Uniqueness of equilibria in atomic splittable polymatroid congestion games

Tobias Harks, Veerle Timmermans*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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.

Original languageEnglish
Pages (from-to)812-830
Number of pages19
JournalJournal of Combinatorial Optimization
Issue number3
Publication statusPublished - 1 Oct 2018


  • Polymatroid
  • Congestion game
  • Uniqueness of equilibria


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