Abstract
Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games , where the payoffs are computed by a given boolean circuit, and graph games , where each agent’s payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.
Highlights
In recent years, there has been a surge of interest at the interface between computer science and game theory
Game theory and its notions of equilibria provide a rich framework for modeling the behavior of selfish agents in the kinds of distributed or networked environments that often arise in computer science and offer mechanisms to achieve efficient and desirable global outcomes in spite of the selfish behavior
Game-theoretic characterizations of complexity classes have proved to be extremely useful even in addressing questions that a priori have nothing to do with games, of particular note being the work on interactive proofs and their applications to cryptography and hardness of approximation [GMR89, GMW91, FGL+96, AS98, ALM+98]
Summary
There has been a surge of interest at the interface between computer science and game theory. A central topic at the interface of computer science and economics is understanding the complexity of computational problems involving equilibria in games While these types of questions are already interesting (and often difficult) for standard two-player games presented in explicit “bimatrix” form [MP91, CS08, GZ89, LMM03], many of the current motivations for such games come. Circuit games were previously studied in the setting of two-player zerosum games, where computing (resp., approximating) the “value” of such a game is shown to be EXP-complete [FKS95] (resp., S2P-complete [FIKU08]) They are a very general model, capturing essentially any representation in which the payoffs are efficiently computable. Greco, and Scarcello [GGS03] recently showed that the problem of deciding if a degree-4 graph game has a pure-Nash equilibrium is NP-complete In each of these two models (circuit games and graph games), we study 4 problems: 1. FindNash: Given a game G, find a Nash equilibrium in G, and
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