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 an 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 an 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 Sigma_2^p-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).
CitationÁlvarez, C., Gabarró, J., Serna, M. "Pure Nash equilibria in games with a large number of actions". 2004.
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