Convergence Issues in Congestion Games

Congestion games are a widely studied class of non-cooperative games. In fact, besides being able to model many practical settings, they constitute a framework with nice theoretical properties: Congestion games always converge to pure Nash Equilibria by means of improvement moves performed by the players, and many classes of congestion games guarantee a low price of anarchy, that is the ratio between the worst Nash Equilibrium and the social optimum. Unfortunately, the time of convergence to Nash Equilibria, even under best response moves of the players, can be very high, i.e., exponential in the number of players, and in many setting also computing a Nash equilibrium can require a high computational complexity. Motivated by the above facts, in order to guarantee a fast convergence to Nash Equilibria, in the last decade many computer science researchers focused on special classes of congestion games (e.g., with linear or polynomial delay functions), on simplified structures of the strategy space (e.g., on symmetric games in which all players share the same set of strategies or on matroid congestion games in which the set of strategies constitutes a matroid) and on the relaxation of the notion of Nash Equilibria (e.g., exploiting the notion of -Nash Equilibria). We survey such attempts that, however, only in some very specific cases have led to satisfactory results on the speed of convergence to Nash Equilibria. If we relax the constraint of reaching a Nash Equilibrium, and our goal becomes that of reaching states approximating the social optimum by a “low” factor, i.e., a factor being order of the price of anarchy, significantly better results on the speed of convergence under best response dynamics can be achieved. Interestingly, in the more general asymmetric setting, fairness among players influences the speed of convergence. For instance, considering the fundamental class of linear congestion games, if each player is allowed to play at least once and at most β times every T best responses, states with approximation ratio O(β) times the price of anarchy are reached after ∗Department of Economic Studies, University of Chieti-Pescara, Viale Pindaro 42, 65127 Pescara, Italy. Email: [email protected] T dlog log ne best responses, and such a bound is essentially tight also after exponentially many ones. It is worth noticing that the structure of the game implicitly affects its performances in terms of convergence speed: In particular, in the symmetric setting the game always converges to an efficient state after a polynomial number of best responses, regardless of the frequency each player moves with. Most of these results extend to polynomial and weighted congestion games.