Nash Equilibrium in Smoothed Polynomial Time for Network Coordination Games
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Extensive work in the last two decades has led to deep insights into the worst case complexity of computing Nash equilibria (NE). However, largely negative results have raised the need for beyond worst-case analysis of these problems. In this paper, we study the smoothed complexity of finding pure NE in Network Coordination Games, a PLS-complete problem in the worst case. First, we prove (quasi-) polynomial smoothed complexity when the underlying game graph is an (arbitrary) complete graph, and every player has constantly many strategies. To the best of our knowledge, ours is the first smoothed efficient algorithm for a Nash equilibrium problem that is hard in the worst case. We note that the complete graph case is reminiscent of perturbing all parameters, a common assumption in almost all known smoothed efficient analysis. Second, we define a notion of smoothness-preserving reduction among search problems, and obtain reductions from 2-strategy network coordination games to local-max-cut, and from k-strategy games (with arbitrary k) to local-max-cut up to two flips. The former, together with the known smoothed efficient algorithm for the local-max-cut problem, gives an alternate efficient smoothed algorithm for the 2-strategy games. This notion of reduction may be utilized to extend smoothed efficient algorithms from one problem to another. For the first set of results, we develop techniques to analyze the probability of a slow increase in potential during the better-response algorithm for a perturbed game, by building on and significantly extending the recent approaches for smoothed analysis of local-max-cut [ER'14,ABPW'17]. We note that network coordination games are far more general than local-max-cut since the latter reduces to a 2-strategy case of the former. Our techniques may be of independent interest to perform smoothed analysis of other potential/congestion games.
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