Notes on Bias Uncertainty and Communication in Committees 1

Transcription

1 Notes on Bas Uncertanty and Communcaton n Commttees 1 Davd Austen-Smth MEDS, Kellogg Graduate School of Management Northwestern Unversty Evanston, IL Tm Feddersen MEDS, Kellogg Graduate School of Management Northwestern Unversty Evanston, IL March Prepared for presentaton to the Roy Semnar, Pars, March Part of these Notes rely on Austen-Smth and Feddersen (2006).

2 1 Introducton It s well understood that when commttee members vote under ncomplete nformaton, the resultng commttee decson need not re ect the decson that would have been made under fully shared nformaton (eg Austen-Smth and Banks 1996; Feddersen and Pesendorfer 1998). But people n commttees often talk before votng and so have an opportunty to share decson-relevant nformaton. Over the past few years there has been a growng strategc game-theoretc lterature concerned to understand better what mplcatons such communcaton mght have for the character and qualty of collectve decsons under ncomplete nformaton. 1 One ssue here concerns how d erent votng rules for reachng a nal collectve choce n uences the nformaton that mght be shared n any pror debate. Gerard and Yarv (2004) suggest that the votng rule (essentally) s rrelevant, but ther argument hnges on usng domnated (debate-condtonal) votng strateges. Assumng undomnated votng changes the pcture (Austen-Smth and Feddersen 2005). And n ths case, t turns out that whether or not ndvduals bases, that s, d erences n full nformaton preferences over avalable collectve choces, are common knowledge plays a crtcal role. 2 In these Notes, we revew some of the postve ssues arsng from bas uncertanty and suggest some ntutons regardng the welfare mplcatons of bases beng prvate nformaton. 2 An example Consder a commttee of three people, = 1; 2; 3, that has to choose between a xed par of alternatves fx; Y g. Each ndvdual has prvate nformaton (b ; s ) 2 fx; yg fx; yg, where b s a preference parameter, or bas, and s s a nosy but nformatve sgnal regardng the 1 For example, see: Austen-Smth 1990; Calvert and Johnson 1998; Coughlan 2000; Doraszelsk et al 2003; Gerard and Yarv 2004; Merowtz 2006, 2007; Hafer and Landa 2007; Austen-Smth and Feddersen 2005, 2006; Callaud and Trole An ntuton noted too n Merowtz (2006). 1

3 alternatves. Let (b; s) = ((b 1 ; b 2 ; b 3 ); (s 1 ; s 2 ; s 3 )) denote a pro le of realzed bases and sgnals. For ths example, assume sgnals are uncorrelated wth bases. Let fa 1 ; A 2 g be states such that, for all, Pr[s = xja 1 ] = Pr[s = yja 2 ] = p 2 (1=2; 1) wth the common pror probablty on A 1 beng 1=2; and, for all, assume Pr[b = x] = r 2 (0; 1). Wrte u(z; b; s) for an ndvdual s payo from the collectve choce Z 2 fx; Y g, gven that ndvdual s bas s b 2 fx; yg and the pro le of sgnals s s 2 fx; yg 3. Assume 8 < 0 f s 2 f (y; y; y)g u(x; x; s) = : 1 otherwse u(y; x; s) = 1 u(x; x; s) 8 < 1 f s 2 f (x; x; x)g u(x; y; s) = : 0 otherwse u(y; y; s) = 1 u(x; y; s) Suppose rst that the commttee s choce s determned by a majorty vote. In the absence of any pre-vote communcaton, there s no equlbrum n whch all ndvduals vote nformatvely, that s, wth ther sgnals (eg Austen-Smth and Banks, 1996). As a result, the collectve choce may favour a mnorty ex post. More formally, let v : fx; ygfx; yg! [0; 1] be a vote strategy, where v(b; s) s the probablty an ndvdual wth bas b and sgnal s votes for X (we assume strateges are anonymous throughout). equlbrum n undomnated strateges n whch v(b; x) = 1 Then under majorty rule, there s no (Bayesan) v(b; y) = 1 for all b. To see ths suppose not and, wthout loss of generalty, consder an ndvdual wth bas b = y. Gven the other commttee members are votng wth ther sgnals, s pvotal only n the event that one ndvdual has a sgnal x and the other has a sgnal y; but then s unque best response s to vote surely for Y rrespectve of hs sgnal. There s, however, a mxed strategy equlbrum when q s not too extreme: n ths equlbrum, v(x; x) = 1 2 v(y; y) = 1, v(x; y) 2 (0; 1) and

4 v(y; x) 2 (0; 1). And clearly, the nal decson may not re ect the full nformaton majorty preference. Now suppose that the commttee can delberate before votng; spec cally, suppose that votng follows one round of smultaneous cheap talk sgnalng. Note that n ths context, and n contrast to n-person sender-recever games (eg Ottavan and Sorenson 2001; Glazer and Rubnsten 2001), every commttee member s both a "sender" and a "recever" and no one ndvdual has the rght to dctate the nal decson. Let : fx; yg fx; yg! M be a message strategy for, where M s n general an arbtrarly large set of messages. For current purposes, t su ces to take M = fx; yg wth the nterpretaton that (b; s) s the probablty an ndvdual wth bas b and sgnal s declares "s = x". 3 Wth talk, ndvduals votng behavour depend on the realzed lst of messages from debate: wth an abuse of notaton, wrte v : fx; yg fx; yg fx; yg 3! fx; Y g be a vote strategy for, where v (b; s; m) s the probablty an ndvdual wth bas b and sgnal s who hears debate m = (m 1 ; m 2 ; m 3 ) 2 fx; yg 3 votes for X. A perfect Bayesan equlbrum n undomnated strateges for ths settng s a pro le of strateges (; v) such that, for every trple (b; s; m) and all : (1) v (b; s; m) s an undomnated best response to (; v ); (2) (b; s) s an undomnated best response votng strategy to ( ; v); and (3) belefs are consstent wth the strategy pro le and Bayes rule where de ned. Assume all ndvduals truthfully reveal ther sgnals: (b; x) = 1 (b; y) = 1 for all and bases b. Then there s fully shared nformaton at the votng stage and the unque undomnated votng equlbrum condtonal on these messages s for each ndvdual to vote sncerely for ther most preferred alternatve as de ned by (b ; (s 1 ; s 2 ; s 3 )). Wth ths fact n mnd, consder whether n fact truthful nformaton sharng n debate s ncentve compatble. Say that an ndvdual s message pvotal n debate f the nal colllectve choce s senstve to whch sgnal the ndvdual reveals he or she has observed. Then t s mmedate that any 3 Later, n a smpler verson of ths example, we explctly consder allowng bases to be revealed n debate. Dealng wth ths possblty here s not essental. 3

5 x-based ndvdual wth an x sgnal, or y-based ndvdual wth y sgnal, has a domnant strategy to reveal ther sgnal truthfully. So, wthout loss of generalty, consder an x-based ndvdual wth a y sgnal: (b ; s ) = (x; y). Then there are essentally two events n whch s message pvotal: (1) both of the other commttee members are x-based and both have observed a y sgnal, whch occurs wth (condtonal) probablty 1 2 r2 p 3 + (1 p) 3 ; (2) both of the other commttee members are y-based and both have observed an x sgnal, whch occurs wth (condtonal) probablty 1 2 (1 r)2 (1 p) 2 p + (1 p)p 2 = 1 2 (1 r)2 p(1 p): In event (1), ndvdual strctly prefers to tell the truth whle, n event (2), strctly prefers to dssemble. Makng the requste calculatons, tellng the truth n debate condtonal on all others fully revealng ther sgnals s ncentve compatble for an x-based ndvdual wth a y sgnal f and only f r 2 1 r p(1 p) (1 p) 3 + p 3 : Smlarly, ncentve compatblty of truthtellng for a y-based ndvdual wth an x sgnal s nsured f 1 r 2 r p(1 p) (1 p) 3 + p 3 : For every r 2 (0; 1), therefore, there exsts some p < 1 for whch there exsts an equlbrum n whch all ndvduals reveal ther sgnals truthfully n debate and subsequently vote under full nformaton to yeld the ex post rst best de facto majorty preferred outcome. It s mportant to note that the left sde of these nequaltes descrbes the relatve lkelhood of the speaker s bas-type beng the majorty bas n the commttee. Moreover, f r 2 f0; 1g then truthtellng n debate cannot be an equlbrum for any p 2 (1=2; 1): bas uncertanty 4

6 s essental for fully revealng debate. In partcular, wth bas uncertanty there are multple message pvotal events under majorty rule; whether or not any ndvdual tells the truth n debate thus depends on the relatve lkelhood of those events n whch truthtellng s a best response; that s, on those events n whch the ndvdual s a member of the full nformaton wnnng coalton. And because message pvotal events are de ned both n terms of a partcular pars of bas and sgnal pro les, the fact that sgnals are nformatve, coupled wth the event of beng message pvotal, provdes the ndvdual wth nformaton regardng the realzed bas pro le, even when bases and sgnals are statstcally ndependent. Now suppose commttee decson makng s by unanmty rule wth status quo X: that s, X s the outcome unless all commttee members vote for Y. Wthout debate, ndvduals condton ther vote n equlbrum on beng vote pvotal, whch occurs only f both of the other commttee members are votng for the alternatve, Y. Gven all others are votng wth ther sgnal, a y-based ndvdual s best response s therefore to vote for Y rrespectve of her sgnal. Hence, as wth majorty rule, the ex post commttee choce may not be the decson adopted under full nformaton. Suppose there s a debate stage n whch all ndvduals are supposed to report ther sgnals truthfully and consder a y-based ndvdual wth an x sgnal. Under unanmty rule, any ndvdual s best votng decson s ndependent of her own message n debate. Thus the only message pvotal event s when at least one of the other two ndvduals s x-based and both have y sgnals (any b-based ndvdual wth an s = b sgnal surely votes wth hs sgnal). But then the unque best response message s to le and report "y": bas uncertanty s nsu cent to promote fully revealng debate under unanmty rule. In other words, there can only be a sngle message pvotal event under unanmty rule and ths event nduces dssemblng as a best response for at least some types of commttee member. In sum, the example su ces to show that bas uncertanty can promote full nformaton sharng n debate under majorty rule, but does not demonstrate that the same s false under unanmty rule. Furthermore, whle the presence of bas uncertanty can a ect the extent of nformaton sharng n debate, ths n tself does not say anythng about the welfare propertes 5

7 of commttee decson makng under bas uncertanty. In the next secton we address the rst ssue regardng bas uncertanty and unanmty rule qute generally and, n the subsequent secton, use a consderably less general model to consder some welfare mplcatons of bas uncertanty. 3 Model and theorems The commttee N = f1; 2; :::; ng, n 2, has to choose an alternatve Z 2 fx; Y g; let X be the status quo polcy. Each ndvdual 2 N has prvate nformaton (b ; s ) 2 B S, where b s a preference parameter, or bas, and s s a sgnal regardng the alternatves. Assume the sets B and S are nte and common across ndvduals 2 N. Wrte B n B and S n S; a stuaton s any par (b; s) 2 B S, where b = (b 1 ; :::; b n ), s = (s 1 ; :::; s n ). And wth a convenent abuse of language, call any pro le of ndvdual sgnals s 2 S a state, as such a pro le exhausts all of the relevant nformaton for the collectve decson. Let p(b; s) be the probablty that stuaton (b; s) 2 Bl commttee member 2 N, s preferences over fx; Y g depend exclusvely on s own bas b 2 B and on the state s 2 S: gven a bas b and state s, an ndvdual s payo from a commttee decson Z 2 fx; Y g s wrtten u(z; b; s). We assume there are no dogmatc or partsan types; that s, for any bas b 2 B there s a nonempty subset of states S b S such that s 2 S b mples u(y; b; s) > u(x; b; s) and s =2 S b mples u(y; b; s) < u(x; b; s): n other words, all ndvduals preferences over the two alternatves are subject to change. To avod trvaltes we assume that every stuaton occurs wth postve probablty and, because the concern here s wth unanmty rule, that there always exst states at whch all members prefer alternatve Y. The rst two axoms, respectvely, formalze these two essentally techncal assumptons. Full Support For all (b; s) 2 B S, p(b; s) > 0. Consensus For all b = (b 1 ; :::; b n ) 2 B, S(b) \ 2N S b 6= ;. The nal, most substantve, axom mposes some structure on the set of sgnals. We rst 6

8 state the axom formally and then dscuss ts motvaton. Assume that the set of sgnals S s ordered by a bnary relaton,, such that the followng monotoncty condton obtans. Monotoncty For any s; s 0 2 S such that s s 0 and s 2 S n 1, let s = (s ; s) and s 0 = (s ; s 0 ) 2 S. Then u(y; b; s) > u(y; b; s 0 ) and u(x; b; s) < u(x; b; s 0 ) for any b 2 B. In words, suppose there s a par of states that d er only n that some member has observed s 2 S n the rst state and s 0 2 S; then s s 0 mples that s s stronger nformaton than s 0 n favor of Y and aganst X. And notce that ths axom also bulds n a degree of symmetry: any ndvdual s relatve evaluaton of the two alternatves s monotone n sgnals whatever the ndvdual s bas b and rrespectve of exactly whch commttee member receves what sgnal. Thus, the axom nssts that f one ndvdual consders a partcular sgnal to be better evdence n favor of choosng X over Y than some other sgnal, then all ndvduals share ths relatve evaluaton, but they may dsagree about how much better s the evdence. Although a lmtaton, the axom s conservatve for provng a negatve result, as the possbltes for nformaton sharng are clearly greater when all ndvduals share a common vew of what any partcular pece of nformaton mght mean. The commttee chooses an outcome by votng under unanmty rule. That s, X s the outcome unless every member of the commttee votes for y. Pror to votng we assume there s a delberaton phase n whch every member of the commttee can smultaneously send a message m to every other member of the commttee. The focus here s on delberaton that yelds nformaton relevant to the collectve choce beng shared pror to the votng stage, so there s no loss of generalty n assocatng messages drectly wth the nformaton they are presumed to report. Therefore we take the set of avalable messages to any ndvdual to be a set M such that S M. A message strategy for 2 N s a functon, : B S! M. A message pro le m = (m 1 ; m 2 ; :::; m n ) 2 M n M s a debate. De nton 1 A message strategy pro le s fully revealng f, for all 2 N, for all pars of dstnct sgnals s; s 0 2 S, [[ b2b (b; s)] \ [[ b2b (b; s 0 )] = ;. 7

9 As de ned here, fully revealng message strateges may or may not reveal nformaton about ndvdual bases. 4 Because ndvduals preferences depend only on the state and on ther own bas, f a debate fully reveals the state then addtonal nformaton about others bases s decson-rrelevant. Thus the key feature of a fully revealng message strategy s that t provdes full nformaton about the speaker s sgnal. That s, f s fully revealng then, for all ndvduals 2 N and all bas and sgnal pars (b; s) 2 B S, the message (b; s) 2 M unambguously reveals that s prvate sgnal s s. De nton 2 A commttee s mnmally dverse f and only f there exst b; b 0 2 B such that S b 6= S b 0. Note that under the full support assumpton, t s possble for all ndvduals to exhbt the same bas and, therefore, the only commttees that are not mnmally dverse are commttees n whch there s never any dsagreement about when alternatve y s the best choce (S b = S b 0 for all b; b 0 2 B). A votng strategy for member 2 N s a functon v : B S M! fx; yg that maps every debate nto a votng decson. A fully revealng debate equlbrum s a Perfect Bayesan Equlbrum (; v) = (( 1 ; : : : ; n ); (v 1 ; : : : ; v n )) such that s fully revealng and v s a pro le of weakly undomnated votng strateges. Before presentng the result on unanmty rule, t s useful to recall a (slght generalzaton of a) result due to Coughlan (2000). For any q 2 f1; 2; : : : ; ng, a q-rule s a votng rule such that f at least q 1 commttee members vote for Y aganst X, then Y s the commttee decson. Unanmty rule s a q-rule wth q = n. Theorem 1 (Coughlan 2000) Assume the realzed bas pro le s common knowledge and that every state s 2 S occurs wth postve probablty. Assume consensus and monotoncty. Then, 4 At rst glance, the de nton here mght seem unnecessarly awkward wth a smpler verson beng to requre only that f, for all, all b and any s 6= s 0, (b; s) 6= (b; s 0 ). But ths does not work, as t admts the possblty that (b 0 ; s) = (b; s 0_ ) for some b 0 6= b, n whch case s sgnal regardng the state s not revealed. 8

10 for all q rules, n=2 < q < n, there exsts a FRDE f and only f the commttee s not mnmally dverse. Therefore, bas uncertanty s typcally necessary for full nformaton sharng n debate under any rule shy of unanmty. And whle the example showed that bas uncertanty can support a FRDE, ths s by no means guaranteed. That bas uncertanty can never help wth unanmty, however, s the content of the next result. Theorem 2 Assume full support, consensus and monotoncty. There exsts a fully revealng debate equlbrum under unanmty rule f and only f the commttee s not mnmally dverse. Thus the crcumstances under whch unanmty rule promotes fully revealng delberaton are con ned to those n whch t s common knowledge that the commttee s homogenous wth respect to preferences over alternatves, whether or not there s bas uncertanty. A proof for ths result s n the Appendx (where we also con rm that the theorem extends to the case that the true bas pro le b 2 B s common knowledge). 4 Welfare An mportant problem s to dentfy the optmal rule for commttee choce under ncomplete nformaton wth debate. 5 Ths problem s partcularly challengng f we nsst that ndvduals votng strateges are undomnated condtonal on the realzed debate, a requrement that opens up the possblty, as llustrated by precedng theorems, that votng rules a ect the ncentves for delberaton. 6 A more lmted welfare queston concerns the extent to whch the de facto 5 Chwe (2006) derves the optmal votng rule when there s no debate. Interestngly, ths rule turns out to be nonmonotonc n votes and s thus not a q-rule. 6 Gerard and Yarv (2004) show, rst, that all non-unanmous q rules are equvalent n that the sets of sequental equlbrum outcomes nduced by use of any q rules wth q 6= 1; n are dentcal once votng s preceded by delberaton and, second, that those outcomes nduced by unanmty rule are a (not necessarly proper) 9

11 commttee decson under a gven votng rule (typcally, a q-rule) wth ncomplete nformaton and debate re ects the commttee s decson condtonal on all prvate nformaton beng common knowledge at the tme of the vote (eg McLennan 1998; Federsen and Pesendorfer 1997). Equvalently, gven the rule, the goal s to maxmze the aggregate expected payo of those n the full nformaton wnnng coalton. And the ntuton suggested by the results above s that, at least for nonunanmous q-rules, bas uncertanty can mprove the welfare of the full nformaton wnnng coalton by facltatng more nformaton sharng n debate: when ndvduals are not sure ex ante whether they are members of the full nformaton wnnng coalton, then they have an ncentve to reveal prvate nformaton n debate that s absent when they are con dent they are not members of ths coalton. Unfortunately, even restrctng attenton to q-rules, addressng the latter welfare ssue drectly s complcated by havng to descrbe the full equlbrum set to any more or less general ncomplete nformaton (debate and votng) game nduced by the rule (see for example Austen- Smth and Feddersen (2005) on ths ssue). At ths stage, no such characterzaton s avalable. 7 So rather than attempt a general result here, we nstead llustrate the ntuton above wth a smpler varant of the openng example. In the varant of the example, we assume two ndvduals n a commttee n whch one xed ndvdual has the rght to make the commttee decson. Indvduals preferences and subset of those of nduced by any non-unanmous rule. Thus any non-unanmous q rule can be chosen wthout a ectng what s possble n a pre-vote debate. However, Gerard and Yarv do not requre votng strateges to be undomnated condtonal on the realzed debate. Instead, ther result explots the fact that the rules governng debate are unconstraned and that (at least on the equlbrum path) all voters votng unanmously s consstent wth sequental ratonalty for non-unanmous q rules. Such consstency s obtaned n ther analyss ether by admttng weakly domnated strateges or, wth a mld doman restrcton, precludng domnated strateges de ned n terms of (ex ante or nterm) expectatons formed pror to any ndvdual hearng any debate. 7 Lkewse, the optmal mechansm here s as yet unavalable n general. Chwe s (2006) result for majorty votng wth ncomplete nformaton suggests such a mechansm wll be complex. 10

12 the nformatonal assumptons on sgnals s 2 fx; yg are exactly as spec ed above. However, we perturb the assumptons on the dstrbuton of bases: for each = 1; 2, assume that whle the pror probablty that b = x s 1=2, realzed bas types are correlated; that s, for dstnct ; j 2 f1; 2g Pr[b = b j jb ] = 2 [0; 1]: If = 0 they have con ctng preferences wth probablty 1 whereas f = 1 both have dentcal preferences. The dea here, s that the decson-maker s a proxy for the pvotal voter de nng the wnnng coalton under a q-rule n a larger commttee, and the other ndvdual (the advsor) s unsure whether she too s a member of that wnnng coalton. Hereafter, let ndvdual = 1 be the advsor and ndvdual = 2 be the decson-maker. We consder two communcaton protocols. In the rst protocol there s a sngle debate stage exactly as htherto, n whch both ndvduals make cheap talk speeches regardng ther sgnals, followng whch the decson-maker chooses X or Y. For the second protocol, we add an earler debate stage n whch both players can declare ther bases followng whch they both send messages regardng ther sgnals and, nally, the decson-maker makes a decson. Equlbra are agan perfect Bayesan wth undomnated strateges. Note that, because an ndvdual s payo s, gven her bas, depend exclusvely on the nal decson and the pro le of realzed sgnals s, there can be no value (n n sgnalng both bas and sgnal smultaneously or of havng a sgnal debate pror to makng any statements regardng bases. Thus the role of permttng bas revelaton early n any commttee dscusson s to mprove the possblty of coordnaton between commttee members. Begn wth assumng only a sngle debate stage n whch ndvduals make speeches about ther sgnals. Strateges are as before: : fx; yg fx; yg! fx; yg s a debate strategy wth (b; s) beng the probablty that ndvdual wth bas b and sgnal s declares "s = x"; and v : fx; yg fx; yg fx; yg 2! fx; Y g s the decson-maker s decson (vote) strategy wth v(b; s; m) beng the probablty that the decson-maker (ndvdual 2) wth bas b and sgnal 11

13 s who hears debate m = (m 1 ; m 2 ) 2 fx; yg 2 chooses X. Hereon, we economze on notaton by wrtng b;s for the probablty an advsor wth bas type b and sgnal s sends message x; and wrtng v b s(n) for the probablty the decson-maker wth bas b; sgnal s who hears message n chooses X. Proposton 1 There exsts a FRDE (1 p). If < (1 p) the advsor cannot reveal any nformaton n equlbrum and the decson-maker chooses wth hs sgnal. Proof: In a fully revealng equlbrum we have, for all, b;x = 1 b;y = 1 for every b 2 fx; yg and = 1; 2. In ths case the best response strategy for a decson-maker wth bas b who observes sgnal s 0 and hears message s from the advsor s 8 1 f (s; s >< 0 ) 2 fx; xg 8b vs b 0(s) 1 f (s; s 0 ) 2 ffx; yg; fy; xgg and b = x >: 0 otherwse The ncentve compatblty condtons for a FRDE are obvously sats ed for the decsonmaker snce he s correctly choosng as f he were fully nformed. So, we only need to check the IC constrants for the advsor. Wthout loss of generalty, assume the sender has bas b 1 = x. Then t s trvally the case (mod nversons of natural language) that she surely reveals a sgnal s 1 = x. So suppose she has observed s 1 = y. There are only two pvotal events n whch the sender s message a ects the outcome here: ether the recever shares the sender s bas (x) and has observed a sgnal s 2 = y; or the recever has the opposte bas (y) and has observed a sgnal s 2 = x. In the former event, the sender s best response s to tell the truth and reveal s 1 = y; n the latter event, the sender s best response s to le and clam s 1 = x. The probablty of these two events s, respectvely, p and (1 )(1 p). Therefore the sender s wllng to tell the truth when she has sgnal y p (1 )(1 p); 12

14 that s, (1 p). If < (1 p), then clearly the advsor has a strct best response to announce "s = b" rrespectve of her sgnal; hence no nformaton s credbly conveyed n debate (the decson-maker can always reveal hs sgnal n debate, but ths s of no consequence for the nal outcome). Hence the decson-maker must choose on the bass of hs own nformaton. Suppose, wthout loss of generalty, that b 2 = x. Then player 2 has a domnant strategy to choose X condtonal on s 2 = x. If s 2 = y, however, hs payo from choosng X s (1 p)=2 whereas that from choosng Y s p=2. Snce p > 1=2, the best response s to choose Y. It s worth emphassng that a FRDE can exst here for very small values of, dependng on just how nformatve s the sgnal, p: the more nformaton that any gven sgnal provdes, the less concern the advsor exhbts about the probablty of not beng a member of the de facto wnnng coalton. Ths result s clearly a drect analogue of the condtons de nng exstence of a FRDE for the majorty rule example wth whch we began. The analytcal advantage here, however, s that the proposton completely descrbes the equlbrum set up to the mxed equlbrum for the nongenerc event = (1 here are therefore easy to descrbe. p). The welfare mplcatons of bas uncertanty Proposton 2 Assume the most nformatve equlbrum s played for every parameterzaton (p; ). (a) If > (1 p) then, ex ante, the decson-maker (respectvely, advsor) strctly prefers bases to reman secret (respectvely, revealed) before the game s played. (b) If < (1 p) then, ex ante, both players strctly prefer bases to be revealed pror to the game beng played. Proof: Suppose that both players bases are made common knowledge pror to the game beng played. Then ether they share the same bas, n whch case the most nformatve equlbrum s a FRDE, or they have opposng bases n whch case (as s easy to con rm) no nformaton s revealed by the advsor, player 1. The payo s to the two players are therefore: 1 + p EU1 rev = EU2 rev = + (1 ) : 2 13

15 To see ths, rst note that f bases are the same, all nformaton s revealed by player 1 and both players are assured a payo of one. Next, suppose the bases are revealed to be dstnct. Then no nformaton s revealed and the decson-maker can do no better than choose accordng to hs sgnal; that s, choose X s 2 = x. In ths case the sender (player 1) receves one ether s 1 = s 2, whch occurs wth probablty p, or her bas s x (respectvely, y) and s 2 = x (respectvely, y), whch occurs wth probablty (1 p)=2. These facts yeld the expresson for EU rev 1. Smlarly, gven the bases are dstnct, the decson-maker obtans a payo equal to one ether s 1 = s 2 or her bas s x (respectvely, y) and s 2 = x (respectvely, y). Ths just es EU rev 2. Now suppose both players bases reman secret. If > (1 p) then there exsts a FRDE and the decson-maker obtans a payo EU pvt 2 = 1 surely. The advsor, however, receves payo one surely only f the bases are the same; f bases are d erent, then the advsor reveals her sgnal and receves payo one s 1 = s 2. Thus EU pvt 1 = +(1 )p. On the other hand, f < (1 p) then the only equlbrum nvolves no nformaton revelaton and the decson-maker chooses accordng to hs sgnal. Hence, followng the reasonng for the case n whch bases are revealed, we obtan EU pvt 1 = EU pvt 2 = 1 + p 2 : Therefore, f > (1 p) the decson-maker gans (1 )(1 + p)=2 when bases are secret but the advsor loses (1 )(1 p)=2. And when < (1 p), both players strctly prefer bases to be revealed. If the lkelhood that the advsor s n fact a member of the wnnng coalton n that she shares a common bas wth the decson-maker, then t the presence of bas uncertanty strctly mproves the welfare of that coalton (.e. the decson-maker) snce all decsonrelevant nformaton can be shared n equlbrum. Because the two players agree about the desrablty of bas-revelaton when s su cently low relatve to p, however, t s nterestng to ask whether provdng an opportunty to coordnate drectly by revealng bases pror to 14

16 debatng sgnals can mprove commttee performance. To address ths ssue, we turn to the second protocol descrbed earler, whereby there s a pror debate stage n whch (e ectvely) ndvduals can reveal ther bases. 8 Strateges for ths protocol are as follows. Let b;s 2 [0; 1] denote the probablty that ndvdual wth bas b and sgnal s announces her bas s x; gven bas-stage messages (m; m 0 ) 2 fx; yg 2 ; let b;s (m; m 0 ) 2 [0; 1] s the probablty ndvdual wth bas b and sgnal s who has announced her bas s m and heard a message that the other player s bas s m 0, announces that her sgnal s x; and let v b s((m; m 0 ); (n; n 0 )) be the probablty the decson-maker wth bas b and sgnal s who announces bas m and sgnal n, hears the advsor s bas message m 0 and sgnal message n 0, chooses outcome X. Proposton 3 Assume each player has observed hs or her partcular bas and sgnal. If 1=2, there exsts an equlbrum n whch each player truthfully reveals ther bas n the rst message round and (1) f the bases are the same, players truthfully reveal ther sgnal n the second round and the recever chooses on the bass of full nformaton; (2) f the bases are d erent, players smply announce ther true bas ndependently of ther sgnal and the decson-maker chooses wth hs sgnal. Proof: Consder the followng strateges (spec ed only for an x-based ndvdual; those for a y-based ndvdual are symmetrc): x;x = x;y = 1; = 1; 2 x;x (m; m 0 ) = 1; 8(m; m 0 ); = 1; 2 x;y (m; x) = 1 x;y (m; y) = 0; 8m; = 1; 2 8 Of course ndvduals can talk about anythng at any stage. However, gven messages are cheap talk, t s not hard to see that assumng that bases are the subject of the rst communcaton round and sgnals the subject of second s, at least n equlbrum, wthout loss of generalty. 15

17 v x x( m; m 0 ; n; n 0 ) = 1; 8( m; m 0 ; n; n 0 ) v x y ((m; m) ; (; x)) = v x y ((x; y) ; (x; x)) = 1; 8m v x y ( m; m 0 ; n; n 0 ) = 0 otherwse To con rm that ths strategy pro le consttutes an equlbrum f 1=2, we calculate where E[U b;s E[U x;x j x;x = 1; ] = E[U x;x E[U x;y j x;y (1 + p) = 1; ] = + (1 ) 2 E[U x;y j x;y (1 + p) = 0; ] = + (1 ) 2 j b;s j x;x (1 + p) = 0; ] = + (1 ) 2 ; ] denotes the equlbrum expected payo for ndvdual wth bas b and sgnal s from adoptng bas-debate strategy b;s. Hence tellng the truth at the bas revelaton stage s ncentve compatble E[U x;y j x;y = 1; ] E[U x;y j x;y = 0; ], whch obtans f 1=2. And gven bases are revealed truthfully, the subsequent debate and decson strateges are easly checked to be best responses. Thus allowng an opportunty to coordnate through sharng bas nformaton before revealng anythng about decson-relevant sgnals, cannot mprove the welfare propertes for the decsonmaker when < 1 p. 9 Note that n the dent ed equlbrum (say, the bas revelaton equlbrum), so long as 1=2, both ndvduals reveal ther bases n the rst round of talk; f ther bases are 9 For completeness, we note that there s an essentally dentcal equlbrum wth the three person majorty rule commttee ntroduced at the start of these Notes. Suppose n that settng too we ntroduced two rounds of debate, n whch ndvduals smultaneously send messages regardng ther bases n the rst round of talk. Then there s an equlbrum n whch all bases are revealed n the rst stage; those ndvduals n the majorty then reveal ther sgnals truthfully n the second round of debate, and the mnorty player smply announces hs bas agan; and nally, the majorty votes together for the alternatve that maxmzes ts expected welfare gven the nformaton revealed n debate. The mnorty player votes sncerely under full nformaton. Clearly, just as wth the current two player example, the de facto decson does not always re ect the full nformaton welfare maxmzng decson for the majorty coalton. 16

18 the same then all nformaton s revealed n the second round whereas, f bases are d erent, no further decson-relevant nformaton sharng takes place. It s mmedate, therefore, that the expected payo to any (b; s)-type ndvdual from playng the bas revelaton equlbrum dent ed n Proposton 3 equals the expected payo they acheve f bases could be revealed ex ante by some external agent. Spec cally, for all ; b; s, E[U b;s j bases revealed] = + (1 ) (1 + p) : 2 Furthermore, because p > 1=2, Propostons 1 and 3 together mply there s also a FRDE wth a bas debate stage when 1 p: both ndvduals babble durng the rst round of talk and, subsequently, all decson-relevant nformaton s revealed durng the second round of communcaton exactly as descrbed n Proposton 1; that s, for all ; b; s, b;s = 1=2 and b;x = 1 b;y = 1. Hence we have the same welfare comparsons as descrbed n Proposton 2, wth the further observaton that, when 1=2 > 1 p, the advsor strctly prefers to play the bas revelaton equlbrum to playng the FRDE wth no bas revelaton (of course, the decson-maker has the opposte preferences). In sum, bas uncertanty can mprove the welfare of the de facto full nformaton wnnng coalton. And even n those crcumstances n whch bas revelaton s avalable n equlbrum, there exsts a superor (from the decson-maker s perspectve) equlbrum n whch bases reman prvate nformaton. 5 Concluson There s clearly a great deal left to be learned regardng communcaton n commttees. In partcular, dentfyng the optmal votng rule n the presence of debate s an mportant and open ssue. 17

19 Appendx For all b 2 B, let T 0 (b) S(b) and, for any k = 1; 2; : : :, recursvely de ne the sets T k (b) = fs =2 [ l=k l=1 Tk l (b)j9s; s 0 2 S : s 0 s, (s ; s) = s; (s ; s 0 ) = s 0 and s 0 2 T k 1 (b)g: Thus T 1 (b) s the set of states not n T 0 (b) such that, gven the realzed bas pro le b, changng any one person s nformaton from s to s 0 results n a state n T 0 (b) S(b); T 2 (b) s the set of states not n T 1 (b) such that changng any one person s nformaton from s to s 0 results n a state n ; and so on. Informally, the set T k (b) s the set of states such that there s a path of k sngle coordnate changes of nformaton that lead to a state at whch y s preferred unanmously. Snce S and N are nte t follows that [ k=0;1;:::;n T k (b) = S: For example, suppose n = 3, S = f0; 1g 3 and S(b) = f(1; 1; 1)g. Then T 0 (b) = f(1; 1; 1)g T 1 (b) = f(0; 1; 1); (1; 0; 1); (1; 1; 0)g T 2 (b) = f(1; 0; 0); (0; 1; 0); (0; 0; 1)g T 3 (b) = f(0; 0; 0)g: The followng property of mnmally dverse commttees n envronments satsfyng the three axoms s useful for provng the man theorem. The lemma nsures that n mnmally dverse commttees, there must exst two bas types and a state such that, rst, the two bas types have strctly opposng preferences at the state and, second, that the state d ers n only one component from another state at whch all bas types strctly prefer y to x. Lemma Assume full support, consensus and monotoncty. In a mnmally dverse commttee there exsts a bas pro le b = (b ; b; b 0 ) 2 B and a state s 2 T 1 (b) such that s =2 S b but s 2 S b 0: 18

20 Proof Let b = (b ; b; b 0 ) 2 B (where, by an abuse of notaton, t s understood that b 2 B n 2 ); by consensus, S(b) 6= ;. Frst assume there s a state s 2 S b \T k+1 (b). By full support and de nton of T k (b), there exsts a sgnal s 0 s such that (s ; s 0 ) = s 0 2 T k (b); moreover, by monotoncty, s 0 2 S b. Hence, s 2 S b \ T k+1 (b) mples there exsts a state s 0 2 S b \ T k (b). Now suppose b s such that, for any s 2 T k (b), s =2 S b. Then by the prevous argument, there can be no s 2 T k+1 (b) such that s 2 S b. Hence, S b \ T 1 (b) = ; mples S b \ T k (b) = ; for all k > 1 n whch case, because [ k=0;1;:::;n T k (b) = S, t must be that S b = S(b). It follows that f, contrary to the lemma, for all b 2 B there exsts no s 2 T 1 (b) and components b; b 0 of b such that s =2 S b but s 2 S b 0, then S b = S(b) for all components of b, volatng mnmal dversty. Proof of Theorem 2 (Necessty) In any fully revealng debate equlbrum, the restrcton to weakly undomnated votng strateges mples v (b; s; m) = y f and only f (s ; s) 2 S b, where s = m for every 2 N and b 2 B. It follows that a member s votng strategy does not depend on the message she sends n debate. Consder the delberaton stage and, by way of contradcton, suppose s fully revealng yet the commttee s mnmally dverse. Then, gven the behavor at the votng stage, fully revealng message strateges consttute an equlbrum f and only f, for every 2 N and every (b ; s ) 2 B S, t s the case that EU(m = s ; b ; s ) EU (m = s 0 ; b ; s ) 0 for any s 0 2 Mnfs g (1) where EU(m ; b ; s ) = X X b 2B n 1 s 2S n 1 p(b ; s jb ; s ) [Pr(xjb; s; m )u(x; b ; s) + Pr(yjb; s; m )u(y; b ; s)] and Pr(zjb; s; m ) s the probablty that z 2 fx; yg s the commttee decson gven bas pro le b = (b ; b ), state s = (s ; s ) and debate (m ; m ) = (s ; m ). (b ; s ) = (b; s); for any s 0 2 Mnfsg, de ne the functon Fx 2 N and let ' (b;s) (s; s 0 ; b ; s ) Pr(xjb; s; s) Pr(xjb; s; s 0 ) [u(x; b; s) u(y; b; s)] 19

21 wth b = (b ; b) and s = (s ; s). Then we can rewrte (1) equvalently as requrng that for all (b; s) 2 B S and all s 0 2 Mnfsg, X X b 2B n 1 s 2S n 1 p(b ; s jb; s)' (b;s) (s; s 0 ; b ; s ) 0: (2) By assumpton, s fully revealng of all others sgnals and, by the precedng argument on v, for all messages m 2 M and all bas pro les (b ; b) 2 B, (s ; s) 2 SnS b mples Pr(xj(b ; b); (s ; s); m ) = 1. Smlarly, for any state (s ; s) 2 Sn(S(b) [ T 1 (b)) t must be that Pr(xj(b ; b); (s ; s); m ) = 1. Gven (b ; s ) = (b; s), therefore, for all s 0 2 Mnfsg and all b 2 B n 1, (s ; s) 2 Sn S(b) [ T 1 (b) [ S b ) '(b;s) (s; s 0 ; b ; s ) = 0: (3) The precedng argument mples that an ndvdual wth bas b can change the outcome by swtchng from message s to some s 0 6= s only n stuatons (b; s) such that (s ; s) 2 S(b) [ T 1 (b) \ S b. For all b 2 B, de ne x (b; s; s 0 ) = f(s ; s) 2 S(b)j(s ; s 0 ) =2 S(b)g to be the set of states such that f an ndvdual who s supposed to report s nstead reports s 0 then, condtonal on b, the outcome changes from y to x. Smlarly, de ne y (b; s; s 0 ) = f(s ; s) 2 T 1 (b) \ S b j(s ; s 0 ) 2 S(b)g to be the set of states n whch prefers y and, f s supposed to report s but nstead reports s 0 at b, the outcome changes from x to y. Note that, by monotoncty, f y (b; s; s 0 ) 6= ; for some b 2 B, then x (b; s; s 0 ) = ; for all b 2 B and, f x (b; s; s 0 ) 6= ; for some b 2 B, then y (b; s; s 0 ) = ; for all b 2 B: That s, y (b; s; s 0 ) 6= ; for some b 2 B mples that s 0 s stronger evdence for y than s, whereas x (b; s; s 0 ) 6= ; for some b 2 B mples s 0 s weaker evdence for y than s. By monotoncty both statements cannot be true. For any b 2 B and s; s 0 2 S, let Z (b; s; s 0 ) fs 2 S n 1 j(s ; s) 2 y (b; s; s 0 ) [ x (b; s; s 0 ) g: 20

22 Collectng terms and usng (3), we can rewrte the ncentve compatblty constrant (2) as requrng, for all 2 N, (b; s) 2 B S and s 0 2 Mnfsg, X b 2B X s 2Z (b;s;s 0 ) p(b ; s ; jb; s)' (b;s) (s; s 0 ; b ; s ) 0: By the Lemma and full support, mnmal dversty mples there s a (b ; b) 2 B and a par of sgnals s; s 0 2 S such that y ((b ; b); s; s 0 ) 6= ; and x ((b ; b); s; s 0 ) = ;. By de nton, (s ; s) 2 y ((b ; b); s; s 0 ) mples u(x; b; (s ; s)) < u(y; b; (s ; s)) and Pr(xj(b ; b); s; s) Pr(xj(b ; b); s; s 0 ) = 1. Hence, for all (b ; b) 2 B, s 2 Z (b; s; s 0 ) ) ' (b;s) (s; s 0 ; b ; s ) < 0: But then the ncentve compatblty condtons are surely volated, contradctng the exstence of a fully revealng debate equlbrum n any mnmally dverse commttee. Ths proves necessty. (Su cency) Assume the commttee s not mnmally dverse. Then for all b 2 B and all b = (b ; b) 2 B, S b = S(b). In ths case there s no b 2 B and par of sgnals s; s 0 2 S such that y (b; s; s 0 ) 6= ; for any 2 N. Snce ncentve compatblty s assured for any 2 N, b 2 B and par of sgnals s; s 0 2 S such that x (b; s; s 0 ) 6= ; and y (b; s; s 0 ) = ;, full revelaton s an equlbrum strategy. Ths completes the proof. Fnally, to see that the theorem goes through under complete nformaton regardng ndvduals bases, x a bas pro le b = (b 1 ; : : : ; b n ), suppose b s common knowledge and let B = fbg. Then the de ntons and the argument drectly apply on replacng references to bases b; b 0 2 B wth references to ndvduals ; j 2 N wth bases b ; b j, and so on. 21

23 References Austen-Smth, Davd. 1990a. Informaton transmsson n debate. Amercan Journal of Poltcal Scence 34 (February): Austen-Smth, Davd and Je rey S. Banks Informaton aggregaton, ratonalty, and the Condorcet Jury Theorem. Amercan Poltcal Scence Revew 90 (March): Austen-Smth, Davd and Tmothy J. Feddersen Delberaton and votng rules. In Socal Choce and Strategc Decsons: Essays n Honor of Je rey S. Banks, eds. Davd Austen-Smth and John Duggan. Hedelberg:Sprnger. Austen-Smth, Davd and Tmothy J. Feddersen Delberaton, preference uncertanty and votng rules. Amercan Poltcal Scence Revew 100 (May): Callaud, Bernard and Jean Trole "Consensus buldng: how to persuade a group" CNRS Workng Paper. Calvert, Randall L. and James Johnson Ratonal argument, poltcal argument and democratc delberaton. Presented at the Annual Meetng of the Amercan Poltcal Scence Assocaton. Chwe, Mchael "A robust and optmal anonymous procedure for Condorcet s model." UCLA, Workng Paper. Coughlan, Peter J In defense of unanmous jury verdcts: mstrals, communcaton and strategc votng. Amercan Poltcal Scence Revew 94 (June): Doraszelsk, Ulrch, Dno Gerard and Francesco Squntan Communcaton and votng wth double-sded nformaton. Contrbutons to Theoretcal Economcs 3(1), Artcle 6. Feddersen, Tmothy J. and Wolfgang Pesendorfer Votng behavor and nformaton aggregaton n large electons wth prvate nformaton. Econometrca 65(5):

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