On approximate pure Nash equilibria in weighted congestion games with polynomial latencies

Abstract

We consider the problem of the existence of natural improvement dynamics leading to approximate pure Nash equilibria, with a reasonable small approximation, and the problem of bounding the efficiency of such equilibria in the fundamental framework of weighted congestion game with polynomial latencies of degree at most d ≥ 1. In this work, by exploiting a simple technique, we firstly show that the game always admits a d-approximate potential function. This implies that every sequence of d-approximate improvement moves by the players always leads the game to a d-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, the game always admits a constant approximate potential function. Secondly, by using a simple potential function argument, we are able to show that in the game there always exists a (d + δ)-approximate pure Nash equilibrium, with δ ∈ [0, 1], whose cost is 2/(1 + δ) times the cost of an optimal state. 2012 ACM Subject Classification Theory of computation → Algorithmic game theory; Theory of computation → Convergence and learning in games Keywords and phrases Congestion games, approximate pure Nash equilibrium, potential functions, approximate price of stability

Natural and efficient dynamics through convex optimization

“…The latter implies that any local minimum of the potential function is also a Nash Equilibrium and that best response dynamics always converges to equilibrium. Although computing Nash Equilibrium in congestion games proved to be a computationally hard problem (PLS-complete) [68] meaning that best response dynamics can take exponentially many rounds before reaching an equilibrium, there are many positive results for its convergence properties to approximate Nash equilibrium [112,49,46,31,30]. Moreover best response dynamics is known to converge to Nash Equilibrium in polynomial number of rounds for many important special cases of congestion games [66,123] or when the instance of the game is contaminated with random noise [65,4,22].…”