Motivated by the fact that in many game-theoretic settings, the game analyzed is only an approximation to the game being played, in this work we analyze equilibrium computation for the broad and natural class of bimatrix games that are stable to perturbations. We specifically focus on games with the property that small changes in the payoff matrices do not cause the Nash equilibria of the game to fluctuate wildly. For such games we show how one can compute approximate Nash equilibria more efficiently than the general result of LMM03 , by an amount that depends on the degree of stability of the game and that reduces to their bound in the worst case. Furthermore, we show that for stable games the approximate equilibria found will be close in variation distance to true equilibria, and moreover this holds even if we are given as input only a perturbation of the actual underlying stable game. For uniformly-stable games, where the equilibria fluctuate at most quasi-linearly in the extent of the perturbation, we get a particularly dramatic improvement. Here, we achieve a fully quasi-polynomial-time approximation scheme: that is, we can find @math -approximate equilibria in quasi-polynomial time. This is in marked contrast to the general class of bimatrix games for which finding such approximate equilibria is PPAD-hard. In particular, under the (widely believed) assumption that PPAD is not contained in quasi-polynomial time, our results imply that such uniformly stable games are inherently easier for computation of approximate equilibria than general bimatrix games.
@cite_11 analyzed the question of finding an approximate Nash in 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 bounds on approximate equilibria can be viewed as generalizing the corresponding result of @cite_11 and it is significantly more challenging technically. Moreover, our lower bounds also apply to the stability notion of @cite_11 and provide the first (nontrivial) results about the interesting range of parameters for that stability notion as well.
@cite_11
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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 e-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 e-approximate equilibria are contained inside a small ball of radius Δ 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 (e,Δ) approximation-stable games must have an e-equilibrium of support O(Δ2-o(1) e2 log n), yielding an immediate nO(Δ2-o(1) e2 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 Δ and e 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. We analyze the number of payoff queries needed to compute approximate equilibria of multi-player games. We find that query complexity is an effective tool for distinguishing the computational difficulty of alternative solution concepts, and we develop new techniques for upper- and lower bounding the query complexity. For binary-choice games, we show logarithmic upper and lower bounds on the query complexity of approximate correlated equilibrium. For well-supported approximate correlated equilibrium (a restriction where a player’s behavior must always be approximately optimal, in the worst case over draws from the distribution) we show a linear lower bound, thus separating the query complexity of well supported approximate correlated equilibrium from the standard notion of approximate correlated equilibrium. Finally, we give a query-efficient reduction from the problem of computing an approximate well-supported Nash equilibrium to the problem of verifying a well supported Nash equilibrium, where the additional query overhead is proportional to the description length of the game. This gives a polynomial-query algorithm for computing well supported approximate Nash equilibria (and hence correlated equilibria) in concisely represented games. We identify a class of games (which includes congestion games) in which the reduction can be made not only query efficient, but also computationally efficient. We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the well- known multiplicative-weights learning algorithm results in convergence to pure equilibria. Our results show that nat- ural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the load-balancing game introduced by Koutsoupias and Pa- padimitriou, which has super-constant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined game-theoretically, but we show that this property holds whenever a stability criterion from the theory of dy- namical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribu- tion). We further extend our results to show that players can use algorithms with different (sufficiently small) learn- ing rates, i.e. they can trade off convergence speed and long term average regret differently. A recent sequence of results established that computing Nash equilibria in normal form games is a PPAD-complete problem even in the case of two players [11,6,4]. By extending these techniques we prove a general theorem, showing that, for a far more general class of families of succinctly representable multiplayer games, the Nash equilibrium problem can also be reduced to the two-player case. In view of empirically successful algorithms available for this problem, this is in essence a positive result — even though, due to the complexity of the reductions, it is of no immediate practical significance. We further extend this conclusion to extensive form games and network congestion games, two classes which do not fall into the same succinct representation framework, and for which no positive algorithmic result had been known.