  A central question in routing games has been to establish conditions for the
uniqueness of the equilibrium, either in terms of network topology or in terms
of costs. This question is well understood in two classes of routing games. The
first is the non-atomic routing introduced by Wardrop on 1952 in the context of
road traffic in which each player (car) is infinitesimally small; a single car
has a negligible impact on the congestion. Each car wishes to minimize its
expected delay. Under arbitrary topology, such games are known to have a convex
potential and thus a unique equilibrium. The second framework is splitable
atomic games: there are finitely many players, each controlling the route of a
population of individuals (let them be cars in road traffic or packets in the
communication networks). In this paper, we study two other frameworks of
routing games in which each of several players has an integer number of
connections (which are population of packets) to route and where there is a
constraint that a connection cannot be split. Through a particular game with a
simple three link topology, we identify various novel and surprising properties
of games within these frameworks. We show in particular that equilibria are non
unique even in the potential game setting of Rosenthal with strictly convex
link costs. We further show that non-symmetric equilibria arise in symmetric
networks. I. INTRODUCTION A central question in routing games has been to
establish conditions for the uniqueness of the equilibria, either in terms of
the network topology or in terms of the costs. A survey on these issues is
given in [1]. The question of uniqueness of equilibria has been studied in two
different frameworks. The first, which we call F1, is the non-atomic routing
introduced by Wardrop on 1952 in the context of road traffic in which each
player (car) is infinitesimally small; a single car has a negligible impact on
the congestion. Each car wishes to minimize its expected delay. Under arbitrary
topology, such games are known to have a convex potential and thus have a
unique equilibrium [2]. The second framework, denoted by F2, is splitable
atomic games. There are finitely many players, each controlling the route of a
population of individuals. This type of games have already been studied in the
context of road traffic by Haurie and Marcotte [3] but have become central in
the telecom community to model routing decisions of Internet Service Providers
that can decide how to split the traffic of their subscribers among various
routes so as to minimize network congestion [4]. In this paper we study
properties of equilibria in two other frameworks of routing games which exhibit
surprising
