A note on the one-deviation property in extensive form games

Transcription

1 Games and Economc Behavor 40 (2002) Note A note on the one-devaton property n extensve form games Andrés Perea Departamento de Economía, Unversdad Carlos III de Madrd, Calle Madrd 126, Getafe, Madrd, Span Receved 7 December 2000 Abstract In an extensve form game, an assessment s sad to satsfy the one-devaton property f for all possble payoffs at the termnal nodes the followng holds: f a player at each of hs nformaton sets cannot mprove upon hs expected payoff by devatng unlaterally at ths nformaton set only, he cannot do so by devatng at any arbtrary collecton of nformaton sets. Hendon et al. (1996. Games Econom. Behav. 12, ) have shown that preconsstency of assessments mples the one-devaton property. In ths note, t s shown that an approprate weakenng of pre-consstency, termed updatng consstency, s both a suffcent and necessary condton for the one-devaton property. The result s extended to the context of ratonalzablty Elsever Scence (USA). All rghts reserved. JEL classfcaton: C72 Keywords: Extensve form games; Sequental ratonalty; One-devaton property; Ratonalzablty 1. Introducton In dynamc one-person and mult-person decson makng, the one-devaton property (also called one-shot devaton prncple) reflects the phenomenon that a stream of locally optmal decsons consttutes a globally optmal decson stream. By locally optmal, we mean that the decson for an ndvdual at a partcular stage maxmzes hs expected payoff, takng as gven the decsons E-mal address: perea@eco.uc3m.es /02/$ see front matter 2002 Elsever Scence (USA). All rghts reserved. PII: S (02)

2 Note / Games and Economc Behavor 40 (2002) chosen at all other stages (ncludng hs own decsons at other stages). Globally optmal refers to the fact that the decson maker cannot mprove upon hs expected payoff by changng hs decsons at any arbtrary subset of stages. The one-devaton property thus reflects a knd a tme-consstency, statng that for optmal decson makng t should be suffcent to check the optmalty of each of the decsons on a one-by-one bass. It s a well-known fact that the one-devaton property holds generally for the context of one-person decson makng; a result whch s known as the optmalty prncple for dynamc programmng. If more than one decson maker s nvolved, the fact whether the one-devaton property holds or not depends crucally on the way decson makers (from now on called players) update ther conjectures about the opponents behavor as tmes passes by. It s the am of ths note to fgure out whch condtons on the players updatng behavor are necessary and suffcent n order for the one-devaton property to hold. To ths purpose, we focus on two dfferent contexts whch have both been mportant for the development of ratonalty concepts for extensve form games. In the frst t s assumed that players, at each of ther nformaton sets, hold conjectures about the opponents future behavor that concde exactly wth the real behavor of the opponents. The uncertanty of a player at an nformaton set about the actual play of the game thus reduces to uncertanty about the past play, captured formally by the noton of belefs at nformaton sets. The conjecture about future play s completely determned by a fxed behavor strategy profle, whch prescrbes a randomzaton over actons at each nformaton set. The above mentoned assumpton mples that a player at an nformaton set always beleves that future play wll be accordng to ths strategy profle, also f the event of reachng ths nformaton set actually contradcts ths strategy profle. Ths assumpton s used n the backward nducton concept for games wth perfect nformaton and most of the extensve form equlbrum refnements, such as subgame perfect equlbrum (Selten, 1965), sequental equlbrum (Kreps and Wlson, 1982), dfferent versons of perfect Bayesan equlbrum and extensve form perfect equlbrum (Selten, 1975). In ths partcular settng, the players choces and conjectures about the play by opponents are represented by a so-called assessment: a combnaton of a behavor strategy profle and a system of belefs at nformaton sets. Consder an extensve form structure, that s, a combnaton of the game tree, the nformaton sets, the actons and possbly chance moves, together specfyng how the game s to be played. An extensve form structure s extended to an extensve form game by assgnng a vector of payoffs to each of the termnal nodes. Formally, an assessment for a gven extensve form structure s sad to satsfy the one-devaton property f for every extensve form game havng ths extensve form structure the followng holds: f a player at each of hs nformaton sets cannot mprove upon hs expected payoff by devatng unlaterally at ths nformaton set, whle leavng hs behavor at other nformaton sets unchanged, he cannot do so by devatng

3 324 Note / Games and Economc Behavor 40 (2002) at any arbtrary collecton of nformaton sets. The fact that ths property should holdfor all extensve form games havng ths extensve form structure mples that the one-devaton property puts restrctons on an assessment that solely depend on the extensve form structure, and not on the partcular choce of payoffs at the termnal nodes. In games wth perfect nformaton t s well-known that every strategy profle, together wth the trval belefs at the sngleton nformaton sets, satsfes the one-devaton property. For games wth mperfect nformaton the consstency condton on assessments, whch s part of Kreps and Wlson s defnton of sequental equlbrum, turns out to be suffcent for the one-devaton property. Hendon et al. (1996) show that some weakenng of consstency, termed preconsstency, s enough to mply the one-devaton property. In Theorem 2.2 we prove that a further weakenng, called updatng consstency, s both suffcent and necessary for the one-devaton property to hold. Intutvely, updatng consstency states that player s conjecture at nformaton set B about the opponents behavor should be nduced by hs conjecture at nformaton set A whenever B comes after A and the conjecture at A does not exclude reachng B. Important s that ths condton should hold also f player s own strategy choce at A prevents B from beng reached. The second context we focus on leaves more freedom to the players conjectures about the opponents behavor, snce t s now no longer assumed that players hold correct conjectures about the opponents future strategy choces. Ths more flexble settng corresponds to ratonalzablty concepts for extensve form games, such as extensve form ratonalzablty (Pearce, 1984; see also Battgall, 1997), subgame perfect ratonalzablty (Bernhem, 1984) and weak extensve form ratonalzablty (Ben-Porath, 1997; Battgall and Bonanno, 1999), among others. It also apples to ntermedate models that place restrctons on the players conjectures that are weaker than n the frst context dscussed above, but stronger than n ratonalzablty. For nstance, the concept of self-confrmng equlbrum (Fudenberg and Levne, 1993) requres the players conjectures to concde wth the actual behavor on the equlbrum path, but allows them to dffer from the actual behavor at unreached nformaton sets. In Dekel et al. (1999, 2000), the concept of self-confrmng equlbrum s refned to the case where conjectures about the opponents behavor at unreached nformaton sets should, n addton, be ratonalzable. Greenberg (1996) proposes a model n whch the players conjectures about the play are assumed to agree at some, but not necessarly all nformaton sets, and defnes a correspondng noton of stablty. Wthn ths context, players may thus have dfferent conjectures about the play of the game at nformaton sets for whch no agreement s requred. As a prmtve to model the players conjectures about the opponents behavor we use the noton of updatng systems (cf. Battgall, 1997), whch specfes for each player and each nformaton set controlled by ths player a subjectve randomzaton on the set of opponents strategy profles that are compatble

4 Note / Games and Economc Behavor 40 (2002) wth reachng ths nformaton set. In order to avod the ssue whether such randomzatons should be correlated or uncorrelated, we restrct our attenton to the case of two players. There should be no problem, however, n extendng the result to games wth more than two players, once t s decded whch class of conjectures (correlated or uncorrelated) s to be used. For a gven extensve form structure, an updatng system for a player s sad to satsfy the one-devaton property f for all extensve form games havng ths extensve form structure and all strateges for ths player the followng holds: f at each of hs nformaton sets the player cannot mprove upon hs expected payoff by devatng at ths nformaton set only, gven hs conjecture about the opponent s behavor and gven hs decsons at other nformaton sets, then he cannot mprove by devatng at any arbtrary collecton of nformaton sets. We present a condton on updatng systems, termed updatng consstency, whch s a weakenng of the noton of consstent updatng systems, used by Battgall (1997). The ntuton of updatng consstency s the same as n the frst context: f the player holds a certan conjecture at an nformaton set A, then conjectures at future nformaton sets should be derved from ths by Bayesan updatng, as long as reachng these nformaton sets does not contradct the conjecture at A. What dstngushes t from consstent updatng systems s that, unlke the latter, players are allowed to reshuffle conjectures at nformaton sets as long as t does not affect the expected outcome condtonal on reachng ths nformaton set. It thus leaves some more freedom than updatng consstency. In Theorem 3.1 t s shown that updatng consstency s both a necessary and suffcent condton for the one-devaton property. The note s organzed as follows. Secton 2 deals wth the context n whch players are requred to hold correct conjectures about the opponents future behavor. It frst provdes some notaton and defntons, and then presents the result whch characterzes the assessments that satsfy the one-devaton property. Secton 3 procedes dentcally for the context of updatng systems. 2. One-devaton property for assessments 2.1. Notaton n extensve form games An extensve form structure S specfes a fnte set of players, a fnte game tree, a collecton of nformaton sets for each player, a set of actons at each nformaton set and the probabltes of each of the chance moves. Let I be the set of players. For every I, let H be the collecton of nformaton sets controlled by player, and let H be the collecton of all nformaton sets n the game. For every h H denote by A(h) the set of actons avalable at h. We assume that A(h) contans at least two actons for every h. Suppose that two actons avalable at dfferent nformaton sets are labelled dfferently, that s, A(h) A(h ) =

5 326 Note / Games and Economc Behavor 40 (2002) f h h. It s assumed, moreover, that S satsfes perfect recall (Kuhn, 1953), whch means that two dfferent paths leadng to the same player nformaton set h contan the same player actons. Snce actons at dfferent nformaton sets are, by assumpton, dfferent, perfect recall mples n partcular that two paths leadng to the same player nformaton set h pass through the same collecton of precedng player nformaton sets. The set of termnal nodes s denoted by Z. An extensve form game s a par Γ = (S,u) where S s an extensve form structure and u s the payoff functon assgnng to every termnal node z Z a vector u(z) = (u (z)) I R I of payoffs Strateges and belefs A behavor strategy for player s a vector σ = (σ h ) h H that assgns to every nformaton set h H some probablty dstrbuton σ h on A(h). A vector σ = (σ ) I of behavor strateges s called a behavor strategy profle. A belef system s a vector β = (β h ) h H where β h s a probablty dstrbuton on the set of nodes n h for all h H. Apar(σ, β) s called an assessment. Note that the set of assessments n a game depends only on the extensve form structure Sequental ratonalty Let σ be a behavor strategy profle, x a node and Z(x) the collecton of termnal nodes that follow x. For every z Z(x), let P σ (z x) be the probablty that z s reached under σ, condtonal on the event that the game has reached x. By u (σ x) = z Z(x) P σ (z x)u (z) we denote the expected payoff for player, condtonal on x beng reached. For a gven assessment (σ, β) and an nformaton set h H, let u (σ h, β h ) = x h β h(x)u (σ x) be the expected payoff for player condtonal on h beng reached, gven the belefs β h at h. The assessment (σ, β) s called sequentally ratonal f for every player and every h H t holds that u (σ h, β h ) = max σ u ((σ,σ ) h, β h ). Here, (σ,σ ) s the behavor strategy profle n whch player plays σ and the other players act accordng to σ. The assessment s called locally sequentally ratonal f for every player and every h H t holds that u (σ h, β h ) = max σ h u ((σ h,σ h) h, β h ). Here, (σ h,σ h) s the behavor strategy profle n whch player plays the local strategy σ h at nformaton set h and σ s played at all other nformaton sets (ncludng the other player nformaton sets). The dfference between sequental ratonalty and local sequental ratonalty s thus that the former takes nto account all possble devatons by a player, whereas the latter concentrates on those devatons n whch a player changes hs behavor at only one nformaton set.

6 Note / Games and Economc Behavor 40 (2002) One-devaton property Let S be an extensve form structure and (σ, β) an assessment n S. Wesay that (σ, β) satsfes the one-devaton property f for every payoff functon u the followng holds: (σ, β) s sequentally ratonal n the game Γ = (S,u)f and only f t s locally sequentally ratonal n Γ Updatng consstency of assessments In Hendon et al. (1996) t has been shown that the set of so-called preconsstent assessments satsfes the one-devaton property. Ther defnton of pre-consstency conssts of two parts. The frst part, whch we call updatng consstency, states that a player should update hs belefs n some consstent manner to be specfed below. The second part, called Bayesan consstency, s an equlbrum condton whch assures that every player holds a correct conjecture about the opponents past behavor at nformaton sets reached wth postve probablty under σ. Snce Bayesan consstency s not needed n ther proof, t follows that the larger set of updatng consstent assessments satsfes the onedevaton property as well. Formally, an assessment (σ, β) s called updatng consstent f for every player, every two nformaton sets h 1,h 2 H where h 2 comes after h 1, and every behavor strategy σ for player, β h 2(x) = P (σ,σ )(x h 1,β h 1) P (σ,σ )(h 2 h 1,βh 1) for all x h 2, whenever P (σ,σ )(h 2 h 1,β h 1)>0. Here, P (σ,σ )(x h 1,β h 1) s the probablty that the node x s reached, condtonal on h 1 beng reached and gven the belefs β h 1 at h 1. By P (σ,σ )(h 2 h 1,βh 1) = y h 2 P (σ,σ )(y h 1,β h 1) we denote the probablty that h 2 s reached, condtonal on h 1 beng reached and gven the belefs at h 1. By perfect recall, every path from a node n h 1 to a node n h 2 contans the same player actons. Consequently, the rato n the defnton of updatng consstency does not depend on the partcular choce of σ, as long as P (σ,σ )(h 2 h 1,β h 1)>0. The ntuton behnd updatng consstency s the followng. Consder two nformaton sets h 1 and h 2 whch are controlled by the same player, and assume that h 2 comes after h 1. Player s conjecture about the opponents past behavor at h 1 s reflected by the belefs β h 1. If we assume that players hold correct conjectures about the opponents future behavor, also at nformaton sets whch should actually have been avoded by σ, t follows that player at h 1 beleves that the opponents future behavor s determned by σ. Updatng consstency states that player s conjecture about the opponents past behavor at h 2 should be nduced by hs conjectures about past and future behavor at h 1

7 328 Note / Games and Economc Behavor 40 (2002) whenever the event of reachng h 2 s compatble wth hs conjectures at h 1 (.e., whenever there s some σ wth P (σ,σ )(h 2 h 1,β h 1)>0). Important s that ths condton should also hold when P σ (h 2 h 1,β h 1) = 0. Hence, even f player s own behavor after h 1 precludes the nformaton set h 2 from beng reached, hs belefs at h 2 should be nduced by hs conjecture about past and future behavor at h 1. Ths property s satsfed n concepts such as sequental equlbrum and extensve form perfect equlbrum. The reason s that n both concepts, the players belefs are derved from takng a sequence of strctly postve behavor strategy profles convergng to the orgnal one. 1 Along the sequence, t s clear that the belefs of a player at two consecutve nformaton sets are always n accordance wth each other, snce all nformaton sets are reached wth postve probablty. As may be verfed easly, ths property remans vald n the lmt, and hence every consstent assessment s updatng consstent. The followng result s due to Hendon et al. Theorem 2.1 (Hendon et al., 1996). Let S be an extensve form structure. Then, every updatng consstent assessment n S satsfes the one-devaton property. The theorem below shows that updatng consstency s not only suffcent, but also necessary for the one-devaton property. Theorem 2.2. Let S be an extensve form structure. Then, an assessment (σ, β) n S satsfes the one-devaton property f and only f t s updatng consstent. Proof. In vew of Theorem 2.1, t suffces to show that every assessment whch s not updatng consstent fals to satsfy the one-devaton property. Let (σ, β) be an assessment n S whch s not updatng consstent. We show that there s a payoff vector u for the termnal nodes such that n the extensve form game Γ = (S,u) the assessment (σ, β) s locally sequentally ratonal but not sequentally ratonal. Snce (σ, β) snotupdatngconsstent, theressomeplayer, two nformaton sets h 1,h 2 H where h 2 follows h 1, and some behavor strategy σ such that P (σ,σ )(h 2 h 1,β h 1)>0but β h 2(x ) P (σ,σ )(x h 1,β h 1) P (σ,σ )(h 2 h 1,β h 1) 1 These belefs play an explct role n sequental equlbrum, whereas used mplctly n extensve form perfect equlbrum.

8 Note / Games and Economc Behavor 40 (2002) for some node x h 2. Snce both β h 2( ) and P (σ,σ )( h 1,β h 1)/P (σ,σ )(h 2 h 1,β h 1) are probablty dstrbutons on the set of nodes at h 2, we can choose x such that β h 2(x )< P (σ,σ )(x h 1,β h 1) P (σ,σ )(h 2 h 1,β h 1). (2.1) The reader may verfy that h 1 and h 2 can always be chosen n such a way that Eq. (2.1) holds and there s no further player nformaton set between h 1 and h 2. By perfect recall, there s a unque sequence h 1,...,h K of player nformaton sets wth the followng propertes: (1) h k follows h k 1 for all k, (2) there s no player nformaton set between h k 1 and h k for all k, (3) there s no player nformaton set before h 1,and(4)h K 1 = h 1 and h K = h 2. We defne the player payoffs followng h k by nducton on k. We frst defne the player payoffs followng h K = h 2. Let a K be some acton at h K wth σ h 2(a K )<1. Such an acton exsts snce by assumpton there are at least two actons at h K. For every termnal node z followng node x (see (2.1)) and acton a K, set u (z) = 1. For all termnal nodes z followng acton a K but not followng node x, set u (z) = 0. For every termnal node z followng h K but not followng acton a K, set u (z) = β h 2(x ). Now, suppose that k<kand that the player payoffs u (z) have been defned for all termnal nodes z followng h k+1. We defne the player payoffs followng h k but not followng h k+1 n the followng way. Let a k be the unque acton at h k that leads to h k+1. For every termnal node z followng a k but not followng h k+1, set u (z) = 0. By (a k,σ hk ) we denote the strategy profle n whch player chooses acton a k wth probablty one at h k, and players act accordng to σ at all other nformaton sets. Let u ((a k,σ hk ) h k,β hk ) be the expected payoff nduced by (a k,σ hk ) at h k, gven the belefs β hk and the payoffs followng a k, whch have already been defned above. For all termnal nodes z followng h k but not acton a k, we set u (z) = u ((a k,σ hk ) h k,β hk ). Fnally, for all termnal nodes not covered by the procedure above, we set u (z) = 0. For all players j, we set u j (z) = 0 for all termnal nodes z. It can be verfed easly that the assessment (σ, β) s locally sequentally ratonal, gven the payoff vector u. Note that the payoffs are constructed n such a way that at every nformaton set h k, for k = 1,...,K, player s ndfferent between acton a k and all other actons avalable at h k, gven hs belefs β hk, and gven σ hk. If a player nformaton set h does not belong to {h 1,...,h K }, then, by constructon of the payoffs, for every node x h all payoffs followng x are equal, and hence local sequental ratonalty follows trvally. We fnally show that (σ, β) s not sequentally ratonal at h 1 = h K 1. For every acton a at h 1 we have, by constructon of the payoffs followng h 1, that u ((a, σ h 1) h 1,β h 1) = u ((a K 1,σ h 1) h 1,β h 1). Hence, u (σ h 1,β h 1) =

9 330 Note / Games and Economc Behavor 40 (2002) u ((a K 1,σ h 1) h 1,β h 1). After choosng a K 1 at h 1, the only feasble payoffs dfferent from zero are the ones followng h 2. By defnton of the payoffs followng h 2 = h K, we have that u ((a K 1,σ h 1) h 1,β h 1) equals P (ak 1,σ h 1)( x h 1,β h 1) σh 2(a K )1 + P (ak 1,σ h 1)( h 2 h 1,β h 1)( 1 σh 2(a K ) ) β h 2(x ). (2.2) We may thus conclude that u (σ h 1,β h 1) s equal to (2.2). Let σ be the player strategy defned as follows: (1) at h 2, t chooses wth probablty one the acton a K defned above, (2) at nformaton set h 1 t chooses wth probablty one the acton a K 1 leadng to h 2, and (3) at all other player nformaton sets t concdes wth σ. It can be verfed that u ((σ,σ ) h 1,β h 1) equals P (ak 1,σ h 1 )(x h 1,β h 1). By Eq. (2.1), there exsts a strategy σ wth P (σ,σ )(h 2 h 1,β h 1)>0. Snce there s no player nformaton set between h 1 and h 2, and a K 1 s the unque acton that leads from h 1 to h 2, t follows that P (ak 1,σ h 1)(h 2 h 1,β h 1)>0. We know that the rato n (2.1) does not depend upon the choce of σ, as long as P (σ,σ )(h 2 h 1,β h 1)>0. Hence, β h 2(x )< P (σ,σ )(x h 1,β h 1) P (σ,σ )(h 2 h 1,β h 1) = P (ak 1,σ h 1)(x h 1,β h 1) P (ak 1,σ h 1)(h 2 h 1,β h 1), whch mples that P (ak 1,σ h 1)( x h 1,β h 1) >βh 2(x )P (ak 1,σ h )( h 2 h 1,β 1 h 1). (2.3) Snce σ h 2(a K )<1, t follows from (2.3) that ( u (σ,σ ) h 1,β h 1) = P(aK 1,σ h )( x h 1 (,β 1 h 1) > (2.2) = u σ h 1,β h 1). Hence, (σ, β) s not sequentally ratonal at h One-devaton property for updatng systems In ths secton, we turn to the context n whch players are no longer assumed to hold correct conjectures about the opponents future behavor. As mentoned n the ntroducton, we restrct our attenton to the case of two players. For the sake of convenence, we further assume that there are no chance moves. Before statng the result, we need some termnology Updatng systems A pure strategy for player s a vector s = (s (h)) h H,wheres (h) A(h) for all h H. Let S be the set of pure strateges for player. Every player

10 Note / Games and Economc Behavor 40 (2002) holds at each of hs nformaton sets a conjecture about the opponent s behavor that s compatble wth the event of reachng ths nformaton set. Such vectors of conjectures are called updatng systems (cf. Battgall, 1997). Formally, for every h H and both players, let S (h) ={s S s j S j such that (s,s j ) reaches h} be the set of player strateges that are compatble wth the event of reachng h. Here, we always assume that j. By perfect recall, t holds that astrategyprofle(s 1,s 2 ) reaches h f and only f s 1 S 1 (h) and s 2 S 2 (h). An updatng system for player s a vector c = (c h ) h H where c h s a probablty dstrbuton on S j (h) for every h H Sequental ratonalty Let s be a player strategy and h H. By s h we denote the strategy that at every h H precedng h chooses the unque acton at h leadng to h, and at all other nformaton sets concdes wth s. By constructon, s h S (h). We say that the strategy s s sequentally ratonal wth respect to the updatng system c f for all h H t holds that u (s h,c h ) = max s u (s h,c h ). Here, u (s h,c h ) s the expected payoff nduced by s h and c h. For every s S, nformaton set h H and acton a A(h), let (a, s h ) be the player strategy that chooses a at h and concdes wth s at all other nformaton sets. We say that s s locally sequentally ratonal wth respect to the updatng system c f for all h H t holds that u (s h,c h ) = max a A(h) u ((a, s h ) h,c h ) One-devaton property Let S be an extensve form structure and c an updatng system for player n S. We say that c satsfes the one-devaton property f for every payoff functon u and every strategy s S the followng holds: s s sequentally ratonal wth respect to c n the game Γ = (S,u) f and only f s s locally sequentally ratonal wth respect to c n Γ Updatng consstency Let µ 1,µ 2 (S ) be two mxed strateges for player and let T j S j. We say that µ 1 s equvalent to µ 2 on T j f for every s j T j the probablty dstrbutons on the termnal nodes nduced by (µ 1,s j ) and (µ 2,s j ) are dentcal. Let c be an updatng system and h 1,h 2 H be such that h 2 follows h 1 and c h 1(S j (h 2 )) > 0. Here, we use the conventon c h 1(S j (h 2 )) = s j S j (h 2 ) c h 1(s j ). By c h 1 h2 we denote the condtonal probablty dstrbuton on S j (h 2 ) gven by c h 1 h 2(s j ) = c h 1(s j ) c h 1(S j (h 2 ))

11 332 Note / Games and Economc Behavor 40 (2002) for all s j S j (h 2 ). We say that an updatng system c s updatng consstent f for every two nformaton sets h 1,h 2 H where h 2 follows h 1 and c h 1(S j (h 2 )) > 0 t holds that c h 2 s equvalent to c h 1 h 2 on S (h 2 ). The ntuton of updatng consstency s bascally the same as n Secton 2: f player s conjecture at h 1 about the opponent s behavor s compatble wth the event of reachng h 2, then hs conjecture at h 2 should be nduced by hs conjecture at h 1, up to nessental dfferences. By the latter we mean that player, when updatng hs conjecture, s allowed to shft weght from one opponent s strategy to some other, as long as t does not affect the expected outcome condtonal on h 2 beng reached. Updatng consstency s somewhat weaker than the noton of consstent updatng systems, as used by Battgall (1997). An updatng system c s called consstent f for all h 1,h 2 H t holds that c h 2 s equal to c h 1 h 2 whenever h 2 comes after h 1 and c h 1(S j (h 2 )) > 0. Clearly, consstency mples updatng consstency, but the reverse s not true. In order to llustrate the dfference between updatng consstency and consstency, consder the example n Fg. 1. Let h 1, h 2 be the frst and the second nformaton set controlled by player 2, respectvely. Let c 2 = (c 2h 1,c 2h 2) be player 2 s updatng system gven by c 2h 1 = 1 2 (a,e,g,k)+ 1 2 (b,e,h,k) and c 2h 2 = (a,e,g,l). Here, 1 2 (a,e,g,k)+ 1 2 (b,e,h,k) denotes the probablty dstrbuton whch assgns equal probablty to the strateges (a,e,g,k)and (b,e,h,k).the updatng system s not consstent, snce c 2h 1 h 2 = (a,e,g,k) c 2h2. However, the updatng system s updatng consstent snce c 2h 1 h 2 and c 2h 2 are equvalent f player 2 chooses from S 2(h 2 ), that s, f player 2 plays c. Beforng statng the theorem, we brefly outlne how the setup could be generalzed to games wth more than two players. In ths case, player s updatng system would be a vector c = (c h ) h H assgnng to every nformaton set h H some probablty dstrbuton c h (S (h)) on the set of opponents strategy Fg. 1.

12 Note / Games and Economc Behavor 40 (2002) profles S (h) leadng to h. If we assume that conjectures are uncorrelated then c h should be the product of probablty dstrbutons on the opponents ndvdual strategy spaces. If conjectures are allowed to be correlated then c h may be any probablty dstrbuton on S (h). The remanng concepts can be generalzed n a straghtforward fashon to games wth more than two players n both the correlated and the uncorrelated case. (See also Battgall, 1997). Theorem 3.1. Let S be an extensve form structure wth two players. Then, an updatng system n S satsfes the one-devaton property f and only f t s updatng consstent. Proof. (a) We frst show that every updatng system whch s updatng consstent satsfes the one-devaton property. Let the updatng system c be updatng consstent and let the strategy s be locally sequentally ratonal wth respect to c n some game Γ = (S,u). We show that s s sequentally ratonal wth respect to c.lets be an arbtrary pure strategy for player. We prove that u (s h,c h ) u (s h,c h ) (3.1) at every nformaton set h H. We procede by nducton on the number of player nformaton sets that follow h. If h s not followed by any other nformaton set of player then the above nequalty holds by local sequental ratonalty and the observaton that u (s h,c h ) depends only on the acton prescrbed by s at h. Now, let k N and assume that (3.1) holds for all player nformaton sets that are followed by at most k other player nformaton sets. Let h H be followed by at most k + 1 player nformaton sets. Let H (s ) be the set of player nformaton sets h wth the followng propertes: (1) h follows h, (2) s h S (h ) and (3) there s no player nformaton set between h and h. Let Sj 0(h, s ) be the set of strateges s j S j (h) for whch (s h,s j ) does not reach any h H (s ). For every s j S j (h)\sj 0(h, s ), the strategy profle (s h,s j ) reaches exactly one h H (s ). Usng perfect recall, t may be checked that (s h,s j ) reaches h H (s ) f and only f s j S j (h ). Moreover, we clam that the sets S j (h ) are dsjont for h H (s ). In order to see ths, assume that s j S j (h 1 ) S j (h 2 ) for two dfferent h 1,h 2 H (s ). Hence, there exst s 1,s2 such that (s 1,s j ) reaches h 1 and (s 2,s j ) reaches h 2. By constructon of the set H (s ), all paths to h1 and h 2 pass through h and contan the acton prescrbed by s at h. Hence, all paths to h1 and h 2 contan the same sequence of player actons. But ths mples that (s 1,s j ) and (s 2,s j ) should lead to the same nformaton set n H (s ), whch s a contradcton. We may therefore conclude that every s j S j (h) ether belongs to Sj 0(h, s ) or belongs to exactly one S j (h )

13 334 Note / Games and Economc Behavor 40 (2002) wth h H (s ). Consequently, u (s h,c h ) = c h (s j )u (s h,s j ) s j S j (h) = = = h H (s ) s j S j (h ) + c h (s j )u (s h,s j ) s j S 0 j (h,s ) c h (s j )u (s h,s j ) h H (s ) c h (S j (h ))>0 + c h ( Sj (h ) ) s j S j (h ) c h (s j )u (s h,s j ) s j Sj 0(h,s ) ( c h Sj (h ) ) u (s h,c h h ) h H (s ) c h (S j (h ))>0 + s j S 0 j (h,s ) c h (s j )u (s h,s j ). c h (s j ) c h (S j (h )) u (s h,s j ) Snce c s updatng consstent, we have that u (s h,c h h ) = u (s h,c h ) for all h H (s ) wth c h(s j (h )) > 0. Hence, u (s ( h,c h ) = c h Sj (h ) ) u (s h,c h ) h H (s ) c h (S j (h ))>0 + c h (s j )u (s h,s j ) s j Sj 0(h,s ) = ( Sj (h ) ) u (s h,c h ) h H (s ) c h + s j S 0 j (h,s ) c h (s j )u (s h,s j ). For all h H (s ) t holds, by defnton, that s h S (h ), and hence s h = s h for all h H (s ). Consequently, u (s h,c h ) = ( Sj (h ) ) u (s h,c h ) h H (s ) c h

14 Note / Games and Economc Behavor 40 (2002) s j S 0 j (h,s ) c h (s j )u (s h,s j ). (3.2) Snce every h H (s ) s followed by at most k player nformaton sets, we know by nducton assumpton that u (s h,c h ) u (s h,c h ) for all h H (s ). Ths mples that u (s h,c h ) ( Sj (h ) ) u (s h,c h ) h H (s ) c h + s j S 0 j (h,s ) c h (s j )u (s h,s j ). (3.3) Let s be the player strategy whch concdes wth s at h and concdes wth s at all other nformaton sets. It can be checked that H (s ) = H (s ) and that Sj 0(h, s ) = S0 j (h, s ). Moreover, u (s h,c h ) = u (s h,c h ) for all h H (s ) and u (s h,s j ) = u (s h,s j ) for all s j Sj 0 (h, s ). Together wth (3.3) we obtan that u (s h,c h ) ( Sj (h ) ) u (s h,c h ) c h h H (s ) + c h(s j )u (s h,s j ) s j Sj 0(h,s ) = u (s h,c h ), where the last equalty follows from substtutng s by s n (3.2). Snce s dffers only at h from s and s s locally sequentally ratonal wth respect to c, t holds that u (s h,c h ) u (s h,c h ). Ths mples that u (s h,c h ) u (s h,c h ), whch completes the proof of (a). (b) Next, we prove that every updatng system whch s not updatng consstent does not satsfy the one-devaton property. Let c be an updatng system n S whch s not updatng consstent. We show that there s a payoff vector u for the termnal nodes and a pure strategy s for player such that s s locally sequentally ratonal wth respect to c n the extensve form game Γ = (S,u) but not sequentally ratonal. Snce c s not updatng consstent, there are h 1,h 2 H where h 2 follows h 1 and c h 1(S j (h 2 )) > 0 such that c h 2 s not equvalent to c h 1 h 2 on S (h 2 ). Hence, there s some s S (h 2 ) such that the probablty dstrbutons on the termnal nodes nduced by (s,c h 2) and (s,c h 1 h2) are dfferent. Let these probablty dstrbutons be denoted by P (s,c h 2 ) and P (s,c h 1 h 2 ), respectvely. We thus can fnd a termnal node z wth P (s,c h 2 )(z )<P (s,c h 1 h 2 )(z ). (3.4)

15 336 Note / Games and Economc Behavor 40 (2002) The above nequalty mples that P (s,c h 1 h 2 )(z )>0, and hence s necessarly chooses all the player actons on the path to z. Snce s S (h 2 ) and c h 1 h 2 (S j (h 2 )), t follows that (s,c h 1 h 2) passes h2 wth probablty one, and hence z follows nformaton set h 2. By perfect recall, there s a unque sequence h 1,...,h K of player nformaton sets wth the followng propertes: (1) h k follows h k 1 for all k, (2) there s no player nformaton set between h k 1 and h k for all k, (3) there s no player nformaton set before h 1 and (4) h K = h 2. For every k<k,let a k be the unque acton at h k whch leads to h k+1, and let a K be the unque acton at h K = h 2 whch leads to z. Snce we know that s chooses all the player actons that lead to z, t holds that s chooses acton a k at h k for all k = 1,...,K. Snce h 1 precedes h 2 = h K, t must hold that h 1 = h k for some k {1,...,K 1}. Let b k be some acton dfferent from a k at h 1, and let b K be some acton dfferent from a K at h 2. Let s be the player strategy whch chooses b k at h 1, chooses b K at h 2 and concdes wth s at all other nformaton sets. We now defne the player payoffs u followng h k by nducton on k. We start wth the player payoffs followng h K = h 2. Set u (z ) = 1. For every termnal node z followng h K but not followng acton a K, set u (z) = P (s h 2,c h 2 )(z ). For every termnal node z z followng acton a K, we set u (z) = 0. Now, suppose that the player payoffs u (z) have been defned for all termnal nodes z followng h k+1. We defne the player payoffs followng h k but not followng h k+1 n the followng way. For every termnal node z followng acton a k but not followng h k+1, set u (z) = 0. Let (a k,s h k ) be the player strategy that chooses a k at h k and concdes wth s at all other player nformaton sets. Snce all player payoffs followng a k have already defned, the expresson u ((a k,s h k ) hk,c hk ) s well-defned. For all termnal nodes z followng h k but not followng acton a k, set u (z) = u ((a k,s h k ) hk,c hk ). Fnally, for all termnal nodes not covered by the procedure above, we set u (z) = 0. For player j, we set u j (z) = 0 for all termnal nodes z. It may be verfed easly that s s locally sequentally ratonal wth respect to c n the game Γ = (S,u). The payoffs are constructed n such a way that at every nformaton set h k, wth k = 1,...,K, player s ndfferent between acton a k and any other acton avalable, gven that s s played at all other player nformaton sets, and gven the conjecture c hk. If h s a player nformaton set that follows h 2, we dstngush two cases. If h les on the path to z, then there s a unque acton a at h that leads to z. By constructon of the payoffs followng acton a K at h 2, acton a at h leads to payoff 1, f z s reached, and leads to payoff zero otherwse, and all other actons at h lead to payoff zero for sure. Snce s concdes wth s at h, and s chooses a here, we have that s chooses a at h, whch s optmal.

16 Note / Games and Economc Behavor 40 (2002) If h does not le on the path to z, then, by constructon of the payoffs, all payoffs followng h are equal, and hence local sequental ratonalty follows trvally. If a player nformaton set h does not follow h 2 nor precede h 2, all payoffs followng h are equal, and local sequental ratonalty follows trvally. We fnally show that s s not sequentally ratonal. To ths purpose, we prove that u (s h 1,c h 1)>u (s h 1, c h 1). Snce s chooses b k at h 1, we have, by defnton of the payoffs followng b k, that u (s h 1, c h 1) = u ( (ak,s h 1 ) h 1,c h 1). (3.5) Note that (a k,s ) chooses all the actons a h 1 k, for k = 1,...,K 1, that lead to nformaton set h 2. Hence, by constructon of the payoffs, the only termnal nodes whch are feasble for ((a k,s ) h 1 h 1,c h 1) and have payoffs dfferent from zero, are the ones followng h 2. Snce (a k,s ) S h 1 (h 2 ), the probablty of ((a k,s ) h 1 h 1,c h 1) reachng h 2 s equal to c h 1(S j (h 2 )). Recall that (a k,s ) chooses b h 1 K at h 2 and that all termnal nodes followng b K have payoff P (s h 2,c h 2 )(z ). Gatherng all these nsghts leads to the observaton that u ( (ak,s h 1 ) h 1,c h 1) = ch 1( Sj ( h 2 )) P (s h 2,c h 2 )(z ). Snce s S (h 2 ) we have that s h 2 = s. Together wth (3.5) t mples that u (s h 1, c h 1) = c h 1 ( Sj ( h 2 )) P (s,c h 2 )(z ). (3.6) On the other hand, s chooses all the player actons a k, for k = 1,...,K 1, that lead to h 2. Hence, the only termnal nodes feasble for (s h 1,c h 1) and havng payoffs dfferent from zero are the ones followng h 2. Recall that s chooses a K at h 2, and that the only termnal node followng a K wth non-zero payoff s z, wth u (z ) = 1. Hence, u (s h 1,c h 1) = u (s,c h 1) = P (s,c h 1 )(z ). Snce z follows h 2, t holds that only player j strateges s j S j (h 2 ) can lead to z, and therefore P (s,c h 1 )(z ) = c h 1(S j (h 2 )) P (s,c h 1 h 2 )(z ). It follows that u (s h 1,c h 1) = c h 1( Sj ( h 2 )) P (s,c h 1 h 2 )(z ). (3.7) Snce, by assumpton, c h 1(S j (h 2 )) > 0, and P (s,c h 2 )(z )<P (s,c h 1 h 2 )(z ), t follows from (3.6) and (3.7) that u (s h 1, c h 1)<u (s h 1,c h 1). Acknowledgments Ths paper has been wrtten whle the author was vstng the Department of Quanttve Economcs at Maastrcht Unversty. The author wshes to thank ther hosptalty. He would also lke to thank an anonymous referee for hs suggestons.

17 338 Note / Games and Economc Behavor 40 (2002) References Battgall, P., On ratonalzablty n extensve games. J. Econ. Theory 74, Battgall, P., Bonanno, B., Recent results on belef, knowledge and the epstemc foundatons of game theory. Res. Econ. 53, Ben-Porath, E., Ratonalty, Nash equlbrum, and backwards nducton n perfect nformaton games. Rev. Econ. Stud. 64, Bernhem, D., Ratonalzable strategc behavor. Econometrca 52, Dekel, E., Fudenberg, D., Levne, D., Payoff nformaton and self-confrmng equlbrum. J. Econ. Theory 89, Dekel E., Fudenberg D., Levne D., Subjectve Uncertanty over Behavor Strateges: A Correcton. Harvard Unversty. Fudenberg, D., Levne, D., Self-confrmng equlbrum. Econometrca 61, Greenberg, J., Stable (Incomplete) Agreements n Dynamc Games: Worlds Apart but Actng Together. McGll Unversty. Hendon, E., Jacobsen, J., Sloth, B., The one-shot-devaton prncple for sequental ratonalty. Games Econ. Behav. 12, Kreps, D., Wlson, R., Sequental equlbra. Econometrca 50, Kuhn, H.W., Extensve form games and the problem of nformaton. Ann. Math. Stud. 28, Pearce, D., Ratonalzable strategc behavor and the problem of perfecton. Econometrca 52, Selten, R., Speltheoretsche Behandlung enes Olgopolmodells mt Nachfragezet. Z. Gesam. Staatswss. 121, , Selten, R., Reexamnaton of the perfectness concept for equlbrum ponts n extensve games. Internat. J. Game Theory 4,