Characterizing Solution Concepts in Games Using Knowledge-Based Programs

Transcription

1 Characterzng Soluton Concepts n Games Usng Knowledge-Based Programs Joseph Y. Halpern Computer Scence Department Cornell Unversty, U.S.A. e-mal: halpern@cs.cornell.edu Abstract We show how soluton concepts n games such as Nash equlbrum, correlated equlbrum, ratonalzablty, and sequental equlbrum can be gven a unform defnton n terms of knowledge-based programs. Intutvely, all soluton concepts are mplementatons of two knowledge-based programs, one approprate for games represented n normal form, the other for games represented n extensve form. These knowledge-based programs can be vewed as embodyng ratonalty. The representaton works even f (a) nformaton sets do not capture an agent s knowledge, (b) uncertanty s not represented by probablty, or (c) the underlyng game s not common knowledge. 1 Introducton Game theorsts represent games n two standard ways: n normal form, where each agent smply chooses a strategy, and n extensve form, usng game trees, where the agents make choces over tme. An extensve-form representaton has the advantage that t descrbes the dynamc structure of the game t explctly represents the sequence of decson problems encountered by agents. However, the extensveform representaton purports to do more than just descrbe the structure of the game; t also attempts to represent the nformaton that players have n the game, by the use of nformaton sets. Intutvely, an nformaton set conssts of a set of nodes n the game tree where a player has the same nformaton. However, as Halpern [1997] has ponted out, nformaton sets may not adequately represent a player s nformaton. Halpern makes ths pont by consderng the followng sngle-agentgame of mperfect recall, orgnally presented by Pccone and Rubnsten [1997]: The game starts wth nature movng ether left or rght, each wth probablty 1/2. The agent can then ether stop the game (playng move S) and get Supported n part by NSF under grants CTC and ITR , by ONR under grant N , by the DoD Multdscplnary Unversty Research Intatve (MURI) program admnstered by the ONR under grants N and N , and by AFOSR under grant F Yoram Moses Department of Electrcal Engneerng Technon Israel Insttute of Technology Hafa, Israel emal: moses@ee.technon.ac.l z 2 S.5.5 x x S z B x z z z z x 0 B 3 4 L R L R Fgure 1: A game of mperfect recall. a payoff of 2, or contnue, by playng move B. If he contnues, he gets a hgh payoff f he matches nature s move, and a low payoff otherwse. Although he orgnally knows nature s move, the nformaton set that ncludes the nodes labeled x 3 and x 4 s ntended to ndcate that the player forgets whether nature moved left or rght after movng B. Intutvely, when he s at the nformaton set X, the agent s not supposed to know whether he s at x 3 or at x 4. It s not hard to show that the strategy that maxmzes expected utlty chooses acton S at node x 1, acton B at node x 2, and acton R at the nformaton set X consstng of x 3 and x 4. Call ths strategy f. Letf be the strategy of choosng acton B at x 1, acton S at x 2,andL at X. Pccone and Rubnsten argue that f node x 1 s reached, the player should reconsder, and decde to swtch from f to f. As Halpern ponts out, ths s ndeed true, provded that the player knows at each stage of the game what strategy he s currently usng. However, n that case, f the player s usng f at the nformaton set, then he knows that he s at node x 4 ;fhehas swtched and s usng f, then he knows that he s at x 3.So, n ths settng, t s no longer the case that the player does not know whether he s at x 3 or x 4 n the nformaton set; he can nfer whch state he s at from the strategy he s usng. In game theory, a strategy s taken to be a functon from nformaton sets to actons. The ntuton behnd ths s that, snce an agent cannot tell the nodes n an nformaton set x

2 apart, he must do the same thng at all these nodes. But ths example shows that f the agent has mperfect recall but can swtch strateges, then he can arrange to do dfferent thngs at dfferent nodes n the same nformaton set. As Halpern [1997] observes, stuatons that [an agent] cannot dstngush and nodes n the same nformaton set may be two qute dfferent notons. He suggests usng the game tree to descrbe the structure of the game, and usng the runs and systems framework [Fagn et al., 1995] to descrbe the agent s nformaton. The dea s that an agent has an nternal local state that descrbes all the nformaton that he has. A strategy (or protocol n the language of [Fagn et al., 1995]) sa functon from local states to actons. Protocols capture the ntuton that what an agent does can depend only what he knows. But now an agent s knowledge s represented by ts local state, not by an nformaton set. Dfferent assumptons about what agents know (for example, whether they know ther current strateges) are captured by runnng the same protocol n dfferent contexts. If the nformaton sets approprately represent an agent s knowledge n a game, then we can dentfy local states wth nformaton sets. But, as the example above shows, we cannot do ths n general. A number of soluton concepts have been consdered n the game-theory lterature, rangng from Nash equlbrum and correlated equlbrum to refnements of Nash equlbrum such as sequental equlbrum and weaker notons such as ratonalzablty (see [Osborne and Rubnsten, 1994] for an overvew). The fact that game trees represent both the game and the players nformaton has proved crtcal n defnng soluton concepts n extensve-form games. Can we stll represent soluton concepts n a useful way usng runs and systems to represent a player s nformaton? As we show here, not only can we do ths, but we can do t n a way that gves deeper nsght nto soluton concepts. Indeed, all the standard soluton concepts n the lterature can be understood as nstances of a sngle knowledge-based (kb) program [Fagn et al., 1995; 1997], whch captures the underlyng ntuton that a player should make a best response, gven her belefs. The dfferences between soluton concepts arse from runnng the kb program n dfferent contexts. In a kb program, a player s actons depend explctly on the player s knowledge. For example, a kb program could have a test that says If you don t know that Ann receved the nformaton, then send her a message, whch can be wrtten f B (Ann receved nfo) then send Ann a message. Ths kb program has the form of a standard f...then statement, except that the test n the f clause s a test on s knowledge (expressed usng the modal operator B for belef; see Secton 2 for a dscusson of the use of knowledge vs. belef). Usng such tests for knowledge allows us to abstract away from low-level detals of how the knowledge s obtaned. Kb programs have been appled to a number of problems n the computer scence lterature (see [Fagn et al., 1995] and the references theren). To see how they can be appled to understand equlbrum, gven a game Γ n normal form, let S (Γ) consst of all the pure strateges for player n Γ. Roughly speakng, we want a kb program that says that f player beleves that she s about to perform strategy S (whch we express wth the formula do (S)), and she beleves that she would not do any better wth another strategy, then she should ndeed go ahead and run S. Ths test can be vewed as embodyng ratonalty. There s a subtlety n expressng the statement she would not do any better wth another strategy. We express ths by sayng f her expected utlty, gven that she wll use strategy S, sx, then her expected utlty f she were to use strategy S s at most x. The f she were to use S sacounterfactual statement. She s plannng to use strategy S, but s contemplatng what would happen f she were to do somethng counter to fact, namely, to use S. Counterfactuals have been the subject of ntense study n the phlosophy lterature (see, for example, [Lews, 1973; Stalnaker, 1968]) and, more recently, n the game theory lterature (see, for example, [Aumann, 1995; Halpern, 2001; Samet, 1996]). We wrte the counterfactual If A were the case then B would be true as A B. Although ths statement nvolves an f... then, the semantcs of the counterfactual mplcaton A B s qute dfferent from the materal mplcaton A B. In partcular, whle A B s true f A s false, A B mght not be. Wth ths background, consder the followng kb program for player : for each strategy S S (Γ) do f B (do (S) x(eu = x S S (do (Γ) (S ) (EU x)))) then S. Ths kb program s meant to capture the ntuton above. Intutvely, t says that f player beleves that she s about to perform strategy S and, f her expected utlty s x, thenf she were to perform another strategy S, then her expected utlty would be no greater than x, then she should perform strategy S. Call ths kb program EQNF Γ (wth the ndvdual nstance for player denoted by EQNF Γ ). As we show, f all players follow EQNF Γ, then they end up playng some type of equlbrum. Whch type of equlbrum they play depends on the context. Due to space consderatons, we focus on three examples n ths abstract. If the players have a common pror on the jont strateges beng used, and ths common pror s such that players belefs are ndependent of the strateges they use, then they play a Nash equlbrum. Wthout ths ndependence assumpton, we get a correlated equlbrum. On the other hand, f players have possbly dfferent prors on the space of strateges, then ths kb program defnes ratonalzable strateges [Bernhem, 1984; Pearce, 1984]. To deal wth extensve-form games, we need a slghtly dfferent kb program, snce agents choose moves, not strateges. be the followng program, where a PM de- Let EQEF Γ notes that a s a move that s currently possble. for each move a PM do f B (do (a) x((eu = x) a PM (do (a ) (EU x)))) then a. Just as EQNF Γ characterzes equlbra of a game Γ represented n normal form, EQEF Γ characterzes equlbra of a game represented n extensve form. We gve one example here: sequental equlbrum. To capture sequental equlbrum, we need to assume that nformaton sets do correctly 1301

3 descrbe an agent s knowledge. If we drop ths assumpton, however, we can dstngush between the two equlbra for the game descrbed n Fgure 1. All these soluton concepts are based on expected utlty. But we can also consder soluton concepts based on other decson rules. For example, Boutler and Hyafl [2004] consder mnmax-regret equlbra, where each player uses a strategy that s a best-response n a mnmax-regret sense to the choces of the other players. Smlarly, we can use maxmn equlbra [Aghass and Bertsmas, 2006]. As ponted out by Chu and Halpern [2003], all these decson rules can be vewed as nstances of a generalzed noton of expected utlty, where uncertanty s represented by a plausblty measure, a generalzaton of a probablty measure, utltes are elements of an arbtrary partally ordered space, and plausbltes and utltes are combned usng and, generalzatons of + and. We show n the full paper that, just by nterpretng EU = u approprately, we can capture these more exotc soluton concepts as well. Moreover, we can capture soluton concepts n games where the game tself s not common knowledge, or where agents are not aware of all moves avalable, as dscussed by Halpern and Rêgo [2006]. Our approach thus provdes a powerful tool for representng soluton concepts, whch works even f (a) nformaton sets do not capture an agent s knowledge, (b) uncertanty s not represented by probablty, or (c) the underlyng game s not common knowledge. The rest of ths paper s organzed as follows. In Secton 2, we revew the relevant background on game theory and knowledge-based programs. In Secton 3, we show that EQNF Γ and EQEF Γ characterze Nash equlbrum, correlated equlbrum, ratonalzablty, and sequental equlbrum n a game Γ n the approprate contexts. We conclude n Secton 4 wth a dscusson of how our results compare to other characterzatons of soluton concepts. 2 Background In ths secton, we revew the relevant background on games and knowledge-based programs. We descrbe only what we need for provng our results. The reader s encouraged to consult [Osborne and Rubnsten, 1994] for more on game theory, [Fagn et al., 1995; 1997] for more on knowledge-based programs wthout counterfactuals, and [Halpern and Moses, 2004] for more on addng counterfactuals to knowledgebased programs. 2.1 Games and Strateges Agamenextensve form s descrbed by a game tree. Assocated wth each non-leaf node or hstory s ether a player the player whose move t s at that node or nature (whch can make a randomzed move). The nodes where a player moves are further parttoned nto nformaton sets. Wth each run or maxmal hstory h n the game tree and player we can assocate s utlty, denoted u (h), f that run s played. A strategy for player s a (possbly randomzed) functon from s nformaton sets to actons. Thus a strategy for player tells player what to do at each node n the game tree where s supposed to move. Intutvely, at all the nodes that player cannot tell apart, player must do the same thng. A jont strategy S =(S 1,...,S n ) for the players determnes a dstrbuton over paths n the game tree. A normal-form game can be vewed as a specal case of an extensve-form game where each player makes only one move, and all players move smultaneously. 2.2 Protocols, Systems, and Contexts To explan kb programs, we must frst descrbe standard protocols. We assume that, at any gven pont n tme, a player n agamesnsomelocal state. The local state could nclude the hstory of the game up to ths pont, the strategy beng used by the player, and perhaps some other features of the player s type, such as belefs about the strateges beng used by other players. A global state s a tuple consstng of a local state for each player. A protocol for player s a functon from player s local states to actons. For ease of exposton, we consder only determnstc protocols, although t s relatvely straghtforward to model randomzed protocols correspondng to mxed strateges as functons from local states to dstrbutons over actons. Although we restrct to determnstc protocols, we deal wth mxed strateges by consderng dstrbutons over pure strateges. A run s a sequence of global states; formally, a run s a functon from tmes to global states. Thus, r(m) s the global state n run r at tme m. A pont s a par (r, m) consstng of a run r and tme m. Let r (m) be s local state at the pont (r, m); thats,fr(m) =(s 1,...,s n ),then r (m) =s.ajont protocol s an assgnment of a protocol for each player; essentally, a jont protocol s a jont strategy. At each pont, a jont protocol P performs a jont acton (P 1 (r 1 (m)),...,p n (r n (m))), whch changes the global state. Thus, gven an ntal global state, a jont protocol P generates a (unque) run, whch can be thought of as an executon of P. The runs n a normal-form game nvolve only one round and two tme steps: tme 0 (the ntal state) and tme 1, after the jont strategy has been executed. (We assume that the payoff s then represented n the player s local state at tme 1.) In an extensve-form game, a run s agan characterzed by the strateges used, but now the length of the run depends on the path of play. A probablstc system s a tuple PS =(R, μ),wherer s a set of runs and μ =(μ 1,...,μ n ) assocates a probablty μ on the runs of R wth each player. Intutvely, μ represents player s pror belefs. In the specal case where μ 1 = = μ n = μ, the players have a common pror μ on R. In ths case, we wrte just (R,μ). We are nterested n the system correspondng to a jont protocol P. To determne ths system, we need to descrbe the settng n whch P s beng executed. For our purposes, ths settng can be modeled by a set G of global states, a subset G 0 of G that descrbes the possble ntal global states, a set A s of possble jont actons at each global state s,andnprobabl- ty measures on G 0, one for each player. Thus, a probablstc context s a tuple γ =(G, G 0, {A s : s G}, μ). 1 A jont 1 We are mplctly assumng that the global state that results from 1302

4 protocol P s approprate for such a context γ f, for every global state s, the jont actons that P can generate are n A s. When P s approprate for γ, we abuse notaton slghtly and refer to γ by specfyng only the par (G 0, μ). A protocol P and a context γ for whch P s approprate generate a system; the system depends on the ntal states and probablty measures n γ. Snce these are all that matter, we typcally smplfy the descrpton of a context by omttng the set G of global states and the sets A s of global actons. Let R( P,γ) denote the system generated by jont protocol P n context γ. If γ =(G 0, μ),thenr( P,γ)=(R, μ ),whererconssts of atherunr s for each ntal state s G 0,wherer s s the run generated by P when started n state s, andμ (r s) =μ ( s), for =1,...,n. A probablstc system (R, μ ) s compatble wth a context γ =(G 0, μ) f (a) every ntal state n G 0 s the ntal stateofsomerunnr, (b) every run s the run of some protocol approprate for γ, and(c)fr( s) s the set of runs n R wth ntal global state s, thenμ j (R( s)) = μ j( s), for j =1,...,n. Clearly R( P,γ) s compatble wth γ. We can thnk of the context as descrbng background nformaton. In dstrbuted-systems applcatons, the context also typcally ncludes nformaton about message delvery. For example, t may determne whether all messages sent are receved n one round, or whether they may take up to, say, fve rounds. Moreover, when ths s not obvous, the context specfes how actons transform the global state; for example, t descrbes what happens f n the same jont acton two players attempt to modfy the same memory cell. Snce such ssues do not arse n the games we consder, we gnore these facets of contexts here. For smplcty, we consder only contexts where each ntal state corresponds to a partcular jont strategy of Γ. Thats,Σ Γ s a set of local states for player ndexed by (pure) strateges. The set Σ Γ can be vewed as descrbng s types; the state s S can the thought of as the ntal state where player s type s such that he plays S (although we stress that ths s only ntuton; player does not have to play S at the state s S ). Let G0 Γ =ΣΓ 1... ΣΓ n. We wll be nterested n contexts where the set of ntal global states s a subset G 0 of G0 Γ. In a normal-form game, the only actons possble for player at an ntal global state amount to choosng a pure strategy, so the jont actons are jont strateges; no actons are possble at later tmes. For an extensve-form game, the possble moves are descrbed by the game tree. We say that a context for an extensve-form game s standard f the local states have the form (s, I), wheres s the ntal state and I s the current nformaton set. In a standard context, an agent s knowledge s ndeed descrbed by the nformaton set. However, we do not requre a context to be standard. For example, f an agent s allowed to swtch strateges, then the local state could nclude the hstory of strateges used. In such a context, the agent n the game of Fgure 1 would know more than just what s n the nformaton set, and would want to swtch strateges. performng a jont acton n A s at the global state s s unque and obvous; otherwse, such nformaton would also appear n the context, as n the general framework of [Fagn et al., 1995]. 2.3 Knowledge-Based Programs A knowledge-based program s a syntactc object. For our purposes, we can take a knowledge-based program for player to have the form f κ 1 then a 1 f κ 2 then a 2..., where each κ j s a Boolean combnaton of formulas of the form B ϕ,nwhchtheϕ s can have nested occurrences of B l operators and counterfactual mplcatons. We assume that the tests κ 1,κ 2,...are mutually exclusve and exhaustve, so that exactly one wll evaluate to true n any gven nstance. The program EQNF Γ can be wrtten n ths form by smply replacng the for... do statement by one lne for each pure strategy n S (Γ); smlarly for EQEF Γ. We want to assocate a protocol wth a kb program. Unfortunately, we cannot execute a kb program as we can a protocol. How the kb program executes depends on the outcome of tests κ j. Snce the tests nvolve belefs and counterfactuals, we need to nterpret them wth respect to a system. The dea s that a kb program Pg for player and a probablstc system PS together determne a protocol P for player. Rather than gvng the general defntons (whch can be found n [Halpern and Moses, 2004]), we just show how they work n the two kb programs we consder n ths paper: EQNF and EQEF. Gven a system PS =(R, μ), we assocate wth each formula ϕ aset[ϕ] PS of ponts n PS. Intutvely, [ϕ] PS s the set of ponts of PS where the formula ϕ s true. We need a lttle notaton: If E s a set of ponts n PS,letR(E) denote the set of runs gong through ponts n E; thatsr(e) ={r : m((r, m) E)}. Let K (r, m) denote the set of ponts that cannot dstngush from (r, m): K (r, m) ={(r,m ):(r (m )= r (m)}. Roughly speakng, K (r, m) corresponds to s nformaton set at the pont (r, m). Gven a pont (r, m) and a player, letμ (,r,m) be the probablty measure that results from condtonng μ on K (r, m), s nformaton at (r, m). We cannot condton on K (r, m) drectly: μ s a probablty measure on runs, and K (r, m) s a set of ponts. So we actually condton, not on K (r, m), but on R(K (r, m)),the set of runs gong through the ponts n K (r, m). Thus, μ,r,m = μ R(K (r, m)). (For the purposes of ths abstract, we do not specfy μ,r,m f μ (R(K (r, m))) = 0. It turns out not to be relevant to our dscusson.) The kb programs we consder n ths paper use a lmted collecton of formulas. We now can defne [ϕ] PS for the formulas we consder that do not nvolve counterfactuals. In a system PS correspondng to a normal-form game Γ, fs S (Γ), then[do (S)] PS s the set of ntal ponts (r, 0) such that player uses strategy S n run r. Smlarly, f PS corresponds to an extensve-form game, then [do (a)] PS s the set of ponts (r, m) of PS at whch performs acton a. 1303

5 Player beleves a formula ϕ at a pont (r, m) f the event correspondng to formula ϕ has probablty 1 accordng to μ,r,m. That s, (r, m) [B ϕ] PS f μ (R(K (r, m)) 0(so that condtonng on K (r, m) s defned) and μ,r,m ([ϕ] PS K (r, m)) = 1. Wth every run r n the systems we consder, we can assocate the jont (pure) strategy S used n r. 2 Ths pure strategy determnes the hstory n the game, and thus determnes player s utlty. Thus, we can assocate wth every pont (r, m) player s expected utlty at (r, m), where the expectaton s taken wth respect to the probablty μ,r,m.ifusareal number, then [EU = u] PS s the set ofponts whereplayer s expected utlty s u; [EU u] PS s defned smlarly. Assume that ϕ(x) has no occurrences of. Then [ xϕ(x))] PS = a IR [ϕ[x/a]] PS,whereϕ[x/a] s the result of replacng all occurrences of x n ϕ by a. That s, x s just unversal quantfcaton over x, where x ranges over the reals. Ths quantfcaton arses for us when x represents a utlty, so that xϕ(x) s sayng that ϕ holds for all choces of utlty. We now gve the semantcs of formulas nvolvng counterfactuals. Here we consder only a restrcted class of such formulas, those where the counterfactual only occurs n the form do (S) ϕ, whch should be read as f were to use strategy S, thenϕ would be true. Intutvely, do (S) ϕ s true at a pont (r, m) f ϕ holds n a world that dffers from (r, m) only n that uses the strategy S. Thats,do (S) ϕ s true at (r, m) f ϕ s true at the pont (r,m) where, n run r, player uses strategy S and all the other players use the same strategy that they do at (r, m). (Ths can be vewed as an nstance of the general semantcs for counterfactuals used n the phlosophy lterature [Lews, 1973; Stalnaker, 1968] where ψ ϕ s taken to be true at a world w f ϕ s true at all the worlds w closest to w where ψ s true.) Of course, f actually uses strategy S n run r, then r = r. Smlarly, n an extensve-form game Γ, the closest pont to (r, m) where do (a ) s true (assumng that a s an acton that can perform n the local state r (m)) s the pont (r,m) where all players other than player use the same protocol n r and r,and s protocol n r agrees wth s protocol n r except at the local state r (m), where performs acton a. Thus, r s the run that results from player makng a sngle devaton (to a at tme m) from the protocol she uses n r, and all other players use the same protocol as n r. There s a problem wth ths approach. There s no guarantee that, n general, such a closest pont (r,m) exsts n the system PS. To deal wth ths problem, we restrct attenton to a class of systems where ths pont s guaranteed to exst. A system (R, μ) s complete wth respect to context γ f R ncludes every run generated by a protocol approprate for context γ. In complete systems, the closest pont (r,m) s guaranteed to exst. For the remander of the paper, we 2 If we allow players to change strateges durng a run, then we wll n general have dfferent jont strateges at each pont n a run. For our theorems n the next secton, we restrct to contexts where players do not change strateges. evaluate formulas only wth respect to complete systems. In a complete system PS,wedefne[do (S) ϕ] PS to consst of all the ponts (r, m) such that the closest pont (r,m) to (r, m) where uses strategy S s n [ϕ] PS. The defnton of [do (a) ϕ] PS s smlar. We say that a complete system (R, μ ) extends (R, μ) f μ j and μ j agree on R (so that μ j (A) =μ j(a)) foralla R)forj =1,...,n. Snce each formula κ that appears as a test n a kb program Pg for player s a Boolean combnaton of formulas of the form B ϕ, t s easy to check that f (r, m) [κ] PS,then K (r, m) [κ] PS. In other words, the truth of κ depends only on s local state. Moreover, snce the tests are mutually exclusve and exhaustve, exactly one of them holds n each local state. Gven a system PS, we take the protocol Pg PS to be such that Pg PS (l) =a j f, for some pont (r, m) n PS wth r (m) =l, wehave(r, m) [κ j ] PS.Snceκ 1,κ 2,... are mutually exclusve and exhaustve, there s exactly one acton a j wth ths property. We are manly nterested n protocols that mplement a kb program. Intutvely, a jont protocol P mplements a kb program Pg n context γ f P performs the same actons as Pg n all runs of P that have postve probablty, assumng that the knowledge tests n Pg are nterpreted wth respect to the complete system PS extendng R( P,γ). Formally, a jont protocol P (de facto) mplements a jont kb program Pg [Halpern and Moses, 2004] n a context γ =(G 0, μ) f P (l) =Pg PS (l) for every local state l = r (m) such that r R( P,γ) and μ (r) 0,wherePS s the complete system extendng R( P,γ). We remark that, n general, there may not be any jont protocols that mplement a kb program n a gven context, there may be exactly one, or there may be more than one (see [Fagn et al., 1995] for examples). Ths s somewhat analogous to the fact that there may not be any equlbrum of a game for some notons of equlbrum, there may be one, or there may be more than one. 3 The Man Results Fx a game Γ n normal form. Let P nf be the protocol that, n ntal state s S Σ Γ, chooses strategy S; let P nf = (P nf 1,...,Pnf n ). Let STRAT be the random varable on ntal global states that assocates wth an ntal global state s player s strategy n r. As we sad, Nash equlbrum arses n contexts wth a common pror. Suppose that γ =(G 0,μ) s a context wth a common pror. We say that μ s compatble wth the mxed jont strategy S f μ s the probablty on pure jont strateges nduced by S (under the obvous dentfcaton of ntal global states wth jont strateges). Theorem 3.1: The jont strategy S s a Nash equlbrum of the game Γ ff there s a common pror probablty measure μ on G0 Γ such that STRAT 1,...,STRAT n are ndependent wth respect to μ, μ s compatble wth S, and P nf mplements EQNF Γ n the context (G0 Γ,μ). Proof: Suppose that S s a (possbly mxed strategy) Nash equlbrum of the game Γ. Letμ S be the unque probablty 1304

6 on G0 Γ compatble wth S. If S s played, then the probablty of a run where the pure jont strategy (T 1,...,T n ) s played s just the product of the probabltes assgned to T by S,soSTRAT 1,...,STRAT n are ndependent wth respect to μ S. To see that P nf mplements EQNF Γ n the context γ =(G0 Γ,μ S ),letl = r (0) be a local state such that r = R( P nf,γ) and μ(r) 0.Ifl = s T,thenP nf (l) =T, so T must be n the support of S. Thus, T must be a best response to S, the jont strategy where each player j plays ts component of S.Snce uses strategy T n r,theformula B (do (T )) holds at (r, 0) ff T = T. Moreover, snce T s a best response, f u s s expected utlty wth the jont strategy S, then for all T, the formula do (T ) (EU u) holds at (r, 0). Thus, (EQNF Γ )PS (l) =T,wherePS s the complete system extendng R( P nf,γ). It follows that P nf mplements EQNF Γ. For the converse, suppose that μ s a common pror probablty measure on G0 Γ, STRAT 1,...,STRAT n are ndependent wth respect to μ, μ s compatble wth S,and P nf mplements EQNF Γ n the context γ =(G0 Γ,μ). Wewantto show that S s a Nash equlbrum. It suffces to show that each pure strategy T n the support of S s a best response to S.Snceμs compatble wth S, there must be a run r such that μ(r) > 0 and r (0) = s T (.e., player chooses T n run r). It snce P nf mplements EQNF Γ, and n the context γ, EQNF Γ ensures that no devaton from T can mprove s expected utlty wth respect to S, t follows that T s ndeed a best response. As s well known, players can sometmes acheve better outcomes than a Nash equlbrum f they have access to a helpful medator. Consder the smple 2-player game descrbed n Fgure 2, where Alce, the row player, must choose between top and bottom (T and B), whle Bob, the column player, must choose between left and rght (L and R): L R T (3, 3) (1, 4) B (4, 1) (0, 0) Fgure 2: A smple 2-player game. It s not hard to check that the best Nash equlbrum for ths game has Alce randomzng between T and B, andbob randomzng between L and R; ths gves each of them expected utlty 2. They can do better wth a trusted medator, who makes a recommendaton by choosng at random between (T,L), (T,R), and(b,l). Ths gves each of them expected utlty 8/3. Thssacorrelated equlbrum snce, for example, f the medator chooses (T,L), and thus sends recommendaton T to Alce and L to Bob, then Alce consders t equally lkely that Bob was told L and R, and thus has no ncentve to devate; smlarly, Bob has no ncentve to devate. In general, a dstrbuton μ over pure jont strateges s a correlated equlbrum f players cannot do better than followng a medator s recommendaton f a medator makes recommendatons accordng to μ. (Note that, as n our example, f a medator chooses a jont strategy (S 1,...,S n ) accordng to μ, the medator recommends S to player ; player s not told the jont strategy.) We omt the formal defnton of correlated equlbrum (due to Aumman [1974]) here; however, we stress that a correlated equlbrum s a dstrbuton over (pure) jont strateges. We can easly capture correlated equlbrum usng EQNF. Theorem 3.2: The dstrbuton μ on jont strateges s a correlated equlbrum of the game Γ ff P nf mplements EQNF Γ n the context (G0 Γ,μ). Both Nash equlbrum and correlated equlbrum requre a common pror on runs. By droppng ths assumpton, we get another standard soluton concept: ratonalzablty [Bernhem, 1984; Pearce, 1984]. Intutvely, a strategy for player s ratonalzable f t s a best response to some belefs that player may have about the strateges that other players are followng, assumng that these strateges are themselves best responses to belefs that the other players have about strateges that other players are followng, and so on. To make ths precse, we need a lttle notaton. Let S =Π j S j. Let u ( S) denote player s utlty f the strategy tuple S s played. We descrbe player s belefs about what strateges the other players are usng by a probablty μ on S. A strategy S for player s a best response to belefs descrbed by a probablty μ on S (Γ) f T S u (S, T )μ ( T ) T S u (S, T )μ ( T ) for all S S. Followng Osborne and Rubnsten [1994], we say that a strategy S for player n game Γ s ratonalzable f, for each player j, theresa set Z j S j (Γ) and, for each strategy T Z j, a probablty measure μ j,t on S j (Γ) whose support s Z j such that S Z ;and for each player j and strategy T Z j, T s a best response to the belefs μ j,t. For ease of exposton, we consder only pure ratonalzable strateges. Ths s essentally wthout loss of generalty. It s easy to see that a mxed strategy S for player s a best response to some belefs μ of player ff each pure strategy n the support of S s a best response to μ. Moreover, we can assume wthout loss of generalty that the support of μ conssts of only pure jont strateges. Theorem 3.3: ApurestrategyS for player s ratonalzable ff there exst probablty measures μ 1,...,μ n,asetg 0 G0 Γ, and a state s G 0 such that P nf (s ) = S and P nf mplements EQNF Γ n the context (G 0, μ). Proof: Frst, suppose that P nf mplements EQNF Γ n context (G 0, μ). We show that for each state s G 0 and player, the strategy S s, = P nf (s ) s ratonalzable. Let Z = {S s, : s G 0 }.ForS Z,letE(S) ={ s G 0 : s = s S }; that s, E(S) conssts conssts of all ntal global states where player s local state s s S ;letμ,s = μ ( E(S)) (under the obvous dentfcaton of global states n G 0 wth jont strateges). Snce P nf mplements EQNF Γ, t easly follows that S best response to μ,s. Hence, all the strateges n Z are ratonalzable, as desred. 1305

7 For the converse, let Z consst of all the pure ratonalzable strateges for player. It follows from the defnton of ratonalzablty that, for each strategy S Z, there exsts a probablty measure μ,s on Z such that S s a best response to μ,s.forasetz of strateges, we denote by Z the set {s T : T Z}. SetG 0 = Z 1... Z n, and choose some measure μ on G 0 such that μ ( E(S)) = μ,s for all S Z. (We can take μ = S Z α S μ,s,whereα S (0, 1) and S Z α S =1.) Recall that P nf (s S )=S for all states s S. It mmedately follows that, for every ratonalzable jont strategy S =(S 1,...,S n ), both s =(s S1,...,s Sn ) G 0, and S = P nf ( s). Snce the states n G 0 all correspond to ratonalzable strateges, and by defnton of ratonalzablty each (ndvdual) strategy S s a best response to μ,s,ts easy to check that P nf mplements EQNF Γ n the context, μ), as desred. (G Γ 0 We remark that Osborne and Rubnsten s defnton of ratonalzablty allows μ j,t to be such that j beleves that other players strategy choces are correlated. In most of the lterature, players are assumed to beleve that other players choces are made ndependently. If we add that requrement, then we must mpose the same requrement on the probablty measures μ 1,...,μ n n Theorem 3.3. Up to now we have consdered soluton concepts for games n normal form. Perhaps the best-known soluton concept for games n extensve form s sequental equlbrum [Kreps and Wlson, 1982]. Roughly speakng, a jont strategy S s a sequental equlbrum f S s a best response to S at all nformaton sets, not just the nformaton sets that are reached wth postve probablty when playng S. To understand how sequental equlbrum dffers from Nash equlbrum, consder the game shown n Fgure 3. Fgure 3: A game wth an unreasonable Nash equlbrum. One Nash equlbrum of ths game has A playng down A and B playng across B. However, ths s not a sequental equlbrum, snce playng across s not a best response for B f B s called on to play. Ths s not a problem n a Nash equlbrum because the node where B plays s not reached n the equlbrum. Sequental equlbrum refnes Nash equlbrum (n the sense that every sequental equlbrum s a Nash equlbrum) and does not allow solutons such as (down A, across B ). Intutvely, n a sequental equlbrum, every player must make a best response at every nformaton set (even f t s reached wth probablty 0). In the game shown n Fgure 3, the unque jont strategy n a sequental equlbrum has A choosng across A and B choosng down B. The man dffculty n defnng sequental equlbrum les n capturng the ntuton of best response n nformaton sets that are reached wth probablty 0. To deal wth ths, a sequental equlbrum s defned to be a par ( S,β), consstng of a jont strategy S and a belef system β, whch assocates wth every nformaton set I a probablty β(i) on the hstores n I. There are a number of somewhat subtle consstency condtons on these pars pars; we omt them here due to lack of space (see [Kreps and Wlson, 1982; Osborne and Rubnsten, 1994] for detals). Our result depends on a recent characterzaton of sequental equlbrum [Halpern, 2006] that uses nonstandard probabltes, whch can assgn nfntesmal probabltes to hstores. By assumng that every hstory gets postve (although possbly nfntesmal) probablty, we can avod the problem of dealng wth nformaton sets that are reached wth probalty 0. To every nonstandard real number r, there s a closest standard real number denoted st (r), and read the standard part of r : r st (r) s an nfntesmal. Gven a nonstandard probablty measure ν, we can defne the standard probablty measure st (ν) by takng st (ν)(w) =st (ν(w)). A nonstandard probablty ν on G 0 s compatble wth jont strategy S f st (ν) s the probablty on pure strateges nduced by S. When dealng wth nonstandard probabltes, we generalze the defnton of mplementaton by requrng only that P performs the same actons as Pg n runs r of P such that st (ν)(r) > 0. Moreover, the expresson EU = x n EQEF Γ s nterpreted as the standard part of s expected utlty s x (sncex ranges over the standard real numbers). Theorem 3.4: If Γ s a game wth perfect recall 3 there s a belef system β such that ( S,β) s a sequental equlbrum of Γ ff there s a common pror nonstandard probablty measure ν on G0 Γ that gves postve measure to all states such that STRAT 1,...,STRAT n are ndependent wth respect to ν, ν s compatble wth S, and P ef mplements EQEF Γ n the standard context (G0 Γ,ν). Ths s very smlar n sprt to Theorem 3.1. The key dfference s the use of a nonstandard probablty measure. Intutvely, ths forces S to be a best response even at nformaton sets that are reached wth (standard) probablty 0. The effect of nterpretng EU = x as the standard part of s expected utlty s x s that we gnorenfntesmal dfferences. Thus, for example, the strategy P ef ( s 0 ) mght not be a best response to S ; t mght just be an ɛ-best response for some nfntesmal ɛ. As we show n the full paper, t follows from Halpern s [2006] results that we can also obtan a characterzaton of (tremblng hand) perfect equlbrum [Selten, 1975], another standard refnement of Nash equlbrum, f we nterpret EU = x as the expected utlty for agent s x and allow x to range over the nonstandard reals nstead of just the standard reals. 3 These are games where players remember all actons made and the states they have gone through; we gve a formal defnton n the full paper. See also [Osborne and Rubnsten, 1994]. 1306

8 4 Conclusons We have shown how a number of dfferent soluton concepts from game theory can be captured by essentally one knowledge-based program, whch comes n two varants: one approprate for normal-form games and one for extensveform games. The dfferences between these soluton concepts s captured by changes n the context n whch the games are played: whether players have a common pror (for Nash equlbrum, correlated equlbrum, and sequental equlbrum) or not (for ratonalzablty), whether strateges are chosen ndependently (for Nash equlbrum, sequental equlbrum, and ratonalzablty) or not (for correlated equlbrum); and whether uncertanty s represented usng a standard or nonstandard probablty measure. Our results can be vewed as showng that each of these soluton concepts sc can be characterzed n terms of common knowledge of ratonalty (snce the kb programs EQNF Γ and EQEF Γ embody ratonalty, and we are nterested n systems generated by these program, so that ratonalty holds at all states), and common knowledge of some other features X sc captured by the context approprate for sc (e.g., that strateges are chosen ndependently or that the pror). Roughly speakng, our results say that f X sc s common knowledge n a system, then common knowledge of ratonalty mples that the strateges used must satsfy soluton concept sc; conversely, f a jont strategy S satsfes sc, then there s a system where X sc s common knowledge, ratonalty s common knowledge, and S s beng played at some state. Results smlar n sprt have been proved for ratonalzablty [Brandenburger and Dekel, 187] and correlated equlbrum [Aumann, 1987]. Our approach allows us to unfy and extend these results and, as suggested n the ntroducton, apples even to settngs where the game s not common knowledge and n settngs where uncertanty s not represented by probablty. We beleve that the approach captures the essence of the ntuton that a soluton concept should embody common knowledge of ratonalty. References [Aghass and Bertsmas, 2006] M. Aghass and D. Bertsmas. Robust game theory. Mathematcal Programmng, Seres B, 107(1 2): , [Aumann, 1974] R. J. Aumann. Subjectvty and correlaton n randomzed strateges. Journal of Mathematcal Economcs, 1:67 96, [Aumann, 1987] R. J. Aumann. Correlated equlbrum as an expresson of Bayesan ratonalty. Econometrca, 55:1 18, [Aumann, 1995] R. J. Aumann. Backwards nducton and common knowledge of ratonalty. Games and Economc Behavor, 8:6 19, [Bernhem, 1984] B. D. Bernhem. Ratonalzable strategc behavor. Econometrca, 52(4): , [Boutler and Hyafl, 2004] C. Boutler and N. Hyafl. Regret mnmzng equlbra and mechansms for games wth strct type uncertanty. In Proc. Twenteth Conf. on Uncertanty n AI (UAI 2004), pages , [Brandenburger and Dekel, 187] A. Brandenburger and E. Dekel. Ratonalzablty and correlated equlbra. Econometrca, 55: , 187. [Chu and Halpern, 2003] F. Chu and J. Y. Halpern. Great expectatons. Part I: On the customzablty of generalzed expected utlty. In Proc. Eghteenth Internatonal Jont Conf. on AI (IJCAI 03), pages , [Fagn et al., 1995] R. Fagn, J. Y. Halpern, Y. Moses, and M. Y. Vard. Reasonng About Knowledge. MIT Press, (Slghtly revsed paperback verson, 2003.) [Fagn et al., 1997] R. Fagn, J. Y. Halpern, Y. Moses, and M. Y. Vard. Knowledge-based programs. Dstrbuted Computng, 10(4): , [Halpern and Moses, 2004] J. Y. Halpern and Y. Moses. Usng counterfactuals n knowledge-based programmng. Dstrbuted Computng, 17(2):91 106, [Halpern and Rêgo, 2006] J. Y. Halpern and L. C. Rêgo. Extensve games wth possbly unaware players. In Proc. Ffth Internatonal Jont Conf. on Autonomous Agents and Multagent Systems, pages , [Halpern, 1997] J. Y. Halpern. On ambgutes n the nterpretaton of game trees. Games and Economc Behavor, 20:66 96, [Halpern, 2001] J. Y. Halpern. Substantve ratonalty and backward nducton. Games and Economc Behavor, 37: , [Halpern, 2006] J. Y. Halpern. A nonstandard characterzaton of sequental equlbru, perfect equlbrum, and proper equlbrum. Unpublshed manuscrpt, [Kreps and Wlson, 1982] D. M. Kreps and R. B. Wlson. Sequental equlbra. Econometrca, 50: , [Lews, 1973] D. K. Lews. Counterfactuals. HarvardUnversty Press, [Osborne and Rubnsten, 1994] M. J. Osborne and A. Rubnsten. A Course n Game Theory. MIT Press, [Pearce, 1984] D. G. Pearce. Ratonalzable strategc behavor and the problem of perfecton. Econometrca, 52(4): , [Pccone and Rubnsten, 1997] M. Pccone and A. Rubnsten. On the nterpretaton of decson problems wth mperfect recall. Games and Economc Behavor, 20(1):3 24, [Samet, 1996] D. Samet. Hypothetcal knowledge and games wth perfect nformaton. Games and Economc Behavor, 17: , [Selten, 1975] R. Selten. Reexamnaton of the perfectness concept for equlbrum ponts n extensve games. Internatonal Journal of Game Theory, 4:25 55, [Stalnaker, 1968] R. C. Stalnaker. A semantc analyss of condtonal logc. In N. Rescher, edtor, Studes n Logcal Theory, pages Oxford Unversty Press,