Computing Equilibria in Atomic Splittable Polymatroid Congestion Games with Convex Costs

TL;DR

This paper presents a method to compute approximate Nash equilibria in complex congestion games with convex costs by leveraging integrally-splittable game solutions, extending to oligopoly models with polynomial-time approximations.

Contribution

It introduces a novel approach to find epsilon-approximate equilibria in atomic splittable polymatroid congestion games using integrally-splittable game solutions, applicable to oligopoly models.

Findings

Computing pure Nash equilibria for integrally-splittable games is pseudo-polynomial.

The associated k_epsilon-splittable equilibrium approximates the original game within epsilon.

The method applies to oligopoly models, enabling polynomial-time approximation of Cournot-Nash equilibria.

Abstract

In this paper, we compute $\epsilon$-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 $\epsilon>0$, a packet size $k_{\epsilon}$ and prove that the associated $k_{\epsilon}$-splittable Nash equilibrium is an $\epsilon$-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 $\epsilon$-approximate Cournot-Nash equilibria within pseudo-polynomial time.