Performances of One-Round Walks in Linear Congestion Games

Abstract

We investigate the approximation ratio of the solutions achieved after a one-round walk in linear congestion games. We consider the social functions ${\mathrm{S}\textsc{um}}$, defined as the sum of the players’ costs, and ${\mathrm{M}\textsc{ax}}$, defined as the maximum cost per player, as a measure of the quality of a given solution. For the social function ${\mathrm{S}\textsc{um}}$ and one-round walks starting from the empty strategy profile, we close the gap between the upper bound of $2+\sqrt{5}\approx 4.24$ given in [8] and the lower bound of 4 derived in [4] by providing a matching lower bound whose construction and analysis require non-trivial arguments. For the social function ${\mathrm{M}\textsc{ax}}$, for which, to the best of our knowledge, no results were known prior to this work, we show an approximation ratio of $\Theta(\sqrt[4]{n^3})$ (resp. $\Theta(n\sqrt{n})$), where n is the number of players, for one-round walks starting from the empty (resp. an arbitrary) strategy profile.