A mean field approach to model flows of agents with path preferences over a network

Abstract

In this paper, we address the problem of modeling the traffic flow of a heritage city whose streets are represented by a network. We consider a mean field approach where the standard forward backward system of equations is also intertwined with a path preferences dynamics. The path preferences are influenced by the congestion status on the whole network as well as the possible hassle of being forced to run during the tour. We prove the existence of a mean field equilibrium as a fixed point, over a suitable set of time-varying distributions, of a map obtained as a limit of a sequence of approximating functions. Then, a bi-level optimization problem is formulated for an external controller who aims to induce a specific mean field equilibrium.

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

“…While with the noisy best response dynamics, the agents update their path preferences comparing the difference between the noisy best response function and their current path preferences, in (8) the agents acts in a way to control the error between the answer to the global information about the actual congestion status and the path preferences of agents who previously entered the network. Another possible generalization of the noisy best response dynamics, when λ varies over time, is the one given in [4].…”

“…In [5], [3] a mean field game approach is implemented for studying the optimal behavior of agents flowing on a network having more than one target (vertices of the networks) to be reached (visited). In [4] an origin-destination model with path preferences dynamics as the one here presented is preliminary treated. In the present paper, generalizing the results in [4], we consider the agent's path preferences dynamics in addition to the usual framing of mean field games (typically defined by the pair made of Hamilton-Jacobi-Bellman and mass conservation equations).…”

“…In [4] an origin-destination model with path preferences dynamics as the one here presented is preliminary treated. In the present paper, generalizing the results in [4], we consider the agent's path preferences dynamics in addition to the usual framing of mean field games (typically defined by the pair made of Hamilton-Jacobi-Bellman and mass conservation equations). Specifically, we propose a model in which the agents choose their path having access to global information about the network congestion, but also being influenced by the decision of agents that has already made their decisions.…”

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

“…In [3], [4] a mean field game approach is implemented for studying the optimal behavior of agents flowing on a network having more than one target (vertices of the networks) to be reached (visited). In [5] an origin-destination model with path preferences dynamics as the one here presented is preliminary treated. In the present paper, starting from the similar analysis of the different problem in [4], and generalizing the results in [5], beside the usual framing of mean field games (typically defined by the pair made of Hamilton-Jacobi-Bellman and mass conservation equations), we also consider the agent's path preferences dynamics.…”

“…In [5] an origin-destination model with path preferences dynamics as the one here presented is preliminary treated. In the present paper, starting from the similar analysis of the different problem in [4], and generalizing the results in [5], beside the usual framing of mean field games (typically defined by the pair made of Hamilton-Jacobi-Bellman and mass conservation equations), we also consider the agent's path preferences dynamics. Specifically, we propose a model in which the agents choose their path having access to global information about the network congestion, but also being influenced by the decision of agents that has already made their decisions.…”