Computing Equilibria in Atomic Splittable Polymatroid Congestion Games with Convex Costs
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In this paper, we compute ϵ-approximate Nash equilibria in atomic splittable polymatroid congestion games with convex Lipschitz continuous cost functions. The main approach relies on computing a pure Nash equilibrium for an associated integrally-splittable congestion game, where players can only split their demand in integral multiples of a common packet size. It is known that one can compute pure Nash equilibria for integrally-splittable congestion games within a running time that is pseudo-polynomial in the aggregated demand of the players. As the main contribution of this paper, we decide for every ϵ>0, a packet size k_ϵ and prove that the associated k_ϵ-splittable Nash equilibrium is an ϵ-approximate Nash equilibrium for the original game. We further show that our result applies to multimarket oligopolies with decreasing, concave Lipschitz continuous price functions and quadratic production costs: there is a polynomial time transformation to atomic splittable polymatroid congestion games implying that we can compute ϵ-approximate Cournot-Nash equilibria within pseudo-polynomial time.
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