We consider a nonatomic congestion game on a graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origindestination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine. We show that this problem is polynomially solvable when the number of vertices and the number of classes are fixed. In particular, it shows that the parallel-link case with a fixed number of classes is polynomially solvable. On a more practical side, we propose an extension of Lemke's algorithm able to solve this problem. IntroductionContext. Being able to predict the impact of a new infrastructure on the traffic in a transportation network is an old but still important objective for transport planners. In 1952, Wardrop (1952 noted that after some while the traffic arranges itself to form an equilibrium and formalized principles characterizing this equilibrium. With the terminology of game theory, the equilibrium is a Nash equilibrium for a congestion game with nonatomic users. , Beckmann et al. (1956 translated these principles as a mathematical program which turned out to be convex, opening the door to the tools from convex optimization. The currently most commonly used algorithm for such convex programs is probably the Frank-Wolfe algorithm (Frank and Wolfe, 1956), because of its simplicity and its efficiency, but many other algorithms with excellent behaviors have been proposed, designed, and experimented.One of the main assumptions used by Beckmann to derive his program is the fact that all users are equally impacted by the congestion. With the transportation terminology, it means that there is only one class. In order to improve the prediction of traffic patterns, researchers started in the 70s to study the multiclass situation where each class has its own way of being impacted by the congestion. Each class models a distinct mode of transportation, such as cars, trucks, or motorbikes. Dafermos (1972( , 1980( ) and Smith (1979 are probably the first who proposed a mathematical formulation of the equilibrium problem in the multiclass case. However, even if this problem has been the topic of many research works, an efficient algorithm for solving it is not known, except in some special cases (Florian, 1977, Harker, 1988, Mahmassani and Mouskos, 1988, Marcotte and Wynter, 2004. In particular, there are no general algorithms in the literature for solving the problem when the cost of each arc is in an affine dependence with the flow on it.Our purpose is to discuss the existence of such algorithms.Model. We are given a directed graph D = (V, A) modeling the transportation network. The set of all paths (resp. s-t paths) is denoted by P (resp. P (s,t) ). The population of players is mo...

We consider congestion games on networks with nonatomic users and user-specific costs. We are interested in the uniqueness property defined by Milchtaich [Milchtaich, I. 2005. Topological conditions for uniqueness of equilibrium in networks. as the uniqueness of equilibrium flows for all assignments of strictly increasing cost functions. He settled the case with two-terminal networks. As a corollary of his result, it is possible to prove that some other networks have the uniqueness property as well by adding common fictitious origin and destination. In the present work, we find a necessary condition for networks with several origin-destination pairs to have the uniqueness property in terms of excluded minors or subgraphs. As a key result, we characterize completely bidirectional rings for which the uniqueness property holds: it holds precisely for nine networks and those obtained from them by elementary operations. For other bidirectional rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. Related results are also proven. For instance, we characterize networks having the uniqueness property for any choice of origin-destination pairs. and those obtained from them by elementary operations. For other rings, we exhibit affine cost functions yielding to two distinct equilibrium flows. It allows to describe infinite families of graphs for which the uniqueness property does not hold. For instance, there is a family of ring networks such that every network with a minor in this family does not have the uniqueness property. Preliminaries on graphsAn undirected graph is a pair G = (V, E) where V is a finite set of vertices and E is a family of unordered pairs of vertices called edges. A directed graph, or digraph for short, is a pair D = (V, A) where V is a finite set of vertices and A is a family of ordered pairs of vertices called arcs. A mixed graph is a graph having edges and arcs. More formally, it is a triple M = (V, E, A) where V is a finite set of vertices, E is a family of unordered pairs of vertices (edges) and A is a family of ordered pairs of vertices (arcs). Given an undirected graph G = (V, E), we define the directed version of G as the digraph D = (V, A) obtained by replacing each (undirected) edge in E by two (directed) arcs, one in each direction. An arc of G is understood as an arc of its directed version. In these graphs, loops -edges or arcs having identical endpoints -are not allowed, but pairs of vertices occuring more than onceparallel edges or parallel arcs -are allowed.A walk in a directed graph D is a sequenceIf all v i are distinct, the walk is called a path. If no confusion may arise, we identify sometimes a path P with the set of its vertices or with the set of its arcs, allowing to use the notation v ∈ P (resp. a ∈ P ) if a vertex v (resp. an arc a) occurs in P . An undirected graph G = (V , E ) is a subgraph of an undirected graph G = (V, E) if V ⊆ V and E ⊆ E. An undirected graph G is a minor of an undirected graph G if G is obtained by contracting edges (poss...

Abstract. We consider a nonatomic congestion game on a connected graph, with several classes of players. Each player wants to go from its origin vertex to its destination vertex at the minimum cost and all players of a given class share the same characteristics: cost functions on each arc, and origin-destination pair. Under some mild conditions, it is known that a Nash equilibrium exists, but the computation of an equilibrium in the multiclass case is an open problem for general functions. We consider the specific case where the cost functions are affine and propose an extension of Lemke's algorithm able to solve this problem. At the same time, it provides a constructive proof of the existence of an equilibrium in this case.

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