The Speed of Convergence in Congestion Games under Best-Response Dynamics

We investigate the speed of convergence of congestion games with linear latency functions under best response dynamics. Namely, we estimate the social performance achieved after a limited number of rounds, during each of which every player performs one best response move. In particular, we show that the price of anarchy achieved after krounds, defined as the highest possible ratio among the total latency cost, that is the sum of all players latencies, and the minimum possible cost, is $O(\sqrt[2^{k-1}] {n})$, where nis the number of players. For constant values of ksuch a bound asymptotically matches the $\Omega(\sqrt[2^{k-1}] {n}/k)$ lower bound that we determine as a refinement of the one in [7]. As a consequence, we prove that order of loglognrounds are not only necessary, but also sufficient to achieve a constant price of anarchy, i.e. comparable to the one at Nash equilibrium. This result is particularly relevant, as reaching an equilibrium may require a number of rounds exponential in n. We then provide a new lower bound of $\Omega(\sqrt[2^k-1] {n})$ for load balancing games, that is congestion games in which every strategy consists of a single resource, thus showing that a number of rounds proportional to loglognis necessary and sufficient also under such a restriction. Our results thus solve the important left open question of the polynomial speed of convergence of linear congestion games to constant factor solutions.