We present two simple search methods for computing a sample Nash equilibrium in a normal-form game: one for 2player games and one for n-player games. We test these algorithms on many classes of games, and show that they perform well against the state of the art-- the Lemke-Howson algorithm for 2-player games, and Simplicial Subdivision and Govindan-Wilson for n-player games.

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.

Action-graph games (AGGs) are a fully expressive game representation which can compactly express both strict and context-specific independence between players' utility functions. Actions are represented as nodes in a graph G, and the payoff to an agent who chose the action s depends only on the numbers of other agents who chose actions connected to s. We present algorithms for computing both symmetric and arbitrary equilibria of AGGs using a continuation method. We analyze the worst-case cost of computing the Jacobian of the payoff function, the exponential-time bottleneck step, and in all cases achieve exponential speedup. When the indegree of G is bounded by a constant and the game is symmetric, the Jacobian can be computed in polynomial time.

We describe algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through the space of perturbed games and their equilibria. Our algorithms exploit game structure through fast computation of the Jacobian of the game's payoff function. They are guaranteed to find at least one equilibrium of the game and may find more. Our approach provides the first exact algorithm for computing an exact equilibrium in graphical games with arbitrary topology, and the first algorithm to exploit fine-grain structural properties of MAIDs. We present experimental results for our algorithms. The running time for our graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. Our algorithm for MAIDs can effectively solve games that arc much larger than those that could be solved using previous methods. 1

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We describe an algorithm for computing bestresponse strategies in a class of two-player infinite games of incomplete information, defined by payoffs piecewise linear in agents ’ types and actions, conditional on linear comparisons of agents ’ actions. We show that this class includes many well-known games including a variety of auctions and a novel allocation game. In some cases, the best-response algorithm can be iterated to compute Bayes-Nash equilibria. We demonstrate the efficacy of our approach on existing and new games. 1

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We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call “robust-optimization equilibrium, ” to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.

Strategic games may exhibit symmetries in a variety of ways. A common aspect of symmetry, enabling the compact representation of games even when the number of players is unbounded, is that players cannot (or need not) distinguish between the other players. We define four classes of symmetric games by considering two additional properties: identical payoff functions for all players and the ability to distinguish oneself from the other players. Based on these varying notions of symmetry, we investigate the computational complexity of pure Nash equilibria. It turns out that in all four classes of games equilibria can be found efficiently when only a constant number of actions is available to each player, a problem that has been shown intractable for other succinct representations of multi-player games. We further show that identical payoff functions simplify the search for equilibria, while a growing number of actions renders it intractable. Finally, we show that our results extend to wider classes of threshold symmetric games where players are unable to determine the exact number of players playing a certain action.

Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multi-player sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably close-to-optimal strategies.