Parametrized Nash Equilibria in Atomic Splittable Congestion Games via Weighted Block Laplacians
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We consider atomic splittable congestion games with affine cost functions and develop an algorithm that computes all Nash equilibria of the game parametrized by the players' demands. That is, given a game where the players' demand rates are piece-wise linear functions of some parameter λ≥ 0, we compute a family of multi-commodity flows x(λ) parametrized in λ such that x(λ) is a Nash equilibrium for the corresponding demand rate vector r(λ). Our algorithm is based on a novel weighted block Laplacian matrix concept for atomic splittable games. We show that the weighted block Laplacians have similar properties as ordinary weighted graph Laplacians which allows to compute the parametrized Nash equilibria by matrix pivot operations. Our algorithm is output-polynomial on all instances, and each pivot step needs only O((nk)^2.4) where k is the number of players and n is the number of vertices.
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