Finding equilibria in games of no chance (original) (raw)

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Optimal Behavioral Strategies In 0-Sum Games with Almost Perfect Information

Mathematics of Operations Research, 1982

This paper provides the general construction of the optimal strategies in a special class of zero sum games with incomplete information, those in which the players move sequentially. It is shown that at any point of the game tree, a player's optimal behavioral strategy may be derived from a state variable involving two components: the first one keeps track of the information he revealed, the second one keeps track of the (vector) payoff he should secure over his opponent's possible position. This construction gives new insights on earlier results obtained in the context of sequential repeated games. Several examples are discussed in detail. I. Introduction. This paper is concerned with the recursive construction of optimal behavioral strategies for a class of 0-sum extensive games with incomplete information, defined as follows: (i) Let G be a two-person finite game tree with perfect information. (ii) Let AH be the zero-sum payoff associated with a play H of G, AH is a discrete random variable such that Prob(AH = as) = prqS where p = (Pr)rR q = (qS)S are two independent probability distributions over two given finite sets R and S. (iii) The game is played the following way: chance selects r E R and s E S according to p and q and reveals r to Player I and s to Player II, then the players proceed along the game tree until they reach a play H; at that stage, Player II (the minimizer) pays a'7 to Player I (the maximizer) and the game ends. All the preceding description is common knowledge (i.e., r is revealed to Player I but not to Player II, similarly for s, but both players know that, etc.) This class of games called games with almost perfect information was first studied in Ponssard (1975). It overlaps with the class of repeated games with incomplete information (the study of which was initiated by Aumann and Maschler (1966)), since it includes the sequential finitely repeated games (Ponssard and Zamir, 1973). Briefly speaking, in all these games, the strategic problem of information usage may be stated in the following terms. First, the players are concerned with using their private information (i.e., for Player I to correlate his moves with the event r selected by chance and only known by himself) but this exposes them to reveal it to their opponent (i.e., through the observation of Player I's past moves, Player II may infer something about the selected r) and possibly reduce its value for the future moves of the game. Second, the players cannot take at its face value the information apparently revealed by their opponent without being subject to bluffing. These two aspects have been developed in the context of two persons zero-sum infinitely repeated games in order to find "maxmin" strategies (Aumann, Maschler (1966); Kohlberg (1975); Mertens, Zamir (1980)). In this context the main tools to *

PERFECT RECALL AND PRUNING IN GAMES WITH IMPERFECT INFORMATION

Computational Intelligence, 1996

Games with imperfect information are an interesting and important class of games. They include most card games (e.g., bridge and poker), as well as many economic and political models. Here, we investigate algorithms for solving imperfect information games expressed in their extensive (game-tree) form. In particular, we consider algorithms for the simplest form of solution | a pure-strategy equilibrium point. We introduce to the arti cial intelligence (AI) community a classic algorithm due to Wilson, that nds a pure-strategy equilibrium point in one-player games with perfect recall. Wilson's algorithm, which we call IMP-minimax, runs in time linear in the size of the game-tree searched. In contrast to Wilson's result, Koller & Meggido have shown that nding a pure-strategy equilibrium point in one-player games without perfect recall is NP-hard. Here, we provide another contrast to Wilson's result | we show that in games with perfect recall, nding a pure-strategy equilibrium point, given that such an equilibrium point exists, remains NP-hard, in games with more than a single player.

Computing sequential equilibria for two-player games

2006

Koller, Megiddo and von Stengel showed how to efficiently compute minimax strategies for two-player extensive-form zero-sum games with imperfect information but perfect recall using linear programming and avoiding conversion to normal form. Their algorithm has been used by AI researchers for constructing prescriptive strategies for concrete, often fairly large games. Koller and Pfeffer pointed out that the strategies obtained by the algorithm are not necessarily sequentially rational and that this deficiency is often problematic for the practical applications. We show how to remove this deficiency by modifying the linear programs constructed by Koller, Megiddo and von Stengel so that pairs of strategies forming a sequential equilibrium are computed. In particular, we show that a sequential equilibrium for a two-player zero-sum game with imperfect information but perfect recall can be found in polynomial time. In addition, the equilibrium we find is normal-form perfect. We also describe an extension of our technique to general-sum games which is likely to be prove practical, even though it is not polynomial-time.

Random perfect information games

2021

The paper proposes a natural measure space of zero-sum perfect information games with upper semicontinuous payoffs. Each game is specified by the game tree, and by the assignment of the active player and of the capacity to each node of the tree. The payoff in a game is defined as the infimum of the capacity over the nodes that have been visited during the play. The active player, the number of children, and the capacity are drawn from a given joint distribution independently across the nodes. We characterize the cumulative distribution function of the value v using the fixed points of the so-called value generating function. The characterization leads to a necessary and sufficient condition for the event v ≥ k to occur with positive probability. We also study probabilistic properties of the set of Player I’s k-optimal strategies and the corresponding plays.

Credible equilibria in non-finite games and in games without perfect recall

1998

Credible equilibria were defined in Ferreira et al. [6] to handle situations of preferences changing along time in a model given by an extensive form game. This paper extends the definition to the case of infinite games and, more important, to games with non-perfect recall. These games are of great interest in possible applications of the model, but the original definition was not applicable to them. The difficulties of this extension are solved by using some ideas in the literature of abstract systems and by proposing new ones that may prove useful in more general settings.

Information patterns and Nash equilibria in extensive games — II

Mathematical Social Sciences, 1985

In Part I of this paper we introduced extensive games with a non-atomic continuum of players. It was shown that the Nash plays (outcomes) are invariant of the information patterns on the game, provided that no player's unilateral change in moves can be observed by others. This led to an enormous reduction in the Nash plays of the these games, as exemplified in the anti-folk theorem. Our concern in this sequel is to develop a finite version of these results.

On the Existence of Markov Perfect Equilibria in Perfect Information Games

2011

We study the existence of pure strategy Markov perfect equilibria in two-person perfect information games. There is a state space X and each period player's possible actions are a subset of X. This set o f f e a s i b l e a c t i o n s d e p e n d s o n t h e c u r r e n t s t a t e , w h i c h i s determined by the choice of the other player in the previous period. We assume that X is a compact Hausdorff space and that the action correspondence has an acyclic and asymmetric graph. For some states there may be no feasible actions and then the game ends. Payoffs are either discounted sums of utilities of the states visited, or the utility of the state where the game ends. We give sufficient conditions for the existence of equilibrium e.g. in case when either feasible action sets are finite or when players' payoffs are continuously dependent on each other. The latter class of games includes zero-sum games and pure coordination games. JEL Classification: C72, C73

Games with Imperfect Information

1993

An information set in a game tree is a set of nodes from which the rules of the game require that the same alternative (i.e., move) be selected. Thus the nodes an information set are indistinguishable to the player moving from that set, thereby reflecting imperfect in- formation, that is, information hidden from that player. Information sets arise naturally in (for example) card gaines like poker and bridge. IIere we focus not on the solution concept for im- perfect information games (which has been studied at length), but rather on the computational aspects of such games: how hard is it to compute solutions? We present two fundainental results for imperfect informa- tion games. The first result shows that even if there is only a single player, we must seek special cases or heuristics. The second result complements the first, providing an efficient algorithm for just such a special case. Additionally, we show how our special case algo- rithm can be used as a heuristic in the general...

On non-existence of pure strategy Markov perfect equilibrium

Economics Letters, 2002

This paper provides a simple counterexample to existence of pure strategy (stationary) Markov perfect equilibrium for a class of infinite-horizon games with complete information, finitely many actions, and finitely many ordered states. For the given example, the proof of non-existence is provided. 