Smoothed Efficient Algorithms and Reductions for Network Coordination Games

Abstract

Worst-case hardness results for most equilibrium computation problems have raised the need for beyond-worst-case analysis. To this end, we study the smoothed complexity of finding pure Nash equilibria in Network Coordination Games, a PLS-complete problem in the worst case. This is a potential game where the sequential-better-response algorithm is known to converge to a pure NE, albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial) smoothed complexity when the underlying game graph is a complete (resp. arbitrary) graph, and every player has constantly many strategies. We note that the complete graph case is reminiscent of perturbing all parameters, a common assumption in most known smoothed analysis results.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 kstrategy games (with arbitrary k) to local-max-cut up to two flips. The former together with the recent result of Bibak et al. [2018] gives an alternate O(n 8 )-time smoothed algorithm for the 2-strategy case. This notion of reduction allows for the extension of smoothed efficient algorithms from one problem to another.For the first set of results, we develop techniques to bound the probability that an (adversarial) better-response sequence makes slow improvements on the potential. Our approach combines and generalizes the local-max-cut approaches of Etscheid and Röglin [2017], Angel et al. [2017] to handle the multi-strategy case: it requires a careful definition of the matrix which captures the increase in potential, a tighter union bound on adversarial sequences, and balancing it with good enough rank bounds. We believe that the approach and notions developed herein could be of interest in addressing the smoothed complexity of other potential and/or congestion games.

Heterogeneous Beliefs and Multi-Population Learning in Network Games

“…Note that this theorem applies to all 2-population coordination games, as network games with or without star structure are essentially the same when there are only two vertices. We also remark that pure or mixed Nash equilibria in coordination network games are complex; as reported in recent works [5,4,1], finding a pure Nash equilibrium is PLS-complete. Hence, learning in the general case of network coordination is difficult and generally requires some conditions for theoretical analysis [34,35].…”

“…In general, the mean belief dynamics is under the joint effects of the mean, variance, and infinitely many higher moments of the belief distribution. To allow for more conclusive results, we apply the moment closure approximation 4 and assume the effects of the third and higher moments to be negligible. Now, just for a moment, suppose that the system beliefs are homogeneous --the beliefs of every individuals are the same.…”