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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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03729443
Contributor : Claudio Marchi Connect in order to contact the contributor
Submitted on : Wednesday, July 20, 2022 - 3:45:04 PM
Last modification on : Friday, August 5, 2022 - 12:02:00 PM

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  • HAL Id : hal-03729443, version 1

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Yves Achdou, Paola Mannucci, Claudio Marchi, Nicoletta Tchou. First order Mean Field Games on networks. 2022. ⟨hal-03729443⟩

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