UNIVERSITY OF NOTTINGHAM. Extensive Games of Imperfect Recall and Mind Perfection

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1 UNIVERSITY OF NOTTINGHAM DISUSSION PAPERS IN EONOMIS No. 98/ Extensve Games of Imperfect Recall and Mnd Perfecton Matt Ayres Abstract In ths paper we examne how the addton of mperfect recall as a perturbaton to a perfect recall game can be used as an equlbrum refnement. We dscuss the propertes of two such concepts, from the addton of complete confuson between smlar hstores to consderng small trembles n a players belefs. entral to our dscusson s the noton of whch decsons can reasonably be confused and we suggest that modellng nformatonal confuson may be a useful way of measurng strategc complexty.

2 2. Introducton Game theoretc modellng of economc agents as ratonal players has led to paradoxes and serous dscrepances between observaton and theory. One way we mght hope to gan further nsghts nto behavour and exstng results s by addng psychologcal elements to the reasonng process of players. Such models are n the class of those dealng wth bounded ratonalty. The smple psychologcal addton we consder here s to model players wth bad memores or mperfect recall. Due to recent work by Pccone and Rubnsten (997), games and decson problems wth mperfect recall have been re-examned. Ths re-examnaton has ndcated dffcultes both n modellng and the possblty of a new type of tme nconsstency problem, as demonstrated by ther example of the absentmnded drver. Most of the papers followng Pccone and Rubnsten, have concentrated on ths tme consstency problem, suggestng a varety of nterpretatons and resolutons (Aumann et al (997a, 997b), Battgall (997), Bnmore (forthcomng), Glboa (997), Grove and Halpern (997), Halpern (997) and Lpman (997)). In ths paper we take a dfferent approach, ntroducng mperfect recall as a perturbaton to a perfect recall game and examne how the addton of bad memory affect the set of equlbrum predctons. Our ratonale s that players should, ceters parbus, prefer equlbra n whch they use less cogntvely demandng strateges, n our case ones that requre less memory. Note the lnk between memory lmtatons and strategc complexty has been made n the lterature on fnte automata play n nfntely repeated games, (for example, Abreu and Rubnsten (988) and Rubnsten (987)). Our frst refnement, whch we call mnd perfecton, deals wth a complete reducton of a player's ablty to dstngush between smlar decson nodes. By ntroducng as much mperfect recall and mperfect nformaton as possble we ask the queston Whch, f any, equlbra survve? Thus, a mnd perfect strategy corresponds to a very smple strategy n whch the player does not have to dstngush smlar decson nodes. The general acceptance of bounded ratonalty n economcs s largely s due to the work of Smon (978 and others). In partcular Smon s emphasses the potental lnkages between economc and psychologcal research.

3 3 As an example, consder the perfect recall game n fgure 2. A ratonal player would contnue at the frst decson node and at the thrd decson node he s ndfferent between actons E and (both {,,E} and {,,} are subgame perfect). Under mnd perfecton, we assume that player confuses the smlar decsons he makes at the frst and thrd decson node. A decson node or hstory s sad to be smlar to another f t satsfes the mnmal requrements for beng n the same nformaton set. 3 For an absentmnded player, confusng the frst and thrd decson node, acton s strctly better than acton E. Our nterpretaton s that for player the strategy of always contnung {,} requres less memory than the strategy {,E},.e. strategy {,} s less complex than strategy {,E}. Fundamental to ths example and the rest of ths paper s the noton of whch decson nodes (hstores) can reasonably be confused, a noton we descrbe as smlarty. Fgure 2, E E E 0,0 0,0, Mnd perfecton s a very strct requrement and consequently t lacks applcaton n all but a few games. However, a mnd perfect equlbrum s stable even when the game s perturbed to the extent that all smlar hstores are n the same nformaton set. Followng Selten s (978, p47-52) dstncton, we wsh to classfy solutons that requre only mnmal nformaton as correspondng to problem of a routne nature. An equlbrum whch fals to meet the requrements of a mnd perfect equlbrum needs applcaton of some further reasonng. 2 The ntal node, where h=, s represented by a bold crcle. 3 As a mnmum we requre that players always know the set of actons from whch they are choosng and whose move t s.

4 4 Our second refnement consders the possblty of mperfect recall and mperfect nformaton as a small perturbaton n a player s belef about what decson s beng made. Equlbra whch survve such perturbatons we call tremblng mnd perfect, snce n a smlar way to tremblng hand (perfect) equlbrum we wsh to fnd equlbra whch survve the addton of mstakes. The ratonale here s that these mstakes are based around perturbatons n belefs (mstakes confusng smlar hstores) rather than mstakes n takng actons. Intutvely we fnd mstakes based on perturbatons n belefs to be more appealng snce they are determned by the player's percepton of the games structure,.e. we can provde a psychologcal explanaton. 4 Secton 2 ntroduces the requred notaton and formal defntons of mperfect recall. Secton 3 defnes our noton of smlarty and notes the problems n such an explct defnton. Our two equlbrum refnements and dscusson of ther propertes are gven n secton 4. We also note the concept of a tremblng mnd and ndependent. 2. An Extensve Game wth Imperfect Recall We defne a fnte extensve game Γ wth mperfect nformaton and mperfect recall as Γ =< H, N, P, f, ρ ( h), A( h) h H,(I ) N, X ( h), where: h H N > H s a set of fnte hstores, such that H and f a sequence of actons (a k ) k=...k H then (a k ) k=...l H for all L<K. All hstores begn wth and contan the moves made n sequental order. A hstory s nterpreted as a physcal descrpton of all the moves made by the players (ncludng chance) and we label each node wth ths hstory. We defne Z to be the set of termnal hstores, where a hstory s sad to be termnal f there s no a K+ such that (a k ) k=...k+ H. The node wth a hstory of just, s called the ntal node. Graphcally we represent the ntal node of any game wth a bold crcle. 4 Myerson s (978) Proper equlbrum takes a dfferent approach to ratonalsng trembles, but mstakes stll reman based on actons rather than belefs.

5 5 N s a fnte set of players, not ncludng the chance (nature) player whch we denote c. P the player functon assgns a player n N { c} to move after every non-termnal hstory H\Z. For each player n N there s a preference relaton f on Z. When P(h)=c the next move s made by the chance player and ρ( h ) assgns a probablty of occurrence to each acton a A( h). After every non-termnal hstory h H \ Z player P(h) chooses an acton from A(h)={a:(h,a) H}. To avod degeneracy we assume A(h) contans at least two elements. For each player N there s a partton I of hstores h H at whch P(h)=. For each I I (an nformaton set), any two hstores h and h n I must satsfy the property that A( h) = A( h ). We label the actons avalable at the nformaton set I, as A( I ) such that A( I ) = A( h) for all h I. The nterpretaton of an nformaton set s that all hstores (nodes) n I cannot be dstngushed from one another. Graphcally we depct nformaton sets as a shaded box lnkng two or more decson nodes. { } I a X (h) s the players experence at the hstory h, consstng of a par X ( h), X ( h) such that: X I ( h) s a sequental orderng of nformaton sets player has vsted n order to reach h (player s experence of nformaton sets at the hstory h). X a ( h) s a sequental orderng of actons player has taken n order to reach h (player s experence of actons at the hstory h). Usng our defnton of an extensve game we can defne stuatons of mperfect recall.

6 6 Defnton: Imperfect Recall A player s sad to have mperfect recall f X ( h) X ( h ) for any hstores h and h that are n the same nformaton set I. In lne wth the Pccone and Rubnsten (997), we dstngush three types of recall problems. Imperfect Recall of Informaton Sets: where a player forgets the sequence of nformaton sets through whch play has past. Imperfect Recall of Actons: where the player recalls he has made a pror move but not what acton he chose. Absentmndedness: where the player cannot recall whether he has made a pror move or not. More formally, Defnton: Imperfect Recall of Informaton Sets I I If for some h and h n the same nformaton set, X ( h) X ( h ) then player s sad to have mperfect recall of nformaton sets. Defnton: Imperfect Recall of Actons a a If for some h and h n the same nformaton set, X ( h) X ( h ) then player has mperfect recall of actons. Defnton: Absentmndedness If for some h and h n the same nformaton set, a A( h) s part of the sequence of actons X a ( ) then player s sad to suffer from absentmndedness. For h and h n the same nformaton set, the statement a A( h) s part of the sequence of actons X a ( h ') s equvalent to h beng a subhstory of h. Thus the defnton presented above s equvalent to that of Pccone and Rubnsten.

7 7 Defnton: Absentmndedness (Pccone and Rubnsten) A player s absentmnded f for some h h =(a k ) k=...k H and h=(a k ) k=...l H for L<K. and h are n the same nformaton set where Absentmndedness represents a specal case of mperfect recall where the player fals to recall both vstng an nformaton set and the acton he took there. From our defnton of absentmndedness snce a A( h) s part of the sequence of actons X a ( ) the I I sequence of nformaton sets must be such that X ( h) X ( h ) I ( ) ( ) I X h X h (our defnton of mperfect recall of nformaton sets). and hence Snce, under absentmndedness we allow h to be a subhstory of h, for h and h, the same nformaton set can be vsted more than once. Whlst we allow an nformaton set to be vsted more than once t must be to a dfferent decson node,.e. we do not allow nfnte cycles or H (the set of hstores) to be nfnte. Fgure 2 shows a game wth both absentmndedness and mperfect recall of actons. At hs frst decson node player suffers from absentmndedness, that s he s unsure whether he s ndeed at the ntal node or he has already chosen the acton and s at node c. The player s smlarly confused at node c. In the nformaton set jonng nodes a and b player has mperfect recall of actons, that s he knows he has chosen ether A or B at the ntal node but cannot recall whch.

8 8 Fgure 2: An Extensve Game wth Absentmndedness and Imperfect Recall of Actons (Isbell 957, p85) A B a b a b a b A c,- 0,0 B,- 0, ,0 0,0,-,- 0,0 It should that whlst mperfect recall s clearly defned n the extensve form, many defntons preclude cases of mperfect recall due to the dffcultes t mposes. For example, Kreps and Wlson s (982) sequental equlbrum extends only to cover perfect recall games. hoce of strategy n games of mperfect recall s problematc because the possble structure of nformaton sets renders some results concernng strategc equvalence nvald. Defnton: A pure strategy for player N n an extensve game s a functon assgnng a sngle acton n A( I ) to each nformaton set I I. A mxed strategy for player N s a probablty measure π over the set of player s pure strateges. A behavoural strategy for player N s a collecton of ndependent probablty measures ( I ) for all I I, where ( ) β I assgns a probablty to each acton n A I ( ). β We nterpret a pure strategy as a plan of acton formulated before play begns. A mxed strategy s thus a randomsaton over such plans of acton and a behavoural strategy s

9 9 a sngle plan of acton, wth nstructons for the player to randomse on reachng an nformaton set. In the case of absentmndedness we allow the player to vst an nformaton set more than once n any play of the game. The player s restrcted to followng the same behavoural randomsaton at all decson nodes n the nformaton set and ths randomsaton s assumed to be realsed each tme he/she vsts that nformaton set. Further dscusson of ths pont can be found n Pccone and Rubnsten (997). In games nvolvng mperfect recall behavoural and mxed strateges are not outcome equvalent (see for example, Osborne and Rubnsten 994, p ). In ths paper we restrct strategy choce to behavoural strateges. We gnore mxed strateges wthout behavoural equvalents snce they necessarly nvolve the player recallng the result of a centralsed randomsaton chosen at the begnnng of the game. Such a devce can be used to overcome some of the problems of mperfect recall. Snce behavoural strateges are defned at the level of the nformaton set, no such problem arses. 3. Smlarty Our formal defnton of (strong) smlarty, states that one hstory s smlar to another f they satsfy the mnmal requrements for beng n the same nformaton set. We are only nterested n the number of dstnct smlar hstores and thus defne smlarty to exclude a hstory beng smlar to tself. Further we assume that no hstory at whch nature moves can be smlar to any other. The latter requrement s mposed snce we fnd no clear nterpretaton of what t means for nature to become confused. Defnton: (Strongly) Smlar Hstores A hstory h H \ Z s smlar to the hstory h H \ Z f and only f h h and A( h) = A( h ) and P( h) = P( h ) c.

10 0 In ths paper we only consder smlarty as defned above, but note that decdng what nformaton s relevant for decsons to be classed smlar s not an easy task.. Frstly, we requre that actons are not solely defned n terms of what choces are taken (e.g. whether to go left or rght) but also reflect the state n whch the choce s made (e.g. whether t s lght or dark). Thus f a player s able to dstngush between lght and dark, the acton left when lght and left when dark should be labelled dfferently n the extensve form (.e. the decsons cannot be smlar). Secondly, f we allow players to have small doubts about ther memores rather than complete memory loss, t may be more approprate to consder a wder range of possble confusons. For example, consder that f at one decson node a boundedly ratonal player chooses between a set of actons {a,b,c} and at another {a,b,c,d}. We may wsh to model the stuaton where on arrvng at the frst decson node the player has some postve belef he s at the second decson node but has faled to notce the acton {d}.

11 4. Equlbrum Refnements For smplcty we state our formal defnton of mnd perfecton as a refnement of subgame perfecton (.e. applcable to perfect nformaton games). Let S(h) be the set of hstores strongly smlar to h. Defnton: Subgame Perfect Equlbrum (SPE) For Γ a game wth perfect recall and perfect nformaton, let Γ(h) be a subgame of Γ begnnng at the hstory h. A subgame perfect equlbrum s a strategy profle s * (consstng of a pure strategy s * for each player ) whch, for all hstores h H \ Z, the strategy profle begnnng at h, s a Nash equlbrum of the subgame Γ(h). Defnton: Mnd Perfect Equlbrum (MPE) For Γ a game wth perfect recall and perfect nformaton, a mnd perfect equlbrum s one whch s both subgame perfect and satsfes the property that f h and h are smlar hstores then s * (h)= s * (h ), where s * (h) s the choce of acton n the pure strategy s * at the hstory h. We consder the followng propertes of mnd perfecton n game Γ wth perfect recall and perfect nformaton. As prelmnary we defne: Defnton: Strct Domnance n Smlar Hstores An acton a A( h) s sad to be strctly domnant n hstores smlar to h f n at least one hstory h S( h) h, a f a and for all other hstores h S( h) h, af, for all a A( h) and a a. a We obtan the followng results. Theorem 4. All games Γ that contan no smlar hstores have at least one mnd perfect equlbrum. Proof: If Γ all hstores h H \ Z, S ( h) = by defnton mnd perfecton and subgame perfecton are dentcal. Snce every game has at least one subgame perfect

12 2 equlbrum, f there are no smlar hstores, every game must have a mnd perfect equlbrum. Theorem 4.2 In a game Γ wth at least one par of smlar hstores there may exst one mnd perfect equlbrum, there may exst no mnd perfect equlbrum or there may exst multple mnd perfect equlbra. Proof: onsder the sngle player extensve form game (a decson problem) n fgure 3. All three hstores at whch the player moves have smlar hstores. If the payoffs are such that γ > β and γ > α then there s a unque mnd perfect equlbrum {,}. In the case where β > γ and γ > α then there are no mnd perfect equlbrum. If α = β = γ then there are two mnd perfect equlbra {,} and {E,E}. Hence there may exst a unque, there may exst no or there may exst multple mnd perfect equlbra. Fgure 3 γ E E α β Theorem 4.3 A game Γ has a unque mnd perfect equlbrum f for every hstory, (ncludng those where S ( h) = ), there exsts an acton whch satsfes strct domnance n smlar hstores. Proof: In the case where Γ a game wth perfect recall and perfect nformaton has no smlar hstores, S( h) = h. If strct domnance n smlar hstores s satsfed at the hstory h, there exsts a unque a f a for all a A( h) and a a. Snce at each hstory there s a sngle domnant acton there s a unque subgame perfect equlbrum and by 4. a unque mnd perfect equlbrum.

13 3 In the case where Γ, a game wth perfect recall and perfect nformaton, has at least some hstores smlar to h H \ Z. For each player we assume for all hstores h strct domnance n smlar hstores s satsfed by some acton a A( h). Ths acton must be unque for all h S( h) h, snce for some, a A( h ) f a A( h ) for all a a excludes a A( h) f a A( h) for any a a. The acton a A( h ) must be part of a strategy s * that for each player makes up a subgame perfect equlbrum of Γ (snce a s ether strctly or weaker domnant at ). Snce ths acton s unque and the same at all hstores smlar to h, s * wll also be part of the unque mnd perfect equlbrum of Γ. As an example we consder the ultmatum game wth a sngle ndvsble good as shown n fgure 4. In ths game, player can choose to gve the good to player 2 or choose to keep t for hmself, where the payoffs reflect the fnal holdngs of the good. Ths game has two subgame perfect equlbra {keep,y}, (keep,n} and a sngle mnd perfect equlbrum {keep,y}. Whlst subgame perfecton removes the possblty of ncredble threats (such as player 2 choosng {n} followng {gve}), mnd perfecton can be nterpreted as removng weak credble threats. Mnd perfecton uses strct preferences n smlar parts of the game tree to rule out some weakly domnated actons (a weakly credble threat). In the example n fgure 4 player 2 s domnant acton of {y} followng {gve} and ndfference between {y} and {n} followng {keep}, result n the sngle mnd perfect outcome {keep,y}. Ths strategy does not requre player 2 to dstngush hs two decson nodes. onsequently, such a strategy may be seen as one of lower complexty and whch has a lower memory requrement (player 2 does not have to remember what player chose).

14 4 Fgure 4 Keep Gve 2 2 y n y n,0 0,0 0, 0,0 Although we have stated mnd perfect equlbrum as a refnement of subgame perfecton, we can use the same dea to form a crteron that can be appled to any strategy (possbly behavoural) or equlbrum set of strateges. Defnton: Mnd Perfecton rteron A strategy β s sad to be mnd perfect f for all hstores h β and h whch are smlar, ( h) = β ( h ), where β ( h) s behavoural acton n the strategy β at the hstory h. Thus takng the example gven n fgure 3, the strateges {E,E}, {,} and behavoural randomsaton assgnng the same probablty to takng acton {} at both decson nodes satsfes the mnd perfect crteron. In the case where α > β, γ and γ > β, the strategy {E,E} s not a mnd perfect equlbrum accordng to our defnton, t s, however, a Nash equlbrum whch satsfes the mnd perfect crteron. Our second concept looks at the stuaton where each player assgns a postve, but possbly small probablty ε to confusng smlar hstores. Equlbrum strateges whch survve such perturbatons are sad to be tremblng mnd perfect. As a prelmnary we defne a game nvolvng such perturbatons.

15 5 Defnton: The perturbed game Γ For a fnte extensve game Γ wth perfect recall, let Γ be perturbaton of Γ where for all hstores h H \ Z and P(h)=, nature causes player to beleve they are at a hstory h when he/she s at the hstory h, wth probablty ε h ε h >0 for all h S( h),, such that ε h < S ( h) and, for smplcty, we assume that play can never reach a path n whch player makes more than one mstake. Thus any path of the perturbed game where h s confused wth a hstory h and h s a subhstory of h wll be one nvolvng a stuaton of absentmndedness. Any path where h s confused wth a hstory h such that h s not a subhstory of h and h s not a subhstory of h wll be one nvolvng ether mperfect recall of actons. Usng our defnton of a perturbed game we can defne a tremblng mnd perfect equlbrum. Defnton: Tremblng Mnd Perfecton Let β * be a behavoral strategy for player that s a best response gven the actons * of the other players β n the unperturbed game Γ. The behavoral strategy β * sad to tremblng mnd perfect f there exsts a perturbed game Γ n whch β * s also a best response to β *. A tremblng mnd perfect equlbrum conssts of a behavoral strategy for each player whch s a best response n the game Γ. Note that the perturbed game Γ conssts of one path where player makes no mstakes, occurrng wth probablty ε h S ( h) and a further n paths for each hstory h, that has n smlar hstores h each occurrng wth probablty ε h (.e. n s the

16 6 cardnalty of the set S(h)). For example: onsder the one player extensve game (decson problem) Γ, shown n fgure 5, wth a sngle par of smlar hstores h (the players frst move) and h (the player's second move). Ths gves a perturbed game Γ (fgure 6) where nature selects between three possble paths. One path where no h mstakes are made (occurrng wth probablty ( ε ε ) ), one when reachng h the player thnks he s at h (occurrng wth probablty ε h ) and one when reachng h the player thnks he s at h (occurrng wth probablty ε h ). Note n the dark shaded nformaton set the player beleves he s at h and n the lght nformaton set thnks he s at h. h A game Γ wth two hstores h and h both smlar to h and all other hstores dstnct (non-smlar) gves a perturbed game where nature selects between seven possble paths. Fgure 5 γ Fgure 6 α β α β γ h ε h α β N ε h h γ h ( h ε h ε h ) γ α β

17 7 From our defnton of tremblng mnd perfecton, t follows: Theorem 4.4 For any perfect recall game Γ there exsts at least one tremblng mnd perfect equlbrum. The proof of Theorem 4.4 s gven n the Appendx From Theorem 4.4 t follows that any extensve game wth perfect recall has at least one tremblng mnd perfect equlbrum. Note that f Γ contans smlar hstores and there are two or more equlbra n behavoral strateges, equlbra wll be excluded from beng perfect to a tremblng mnd f they nvolve playng actons that are strctly domnated n smlar hstores. More formally, Theorem 4.5 Strateges whch nvolve playng actons whch are strctly domnated n smlar hstores wll not be tremblng mnd perfect. Proof: onsder that under an equlbrum strategy β * the acton a s chosen at the hstory h and that a s strctly domnated n smlar hstores by the acton a. In partcular suppose, that a f a at hstory, where h S ( h). Then t follows that a, cannot be part of a tremblng mnd perfect strategy snce n all possble perturbatons confusng h and wth postve probablty followng a s strctly better. It should be obvous that all mnd perfect strateges wll also be tremblng mnd perfect. onsder two examples. The extensve game shown n fgure the equlbrum n whch player plays {,} and player 2 plays {} s both the unque mnd perfect and unque tremblng mnd perfect equlbrum. The extensve game shown n fgure 7 s more nterestng. Players makng repeated choces between and E, all hstores for player are smlar to three other hstores and all hstores for player 2 smlar to two other hstores. Note that player s second and thrd move mmedately follow one another. Equvalence prncples suggest that such moves could be coalesced nto a sngle choce.e. player selects between the

18 8 actons E, E and. It should be obvous that n ths paper we requre such decsons to reman separate so as to allow the possblty of confuson between these two moves. Fgure 7 2 0,0 E 2 2, E E E E E 0,0 0,0 0,0,, The game n fgure 7 has no mnd perfect equlbra (snce no acton strctly domnates n smlar hstores for player 2) and two tremblng mnd perfect equlbra {E,,,,,} and {E,,,,,E}. Note that for some values of ε h player 2 prefers to play and for other values prefers E, thus both equlbra are tremblng mnd perfect. Note also that player strctly prefers to play at hs last decson node gven he holds some postve belef about beng at a smlar hstory. Fnally t should be noted that not all tremblng mnd perfect equlbra are tremblng hand perfect and not all tremblng hand perfect equlbra are tremblng mnd perfect. Note that n the case where a game has a unque Nash equlbrum, t must also be tremblng hand perfect, sequental and tremblng mnd perfect. The example n fgure 8 has a unque tremblng mnd perfect equlbrum {R,r} and two sequental equlbra, {L,l} and {R,r}. Whlst {L,l} s sequental t s not a tremblng hand perfect equlbrum. Ths s one of the specal cases where tremblng hand and sequental equlbrum dverge. Both concepts have a consstency requrement, but only the former consders trembles (see Kreps and Wlson, 982 p882). Our concept of tremblng mnd perfecton does not have ths same consstency requrement but does ntroduce trembles.

19 9 If we change the game depcted n fgure 8 so that player 2 s decson nodes are all sngletons, {L,l} s stll not a tremblng mnd perfect equlbrum although t s both a sequental equlbrum and s tremblng hand perfect. Smlarly n fgure, {,,} and {,,E} are tremblng hand perfect but only {,,} s tremblng mnd perfect. Not all tremblng hand perfect equlbra are tremblng mnd perfect. Ths s because tremblng mnd perfecton allows for errors to be made between decson nodes that are not n the same nformaton set but satsfy our defnton of smlarty. Fgure 8 L M R l m r l m r l m r 0,0 0,0 0,0 0,0, 0,2 0,0 2,0 2,2 Fgure 9 (Selten's Horse) shows a three player game. Play begns at the top left node and contnues ether across to player 2's node or downwards to player 3's nformaton set. Ths example llustrates that not all tremblng mnd perfect equlbrum are tremblng hand perfect. The unque tremblng hand perfect (and sequental) equlbrum of the game s {A,a,r}, where r s played wth at least a probablty of three quarters. However, {D,a,L} s also a tremblng mnd perfect equlbrum. Note that, we do not clam that ths last equlbrum s necessarly sensble, merely that t cannot be ruled out by a tremblng mnd (although t can by a tremblng hand). 5 5 Snce {D,a,L} s a Nash equlbrum whch does not nvolve player 3 choosng an acton whch s strctly domnated n smlar hstores and hence t must be tremblng mnd perfect. Note that players and 2 have no smlar hstores so for them ths requrement s trvally satsfed.

20 20 Fgure 9 3 D A a 2,, d 3 L R L R 3,3,2 0,0,0 4,4,0 0,0, We summarze the relatonshps between equlbra n the followng theorem, Theorem 4.6 There exst extensve games such that: I. not all tremblng mnd perfect equlbra are tremblng hand perfect II. not all tremblng hand perfect equlbra are tremblng mnd perfect III.those equlbra whch are tremblng hand perfect but not sequental equlbra, are tremblng mnd perfect. 5. oncludng Remarks We fnd the concepts presented n ths paper to be appealng n that they use stablty to the addton of confuson (mstakes n belefs) as a selecton crteron. Part of ths appeal s that our models can be nterpreted as gvng a psychologcal explanaton of mstakes. The extreme case of mnd perfecton corresponds to lmtng the players to very smple strateges. The problem wth such severe restrcton on strategy choce s that s possbly further from a desred model of descrptve behavor than s that of the fully ratonal player. At the other extreme we have consdered strateges that are perfect to a tremblng mnd. By addng only an arbtrarly small amount of confuson we can guarantee exstence, although the power of the concept s lmted to removng only some weakly domnated strateges. Thus whlst we fnd the dea of confuson to have psychologcal appeal, we do not see ether of our concepts as presentng a model that s suffcently rch to be appealng n descrbng actual behavour. The purpose of consderng the extremes of mnd and tremblng mnd perfecton s largely nstructve; we consder players should play mnd

21 2 perfect strateges where possble and avod those whch are not tremblng mnd perfect. To move towards a model wth more descrptve appeal we need to consder levels of confuson between our two extremes. In such a model the confuson or the fear of confuson between smlar decsons becomes an ntegral part of the decson makng process. We also suggest that such a model may gve us a way of rankng strateges on grounds of complexty. A smple strategy beng one whch s optmal rrespectve of the players fear about becomng confused n carryng out hs strategy. It s also mportant to note that the power of our concepts s dependent upon our defnton of smlarty,.e. what hstores can reasonably be confused. There s unlkely to be any sngle correct defnton of smlarty, but rather t s lkely to depend upon the stuaton beng modeled and n partcular the player's percepton of the dfferent decsons beng made. Fnally, all the models consdered here have assumed memory lmtatons are exogenously determned. An alternatve would be to try formally ncorporate the costs of recall nto the games structure. Such models would allow players a more actve cognton and seem appealng f the costs of recall are easy to dentfy, for example, where players are frms we may magne recall costs as the cost of searchng through a flng system or database to extract hstorc nformaton. Our prmary concern n ths paper has been wth ndvdual human decson makng agents where problems of recall are real but the costs of such are not easy to quantfy. Appendx To show the exstence of tremblng mnd perfecton we characterze three stuatons (). The frst s trval, formal proofs of the other two are gven below. (). The game Γ has no smlar hstores, thus for all players the games Γ and Γ are dentcal. All equlbrum strateges β * n Γ are tremblng mnd perfect. (). The game Γ contans smlar hstores and a unque equlbrum n whch each player follows the behavoral strategy β * whch s a best response to the strateges of

22 22 the other players β *. Ths wll also be the unque best response for the game Γ and hence tremblng mnd perfect for the game Γ. Proof: We want to show for all players, that there exsts an ε h > 0 (and ε h < ) S ( h) such that β * s a best response for player n the perturbed game Γ. Let Π * payoff to player from the strategy β * n the game Γ Π Π Π ' mnmum possble payoff for player, n the game Γ maxmum possble payoff for player, n the game Γ * maxmum possble payoff for player gven the strategy β s used by players -, n the game Γ from any behavoral strategy β β *. Snce β * * ' s unque, ( Π Π ) and ( Π Π ) must be greater than zero. * In the game Γ all the players - all follow ther equlbrum strateges β for the orgnal game Γ. Snce the payoffs are not changed n movng from Γ to Γ, player cannot receve less than Π and not more than Π. Wth probablty ε h S ( h) any strategy β nature selects a path where player makes no mstakes and on ths path β * gves at most Π < Π ' * Thus β * * must be a best response to β n the game Γ f the followng condton s satsfed: * ' ε h Π + Π ε h > ε h Π + Π ε h S ( h) S ( h) S ( h) S ( h)

23 23 Where the left hand sde represents the mnmum payoff possble from β * rght hand sde represents the maxmum possble from any other strategy β ', and the. Snce all the probabltes and payoff dfferentals are non zero, we can re-wrte ths as 0 < ε h S ( h) ε h S ( h) * ' ( Π Π ) < ( Π Π ) Snce for any payoffs we can fnd values of ε h such that ths s satsfed. Repeatng ths argument for all players N, we can fnd values of ε h, 0 < ε h S ( h) ε h S ( h) * ' ( Π Π ) < mn ( Π Π ) for all players N such that all players behavoral strateges β * are best responses. (). The game Γ contans smlar hstores and there are two or more equlbra n behavoral strateges. At least one of these wll be tremblng mnd perfect. Proof: It can be shown that any equlbrum of the orgnal game Γ s for some arbtrarly small value of ε h h >0 a canddate for tremblng mnd perfecton. Where ( Π Π ) =0, all behavoral strateges for player are equally good n all possble perturbed games and hence all are tremblng mnd perfect. * ' Followng a smlar argument to that presented n (), f ( Π Π ) part of a tremblng mnd perfect equlbrum. >0 then β * can be

24 24 * ' In the case where ( Π Π ) =0 the strategy β ' * s a best response to β (and hence also part of an equlbrum of the game Γ ). The strategy β * wll be tremblng mnd perfect unless β ' s strctly better n all possble perturbed games. If ths s the case then β ' wll be tremblng mnd perfect. Bblography Abreu, D. and Rubnsten, A. (988),'The Structure of Nash Equlbra n Repeated Games wth Fnte Automata', Econometrca 56, Aumann, R., Hart, S. and Perry, M. (997a), The Absentmnded Drver, Games and Economc Behavor 20, Aumann, R., Hart, S. and Perry, M. (997b), The Forgetful Passenger, Games and Economc Behavor 20, Battgall, P. (997) Dynamc onsstency and Imperfect Recall, Games and Economc Behavor 20, Bnmore, K. (forthcomng) A Note on Imperfect Recall, n Understandng Strategc Interacton - Essays n Honour of Renhard Selten, W.Guth et al. (edtors), Sprnger- Verlag Glboa, I. (997) A omment on the Absent Mnded Drver Paradox. Games and Economc Behavor 20, Grove, A. and Halpern, J. (997) On the Expected Value of Games wth Absentmndedness, Games and Economc Behavor 20, 5-65 Halpern, J. On Ambgutes n the Interpretaton of Game Trees, Games and Economc Behavor 20,

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