Pure nash equilibria in games with a large number of actions

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 ∑2p-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). © Springer-Verlag Berlin Heidelberg 2005.