We introduce a compact graph-theoretic representation for multi-party game theory. Our main result is a provably correct and efficient algorithm for computing approximate Nash equilibria in one-stage games represented by trees or sparse graphs.

We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomialtime solvability of normal-form games and graphical games, and shows that these kinds of games can implement arbitrary members of a PPAD-complete class of Brouwer functions. 1

Noncooperative game theory provides a normative framework for analyzing strategic interactions.

Interactions among agents can be conveniently described by game trees. In order to analyze a game, it is important to derive optimal (or equilibrium) strategies for the different players. The standard approach to finding such strategies in games with imperfect information is, in general, computationally intractable. The approach is to generate the normal form of the game (the matrix containing the payoff for each strategy combination), and then solve a linear program (LP) or a linear complementarity problem (LCP). The size of the normal form, however, is typically exponential in the size of the game tree, thus making this method impractical in all but the simplest cases. This paper describes a new representation of strategies which results in a practical linear formulation of the problem of two-player games with perfect recall (i.e., games where players never forget anything, which is a standard assumption). Standard LP or LCP solvers can then be applied to find optimal randomized strategies. The resulting algorithms are, in general, exponentially better than the standard ones, both in terms of time and in terms of space.

In this work, we study the combinatorial structure and the computational complexity of Nash equilibria for a certain game that models sel sh routing over a network consisting of m parallel links. We assume a collection of n users, each employing a mixed strategy, which is a probability distribution over links, to control the routing of its own assigned trac. In a Nash equilibrium, each user sel shly routes its trac on those links that minimize its expected latency cost, given the network congestion caused by the other users. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.

. The Nash equilibria of a two-person, non-zero-sum game are the solutions of a certain linear complementarity problem (LCP). In order to use this for solving a game in extensive form, it is first necessary to convert the game to a strategic description such as the normal form. The classical normal form, however, is often exponentially large in the size of the game tree. In this paper we suggest an alternative approach, based on the sequence form of the game. For a game with perfect recall, the sequence form is a linear sized strategic description, which results in an LCP of linear size. For this LCP, we show that an equilibrium is found by Lemke's algorithm, a generalization of the Lemke-Howson method. Keywords. Equilibrium, extensive game, Lemke-Howson algorithm, linear complementarity, sequence form. Computer Science Division, University of California, Berkeley, CA 94720; and IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120 y IBM Almaden Research Center, 650 Harr...

We prove the existence of #-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payo#s to all players in any (exact) Nash equilibrium can be #-approximated by the payo#s to the players in some such logarithmic support #-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an #-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an #-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payo#s to all players in any m-tuple of mixed strategies can be #-approximated by the payo#s in some m-tuple of constant support strategies.

Two minimal requirements for a satisfactory multiagent learning algorithm are that it 1. learns to play optimally against stationary opponents and 2. converges to a Nash equilibrium in self-play. The previous algorithm that has come closest, WoLF-IGA, has been proven to have these two properties in 2-player 2-action (repeated) games -- assuming that the opponent's mixed strategy is observable. Another algorithm, ReDVaLeR (which was introduced after the algorithm described in this paper), achieves the two properties in games with arbitrary numbers of actions and players, but still requires that the opponents' mixed strategies are observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have the two properties in games with arbitrary numbers of actions and players. It is still the only algorithm that does so while only relying on observing the other players' actual actions (not their mixed strategies). It also learns to play optimally against opponents that eventually become stationary. The basic idea behind AWESOME (Adapt When Everybody is Stationary, Otherwise Move to Equilibrium) is to try to adapt to the others' strategies when they appear stationary, but otherwise to retreat to a precomputed equilibrium strategy. We provide experimental results that suggest that AWESOME converges fast in practice. The techniques used to prove the properties of AWESOME are fundamentally different from those used for previous algorithms, and may help in analyzing future multiagent learning algorithms as well.

We present GAMUT 1, a suite of game generators designed for testing game-theoretic algorithms. We explain why such a generator is necessary, offer a way of visualizing relationships between the sets of games supported by GAMUT, and give an overview of GAMUT’s architecture. We highlight the importance of using comprehensive test data by benchmarking existing algorithms. We show surprisingly large variation in algorithm performance across different sets of games for two widely-studied problems: computing Nash equilibria and multiagent learning in repeated games. 2 1.

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)