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

We consider weighted congestion games with polynomial latency functions of maximum degree d≥1. For these games, we investigate the existence and efficiency of approximate pure Nash equilibria which are obtained through sequences of unilateral improvement moves by the players. By exploiting a simple technique, we firstly show that these games admit an infinite set of d-approximate potential functions. This implies that there always exists a d-approximate pure Nash equilibrium which can be reached through any sequence of d-approximate improvement moves by the players. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, these games also admit a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a (d+δ)-approximate pure Nash equilibrium of cost at most (d+1)/(d+δ) times the cost of an optimal state always exists, for every δ∈[0,1].