Lipschitz Continuity and Approximate Equilibria

TL;DR

This paper explores how Lipschitz continuity in payoff functions enables the development of efficient algorithms for finding approximate equilibria in various classes of continuous action games, including Lipschitz, penalty, and distance biased games.

Contribution

It introduces new algorithms and approximation schemes for computing equilibria in continuous action games with Lipschitz continuous payoffs, covering a range of game types.

Findings

Efficient algorithm for approximate equilibria in Lipschitz games.

Quasi-polynomial time approximation scheme for penalty games with Lipschitz penalty functions.

Strongly polynomial algorithms for best responses and approximations in distance biased games.

Abstract

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 $L_1$, $L_2^2$, and $L_\infty$ 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.