Meaningful cheap talk must improve equilibrium payoffs

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1 Mathematcal Socal Scences 37 (1999) Meanngful cheap talk must mprove equlbrum payoffs Lanny Arvan, Luıs Cabral *, Vasco Santos a b, c a Unversty of Illnos at Urbana-Champagn, Department of Economcs, 106 South Sxth Street, Champagn, Illnos 6180, USA b London Busness School, Sussex Place, Regent s Park, London NW1 4SA, UK Unversdade Nova de Lsboa, Faculdade de Economa, Travessa Estevao ˆ Pnto, P-1070 Lsbon, Portugal Receved November 1997; receved n revsed form 31 December 1997; accepted 31 March 1998 c Abstract We generalze Farrell s (1987) dea of coordnaton by means of cheap talk. We show that f cheap talk s meanngful (n the sense that babblng equlbra are ruled out) and f there s room for cooperaton (namely f there exsts at least one pure-strategy equlbrum Pareto superor to the default equlbrum), then cheap talk must ncrease the equlbrum expected payoff relatve to the play of the game wthout preplay communcaton. The result s lmted to proper equlbra of the communcaton game and to games wth two players. It s shown that 1999 Elsever Scence B.V. All rghts reserved. Keywords: Cheap talk; Communcaton; Equlbrum 1. Introducton In a semnal paper, Farrell (1987) has shown that nonbndng, costless preplay communcaton (cheap talk) may mprove the equlbrum payoff n a game wth the structure of the battle of the sexes. The battle of the sexes s a game that entals both elements of competton and of coordnaton. It possesses two pure-strategy equlbra whch are not Pareto ranked, and a thrd equlbrum n mxed strateges whch s Pareto nferor to the pure-strategy equlbra. The frst fact nduces the compettve element of the game, whle the second one adds the cooperatve element. Farrell s (1987) dea s that cheap talk may help players coordnate n playng one of the pure-strategy equlbra: whle players preferences are opposte regardng the pure-strategy equlbra, they both prefer any of these equlbra to the mxed-strategy equlbrum, whch, by * Correspondng author. Tel.: ; fax: ; e-mal: lcabral@lbs.ac.uk / 99/ $ see front matter 1999 Elsever Scence B.V. All rghts reserved. PII: S (98)

2 98 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) assumpton, s the default equlbrum to be played n case no agreement s reached at the communcaton stage. Specfcally, Farrell consders an extended game whereby, before any acton s taken, players smultaneously announce whch acton they ntend to take. Although announcements are not bndng (cheap talk), they serve as a means for coordnatng on a Nash equlbrum superor to the default equlbrum. In fact, Farrell shows that there exsts an equlbrum of the extended game whch yelds a hgher payoff than the default 1 equlbrum. We generalze and strengthen Farrell s result n two ways. Frst, we consder games where each player chooses between n $ pure-strateges, whereas Farrell only consd- ered the case n 5. Second, and more mportantly, our results show that, provded some condtons are satsfed, cheap talk must mply an mprovement n equlbrum payoffs, whereas Farrell only argues that t may mply an mprovement n equlbrum payoffs. A central assumpton underlyng our result refers to the relaton between communcaton and acton. Followng Farrell, we assume that, f players announcements correspond to a Nash equlbrum of the game to be played, then such equlbrum becomes focal and s ndeed played; f, on the other hand, communcaton does not lead to any partcular equlbrum, then a default equlbrum s played. Moreover, we assume that there exst pure-strategy Nash equlbra whch are Pareto superor to the default equlbrum,.e., we assume there s scope for mprovement n the equlbrum payoff. The ntuton behnd our result s as follows. At the communcaton stage, players have no ncentve to announce a strategy assocated wth an equlbrum that yelds the other player a low payoff. Ths s so because, n equlbrum, such players wll not announce the acton correspondng to that equlbrum. In fact, players wll only make announcements assocated wth superor equlbra (equlbra benefcal to both players), whch n turn mples that cheap talk must mprove expected equlbrum payoffs. In addton to the assumptons spelled out above, our result requres that the number 3 of players be two and that the equlbrum of the augmented game be proper. In Secton 3, we show, by means of examples, that both these requrements are necessary. In fact, the restrcton to the case of two players s necessary even when we consder a rcher message space whereby players announce acton profles nstead of actons (cf. Secton 4). 1 The extended game conssts of a communcaton stage (comprsng one or more rounds) followed by play of the orgnal game. However, we requre all pure-strategy Nash equlbra of the orgnal game to be strct. Alternatvely, we may assume nstead that there s only one pure-strategy Nash equlbrum per row, a weaker hypothess than strctness. Fnally, we can also consder a dfferent message space than the one proposed by Farrell; cf. Secton Cf. Myerson (1978). Essentally, properness mples that, for each two player s actons, a and a, fa 9 yelds a lower payoff than a, then, n the play of the perturbed game, a should be chosen wth a probablty 9 that s at least one order of magntude lower than the probablty that a s chosen.

3 . Basc results L. Arvan et al. / Mathematcal Socal Scences 37 (1999) Consder the followng two-player game n normal form, denoted by G: S s the space of player s pure strateges and s s a generc element of S, 5 1, ; S ; S 3 S and 1 s ; (s, s ); fnally, u (s) denotes player s utlty. 1 Based on game G, we now defne an extended game, denoted by G, whch nvolves a communcaton stage pror to playng G. At the communcaton stage, players smultaneously announce actons from ther own pure-strategy sets S (the terms acton and pure strategy meanng the same thng). After all messages are sent and receved, game G s played. Games of ths sort always admt equlbra n whch cheap talk has no nfluence on the 4 outcome of G ( babblng equlbra ). Followng Farrell (1987), we concentrate on no-babblng equlbra, that s, equlbra wth the followng propertes: 1. If announcements correspond to a Nash equlbrum of G, then that equlbrum becomes focal and s thus played.. If announcements do not correspond to a Nash equlbrum of G, then G s played as f no communcaton has taken place: a default equlbrum s played ndependently of whch announcements are made. Payoffs n ths default equlbrum are gven by ū. Notce that propertes () () nduce a well-defned reduced game n the communcaton stage, so the set of no-babblng equlbra (Nash, perfect Nash, or proper Nash) s nonempty. Fnally, we defne s to be a superor equlbrum of G f and only f s s a pure-strategy equlbrum and u (s). u, ;. Our man result wll be based on the hypothess that there exsts at least one superor equlbrum. Ths hypothess mples that G s to some extent a game of coordnaton, specfcally, a game wth at least two Nash equlbra whch are Pareto ranked. Theorem 1. Assume that all pure-strategy Nash equlbra of G are strct. If there exsts some superor equlbrum of the orgnal game, then, n a proper no-babblng equlbrum of the extended game, only actons correspondng to superor equlbra are announced, and the expected payoff of both players s strctly greater than n the game wth no communcaton. Proof. Consder a perturbed game n the communcaton stage. In ths game, every message s announced wth postve probablty. Therefore, t s a strctly domnated strategy for player to announce an acton correspondng to a pure-strategy Nash 4 See, for example, Farrell and Gbbons (1989). Sedmann (199) shows that the same s not necessarly true n games of ncomplete nformaton.

4 100 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) equlbrum, say s*, yeldng hm or her a payoff less than u. In fact, the expected payoff of announcng such an acton would be lower than u, by the assumpton that all pure-strategy Nash equlbra are strct; whereas announcng an acton correspondng to a superor equlbrum, say s**, would yeld player an expected payoff greater than u. By the requrement of properness (Myerson, 1978), the weght assgned by player to the acton correspondng to s* must be at least one order of magntude lower than the weght assgned to the acton correspondng to s**. Now, gven the above, t s also a domnated strategy for player j to announce the acton correspondng to s*, assumng that payoffs are bounded and that the perturbed game s suffcently close to the unperturbed game. To see why, notce that player j s expected payoff from announcng the acton assocated to s* s A 5 P(s *)u (s*) 1 (1 P(s *))u, (1) j j where P(s * ) s the probablty that acton s* s announced by player. On the other hand, announcng the acton correspondng to s** yelds player j an expected payoff of B 5 P(s **)u (s**) 1 (1 P(s **))u. () j j Snce u j(s**).u j and P(s * )#ep(s** ) (the latter by properness), we have A,B. Fnally, a smlar argument, f somewhat smpler, apples to actons correspondng to no pure-strategy Nash equlbrum. Lkewse, t s straghtforward to show, by contradcton, that at least one superor equlbrum must be played wth postve probablty. j It should be remarked that the argument extends to games wth T.1 rounds of cheap talk (a case also consdered n Farrell (1987)). It suffces to note that the proof apples to any perod t,t, wth the dfference that payoffs n case of no agreement at stage t are t t t now gven by (u, u ), where u.u Counterexamples In ths secton, we present a seres of counterexamples whch elucdate the necessty of some of the assumptons and hypotheses underlyng the man result of the prevous secton. All examples nvolve a communcaton stage wth a sngle round of cheap talk. 1. The frst example shows how the assumpton that all pure-strategy Nash equlbra are strct s necessary for the result. In ths example, the orgnal game, shown n Fg. 1, has two pure-strategy equlbra, (T, L) and (T, R), nether of whch s strct. There also exsts a mxed-strategy equlbrum n whch both players choose each acton wth equal probablty. The expected payoff n ths equlbrum, whch we assume to be the default equlbrum, s / 3 for each player. Snce we restrct our nterests to no-babblng equlbra, expected payoffs at the communcaton stage are summarzed by the payoff matrx n Fg.. It can be seen that t s a proper equlbrum for the row player to choose M and the column player to choose L wth probablty a and R wth probablty

5 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) Fg. 1. Game G n frst example. Fg.. Game G n frst example. 1a, wth a,1/3. Although there exsts a superor equlbrum, namely (T, L), the row player never announces the strategy correspondng to ths equlbrum. The ntuton behnd the frst example s smple. By beng ndfferent between two equlbra ((T, L) and (T, R)) and announcng more often the one least desred by ts opponent ((T, R)), one player may end up preventng coordnaton.. The second example shows that perfecton s not a suffcent refnement to produce the man result. The orgnal game n ths example s descrbed n Fg. 3. The game has three pure-strategy equlbra, (T, R), (M, C) and (B, L). There also exsts a mxed-strategy equlbrum n whch players choose the frst two actons wth equal probablty. The expected value n ths equlbrum, whch we assume to be the default equlbrum, s 3/ for each player. The payoff structure at the communcaton stage s gven by the matrx n Fg. 4. We wll now show that t s a perfect equlbrum, although not a proper equlbrum, Fg. 3. Game G n second example. Fg. 4. Game G n second example.

6 10 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) Fg. 5. Game G n thrd example. for the row player to announce T and the column player to announce L. For ths purpose, consder the followng strateges n an e-perturbed game: each player chooses the frst strategy wth probablty 1e and the two remanng strateges wth probablty e each. Smple nspecton reveals that ths consttutes a Nash equlbrum gven the constrant that all actons be chosen wth probablty greater than e. Hence, the desgnated equlbrum s ndeed perfect. However, the strateges n the perturbed game do not satsfy the requrement for properness, namely that the thrd strategy be chosen by each player wth a probablty whch s one order of magntude lower than the second one. In fact, our man result states that the only equlbrum at the communcaton stage conssts of players announcng the mddle strategy wth probablty one. The ntuton behnd the second example s also smple. Perfectness allows actons assocated wth superor and nonsuperor equlbra to be announced wth probabltes of the same order of magntude. Ths, n turn, may lead players to try to coordnate on nonsuperor but, to them, partcularly favorable equlbra, a behavor that may end up precludng coordnaton. Properness rules ths out by ensurng that the probablty of coordnatng on a nonsuperor equlbrum s at least one order of magntude lower than on a superor equlbrum. 3. Fnally, the thrd example shows that the result does not extend to games wth more than two players. Fg. 5 depcts the orgnal game n ths example. As before, players 1 and choose rows and columns, respectvely. Now we add a thrd player who pcks one of the two matrces, LM or RM. The game has two pure-strategy equlbra, (B, L, LM) and (B, R, RM), both of whch are strct. In addton, there exsts a mxed-strategy equlbrum n whch the frst two players mx wth equal probablty each of ther actons and the thrd player chooses RM. Expected payoff under ths equlbrum, whch we assume to be the default equlbrum, s gven by for all players. The payoff matrx at the communcaton stage s thus the one descrbed n Fg. 6. Fg. 6. Game G n thrd example.

7 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) We wll now argue that (T, L, LM) consttutes a proper equlbrum of G, even though one of the players, player 1, chooses an acton not assocated wth any Nash equlbrum. Consder the followng equlbrum of the perturbed game. Player 1 chooses T wth probablty 1e and B wth probablty e. Player chooses L wth probablty 1e and R wth probablty e. Fnally, player 3 chooses LM wth probablty 1e and RM wth probablty e. Clearly ths equlbrum converges to the desgnated Nash equlbrum. We 5 wll now show that t consttutes an equlbrum of the properly perturbed game. If player 1 devates, t gets 1(1e) 14e 1?e(1e), whch s lower than, ts equlbrum payoff. If player devates, t gets 4e 1(1e ), whch s lower than 3e(1e)1(1e 1e ), ts equlbrum payoff. Fnally, f player 3 devates, t gets 4e 1(1e ), whch s lower than 3e(1e)1(1e 1e ), ts equlbrum payoff. The dea of ths example s that, wth three or more players, there appear coordnaton problems whch are absent n the case of two players only. Players and 3 would prefer to swtch from playng (T, L, LM) to playng (B, R, LM). Unlateral moves, however, can only reduce expected payoff. At (T, L, LM), both player and player 3 look forward to a mstake by player 1, a mstake that wll nduce a payoff of 3 nstead of. If player or player 3 unlaterally devate, then t would requre two smultaneous mstakes to ncrease payoff, a possblty nfntely less lkely. 4. A dfferent message space In the prevous sectons, we have consdered a game wth the structure proposed n Farrell (1987): startng from a normal-form game, we augment ths by addng a pror stage of communcaton. The set of messages sent by player n the communcaton stage conssts of the names of hs or her actons n the ntal normal-form game. In other words, each player announces what acton he or she ntends to play. An alternatve message space conssts of players announcng acton profles nstead of 6 actons. Ths message space s perhaps as natural as the one assumed n Farrell (1987) and by ourselves n the prevous sectons. It allows for a somewhat sharper result: Theorem. If there exsts some superor equlbrum of the orgnal game, then, n a proper no-babblng equlbrum of the extended game, only actons correspondng to superor equlbra are announced, and the expected payoff of both players s strctly greater than n the game wth no communcaton. Proof. Consder a perturbed game n the communcaton stage. In ths game, every acton profle s announced wth postve probablty. Therefore, t s a strctly domnated strategy for player to announce an acton profle correspondng to a pure-strategy Nash 5 Snce each player has only two pure strateges, properness corresponds to tremblng-hand perfecton. 6 Ths extenson follows a referee s suggeston that we consder a message space of the type hm (s)j, where denotes the player and s an acton profle, together wth the assumpton that m (s)±m (s9) for s±s9. The message space we propose s a natural partcular case that satsfes ths assumpton.

8 104 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) equlbrum, say s*, yeldng hm or her a payoff less or equal to u. (For smplcty, we wll refer to ths move n the communcaton stage as proposng equlbrum s*. ) In fact, the expected payoff from proposng s* would be strctly lower than u, whereas proposng a superor equlbrum, say s**, would yeld player an expected payoff strctly greater than u. By the requrement of properness (Myerson, 1978), the weght assgned by player to s* must be at least one order of magntude lower than the weght assgned to s**. Now, gven the above, t s also a domnated strategy for player j to announce s*, assumng that payoffs are bounded and that the perturbed game s suffcently close to the unperturbed game. To see why, notce that player j s expected payoff from announcng s* s A 5 P(s *)u (s*) 1 (1 P(s *))u, (3) j j where P(s * ) s the probablty that s* s announced by player. On the other hand, announcng s** yelds player j an expected payoff of B 5 P(s **)u (s**) 1 (1 P(s **))u. (4) j j Snce u j(s**).u j and P(s * )#ep(s* *) (the latter by properness), we have A,B. Fnally, a smlar argument, f somewhat smpler, apples to actons profles correspondng to no pure-strategy Nash equlbrum. Lkewse, t s straghtforward to show, by contradcton, that at least one superor equlbrum must be played wth postve probablty. j The man dfference wth respect to Theorem 1 s that Theorem dspenses wth the assumpton that all pure-strategy Nash equlbra of G are strct. However, the assumpton that there are only two players remans a necessary condton, as the followng example shows. The example features a three-player game wth payoffs as n Fg. 7. As before, players 1, and 3 chose rows, columns and matrces, respectvely. Ths game has two pure-strategy Nash equlbra, (T, L, LM) and (B, R, RM), and one mxed-strategy equlbrum where players 1 and mx wth equal probablty and player 3 plays RM. The latter equlbrum yelds all players a payoff of and s assumed to be the default equlbrum. As n Theorem, we consder an extended game, G, n whch players announce acton profles. If the acton profle announced by all three players concdes and f ths acton Fg. 7. Game G n fourth example.

9 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) Fg. 8. Summary of payoffs of game G n fourth example. profle consttutes a Nash equlbrum, then that equlbrum s played; otherwse the default equlbrum s played. The normal form of the extended game s qute complex: t comprses eght matrces of eght by eght. Instead of wrtng out the complete normal form, Fg. 8 presents a summary of the payoffs of the extended game when all players propose the same acton profle. Whenever announcements dffer, payoffs are (,, ). Player 1 prefers to propose (B, R, RM) (expected payoff greater than ) then to propose an acton profle that s not an equlbrum (expected payoff of ); and ths, n turn, player 1 prefers to announcng (T, L, LM) (expected payoff lower than ). Assume that player 1 proposes (B, R, RM) wth probablty 16e e, (T, L, LM) wth probablty e, and all other acton profles wth probablty e. Players and 3 prefer to propose (B, R, RM) and (T, L, LM) (expected payoff greater than ) then to propose any other acton profle (expected payoff of ). Assume that 3 4 players and 3 propose (T, L, LM) wth probablty 1e 6e,(B, R, RM) wth 3 4 probablty e and all other acton profles wth probablty e. To show that ths consttutes an equlbrum of the perturbed game, notce that, by proposng (T, L, LM), players and 3 receve a payoff that exceeds by a factor of order e (the probablty of a mstake by player 1). However, by unlaterally devatng to 3 proposng (B, R, RM), expected payoff exceeds by a factor of order e (the probablty of a mstake by player or 3, whchever s not devatng). 5. Fnal remarks As Farrell (1988) noted, cheap talk s notorously hard to model: there are no obvously rght rules about who speaks when, what he may say, and when dscusson ends. Not surprsngly, dfferent structures of preplay communcaton have been attempted n the lterature, some of whch present results related to ours. Farrell (1988) and Watson (1991) consder the case n whch one of the players unlaterally suggests to the other whch set of strateges to play. Watson (1991) shows that f the orgnal game has a sngle Pareto-effcent outcome, then ths s the only sensble outcome of the extended game. Matsu (1991) assumes a structure of preplay communcaton smlar to ours, wth players sendng messages smultaneously. However, the equlbrum concept he consders s qute dfferent from ours. He consders a large populaton matched to play a game

10 106 L. Arvan et al. / Mathematcal Socal Scences 37 (1999) of common nterest wth cheap talk. He shows that a unque cyclcally stable set exsts and ths contans only Pareto optmal outcomes. The paper whch s closest to ours s Rabn (1994). As n our paper, he consders a two-player game and smultaneous message exchange. However, he consders a wder set of communcaton possbltes than our paper (and Farrell, 1987). Rabn shows that wth enough rounds of cheap talk, each player s expected payoff s at least as great as the payoff from hs worst Pareto-effcent Nash equlbrum. Our paper partly confrms the communcaton-yelds-effcency hypothess, to borrow Rabn s (1994) expresson, although we show that some mportant qualfcatons need to be made. Fnally, although our results mply that, under some condtons, equlbrum payoffs must mprove through cheap talk, t can be shown that, genercally, there exsts no lower bound to the sze (or the probablty) of the mprovement n payoff resultng from preplay communcaton (cf. Farrell, 1987). It thus seems that our result s tght both n terms of ts extent and n terms of the requred condtons. Acknowledgements We are grateful to a referee for comments and suggestons that mproved the paper consderably. References Farrell, J., Cheap talk, coordnaton, and entry. Rand Journal of Economcs 18, Farrell, J., Communcaton, coordnaton and Nash equlbrum. Economcs Letters 7, Farrell, J., Gbbons, R., Cheap talk can matter n barganng. Journal of Economc Theory 48, Matsu, A., Cheap talk and cooperaton n a socety. Journal of Economc Theory 54, Myerson, R., Refnements of the Nash equlbrum concept. Internatonal Journal of Game Theory 7, Rabn, M., A model of pregame communcaton. Journal of Economc Theory 63, Sedmann, D., 199. Cheap talk games may have unque nformatve equlbrum outcomes. Games and Economc Behavor 4, Watson, J., Communcaton and superor cooperaton n two-player normal form games. Economcs Letters 35,