On Nash-Equilibria of Approximation-Stable Games

Abstract

Abstract. One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ǫ-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ǫ-approximate equilibria are contained inside a small ball of radius ∆ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximationstable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ǫ, ∆) approximation-stable games must have an ǫ-equilibrium of support O(log n) -time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that ∆ and ǫ are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.

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“…We also connect our perturbation-stability condition to a seemingly very different approximationstability condition introduced by Awasthi et al [5,6]. Formally, a game satisfies the strong (ε, ∆)-approximation-stability condition if all ε-approximate equilibria are contained inside a small ball of radius ∆ around a single equilibrium.…”

“…Awasthi et al [5,6] analyzed the question of finding an approximate Nash equilibrium in games that satisfy stability with respect to approximation. However, their condition is quite restrictive in that it focuses only on games that have the property that all the Nash equilibria are close together, thus eliminating from consideration most common games.…”

“…By contrast, our perturbation stability notion, which (as mentioned above) can be shown to be a generalization of their notion, captures many more realistic situations. Our upper bound on approximate equilibria can be viewed as generalizing the corresponding result of [5,6] and it is significantly more challenging technically. Moreover, our lower bounds also apply to the stability notion of [5,6] and provide the first (nontrivial) results about the interesting range of parameters for that stability notion as well.…”

“…Interestingly, approximation stability (which is a restriction of well-supported approximation stability and perturbation stability) is itself a relaxation of the stability condition considered by Awasthi et al [5] which requires that all approximate equilibria be contained in a ball of radius ∆ around a single Nash equilibrium. In this direction, we define the strong version of the stability conditions given in Definitions 3.1 and 3.2 to be a reversal of quantifiers that asks that there be a single (p * , q * ) such that each relevant (p, q) (equilibrium in an ε-perturbed game, ε-approximate equilibrium, or well-supported ε-approximate equilibrium) is ∆-close to (p * , q * ).…”

“…If the game happens to be reasonably stable, then we get improved running times over the Lipton et al [28] guarantees; if this is not the case, then we fall back to the Lipton et al [28] bound. 6 However, given a game, it might be interesting to know how stable it is. In this direction, we provide a characterization of stable constant-sum games along with an algorithm for computing the strong stability parameters of a given constant-sum game.…”