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

Abstract

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.

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

“…In the next section we are going to make a suitable approximation of the problem, in order to be able to work with piecewise constant functions. Moreover, in that case, we will see a possible direct construction of such functions λ also explaining their presence and roles in (5), and then the construction of the functions ρ. Actually, we will not use the formal equations ( 5) but directly construct step-by-step (switch-by-switch) the solutions.…”

“…In the the limit as ε → 0, we get instead a possible sum of functions λ i,j (t, •), defined on the whole interval [τ − , τ + ] and other sums of delta functions. Hence the situation is more complex, including the interpretation of system (5). A rigorous investigation of this situation is going to be the subject of future works.…”

“…(ii) V is inserted in (3) and the variable P , which is not necessarily unique (that is, a priori, there may exist more than one optimal switching instant and more than one admissible subsequent node where it is optimal to switch), is derived; (iii) we suitably approximate the optimal switching instants given by P at point (ii) with the nodes of the partition P ε ; (iv) with such approximated variables P ε as in (iii), we construct all the possible optimal switching paths with their decision and switching times; (v) for each optimal switching path π of point (iv), we construct the corresponding functions λ in (5), as all the agents were following π, that is…”

“…Remark 5. The functions λ i,j and their products as shown in the example above, together with the decsional and switching instants, give the coefficients λ i,j in the formal equations (5), for the decision-making part ρ dm of the evolution.…”

“…The model for a crowd of indistinguishable players is taken from the framework of meanfield games [18,17,15,10], while the adaptation of the same hybrid structure to networks has been only very recently attempted, as in [1,5] and, more generally, in [8,9]. Some works which share the same ideas to treat the mean-field case in the presence of switches in the dynamics of the problem are [7,6] and [3], where a mean-field optimal stopping problem, possibly with sinks and sources, is discussed, and [14], where a hybrid mean-field game is presented to model a multi-lane traffic flux of vehicles.…”