Chapter

Mathematical Foundations of Computer Science 2005

Volume 3618 of the series Lecture Notes in Computer Science pp 95-106

Pure Nash Equilibria in Games with a Large Number of Actions

  • Carme ÀlvarezAffiliated withALBCOM Research Group., Universitat Politècnica de Catalunya
  • , Joaquim GabarróAffiliated withALBCOM Research Group., Universitat Politècnica de Catalunya
  • , Maria SernaAffiliated withALBCOM Research Group., Universitat Politècnica de Catalunya

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Abstract

We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game? and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in a strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that deciding the existence of a Nash equilibrium in a strategic game is NP-complete when the number of players is large and the number of strategies for each player is constant, while the problem is Σ\(^{p}_{\rm 2}\)-complete when the number of players is a constant and the size of the sets of strategies is exponential (with respect to the length of the strategies).

Keywords

Strategic games Nash equilibria complexity classes