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).
Work partially supported by the EU IST-2001-33116 (Flags) and IST-2004-15964 (AEOLUS) and by Spanish CICYT TIC2002-04498-C05-03 (Tracer).
Due to space restrictions some proofs are omitted, we refer the reader to for further details.
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Àlvarez, C., Gabarró, J., Serna, M. (2005). Pure Nash Equilibria in Games with a Large Number of Actions. In: Jȩdrzejowicz, J., Szepietowski, A. (eds) Mathematical Foundations of Computer Science 2005. MFCS 2005. Lecture Notes in Computer Science, vol 3618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11549345_10
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DOI: https://doi.org/10.1007/11549345_10
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