First order Mean Field Games on networks

Abstract

We study deterministic mean field games in which the state space is a network. Each agent controls its velocity; in particular, when it occupies a vertex, it can enter in any edge incident to the vertex. The cost is continuous in each closed edge but not necessarily globally in the network. We shall follow the Lagrangian approach studying relaxed equilibria which describe the game in terms of a probability measure on admissible trajectories. The first main result of this paper establishes the existence of a relaxed equilibrium. The proof requires the existence of optimal trajectories and a closed graph property for the map which associates to each point of the network the set of optimal trajectories starting from that point. Each relaxed equilibrium gives rise to a cost for the agents and consequently to a value function. The second main result of this paper is to prove that such a value function solves an Hamilton-Jacobi problem on the network.