First order Mean Field Games on networks
speaker: Claudio Marchi (Università degli Studi di Padova)abstract: We focus our attention on deterministic Mean Field Games with finite horizon in which the states of the players are constrained in a network (in our setting, a network is given by a finite collection of vertices connected by continuous edges) and the cost may change from edge to edge. As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria which describe the game in terms of probability measures on trajectories instead of time-dependent probability measures on the network. Our first main result is to establish the existence of such a MFG equilibrium. Afterward, to each MFG equilibrium, can be naturally associated a cost, the corresponding value function and optimal trajectories (chosen by the agents). We prove that optimal trajectories starting at time t=0 are Lipschitz continuous, locally uniformly with respect to the initial position. As a byproduct, we obtain a “Lipschitz” continuity of the MFG equilibrium: its push-forward through the evaluation-function at each time gives rise to a Lipschitz continuous function from the time interval to the space of probability measures on the network. The second main result is to prove that this value function is Lipschitz continuous and solves a Hamilton-Jacobi partial differential equation in the network. This is a joint work with: Y. Achdou (Univ. of Paris), P. Mannucci (Univ. of Padova) and N. Tchou (Univ. of Rennes).
timetable:
Tue 9 May, 11:20 - 11:50, Aula Dini
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