Analogy-Based Expectation Equilibrium

Transcription

1 Analogy-Based Expectaton Equlbrum Phlppe Jehel March 2001 Abstract It s assumed that players bundle nodes n whch other players must move nto analogy classes, and players only have expectatons about the average behavor n every class. A soluton concept s proposed for mult-stage games wth perfect nformaton: at every node players choose best-responses to ther analogy-based expectatons, and expectatons are correct on average over those varous nodes pooled together nto the same analogy classes. The approach s appled to a varety of games. It s shown that a player may beneþt from havng a coarse analogy parttonng. And for smple analogy parttonng, (1) ntal cooperaton followed by an end opportunstc behavor may emerge n the Þntely repeated prsoner s dlemma (or n the centpede game), (2) an agreement need not be reached mmedately n barganng games wth complete nformaton. Key words: Game theory, bounded ratonalty, reasonng by analogy. JEL numbers: C72, D81. I would lke to thank O. Compte, D. Ettnger, I. Glboa, B. MacLeod, E. Maskn, G. Nöldeke, R. Radner, A. Rubnsten, L. Samuelson, R. Spegler, and E. Dekel and three anonymous referees for many helpful comments made at varous stages of ths research. I have also beneþtted fromthecommentsmadensemnarsatprnceton Unversty, Bonn Unversty, Pars (ENS), NYU, Penn, the Insttute for Advanced Study (Prnceton), Yale, and the ATT Labs. CERAS, Pars and UCL, London. malng address: C.E.R.A.S.-E.N.P.C., C.N.R.S. (URA 2036), 28 rue des Sants-Pères, Pars, France; e-mal: jehel@enpc.fr. 1

2 1 Introducton Receved game theory assumes that players are perfectly ratonal both n ther ablty to form correct expectatons about other players behavor and n ther ablty to select best-responses to ther expectatons. The game of chess s a strkng example n whch the standard approach s napproprate. In chess, t s clearly mpossble to know (learn) what the opponent mght do for every board poston. In ths paper, we nvestgate stuatons n whch players form ther expectatons about others behavor by analogy between several contngences as opposed to for every sngle contngency n whch each of these other players must move. 1 More precsely, each player bundles nodes at whch players other than must move - a bundle s called an analogy class. And player only forms expectatons about the average behavor n each analogy class that he consders. In other words, player s vewed here as smplfyng what he wants to know (learn) about others behavor: 2 Player categorzes nodes n whch other players must move nto analogy classes. And only the average behavor n each analogy class s beng consdered by player. We use the word analogy because n two nodes belongng to a same class, the expectaton formed by the player s the same. Besdes, the equlbrum expectaton n an analogy class wll be assumed to concde wth the effectve average behavor n the class. Accordngly, nodes whch are vsted more often wll contrbute more to the expectaton, and the behavors n those nodes wll contamnate the expectaton used n all nodes of the analogy class (no matter how often they are vsted). The extrapolaton (here of the expectaton) from more vsted to less vsted contngences s - we beleve - a key feature of the analogy dea. 3 The am of ths paper s twofold. The Þrst objectve s to propose a soluton concept to descrbe the nteracton of players formng ther expectatons by analogy. Ths wll be called the analogy-based expectaton equlbrum. The second objectve s to analyze the propertes of analogy-based expectaton equlbra n varous strategc nteracton contexts. The games we consder are mult-stage games wth almost perfect nformaton and perfect recall. That s, smultaneous moves and moves by Nature are allowed. But, n any stage, all prevous moves are assumed to be known to every player. 4 1 Ths approach seems partcularly approprate n stuatons wth many contngences (lke chess) so that learnng behavor for every possble contngency seems too hard (mpossble). 2 Ths makes learnng easer and successful learnng more plausble. 3 It should be noted that what s consdered here s the dea of formng expectatons by analogy as opposed to actng by analogy (see dscusson secton). 4 Extensons to ncomplete nformaton setups rase no conceptual dffcultes, but make the exposton 2

3 The parttonng nto analogy classes used by the players s gven exogenously. 5 It s vewed as part of the descrpton of the strategc envronment. An analogy class α of player s a set of pars (j, h) such that player j, j 6=, mustmoveatnodeh. We requre that f two elements (j, h) and(j 0,h 0 ) belong to the same analogy class, the acton spaces of player j at node h andofplayerj 0 at node h 0 are dentcally labelled. 6 Player s analogy-based expectaton β s player s expectaton about the average behavor of other players n every analogy class α consdered by player - we wll denote by β (α ) the expectaton n the analogy class α. An analogy-based expectaton equlbrum s a par (σ, β) whereσ s a strategy proþle and β s an analogy-based expectaton proþle such that two condtons are satsþed. Frst, for each player and for each node at whch player must move, player s strategy σ s a best-response to hs analogy-based expectaton β. 7 Second, for each player and analogy class α,player s expectaton β (α )sconsstent wth the average behavor n class α as nduced by the strategy proþle σ (where the behavor of player j n node h,(j, h) α, s weghted by the frequency wth whch (j, h) s vsted accordng to σ - relatve to other elements n α ). 8 Clearly, f all players use the Þnest parttonng as ther analogy devces, the strategy proþle of an analogy-based expectaton equlbrum concdes wth a Subgame Perfect Nash equlbrum. However, when at least one player does not use the Þnest parttonng, the play of an analogy-based expectaton equlbrum wll n general dffer from that of a Subgame Perfect Nash equlbrum (or even from that of a Nash equlbrum). We also note that n Þnte envronments an analogy-based expectaton equlbrum always exsts. In the second part of the paper, we nvestgate a few propertes of analogy-based expectaton equlbra n a varety of games. We Þrst observe that sometmes a player may beneþt from havng a coarse analogy parttonng as compared wth the Þnest parttonng. notatonally heavy. 5 One mght thnk of the parttonng as resultng from the past experences of the players and also from the way the strategc nteracton s framed to the players thus trggerng some connectons wth past experences (the so called framng effect, see Tversky-Khaneman 1981). 6 Strctly speakng, t s enough to requre that there s a bjecton between the two acton spaces. Note also that our formalsm allows for analoges accross dfferent players. 7 More precsely, player s strategy σ s a best-response (after every node where player must move) to the behavoral strategy that assgns player j toplayaccordngtotheexpectatonβ (α )atnodeh, forevery (j, h) n the analogy class α and for every analogy class α. 8 We thnk of the consstency requrement as resultng from a learnng process n whch players would eventually manage to have correct analogy-based expectatons (and not as resultng from ntrospecton or calculatons on the part of the players). And f no node h such that (j, h) belongs to α s ever vsted accordng to σ, (strong) consstency s deþnedwthrespecttoasmallperturbatonofσ. (Ths s n sprt of the deþnton of sequental equlbrum.) 3

4 Clearly, ths s not so f ths player plays aganst Nature or f other players have a domnant strategy. Then a coarse parttonng has the sole effect of makng ths player s choce of strategy possbly suboptmal wthout affectng the behavors of others. But, otherwse, a coarse parttonng of, say, player may well nduce (n equlbrum) a change of strateges of players other than (as a response to a change of strategy of player ). When such a change of strateges s good for player, player may n equlbrum end up wth a strctly hgher payoff. We next apply the analogy-based expectaton approach to the so called Þnte horzon paradoxes. For smple analogy parttonng, we show both n the centpede game and n the Þntely repeated prsoner s dlemma that there may be equlbra n whch there s a far amount of cooperaton throughout the game except possbly toward reachng the end of the game at whch tme some opportunstc behavor may occur. To llustrate the clam, consder a varant of the Þntely repeated prsoner s dlemma n whch there are many perods, there s no dscountng and the exact values of the stage game prsoner s dlemma payoffs are ndependently drawn from perod to perod accordng to some pre-specþed dstrbuton (wth Þnte support). And assume that both players categorze hstores nto two analogy classes accordng to whether or not some opportunstc behavor was prevously observed (wthn the game). Playng cooperatvely most of the tme except f some opportunstc behavor prevously occurred or toward the end of the game (f the mmedate gan from swtchng to an opportunstc behavor s suffcently hgh) s part of an analogy-based expectaton equlbrum. To see ths, consder the expectatons nduced by the behavors just descrbed. Each player should expect the other player (1) to play opportunstcally whenever some opportunstc behavor prevously occurred and (2) to play cooperatvely (on average) wth a large probablty otherwse (f the number of repettons s large). Gven such expectatons, playng opportunstcally s optmal whenever some opportunstc behavor prevously occurred. And, when no opportunstc behavor prevously occurred, playng cooperatvely n all but a few perods toward the end s also optmal because players perceve that by playng opportunstcally they wll trgger a non-cooperatve phase whereas by playng cooperatvely they expect the other player to contnue playng cooperatvely wth a large probablty. The key reason why the logc of backward nducton fals here s that players do not perceve exactly when the other player wll start havng an opportunstc behavor. As a result of ths fuzzy percepton (whch s due to ther analogy parttonng), players play cooperatvely most of the tme because on average t s true that by playng cooperatvely the other player keeps playng cooperatvely wth a large probablty. 4

5 It should be noted that players do consder playng opportunstcally toward reachng the end of the game, even f no opportunstc behavor prevously occurred. Ths s so whenever the mmedate gan from playng opportunstcally offsets the cost of trggerng a non-cooperatvephasetlltheendofthegame(asopposedtomantanngthecooperatve phase). In ths sense, players do perceve the tme structure of the nteracton even though they do not perceve the exact tme structure of the strategy employed by ther opponent. 9 We also breßyconsderthenþntely repeated prsoner s dlemma. We observe that strategy proþles n whch some devatons are not punshed can be sustaned wth the analogybased expectaton approach. The pont s that, whle such a devaton would be proþtable, t need not perceved as such f the correspondng node s bundled wth nodes n whch there wouldbeaneffectve punshment. As a result (of such an analogy parttonng), the nvolved player perceves an average punshment, whch deters hm from devatng. Thus, n repeated games, the analogy-based approach permts less systematc punshments than the standard approach does. Our next applcaton deals wth ultmatum and barganng games. Suppose that players canmakeanypossbleoffer, but that they have expectatons about the acceptance probablty only accordng to whether the offer s above or below a threshold (.e., whether or not ther offer s generous). We show that (1) the responder n a take-t-or-leave-t offer game may get a payoff that les strctly above hs reservaton utlty (.e. hs payoff from refusng any agreement), (2) there may be no mmedate agreement n a (complete nformaton) barganng game n whch players alternate makng offers. The effect of analogy reasonng here s to reduce the set of offers that players consder n equlbrum. If a player makes a generous offer, he wll always consder the least generous offers among these. Ths s because (due to hs analogy parttonng) he has the same (acceptance) expectaton for all such offers, and the least generous offer among these s clearly the one he lkes best gven such an expectaton. The analyss of ultmatum and barganng games follows. In the last part of the paper, we provde some general dscusson. We Þrst dfferentate the analogy-based expectaton equlbrum from other soluton concepts, n partcular related to the dea of mperfect recall (and of mperfect nformaton). 10 Second, we suggest two prncples that may help structure analogy parttonng. The Þrst prncple apples to those games n whch all players must move n the same nodes, and we consder the extra requrement that a player should hmself behave n the same way n all nodes assocated wth the same 9 The analogy approach thus permts an endogenous treatment of the end effect dentþed n experments (see Selten-Stocker 1986). 10 An alternatve nterpretaton for the concept s also proposed and dscussed. 5

6 analogy class. The second prncple s that all analogy classes consdered by every player should be reached wth postve probablty along the played path. For both prncples, we provde examples n whch the predcton of the analogy-based approach s n sharp contrast wth the conventonal approach. We also analyze the ssue of multplcty of equlbra, and we dscuss some of the related lterature on bounded ratonalty. 2 A general framework 2.1 The class of games We consder mult-stage games wth almost perfect nformaton and perfect recall. That s, n each stage every player knows all the actons that were taken at any prevous stage (ncludng those exogenous events determned by Nature at any prevous stage), and no nformaton set contaned n the current stage provdes any knowledge of play n that stage. 11 In the man part of the paper, we wll restrct attenton to games wth a Þnte number of stages such that, at every stage and for every player (ncludng Nature), the set of pure actons s Þnte. Ths class of (Þnte) mult-stage games wth almost perfect nformaton s referred to as Γ. 12 ThestandardrepresentatonofanextensveformgamenclassΓ ncludes the set of players =1,...n denoted by N, thegametreeυ (specfyng who moves when and over whch space, ncludng the exogenous events chosen by Nature), and the preferences % of every player over outcomes n the game. AnodenthegametreeΥ wll be denoted by h; t contans nformaton about all the actons, ncludng those by Nature, that were taken at any stage pror to node h. Thesetof nodes h wll be denoted by H. Thesetofnodesatwhchplayer must move wll be denoted by H.Foreverynodeh H,weletA (h) denote player s acton space at node h. Remark: When nterpretng experments, t may be meanngful to vew the players as beng engaged n a varety of games as opposed to only one game. 13 One can represent ths as a metagame made of an extra move by Nature n stage 0 whch would determne the effectve game to be played (accordng to the frequency wth whch each (orgnal) game was played). 11 Also, smultaneous moves are allowed, but each player moves at most once wthn a gven stage. 12 In some applcatons, we wll consder nþnte acton spaces and/or nþntely many stages. The soluton concept wll easly generalze to these applcatons. 13 For example, barganng and ultmatum games or centpede games of varous lengths... 6

7 Classes of analogy: Each player forms an expectaton about the behavor of other players j, j 6=. However, player does not form an expectaton about every player j s behavor n every contngency h H j n whch player j must move. He pools together several contngences n whch other players must move, and he forms an expectaton about the average behavor n these pooled contngences. Such a pool s referred to as a class of analogy. Formally, each player parttons the set {(j, h) N H j,j6= } nto subsets α referred to as analogy classes. 14 The collecton of player s analogy classes α s referred to as player s analogy partton, and t s denoted by An. When (j, h) and(j 0,h 0 ) are n the same analogy class α,werequrethata j (h) =A j 0(h 0 ). That s, n two contngences (j, h) and (j 0,h 0 )thatplayer treats by analogy, the acton space of the nvolved player(s) should be the same. 15 The common acton space n the analogy class α wll be denoted by A(α ). The proþle of analogy parttons (An ) N wll be denoted by An. Remark: At Þrst glance, there s some resemblance between an analogy class and an nformaton set n an extensve form game wth ncomplete nformaton. However, note that An refers to a parttonng of the nodes where players other than must move (as opposed to a parttonng of the nodes where player hmself must move as n the noton of player s nformaton set). 16 Strategc envronment: A strategc envronment n our setup not only specþes the set of players N, thegame tree Υ and players preferences %.ItalsospecÞes how the varous players partton the set of nodes at whch other players must move nto classes of analogy, whch s summarzed n An. A strategc envronment s thus formally gven by (N,Υ, %,An). 2.2 Concepts Analogy-based expectatons: An analogy-based expectaton for player s denoted by β.itspecþesfor every player s analogy class α, a probablty measure over the acton space A(α ). Ths probablty measure s denoted by β (α ), and β (α ) should be nterpreted as player s expectaton about the 14 A partton of a set X s a collecton of subsets x k X such that S x k = X and x k x k 0 = for k 6= k k More generally, we could allow the players to relabel the orgnal actons of the varous players as they wsh. From that prespectve, A j (h) should only be requred to be n bjecton wth A j 0(h 0 )(asopposedto beng equal). Descrbng ths and the subsequent noton of consstency would requre heavy notatons wthout addng anythng to the concept. It s therefore gnored for expostonal reasons. 16 We wll offer more dscusson throughout the paper on the relatonshp between analogy reasonng and ncomplete nformaton (and mperfect recall) n extensve form games. 7

8 average behavor n class α. Remark: Note agan the dfferent nature of β ( )andofplayer s belef system n extensve form games wth ncomplete nformaton. Here β (α ) s an expectaton (or belef) about the average behavor of players other than n class α. (It s not a belef, say, about the lkelhood of the varous elements (j, h) poolednα.) Strategy: A behavor strategy for player s a mappng that assgns to each node h H at whch player must move a dstrbuton over player s acton space at that node. 17 Formally, a behavor strategy for player s denoted by σ.itspecþes for every h H a dstrbuton - denoted σ (h) A (h) -accordngtowhchplayer selects actons n A (h) when at node h. Wealsoletσ denote the strategy proþle of players other than, and we let σ denote the strategy proþle of all players. Sequental ratonalty: The crteron used by the players to choose ther strateges gven ther analogy-based expectatons s as follows. Gven hs analogy-based expectaton β,player constructs a strategy proþle for players other than that assgns player j to play accordng to β (α )at node h whenever (j, h) α. (Ths s the most natural strategy proþle compatble wth player s partal expectaton β. 18 ) And the crteron consdered by player s that of best-response aganst ths nduced strategy proþle after every node where player must move. Formally, for every β and j 6=, wedeþne the β -perceved strategy of player j, σ β j,as19 σ β j (h) =β (α ) whenever (j, h) α. Gven player s strategy σ and gven node h, weletσ h denote the contnuaton strategy of player nduced by σ from node h onwards. Smlarly, we let σ h and σ h be the strategy proþles nduced by σ and σ, respectvely, from node h onwards. We also let u h (σ h, σ h ) denote the expected payoff obtaned by player when the play has reached node h, and players behave accordng to the strategy proþle σ Mxed strateges and behavor strateges are equvalent snce we consder games of perfect recall. The behavor strategy formulaton s better suted to deþne the consstency condton (see below). 18 If for every h such that (j, h) α, the behavor strategy of player j at node h s gven by β (α ), then the average of these (whatever the weghtng of the varous elements of α )mustbeβ (α ). A rcher setup would allow player to consder any strategy proþle for players other than that s compatble wth hs partal knowledge β (see Remark 2 after the deþnton of analogy-based expectaton equlbra). 19 Ths deþnes a strategy for player j because all (j, h) whereh H j belong to one and only one α snce the set of α s a partton of {(j, h) N H j, j 6= }. 20 These functons can formally be derved from % and the dstrbutons over outcomes nduced by σ h. 8

9 Defnton 1 (Crteron) Player s strategy σ s a sequental best-response to the analogybased expectaton β f and only f for all strateges σ 0 and all nodes h H, u h (σ h, σ β h) u h (σ 0 h, σ β h). Consstency: In equlbrum, we requre the analogy-based expectatons of the players to be consstent. That s, to correspond to the real average behavor n every consdered class where the weght gven to the varous elements of an analogy class must tself be consstent wth the real probabltes of vsts of these varous elements. We thnk of the consstency requrement as resultng from a learnng process n whch players would eventually manage to have correct analogy-based expectatons. In lne wth the lterature on learnng n games (see Fudenberg-Levne 1998), we dstngush accordng to whether or not consstency s also requred for analogy classes that are reached wth probablty 0 n equlbrum. 21 To present formally the consstency dea, we denote by P σ (h) the probablty that node h s reached accordng to the strategy proþle σ. Defnton 2 (Weak Consstency) Player s analogy based expectaton β s consstent wth the strategy proþle σ f and only f for all α An : β (α )= X P σ (h) σ j (h) / X P σ (h) (1) (j,h) α (j,h) α whenever P σ (h) > 0 for some h and j such that (j, h) α. Ths deþnton deserves a few comments. The vew s that each player happens to make consstent (or correct) analogy-based expectatons as a result of learnng. Supposeplayers repeatedly act n the envronment as descrbed above. Suppose further that the true pattern of behavor adopted by the players s that descrbed by the strategy proþle σ. Consder player who tres to forecast the average behavor n the analogy class α, assumed to be reached wth postve probablty (accordng to σ). The actual behavor n the analogy class α s an average of what every player j actually does n each of the nodes h where (j, h) α,thats,σ j (h). The correct weghtng of σ j (h) should concde wth the frequency wth whch node (j, h) s vsted (accordng to σ) relatveto 21 When t s requred for unreached classes, the underlyng learnng model should nvolve some form of tremblng (or exogenous expermentaton). When t s not, trembles are not necessary. 9

10 Ã! P other elements n α. The correct weghtng of σ j (h) should thus be P σ (h)/ P σ (h), (j,h) α whch n turn yelds expresson (1). It should be noted that DeÞnton2placesnorestrctonsonplayer s expectatons about those analogy classes that are not reached accordng to σ. The next deþnton proposes a stronger noton of consstency (n the sprt of tremblng hand or sequental equlbrum) that places restrctons also on those expectatons. Formally, we deþne Σ 0 to be the set of totally mxed strategy proþles,.e. strategy proþles σ such that for every player j, for every node h H j at whch player j must move, any acton a j n the acton space A j (h) s played wth strctly postve probablty (thus mplyng that σ j (h) has full support on A j (h) for all j, h H j ). For every strategy proþle σ Σ 0, all analogy classes are reached wth postve probablty. Thus, there s a unque analogy-based expectaton β that s consstent wth σ n the sense of satsfyng condton (1) for all analogy classes α. Denote ths analogy-based expectaton by β hσ. Defnton 3 (Strong consstency) Player s analogy-based expectaton β s strongly consstent wth σ f and only f there exsts a sequence of totally mxed strategy proþles σ k k=1 that converges to σ such that the sequence β σ k k=1 converges to β. Soluton concepts: In equlbrum, we requre that at every node players play best-responses to ther analogybased expectatons (sequental ratonalty) and that expectatons are consstent. We deþne two soluton concepts accordng to whether or not consstency s mposed for analogy classes that are not reached along the played path. And we refer to a par (σ, β) ofstrategyproþle and analogy-based expectaton proþle as an assessment. Defnton 4 An assessment (σ, β) s a Self-ConÞrmng Analogy-Based Expectaton Equlbrum f and only f for every player N, 1. σ s a sequental best-response to β and 2. β s consstent wth σ. Defnton 5 An assessment (σ, β) s an Analogy-Based Expectaton Equlbrum f and only f for every player N, 1. σ s a sequental best-response to β and 2. β s strongly consstent wth σ. 10

11 Remark 1: To the extent that the number of analogy classes α consdered by player s small, player has few features of the other players behavor to learn, whch makes the consstency requrement more plausble from a learnng perspectve than n the perfect ratonalty paradgm. Remark 2: A pror there are strateges other than σ β that could generate the analogybased expectaton β. 22 AmoreelaboratecrteronthantheoneconsderednDeÞnton 1 would vew player as playng a best-response aganst some strategy proþle σ 0 compatble wth β but not necessarly σ β. The correspondng soluton concepts would be somewhat more complcated to present (but most of the nsghts developed below would contnue to hold for such alternatve specþcatons). Remark 3: We have assumed that player s analogy classes are parttons of the nodes where players other than must move. In some cases, t may be meanngful to allow players to predct the behavor of other players also based on ther own behavor. There s no dffculty wth allowng the analogy classes α to also nclude nodes (, h) suchthatatnodeh player must choose an acton n A(α ) (the same acton space as the one faced by the other players nvolved n α ). However, t should be understood that the correspondng analogy-based expectaton β (α )susedbyplayer only to construct a strategy proþle for players other than (see DeÞnton 1). 23 Remark 4: The setup could easly be extended to cover the case where players have prvate nformaton. However, ths would sgnþcantly complcate the descrpton of the setup. For expostonal (rather than conceptual) reasons, we have chosen to focus on games wth almost perfect nformaton. 2.3 Prelmnary results We conclude ths general presentaton by makng two smple observatons. The Þrst one shows the relaton to subgame perfecton when all players use the Þnest parttonng as ther analogy devce. The second one shows the exstence of analogy-based expectaton equlbra n Þnte envronments. Formally, we say that all players use the Þnest analogy parttonng f there are no, (j, h), (j 0,h 0 ) 6= (j, h) andα An such that (j, h) α and (j 0,h 0 ) α.wehave: Proposton 1 Consder an envronment (N,Υ, %,An) n whch all players use the Þnest 22 In general (except for σ β ), to check that σ0 generates β t s ndspensable to know the frequency of vsts of every node h α (as gven by the canddate strategy proþle σ). 23 We have chosen not to present the concept wth that extenson because t could create an extra source of confuson (wth the noton of nformaton set). 11

12 analogy parttonng. Then f (σ, β) s an analogy-based expectaton equlbrum of (N,Υ, %,An), σ sasubgameperfectnashequlbrumof(n,υ, % ). Proof. When players use the Þnest analogy parttonng, strong consstency of β wth respect to σ mples that σ β = σ. Proposton 1 then follows from DeÞnton 1. Remark: When at least one player, say player, doesnotusetheþnest partton as hs analogy devce, the play of an analogy-based expectaton equlbrum need not correspond to that of a Subgame Perfect Nash Equlbrum. Ths s because n an analogy-based expectaton equlbrum (σ, β), player s strategy σ s requred to be a best-response to σ β (after every node h). But, σ β need not (n general) concde wth σ as n a Subgame Perfect Nash equlbrum. Ths wll be further llustrated throughout the paper. Proposton 2 (Exstence) Every Þnte envronment (N,Υ, %,An) has at least one analogybased expectaton equlbrum. Proof. Thestrategyofproofsthesameasthatfortheexstenceproofofsequental equlbra (Kreps and Wlson 1982). We menton the argument, but for space reasons we do not gve the detals of t. Frst, assume that n every node h H,player must choose every acton a A (h)wth probablty no smaller than ε (thssnsprtofselten1975). 24 Itsclearthanananalogybased expectaton equlbrum wth such addtonal constrants must exst. Call (σ ε, β ε )one such proþle of strateges and analogy-based expectatons. By compactness propertes (whch hold n the Þnte envronment case), some subsequence must be convergng to say (σ, β), whch s an analogy-based expectaton equlbrum. 3 Varous effects of analogy reasonng 3.1 Analogy reasonng can be good or bad We wsh to llustrate that bundlng contngences by analogy can ether beneþt orhurta player. To ths end, we consder the followng envronment. Two normal form games G and G 0 are beng played n parallel. Game G s played wth probablty ν and game G 0 s played wth probablty 1 ν. (In the formulaton of Secton 2, the game tree Υ conssts of a Þrst move by Nature about the selecton of the game - accordng to the probabltes ν and 1 ν - then followed by the normal form game G or G 0 accordngly.) There are two players =1, 2 n G and G 0.InbothGand G 0,player must choose an acton a n the same Þnte acton space A. 24 Ths requres amendng DeÞnton 1 to ncorportate such constrants n the maxmzaton programmes. 12

13 InthegametreeΥ, a node can be dentþed wth a normal form game G or G 0. We assume that player 2 uses the Þnest parttonng (.e., player 2 uses two analogy classes {(1,G)} and {(1,G 0 )}). We wsh to compare the equlbrum payoff obtaned by player 1 n each of the subgames G, G 0 accordngtowhetherplayer1usestheþnest parttonng or the coarsest parttonng (n the latter case player 1 pools together the two subgames G and G 0 nto a sngle class of analogy {(2,G),(2,G 0 )}). Clam 1: Suppose player 2 has a domnant strategy 25 n both games G and G 0. Player 1 s equlbrum payoff - n both G and G 0 - s no smaller when player 1 uses the Þnest parttonng as opposed to the coarsest parttonng. Proof. Whatever the parttonng of player 1, player 2 wll n equlbrum select hs domnant strategy n both G and G 0.TheÞnest parttonng of player 1 allows player 1 to pck a bestresponse to player 2 s domnant strategy n both G and G 0, whch s the hghest payoff player 1canhopetoget(nbothG and G 0 ) gven player 2 s behavor. Wthn the context of Clam 1, t s mmedate to construct an example n whch player 1 s equlbrum payoff s strctly lower when he uses the coarse parttonng as opposed to the Þnest parttonng. (Such an example must be such that player 2 s domnant strategy s not thesamengamesg and G 0, and thus player 1 s analogy-based expectaton s not accurate for games G and G 0 n solaton.) When player 2 has no domnant strategy, however, analogy reasonng may beneþt player 1, as the followng example shows. Example 1: Consder the followng stuaton L M R U 5, 1 0, 1 2, 2 D 3, 1 3, 0 1, 0 Game G L M R U 3, 0 3, 1 1, 0 D 0, 1 5, 1 2, 2 Game G 0 where n each cell the left and rght numbers ndcate players 1 and 2 s payoffs, respectvely. Both games are assumed to be played wth equal probablty,.e. ν = 1 2.InbothGand G0, the acton space of players 1 and 2 are A 1 = {U, D} and A 2 = {L, M, R}, respectvely. The example s such that both G and G 0 have a unque Nash equlbrum, whch s UR n game G and DR n game G 0. Thus, when both players use the Þnest parttonng, player 1getsapayoff of2nbothsubgames. Suppose now that player 1 uses the coarsest parttonng (whle player 2 uses the Þnest). The followng assessment s an analogy-based expectaton equlbrum. 25 Ths domnant strategy need not be the same n both games G and G 0. 13

14 Strategy proþle: Player1playsD n game G and U n game G 0.Player2playsL n game G and M n game G 0. Analogy-based expectatons: Player 1 expects player 2 to play L and M each wth probablty 1 2 (n hs unque analogy class {(2,G),(2,G0 )}). Player 2 expects player 1 to play D n game G and U n game G 0. To check that the above assessment s an equlbrum, note that gven the strategy proþle, players analogy-based expectatons are consstent. Then gven player 1 s analogy-based expectaton, player 1 chooses D (resp. U) rather than U (resp. D) ngameg (resp. G 0 ) because 1 2 (3 + 3) > 1 2 (0 + 5). Gven player 1 s strategy, player 2 s best-response s L n game G and M n game G 0. Fnally, note that accordng to the above strategy proþle player 1 gets a payoff of 3 n both G and G 0, whch s strctly larger than 2 - the equlbrum payoff obtaned by player 1 when he uses the Þnest parttonng. The key feature of Example 1 s that player 2 does not play n the same way when player 1usestheÞnest parttonng and when he uses the coarsest parttonng. It s stll the case that the coarseness of player 1 s parttonng nduces player 1 not to optmze aganst player 2 s behavor n G and G 0 (because the best-response would be U (and not D) ngameg and D (and not U) n game G 0 ). However, t allows player 1 to Þnd t optmal to play D (resp. U) n game G (resp. G 0 ),whchnturnnducesplayer2toplayanactonthatsmorefavorable to player 1. Remark 1: In the analogy-based expectaton equlbrum shown n Example 1, both players 1 and 2 behave dfferently n games G and G 0. Thus, even by varyng the payoff matrx of games G and G 0, t s not possble to nterpret the equlbrum outcome as emergng from an mperfect nformaton (of ether player) as to whch game (G or G 0 ) s beng played. Remark 2: In Example 1, when player 1 uses the coarsest parttonng, there s also an equlbrum n whch UR s played n game G and DR s played n game G 0 as n the Þnest parttonng case. Modfy the specþcaton of game G so that player 2 has a domnant strategy whch s to play L. It can be checked that when player 1 uses the coarsest parttonng the assessment shown n Example 1 s the only analogy-based expectaton equlbrum n ths modþed example. Thus, n ths modþed setup, player 1 beneþts from the coarse parttonng n subgame G 0 whatever the equlbrum under consderaton. Remark 3: If one nssts on havng equlbra that employ pure strateges, player 2 should have at least three actons for an example of the sort dsplayed n Example 1 to work. Otherwse, smlar conclusons can be derved wth 2x2 games and mxed strategy equlbra. Comment: Intheabovestuatonwehaveassumedthatthesameplayer2weretoplayn 14

15 both games G and G 0. Of course, an alternatve nterpretaton s that the player other than 1nvolvedngameG s not the same as the one nvolved n game G 0, say player 2 n game G and player 2 0 n game G 0. The parttonng of player 1 consdered above corresponds then to {(2,G),(2 0,G 0 )}. For that nterpretaton, t s essental to allow player 1 to treat by analogy nodes n whch several dfferent players (here players 2 and 2 0 ) are nvolved. 3.2 Centpede game Consder the centpede game CP K Fgure 1. (Þrst ntroduced by Rosenthal 1982) and depcted n N (K) 2 N (K) 1 N (1) 2 N (1) 1 P 2 P 1 P 2 P 1 (a 0,b 0 ) T 2 T 1 T 2 T 1 (a 2K,b 2K ) (a 2K 1,b 2K 1 ) (a 2,b 2 ) (a 1,b 1 ) Fgure 1: The centpede game It s a (2K)-perod extensve form game descrbed as follows. There are two players =1, 2 who move n alternate order. In each perod, the player whose turn t s to move, say player, mayethertake or Pass,.e.A = {Pass,Take}. 26 If he Takes, ths s the end ofthegame.ifhepasses,thegameproceedstothenextstagewheretstheotherplayer s turn to move unless the game has reached the last perod 2K n whch case ths s the end of the game. Player 1 moves n the last perod 2K, player 2 moves n the last but one perod and so on. Nodes at whch player 1 must move are labelled N (k) 1, k =1,...K where N (1) 1 desgnates the last node (.e., perod 2K) atwhchplayer1mustmove,n (2) 1 the last but one,andsoontlln (K) 1 the Þrst node (.e, perod 2) at whch player 1 must move. Smlarly, nodes at whch player 2 must move are labelled N (k) 2, k =1,...K where N (1) 2 desgnates the lastnode(nperod2k 1) at whch player 2 must move, and N (K) 2 the Þrst node (n perod 26 We mplctly assume n the followng that the players label these actons the same way. 15

16 1) at whch player 2 must move. If player 2 Takes at node N (k) 2, the payoffs toplayers1and 2area 2k and b 2k, respectvely. If player 1 Takes at node N (k) 1,thepayoffs toplayers1and 2area 2k 1 and b 2k 1, respectvely. If player 1 Passes at node N (1) 1,thepayoffs toplayers1 and 2 are a 0 and b 0, respectvely. All a t and b t, t =0,...2K areassumedtobentegersthat satsfy for all k 1: a 2k 1 > a 2k 2 >a 2k+1 >a 2k (2) b 2k 2 > b 2k 3 >b 2k >b 2k 1 These condtons guarantee that (1) the unque Subgame Perfect Nash Equlbrum (SPNE) of CP K s such that player 2 Takes n the Þrst perod (ths follows from a 2k+1 >a 2k and b 2k >b 2k 1 ), and (2) n any perod t 2K 2, both players are better off f Take occurs two perods later,.e. n perod t + 2, than f t occurs n the current perod t (ths follows from a t >a t+2 and b t >b t+2 for all t 2K 2). The predcton of the SPNE sounds relatvely unntutve, especally when the number of perods 2K s large (because then takng n the Þrst node seems to nduce very severe losses for both players). As we now llustrate, the analogy approach explans why players may Pass most of the tme n the centpede game, at least for long enough versons of the game. In order to deal wth the effect of ncreasng the number of perods n CP K, we wll consder nþnte sequences of ntegers (a k ) k=0,(b k) k=0 satsfyng (2). We wll also assume that the dfference between two consecutve payoffs s unformly bounded from above. That s, there exsts > 0 such that for all t 0, a t a t+1 < and b t b t+1 <. (3) Regardng analogy parttonng, we wll mostly consder the case n whch both players use the coarsest parttons as ther analogy devce. That s, each player s assumed to pool together all the nodes N (k) j at whch player j, j 6= must move nto a sngle class of analogy α : n o α = (j, N (k) j ), 1 k K. The strategc envronment s thus descrbed by the set of players N = {1, 2}, thegametree CP K, players preferences % as deþned by a t, b t, and the analogy parttonng structure An as descrbed by α 1 and α 2 :(N,CP K, %,An). Player s analogy-based expectaton β reduces here to a sngle probablty measure β (α )=λ Pass+(1 λ ) Take A j, whch stands for player s expectaton about the average behavor of player j throughout the game. 16

17 We Þrst consder assessments (σ, β) such that the strategy of every player s pure (.e. for every, h H, σ (h) A ). And we consder the followng condton: K 1 K b 2k + 1 K b 2k+1 >b 2k+2 for all k 0. (4) Proposton 3 Suppose that condton (4) holds, and consder the envronment (N,CP K %,An). There are two possble equlbrum paths correspondng to self-conþrmng analogy-based expectaton equlbra n pure strateges: Ether player 2 Takes n the Þrst perod or the game reaches perod 2K n whch player 1 Takes. Proof. (a) We Þrst provethatthetwomentonedoutcomes canbeobtanedasanalogy-based expectaton equlbrum outcomes. () Observe Þrst that the Subgame Perfect Nash Equlbrum outcome corresponds to the analogy-based expectaton equlbrum (σ, β) nwhchfor =1, 2, β (α )=Take and σ (N (k) )=Take for all k =1,...K. () Consder the strategy proþle σ such that player 2 Passes always and player 1 Takes n the last perod 2K. To be consstent wth σ, the analogy-based expectaton of player 1 must be that player 2 Passes wth probablty 1,.e. β 1 (α 1 )=Pass (snce player 2 Passes always when he has to move). To be consstent wth σ, the analogy-based expectaton of player 2 must be that player 1 Passes wth probablty K 1 (k) K, snce (accordng to σ) each node N 1, k = K,...1 s reached wth probablty 1,.e. P σ (N (k) 1 ) = 1, (so that they have equal weghtng), and player 1 Passes (wth probablty 1) n all nodes N (k) 1, k = K,...2 andtakesnnoden (1) 1. Thus, β 2 (α 2 )= K 1 K Pass+ 1 K Take. The (sequental) best-response of player 1 to the analogy-based expectaton β 1 s to Take n the last node N (1) 1. Thus, t s to play accordng to σ 1. When condton (4) holds, the best-response of player 2 to the analogy-based expectaton β 2 s to Pass always (snce by Takng at N (k+1) 2, player 2 would only get b 2k+2,whchs less than the expected payoff he gets by Passng at N (k+1) 2 and Takng at N (k) 2,say,.e. K 1 K b 2k + K 1 b 2k+1 >b 2k+2 ). Thus, t s to play accordng to σ 2. Altogether the above consderatons show that the assessment (σ, β) sananalogy-based expectaton equlbrum. (b) It remans to show that there are no other possble outcomes n any pure strategy self-conþrmng analogy-based expectaton equlbrum. Observe Þrst that f an outcome other than Player 2 Takes n the Þrst perod emerges (as a self-conþrmng analogy-based expectaton equlbrum outcome), t must correspond to an analogy-based expectaton equlbrum 17

18 outcome. (Ths s because the unque analogy class of every player s then reached wth strctly postve probablty.) Consder the outcome n whch player Takes at node N (k),andn (k) dffers from N (K) 2.If a pure strategy analogy-based expectaton equlbrum leads to that outcome, t must be (by consstency) that player s analogy based expectaton satsþes β (α )=Pass,snceonthe equlbrum path, player j would always Pass. Player s best response to such a β depends on whether = 1 or 2. If = 1, player 1 s best response to β 1 (α 1 )=Pass s to Take at node N (1) 1 (whch corresponds to an outcome already dentþed as a possble analogy-based expectaton equlbrum outcome). If = 2, player 2 s best response to β 2 (α 2 )=Pass s to Pass always, whch s n contradcton wth player 2 Takng at node N (k) 2. Fnally, the outcome n whch both players Pass n every perod cannot be an analogybased expectaton equlbrum outcome because whatever player 1 s expectaton, player 1 strctly prefers Takng at node N (1) 1 to Passng always. Comment 1: The two outcomes mentoned n Proposton 3 reman equlbrum outcomes even f player 1 uses a parttonng other than the coarsest, as long as player 2 uses the coarsest parttonng. 27 Comment 2: Consder the case where player 2 uses the coarsest parttonng and player 1 uses the Þnest parttonng (and condton (4) holds). As mentoned n Comment 1, Take by player 1 n the last node can be sustaned as the equlbrum outcome of an analogy-based expectaton equlbrum. Note that n ths equlbrum, player 2 behaves n the same way n every node where he must move, whch s to be related to hs bundlng of all nodes n whch player 1 must move nto a sngle class of analogy. We wll suggest such a prncple for reþnng analogy-based expectaton equlbra n Secton 4. Proposton (3) leaves open what happens when condton (4) does not hold. 28 And t does not deal wth equlbra n mxed strateges. The next Proposton provdes the man mssng nformaton (stll assumng that condtons (2) and (3) hold): Proposton 4 There exsts an nteger m such that for all K>m: (1)(N,CP K, %,An) has an analogy-based expectaton equlbrum (σ, β) n whch each player Passes wth probablty 27 If one addtonally requres that for all k, b 2k < b 2k 1+b 2k 2 2, then these are the only possble outcomes of self-conþrmng analogy-based expectaton equlbra n pure strateges. (The pont s that for a pure outcome other than that of the SPNE to emerge as a self-conþrmng analogy-based expectaton equlbrum, t should be that λ 2 1. But, then the best-response of player 2 to β2(α2) 2 =λ2 Pass+(1 λ 2 )Take s to Pass always, thus leadng to the wshed concluson.) 28 Take at the last node may then fal to be the outcome of an analogy-based expectaton equlbrum n pure strateges. Ths s, for example, the case when K 1 K b0 + 1 K b1 >b2 (because then player 2 would strctly prefer Takng n the last but one node rather than Passng always). 18

19 1 n the Þrst K m nodes,.e. n every N (k), k = K,...K m. (2) Any self-conþrmng analogy-based expectaton equlbrum of (N,CP K, %,An) n whch each player Passes wth probablty 1 n the Þrst node N (K) s such that each player Passes wth probablty 1 n the Þrst K m nodes,.e. n every N (k), k = K,...K m. Proof. (1) Suppose β (α )=λ.pass +(1 λ ).T ake wth λ 1 2 for =1, 2. Under condton (3), 29 t s readly verþed that there exsts an nteger m such that for all K>m, player s best-response to β requres Passng (wth probablty 1) n the Þrst K m moves (atleast)(becauseforsomeappropratelyspecþed m, Takng earler s domnated by never Takng). Suppose that players 1 and 2 Pass wth probablty 1 n the Þrst node where they must move. The consstency condton mples that the analogy-based expectaton of player, β (α )=λ.pass +(1 λ ).Take, should satsfy λ 1 2. Together these two observatons mply that the mappng β =(β 1, β 2 ) σ =(σ 1, σ 2 ) (β 1 hσ, β 2 hσ) Best-response Consstency has a Þxed pont such that λ 1 2 for =1, 2. Gven the best-response to such analogy-based expectatons, we may conclude. (2) Suppose player s strategy requres hm to Pass wth probablty 1 n node N (K) for = 1, 2. By the consstency requrement t should be that player s analogy-based expectaton β (α )=λ.p ass +(1 λ ).T ake satsþes λ 1 2 for =1, 2. The best-response to β s to Pass at least n the Þrst K m where he must move. Proposton 4 (1) shows that rrespectve of the length 2K of the game, there s an equlbrum (possbly n mxed strateges) n whch Take occurs n a Þnte number of perods toward the end of the game. 30 Proposton 4 (2) shows that there cannot be equlbra n whch Take occurs n the mddle phase of the game (.e. between perod 3 and perod 2K 2). Comment 1: A predcton of the analogy setup (at least wth the coarsest parttonng and restrctng attenton to equlbra n whch Take never occurs n the Þrst two perods) s that, by ncreasng the length of CP K, the length of the end phase - n whch Take may occur - can never grow above some Þxed and bounded value. Comment 2: It should be noted that the Subgame Perfect Nash Equlbrum outcome s also an analogy-based expectaton equlbrum outcome (n whch Player 1 Takes n N (K) 1 expectng player 2 to Take n α 1 ). And that there s another equlbrum n mxed strateges 29 Snce all payoffs are ntegers satsfyng (2), the dfferences a t a t+2, b t b t+2 are no smaller than When condton (4) does not hold, ths may nvolve an equlbrum n mxed strateges. 19

20 n whch Take may occur n the Þrst two perods (t s such that each player =1, 2 plays n mxed strateges n N (K) and Takes wth probablty 1 n all other nodes). We now consder a slght modþcaton of the envronment n whch the Subgame Perfect Nash Equlbrum s no longer an equlbrum and Take can only occur toward the end of the game. SpecÞcally, assume the players not only play game CP K, but also another game that s the same as game CP K except that there are only two moves correspondng to Player 1 passng or not to the mddle of the game and Player 2 passng or not from the mddle to theendofthegame. Formally,letK be an odd number. Consder the game tree Υ such that n stage 0 Nature selects ether game CP K wth probablty ν CP > 0orgameF wth probablty ν F > 0 where game F s descrbed as follows. Game F has the same two players =1, 2asCP K and two moves. Player 2 moves n the Þrst node denoted by M 2.AtnodeM 2, player 2 must choose an acton n A 2 = {Pass,Take}. If player 2 Takes, the game ends, players 1 and 2 payoffs area 2K and b 2K, respectvely. If player 2 Passes, the game moves to node M 1 where t s player 1 s turn to move. Player 1 must choose an acton n A 1 = {Pass,Take}. If player 1 Takes, ths s the end of the game and the payoffs of players 1 and 2 are a K and b K, respectvely; f he Passes, ths s also the endofthegameandthepayoffs toplayers1and2area 0 and b 0, respectvely. We assume that K s larger than 2 so that a 0 >a K >a 2K and b 0 >b K >b 2K. Also, we assume that each player uses a sngle class of analogy. That s, n α = (j, N (k) [ j ), 1 k Ko {(j, Mj )} and we call (N,Υ, %,An) the assocated envronment. Proposton 5 Suppose that condtons (2) and (3) hold. There exsts an nteger m such that for all K>m, all self-conþrmng analogy-based expectaton equlbra (σ, β) of (N,Υ, %,An) are such that player Passes wth probablty 1 n M and n every N (k), k = K,...K m. Proof. In game F, whatever ther analogy-based expectaton, each player chooses optmally to Pass. Ths ensures that the analogy-based expectaton of player, β (α )=λ.pass + (1 λ ).T ake satsþes λ ν F > 0for =1, 2. Gven condton (3), ths ensures that, for K large enough, the best-response n CP K of each player s at least to Pass n the Þrst node where he must move, thus ensurng that λ > 1 2 for =1, 2. We may then conclude usng the best-responseargumentntheproofofproposton4. In the above analyss of the centpede game CP K, we assumed that players use the coarsest analogy parttonng. However, the nsght that analogy reasonng may lead players to Pass mostofthetmenlongenoughcp K would n general carry over, even when players use more than one analogy class. 20

21 Suppose, for example, that each player consders two classes: α end = α man = n o (j, N (k) j )suchthatk k n o (j, N (k) j )suchthatk<k accordng to whether the end phase or the man phase of the game s beng consdered, and call (N,CP K, %,An) the correspondng envronment. Proposton 6 There exst m and an analogy-based-expectaton equlbrum of (N,CP K, %,An) such that, for all K, eachplayer Passes wth probablty 1 at least n the Þrst K m nodes where he must move. Proof. If k 1b k 2k + 1b k 2k+1 >b 2k+2 for all k k, then Player 1 Takng n the last node N (1) 1 s an analogy-based expectaton equlbrum outcome (ths follows from the analyss n Proposton 3). Otherwse, usng the argument n the proof of Proposton 4, t s readly verþedthat there s m such that for, K large enough, there s an equlbrum (σ, β) satsfyng (1) β (α man )= λ,man Pass+(1 λ,man ) Takewth λ,man 1 2 for =1, 2, and (2) player s best-response to β s to Pass wth probablty 1 at least n the Þrst K m moves. 3.3 (Fntely) Repeated Prsoner s Dlemma Consder the Prsoner s Dlemma PD whose matrx payoff s represented as: D C D 0, 0 1+g 1, l 2 C l 1, 1+g 2 1, 1 Game PD wth l, g > 0for =1, 2, where each player =1, 2 has to choose smultaneously an acton n {D, C}. We now consder several varants of repeated PD. TheÞrst two varants llustrate how analogy reasonng may gve rse to (non-trval) end effects n the Þntely repeated PD. The thrd varant deals wth the nþnte repetton. T-repetton: We Þrst consder the T repetton of PDwth no dscount factor, and we denote by PD T the correspondng game tree. Nodes n PD T correspond to hstores of length 0toT specfyng the acton proþles played n earler perods (f any). The hstory ³ of length 0 s denoted by, andahstoryh of length t>0s(a (1),..., a (t) )wherea (k) = a (k) 1,a(k) 2 and a (k) {D, C} stands for the acton played by player n perod k. 21