Subjective Uncertainty Over Behavior Strategies: A Correction

Transcription

1 Subjectve Uncertanty Over Behavor Strateges: A Correcton The Harvard communty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton Publshed Verson Accessed Ctable Lnk Terms of Use Dekel, Edde, Drew Fudenberg, and Davd K. Levne Subjectve uncertanty over behavor strateges: A correcton. Journal of Economc Theory 104, no. 2: do: /jeth Aprl 21, :15:57 PM EDT Ths artcle was downloaded from Harvard Unversty's DASH repostory, and s made avalable under the terms and condtons applcable to Other Posted Materal, as set forth at (Artcle begns on next page)

2 Subjectve Uncertanty over Behavor Strateges: A Correcton * Runnng Ttle: Uncertanty over Behavor Strateges Edde Dekel, Drew Fudenberg, and Davd K. Levne November 2000 Revsed May 2001 Edde Dekel: Departments of Economcs, Northwestern and Tel Avv Unverstes, dekel@northwestern.edu Drew Fudenberg: Department of Economcs, Harvard Unversty, dfudenberg@harvard.edu Davd K. Levne: Department of Economcs, ULCA, dlevne@ucla.edu Send manuscrpt correspondence to Drew Fudenberg, Department of Economcs, Harvard Unversty, Cambrdge MA 02138; phone (617) , fax (617) * Ths work s supported by the Natonal Scence Foundaton under Grants , , and

3 2 Abstract In order to model the subjectve uncertanty of a player over the behavor strateges of an opponent, one must consder the player s belefs about the opponents play at nformaton sets that player thnks have probablty 0. Ths erratum uses trembles to provde a defnton of the convex hull of a set of behavor strateges. Ths corrects a defnton we gave n [5], whch led two of the soluton concepts we defned there to not have the propertes we ntended.

4 3 1. Introducton Ratonalzablty and related concepts are defned and characterzed n terms of sets of strateges for a player that other players thnk he mght use. In the strategc form each player s belefs about the play of an opponent are gven by a probablty measure over ths set, and each such measure maps to a pont n the convex hull of the set of possble strateges, as n Bernhem [3] and Pearce [9]. Consequently, we can take ths convex hull as a model of what players mght thnk about other players. In extensveform models that use behavor strateges, the correct way to model belefs and map them to strateges s less transparent. Ger Ashem has ponted out to us that n Dekel, Fudenberg and Levne [5] we gve an ncorrect defnton of convex combnatons of behavor strateges. As a result two of the concepts that we defned (sequental ratonalzablty and sequentally ratonalzable self-confrmng equlbrum) do not have the propertes that the paper mples and ntended. Ths erratum uses trembles to provde a defnton of mxtures that, when embedded n our defntons of sequental ratonalzablty and sequentally ratonalzable self-confrmng equlbrum, makes them functon n the way we ntended. 1 In partcular, wth the corrected defnton t wll be the case that when a player thnks that only a sngle behavor strategy s consstent wth ratonal play by hs opponent, hs belefs about that opponent correspond to that unque behavor strategy. Consequently, sequental ratonalzablty mples backwards nducton n fnte games of perfect nformaton wth generc payoffs. Instead of usng trembles, Ashem and Perea [1] use lexcographc probablty systems (extendng Blume, Brandenburger and Dekel [4]) to model players belefs n extensve-form games; among other thngs they use these systems to provde a correct defnton of sequental ratonalzablty for two-player games. Battgall [3] models belefs n extensve-form games usng Myerson's [8] condtonal probablty systems, to whch lexcographc probablty systems are closely related. We prefer to use trembles nstead because we already used them n another part of our 1999 paper, and because for 1 As we explan below, the error does not matter for the soluton concept of ratonalzable self-confrmng equlbrum, whch was the prmary focus of that paper.

5 4 our purposes t s not necessary to track the relatve lkelhoods of varous zeroprobablty events Prelmnares To save space we wll assume that the reader s famlar wth most of the notaton and termnology of Dekel, Fudenberg, and Levne [5], and so we wll only restate a few of the most relevant defntons. An assessment a for player s a probablty dstrbuton over nodes at each of hs nformaton sets. A belef for player s a par b (a, π -), consstng of s assessment over nodes a, and 's expectatons of opponents strateges π - = (π j) j. 3 In that paper we defned belef closed as follows: Defnton 2.2: A belef model V s belef closed f for every ( π,( a, π )) V, π j arses from a mxture over strateges n the set { π ( π, b ) V for some belef b }. j j j j j Ths defnton s slent on what t means to say that a behavor strategy arses from a mxture over other behavor strateges; the paper elaborates n footnote 11, whch says: A behavor strategy π j s generated by a mxture ( a, 1- a) over π j and π j f for every π j, the dstrbuton over termnal nodes nduced by ( π j, π j) equals the ( a, 1- a) mxture over the dstrbutons nduced by ( π j, π j) and ( π j, π j) respectvely. Ths clarfcaton s ncorrect. The problems arse n defnng the behavor of π j at nformaton sets for j that are not reachable under any of the strateges n V j. 4 Such nformaton sets are rrelevant for concepts that place no restrctons on play at 2 Contemporaneously wth ths paper, Ashem and Perea developed a trembles-based alternatve to ther use of lexcographc probablty systems. 3 The assumpton that player s expectatons about an opponents play corresponds to a strategy profle ncorporates the mplct restrcton that opponents randomze ndependently. Note that what we call an assessment s what Kreps and Wlson [8] call a system of belefs for player, and that our belef s smlar to what they call an assessment. 4 That s, an nformaton set s unreachable under Q f there s no profle J the nformaton set s reached wth postve probablty. Q for j s opponents such that J

6 5 nformaton sets that the strategy precludes, but the mstake s mportant for concepts such as our sequental ratonalzablty that mpose restrctons on play at all nformaton sets. For example consder the game n Fgure 1: <put Fgure 1 here> The backwards nducton profle s π 1 = (n, up) and π 2 = (across); the profle ((out, down), (down)) s an mperfect Nash equlbrum. Only up s sequentally ratonal at player 1 s second nformaton set, so the set of sequentally ratonal behavor strateges for player 1 must contan only strateges that play up at ths nformaton set. Consder the sets 6 OUT UP DOWN, 6 DOWNOUT DOWN. Snce the strategy (out, down) s equvalent n the strategc form to (out, up), t arses as a mxture over the set of player 1 s strateges n 6. Consequently, ths par s sequentally ratonal and belef closed when arses from s defned as n footnote 11. In partcular, sequental ratonalty and belef-closed wth the orgnal defnton does not mply backwards nducton. 3. The Extensve-Form Convex Hull We therefore propose the followng defnton of the convex hull of behavor strateges, whch corrects and bulds from our prevous defnton by usng trembles to make sure that every nformaton set of player s reachable. 5 When workng wth strctly postve behavor strateges, there are no unreached nformaton sets. In ths case the defnton of generated by a mxture from our prevous paper s adequate. Defnton: Strategy π s n the extensve-form convex hull of a set strateges for player f there s an nteger k, strateges { } j,.. π j =1 k n I 1 of behavor I 1, sequences of 5 Here and subsequently, we gve the space of behavor strateges the norm topology, so that a sequence of behavor strateges converges ff t converges pontwse.

7 strctly postve behavor strateges π dstrbutons on [1,, k], such that the behavor strateges combnaton of K I I I jn, 6 Q Q Q wth weghts j π, and a sequence α n α of probablty n π generated by the convex N N KN B B KB converge to π. We let α vary along the sequence so that the extensve-form convex hull wll be closed. To see why the set would not be closed f the defnton used only a fxed α, consder the followng one-player game. Player 1 has two moves n a row: The frst choce s In or Out; Out ends the game, In gves hm a second choce of L or R. Strategy 1 s (Out, L), 2 s (In, R). Now suppose that the defnton of convex hull used only fxed weghts, and let B B be the weghts on strateges 1 and 2. Snce only strategy 2 plays In and enables the move n the second perod, the convex combnaton of the two strateges wth strctly postve weghts s (( B Out, B (Out, R) as In), R), whch approaches B l. However, ths s not a convex combnaton of strateges 1 and 2, even for α =(1, 0). Consequently, the set of convex combnatons by ths defnton s not closed. Our defnton of the extensve-form convex hull ncludes both (Out, L) and (Out, R). Defnton 2.2 (revsed): A belef model V s belef closed f for every ( π,( a, π )) V, π j s n the extensve-form convex hull of the set { π ( π, b ) V for some belef b }. j j j j j All of the other defntons n our 1999 paper stay unchanged, modulo the change n the defnton of belef closed. Note that the dfference between the corrected defnton of belef closed and the prevous one arses when for some players and j, π j nduces the same dstrbuton over outcomes as a mxture over the set { π ( π, b ) V for some belef b }, but dffers from these strateges at an nformaton j j j j j set that the strateges themselves preclude. For ths reason, the changed defnton of belef closed has only a mnor effect on the concept of ratonalzablty at reachable nodes (defnton 2.3), as ths concept does not requre that strateges be optmal at

8 7 nformaton sets that the strateges themselves rule out. In partcular, whle a gven belef model V (such as the par of sngleton belefs V 1, V 2 n the example of the last secton) can be ratonalzable under reachable nodes under the old defnton but not under the new one (because t s not belef-closed), for any such V we can construct a Vˆ that s belef-closed by addng to each agrees wth π j at nodes that are reachable under V j and every π j V j, every strategy ˆ π j that π j. Every added strategy s a best response at reachable nodes to the same belefs that ratonalzed the orgnal π j, and snce the set V was belef closed under the old defnton, ˆ V s belef closed under the new one. 6 In partcular, the change n defntons has no effect on whether a strategy profle ˆ π s a ratonalzable self-confrmng equlbrum, as ths requres that there exst a belef model V that s ratonalzable at reachable nodes, such that for all players, every ( π, b) V has the dstrbuton of outcomes nduced by ˆ π. Thus Theorems 2.1, 4.1, and all of the examples n Secton 3 are unaffected by the change. As we noted earler, the change n defnton does matter for the concept of sequental ratonalzablty, whch requres that strateges n the belef model be ratonalzed at every nformaton set, and t has a smlar mpact on the concept of a sequentally ratonalzable self-confrmng equlbrum. In partcular, because the extensve-form convex hull of a sngleton set conssts solely of the sngle strategy n that set, the unque sequentally ratonalzable profle n fnte games of perfect nformaton wth generc payoffs s the one gven by backwards nducton. 7 Despte ths change, theorem 4.2, whch s the only result n [5] that refers to concepts usng sequental 6 Note that when a model V that s ratonalzable at reachable nodes under the old defnton, the model formed by enlargng the set of strateges n each V j to ts extensve-from convex hull need not be ratonalzable at reachable nodes under the corrected defnton. Although the new model wll be belefclosed, the strateges ntroduced need not be ratonal at reachable nodes, and ndeed they may be strctly domnated. 7 Bernhem [3] defnes subgame ratonalzablty, and argues that t yelds backwards nducton n generc game of perfect nformaton.

9 8 ratonalty, s correct as stated, snce the elaboratons used n the proof have a type that s ndfferent between all actons at every nformaton set. Note fnally that that even f a strategy profle π s sequentally ratonalzable as a sngleton set (.e., there are belefs b for each player such that the sets V1 = {( π1, b1)}, V = {( π, b )}, etc. s sequentally ratonalzable) t need not be a sequental equlbrum Whle we have assumed that each player s assessment over nodes n hs nformaton sets s consstent n the Kreps-Wlson sense of beng dervable from the lmt of Bayesan belefs from full-support strateges, we have not requred that all players assessments be consstent wth a sngle sequence of totally mxed strategy profles, and t s known, see for nstance Example 8.5 n Fudenberg and Trole [6], that the freedom to use dfferent sequences to derve each player s assessment can allow addtonal equlbrum outcomes.

10 9 References 1. G. Ashem and A. Perea, Lexcographc Probabltes and Ratonalzablty n Extensve-Form Games, mmeo, P. Battgall, Strategc Independence and Perfect Bayesan Equlbra, J. Econ. Theory 70 (1976), D. Bernhem, Ratonalzable Strategc Behavor, Econometrca, 52 (1984), L. Blume, L., A. Brandenburger, and E. Dekel, Lexcographc Probabltes and Equlbrum Refnements, Econometrca 59 (1991), E. Dekel, D. Fudenberg, and D.K. Levne, Payoff Informaton and Self- Confrmng Equlbrum, J. Econ. Theory 89 (1999), D. Fudenberg, and J. Trole, Game Theory, MIT Press, Cambrdge, MA, D. Kreps and R. Wlson, Sequental Equlbrum, Econometrca, 50 (1982), R. Myerson, Axomatc Foundatons of Bayesan Decson Theory, Center for Mathematcal Studes n Economcs and Management Scence, Northwestern Unversty Workng Paper No. 671, D. Pearce, Ratonalzable Strategc Behavor and the Problem of Perfecton, Econometrca, 52 (1984),

11 10 Fgure 1 1 out n 2 down across 1 (3,3) up down (1,0) (2,0) (1,2)