Bounding the Potential Function in Congestion Games and Approximate Pure Nash Equilibria

In this talk we study the potential function in congestion games. We consider both games with non-decreasing cost functions as well as games with non-increasing utility functions. We show that the value of the potential function Φ(s) of any outcome s of a congestion game approximates the optimum potential value Φ(s *) by a factor Ψ F which only depends on the set of cost/utility functions F, and an additive term which is bounded by the sum of the total possible improvements of the players in the outcome s. To achieve this result we introduce a transition graph, which is defined on a pair of outcomes s, s * and captures how to transform s into s *. On this graph we define an ordered path-cycle decomposition. We upper bound the change in the potential for every path and cycle in the decomposition, and lower bound their contribution to the potential. The result then follows by summing up over all paths and cycles. The significance of this result is twofold. On the one hand it provides Price-of-Anarchy-like results with respect to the potential function. On the other hand, we show that these approximations can be used to compute (1 + ε) · Ψ F-approximate pure Nash equilibria for congestion games with non-decreasing cost functions with the method of Caragiannis et al. [FOCS 2011]. Our technique significantly improves the approximation for polynomial cost functions. Moreover , our analysis suggests and identifies large and practically relevant classes of cost functions for which approximate equilibria with small approximation factors can be computed in polynomial time. For example, in games where resources have a certain cost offset, e.g., traffic networks, the approximation factor drastically decreases with the increase of offsets or coefficients in delay functions.