Origin-to-destination network flow with path preferences and velocity controls: a mean field game-like approach

In this paper we consider a mean field approach to modeling the agents flow over a transportation network. In particular, beside a standard framework of mean field games, with controlled dynamics by the agents and costs mass-distribution dependent, we also consider a path preferences dynamics obtained as a generalization of the so-called noisy best response dynamics. We introduce this last dynamics to model the fact that the agents choose their path on the basis of both the network congestion state and the observation of the agents' decision that have preceded them. We prove the existence of a mean field equilibrium obtained as a fixed point of a map over a suitable set of time-varying mass-distributions, defined edge by edge in the network. We also address the case where the admissible set of controls is suitably bounded depending on the mass-distribution on the edge itself.