Weakly coupled mean-field game systems

Abstract

Here, we prove the existence of solutions to first-order mean-field games (MFGs) arising in optimal switching. First, we use the penalization method to construct approximate solutions. Then, we prove uniform estimates for the penalized problem. Finally, by a limiting procedure, we obtain solutions to the MFG problem.

One-Dimensional Stationary Mean-Field Games with Local Coupling

“…Heuristically, this assumption means that agents prefer sparsely populated areas. In this case, the existence and uniqueness of smooth solutions to (1.1) is well understood for stationary problems [18,19,20,33], weakly coupled MFG systems [11], the obstacle MFG problem [12] and extended MFGs [13]. In the time-dependent setting, similar results are obtained in [14,15,24] for standard MFGs and in [16,23] for forward-forward problems.…”

Monotone solutions for mean field games master equations: finite state space and optimal stopping

“…In recent years, the study of MFG of optimal stopping or other "singular" controls have been the subject of a growing number of researches, namely because such games have natural applications in Economy. Concerning the case of optimal stopping, let us mention [7,19,35,36], for impulse control we refer to [8] and to [27] for optimal switching. Let us also mention that the approach of [7,8] has been used by P.-L. Lions in [33] to study a case of MFG of singular controls.…”

Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

“…In the stationary setting, existence and regularity of MFGs without congestion were considered in [10], [12], [14], [22], [23] (strong solutions) and [5], [19] (weak solutions). Many other stationary problems are examined in the literature, including obstacle problems [9], weaklycoupled systems [8], multi-populations [4], and logistic problems [13]. MFGs on networks, see [1], [3], [2], are important cases of one-dimensional MFGs.…”