Convergence Dynamics of Graphical Congestion Games - Technical Report

Graphical congestion games provide powerful models for a wide range of scenarios where spatially distributed individuals share resources. Understanding when graphical congestion game dynamics converge to pure Nash equilibria yields important engineering insights into when spatially distributed individuals can reach a stable resource allocation. In this paper, we study the convergence dynamics of graphical congestion games where players can use multiple resources simultaneously. We show that when the players are free to use any subset of resources the game always converges to a pure Nash equilibrium in polynomial time via lazy best response updates. When the collection of sets of resources available to each player is a matroid, we show that pure Nash equilibria may not exist in the most general case. However, if the resources are homogenous, the game can converge to a Nash equilibrium in polynomial time.