Coordination failure in repeated games with almost-public monitoring

Transcription

1 Theoretcal Economcs 1 (2006), / Coordnaton falure n repeated games wth almost-publc montorng GEORGE J. MAILATH Department of Economcs, Unversty of Pennsylvana STEPHEN MORRIS Department of Economcs, Prnceton Unversty Some prvate-montorng games, that s, games wth no publc hstores, have hstores that are almost publc. These games are the natural result of perturbng publc-montorng games towards prvate montorng. We explore the extent to whch t s possble to coordnate contnuaton play n such games. It s always possble to coordnate contnuaton play by requrng behavor to have bounded recall (.e., there s a bound L such that n any perod, the last L sgnals are suffcent to determne behavor). We show that, n games wth general almost-publc prvate montorng, ths s essentally the only behavor that can coordnate contnuaton play. KEYWORDS. Repeated games, prvate montorng, almost-publc montorng, coordnaton, bounded recall. JEL CLASSIFICATION. C72, C73, D INTRODUCTION Intertemporal ncentves often allow players to acheve payoffs that are nconsstent wth myopc ncentves. For repeated games wth publc hstores, the constructon of sequentally ratonal equlbra wth nontrval ntertemporal ncentves s straghtforward. Snce contnuaton play n a publc strategy profle s a functon of publc hstores only, the requrement that contnuaton play nduced by any publc hstory consttute a Nash equlbrum of the orgnal game s both the natural noton of sequental ratonalty and relatvely easy to check (Abreu et al. 1990). These perfect publc equlbra (or PPE) use publc hstores to coordnate contnuaton play. George J. Malath: gmalath@econ.upenn.edu Stephen Morrs: smorrs@prnceton.edu Earler versons of ths materal have appeared under the ttle Fnte State Strateges and Coordnaton n Repeated Games wth Prvate Montorng. Some parts of Sectons 4 and 5 frst appeared n Repeated Games wth Imperfect Prvate Montorng: Notes on a Coordnaton Perspectve. That paper s subsumed by Malath and Morrs (2002) and ths paper. We thank Andrew Postlewate for helpful conversatons and an anonymous referee and especally the edtor, Jeffrey Ely, for valuable suggestons. Malath s grateful for support from the Natonal Scence Foundaton under grants #SES and #SES Morrs s grateful for support from the John Smon Guggenhem Foundaton, the Center for Advanced Studes n the Behavoral Scences, and Natonal Scence Foundaton Grant #SES Copyrght c 2006 George J. Malath and Stephen Morrs. Lcensed under the Creatve Commons Attrbuton-NonCommercal Lcense 2.5. Avalable at

2 312 Malath and Morrs Theoretcal Economcs 1 (2006) Whle games wth prvate montorng (where actons and sgnals are prvate) have no publc hstores to coordnate contnuaton play, some do have hstores that are almost publc. We explore the extent to whch perfect publc equlbrum strateges contnue to be equlbra when hstores are only almost publc. We show that t s always possble to coordnate contnuaton play by requrng behavor to have bounded recall (.e., there s a bound L such that n any perod, the last L sgnals are suffcent to determne behavor). 1 But we also show a partal converse: n games wth general almost-publc prvate montorng, ths s the only behavor that can coordnate contnuaton play under an apparently mld restrcton on strateges. To make ths precse, we must descrbe general but almost-publc prvate montorng and characterze the restrcton on strateges When s a general prvate-montorng technology close to some publc montorng technology? To be close, there must be a sgnalng functon for each player that assgns to each prvate sgnal ether some value of the publc sgnal or a dummy sgnal (wth the nterpretaton that that prvate sgnal cannot be related to any publc sgnal). Usng these sgnalng functons (one for each player), the prvate montorng s close to the publc montorng f the probablty of prvate sgnals mappng to a gven publc sgnal, under the prvate-montorng technology, s close to the probablty of that publc sgnal under the publc montorng (for any gven acton profle). If there exst such sgnalng functons satsfyng ths condton, we say there s almost-publc montorng. If every prvate sgnal s mapped to a publc sgnal, we say the almost-publc-montorng game s strongly close to the publc-montorng game. Usng the sgnalng functons, any strategy profle of the publc-montorng game nduces behavor n strongly-close-by almost-publc-montorng games. Gven a sequence of prvate sgnals for a player, that player s prvate state s determned by the nduced sequence of publc sgnals that are the result of applyng hs sgnalng functon. We show that every strct PPE wth bounded recall nduces equlbrum n every strongly-close-by almost-publc-montorng game; and even f the prvate-montorng games are not strongly close to the publc-montorng game, there s stll a natural sense n whch every strct PPE wth bounded recall nduces equlbrum behavor n every close-by almost-publc-montorng game (Theorem 1). The dea s that wth bounded recall we can always restrct posteror belefs to be suffcently close to the publc montorng by requrng the prvate-montorng technology to be suffcently close to the publc-montorng technology. Ths result generalzes the man result n Malath and Morrs (2002), where the prvate sgnal set was assumed to equal the publc sgnal set. 2 When a strategy profle of the publc-montorng game does not have bounded recall, realzatons of the sgnal n early perods can have long-run mplcatons for behavor. We call profles wth ths property separatng. Whle the propertes of bounded 1 Thus when we refer to strategy profles that coordnate contnuaton play n games wth prvate montorng, we mean strategy profles where players choces are best responses f hstores are suffcently close to beng publc. 2 The extenson s nontrval because the rchness of the prvate sgnals s mportant for the formaton of that player s belefs about the other players prvate states. It turns out that the requrement that the prvatemontorng dstrbuton be close to the publc-montorng dstrbuton places essentally no restrcton on the manner n whch prvate sgnals enter nto the formaton of posteror belefs.

3 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 313 recall and separaton do not exhaust possble behavor, they do appear to cover most behavors of nterest. 3 When the space of prvate sgnals s suffcently rch for some player n the values of posteror-odds ratos (ths s what we mean by general almost publc ), and the profle s separatng, t s possble to manpulate that player s updatng over other players prvate states through an approprate choce of prvate hstory. Ths suggests that t should be possble to choose a prvate hstory wth the property that player s n one prvate state and assgns arbtrarly hgh probablty to all the other players beng n a dfferent common prvate state. A sgnfcant dffculty needs to be addressed n order to make ths argument: The hstory needs to have the property that player s very confdent of the other players state transtons for any gven ntal state. Ths, of course, requres the montorng to be almost-publc. At the same tme, montorng must be suffcently mprecse that player, after an approprate ntal segment of the hstory, assgns postve probablty to the other players beng n a common state dfferent from s prvate state. Ths s the source of the dffculty: Fx a perod t. For any T -length hstory (T > t ), there s an ɛ (decreasng n T ) such that for prvate montorng ɛ-close to the publc montorng, player s suffcently confdent of the perod T prvate states of players j as a functon of ther perod t prvate states (and the hstory). However, ths ɛ puts an upper bound on the pror probablty that player can assgn n perod t to the players j beng n a common state dfferent from s prvate state. Snce the choce of T s decreasng n ths pror (.e., larger T s requred for smaller prors), there s a tenson n the determnaton of T and ɛ. We show, however, that ths tenson can be resolved for separatng profles mplementable usng a fnte number of states. For such profles the hstory can be chosen so that not only do the relevant states cycle, but every other state transts under the cycle to a cyclng state. The cycle allows us to effectvely choose the T above ndependently of the pror, and gves us our man result (Theorem 3): Separatng strct PPE profles of publc-montorng games mplementable usng a fnte number of states do not nduce Nash equlbra n any strongly-close-by games wth rch prvate montorng. Thus, separatng strct PPE of publc-montorng games are not robust to the ntroducton of even a mnmal amount of prvate montorng. Consequently, separatng behavor n prvate-montorng games typcally cannot coordnate contnuaton play (Corollary 1). On the other hand, bounded recall profles are robust to the ntroducton of prvate montorng. The extent to whch bounded recall s a substantve restrcton on the set of payoffs s unknown. 4 Our results do suggest, even for publc-montorng games, that bounded recall profles are partcularly attractve (snce they are robust to the ntroducton of prvate montorng). Moreover, other apparently smple strategy profles are problematc. 3 We provde one example of a non-separatng profle wthout bounded recall n Secton 4 (Example 3). Ths profle s not robust to the ntroducton of prvate montorng. We do not know f there exst nonseparatng profles wthout bounded recall that are robust to prvate montorng. 4 Cole and Kocherlakota (2005) show that for some parameterzatons of the repeated prsoners dlemma, the restrcton to strongly symmetrc bounded recall PPE results n a dramatc collapse of the set of equlbrum payoffs.

4 314 Malath and Morrs Theoretcal Economcs 1 (2006) We have analyzed the robustness of fxed strategy profles to prvate montorng. Our results do not say anythng about the set of all equlbrum payoffs n prvate-montorng games. 5 Whle ths classc queston s mportant, we beleve that there are at least three reasons why t s nonetheless also nterestng to focus on a fxed strategy profle. Frst, researchers usng repeated game theory to understand economc phenomena are nterested n hypotheszng and testng partcular strategy profles. 6 Second, understandng propertes of partcular strategy profles may turn out to be an mportant step n characterzng the set of all equlbrum payoffs. Fnally, one of our fndngs s that fne detals of strategy profles, such as hstory dependence, that are rrelevant for the classc recursve characterzaton of the PPE payoff set are very mportant for the robustness queston we consder, and such fne detals mght turn out to be sgnfcant for other questons as well. Both our postve and negatve results restrct attenton to strct PPE, and the assumpton s mportant for both knds of results. In such equlbra, players are not ndfferent between alternatve actons and are thus coordnated n ther contnuaton play. Such strategy profles capture basc ntutons about how cooperaton can be sustaned n repeated games by the threat of coordnated devaton to punshment paths; they form the bass of emprcal applcatons of repeated game theory (see the references n footnote 6); and we beleve they are nterestng objects of study. However, as noted n footnote 5, the most permssve results n the prvate-montorng lterature have used strateges wth a sgnfcant amount of randomzaton and ndfference. The results n ths paper do not have anythng to say about the robustness of such strateges. 7 Ths paper ntroduces a useful representaton of fnte state strateges for prvatemontorng games. Each player has a fnte set of prvate states, a transton functon mappng prvate sgnals and states nto new states, and decson rules for the players, specfyng behavor n each state. The transton functon and decson rules defne a Markov process on vectors of prvate states. Ths representaton s suffcent to descrbe behavor under the gven strateges, but s not suffcent to verfy that the strateges are 5 Malath and Samuelson (2006, Chapter 12) ntroduces the man ssues and concepts. See Kandor (2002) for a bref survey of ths lterature, as well as the accompanyng symposum ssue of the Journal of Economc Theory on Repeated Games wth Prvate Montorng. For the repeated prsoners dlemma wth almost-perfect prvate montorng, folk theorems have been proved usng both equlbra wth a coordnaton nterpretaton (for example, Sekguch 1997 and Bhaskar and Obara 2002) and those that are belef-free (for example, Pccone 2002, Ely and Välmäk 2002, and Matsushma 2004), where equlbrum strateges are constructed usng randomzaton to ensure that players are ndfferent between some actons at all hstores. Whle folk theorems cannot be proved usng belef-free strateges for general payoff matrces (Ely et al. 2005), varatons on belef-free strategy profles have been used to prove general folk theorems (Hörner and Olszewsk 2005). 6 See, for example, Axelrod (1984), Ellson (1994), and Gref (2005). 7 Bhaskar and van Damme (2002) and Ely (2002) show that trgger strategy profles, whch are strct PPE of a repeated prsoners dlemma wth mperfect publc publc montorng, can be approxmated n nearby games wth prvate montorng wth a strategy profle wth strct mxng. Ths possblty suggests that allowng non-strct equlbra may greatly assst n establshng robustness results. On the other hand, the equlbra wth mxng requre players to randomze dfferently at dfferent payoff-equvalent hstores, whch s arguably mplausble. Bhaskar (1998) and Bhaskar and van Damme (2002) suggest that such strateges often do not survve extensve form purfcaton perturbatons.

5 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 315 optmal. It s also necessary to know how each player s belefs over the other players prvate states evolve. Ths s at the heart of the queston of whether hstores can coordnate contnuaton play, snce, gven a strategy profle, a player s prvate state determnes that player s contnuaton play. The crux of our analyss concerns how to track the evoluton of belefs over other players prvate states durng the course of play. In ths paper, we use ths representaton to analyze prvate-montorng profles constructed from a PPE. However, the method s more general and we beleve that t may be of more general use. Examples can be found n Malath and Samuelson (2006, Secton 12.4 and Chapter 14), where the method s used to analyze the mxed strategy employed n the classc analyss of Sekguch (1997) and defne belef-free equlbra. Fnally, we note that we have not allowed any communcaton beyond that contaned n the equlbrum strateges. We vew our fndngs as underlnng the mportance of publc communcaton n prvate-montorng games as a mechansm to facltate coordnaton. For some recent work on communcaton n prvate-montorng games, see Compte (1998), Kandor and Matsushma (1998), Fudenberg and Levne (2004), and McLean et al. (2002). 2. GAMES WITH IMPERFECT MONITORING 2.1 Prvate-montorng games The nfntely-repeated game wth prvate montorng s the nfnte repetton of a stage game n whch at the end of each perod, each player learns only the realzed value of a prvate sgnal. There are n players, wth the fnte stage-game acton set for player N {1,...,n} denoted A. At the end of each perod, each player observes a prvate sgnal, denoted ω, drawn from a fnte set Ω. The sgnal vector ω (ω 1,...,ω n ) Ω Ω 1 Ω n occurs wth probablty π(ω a ) when the acton profle a A A s chosen. Player does not receve any nformaton other than ω about the behavor of the other players. All players use the same dscount factor, δ. Snce ω s the only sgnal a player observes about opponents play, we assume (as usual) that player s payoff after the realzaton (ω,a ) depends only on (ω,a ). We denote ths payoff by u (ω,a ). Stage game payoffs are then gven by u (a ) ω u (ω,a )π(ω a ). It s convenent to ndex games by the montorng technology (Ω,π), fxng the set of players and acton sets. A pure strategy for player n the prvate-montorng game s a functon s : A, where t =1 (A Ω ) t 1 s the set of prvate hstores for player. 2.2 Publc-montorng games We turn now to the benchmark publc-montorng game for our games wth prvate montorng. The fnte acton set for player N s agan A. The publc sgnal s denoted y and s drawn from a fnte set Y. The probablty that the sgnal y occurs when the acton profle a A A s chosen s denoted ρ(y a ). We refer to (Y,ρ) as

6 316 Malath and Morrs Theoretcal Economcs 1 (2006) the publc-montorng dstrbuton. Player s payoff after the realzaton (y, a ) s gven by u (y,a ). Stage game payoffs are then gven by u (a ) y u (y,a )ρ(y a ). The nfntely repeated game wth publc montorng s the nfnte repetton of ths stage game n whch at the end of each perod each player learns only the realzed value of the sgnal y. Players do not receve any other nformaton about the behavor of the other players. All players use the same dscount factor, δ. A strategy for player s publc f, n every perod t, the acton t prescrbes depends only on the publc hstory h t Y t 1, and not on s prvate hstory. Henceforth, by the term publc profle, we always mean a strategy profle for the publc-montorng game that s tself publc. A perfect publc equlbrum (PPE) s a profle of publc strateges that, after any publc hstory h t, specfes a Nash equlbrum for the repeated game. Under mperfect full-support publc montorng, every publc hstory arses wth postve probablty, and so every Nash equlbrum n publc strateges s a PPE. Any pure publc strategy profle can be descrbed as an automaton as follows: There s a set of states, W, an ntal state, w 1 W, a transton functon σ : W Y W, and a collecton of decson rules, d : W A. In the frst perod, each player chooses acton a 1 = d (w 1 ). The vector of actons, a 1, then generates a sgnal y 1 accordng to the dstrbuton ρ( a 1 ). In the second perod, each player chooses the acton a 2 = d (w 2 ), where w 2 = σ(w 1,y 1 ), and so on. Snce we can take W to be the set of all hstores of the publc sgnal, t 1 Y t, W s at most countably nfnte. A publc profle s fnte f W s a fnte set. Note that, gven a pure strategy profle (and the assocated automaton), contnuaton play after any hstory s determned by the publc state reached by that hstory. Denote the vector of average dscounted expected values of followng the publc profle (W,w,σ,d ) (so that the ntal state s w ) by φ(w ). Defne a functon g : A W W by g (a ;w ) (1 δ)u (a ) + δ y φ(σ(w,y ))ρ(y a ). We have (from Abreu et al. 1990), that f the profle s an equlbrum, then, for all w W, the acton profle (d 1 (w ),...,d n (w )) d (w ) s a pure strategy equlbrum of the statc game wth strategy spaces A and payoffs g ( ;w ) for each and, moreover, φ(w ) = g (d (w ),w ). Conversely, f (W,w 1,σ,d ) descrbes an equlbrum of the statc game wth payoffs g ( ;w ) for all w W, then the nduced pure strategy profle n the nfntely repeated game wth publc montorng s an equlbrum. 8 A PPE (W,w 1,σ,d ) s strct f, for all w W, d (w ) s a strct Nash equlbrum of the statc game g ( ;w ). 9 A mantaned assumpton throughout our analyss s that publc montorng has full support. ASSUMPTION 1. ρ(y a ) > 0 for all y Y and all a A. 8 We have ntroduced a dstncton between W and the set of contnuaton payoffs for convenence. Any pure strategy equlbrum payoff can be supported by an equlbrum where W I and φ(w ) = w (agan, see Abreu et al. 1990). 9 Equvalently, a PPE s strct f each player strctly prefers hs equlbrum strategy to every other publc strategy. For a large class of publc-montorng games, strctness s wthout loss of generalty, n that a folk theorem holds for strct PPE (Fudenberg et al. 1994, Theorem 6.4 and remark).

7 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 317 We extend the doman of σ from W Y to W t =1 Y t by recursvely defnng σ(w 1,h t ) = σ(σ(w 1,h t 1 ),y t ) for all h t Y t 1, where h t = (h t 1,y t ). DEFINITION 1. An automaton (W,w 1,σ,d ) s mnmal f for every state w W there exsts a sequence of sgnals h l such that w = σ(w 1, h l ) and for every par of states w, w W, there exsts a sequence of sgnals h L such that for some, d (σ(w,h L )) d (σ( w,h L )). The restrcton to mnmal automata s wthout loss of generalty: every profle has a mnmal representng automaton. Moreover, ths automaton s essentally unque. 10 Accordngly, we treat a publc strategy profle and ts mnmal representng automaton nterchangeably. 2.3 Almost-publc montorng We now defne what t means for a prvate-montorng dstrbuton to be close to a publc-montorng dstrbuton. DEFINITION 2. The prvate-montorng dstrbuton (Ω, π) s ɛ-close under f to the publc-montorng dstrbuton (Y,ρ), where f = (f 1,..., f n ) s a vector of sgnalng functons f : Ω Y { }, f 1. for each a A and y Y, π({ω : f (ω ) = y for all } a ) ρ(y a ) ɛ, and 2. for all y Y, ω f 1 (y ), and all a A, f π({ω } a ) > 0, then π({ω : f j (ω j ) = y for all j } (a,ω )) 1 ɛ. The prvate-montorng dstrbuton (Ω,π) s strongly ɛ-close under f to the publcmontorng dstrbuton (Y, ρ) f t s ɛ-close under f and, n addton, all the sgnalng functons map nto Y. A prvate-montorng dstrbuton (Ω, π) s (strongly) ɛ-close to the publc-montorng dstrbuton (Y,ρ) f there exsts a vector of sgnalng functons f such that (Ω,π) s (strongly) ɛ-close under f to (Y, ρ). If the prvate montorng s ɛ-close under f, but not strongly ɛ-close under f, then some prvate sgnals are not assocated wth any publc sgnal: there s a sgnal ω satsfyng f (ω ) =. Such an unnterpretable sgnal may contan no nformaton about the sgnals observed by the other players. 10 Suppose (W,w 1,σ,d ) and (W, w 1, σ, d ) are two mnmal automata representng the same publc strategy profle. Defne a mappng ϕ : W W as follows: Set ϕ(w 1 ) = w 1. For w W \ {w 1 }, let h l be a publc hstory reachng w (.e., w = σ(w 1, h l )), and set ϕ( w ) = σ( w 1, h l ). Snce both automata are mnmal and represent the same profle, ϕ does not depend on the choce of publc hstory reachng w. It s straghtforward to verfy that ϕ s one-to-one and onto. Moreover, σ( w,y ) = ϕ(σ(ϕ 1 ( w ),y ), and d (w ) = d (ϕ(w )).

8 318 Malath and Morrs Theoretcal Economcs 1 (2006) The condton of ɛ-closeness n Defnton 2 can be restated as follows. Recall from Monderer and Samet (1989) that an event s p-evdent f, whenever t s true, everyone assgns probablty at least p to t beng true. The followng lemma s a straghtforward applcaton of the defntons, and so we omt the proof. LEMMA 1. Suppose f : Ω Y { }, = 1,...,n, s a collecton of sgnalng functons. The prvate-montorng dstrbuton (Ω, π) s ɛ-close under f to the publc montorng dstrbuton (Y,ρ) f and only f for each publc sgnal y, the set of prvate sgnal profles {ω : f (ω ) = y for all } s (1 ɛ)-evdent (condtonal on any acton profle) and has probablty wthn ɛ of the probablty of y (condtonal on that acton profle). DEFINITION 3. A prvate-montorng game (u,(ω,π)) s ɛ-close (under f ) to the publcmontorng game ( u,(y,ρ)) f (Ω,π) s ɛ-close under f to (Y,ρ) and u (f (ω ),a ) u (ω,a ) < ɛ for all N, a A, and ω f 1 (Y ). We say also that such a prvate-montorng game has almost-publc montorng. Note that because of our mantaned assumpton that publc-montorng games have full support montorng, a prvate-montorng game that has almost-publc montorng relatve to a fxed ρ does not have almost perfect montorng n the sense usually assumed n the lterature. 11 The ex ante stage payoffs of any almost-publc-montorng game are close to the ex ante stage payoffs of the benchmark publc-montorng game (the proof s n the Appendx). LEMMA 2. For all η > 0, there s ɛ > 0 such that f (u,(ω,π)) s ɛ-close to ( u,(y,ρ)), then u (ω,a )π(ω 1,...,ω n a ) u (y,a )ρ(y a ) < η. ω 1,...,ω n Fx a publc profle (W,w 1,σ,d ) of a full-support publc-montorng game ( u, (Y,ρ)), and, under f, a strongly ɛ-close prvate-montorng game (u,(ω,π)). The publc profle nduces a prvate profle n the prvate-montorng game n a natural way: Player s strategy s descrbed by the automaton (W,w 1,σ,d ), where σ (w,ω ) = σ(w, f (ω )) for all ω Ω and w W. The set of states, ntal state, and decson functon are from the publc profle. The transton functon σ s well-defned, because the sgnalng functons all map nto Y, rather than Y { }. Note that by constructon, each player s strategy s acton-free,.e., t depends only on past sgnals and not on past actons of that player. (See Malath and Samuelson 2006, Chapter 12 for more dscusson of acton-free. ) 11 The order of quantfers s mportant: We can construct almost-perfect almost-publc montorng dstrbutons by consderng full-support publc-montorng dstrbutons arbtrarly close to perfect montorng see Malath and Morrs (2002, Secton 6). y

9 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 319 e 2 n 2 e 1 2,2 1,3 n 1 3, 1 0,0 FIGURE 1. The prsoners dlemma. If player beleves that the other players are followng a strategy nduced by a publc profle, a suffcent statstc of h t for the purposes of evaluatng contnuaton strateges s player s prvate state and s belefs over the other players prvate states,.e., (w t,βt ), where β t (W N 1 ). Wth a slght abuse of notaton, we wrte β (w h t ) for the probablty that player assgns to hs opponents beng n prvate states w at hstory h t. We can recursvely calculate the prvate states of player as w 2 = σ(w 1, f (ω 1 )) = σ (w 1,ω 1 ), w 3 = σ (w 2,ω2 ), and so on. For any prvate hstory ht, we wrte w t = σ (h t ) for the prvate state of the player n perod t. REMARK 1. In prvate-montorng games that are ɛ-close, but not strongly so, a publc profle nduces only that part of the prvate profle determned by hstores of sgnals ω f 1 (Y ), wth the remanng specfcaton of behavor not determned by the publc profle. For an example, see part () of Theorem 1; see also footnote Prsoners dlemma examples We llustrate our defntons and results usng the repeated prsoners dlemma under varous montorng assumptons. The ex ante stage game s gven by the normal form n Fgure Frst, consder the leadng example from Malath and Morrs (2002, Secton 3.3). The example llustrates that wthout bounded recall, belefs may vary n extreme ways to prevent a strct PPE from beng an equlbrum n nearby prvate montorng games. EXAMPLE 1. In the benchmark publc-montorng game, the set of publc sgnals s Y = { y, ȳ } and the publc-montorng dstrbuton s p f a 1 a 2 = e 1 e 2 ρ(ȳ a 1 a 2 ) = q f a 1 a 2 = e 1 n 2 or n 1 e 2 r f a 1 a 2 = n 1 n 2. The grm trgger strategy profle for the publc-montorng game s descrbed by the automaton W = {w e,w n }, ntal state w e, decson rules d (w a ) = a, and transton 12 Here (and n other examples) we follow the lterature n assumng the ex ante payoff matrx s ndependent of the montorng dstrbuton. Ths smplfes the dscusson and s wthout loss of generalty: Ex ante payoffs are close when the montorng dstrbutons are close (Lemma 2) and all relevant ncentve constrants are strct.

10 320 Malath and Morrs Theoretcal Economcs 1 (2006) a 1 a 2 y 2 ȳ 2 y 1 (1 α)(1 3ɛ) ɛ ȳ 1 ɛ α (1 3ɛ) ȳ 1 ɛ (α α )(1 3ɛ) FIGURE 2. The probablty dstrbuton of the prvate sgnals for Example 2. The dstrbuton s gven as a functon of the acton profle a 1 a 2, where α = p f a 1 a 2 = e 1 e 2, q f a 1 a 2 = e 1 n 2 or n 1 e 2, and r f a 1 a 2 = n 1 n 2 (analogously, α s gven by p, q, or r as a functon of a 1 a 2 ). All probabltes are strctly postve. rule σ(w,y ) = w e w n f y = ȳ and w = w e otherwse. Grm trgger s a strct PPE f δ > (3p 2q) 1 > 0 (a condton we mantan throughout ths example). We consder the ɛ-close prvate-montorng technology where Ω = Y and the sgnalng functons are the dentty functons. For ɛ small, grm trgger nduces a Nash equlbrum n such games f q < r, but not f q > r. Consder frst the case q > r and the prvate hstory (e y 1,n 1 ȳ 1,n 1 ȳ 1,...,n 1 ȳ 1 ). We now argue that, after a suffcently long such hstory, the grm 1 trgger specfcaton of n 1 s not optmal. Intutvely, whle player 1 has transted to the prvate state w1 n, player 1 always puts strctly postve (but perhaps small) probablty on hs opponent beng n prvate state w2 e. Snce q > r (and ɛ s small), the prvate sgnal ȳ 1 after playng n 1 s an ndcaton that player 2 has played e 2 (rather than n 2 ), and so player 1 s posteror that player 2 s stll n w2 e ncreases. Eventually, player 1 s suffcently confdent of player 2 stll beng n w2 e that he fnds n 1 suboptmal. On the other hand, when q r, such a hstory s not problematc because t renforces 1 s belef that 2 s also n w2 n. Two other hstores are worthy of menton: (e y 1,n y 1,n y 1,...,n y 1 ) and (e 1 ȳ 1,e 1 ȳ 1,e 1 ȳ 1,...,e 1 ȳ 1 ). Under the frst hstory, whle the 1 sgnal 1 y 1 s 1 now a sgnal 1 that 2 had chosen e 2 n the prevous perod, for ɛ small, 1 s confdent that 2 also observed y 2 and so transts to w2 n. For the fnal hstory, the sgnal ȳ 1 contnually reassures 1 that 2 s stll playng e 2, and so e 1 remans optmal. (See Malath and Morrs 2002, Secton 3.3 for the calculatons underlyng ths dscusson.) We now consder a rcher case where the prvate sgnal set s not equal to the publc sgnal set. The example llustrates that allowng rcher sgnal sets may be mportant. EXAMPLE 2. Let Ω 1 = { y 1,ȳ 1 1 } and Ω 2 = { y 2,ȳ 2 }. The probablty dstrbuton of the sgnals s gven n Fgure 2. Ths prvate-montorng dstrbuton s ɛ-close to the publcmontorng dstrbuton of Example 1 under the sgnalng functons f ( y ) = y and f 2 (ȳ 2 ) = f 1 (ȳ 1 ) = f 1(ȳ 1 ) = ȳ, as long as ɛ s suffcently small, relatve to mn{α,α α }. In Example 1, we argued that f q < r, grm trgger nduces Nash equlbrum behavor n close-by prvate-montorng games wth Ω = Y. We now argue that under the rcher prvate-montorng dstrbuton of ths example, even f q < r, grm trgger does,ȳ

11 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 321 not nduce Nash equlbrum behavor n some close-by games. In partcular, suppose 0 < r < q < q < r. Under ths parameter restrcton, the sgnal ȳ 1 after n 1 s ndeed a sgnal that player 2 has also played n 2. However, the sgnal ȳ 1 after n 1 s a sgnal that player 2 has played e 2 and so a suffcently long prvate hstory of the form (e y 1,n 1 ȳ 1 1,n 1ȳ 1,...,n 1ȳ 1 ) leads to a posteror for player 1 at whch n 1 s not optmal. 3. PPE WITH BOUNDED RECALL As we saw n Examples 1 and 2, arbtrary publc equlbra need not nduce equlbra of almost-publc-montorng games, because the publc state n perod t s determned, n prncple, by the entre hstory h t. For profles that have bounded recall, the entre hstory s not needed, and equlbra n bounded recall strateges nduce equlbra n almost-publc-montorng games. 13 DEFINITION 4. A publc profle s has L-bounded recall f for all h t = (y 1,...,y t 1 ) and h t = (y 1,..., y t 1 ), f t > L and y τ = y τ for τ = t L,...,t 1, then s (h t ) = s ( h t ). Let W t be the set of states reachable n perod t, W t {w W : w = σ(w 1,h t ) for some h t, where w 1 s the ntal state}. The followng characterzaton of bounded recall s useful. LEMMA 3. The publc profle nduced by the mnmal automaton (W,w 1,σ,d ) has L- bounded recall f and only f for all t, all w, w W t, and all h Y, σ(w,h L ) = σ(w,h L ). If a publc profle nduced by a fnte automaton (W,w 1,σ,d ), where W has K elements, does not have K (K 1)-bounded recall, then t has unbounded recall. PROOF. The frst clam s proved n the Appendx. For the second clam, suppose that the profle nduced by the fnte automaton (W,w 1,σ,d ), where W has K elements, does not have K (K 1)-bounded recall. From the frst clam, for some t, there exst w, w W t and hstory h Y such that σ(w,h τ ) σ(w,h τ ), for τ = 1,..., K (K 1). The sequence {(σ(w,h τ ),σ(w,h τ K (K 1) ))} τ=0, where (σ(w,h 0 ), σ(w,h 0 )) = (w,w ), conssts of K (K 1)+1 terms of pars of states. Snce pars of dentcal states cannot arse, some par of nondentcal states must be repeated. That s, there 13 Denote a dummy sgnal by. Malath and Morrs (2002) use the term bounded memory for publc profles wth the property that there s an nteger L such that a representng automaton s gven by W = (Y { }) L, σ((y 2,...,y 2,y L ),y ) = (y 2,...,y L,y ) for all y Y, and w 1 = (,..., ). Our earler noton mplctly mposes a tme homogenety condton, snce the caveat n Lemma 3 that the two states should be reachable n the same perod s mssng. The strategy profle n whch play alternates between the same two acton profles n odd and even perods has bounded recall, but not bounded memory.

12 322 Malath and Morrs Theoretcal Economcs 1 (2006) exst w w and perods 0 τ 1 < τ 2 K (K 1) such that (w,w ) = (σ(w,h τ 1 ),σ(w,h τ 1 )) = (σ(w,h τ 2 ),σ(w,h τ 2 )). Now we have w, reachable n the same perod as w nfntely often, such that lettng h be the nfnte repetton of the cycle of outcomes τ 1h τ 2, we have σ(w, h t ) σ(w, h t ) for all t. Fx a strct publc equlbrum wth bounded recall, (W,w 1,σ,d ). Fx a prvatemontorng technology (Ω,π) ɛ-close under f to (Y,ρ). Followng Monderer and Samet (1989), we frst consder a constraned game where behavor after unnterpretable sgnals s arbtrarly fxed. Defne the set of unnterpretable prvate hstores, H u = {h t : ω τ f 1 ( ), some τ satsfyng t L τ t 1}. Ths s the set of prvate hstores for whch n any of the last L perods, a prvate sgnal ω τ satsfyng f (ω τ ) = s observed. We fx arbtrarly player s acton after any prvate hstory h t H u. For any prvate hstory that s not unnterpretable, each of the last L observatons of the prvate sgnal can be assocated wth a publc sgnal by the functon f. Denote by w (h t ) the prvate state so obtaned. That s, w (h t ) = (f (ω t L ),..., f (ω t 1 )), for all h t / H u. We are then left wth a game n whch n perod t 2 player chooses an acton only after a sgnal ω t 1 yelds a prvate hstory not n H u. We clam that for ɛ suffcently small, the profle (s 1,..., s N ) s an equlbrum of ths constraned game, where s s the strategy for player : s t (ht ) = d (w 1 ) f t = 1 d (w (h t )) f t > 1 and ht / H u. But ths follows from arguments almost dentcal to that n the proofs of Malath and Morrs (2002, Theorems 4.2 and 4.3): snce a player s behavor depends only on the last L sgnals, for small ɛ, after observng a hstory h t / H u, player assgns a hgh probablty to player j observng a sgnal that leads to the same prvate state (recall Lemma 1). The crucal pont s that for ɛ small, the specfcaton of behavor after sgnals ω satsfyng f (ω ) = s rrelevant for behavor at sgnals ω satsfyng f (ω ) Y. It remans to specfy optmal behavor after sgnals ω satsfyng f (ω ) =. So, consder a new constraned game where player s requred to follow s where possble. Ths constraned game has an equlbrum, and so by constructon, we thus have an equlbrum of the unconstraned game. We have thus proved: THEOREM 1. Fx a full-support publc-montorng game ( u,(y,ρ)) and a strct perfect publc equlbrum, s, wth bounded recall L. There exsts ɛ > 0 such that for all prvatemontorng games (u,(ω,π)) ɛ-close under f to ( u,(y,ρ)), () f f (Ω ) = Y for all, the nduced prvate profle s a Nash equlbrum; and

13 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 323 () f f (Ω ) Y for some, there s a Nash equlbrum of the prvate-montorng game, s, such that, for all h t = (y 1,...,y t 1 ) and h t j = (ω 1 1 j,...,ωt j ), f t > L and y τ = f j (ω τ j ) for τ = t L,...,t 1, then s j (h t j ) = s j (h t ) for all j. Moreover, for all κ > 0, ɛ can be chosen suffcently small that the expected payoff to each player under s s wthn κ of ther publc equlbrum payoff. We could smlarly extend our results on patently-strct, connected, fnte publc profles (Malath and Morrs 2002, Theorem 5.1) and on the almost-publc almostperfect mutual mnmax folk theorem to ths more general noton of nearby prvatemontorng dstrbutons FAILURE OF COORDINATION Examples 1 and 2 llustrate that updatng n almost-publc-montorng games can be very dfferent than would be expected from the underlyng publc-montorng game. In ths secton, we buld on that example to show that when the set of sgnals s suffcently rch (n a sense to be defned), many profles fal to nduce equlbrum behavor n almost-publc-montorng games. Our negatve results are based on the followng converse to Theorem 1 (the proof s n the Appendx). Snce the theorem s negatve, the assumpton of strong ɛ-closeness (rather than ɛ-closeness) does not lmt ts usefulness. The assumpton clarfes the source of the falure of the nduced profle to be a Nash equlbrum, whch s not due to a dffculty wth nterpretng unnterpretable sgnals. Moreover, ths falure arses n any strongly ɛ-close game n whch the belef hypothess holds. Recall also that the publc profle completely determnes a strategy profle n a prvate-montorng game only when the prvate-montorng game s strongly ɛ-close (Secton 2.3). 15 THEOREM 2. Suppose the publc profle (W,w 1,σ,d ) s a strct equlbrum of the fullsupport publc-montorng game ( u,(y,ρ)) for some δ and W <. There exsts η > 0 and ɛ > 0 such that for any game wth prvate montorng (u,(ω,π)) strongly ɛ-close to ( u,(y,ρ)), f there exsts a player, a prvate hstory for that player h t, and a state w such that d (w ) d (σ (h t )) and β (w 1 h t ) > 1 η, then the nduced prvate profle s not a Nash equlbrum of the game wth prvate montorng for the same δ. 14 We ncorrectly clamed that the profle descrbed n the proof of the almost-publc almost-perfect folk theorem (Malath and Morrs 2002, Theorem 6.1) has bounded recall. See Malath and Samuelson (2006, Proposton ) for a proof of the weaker result reported n the text. 15 The result does extend to prvate-montorng games that are ɛ-close, but not strongly so. Any pure prvate strategy for can be represented as an automaton (W, w 1, σ, d ), where σ : W A Ω W and (as usual) d : W A. Say a prvate profle (W, w 1, σ, d ) reflects the publc profle (W,w 1,σ,d ) f for all (perhaps after relabelng states, see footnote 10) W W, w 1 = w 1, σ (w,a,ω ) = σ(w, f (ω )) for all (w,a,ω ) W A f 1 (Y ), and fnally, d (w ) = d (w ) for all w W. Then, there exsts η > 0 and ɛ > 0 such that for any close-by prvate-montorng game and any prvate profle reflectng the publc profle, f there s a player and a prvate hstory, and a state w W W wth the specfed propertes, then the prvate profle s not a Nash equlbrum.

14 324 Malath and Morrs Theoretcal Economcs 1 (2006) We mplctly used ths result n our dscussons of the repeated prsoners dlemma. For example, n Example 1, we argued that there was a prvate hstory for player 1 that leaves hm n the prvate state w1 n, but hs posteror after that hstory assgns probablty close to 1 that player 2 s prvate state s w2 e. Our approach s to ask when t s possble to so manpulate a player s belefs through the selecton of a prvate hstory that the hypotheses of Theorem 2 are satsfed. In partcular, we are nterested n the weakest ndependent condtons on the prvate-montorng dstrbutons and on the strategy profles that would allow such manpulaton. Fx a PPE of the publc-montorng game and a close-by almost-publc-montorng game. The logc of Example 1 runs as follows: Consder a player n a prvate state w who assgns strctly postve (albet small) probablty to all the other players beng n some other common prvate state w w. (Full-support prvate montorng ensures that such an occurrence arses wth postve probablty.) Let a = (d ( w ),d ( w )) be the acton profle that results when s n state w and all the other players are n state w. Suppose that f any other player s n a dfferent prvate state w w, then the resultng acton profle dffers from a. Suppose, moreover, there s a sgnal y such that w = σ( w,y ) and w = σ( w,y ), that s, any player n the state w or w observng a prvate sgnal consstent wth y stays n that prvate state (and so the profle cannot have bounded recall, see Lemma 3). Suppose fnally there s a prvate sgnal ω for player consstent wth y that s more lkely to have come from a than any other acton profle,.e., ω f 1 (y ) and π (ω a ) > π (ω (d ( w ),a )) a d ( w ) (where π (ω a ) s the probablty that player observes the sgnal ω under a ). Then, after observng the prvate sgnal ω, player s posteror probablty that all the other players are n w should ncrease (ths s not mmedate, however, snce the montorng s prvate). Moreover, snce players n w and w do not change ther prvate states, we can eventually make player s posteror probablty that all the other players are n w as close to one as we lke. If d ( w ) d ( w ), an applcaton of Theorem 2 shows that the nduced prvate profle s not an equlbrum. The suppostons n the above logc can be weakened n two ways. Frst, t s not necessary that the same prvate sgnal ω be more lkely to have come from a than any other acton profle. It should be enough f for each acton profle dfferent from a, there s a prvate sgnal more lkely to have come from a than that profle, as long as that sgnal does not mess up the other nferences too badly. In that case, realzatons of the other sgnals could undo any damage done wthout negatvely mpactng on the overall nferences. For example, suppose there are two players, wth player 1 the player whose belefs we are manpulatng, and n addton to state w, player 2 could be n state w or w. Suppose also A 2 = { a 2, a 2,a 2 }. As before, suppose there s a sgnal y such that w = σ( w,y ), w = σ( w,y ), and w = σ(w,y ), that s, any player n the state w, w, or w observng a prvate sgnal consstent wth y stays n that prvate state. We would lke the odds rato Pr(w 2 w h t 1 )/Pr(w 2 = w h t 1 ) to converge to zero as t, for approprate

15 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 325 prvate hstores. Let a 1 = d 1 ( w ), a 2 = d 2 ( w ), a 2 = d 2 ( w ), and a 2 = d 2(w ), and suppose there are two prvate sgnals, ω 1 and ω 1 consstent wth y, satsfyng and π 1 (ω 1 a 1,a 2 ) > π 1(ω 1 a ) > π 1(ω 1 a 1, a 2 ) π 1 (ω 1 a 1, a 2 ) > π 1 (ω 1 a ) > π 1(ω 1 a 1,a 2 ). Then, after observng the prvate sgnal ω 1, we have Pr(w 2 = w h t 1,ω 1 ) Pr(w 2 = w h t 1,ω 1 ) = π 1(ω 1 a 1, a 2 ) Pr(w 2 = w h t 1 ) π 1 (ω 1 a ) Pr(w 2 = w h t 1 ) < Pr(w 2 = w h t 1 ) Pr(w 2 = w h t 1 ) as desred, but Pr(w 2 = w h t 1,ω 1 )/Pr(w 2 = w h t 1,ω 1 ) ncreases. On the other hand, after observng another prvate sgnal ω 1, also consstent wth y, whle the odds rato Pr(w 2 = w h t 1,ω 1 )/Pr(w 2 = w h t 1,ω 1 ) falls, Pr(w 2 = w h t 1,ω 1 )/Pr(w 2 = w h t 1,ω 1 ) ncreases. However, t may be that the ncreases can be offset by approprate decreases, so that, for example, ω 1 followed by two realzatons of ω 1 results n a decrease n both odds ratos. If so, a suffcently hgh number of realzatons of ω 1 ω 1 ω 1 result n Pr(w 2 w h t 1 )/Pr(w 2 = w h t 1 ) beng close to zero. In terms of the odds ratos, the sequence of sgnals ω 1 ω 1 ω 1 lowers both odds ratos f, and only f, π 1 (ω 1 a 1, a 2 ) π1 (ω 1 a 2 1, a 2 ) π 1 (ω 1 a ) π 1 (ω 1 a ) < 1 and π 1 (ω 1 a 1,a 2 ) π1 (ω 1 a 1,a 2 ) 2 π 1 (ω 1 a ) π 1 (ω 1 a ) < 1. Our rchness condton on prvate-montorng dstrbutons captures ths dea. For a prvate-montorng dstrbuton (Ω, π), defne γ a a (ω ) logπ (ω a,a ) logπ (ω a,a ) and let γ a (ω ) = (γ a a (ω )) a A,a a denote the vector n A 1 of the log odds ratos of the sgnal ω assocated wth dfferent acton profles. The last two dsplayed equatons can then be wrtten as 1 3 γ a (ω 1 )+ 2 3 γ a (ω 1 ) > 0, where 0 s the 2 1 zero vector.16 DEFINITION 5. A prvate-montorng dstrbuton (Ω, π) s rch for player, gven hs sgnalng functon f, f for all a A and all y Y, the convex hull of the set of vectors {γ a (ω ) : ω f 1 (y ) and π (ω a,a ) > 0 for all a A } has a nonempty ntersecton wth A The convex combnaton s strctly postve (rather than negatve) because the defnton of γ a a nverts the odds ratos from the dsplayed equatons.

16 326 Malath and Morrs Theoretcal Economcs 1 (2006) Note that we requre only that prvate montorng be rch for one player. It s useful to quantfy the extent to whch the condtons of Defnton 5 are satsfed. Snce the spaces of sgnals and actons are fnte, the number of constrants n Defnton 5 s fnte, and so for any rch prvate-montorng dstrbuton, the set of ζ over whch the supremum s taken n the next defnton s non-empty. 17 DEFINITION 6. Gven f, the rchness of a rch prvate-montorng dstrbuton (Ω, π) for s the supremum of all ζ > 0 satsfyng: for all a A and all y Y, the convex hull of the set of vectors {γ a (ω ) : ω f 1 (y ) and π (ω a,a ) ζ for all a A } has a nonempty ntersecton wth A 1 ζ {x A 1 ++ : x k ζ for k = 1,..., A 1}. The second weakenng of the logc of Example 1 descrbed above concerns the nature of the strategy profle. The logc assumed that there s a sgnal y such that w = σ( w,y ) and w = σ( w,y ). Thus along the hstory (y,y,...), f the player started out n dstnct states w or w, he would reman n those dstnct states and would contnue to play n dstnct ways. But the logc contnues to hold f there exsts an arbtrary hstory h such that some dstnct ntal states lead to dstnct states forever and f, from such dstnct states, play s dstnct along that partcular hstory nfntely often. Ths s the dea behnd the followng defnton of a separatng strategy profle. Defne R( w ) as the set of states that are repeatedly reachable n the same perod as w (.e., R( w ) = {w W : {w, w } W t nfntely often}). Gven an outcome path h (y 1,y 2,...) Y, let τ h (y τ,y τ+1,...) Y denote the outcome path from perod τ, so that h = (h τ, τ h) and τ h τ+t = (y τ,y τ+1,...,y τ+t 1 ). Consder a contnuaton path ( w,h) consstng of an ntal state w followed by an outcome path h. The contnuaton path ( w,h) satsfes state-separaton f there s another state w R( w ) such that startng n state w nstead of w would lead to dstnct states nto the nfnte future: formally, there exsts another state w R( w ) that satsfes σ(w,h t ) σ( w,h t ) for all t. In ths case, state w s separated from w along hstory h. Recall from the proof of the second clam n Lemma 3 that every unbounded recall profle nduced by a fnte automaton has a contnuaton path ( w,h) satsfyng state-separaton. The logc of our proof requres not only state-separaton, but n addton dstnct behavor on the contnuaton path satsfyng state-separaton. The contnuaton path ( w,h) satsfes behavor-separaton f whenever state w R(σ( w,h τ )) s separated from σ( w,h τ ), then all players choose dfferent actons along the outcome path τ h nfntely often. Formally, for all τ and w R(σ( w,h τ )), f σ(w, τ h τ+t ) σ( w,h τ+t ) for all t 0, then d (σ(w, τ h τ+t )) d (σ( w,h τ+t )) nfntely often, for all. Notce that every contnuaton path satsfes behavor-separaton f, for each player, dstnct states always lead to dstnct actons. The need to behavor-separate the state w from every other state that can be reached nfntely often s llustrated by our earler dscusson: because prvate montorng mples all such states are assgned postve 17 The bound ζ appears twce n the defnton. Its frst appearance ensures that for all ζ > 0, there s a unform upper bound on the number of prvate sgnals satsfyng π (ω a,a ) ζ n any prvatemontorng dstrbuton wth a rchness of at least ζ.

17 Theoretcal Economcs 1 (2006) Coordnaton falure n repeated games 327 A B C A 3,3 0,0 0,0 B 0,0 3,3 0,0 C 0,0 0,0 2,2 FIGURE 3. The normal form for Example 3. probablty by a player s belefs, we need to have sgnals that are nformatve about these states relatve to w. Now we have: DEFINITION 7. The publc strategy profle s separatng f there s a state w and an outcome path h Y such that ( w,h) satsfes state-separaton and behavor-separaton. Clearly, a separatng profle cannot have bounded recall. The key queston s how much stronger s ths property than havng unbounded recall under the restrcton to fnte state strateges. Snce every fnte unbounded recall profle has a state-separatng path, the only way a fnte state strategy profle wth unbounded recall can fal separaton s f every contnuaton path satsfyng state-separaton fals behavor-separaton. The followng example llustrates ths possblty. EXAMPLE 3. The stage game s gven n Fgure 3. In the publc-montorng game, there are two publc sgnals, y and y, wth dstrbuton (0 < q < p < 1) ρ(y a 1 a 2 ) = p f a 1 = a 2 q otherwse. Fnally, the publc profle s llustrated n Fgure 4. Under any outcome path n whch the sequence y y or y y occurs, all states transt to the same state. Under any outcome path n whch only y appears, the state eventually cycles between w A and w A. Thus contnuaton path (w,h) s state-separatng only f h = (y,y,...). But ths contnuaton path s not behavor separatng, snce acton A s then played forever. We thnk of ths falure as pathologcal. In ths example, t s easy to see that the profle s not robust. After enough realzatons of prvate sgnals correspondng to y, belefs must assgn roughly equal probablty to w A and w A, 18 and so after the frst realzaton of a prvate sgnal correspondng to y, B s the only best reply (even f the current state s w C ). We do not have an example of a fnte state strategy profle wth unbounded recall that fals separaton but s robust. Example 3 suggests an ntuton why such an example mght be hard to fnd: a strategy profle wth unbounded recall can fal separaton only f all state-separated states gve rse to dentcal behavor most of the tme. Wth the 18 The detals of ths calculaton can be found n Malath and Samuelson (2006, Example ).

18 328 Malath and Morrs Theoretcal Economcs 1 (2006) y A w y wˆ A y y y y B w y C w y FIGURE 4. The strategy profle for Example 3. In states w A and w A the acton A s played, whle n w B the acton B and n w C, the acton C s played. possblty of belef drft, as n the example, t seems hard to make ths consstent wth equlbrum. Moreover, ths possblty of drft mples also that showng that these unbounded recall strategy profles are not robust requres a qute dfferent proof strategy than that pursued n ths paper. It remans to ensure that, under prvate montorng, players may transt to dfferent states. It suffces to assume the followng, weaker than full-support, condton: 19 DEFINITION 8. A prvate-montorng dstrbuton (Ω, π) that s ɛ-close to a publc-montorng dstrbuton (Y,ρ) has essentally full support f for all (y 1,...,y n ) Y n, π{(ω 1,...,ω n ) Ω : f (ω ) = y, = 1,...,n} > 0. THEOREM 3. Fx a separatng strct fnte PPE of a full-support publc-montorng game ( u,(y,ρ)). For all ζ > 0, there exsts ɛ > 0 such that for all ɛ < ɛ, f (u,(ω,π)) s a prvate-montorng game strongly ɛ-close under some sgnalng functon f to ( u,(y,ρ)) wth (Ω,π) havng rchness, gven f, for some player of at least ζ and essentally full support, then the nduced prvate profle s not a Nash equlbrum of the prvate-montorng game. It s worth notng that the bound on ɛ s a functon only of the rchness of the prvate montorng. It s ndependent of the probablty that a dsagreement n prvate states arses. By consderng fnte state profles that are separatng, not only s the dffculty dentfed n the Introducton dealt wth (as we dscuss at the end of the next secton), but we can accommodate arbtrarly small probabltes of dsagreement. 19 If an essentally-full-support prvate montorng dstrbuton does not have full support, Nash equlbra of the prvate-montorng game may not have realzaton-equvalent sequentally-ratonal strategy profles.