Equilibrium Results for Dynamic Congestion Games

Consider the following game. Given a network with a continuum of users at some origins, suppose users wish to reach specific destinations but they are not indifferent to the cost to reach them. They may have multiple possible routes but their choices modify the travel costs on the network. Hence, each user faces the following problem: Given a pattern of travel costs for the different possible routes that reach the destination, find a path of minimal cost. This kind of game belongs to the class of congestion games. In the traditional static approach, travel times are assumed constant during the period of the game. In this paper, we consider the so-called dynamic case where the time-varying nature of traffic conditions is explicitly taken into account. In transportation science, the question of whether there is an equilibrium and how to compute it for such a model is referred to as the dynamic user equilibrium problem. Until now, there was no general model for this problem. Our paper attempts to resolve this issue. We define a new class of games, dynamic congestion games, which capture this time-dependency aspect. Moreover, we prove that under some natural assumptions there is a Nash equilibrium. When we apply this result to the dynamic user equilibrium problem, we get most of the previous known results.