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

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 $\d \geq 1$.\\ 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 $(\d+\delta)$-approximate pure Nash equilibrium, with $\delta\in [0,1]$, whose cost is $2/(1+\delta)$ times the cost of any optimal state.