Lipschitz Continuity and Approximate Equilibria

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 and L 2 2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

Equilibrium computation in resource allocation games

“…They developed a convex potential function that can be minimized within arbitrary precision in polynomial time. Deligkas et al [17] considered general concave games with compact action spaces and investigated algorithms computing an approximate equilibrium. Roughly speaking, they discretized the compact strategy space and use the Lipschitz constants of utility functions to show that only a finite number of representative strategy profiles need to be considered for obtaining an approximate equilibrium (see also Lipton et al [30] for a similar approach).…”

Approximating the existential theory of the reals

“…It was initially used by Althöfer [3] in zero-sum games, before being applied to non-zero sum games by Lipton, Markakis, and Mehta [2]. Subsequently, it was used to produce algorithms for finding approximate equilibria in normal form games with many players [4], sparse bimatrix games [5], tree polymatrix [6], and Lipschitz games [7]. It has also been used to find constrained approximate equilibria in polymatrix games with bounded treewidth [8].…”

“…, x m such that all of the n • m events of (5) are realized with positive probability, therefore the events of (4) are realized with positive probability and thus the lemma follows. By requiring (7) to be strictly less than 1, and solving for k…”

Distance-Based Equilibria in Normal-Form Games

“…In biased games (Caragiannis, Kurokawa, and Procaccia 2014), a subclass of penalty games (Deligkas, Fearnley, and Spirakis 2016)), players are equipped with a non-linear utility function; this is due to an additional bias term (or penalty) occurring in the utility function. The bias term itself is a realvalued function defined on the distance (via an L p norm) between the played strategy and a base strategy (a particular strategy e.g., represents a social norm).…”