Calculus of Variations and Geometric Measure Theory

F. Bagagiolo - L. Marzufero

A time-dependent switching mean-field game on networks motivated by optimal visiting problems

created by marzufero on 12 Jan 2022
modified on 23 Mar 2023


Published Paper

Inserted: 12 jan 2022
Last Updated: 23 mar 2023

Journal: Journal of Dynamics and Games
Volume: 10
Number: 2
Pages: 151-180
Year: 2023

ArXiv: 2201.00260 PDF

Early access: September 2022


Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable is at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from $0$ to $1$ of one component of the string $p =(p_1,\ldots, p_n)$ which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by $i=1,\ldots,n$. The goal is to reach the final string $(1, \ldots, 1)$ in the final time $T$, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated $\varepsilon$-mean-field equilibrium and then address the passage to the limit when $\varepsilon$ goes to 0.

Keywords: Mean-Field Games, networks, switching, optimal visiting, optimal path, impulsive continuity equations