We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be 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, we show that there always exists a PSNE and give a polynomial-time algorithm for finding one. We then focus on a specific compact representation: piecewise-linear utility functions. We give polynomial-time algorithms for finding a sample PSNE, counting the number of PSNEs, and also provide an FPTAS for finding social-welfare-maximizing equilibria. We extend our piecewise-linear representation to achieve what we believe to be the first compact representation for parameterized families of (symmetric) games. We provide methods for answering questions about a parameterized family without needing to solve each game from the family separately.
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