Uniqueness of Equilibria in Atomic Splittable Polymatroid Congestion Games

TL;DR

This paper establishes conditions under which Nash equilibria in atomic splittable congestion games are unique, using polymatroid theory, and identifies classes of set systems that guarantee or prevent uniqueness.

Contribution

It introduces bidirectional flow polymatroids and proves their role in ensuring equilibrium uniqueness in congestion games, extending the understanding of matroidal set systems.

Findings

Equilibria are unique when players' strategy spaces are bidirectional flow polymatroids.

Base orderable matroids are a special case of bidirectional flow polymatroids.

Non-matroidal set systems can lead to multiple equilibria in the game.

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.