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Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games

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Combinatorial Optimization (ISCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

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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 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.

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Notes

  1. 1.

    The formal definition of bidirectional flow polymatroids appears in Definition 1.

  2. 2.

    The term “interweaving” has been introduced by Ackermann et al. [1, 2].

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Acknowledgements

We thank Umang Bhaskar, Britta Peis and Satoru Fujishige for fruitful discussions. We also thank Neil Olver for pointing out the connection to base orderable matroids.

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Authors and Affiliations

  1. Institute of Mathematics, University of Augsburg, 86135, Augsburg, Germany

    Tobias Harks

  2. Department of Quantitative Economics, Maastricht University, 6200 MD, Maastricht, The Netherlands

    Veerle Timmermans

Authors
  1. Tobias Harks
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  2. Veerle Timmermans
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Correspondence to Veerle Timmermans .

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Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

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Harks, T., Timmermans, V. (2016). Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_9

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  • DOI: https://doi.org/10.1007/978-3-319-45587-7_9

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