A Note on Preference Uncertanty and Communcaton n Commttees 1 Davd Austen-Smth MEDS, Kellogg Graduate School of Management Northwestern Unversty Evanston, IL 60208 Tm Feddersen MEDS, Kellogg Graduate School of Management Northwestern Unversty Evanston, IL 60208 July 2008 1 We are grateful for the nancal support of the NSF under grant SES-0505818.
Abstract The recent formal lterature on communcaton n commttees ndcates that truthtellng n commttee delberatons s more lkely under nonunanmous votng rules when ndvduals prvate nterests, or bases, are not common knowledge. The result, however, does not mply that such bas uncertanty s necessarly welfare mprovng n some approprate sense. Ths Note suggests that bas uncertanty s n fact welfare mprovng and, further, argues that even when commttee members prefer that bases be shared, there need not exst any cheap talk equlbra n whch ndvduals can credbly reveal such nformaton.
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 decsonrelevant 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. 2 Our focus n ths paper s not on votng rules but rather on the normatve consequences of preference uncertanty for the qualty of collectve choces under ncomplete nformaton. It s useful rst, however, to revew bre y the motvatng postve results on the mplcatons of such uncertanty for nformaton sharng n debate. To x deas, consder the canoncal example of a jury that has to vote over whether to convct or acqut a defendant. The gult or nnocence of the defendant s subject to uncertanty. Each juror prvately observes an nformatve but nosy sgnal from a gven set of possble sgnals regardng the nnocence or gult of the defendant. If the defendant s truly gulty, then the probablty of any juror recevng a sgnal suggestng as much s strctly greater than the probablty the juror receves a sgnal suggestng nnocence, although there s some lkelhood of such an nnocent sgnal s receved; and smlarly wth respect to nnocent sgnals condtonal on the defendant beng truly nnocent. Jurors also have possbly d erent atttudes (bases) regardng just how much such nformaton su ces to convnce them of gult. For example, f there are only two possble sgnals, some juror mght requre all commttee members to have receved the "gulty" sgnal before they are wllng to convct, whereas another could be wllng to convct on the bass of only one such sgnal among the jurors. As wth ther 1
sgnals, ndvduals bases are prvate nformaton. The commttee uses some q-rule to make the nal decson, where a q-rule s a votng rule under whch convcton s chosen f and only f t receves at least q votes, where q s at least a majorty of the jury. The jury may or may not delberate pror to votng, where delberaton conssts of jurors makng costless (so-called cheap talk) speeches regardng any decson-relevant nformaton they may have (n partcular, ther prvate sgnals). In a very general settng of the sort descrbed above, Coughlan (2000) demonstrates that full (collectve decson-relevant) nformaton sharng under complete nformaton about commttee members bases s mpossble under any q-rule unless all ndvduals have dentcal full nformaton ordnal preferences over the alternatves. Austen-Smth and Feddersen (2006) go on to show that Coughlan s result s senstve to the full nformaton assumpton on commttee members preferences: so long as the rule s not unanmty rule (.e. q s less than the total number of commttee members) and there s uncertanty about both ndvduals preferences and sgnals, then there always exst some crcumstances under whch full nformaton sharng of sgnals n debate s possble, thus nducng the same decson as would be made were all sgnals revealed smultaneously to all jurors. On the other hand, f the jury or commttee uses a unanmty votng rule then preference uncertanty no longer provdes any possblty for full nformaton sharng n debate; there s always some type of ndvdual wth a strct ncentve to dssemble n debate and, therefore, the veracty of any nformaton conveyed n ndvduals speeches s generally suspect. The key ntuton underlyng the results s that bas uncertanty admts the possblty that any speaker belongs to the de facto wnnng coalton under complete nformaton, a possblty that provdes an ncentve for truth-tellng n debate. In contrast, when all ndvduals bases are common knowledge pror to any debate, then those n any losng coalton under full nformaton have no ncentve to share nformaton that mproves the chances of the decson that would be most preferred under full nformaton by a wnnng coalton (see also Merowtz 2004, 2007). 2
By themselves, the precedng results do not mply that unanmty rule s n general nferor to any other q-rule. It s possble that unanmty rule performs better on average than other rules despte the fact that full nformaton sharng n debate s (at least, on grounds of strategc ratonalty) mpossble. More generally, an mportant problem s to dentfy the optmal rule for commttee choce under ncomplete nformaton wth debate. 3 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 results sketched above, that votng rules a ect the ncentves for delberaton. 4 In ths paper we address a more lmted welfare queston that concerns the extent to whch the de facto commttee decson under a gven votng 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 (e.g. McLennan 1998; Feddersen 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: as already observed, 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) or Dorazelsk, Gerard and Squntan (2003) on ths ssue). At ths stage, no such characterzaton s avalable. 5 So rather than attempt a general result here, we nstead explore the welfare mplcatons of bas uncertanty n the context of much smpl ed commttee structure predcated on the observaton that when strategcally ratonal ndvduals are choosng a best response strategy for nformaton sharng n debate, they condton on 3
the event that ther partcular speech s message pvotal; that s, that f they say one thng then the resultant commttee vote leads to one alternatve whereas a d erent speech nduces a commttee decson n favour of the competng alternatve. In e ect, therefore, t s as f the speaker were addressng the (possbly hypothetcal) pvotal member of the commttee whose preferences (bas) and nformaton on the relatve value of the two alternatves s prvate nformaton to that pvot. Thus the smpl caton s to reduce the problem to a game between two ndvduals, one of whom s a representatve of a non-decsve coalton (the advsor) and the other a represenatatve of a decsve coalton (the decson-maker). The model Consder a commttee of two people n whch one xed ndvdual has the rght to make the commttee decson between a xed par of alternatves fx; Y g. Each ndvdual = 1; 2 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 alternatves. Let (b; s) = ((b 1 ; b 2 ); (s 1 ; s 2 )) denote a pro le of realzed bases and sgnals. Assume sgnals are uncorrelated wth bases. Let the pror probablty on each sgnal s 2 fx; yg be 1=2 and suppose that, all, for dstnct ; j 2 f1; 2g Pr[s = s j js ] = p 2 (1=2; 1): Assumng p 2 (1=2; 1) nsures sgnals are both nformatve and nosy. For each = 1; 2, suppose 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. As suggested earler, 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 4
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. 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 2. Assume 8 < 0 f s = (y; y) u(x; x; s) = : 1 otherwse u(y; x; s) = 1 u(x; x; s) 8 < 1 f s = (x; x) u(x; y; s) = : 0 otherwse u(y; y; s) = 1 u(x; y; s) Thus, a bas type b = x (respectvely, b = y) strctly prefers alternatve X (respectvely, Y ) unless both sgnals consttute evdence for Y (respectvely, X). We consder two communcaton protocols. In the rst protocol there s a sngle "debate" stage n whch the advsor makes a cheap talk speech regardng her sgnal, followng whch the decson-maker chooses X or Y. Wthout any real loss of generalty, we assume the avalable messages are smply the sgnals, fx; yg. For the second protocol, we add an earler debate stage n whch both players can declare ther bases (agan, through cheap talk speeches) followng whch the advsor sends a message regardng her sgnal as before and, nally, the decson-maker makes a decson. 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 sgnalng both bas and sgnal smultaneously or of havng a debate on sgnals 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 message stage n whch the advsor makes a speech about her sgnal. Strateges are as follows: : fx; yg 2! fx; yg s a debate strategy wth (b; s) beng the probablty that the advsor (ndvdual 1) wth bas b and sgnal s declares "s = x"; 5
and v : fx; yg 3! fx; Y g s the decson-maker s decson (vote) strategy wth v(b; s; n) beng the probablty that the decson-maker (ndvdual 2) wth bas b and sgnal s who hears a message n 2 fx; yg from the advsor chooses X. Let (s; ) fx; yg be the set of messages sent wth strctly postve probablty when sgnal s s observed. De nton 1 A message strategy s fully revealng f (x; ) \ (y; ) =?. As de ned here, fully revealng message strateges may or may not reveal nformaton about the advsor s bas. Because an ndvdual s preferences depend only on the par of sgnals and on ther own bas, f a message fully reveals the state then addtonal nformaton about the other s bas s decson-rrelevant. Thus the key feature of a fully revealng message strategy s that t provdes full nformaton about the speaker s sgnal. A fully revealng debate equlbrum (FRDE) for the one-stage debate protocol s a Perfect Bayesan Equlbrum (; v) such that s fully revealng. 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 vs(n) b for the probablty the decson-maker wth bas b; sgnal s who hears message n chooses X. Proposton 1 There exsts an FRDE (1 p). If < (1 p) the advsor cannot reveal any nformaton n equlbrum and the decson-maker chooses wth hs sgnal. Proof: Wthout loss of generalty, we can suppose that n a fully revealng equlbrum we have, for all, b x = 1 b y = 1 for every b 2 fx; yg. 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 ) = (x; x) 8b vs b 0(s) = 1 f (s; s 0 ) 2 ffx; yg; fy; xgg and b = x >: 0 otherwse 6
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 advsor 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 advsor s message a ects the outcome here: ether the decson-maker shares the advsor s bas (x) and has observed a sgnal s 2 = y; or the the decson-maker has the opposte bas (y) and has observed a sgnal s 2 = x. In the former event, the advsor s best response s to tell the truth and reveal s 1 = y; n the latter event, the advsor s best response s to le and clam s 1 = x. The condtonal 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); that s, (1 p). If < (1 p), then clearly the advsor has a strct best response to announce "s = b" rrespectve of her sgnal and 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. 6 Ths result s a drect analogue of the condtons de nng exstence of a FRDE for the case of q-rules n a commttee of more than two ndvduals (Austen-Smth 7
and Feddersen, 2005, 2006), dscussed nformally n the Introducton, and shares the same fundamental ntuton (see also the Appendx to ths paper). In partcular, as made explct n the argument for the result, when decdng on her best message the sender condtons on beng message pvotal and there are (n ths model) two such events: n one the decson-maker shares the advsor s bas and sgnal; n the other the decson-maker has the opposte bas and sgnal. Although the dstrbutons of bas and sgnal are qute ndependent, the realzaton of bases and sgnals are lnked for the advsor through the event of beng message pvotal. And although whch event s relatvely more lkely depends symmetrcally here on both the nformatveness of the sgnal, p, and the lkelhood of bases beng the same,, n general, f s not too extreme the fact that a sgnal s nformatve shfts weght to the event n whch the advsor has an ncentve to tell the truth. The analytcal advantage of the two-person model studed here, however, s that, at least up to the mxed equlbrum for the nongenerc event = (1 p), the proposton completely descrbes the equlbrum set. Thus unlke commttees wth more than two persons and a q-rule, the welfare mplcatons of bas uncertanty n ths nstance are easy to dentfy. 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: EU rev 1 = EU rev 2 = + (1 ) 1 + p 2 : 8
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, s su cently large then the presence of bas uncertanty strctly mproves the welfare of that coalton (.e. the decson-maker) snce all decson-relevant 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 9
pror to debatng sgnals can mprove commttee performance. As remarked earler, the de nton of a fully revealng message strategy says nothng about whether or not the speaker reveals her bas b n debate; all that matters s that the speaker s collectve decson-relevant datum, her sgnal s, s unequvocally revealed. It s apparent, however, that f bases are revealed they have no e ect on any lstener s decson gven all sgnals are shared. The only reason for beng concerned about a speaker s bas s that such knowledge, or lack thereof, n uences a lstener s belefs about whether or not he or she s n the full nformaton wnnng coalton: f a commttee member knows that they are a member of the majorty n a commttee before any dscusson of sgnals, then that ndvdual s assessment of the lkely message pvotal events s surely a ected and, therefore, so s her wllngness to share collectve decson-relevant nformaton pror to votng. Thus f revealng nformaton regardng personal bas s relevant to the collectve decson, then t must be that bases are revealed pror to the dscusson of sgnals. So we suppose there are two cheap-talk debate stages pror to votng: n the rst, ndvduals send a message regardng ther bases, followng whch there the second debate stage nvolves ndvduals sendng cheap-talk messages about ther sgnals. Strateges for the two-stage debate 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 1, the advsor, 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 and only f 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 10
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 ) x y(m; x) = 1 x y(m; y) = 0 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. 11
Thus allowng an opportunty to coordnate through sharng bas nformaton before revealng anythng about decson-relevant sgnals, cannot mprove the welfare propertes for the decson-maker when < 1 p. 7 One mght also consder the possblty of an alternatve communcaton protocol for the case < 1 p allowng, for example, medated communcaton. A natural extenson of ths model mght take a mechansm desgn approach. Whle a full blown mechansm desgn analyss s beyond the scope of ths paper t s easy to see that no mechansm can yeld outcomes equvalent to a FRDE. As n Proposton 1, the sender who has observed a sgnal d erent from hs bas has an ncentve to msrepresent hs sgnal. Recall that the stylzed commttee model assumes that the advsor s a representatve agent for a mnorty coalton whle the decson-maker stands n for a majorty coalton. In the Appendx we consder an extenson of the model n whch agents are unsure whether they are part of a majorty or a mnorty. In such a settng there s no comparable equlbrum n an extenson of the model to a three or more person commttee wth majorty rule. That s, as demonstrated n the Appendx, the possblty of bas revelaton seems pecular to the advsor (known mnorty)/decson-maker (known majorty) commttee structure. Equlbrum bas revelaton followed by (not necessarly fully) nformatve sgnal sharng s generally mpossble when a commttee uses a two-stage debate protocol followed by majorty rule votng to determne the commttee choce. Note that, so long as 1=2 n the dent ed equlbrum (say, the bas revelaton equlbrum), both ndvduals reveal ther bases n the rst round of talk; f ther bases are 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 12
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. 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, and comparng ths rule wth the optmal rule wthout communcaton, are mportant and open questons. However, they have as yet proved largely ntractable f we nsst that votng be undomnated condtonal on the realzed messages n debate. On the other hand, the dscusson here provdes further support for the ntuton that full knowledge of commttee members underlyng preferences, or bases, comng nto a decson-makng process s a deterrent to credble and useful sharng of nformaton germane to the collectve decson. Moreover, the expected qualty of any commttee choce, at least as re ected by the lkelhood that any such decson concdes wth the decson that would be made under fully shared nformaton, s greater wth bas uncertanty than otherwse. 13
Appendx In ths Appendx we present a varaton of the two-person commttee model of the text, nvolvng a three-person commttee, = 1; 2; 3, usng majorty rule, q = 2, to make nal decsons. 8 We rst provde condtons for a FRDE and then argue that, unlke the two-person case, there exsts no equlbrum n whch all ndvduals can share ther bases credbly n debate to facltate coordnaton and subsequently dscuss ndvduals s sgnals regardng the state, s = (s 1 ; s 2 ; s 3 ). The commttee conssts of three people, = 1; 2; 3, that has to choose between a xed par of alternatves fx; Y g. Exactly as for the two-person case, each ndvdual has prvate nformaton regardng ther bas and a sgnal about the true state, (b ; s ) 2 fx; yg fx; yg; let (b; s) = ((b 1 ; b 2 ; b 3 ); (s 1 ; s 2 ; s 3 )) denote a pro le of realzed bases and sgnals. As before, assume sgnals are uncorrelated wth bases and suppose fa 1 ; A 2 g are 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). Naturally extendng the two-person model, an ndvdual s payo from the collectve choce Z 2 fx; Y g, gven that her bas s b 2 fx; yg and the pro le of sgnals s s 2 fx; yg 3, s gven by: u(x; x; s) = 8 < : 0 f s = (y; y; y) 1 otherwse u(y; x; s) = 1 u(x; x; s) 8 < 1 f s = (x; x; x)g u(x; y; s) = : 0 otherwse u(y; y; s) = 1 u(x; y; s) The commttee s choce s determned by a majorty vote. As observed n the text, there s 14
no equlbrum n whch all ndvduals vote nformatvely wthout any pre-vote communcaton. To see there can be a FRDE for some envronments, assume all ndvduals truthfully reveal ther sgnals; that s, usng the notaton of the text extended n the obvous manner to the three-person settng, (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 )). Clearly, any 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 : 15
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. Although certanly more complcated than the formula of Proposton 1 n the text, the ntuton and comparatve statcs of ths result are dentcal. Proposton 3 n the text showed that sharng nformaton about bases pror to an n uental debate regardng sgnals s possble n equlbrum f the two ndvduals of the commttee are su cently lkely to share the same bas;.e. the advsor s su cently lkely to be a member of the wnnng coalton, de ned by the decson-maker (pvot). Ths turns out not to be a general result wth respect to commttee sze and decson rule. As before, suppose there are two cheap-talk debate stages pror to votng: n the rst, ndvduals send a message regardng ther bases, followng whch there the second debate stage nvolves ndvduals sendng cheap-talk messages about ther sgnals. Although the notaton s perhaps unnecessary to establsh the result of nterest here, to avod ambguty n dscusson, t seems sensble to specfy formal strateges for the protocol. Spec cally: b;s 2 [0; 1] s the probablty that ndvdual wth bas b and sgnal s announces her bas s x; gven bas-stage messages m = (m 1 ; m 2 ; m s ) 2 fx; yg 3 ; let b;s (m; m ) 2 [0; 1] s the probablty ndvdual wth bas b and sgnal s who has announced her bas s m and heard messages m regardng the other players bases, announces that her sgnal s x; and smlarly, let v b;s ((m; m ); (n; n )) be the probablty ths ndvdual chooses outcome X gven the hstory of speeches n debate. Proposton 4 Assume each player has observed hs or her partcular bas and sgnal. There exsts no equlbrum n whch each player truthfully reveals ther bas n the rst message round and all members of the revealed majorty subsequently reveal ther sgnals pror to votng. Proof: Suppose, to the contrary, that all ndvduals reveal ther bases durng the rst debate stage and, n the second debate stage, those commttee members sharng the majorty bas 16
reveal ther sgnals whle those n the mnorty o er no further credble nformaton; all ndvduals then vote for ther most preferred alternatve condtonal on ther nformaton sets at the tme. In partcular, members of the revealed majorty vote dentcally but, assumng some heterogenety n bases across the commttee, not under complete nformaton; the mnorty ndvdual on the other hand, should she exst, votes under complete nformaton. (It s easy to check that, condtonal on full bas revelaton at the rst stage, these subsequent stage message and vote strateges consttute equlbrum play.) Formally, the strateges (spec ed only for an x-based ndvdual; those for a y-based ndvdual are symmetrc) are descrbed by: x;x = x;y = 1; = 1; 2; 3 x;x (m) = 1; 8m x;y (m ; x; x) = 1 x;y (m ; y; y) = 0 v x;x (m; n) = 1; 8(m; n) [9j 6= : m = m j & n j = x] =) v x;y (m; (n ; n )) = 1 v x;y ((x; (y; y)) ; (x; x; x)) = 1 v x;y (m; n) = 0 otherwse. We argue that ths pro le cannot be an equlbrum. To ths end, suppose the spec ed strategy pro le consttutes an equlbrum and consder an x-based ndvdual, say = 1, wth a y sgnal; now suppose that devates from the prescrbed strategy and les about her bas. There are three cases, dependng on the revealed (truthfully, by supposton) bases of the other ndvduals, 2; 3. Case 1: both of the others credbly reveal they are y-based. Gven the declared bas pro le and the spec ed contnuaton strateges, = 1 s messagepvotal at the second debate stage when both have x sgnals: s 2 = s 3 = x. In ths case, 17
ndvdual 1 has a best response to le regardng her sgnal, else Y s chosen surely at the votng stage. Gven s revealed bas s y, her second stage debate message s credble to the (true) majorty of f2; 3g who therefore have a best votng decson n favour of X. However, snce both 2; 3 declare ther x-sgnals n the second stage and sgnals are nformatve, ths coalton would have unanmously chosen X had ndvdual 1 truthfully declared her bas to be x n the rst stage, revealed herself as the mnorty and subsequently played the spec ed strategy n the second debate stage. Thus, n ths case, ndvdual 1 gans no advantage by lyng about her bas n the rst debate stage. Case 2: both of the others credbly reveal they are x-based. In ths case, s message-pvotal when both 2 and 3 have y-sgnals. Hence, = 1 s undomnated best response s to tell the truth about her sgnal, s 1 = y, snce the only crcumstance n whch she strctly prefers the Y decson s when all three sgnals are y. But havng led about her bas n the rst stage and declared a y bas, any message n favour of Y at the second stage s gnored by the majorty as not credble. However, even so, the majorty votes together for Y because, gven the two credble messages that at least two (nformatve) y sgnals have been observed, votng for Y s a best response. As for Case 1, therefore, = 1 cannot pro t from devatng n the rst debate stage and lyng about her bas. Case 3: the others credbly reveal they have opposng bases. Gven that = 1 has led about her bas, the apparent majorty n the commttee s y- based: wthout loss of generalty, assume m 2 = b 2 = x and m 3 = b 3 = y. Consequently, at the second debate stage, the (apparent) mnorty ndvdual 2 speaks n favour of X rrespectve of hs sgnal, and the (apparent) majorty coalton member 3 both reveals her sgnal credbly. Moreover, all three ndvduals beleve the second stage debate messages from ndvduals 1 and 3 and subsequently vote usng ths nformaton. Therefore, 1 s best response message at the second stage s to reveal her sgnal n 1 = s 1 = y truthfully and subsequently vote for X rrespectve of the y-based ndvdual 3 s message. Followng such a second stage debate, 18
ndvdual 3 has (gven b 3 = y and 1 s revealed y sgnal) a best response decson to vote for Y. In turn, ths makes ndvdual 2, the apparent mnorty member of the commttee, vote pvotal n whch case hs best response s to vote for Y f and only f s 2 = s 3 = y. Hence, gven the others follow the prescrbed strateges, the devaton from full truthtellng regardng her bas s strctly pro table ndvdual = 1 n ths case. In sum, because Cases 1, 2 and 3 are exhaustve, an x-based ndvdual wth a y sgnal can never be made worse o by devatng to reveal a y bas n the rst debate stage, and can be make strctly better o n one case. Thus the supposton that the spec ed strategy pro le consttutes and an equlbrum n undomnated strateges s false. 19
Notes 1 For example, Austen-Smth (1990); Calvert and Johnson (1998); Coughlan (2000); Doraszelsk et al (2003); Gerard and Yarv (2007); Merowtz (2006, 2007); Hafer and Landa (2007); Austen-Smth and Feddersen (2005, 2006); Callaud and Trole (2006). 2 See Austen-Smth and Feddersen (2005), Gerard and Yarv (2007) and Merowtz (2007) for a dscusson of votng rules and nformaton aggregaton. 3 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. 4 In a very general framework, Gerard and Yarv (2007) show that all non-unanmous q rules are equvalent n that the sets of sequental equlbrum outcomes nduced by any such rules are dentcal once votng s preceded by delberaton. However, the assumptons supportng ths result do not nsst that votng strateges are undomnated condtonal on the realzed debate. 5 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. 6 We note too that even though there s no reason for the decson-maker to reveal hs sgnal n debate, there s also no reason here for hm not to do so credbly, a fact that motvates our equlbrum termnology. 7 Of course, full bas revelaton n debate s possble f all subsequent messages regardng ndvduals sgnals are gnored. But ths case s clearly equvalent to the stuaton wth no debate at all. 8 The basc example to follow was ntroduced n Austen-Smth and Feddersen (2006) where the condtons for a FRDE were dent ed. For completeness, we summarze ths ndng here. 20
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