Endogenous Interactions *

Transcription

1 Endogenous Interactons * George J. Malath Department of Economcs Unversty of Pennsylvana 378 Locust Walk Phladelpha, PA 904 USA Larry Samuelson Department of Economcs Unversty of Wsconsn 80 Observatory Drve Madson, Wsconsn USA Avner Shaked Department of Economcs Unversty of Bonn Adenauerallee 4-6 D 533 Bonn, Germany October, 997 * Presented at the Tenth Workshop of the Internatonal School of Economc Research on Evoluton and Economcs held at the Certosa d Pontgnano, Sena (Italy), June 7 - July 5, 997. George Malath thanks Professor Ugo Pagano and Dr. Govann Forcon for organzng the workshop. Ths s a revson of Evoluton and Endogenous Interactons, frst draft, December 0, 99. Part of ths work was done whle George Malath and Larry Samuelson were vstng the Unversty of Bonn and whle all three authors were vstng the Insttute for Advanced Studes at the Hebrew Unversty of Jerusalem. We are grateful for the hosptalty of both. We thank Ken Bnmore for helpful dscussons. Ths work has been presented under the ttles The Evoluton of Heterogenety, Evoluton wth Endogenous Interactons, and Evoluton and Correlated Equlbra. Fnancal support from the Natonal Scence Foundaton and the Deutsche Forschungsgemenschaft, SFB 303, at the Unversty of Bonn, s gratefully acknowledged.

2 Endogenous Interactons by George J. Malath, Larry Samuelson, and Avner Shaked Abstract We examne an evolutonary model wth local nteractons, so that agents are more lkely to nteract wth some agents than wth others. We frst revew the result that equlbrum strategy choces wth gven local nteractons correspond to correlated equlbra of the underlyng game. We then allow the pattern of nteractons tself to be shaped by evolutonary pressures. If agents do not have the ablty to avod unwanted nteractons, then heterogeneous outcomes can appear, ncludng outcomes n whch dfferent groups play dfferent Pareto ranked equlbra. If agents do have the ablty to avod undesred nteractons, then we derve condtons under whch outcomes must be not only homogeneous but effcent.

3 Endogenous Interactons by George J. Malath, Larry Samuelson, and Avner Shaked. Introducton It's not what you know, t's who you know that counts. We have all encountered assertons of ths type, stressng the mportance of nteractng wth the rght people. In a smlar ven, how does one explan the tremendous premum prospectve MBA s put on attendng a top busness school f one s unconvnced that such schools dffer greatly n ther value-added or n the ablty of ther students? A common response s that the real reason for obtanng an MBA from the rght school s the chance to network wth future captans of ndustry. In ths paper, we examne groups or populatons of players who nteract n pars to play a game. However, a player may not be equally lkely to meet each member of the opposng populaton. Instead, patterns can arse n whch some pars of agents are more lkely to nteract than others. In addton, players may have both the desre and the ablty to affect the pattern of such nteractons. Hence, the game nvolves choosng a strategy that one plays whenever matched wth an opponent and also choosng an actvty that wll affect whch opponents one s matched wth to play the game. The exstence of large populatons of players who are matched to play the game suggests an evolutonary model. We follow conventonal models n assumng that a player s characterzed by a sngle pure strategy that s played aganst all opponents the player happens to meet, and n allowng that strategy to be adjusted va an evolutonary process. We depart from conventonal models both n assumng that agents are more lkely to meet some opponents than others and n assumng the actvtes that affect these meetng patterns are themselves subject to evolutonary pressures. We say that an evolutonary process exhbtng the frst characterstc has local nteractons and a process that also has the second characterstcs exhbts endogenous local nteractons or smply endogenous nteractons. We are nterested n local nteractons because we beleve such nteractons to be the rule rather than the excepton, and also because such nteractons can have mportant stablty mplcatons n an evolutonary model. In partcular, local nteractons may cause a mutant to face a local envronment that dffers consderably from the aggregate populaton. By supplyng brdgeheads for nvason, prmarly by allowng mutants to ntally face a relatvely hgh concentraton of other mutants, local nteractons may allow mutants to succeed that would otherwse be doomed to extncton. If local nteractons are mportant, however, then our attenton s naturally drected to the forces that determne these nteractons, and hence to endogenous nteractons. We are nterested n three questons. What can we say about the rest ponts of an evolutonary process wth local nteractons? Can an evolutonary process wth endogenous nteractons lead to heterogeneous outcomes, n whch dfferent groups of agents play dfferent strateges and receve dfferent payoffs? Under what condtons wll an evolutonary process wth endogenous nteractons yeld effcent outcomes? We frst revew Malath, Samuelson, and Shaked (997), whch examnes the case where nteractons are local but agents actvtes are fxed, so that agents cannot alter the opponents wth whom they play. That paper also restrcts attenton to Nash equlbra n strategy choces gven the local nteracton pattern. We found there that the Nash equlbra wth fxed (local) nteractons correspond to correlated equlbra of the underlyng game. Conversely, for any correlated equlbrum of the underlyng game, there s a pattern of local nteractons such that, fxng ths pattern, there s a Nash equlbrum n strategy choces that gves the correlated equlbrum outcome. The vscosty model of Myerson, Pollock and Swnkels (99) s n ths ven. 3

4 Snce we want to descrbe the evoluton of a populaton wth local nteractons, we need an approprate noton of state for such a populaton. A complete descrpton of such a state nvolves lstng the strategy chosen by each agent as well as hs pattern of nteractons. Ths s a complcated object, and t s helpful to have a smpler statstc that captures the mportant features of a state. In conventonal evolutonary models, ths smple statstc s the mxed strategy profle played by the total populaton. In our model wth local nteractons, the smple statstc s the correlated equlbrum nduced by the state. Our nterest then turns to the case n whch nteractons are endogenous, wth both strateges and actvtes adjusted by an evolutonary process. What propertes wll stable equlbra of the evolutonary process have, and n partcular wll they be homogeneous or effcent? The answers to these questons depend upon the extent to whch agents can control ther nteractons. The key factor here s whether agents can ensure that they do not meet certan opponents. If agents do not have the ablty to seclude themselves, then heterogeneous stable outcomes are possble, ncludng outcomes n whch agents are separated nto groups, some of whch play a good equlbrum and others of whch play an neffcent, Pareto nferor bad equlbrum. If agents have the ablty to seclude themselves wthout meetng undesred opponents, then stable outcomes must be effcent as well as homogeneous. These results may ntally appear to be counterntutve. Heterogenety would appear to be most lkely when secluded groups can form; whle the coexstence of good and bad equlbrum outcomes would appear to be problematc when those playng the bad outcome cannot be excluded from nteractng wth those players enjoyng the good outcome. However, heterogenety perssts n the latter case, even though agents playng the bad equlbrum can seek nteractons wth agents playng the good outcome, because the former cannot avod other agents playng the bad equlbrum. Because some agents cannot avod others playng the bad equlbrum, t can be a best response to acquesce n playng the bad equlbrum, rather than seekng other partners wth whom to play the good equlbrum and n the process mscoordnatng wth those who play the bad one. Conversely, effcent outcomes are ensured f groups of agents have the ablty to segregate themselves from others. An outcome that s not effcent can be dsplaced by a small group of agents who seclude themselves and play an effcent outcome. Ths group wll attract other agents who then fnd t optmal to swtch to the effcent outcome because they are ensured of meetng only opponents playng that outcome. These results depend heavly on the assumpton that each agent must play the same strategy aganst all opponents he happens to meet. In partcular, an agent could always (at least weakly) mprove hs outcome by makng hs strategy contngent on the strategy played by the opponent he meets. 3 In many cases, however, such dscrmnaton s mpossble, ether because t s mpossble to dscern the opponent's strategy before choosng one's own or because the technology of choosng strateges allows only one choce. Ths nablty to make strateges contngent on opponents s the essence of the local nteracton problem. If contngent strateges could be played, then each nteracton could be treated as a separate game and local nteracton ssues would be unnterestng. To help n nterpretng the process by whch agents affect ther nteractons, we use the analogy of economsts at the annual wnter meetngs of the Alled Socal Scences Assocatons n North Amerca. 4 Interactons or networkng are the heart of these meetngs. Those who attend the meetngs often devote great attenton to arrangng ther actvtes so as to acheve desred contacts. Partcpants nteract n many dfferent ways, ncludng ntervews, paper sessons, cocktal partes and encounters n hotel lobbes. These actvtes provde qute dfferent opportuntes to control one s pattern of meetngs. As we wll dscuss, there s a smlar result n cheap talk games. Ely (996) has obtaned a smlar result for a related model. 3 See Banerjee and Webull (993) for a model of dscrmnatng players. 4 Any large conference wll serve as an analogy, as long as there are ntervews assocated wth the conference. At the ASSA wnter meetngs, graduate students soon to receve ther Ph.D. s are ntervewed by potental employers. 4

5 The followng secton consders fxed nteracton patterns. Our analogy here s wth ntervews at the wnter meetngs, where there s very lttle ablty, once the meetngs have started, to control one s nteractons. Nash equlbrum strategy choces wth fxed local nteractons correspond to correlated equlbra of the underlyng game. Secton 3 ntroduces the dynamcs by whch nteractons evolve. Sectons 4 and 5 examne endogenous nteractons. Secton 4 shows that stable, heterogeneous outcomes can arse f agents do not have the ablty to avod other agents. Our analogy here s wth paper sessons at the meetngs, where one can choose whch sessons to attend but cannot help but nteract wth those who attend one's own sessons. Secton 5 shows that stable outcomes must be effcent f agents have the ablty to nteract n solated groups. Our analogy here s wth cocktal partes and hotel lobbes at the wnter meetngs, where groups can always steal off nto secluson. Indeed, the number of socetes and assocatons (and correspondng cocktal partes) at the meetngs has been growng steadly over the years, presumably reflectng a desre for lke-mnded partcpants to nteract wth each other (though we would hestate to clam that the meetngs are effcent).. Fxed Interactons and Correlated Equlbra: Intervews In ths secton, we explore the mplcatons of equlbrum strategy choces n the presence of local nteractons. The pattern of nteractons between agents s fxed, so that agents have no freedom to affect the dstrbuton of ther opponents. In terms of the wnter meetngs, our analogy here s wth the ntervewng process. Agents nteract accordng to a fxed ntervew schedule, wth very lttle ablty to affect ths pattern by arrangng ntervews at the meetngs. The nteracton pattern s local, however, wth some pars of agents much more lkely to meet than others. In addton, certan aspects of one's strategy are constraned to be the same aganst all opponents. For example, graduate students typcally have only a sngle paper to present to all ntervewers. The nteracton between a par of agents s descrbed by a fnte, two-player normal form game, denoted G = ( S, π ), where S = S S s the jont acton set and π = ( π, π ) s the reward functon for the parwse nteracton. We assume that there s a fnte populaton N of player s and a fnte populaton N of player s. We thnk of these populatons as beng large, and requre each populaton to have at least as many members as there are pure strateges for that player,.e., N S and N S. Each member of each populaton s assocated wth a pure strategy, so that s: N S and s: N S are functons specfyng the strateges of the members of populatons and, wth player N playng s ( ) and player j N playng s j ( ). Players from these populatons are drawn n pars, one from each populaton, to meet and play the game. 5 These meetngs are descrbed by a lst of the number of tmes that each par of agents meets. Wthout loss of generalty, we normalze the total number of meetngs to one. The proporton of meetngs between players and j can then also be descrbed as the probablty that, gven a meetng, t nvolves players and j. The nteractons between the two populatons are thus descrbed by a probablty dstrbuton µ on the fnte space N N, wth µ(, j ) nterpreted as ether the proporton of all matches that are matches between and j or as the probablty that, gven a match, t nvolves player from populaton and player j from populaton. Defnton. A trple ( s, s ; µ ), consstng of an assgnment of strateges to agents ( s, s ) and an nteracton pattern µ, s an equlbrum wth fxed nteractons f, for all N, 5 Notce that we explctly assume asymmetrc nteractons,.e., there s role dentfcaton. We can allow for no role dentfcaton at the cost of addtonal notatonal complexty. 5

6 π ( s ( ), s ( j)) µ (, j) π ($ s, s ( j)) µ (, j) s$ S, () j N j N and for all j N, π ( s ( ), s ( j)) µ (, j) π ( s ( ), s$ ) µ (, j) s$ S. () N N In equlbrum, dfferent members of a populaton may face dfferent opponents, and hence fnd dfferent strateges optmal. Ths has the flavor of a correlated equlbrum, where dfferent sgnals from a referee also result n dfferent actons beng optmal. To llustrate ths noton, consder the Hawk-Dove game n Fgure. Suppose there are three members each of player s and player s populatons, so that N = { a, b, c} and N = { a, b, c}. The nteractons between the two populatons are represented by the matrx n Fgure. It s easy to verfy that the local nteractons wth the assgnment of strateges ndcated n parentheses n Fgure descrbes an equlbrum wth fxed nteractons. L R T 4,4,5 B 5, 0,0 Fgure : The Hawk-Dove game. Player a (L) Player b (L) Player c (R) Player a (T) /6 0 /6 Player b (T) 0 /6 /6 Player c (B) /6 /6 0 Fgure : A dstrbuton of meetngs. We now compare the noton of an equlbrum wth fxed actvtes wth correlated equlbrum. The defnton of correlated equlbrum we use s motvated by the followng nterpretaton. 6 A referee randomly determnes an acton profle ( s, s ) accordng to some dstrbuton ξ, and then prvately recommends the acton s k to player k. The dstrbuton ξ s a correlated equlbrum f t s a best reply for each player to follow the recommendaton: Defnton. A correlated equlbrum s a probablty dstrbuton ξ on S S, such that, for =,, k =,, f ξ( s ) > 0, where ξ( s ) s the probablty that the referee recommends acton s to player, then π ( s, s ) ξ( s s ) π ($ s, s ) ξ( s s ) s$ S. k k k k (3) sk S k sk S k 6 There are several equvalent defntons of correlated equlbrum. An alternatve nvolves specfyng an nformaton structure for the players and acton choces as functons of sgnals receved. The equlbrum condton s that prescrbed choces are optmal gven belefs condtonal on the players nformaton (.e., sgnals). 6

7 The dstrbuton n Fgure 3 s a correlated equlbrum of the Hawk-Dove game n Fgure. L R T /3 /3 B /3 0 Fgure 3: A correlated equlbrum for the Hawk-Dove game. A correlated equlbrum allows a player to contemplate changng hs strategy (and requres such a change to be suboptmal), but does not allow the player to alter the nformaton that s conveyed by the recommendaton receved by the referee. Ths s mportant, because dfferent recommendatons may gve dfferent equlbrum payoffs. 7 The counterpart of ths n the model wth fxed nteractons s that the player can choose a strategy, but cannot affect the mx of opponents wth whom he plays the game. Ths s agan mportant, as the player may well prefer some dfferent mx of opponents. Let strateges be gven by ( s, s ) and the pattern of nteractons by µ. Then the probablty that the strategy par ( *, * * * s s ) s played n a meetng between two agents s denoted ξ ( s, s, µ ) ( s, s ) and s gven by: Defnton 3. The dstrbuton over strategy pars generated by ( s, s ; µ ) s * * ξ ( s, s ; µ ) ( s, s ) = µ (, j). (4) * * { : s ( ) = s } { j: s ( j) = s } It s straghtforward to support some outcomes that are correlated but not Nash equlbra as equlbra wth fxed nteractons: Suppose each populaton has two players (or two groups of players), called α and α' n populaton and β and β' n populaton. Let the matchng be such that players α and β meet and players α' and β' meet. Moreover, suppose the game G s a battle of the sexes. One equlbrum n fxed actvtes s then for agents α and β to play one of the pure strategy equlbra of the game and for agents α' and β' to play the other. Ths equlbrum concdes wth a correlated, non-nash equlbrum of the game G. Moreover, the correlated equlbrum of Fgure 3 s generated by the equlbrum wth fxed nteractons n Fgure. Malath, Samuelson, and Shaked (997) show that any correlated equlbrum can be supported as an equlbrum wth fxed actvtes: Theorem. (.) If a trple ( s, s ; µ ) s an equlbrum wth fxed nteractons, then ξ ( s, s ; µ ) s a correlated equlbrum. (.) If ξ s a correlated equlbrum, then there exsts an equlbrum wth fxed nteractons ( s, s ; µ ) such that ξ = ξ. ( s, s ; µ ) What s the ntuton behnd ths result? Begnnng wth an equlbrum wth fxed nteractons, we construct a correlated equlbrum by recommendng the pure strategy combnaton ( s, s ) wth the same probablty that the equlbrum wth fxed nteractons produces a match n whch ( s, s ) s played. Now 7 Dfferent recommendatons may correspond to dfferent condtonal dstrbutons of the opponent s recommendatons and hence to dfferent behavor. 7

8 suppose player receves a recommendaton to play strategy s *. The dstrbuton over S, descrbng the opponent's strategy condtonal on recevng a recommendaton to play s *, s a weghted average of the dstrbutons over opponent strateges faced by all of the populaton- agents n the equlbrum-wth-fxednteractons who play s *. But f s * s a best reply to the dstrbuton over opponent strateges facng each of these agents, then t s a best reply to the weghted average of these dstrbutons. Hence, t s a best reply to play s * when t s recommended, ensurng that we have a correlated equlbrum. The converse s demonstrated by notng that, gven a correlated equlbrum, we can smply assgn each pure strategy to a sngle player and then construct nteractons between these players so that the probabltes wth whch any two players meet matches the probablty wth whch the correlated equlbrum pars the strateges played by these players. Ths constructon potentally leaves large numbers of players wth no possblty for meetng others and playng the game. It s straghtforward to brng these players nto the game by replacng the ndvdual players n our constructon wth groups of players. In some cases, the pattern of nteractons may be fxed by restrctons nherent n the envronment. For example, populaton may be buyers and populaton sellers of a durable good. Populaton ncludes dealers who buy for later resale as well as consumers who buy for prvate use. Populaton also ncludes dealers and prvate consumers. Examples are the markets for antques, works of art, used cars, or fnancal assets. Dealers buy from and sell to both consumers and dealers, but consumers typcally nteract only wth dealers, yeldng a local nteracton pattern. In other cases, the nteracton pattern may arse endogenously out of the actons taken by agents. We turn to ths n the next secton. 3. Evoluton The equvalence between equlbra wth fxed actvtes and correlated equlbra depends upon the assumpton that the pattern of nteractons s ndeed fxed, so that agents cannot choose or affect the dstrbuton of ther opponents. Ths restrcton s mportant, because some opponents may provde hgher payoffs than others, so that swtchng between opponents would occur f t were possble. We now expand the model to allow agents to affect ther pattern of nteractons. We are nterested n two types of questons. Frst, what patterns of nteractons evolve? Second, we know from Theorem that correlated equlbra are lkely canddates for rest ponts of the evolutonary process. Whch correlated equlbra wll be selected? We now fnd t convenent to assume that the players are drawn from nfnte populatons, so that there s an nfnte populaton of row players (populaton ) and a populaton of column players (populaton ), each of whch has measure. We allow populaton to consst of J dstnct subpopulatons or groups. Smlarly, populaton conssts of J dstnct subpopulatons. We thnk of these groups as beng observable labels that are rrelevant to the payoffs of the game and are exogenously attached to players. For example, the groups may be characterzed by ethncty or culture. At the wnter meetngs, these groups may be felds of specalzaton among economsts or may be departmental afflatons. The role played n the analyss by these groups wll depend upon the detals of the process by whch nteractons evolve. We assume that nteractons are determned by some features, characterstcs, or choces of the players, whch we refer to as actvtes. At the wnter meetngs, these actvtes may nclude attendng partcular paper sessons or cocktal partes. The fnte set of actvtes for group j n populaton ( =, ) s denoted by A j, and the set of avalable actvtes that could be chosen by some member of populaton s A j j A. To smplfy notaton, we assume that the actvtes of dfferent groups have dfferent J 8

9 j j' names, so that A A = for all j j' and =,. 8 Ths ncludes the case n whch dfferent groups can choose the same actvty, snce we can smply gve these actvtes dfferent names. For any actvty α A and acton s S, the proporton of agents of populaton wth actvty α and acton s s denoted qα s. A specfcaton of qα s for each actvty, acton, and populaton comprses a state of the system. We denote a state by θ and the state space by Θ. It s also convenent to defne Qα qα s, s S so that Qα s the proporton of populaton wth actvty α. The actvtes of players affect the denttes of ther opponents. In partcular, nteractons between players are descrbed by a functon ζ: Θ A A [ 0, ]. We nterpret ζ ( θ, α, α ) as the probablty that, gven state θ, a match occurs and s between agents wth actvtes α and α. Hence, for all θ, ( α, α ) A A ζ( θ, α, α ). We assume that the matchng confguraton ζ depends only on the proporton of agents wth each actvty, and not on ther actons. 9 Fnally, we assume that agents choosng a gven actvty are randomly selected to partcpate n matches, so that the probablty that an agent of actvty α s matched s ζ( θ, α, αk ) / α. 0 α k Ak Q We vew ζ ( θ, α, α ) as a technologcal phenomenon, determned by the socety or envronment n whch the agents nteract. The functon ζ ( θ, α, α ) s thus fxed. However, agents can potentally alter the opponents wth whom they nteract by alterng ther choce of actvty, and the pattern of nteractons may evolve as agents choces of actvtes evolve. The evolutonary process by whch agents change both actons and actvtes s descrbed by the tme path θ( t, θ0 ) whch gves the state at tme t f the state at tme zero s θ 0. Tme s contnuous and the path s a dfferentable functon of both tme and ntal state. We often wrte θ( t, θ0 ) as θ( t ). We assume that acton/actvty pars that currently earn hgher payoffs grow proportonately faster than those that earn lower payoffs. Let π be the expected payoff to an agent who chooses acton s and α s actvty α (we suppress the dependence of ths payoff on the state). More specfcally, we assume that θ( t, θ0 ) s monotonc, by whch we mean: dqα θ θ s ( ( t)) dqα$ s$ ( ( t)) πα θ πα θ s ( ( t)) > ( = ) $ s$ ( ( t)) > ( = ) (5) dt q dt q j for all α, α$ A, s, s$ S, j J, =,. The best-known dynamc satsfyng ths condton s the replcator dynamcs. αs α$ s$ 8 We now use the ndex and k for populatons and j for groups. 9 That s, f Q ( θ) = Q ( θ') and Q ( θ) = Q ( θ'), then ζ ( θ, α, α ) = ζ( θ', α, α ), for all α α α α ( α, α ) A A. 0 Hence, for k we must have ζ( θ, α, α ) α A k Qα k k. j Notce that α and α$ are both taken to be elements of A n (5). Hence, we are explctly restrctng each group of agents to playng only actvtes that are avalable to that group. 9

10 There are several possble nterpretatons of these dynamcs. It may be that ndvdual agents play the game only once, wth new agents constantly replacng old ones, but wth new agents choosng ther strateges on the bass of the payoffs n the prevous generaton. Ths may be especally approprate f the matches are taken to be choces of mates. Alternatvely, t may be that agents play the game repeatedly, adjustng ther payoffs over tme n response to ther experence. Ths may be the approprate model f the matchngs represent socal nteractons. For more dscusson of the varous justfcatons of ths type of dynamc analyss, see Malath (99, 997), Samuelson (997), and Selten (99). We defne an evolutonary rest pont to be a state θ wth dθ( t) / dt = 0. Ths defnton leads mmedately to a straghtforward but mportant observaton concernng evolutonary rest ponts: Remark. If all agents n a populaton have access to the same set of actvtes, then n an evolutonary rest pont, all agents (of the same populaton) must receve the same payoff (otherwse at least one agent would change actvty or acton). If actvtes are fxed, then every equlbrum wth fxed actvtes s an evolutonary rest pont. The converse need not hold. Every pure state, meanng every state n whch all of the agents n a gven populaton play the same strategy, s an evolutonary rest pont for the replcator dynamcs for example, but not all of these wll be equlbra wth fxed actvtes. The dffculty s that there are pure states n whch some agents are not playng best reples (because not all pure strategy combnatons are Nash equlbra of the underlyng game). These states are not equlbra wth fxed actvtes, but they are statonary ponts and hence evolutonary rest ponts under the replcator dynamcs. In partcular, the replcator dynamcs nduces no movement toward best reples as long as these best reples are currently played by a zero proporton of the populaton. One suspects that these evolutonary rest ponts wll have poor stablty propertes, wth a small perturbaton n the drecton of a best reply promptng dynamcs that lead away from the evolutonary rest ponts. We accordngly requre that rest ponts be asymptotcally stable. An asymptotcally stable state s a state wth the property that f the process starts nearby, then t stays nearby (so that the state s Lapunov stable) and the process converges to the state n queston. There may be many states that are stable, meanng smply that f the system starts nearby, then t stays nearby (though t may not converge to the state n queston). Asymptotcally stable states are unlkely to exst because the multtude of actvtes allows ample opportunty for the system to drft between states. We therefore consder a set-valued noton, whch s that a set of states, each of whch s ndvdually Lapunov stable, be collectvely asymptotcally stable: Defnton 4. The set Θ Θ s asymptotcally stable f Θ s closed and connected, and f (4.) for every θ Θ, dθ / dt = 0 ; (4.) for every θ Θ and every open set V contanng θ, there exsts an open set U wth θ U such that θ0 U θ( t, θ0 ) V t ; and (4.3) there exsts an open set U wth Θ U such that θ U θ( t, θ ) Θ as t. 0 0 A stable set s thus a collecton of states wth the property that each of the states s an evolutonary equlbrum (4.), the dynamcs surroundng each state n the set cannot lead the system too far away from that state (4.), the dynamcs take states near the set to the set (4.3), and ponts wthn the set can be vewed as connected va a drft process (the latter s the connectedness assumpton). It s the possblty of drft to move the system along a connected set of states, wthout creatng any dynamc forces, that forces us to use a set-valued rather than sngleton valued stablty noton. A number of papers have recently used An alternatve formulaton would be to replace (4.) wth a mnmalty requrement. 0

11 smlar stablty condtons, ncludng Km and Sobel (99), Matsu (99), Nöldeke and Samuelson (993), Sobel (993), and Swnkels (99a, 99b, 993) Endogenous Interactons: Paper Sessons 4.. Interactons We now examne a model wth endogenous nteractons. Our frst model of nteractons s based on the dea that agents may have preferences over the groups from whch ther opponents are drawn, promptng us to refer to ths as the preferences model. These preferences are nduced by the possblty that agents n dfferent groups may choose dfferent actons. There are many ways that a person s actons may affect the set of opponents wth whom the person s lkely to nteract. We examne a partcularly smple model of such actvtes: Each actvty for an agent desgnates one group from the opposng populaton wth whom the agent seeks to nteract. Ths choce wll make t more lkely that one's opponents are drawn from the preferred group. However, each agent belongs to a group that opponents may be seekng, and agent cannot be assured of avodng meetngs wth opponents who are seekng s group, even f prefers to avod such meetngs. Contnung wth our analogy, we thnk of dscussons held at paper sessons at the wnter meetngs. The groups consst of those workng n the varous felds of economcs, such as game theory, macroeconomcs, labor economcs, and so on. Actvtes consst of choces of paper sessons. Hence, a game theorst may prefer to talk wth other game theorsts. Ths objectve can be advanced by attendng and askng questons at game theory sessons. However, the game theorst must present a paper at hs own sesson, and cannot preclude the possblty of questons from macroeconomsts at that sesson. To model ths type of nteracton, we assocate wth each group of agents a dstnct locaton. We wll often speak of these as f they are physcal locatons, though other nterpretatons are possble. An actvty conssts of an attempt to vst a locaton n order to meet members of the group assocated wth that locaton. However, there s a postve probablty that any agent wll be matched at hs home locaton wth agents from the other populaton who have chosen to vst that locaton. More formally, each agent n group j of populaton has avalable J actvtes (correspondng to the locatons or sessons of the other populaton). We can then denote an actvty for populaton agents by α jl, where α jl represents an attempt by a member of group j to be matched wth a member of group l. Smlarly, each member of group l n populaton has avalable J actvtes, denoted β lj, where β lj represents an attempt by a member of group l n populaton to meet an opponent from group j n populaton. We let J j denote the sze of the j th group n populaton and J l the sze of the l th group n populaton. We assume that α jl causes an agent from group j n populaton to be assgned to locaton j n wth probablty /, and to locaton l wth probablty /. The agent then nteracts wth the agents from the other populaton who are also assgned to the locaton to whch he has been assgned. Ths specfcaton s desgned to capture the fact that agents can pursue ther preferences for matchng partners, but cannot completely escape potental matches who are seekng them. Hence, the rch may prefer to meet the rch, but cannot completely escape any poor tryng to meet them. The probablty / that a populaton agent from group j s assgned to locaton j s then nterpreted as statng that wth probablty /, the agent s matched as a result of a populaton agent seekng a match wth group j. The / probablty of 3 Glboa and Matsu (99) consder a stablty concept that shares a smlar sprt of beng set-valued, but allows movements between members of the set to arse out of forces other than genetc drft.

12 beng assgned to a populaton group represents the probablty of beng matched as a result of one s own efforts to meet a certan opponent. 4 Notce that we force agents to seek a match wth some group from the opposng populaton,.e., we do not allow players the opton of endeavorng to not play the game at all. In some cases, ths restrcton may be undesrable from the player's pont of vew, because every possble match may yeld a lower payoff than not playng. Our opnon s that f players are to have the choce of hdng from the world and not playng the game, then ths should be modeled as one of the choces n the game. We return to the mportance of the payoff of not playng the game below. Gven a partcular pattern of actvtes, each locaton has a collecton of populaton agents, of sze H, and of populaton agents, of sze H. The probablty that an agent from populaton s matched s gven by the functon ρ ( H, H ), =,. One mportant example s the proportonal matchng rule, whch has the property that all agents on the short sde of the market are matched: ρ H H = Hk H (, ) mn{, / }. (6) Note that the same matchng functon s used at all locatons, wth the matchng probabltes at a locaton determned only by the relatve szes of the collectons of populaton agents and populaton agents at that locaton. We assume that matchng s anonymous wthn groups. Hence, an agent n populaton can attempt to meet a member of group l n populaton, but has no control over whch member of that group he wll meet. Hence, the dstrbuton of the strateges of the opponents he meets s gven by the dstrbuton of opponents strateges n the group to whch he s assgned. Ths reflects our assumpton that groups represent observable labels, so that agents can readly seek certan groups from the opposng populaton, but that agents cannot dstngush between dfferent members of a group and cannot observe ther opponents strateges before play. Ths s mportant because the members of that group may play dfferent actons. The formulas for the matchng probabltes are n Appendx A Heterogeneous Outcomes In ths secton, we dscuss three examples showng that gven the matchng technology descrbed n the prevous secton, asymptotcally stable states can gve heterogeneous outcomes. Example Consder the followng game: L R T 4,4, B,, Let the matchng rule be proportonal. Each populaton s dvded nto two groups of equal szes, denoted groups a and b. Suppose group a agents play T (n populaton ) and L (n populaton ), and they 4 The / s are motvated by the equal populaton szes. A number of alternatves readly suggest themselves. For example, agents may be able to dentfy a number of groups from the opposng populaton wth whom they would lke to play the game, possbly wth weghts. Agents mght also be able to choose the mx of probabltes between beng assgned to one's own locaton and beng assgned to a locaton n the other populaton. A hgher probablty of beng assgned to the other populaton would be nterpreted as an ncreased ablty to avod matches wth unsought partners.

13 seek each other. Suppose group b agents play B (n populaton ) and R (n populaton ) and they seek each other. 5 We can represent ths schematcally as: ( T) ( L) a ( B) ( R) b As a result, half of the agents earn a payoff of 4 (the rch, or group- a agents from both populatons), whle half earn a payoff of (the poor, or group-b agents from both populatons). Rch agents meet other rch agents (and play the good equlbrum ths s why they are rch ), whle poor agents meet other poor agents (who, n turn, play the bad equlbrum). Ths state, consdered as a sngleton, s asymptotcally stable. Ž How can the equlbrum n Example be stable when poor agents receve an nferor payoff? Gven that poor agents are most lkely to meet other poor agents, t s n ther best nterests to contnue to play the bad equlbrum. But why do the poor agents not smply seek rch agents wth whch to play the good equlbrum of the game? Equvalently, why sn t the movement n the drecton of some group b agents n populaton choosng α ba T rather than α bb B destablzng? The rch agents are ndfferent between playng aganst group a agents who choose T and group b agents who choose T, and hence are happy to meet the poor. The dffculty, however, s that the rch are not seekng the poor, and other poor are seekng the poor. No matter how much a poor agent desres to play the game wth rch agents, he cannot escape the fact that nearly half of hs matches are gong to be wth poor agents who play the bad equlbrum, and t s then not a best response to swtch to seekng rch agents and playng the good equlbrum. In partcular, the payoff to a poor agent n populaton from playng T and seekng rch agents s 4 + ( ), whch falls short of the payoff of garnered from playng B and seekng other poor agents. The key here s that agent can be dentfed as belongng to group b. Other agents expect group b agents to play B. Hence, populaton agents who play R attempt to meet the agent, and cannot avod such meetngs. These meetngs occur often enough that s best response s to surrender to playng B and recevng the bad-equlbrum payoff. Ths example also llustrates the mportance of the assumpton that agents cannot make ther strategy depend on ther opponent's group. Whle a strong assumpton, there are stuatons where code swtchng s dffcult f not mpossble. Anderson (994), for example, descrbes the code of the street, behavor n some nner cty neghborhoods of the U.S. that many resdents fnd dffcult to swtch from when nteractng wth nonresdents of these neghborhoods. Ths example provdes some nsght nto the questons of equalty and ncome dstrbuton. Attenton has recently been devoted to the queston of how an economy consstng of agents who are ex ante dentcal can gve rse to persstent ncome nequalty. 6 Exstng theores have shown that as long as economc outcomes are stochastc, dfferent outcome realzatons can lead ntally dentcal agents to dfferent ncomes. But why do these dfferences persst rather than beng elmnated by the propensty of ndependent random processes to regress to the mean? The conventonal explanaton nvokes externaltes to create a lnk between current ncome and future prospects (e.g., Durlauf (99)). For example, low ncomes can lead to low nvestment n chldren's educaton whch leads to low future ncomes. If there s also a lnk between communty ncome levels and the effectveness of educaton (perhaps because dfferent a b 5 Formally, q = q = q = q = /. αaat βaa L αbbb βbbr 6 See Durlauf (99) for a dscusson and references. 3

14 ncome levels lead to dfferent academc habts and norms of achevement), then ths cycle can be renforced and can create an absorbng poverty trap. Our example suggests that a poverty trap can arse wthout externaltes (or, dependng upon one's nterpretaton, from externaltes n the matchng process). Poor agents are poor, and wll reman poor, smply because most of ther nteractons are wth other poor people who happen to have coordnated on a low-payoff equlbrum; and because one's wealth or socal status s suffcently correlated wth observable characterstcs that one cannot smply decde to hereafter be taken for a rch person and nteract only wth other rch people. Ths example llustrates the argument that a poverty program servng only a few of the poor and leavng them n ther current envronment gnores potentally mportant factors. In partcular, to the extent that our example s capturng a crucal feature, t suggests that breakng the cycle of poverty wll requre ether a massve nterventon to swtch the poor groups to the good equlbrum or wll requre removng poor agents from ther envronments and puttng them nto envronments where they wll be sought by (as well as beng able to seek) rch agents. We need only attach labels such as black and whte to the groups n Example to obtan an outcome n whch one group appears to be the vctm of dscrmnaton. Statstcal theores of dscrmnaton have produced models wth equlbra n whch blacks fare less well then whtes even though the latter entertan no antpathy for the former. 7 Example suggests yet another such theory, wth the statstcal dscrmnaton takng the form of whte agents not seekng nteractons wth blacks because the latter play the bad equlbra, whle blacks play the bad equlbra because whtes do not seek nteractons wth them. Once agan, the key s that certan behavors have come to be assocated wth the labels black and whte. For a black agent, swtchng behavor does not alter the fact that others stll observe smply that s black, and play aganst as they would aganst other blacks. One s frst mpresson s that affrmatve acton polces, by vtatng the statstcal profle of the oppressed group, should be able to elmnate such dscrmnaton. Coate and Loury (993) show that the effectveness of affrmatve actons n such stuatons, and especally the queston of whether a temporary affrmatve acton polcy can have permanent effects, depends crtcally upon how the polcy affects whtes' belefs about blacks. They fnd that n some cases only permanent affrmatve acton polces wll be effectve. In our case, the affrmatve acton would have to nvolve ensurng blacks suffcent access to nteractons wth whtes to allow the good equlbrum to become a best reply for the former. Once blacks were nduced to play the good equlbrum, the polcy would be unnecessary. Example Consder the Hawk-Dove game of Fgure. Agan we suppose that the each populatons s dvded nto two groups, denoted groups a and b. Consder the state n whch group a agents play T (populaton ) and L (populaton ) and seek each other, whle group b agents play B (populaton ) and R (populaton ) and seek group a from the opposng populaton. Ths s descrbed schematcally by: ( T) ( L) a ( R) ( B) b Assume, however, that group a n each populaton s of sze x whle group b s of sze x. 8 It s straghtforward to verfy that f / 4 < x < / 3, then each agent's actvty-acton choce s a strct best a b 7 See Coate and Loury (993) for a dscusson and references. 8 Formally, q = q = x, and q = q = x. αaat βaa L αbab βbar 4

15 reply, and so the state s asymptotcally stable. 9 Condtonal on beng matched, a group a agent n populaton s matched wth a populaton agent choosng L wth probablty x / ( + x). Also, condtonal on beng matched, a group b agent n populaton s matched wth a populaton agent choosng L wth probablty. Thus, for x close to (but less than) /3, the condtonal dstrbutons are close to those n the correlated dstrbuton that places equal probabltes on the three profles TL, TR, and BL. Ž In contrast to the fxed actvtes case of Example, there s a postve probablty that an agent wll not be matched n the equlbrum descrbed n Example 3. For example, group- b agents are not matched when the matchng process allocates them to ther own locaton (whch occurs wth probablty /). In addton, the entre populaton s attemptng to match wth group- a agents of the opposng populaton, but only a fracton x can succeed at that locaton. Thus, group b agents n each populaton are not matched wth a probablty of / + ( x) / = x /. An analogous calculaton shows that group a agents are not matched wth probablty ( x ) /. As long as nteractons are endogenous, so that agents have some ablty to affect the dentty of ther opponents, then we must take serously the possblty that some agents are not matched or, equvalently, that some agents are matched more often than others. The desred matchng plans of all agents may not be compatble or feasble, and the result may be that some agents are frustrated n ther efforts to meet others. In calculatng the payoff to an actvty, agents must then nclude the possblty that they are not matched, wth ts attendant payoff. As a result, the payoffs n a game do not provde a complete descrpton of the strategc stuaton untl the payoff to not beng matched s specfed. Whle t s temptng to set ths payoff to zero, ths s not the only possblty. The followng example shows how the magntude of ths payoff can affect the outcome of the evolutonary process: Example 3 Suppose that we alter the payoff to not beng matched n the Hawk-Dove game of Fgure to be rather than 0. We can then normalze the payoffs n the new game by subtractng from each payoff, ncludng the payoff of not beng matched, to obtan a game n whch the payoff to not beng matched s zero and the other payoffs are: 0 L R T 3,3 0,4 B 4,0, As n Example, suppose populaton s dvded nto two groups, as s populaton. Let x denote the sze of the frst group (assume that t s the same sze for each populaton). Agan suppose all groups wsh to meet the opposng group a s; and the row group a s choose T and group b s choose B ; the column group a s choose L and the group b s choose R. 9 If x / 3, then α aa B s a best reply for group a agents and f x / 4, then α ab T s a best reply for group a agents of populaton (and smlarly for populaton ). Moreover, f x / 5, α bb T s a best reply for group b agents n populaton (and smlarly for populaton ). 0 Ths example s thus equvalent to the game n Fgure wth the payoff to not beng matched set at. 5

16 Is ths an asymptotcally stable state when / 4 < x < / 3, as t was n Example? Consder a group a player from populaton. Choosng α aa T yelds payoffs: The par α aa B yelds: ( x π ( T, L) + ( x) π ( T, R) ) + ( x π ( T, L) ) = ( x 3+ ( x) 0) + ( x 3) = 3x. ( ) ( ) 9. ( x 4 + x ) + ( x 4) = ( x ) The par α ab T yelds (note that the player s now never ratoned): Fnally, α ab B yelds: 0 3 x + ( x) + = x. ( 3 0) ( ) ( ) ( ) 5. ( x 4 + x ) + ( ) = ( x ) The par α aa T yelds a strctly hgher payoff than the other choces f x < / 3. It s also straghtforward to verfy that the group b s strctly prefer α ba B to the other choces (and smlarly for populaton ) f x < / 3. We thus have an equlbrum that resembles that of Example, but ths equlbrum s an asymptotcally stable state as long as x < / 3 (rather than also requrng / 4 < x, as n Example ). The dfference s that the value of not beng matched s hgher here than n Example. Ths makes t more attractve to endure the ratonng assocated wth α aa T n quest of the relatvely hgh payoff π ( T, L) (rather than avodng ratonng by choosng α ab T and settlng for the relatvely low payoff π ( T, R ) ). As a result, the constrant / 4 < x, needed n Example 3 to ensure that agents choosng α aa T are not too severely ratoned, s not needed here. Ž Snce wth probablty /, the player s at hs own locaton, s not ratoned and faces the dstrbuton x of group a and x of group b ; and wth probablty /, the player s at the opponent group a s locaton and only matches wth probablty x. Notce that the payoff to not playng the game does not affect the relatve payoffs to strateges α aa T and α aa B, snce these nvolved dentcal ratonng frequences, so that the equlbra n Examples and 3 share the constrant x < / 3, needed to ensure that the payoff of α aa T exceeds that of α aa B. 6

17 5. Endogenous Interactons: Cocktal Partes and Hotel Lobbes 5.. Interactons In the preferences model of the prevous secton, agents could express preferences over desred matches but could not avod undesred matches. We now examne a model n whch agents can sometmes be assured of avodng certan other agents. We wll fnd t convenent to agan descrbe the nteracton technology n terms of locatons, and wll refer to ths as the locaton model. In our wnter meetngs parable, the locatons are cocktal partes and hotel lobbes. Agents frequent a locaton n an attempt to meet other agents, and nteract wth others who appear at that locaton. By fndng a sutably secluded corner or hotel sute, agents can always ensure that they are not found by those they prefer not to meet. However, ths ablty to avod others carres wth t a lessened ablty to ensure any meetng. There s now no guarantee that any agents of the other populaton wll be at any partcular locaton. One need not attend even the cocktal party of one's own department f one prefers not to, unlke paper sessons. More formally, suppose there are a fnte number of locatons, denoted λ { l, K, l L }. In ts smplest form, a choce of an actvty conssts of a choce of locaton. In general, however, ths choce of locaton may be random, wth an actvty nducng a probablty dstrbuton over locatons rather than correspondng to a sngle locaton. For example, an actvty mght be a decson to frequent one of the conventon hotels on a certan day. Once there, ths may nduce a dstrbuton concernng the lkelhood that one s to be found n varous locatons, ncludng cocktal partes, book dsplays, bars and the hotel lobby. At each locaton, other agents are lkely to be encountered. Fnally, notce that a locaton need not be a physcal locaton n the usual sense of the word, though the latter provdes a convenent nterpretaton that we shall adopt when descrbng the model. A formal descrpton s contaned n Appendx A Effcent Outcomes We now derve condtons under whch the outcome n the locaton model s not only homogeneous, but also satsfes an effcency condton. The suffcent condtons for effcency come n three parts. Frst, we need the matchng technology to be unrestrcted, meanng that all agents have access to the same set of actvtes. Second, we need t to be possble that an solated group of agents can form. Thrd, we requre that nether large nor small groups are penalzed n terms of matchng probabltes, whch holds f the matchng technology exhbts constant returns to scale (see Appendx A. for the precse defnton). 3 Isolated groups of agents wll be able to form f the matchng technology contans pure actvtes, where: Defnton 5. An actvty α s pure f there exsts a locaton l wth the property that the actvty guarantees that the locaton s reached. We then have: Theorem 5. Let all agents (of both populatons) have access to the same set of actvtes n the locaton model. Suppose that game G has a Nash equlbrum whose payoffs strctly domnate the payoffs of all other correlated equlbra. Suppose that the matchng process exhbts constant returns. Let there be at 3 We assume throughout that equlbra exst that gve all agents hgher payoffs than not beng matched, so that effcency requres that the game be played. 7