Journal of Computer and System Sciences ( IF 1.494 ) Pub Date : 2020-11-18 , DOI: 10.1016/j.jcss.2020.10.007
Ioannis Caragiannis; Angelo Fanelli

We consider weighted congestion games with polynomial latency functions of maximum degree $d\ge 1$. 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 $\left(d+\delta \right)$-approximate pure Nash equilibrium of cost at most $\left(d+1\right)/\left(d+\delta \right)$ times the cost of an optimal state always exists, for every $\delta \in \left[0,1\right]$.

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