  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.
