Characterizing Solution Concepts in Games Using Knowledge-Based Programs Joseph Y. Halpern∗ Yoram Moses Computer Science Department Department of Electrical Engineering Cornell University, U.S.A. Technion—Israel Institute of Technology e-mail: [email protected] 32000 Haifa, Israel email: [email protected] Abstract x 0 We show how solution concepts in games such as .5 .5 Nash equilibrium, correlated equilibrium, rational- z S x xzS 1 izability, and sequential equilibrium can be given 0 1 2 a uniform definition in terms of knowledge-based 2 2 programs. Intuitively, all solution concepts are im- plementations of two knowledge-based programs, B B one appropriate for games represented in normal form, the other for games represented in extensive form. These knowledge-based programs can be x 3 x4 viewed as embodying rationality. The representa- L R L R tion works even if (a) information sets do not cap- ture an agent’s knowledge, (b) uncertainty is not zz2 3 z 4 z 5 represented by probability, or (c) the underlying 346 2 game is not common knowledge. Figure 1: A game of imperfect recall. 1 Introduction a payoff of 2, or continue, by playing move B. If he contin- Game theorists represent games in two standard ways: in ues, he gets a high payoff if he matches nature’s move, and a normal form, where each agent simply chooses a strategy, low payoff otherwise. Although he originally knows nature’s and in extensive form, using game trees, where the agents move, the information set that includes the nodes labeled x3 make choices over time. An extensive-form representation and x4 is intended to indicate that the player forgets whether has the advantage that it describes the dynamic structure of nature moved left or right after moving B. Intuitively, when the game—it explicitly represents the sequence of decision he is at the information set X, the agent is not supposed to problems encountered by agents. However, the extensive- know whether he is at x3 or at x4. form representation purports to do more than just describe It is not hard to show that the strategy that maximizes ex- the structure of the game; it also attempts to represent the pected utility chooses action S at node x1, action B at node information that players have in the game, by the use of in- x2, and action R at the information set X consisting of x3 formation sets. Intuitively, an information set consists of a and x4. Call this strategy f.Letf be the strategy of choos- set of nodes in the game tree where a player has the same ing action B at x1, action S at x2,andL at X. Piccione and [ ] information. However, as Halpern 1997 has pointed out, Rubinstein argue that if node x1 is reached, the player should information sets may not adequately represent a player’s in- reconsider, and decide to switch from f to f .AsHalpern formation. points out, this is indeed true, provided that the player knows Halpern makes this point by considering the following at each stage of the game what strategy he is currently using. single-agent game of imperfect recall, originally presented by However, in that case, if the player is using f at the infor- [ ] Piccione and Rubinstein 1997 : The game starts with nature mation set, then he knows that he is at node x4;ifhehas / moving either left or right, each with probability 1 2.The switched and is using f , then he knows that he is at x3.So, S agent can then either stop the game (playing move ) and get in this setting, it is no longer the case that the player does not ∗ know whether he is at x3 or x4 in the information set; he can Supported in part by NSF under grants CTC-0208535 and ITR- 0325453, by ONR under grant N00014-02-1-0455, by the DoD infer which state he is at from the strategy he is using. Multidisciplinary University Research Initiative (MURI) program In game theory, a strategy is taken to be a function from in- administered by the ONR under grants N00014-01-1-0795 and formation sets to actions. The intuition behind this is that, N00014-04-1-0725, and by AFOSR under grant F49620-02-1-0101. since an agent cannot tell the nodes in an information set IJCAI-07 1300 apart, he must do the same thing at all these nodes. But this express with the formula doi(S)), and she believes that she example shows that if the agent has imperfect recall but can would not do any better with another strategy, then she should switch strategies, then he can arrange to do different things indeed go ahead and run S. This test can be viewed as em- at different nodes in the same information set. As Halpern bodying rationality. There is a subtlety in expressing the [1997] observes, ‘ “situations that [an agent] cannot distin- statement “she would not do any better with another strat- guish” and “nodes in the same information set” may be two egy”. We express this by saying “if her expected utility, given quite different notions.’ He suggests using the game tree to that she will use strategy S,isx, then her expected utility if describe the structure of the game, and using the runs and sys- she were to use strategy S is at most x.” The “if she were tems framework [Fagin et al., 1995] to describe the agent’s to use S”isacounterfactual statement. She is planning information. The idea is that an agent has an internal local to use strategy S, but is contemplating what would happen state that describes all the information that he has. A strat- if she were to do something counter to fact, namely, to use egy (or protocol in the language of [Fagin et al., 1995])isa S. Counterfactuals have been the subject of intense study function from local states to actions. Protocols capture the in the philosophy literature (see, for example, [Lewis, 1973; intuition that what an agent does can depend only what he Stalnaker, 1968]) and, more recently, in the game theory lit- knows. But now an agent’s knowledge is represented by its erature (see, for example, [Aumann, 1995; Halpern, 2001; local state, not by an information set. Different assumptions Samet, 1996]). We write the counterfactual “If A were the about what agents know (for example, whether they know case then B would be true” as “A B”. Although this state- their current strategies) are captured by running the same pro- ment involves an “if . then”, the semantics of the counter- tocol in different contexts. If the information sets appropri- factual implication A B is quite different from the material ately represent an agent’s knowledge in a game, then we can implication A ⇒ B. In particular, while A ⇒ B is true if A identify local states with information sets. But, as the exam- is false, A B might not be. ple above shows, we cannot do this in general. With this background, consider the following kb program A number of solution concepts have been considered in the for player i: game-theory literature, ranging from Nash equilibrium and for each strategy S ∈Si(Γ) do correlated equilibrium to refinements of Nash equilibrium if Bi(doi(S) ∧∀x(EUi = x ⇒ such as sequential equilibrium and weaker notions such as S∈S (Γ)(doi(S ) (EUi ≤ x)))) then S. rationalizability (see [Osborne and Rubinstein, 1994] for an i overview). The fact that game trees represent both the game This kb program is meant to capture the intuition above. In- i and the players’ information has proved critical in defining tuitively, it says that if player believes that she is about to S x solution concepts in extensive-form games. Can we still rep- perform strategy and, if her expected utility is ,thenif S resent solution concepts in a useful way using runs and sys- she were to perform another strategy , then her expected utility would be no greater than x, then she should perform tems to represent a player’s information? As we show here, Γ not only can we do this, but we can do it in a way that gives strategy S. Call this kb program EQNF (with the indi- Γ deeper insight into solution concepts. Indeed, all the standard vidual instance for player i denoted by EQNFi ). As we Γ solution concepts in the literature can be understood as in- show, if all players follow EQNF , then they end up play- stances of a single knowledge-based (kb) program [Fagin et ing some type of equilibrium. Which type of equilibrium they al., 1995; 1997], which captures the underlying intuition that play depends on the context. Due to space considerations, a player should make a best response, given her beliefs. The we focus on three examples in this abstract. If the players differences between solution concepts arise from running the have a common prior on the joint strategies being used, and kb program in different contexts. this common prior is such that players’ beliefs are indepen- In a kb program, a player’s actions depend explicitly on dent of the strategies they use, then they play a Nash equilib- the player’s knowledge. For example, a kb program could rium. Without this independence assumption, we get a cor- have a test that says “If you don’t know that Ann received the related equilibrium. On the other hand, if players have pos- information, then send her a message”, which can be written sibly different priors on the space of strategies, then this kb program defines rationalizable strategies [Bernheim, 1984; if ¬Bi(Ann received info) then send Ann a message.