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arXiv · 1512.01375

Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games

Tobias Harks·Veerle Timmermans
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Abstract

We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow polymatroid, then equilibria are unique. Bidirectional flow polymatroids are introduced as a subclass of polymatroids possessing certain exchange properties. We show that important cases such as base orderable matroids can be recovered as a special case of bidirectional flow polymatroids. On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and nonmatroidal set systems per player, there is an isomorphic game having multiple equilibria. Our results leave a gap between base orderable matroids and general matroids for which we do not know whether equilibria are unique.

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Tobias Harks, Veerle Timmermans. 2015-12-04. Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games. https://arxiv.org/abs/1512.01375

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