Mean Field Games of Controls: on the convergence of Nash equilibria

Abstract

In this paper, we investigate a class of mean field games where the mean field interactions are achieved through the joint (conditional) distribution of the controlled state and the control process. The strategies are of open loop type, and the volatility coefficient σ can be controlled. Using (controlled) Fokker-Planck equations, we introduce a notion of measure-valued solution of mean-field games of controls, and through convergence results, prove a relation between these solutions on the one hand, and the ǫN -Nash equilibria on the other hand. It is shown that ǫN -Nash equilibria in the N -player games have limits as N tends to infinity, and each limit is a measure-valued solution of the mean-field games of controls. Conversely, any measure-valued solution can be obtained as the limit of a sequence of ǫN -Nash equilibria in the N -player games. In other words, the measure-valued solutions are the accumulating points of ǫN -Nash equilibria. Similarly, by considering an ǫ-strong solution of mean field games of controls which is the classical strong solution where the optimality is obtained by admitting a small error ǫ, we prove that the measure-valued solutions are the accumulating points of this type of solutions when ǫ goes to zero. Finally, existence of measure-valued solution of mean-field games of controls are proved in the case without common noise. * The author is grateful to Dylan Possamaï and Xiaolu Tan for helpful comments and suggestions.

Large population games with interactions through controls and common noise: convergence results and equivalence between $open$--$loop$ and $closed$--$loop$ controls

“…With the equivalence between our notion of measure-valued MFG equilibrium and the notion considered in [24,Definition 2.7] for open-loop framework that we will prove in Proposition 3.5, thanks to the result proved in [25, Theorem 7.2.4], we have the following result.…”

“…The previous assumptions are standard in the probabilistic approach of mean field game and control problems. The separability condition is more specific to the extended mean field game and control problems (see Carmona and Lacker [15], Cardaliaguet and Lehalle [10], Laurière and Tangpi [46], Possamaï and Tangpi [46], Djete [24]). It is mainly used for technical reasons.…”

“…relatively compact, and when ε N → 0, any limit point is a measure-valued MFG solution (see Definition 2.7). Recall that the notion of measure-valued MFG solution is similar to the one used in [24] (see equivalence in Proposition 3.5 and Remark 3.6) to treat the convergence of open-loop Nash equilibria. While being the first result dealing with convergence of closed-loop Nash equilibria for MFG of controls with common noise, this first result of Theorem 2.10 shows that the limits of open-loop and closed-loop Nash equilibria are exactly the same.…”

“…We will call this formulation (approximate) strong Markovian MFG equilibrium. Second, inspired by [24], we give the notion of measure-valued MFG which generalizes the strong Markovian MFG equilibrium.…”

“…In the situation without the empirical distribution of controls, under relatively general assumptions, a complete picture has been proposed by Fisher [29] and Lacker [38] (for the case with common noise) using the notions of relaxed MFG equilibrium. For the extended MFG, while allowing the volatility to be controlled, a similar study to [38] is provided by Djete [24] thanks to the notion of measure-valued MFG equilibrium. With stronger assumptions but by providing convergence rates, Laurière and Tangpi [46] study these convergence problems in the situation without common noise using notably a notion of backward propagation of chaos.…”