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.
|Title of host publication||International Symposium on Combinatorial Optimization|
|Subtitle of host publication||ISCO 2016: Combinatorial Optimization|
|Editors||R. Cerulli, S. Fujishige, A. Mahjoub|
|Publication status||Published - 2016|
|Series||Lecture Notes in Computer Science|