The computational complexity of nash equilibria in concisely represented games

  • Authors:

  • Grant Schoenebeck;Salil Vadhan

  • Affiliations:

  • University of California, Berkeley, Berkeley, California;Harvard University, Cambridge, MA

  • Venue:
  • EC '06 Proceedings of the 7th ACM conference on Electronic commerce
  • Year:
  • 2006

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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 the players meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.