgure 2 graphcally represents the set of pars (a, b) whch satsfy (4). Accordng to the prevous proposton, there are several equlbra n whch more than P members acqure nformaton wth postve probablty. A natural queston to as 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 a, the number of members who acqure nformaton for sure, then the larger s b, the number of members who randomze, the smaller s the socal welfare generated by the equlbrum. Ths proposton plays an mportant role n pnnng down the optmal sze of the commttee. Proposton 6 Suppose that N, s such that there are both type-(a, b) and type-(a, b +1) equlbra. Then the socal welfare s strctly smaller n the type-(a, b +1) equlbrum. 11

12 In order to prove ths proposton we need the followng results. Lemma 1 () G (r; a, b) >G(r; a, b +1) for r, and () r a,b >r a,b+1,wherer a,b and r a,b+1 are the solutons of G (r; a, b) =c and G (r; a, b +1)=c respectvely. Proof. See the Appendx. Proof of Proposton 6. Suppose that a members collect nformaton wth probablty one, and b members nvest wth probablty r. Letf (r; a, b) denote the beneft of an ndvdual, that s, Clearly f (r; a, b) r = bx =1 f(r; a, b) = bx =0 µ b r 1 (1 r) b η (a + ) µ b r (1 r) b η(a + ). =0 µ b r (b )(1 r) b 1 η (a + ). µ µ µ µ b b 1 b b 1 Notce that = b and (b ) =b. Therefore, the rght-hand sde of the 1 prevous equalty can be rewrtten as bx µ b 1 µ 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 expresson, ths can be further rewrtten: =0 µ b 1 µ b 1 b r (1 r) b 1 η (a + +1) b r (1 r) b 1 η (a + ) =0 =0 µ b 1 = b r (1 r) b 1 η (a + +1) η (a + ). =0 But notce that 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) r = bg (r; a, b). f (r a,b a, b) f (r a,b+1 a, b +1)>b(r a,b r a,b+1 ) c. (8) f (r a,b a, b) f (r a,b+1 a, b +1) = f (r a,b a, b) f (0 a, b) f (r a,b+1 a, b +1) f (0 a, b +1) = b Z ra,b 0 G (r; a, b) dr b Z ra,b+1 0 G (r; a, b +1)dr. 12

13 By part () of Lemma 1, ths last dfference s smaller than b Z ra,b G (r; a, b) dr b 0 0 Z ra,b+1 G (r; a, b) dr = b Z ra,b r a,b+1 G (r; a, b) dr. By part () of Lemma we now that r a,b+1 <r a,b. In addton, snce G s decreasng n r ths last expresson s smaller than b (r a,b r a,b+1 ) G (r a,b ; a, b). Recall, that r a,b s defned as the soluton to G (r; a, b) =r, and hence, we can conclude (8). Let S(a, b) denote the socal welfare n the type-(a, b) equlbrum, that s: S(a, b) =Nf (r a,b a, b) c (a + br a,b ). Then, S (a, b) S (a, b +1) = Nf (r a,b a, b) c (a + br a,b ) [Nf (r a,b+1 a, b +1) c (a + br a,b+1 )] > Nb(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 (8), 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 possble equlbrum yelds hgher socal welfare than the unque equlbrum n the commttee of sze 2. Theorem 1 The optmal commttee sze,,sether P or P +1. We want to 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 equlbrum that maxmzes socal welfare s of type-(a, b). By the defnton of optmal sze, a + b =. If b =0,thenall the commttee members nvest nto nformaton n ths equlbrum. 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 b>0. If there exsts an equlbrum of type-(a, b 1), then, by Proposton 6, s not the optmal commttee sze. Hence, whenever the sze of the commttee s,thereexstsnotype-(a, b 1). By Proposton 5 t can only happen f the par (a, b 1) volates the nequalty chan (4) wth =.Sncethefrst and last nequaltes n (4) hold because there s a type-(a, b) equlbrum, t must be the case that the second nequalty s volated. That s, P a+b 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. 13

14 Theorem 2 In any commttee of sze (> ), all equlbra nduce hgher socal welfare than the unque equlbrum n the commttee of sze 2. The followng lemma plays an mportant role n the proof. We pont out that the statement of ths lemma s a consequence of Assumpton 1. Lemma 2 For all N, g ( 1) {g () g ()} {g () g ( 1)}{η () η ()}, (9) and t holds wth equalty f and only f = or 1. ProofofTheorem2. Let S (a, b) denote the socal welfare generated by the equlbrum of type-(a.b). Thats,S (a, b) =S (a, µ b, r a,b ).Notcethatfexactlya+members collect nformaton, b whch happens wth probablty ra,b (1 r a,b) b, the socal welfare s Nη(a + ) c(a + ). Therefore, bx µ b h S (a, b) = r a,b (1 r a,b ) b Nη(a + ) c (a + ) =0 ( bx µ b h = r a,b (1 r a,b ) b Nη(a + ) c ) ca =0 ( bx µ ) b = N r a,b (1 r a,b ) b η (a + ) c (a + br a,b ). =0 In the last equaton, we used the fact P b b =0 ra,b (1 r a,b ) b = br. From Theorem 1, we now that = P or P +1. Also notce that by Proposton 4 S P 2, 0 <S P 1, 0. Therefore, n order to establsh S ( 2, 0) <S(a, b), tsenough to show that S P 1, 0 <S(a, b), for all pars of (a, b) whch satsfy (4). The prevous nequalty can be rewrtten as Nη P 1 c P 1 ( bx µ ) b <N ra,b (1 r a,b ) b η (a + ) c (a + br a,b ). Snce a P 1 by (4) and b N, the rght hand sde s larger than ( bx µ ) b N ra,b (1 r a,b ) b η (a + ) c ( p 1+Nr a,b ) =0 =0 =0 Hence t suffces to show that Nη P 1 c P 1 ( bx µ ) b <N ra,b (1 r a,b ) b η (a + ) c ( p 1+Nr a,b ). 14

15 After addng c ( p 1) to both sdes and dvdng through by N we have η( P 1) < ( bx =0 µ ) b r a,b(1 r a,b ) b η(a + ) cr a,b. (10) The left-hand sde s a payoff of an ndvdual f P 1 sgnals are acqured by others, whle the rght-hand sde s an ndvdual s 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 Hence (10) s equvalent to µ b 1 r a,b(1 r a,b ) b 1 η(a + ). =0 µ b 1 η( P 1) < r a,b(1 r a,b ) b 1 η(a + ). (11) =0 By Lemma 2 ( µ ) b 1 g ( p 1) r a,b(1 r a,b ) b 1 g(a + ) g ( p ) =0 ( µ ) b 1 > {g ( p ) g ( p 1)} r a,b(1 r a,b ) b 1 η(a + ) η ( p ). =0 (12) Notce that µ b 1 r a,b(1 r a,b ) b 1 g(a + ) =c<g( p 1), =0 where the equalty just guarantees that a member who s randomzng s ndfferent between nvestng and not nvestng, and the nequalty holds by (3). Hence, from (12), g ( p 1) {g ( p 1) g ( p )} ( µ ) b 1 > {g ( p ) g ( p 1)} r a,b(1 r a,b ) b 1 η(a + ) η ( p ). Snce g P 1 g P > 0, the prevous nequalty can be rewrtten as =0 µ b 1 g ( p 1) >η( p ) r a,b(1 r a,b ) b 1 η(a + ). =0 Fnally, snce η ( p ) g ( p 1) = η ( p 1), thsnequaltysjust(11). Now (11) follows and the proof s done. Fgure 3 s the graph of the expected socal welfare, wth parameters N = 100,σ = 1,c =

16 Fgure 3: Socal Welfare, as a functon of the commttee sze 16

17 4 Concluson In ths paper, we have dscussed the optmal commttee sze and the potental welfare losses assocated to overszed commttees. We have focused on envronments where there s no conflct of nterest among ndvduals, but nformaton acquston s costly. Frst, we have confrmed that the optmal commttee sze s bounded. In other words, the Condorcet Jury Theorem fals to hold, that s, larger commttees mght nduce smaller socal welfare. However, we showed that the welfare loss due to overszed commttees s surprsngly small. In an arbtrarly large commttee, even the worst equlbrum generates a hgher welfare than an equlbrum n a commttee where there are two less members than n the optmal one. 5 Appendx Proof of Proposton 1. Part() If q + π =1, then qπ =(1 q)(1 π). Hence η () qπ = Pr [µ (s 1,,s )= 1 ω =1]+ Pr [µ (s 1,,s )=1 ω = 1] s 1,,s s 1,,s Ex post effcent decson rule s µ (s 1,,s )=1 f s s > 0, µ (s 1,,s )= 1 f s s < 0. The sum of normal dstrbuton s also normal; P ³ =1 s N ω, σ. Hence ³ P Pr (s s ω =1) = 1 σ φ =1 s σ ³ P Pr (s s ω = 1) = 1 σ φ =1 s + σ ³ where φ s the pdf of the standard normal dstrbuton; φ(x) =(2π) 1 2 exp η () qπ µ µ = Φ σ +1 Φ σ Ã! = 2Φ. σ x2 2. Therefore Hence Moreover, η 0 () = qπ σ η 0 ( +1) η 0 () Ã! 1 φ σ = for >0 r µ +1 exp 1 2σ 2 17

18 s ncreasng n (> 0). By tang ε =0n the followng lemma, g ( +1)/g () s ncreasng n N. Lemma 3 Let ε 0 and suppose η 0 ( +1)/η 0 () s ncreasng for >ε.theng ( +1)/g () < g ( +2)/g ( +1) for ε. Proof. Fx ( ε). For any t (, +1), Therefore η 0 ( +2) η 0 ( +1) < η0 (t +2) η 0 (t +1) η0 ( +2)η 0 (t +1)<η 0 ( +1)η 0 (t +2) η 0 ( +2) Z +1 η 0 (t +1)dt < η 0 ( +1) Z +1 η 0 (t +2)dt η 0 ( +2)[η ( +2) η ( +1)]<η 0 ( +1)[η ( +3) η ( +2)] η0 ( +2) η ( +3) η ( +2) η 0 < ( +1) η ( +2) η ( +1) Smlarly, for any t (, +1), Therefore Hencewehave η 0 ( +2) η 0 ( +1) > η0 (t +1) η 0 η 0 ( +2)η 0 (t) >η 0 ( +1)η 0 (t +1) (t) η 0 ( +2) Z +1 η 0 (t) dt > η 0 ( +1) Z +1 η 0 (t +1)dt η 0 ( +2)[η ( +1) η ()] >η 0 ( +1)[η ( +2) η ( +1)] η0 ( +2) η ( +2) η ( +1) η 0 > ( +1) η ( +1) η () η ( +2) η ( +1) η ( +1) η () < η0 ( +2) η ( +3) η ( +2) g ( +1) g ( +2) η 0 < < ( +1) η ( +2) η ( +1) g () g ( +1) for any ε. Part() Now, let us consder the asymmetrc model. Ex post effcent decson s a cut-off rule for normal sgnal: µ (s 1,,s ) = 1 f s s >θ, µ (s 1,,s ) = 1 f s s <θ where the cut-off pont s θ = σ2 2 (1 q)(1 π) log. qπ 18

19 Hence η () = qπ Pr [µ (s 1,,s )= 1 ω =1] s 1,,s (1 q)(1 π) Pr [µ (s 1,,s )=1 ω = 1] s 1,,s µ µ θ = qπφ σ (1 q)(1 π) Φ θ + σ. We are gong to show the followng. For any ε>0, there exsts δ>0, such that σ<δmples g ( +1)/g () <g( +2)/g ( +1) for. For that purpose, we clam that for any ε>0, there s a suffcently small σ such that η 0 ( +1)/η 0 () s ncreasng for >ε. ³ r η 0 ( +1) φ θ (+1) σ +1 η 0 = ³ () +1 θ φ σ ³ ³ r exp 1 θ 2 2σ = θ + +1 ³ ³ +1 exp 1 θ 2 2σ 2 2θ + r µ µ 1 = +1 exp θ 2 2σ 2 ( +1) 1 Hence η 0 ( +1)/η 0 () s ncreasng f and only f ϕ () +1 µ exp θ 2 σ 2 ( +1) s decreasng n. µ ϕ 0 θ 2 ( µ +1 θ 2 Ã! ) 1 () =exp σ 2 ( +1) σ ( +1) Therefore ϕ s decreasng +1 µ θ 2 σ 2 θ2 σ 2 < 1 ³ +1 1 σ2 4 µ log Ã! ( +1) < (+1) 2 2 = µ 2 (1 q)(1 π) < qπ ( +1) 2 +1 ( +1) The rght-hand sde s strctly ncreasng n ( 0) and lm 0 { ( +1)/ (2 +1)} =0. Hence for suffcently small σ, ϕ s decreasng for >ε.therefore, η 0 ( +1)/η 0 () s ncreasng for >ε. Then by Lemma 3, g ( +1)/g () <g( +2)/g ( +1) for ε. By tang ε<1, we have shown that g ( +1)/g () <g( +2)/g ( +1) for =1, 2,. What s left to be shown s g (1) /g (0) <g(2) /g (1). As above, η 0 (2) /η 0 (1) <η 0 (t +1)/η 0 (t) for t (1, 2) mples 19

20 η 0 (2) /η 0 (1) <g(2) /g (1). Hence t suffces to show that g (1) /g (0) <η 0 (2) /η 0 (1) for suffcently small σ. Defne L =log{(1 q)(1 π) / (qπ)}. Then for >0, ( Ã! Ã Lσ η () = qπ Φ 2 + e L Φ Lσ!) σ 2 σ and η 0 (2) η 0 (1) ³ 1 2 φ θ 2 σ 2 = φ θ 1 σ Ã ( µθ = 1 exp 1 2 µ )! 2 2 θ σ 2 σ = 1 µ exp 1 µ θ2 2 2σ = 1 µ L 2 exp 2 16 σ2 1 2σ 2. We are gong to show that Then lm σ 0 η()! qπ exp L 2 16 σ2 1 2σ 2 lm σ 0 lm σ 0 g (1) η 0 (2) /η 0 (1) =0 g (1) /g (0) η 0 (2) /η 0 (1) =0 because lm σ 0 g (0) = mn {qπ, (1 q)(1 π)}, whch mples g (1) /g (0) <η 0 (2) /η 0 (1) for suffcently small σ. g (1) lm σ 0 η 0 (2) /η 0 (1) = Ã η(1) qπ 2qπ lm! qπ σ 0 exp. L 2 16 σ2 1 2σ 2 By l Hôptal s theorem, for =1and 2, Ã ³ η() σ qπ = lm σ 0 = lm σ 0 = lm σ 0 σ exp L 2 16 σ2 1 2σ 2 2 σ φ 2 L 2 8 σ + 1 ³ Lσ 2 σ σ exp L 2 3 2σ ³ Lσ φ 2 σ L 2 8 σ4 +1 exp L 2 16 σ σ2 1 2σ 2 2σ 2 20

21 Now ³ lm φ Lσ 2 σ σ 0 exp L 2 16 σ2 1 2σ 2 = = = 1 2π lm σ 0 exp 1 2π lm σ 0 exp à 1 2 µ ( 0 f =2 1 2π e L 2 f =1 Lσ 2 µ ! 2 µ L 2 16 σ2 1 2σ 2 σ L 2 σ 2 + L σ 2 In ether case, Hence lm σ 0 Ã! η() qπ exp L 2 16 σ2 1 2σ 2 lm σ 0 g (1) η 0 (2) /η 0 (1) =0 whch completes the proof. ProofofProposton2. Frst, we clam that the ex post effcent decson rule µ : { 1, 0, 1} { 1, 1} s the followng cut-off rule: where =0 X s > b θ µ (s 1,,s )=1 (13) =1 X s < b θ µ (s 1,,s )= 1 (14) =1 b θ = log ³ (1 q)(1 π) qπ ³ log p 1 p and te s broen by a far con toss. Suppose (s 1,,s ) s a permutaton of 1,, 1, 0,, 0, 1,, 1. {z } {z } {z } a a b b Then µ (s 1,,s )=1f and E ω [u (ω, 1) s 1,,s ] > E ω [u (ω, 1) s 1,,s ] (1 q)(1 π) Pr [s 1,,s ω = 1] > qπ Pr [s 1,,s ω =1] Pr [s 1,,s ] Pr [s 1,,s ] Pr [s 1,,s ω =1] = (pr) a (1 r) a b (r (1 p)) b Pr [s 1,,s ω = 1] = (pr) b (1 r) a b (r (1 p)) a. 21

22 Hence µ (s 1,,s )=1f (1 q)(1 π) p b (1 p) a > qπp a (1 p) b ³ log (1 q)(1 π) qπ a b> = b θ. log ³ p 1 p By defnton, P =1 s = a b, whch gves (13). The opposte sgn of nequalty gves (14). Ex post effcent rule s a cut-off rule. Remember that expected utlty η () s gven by η () = qπ Pr [µ (s 1,,s )= 1 ω =1] s 1,,s (1 q)(1 π) Pr [µ (s 1,,s )=1 ω = 1] s 1,,s Now we consder the case where p converges to 1. Notcethat bθ < 1 when p s close enough to one. There are 3 dstnct cases accordng to the parameter ³ values of q and π. () Frst, suppose that q + π>1. Then log (1 q)(1 π) qπ < 0, whch gves b θ<0. When p s close enough to one, 1 < b θ<0. Then η () = Ã! Pr [a b = 1 ω =1] qπ +Pr[a b< 1 ω =1] Pr [a b =0 ω = 1] (1 q)(1 π) +Pr[a b =1 ω = 1] +Pr[a b>1 ω = 1] and Let ε =1 p. Then Smlarly, and Pr [a b =0 ω = 1] = [ 2 X ]! j!( 2j)!j! (pr)j (1 r) 2j ((1 p) r) j j=0 Pr [a b =0 ω = 1] = (1 r) + ( 1) r 2 (1 r) 2 ε + o ε 2 Pr [a b =0 ω =1] = (1 r) + ( 1) r 2 (1 r) 2 ε + o ε 2 Pr [a b = 1 ω =1] = r (1 r) 1 ε + o ε 2 Pr [a b =1 ω = 1] = r (1 r) 1 ε + o ε 2 Pr [a b< 1 ω =1] = o ε 2 Pr [a b>1 ω = 1] = o ε 2 22

23 Therefore, η () = qπr (1 r) 1 ε Ã! (1 r) + ( 1) r 2 (1 r) 2 ε (1 q)(1 π) +r (1 r) 1 + o ε 2 ε = (1 q)(1 π)(1 r) qπr (1 r)+(1 q)(1 π) ( 1) r 2 + r (1 r) ª (1 r) 2 ε + o ε 2 ³ () Smlarly, f q +π <1, then log (1 q)(1 π) qπ > 0, whch gves b θ>0. When p s close enough to one, 0 < b θ<1. Then, η () = qπ (1 r) (1 q)(1 π) r (1 r)+qπ ( 1) r 2 + r (1 r) ª (1 r) 2 ε + o ε 2 () In the symmetrc case q + π =1, b θ = Pr [a b =0 ω =1] η () = qπ +Pr[a b = 1 ω =1] +Pr[a b< 1 ω =1] 1 2 Pr [a b =0 ω = 1] (1 q)(1 π) +Pr[a b =1 ω = 1] +Pr[a b>1 ω = 1] Snce qπ =(1 p)(1 π), Pr [a b =0 ω =1] η () = qπ +2 Pr [a b = 1 ω =1] +2 Pr [a b< 1 ω =1] Ã! (1 r) + ( 1) r 2 (1 r) 2 ε = qπ +2r (1 r) 1 + o ε 2 ε In the frst 2 cases, η ( +1) η () = qπr (1 r r)(1 r) 1 ε Ã (1 r) + ( 1) r 2 (1 r) 2 ε (1 q)(1 π) +r (1 r) 1 ε = A ()+B() ε + o ε 2! + o ε 2 where A () = (1 q)(1 π)(1 r) B () = qπr (1 r r)(1 r) 1 (1 q)(1 π) (1 2r + r) r (1 r) 2 23

24 Then g ( +1) g () η ( +2) η ( +1) = η ( +1) η () = A ( +1)+B ( +1)ε + o ε 2 A ()+B() ε + o (ε 2 ) ³ A ( +1) 1+ B(+1) A(+1) ε + o 2 ε = ³ A () 1+ B() A() ε + o (ε2 ) Snce g ( +1) g () 1+ B(+1) A(+1) ε + o ε 2 1+ B() A() ε + o (ε2 ) µ B ( +1) =1+ A ( +1) B () ε + o ε 2, A () µ A ( +1) = 1+ A () Ã Ã qπr = (1 r) 1+ (1 q)(1 π)(1 r) µ B ( +1) A ( +1) B () ε + o ε 2 A () + r (1 + 2r r) (1 r) 2!! ε + o ε 2 Therefore, g ( +1)/g () s ncreasng n for suffcently small ε, as long as r s strctly smaller than 1 (whch s what we wanted to show). In the symmetrc case, Ã! r (1 r) + (2 r r) r 2 (1 r) 2 ε η ( +1) η () = qπ +2r (1 r) 1 + o ε 2 (1 r r) ε Hence g ( +1) g () Ã =(1 r) 1+! 2r ( 1+r + r)+2(1 r)( 1+2r +2r) (1 r) 2 ε + o ε 2 Therefore, g ( +1)/g () s ncreasng n for suffcently small ε, as long as r s strctly smaller than 1. ProofofProposton3. Remember that η () = E ω,s [u (ω,µ (s))] = qπ Pr [µ (s) = 1 ω =1] (1 q)(1 π)pr[µ (s) =1 ω = 1] (15) s s Ex post effcent rule s agan the cutoff rule; µ (s) =1 E ω [u (ω, 1) s] E ω [u (ω, 1) s] (1 q)pr ω [ω = 1 s] qpr [ω =1 s] ω (1 q)(1 π)pr[s ω = 1] qπ Pr [s ω =1] s s Pr s [s ω =1] (1 q)(1 π) Pr s [s ω = 1] qπ 24

25 For the bnary sgnal, for both s =1and 1. Hence µ s Pr s [s ω =1] p Pr s [s ω = 1] = 1 p µ (s) =1 µ P s p X 1 p s θ (1 q)(1 π) qπ where log θ = log ³ (1 q)(1 π) qπ ³ p 1 p By symmetry, assume θ 0 wthout loss of generalty. (that s, d = 1 s the status quo.) () Suppose θ>1. Then for any <θ,no sgnal realzaton (s 1,,s ) can upset the status quo, thus η () = qπ. Then the gan g () s zero for <θ 1. Then g ( +1)/g () s not well-defned. () Suppose 0 θ<1. Then we clam that mx µ η (2m) = {qπ +(1 q)(1 π)} 2m p j (1 p) 2m j (16) j j=0 µ 2m +(1 q)(1 π) p m (1 p) m m mx µ η (2m +1) = {qπ +(1 q)(1 π)} 2m +1 p j (1 p) 2m+1 j (17) j Suppose (= 2m) s even. Then µ (s) = 1 ff P s { 2m, 2m +2,, 2, 0}. It happens when there are j ( {0,,m}) sgnals of s =1and the other sgnals are s = 1. Hence mx µ 2m Pr [µ (s) = 1 ω =1]= p j (1 p) 2m j s j j=0 Smlarly, µ (s) =1ff P s {2, 4,, 2m}. It happens when there are j ( {0,,m 1}) sgnals of s = 1 and the other sgnals are s =1. Hence Pr s [µ (s) =1 ω = 1] = m 1 X j=0 j=0 µ 2m j p j (1 p) 2m j Then by (15), (16) follows. Suppose (= 2m +1) s odd. Then µ (s) = 1 ff P s { 2m 1, 2m +1,, 3, 1}. It happens when there are j ( {0,,m}) sgnals of s =1and the other sgnals are s = 1. Hence mx µ 2m Pr [µ (s) = 1 ω =1]= p j (1 p) 2m+1 j s j j=0 25

26 Smlarly, µ (s) =1ff P s {1, 3,, 2m +1}. It happens when there are j ( {0,,m}) sgnals of s = 1 and the other sgnals are s =1. Hence mx µ 2m Pr [µ (s) =1 ω = 1] = p j (1 p) 2m+1 j s j Then by (15), (17) follows. Now, usng (16) and (17), expected gan s µ 2m g (2m) = {pqπ (1 p)(1 q)(1 π)} m µ 2m +1 g (2m +1) = {(1 q)(1 π) qπ} m j=0 p m (1 p) m p m (1 p) m Recall that we assumed 0 θ<1, whch s equvalent to pqπ (1 p)(1 q)(1 π) > 0 and (1 q)(1 π) qπ 0. Now, f θ>0, g (2m +2) pqπ (1 p)(1 q)(1 π) = 2p (1 p) g (2m +1) (1 q)(1 π) qπ s a constant, Assumpton 1 cannot be satsfed. If θ =0,g(2m +2)/g (2m +1)s not well-defned. ProofofLemma1. Part (). Notce that µ 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, µ b 1 r G (r; a, b) = (1 r) b + r +1 (1 r) b 1 g (a + ). =0 Snce g s decreasng µ b 1 µ G (r; a, b) > r (1 r) b b 1 g (a + )+ r +1 (1 r) b 1 g (a + +1) =0 =0 µ b 1 bx µ = r (1 r) b b 1 g (a + )+ r (1 r) b g (a + ) 1 =0 =1 bx µ µ b 1 b 1 = + r (1 r) b g (a + ), 1 =0 where the frstequaltyholdsbecausewehavejustredefned µ theµ notaton n the second summaton, n n and the second equalty holds because, by conventon, = =0for all n N. Fnally, µ µ µ 1 n +1 b 1 b 1 b usng + =,wehave 1 bx µ b G (r; a, b) > r (1 r) b g (a + ) =G (r; a, b +1). =0 26

27 Part (). By the defntons of r a,b and r a,b+1,wehave c = G (r a,b ; a, b) =G (r a,b+1 ; a, b +1), and by part () of ths lemma, 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. Frst, notce that f a j > 0 for j {0,..., } and a j+1 /a j <a j+2 /a j+1 for all j {0,..., 2}, then a a 1 > P j= a j P 1 j= 1 a j for all {1,..., 1}. Assumpton 1 allows us to apply ths result for the sequence a j = g (j), and hence, g () g ( 1) > P j= P g (j) η ( +1) η ( +1) 1 = j= g (j) η () η () for all {1,..., 1}.!!!(equalty f and only f = 1.) Smlarly, f a j > 0 and a j+1 /a j <a j+2 /a j+1 for all j N, then for all >0, a a 1 < P j= g (j) P 1 j= 1 g (j) for all >. Applyng ths result to the sequence a j = g (j), weget g () g ( 1) < P j= P g (j) η ( +1) η ( +1) 1 = j= 1 g (j) η () η () for all >. So far, we have shown that g ( 1) [η ( +1) η ( +1)] g ()[η() η ()], and t holds wth equalty f and only f = 1 or. By subtractng g ( 1) [η () η ()], g ( 1) [g () g ()] [g () g ( 1)] [η () η ()]. 27

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