In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in L1, L22, and Lā biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3, 5/7, and 2/3 approximations for these norms, respectively.
CITATION STYLE
Deligkas, A., Fearnley, J., & Spirakis, P. (2016). Lipschitz continuity and approximate equilibria. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9928 LNCS, pp. 15ā26). Springer Verlag. https://doi.org/10.1007/978-3-662-53354-3_2
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