SYMMETRIC GAMES WITH PIECEWISE LINEAR UTILITIES

Abstract. We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions, where the utilities are compactly represented. Such a representation is able to describe symmetric games whose number of players is exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence of PSNE is NP-complete. For the special case of games with two actions, there always exist a PSNE and we give a polynomial algorithm for finding one. We then focus on a natural representation of utility as piecewise-linear functions, and show that such a representation has nice computational properties. In particular, we give polynomial-time algorithms to count the number of PSNE (thus deciding if such an equilibrium exists) and to find a sample PSNE, when one exists. Our approach makes use of Barvinok and Wood’s rational generating function method [4], which enables us to encode the set of PSNE as a generating function of polynomial size. 1