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Uniqueness of equilibria in atomic splittable polymatroid congestion games

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  • Tobias Harks

    (University of Augsburg)

  • Veerle Timmermans

    (Maastricht University)

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 non-matroidal 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.

Suggested Citation

  • Tobias Harks & Veerle Timmermans, 2018. "Uniqueness of equilibria in atomic splittable polymatroid congestion games," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 812-830, October.
    • Handle: RePEc:spr:jcomop:v:36:y:2018:i:3:d:10.1007_s10878-017-0166-5
    DOI: 10.1007/s10878-017-0166-5
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    References listed on IDEAS

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    1. Roughgarden, Tim & Schoppmann, Florian, 2015. "Local smoothness and the price of anarchy in splittable congestion games," Journal of Economic Theory, Elsevier, vol. 156(C), pages 317-342.
    2. Igal Milchtaich, 2005. "Topological Conditions for Uniqueness of Equilibrium in Networks," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 225-244, February.
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    4. Oran Richman & Nahum Shimkin, 2007. "Topological Uniqueness of the Nash Equilibrium for Selfish Routing with Atomic Users," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 215-232, February.
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    Cited by:

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    2. Kenjiro Takazawa, 2019. "Generalizations of weighted matroid congestion games: pure Nash equilibrium, sensitivity analysis, and discrete convex function," Journal of Combinatorial Optimization, Springer, vol. 38(4), pages 1043-1065, November.

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