Control and optimal stopping Mean Field Games: a linear programming approach

Abstract

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in earlier papers in the pure control case.

Linear Programming Fictitious Play algorithm for Mean Field Games with optimal stopping and absorption

“…In particular, we show the compactness of the set R under the topology of the convergence in measure. This result also allows to prove the existence of an equilibrium under Assumption 1 only (see Theorem 2.11), improving earlier results in [BDT20] and [DLT21], which did not take into account a general dependence of the map f on m (resp. g on µ) nor coefficients with polynomial growth.…”

“…Because of this lack of regularity, MFGs of optimal stopping require a different treatment, and in particular, a different topology than classical MFGs (stochastic control without absorption). For the strong formulation of Optimal Stopping MFG, we refer the reader to [DLT21].…”

“…Finally, the compactification methods consist in relaxing the optimization problems, and often allow to prove existence under weaker assumptions than the first two approaches. We refer here to the controlled martingale problem approach, introduced in the MFG setting by Lacker [Lac15] and the linear programming approach, developed for MFGs of optimal stopping in [BDT20] and extended to a more general framework in [DLT21].…”

“…The results hold under the opposite inequality to the Lasry-Lions monotonicity condition used in this paper. The theory on mean-field games with regular control and absorption has received a lot of interest recently, being developed in [CF18; CGL21; BC21], and using the linear programming approach in [DLT21]. To the best of our knowledge, no algorithm has been proposed so far in this setting and one of our goals is to fill this gap.…”

“…the topology of the convergence in measure in infinitedimensional spaces and providing appropriate estimates using well chosen metrics. Furthermore, the use of this topology allows us to prove the existence of an equilibria in a much more general framework compared to [BDT20] and [DLT21].…”