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In this paper, we compute $\epsilon$-approximate Nash equilibria in atomicsplittable polymatroid congestion games with convex Lipschitz continuous costfunctions. The main approach relies on computing a pure Nash equilibrium for anassociated integrally-splittable congestion game, where players can only splittheir demand in integral multiples of a common packet size. It is known thatone can compute pure Nash equilibria for integrally-splittable congestion gameswithin a running time that is pseudo-polynomial in the aggregated demand of theplayers. As the main contribution of this paper, we decide for every$\epsilon>0$, a packet size $k_{\epsilon}$ and prove that the associated$k_{\epsilon}$-splittable Nash equilibrium is an $\epsilon$-approximate Nashequilibrium for the original game.  We further show that our result applies to multimarket oligopolies withdecreasing, concave Lipschitz continuous price functions and quadraticproduction costs: there is a polynomial time transformation to atomicsplittable polymatroid congestion games implying that we can compute$\epsilon$-approximate Cournot-Nash equilibria within pseudo-polynomial time.

Computing Equilibria in Atomic Splittable Polymatroid Congestion Games with Convex Costs