We investigate the use of standard statistical models for quantal choice in a game theoretic setting. Players choose strategies based on relative expected utility, and assume other players do so as well. We define a Quantal Response Equilibrium (QRE) as a fixed point of this process, and establish existence. For a logit specification of the error structure, we show that as the error goes to zero, QRE approaches a subset of Nash equilibria and also implies a unique selection from the set of Nash equilibria in generic games. We fit the model to a variety of experimental data sets by using maximum likelihood estimatation.

We review the current state of the art of methods for numerical computation of Nash equilibria for finite n-person games. Classical path following methods, such as the Lemke-Howson algorithm for two person games, and Scarf-type fixed point algorithms for n-person games provide globally convergent methods for finding a sample equilibrium. For large problems, methods which are not globally convergent, such as sequential linear complementarity methods may be preferred on the grounds of speed. None of these methods are capable of characterizing the entire set of Nash equilibria. More computationally intensive methods, which derive from the theory of semi-algebraic sets are required for finding all equilibria. These methods can also be applied to compute various equilibrium refinements.

According to conventional wisdom, Nash equilibrium in a game “involves” common knowl-edge of the payoff functions, of the rationality of the players, and of the strategies played. The basis for this wisdom is explored, and it turns out that considerably weaker conditions suffice. First, note that if each player is rational and knows his own payoff function, and the strategy choices of the players are mutually known, then these choices form a Nash equilibrium. The other two results treat the mixed strategies of a player not as conscious randomization of that player, but as conjectures of the other players about what he will do. When n = 2, mutual knowledge of the payoff functions, of rationality, and of the conjectures yields Nash equilibrium. When n ≥ 3, mutual knowledge of the payoff functions and of rationality, and common knowl-edge of the conjectures yield Nash equilibrium when there is a common prior. Examples are provided showing these results to be sharp.

This paper is a survey and exposition of linear methods for finding Nash equilibria. Above all, these apply to games with two players. In an equilibrium of a twoperson game, the mixed strategy probabilities of one player equalize the expected payoffs for the pure strategies used by the other player. This defines an optimization problem with linear constraints. We do not consider nonlinear methods like simplicial subdivision for approximating fixed points, or systems of inequalities for higher-degree polynomials as they arise for noncooperative games with more than two players. These are surveyed in McKelvey and McLennan (1996)

We consider sel sh routing over a network consisting of m parallel links through which n sel sh users route their tra c trying to minimize their own expected latency. Westudy the class of mixed strategies in which the expected latency through each link is at most a constant multiple of the optimum maximum latency had global regulation been available. For the case of uniform links it is known that all Nash equilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) of these strategies de ned as the worst-case ratio of the maximum (over all links) expected latency over the optimum maximum latency. The load balancing aspect of the problem immediately implies a lower bound; lnm ln lnm of the coordination ratio. We give a tight (uptoamultiplicative constant) upper bound. To show the upper bound, we analyze a variant ofthe classical balls and bins problem, in which balls with arbitrary weights are placed into bins according to arbitrary probability distributions. At the heart of our approach is a new probabilistic tool that we call

We introduce Game networks (G nets), a novel representation for multi-agent decision problems. Compared to other game-theoretic representations, such as strategic or extensive forms, G nets are more structured and more compact; more fundamentally, G nets constitute a computationally advantageous framework for strategic inference, as both probability and utility independencies are captured in the structure of the network and can be exploited in order to simplify the inference process. An important aspect of multiagent reasoning is the identification of some or all of the strategic equilibria in a game; we present original convergence methods for strategic equilibrium which can take advantage of strategic separabilities in the G net structure in order to simplify the computations. Specifically, we describe a method which identifies a unique equilibrium as a function of the game payo#s, and one which identifies all equilibria. 1 Introduction The formal analysis of m...

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The language of game theory—coalitions, payo¤s, markets, votes— suggests that it is not a branch of abstract mathematics; that it is motivated by and related to the world around us; and that it should be able to

1992 version is intended as a contribution to a two volume collection honouring Patrick Suppes, to be edited by Paul Humphreys and published by Kluwer Academic Publishers. ABSTRACT. In an extensive form game, whether a player has a better strategy than in a presumed equilibrium depends on the other players ’ equilibrium reactions to a counterfactual deviation. To allowconditioning on counterfactual events with prior probability zero, extended probabilities are proposed and given the four equivalent characterizations: (i) complete conditional probability sys-tems; (ii) lexicographic hierarchies of probabilities; (iii) extended logarithmic likelihood ratios; and (iv) certain ‘canonical rational probability functions ’ representing ‘trembles ’ directly as in-finitesimal probabilities. However, having joint probability distributions be uniquely determined by independent marginal probability distributions requires general probabilities taking values in a space no smaller than the non-Archimedean ordered field whose members are rational functions of a particular infinitesimal. Elinor now found the difference between the expectation of an unpleasant event, however certain the mind may be told to consider it, and certainty itself. — Jane Austen, Sense and Sensibility, ch. 48.... a more attractive and manageable theory may result from a non-Archimedean representation.... One must keep in mind the fact that the refutability of axioms depends both on their mathematical form and their empirical interpretation. — Krantz, Luce, Suppes and Tversky (1971, p. 29).

In Bagwell (1995) it is claimed that, in models of commitment, "the firstmover advantage is eliminated when there is a slight amount of noise associated with the observation of the first-mover's selection." We show that the validity of this claim depends crucially on the restriction to pure strategy equilibria. The game analyzed by Bagwell always has a mixed equilibrium that is close to the Stackelberg equilibrium when the noise is small. Furthermore, an equilibrium selection theory, that combines elements from the theory of Harsanyi and Selten (1988) with elements from the theory of Harsanyi (1995), actually selects this "noisy Stackelberg equilibrium." Journal of Economic Literature Classification Number: C72. Copyright c fl1997 by Academic Press. This material has been published in Games and Economic Behavior, 21, 282308, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Academic ...