Closed-loop convergence for mean field games with common noise

Abstract

This paper studies the convergence problem for mean field games with common noise. We define a suitable notion of weak mean field equilibria, which we prove captures all subsequential limit points, as n → ∞, of closed-loop approximate equilibria from the corresponding n-player games. This extends to the common noise setting a recent result of the first author, while also simplifying a key step in the proof and allowing unbounded coefficients and non-i.i.d. initial conditions. Conversely, we show that every weak mean field equilibrium arises as the limit of some sequence of approximate equilibria for the n-player games, as long as the latter are formulated over a broader class of closed-loop strategies which may depend on an additional common signal.

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

“…Indeed, [41] cannot show that any weak MFG equilibrium is the limit of a sequence of approximate Nash equilibria. In the same spirit, Lacker and Flem [42] provide, first, a result relating to the convergence of closed-loop Nash equilibria in a common noise setting. Second, by considering an extension of the notion of closed-loop Nash equilibrium, unlike [41], they are able to show the converse convergence result.…”

“…This result is achieved by adapting the arguments of [41] in the setting of MFGC with common noise, and by using a delicate estimate of the regularity of the fokker planck equation proved by Aronson and Serrin [2,Theorem 4]. It is worth mentioning that this result contains part of those of [42] in the classical MFG framework but allow in addition σ to be non-constant. Second, similarly to the convergence of closed-loop Nash equilibria, Theorem 2.10 shows the convergence of approximate strong Markovian MFG equilibria to the measure-valued MFG equilibria.…”

“…Even in the framework of classical MFG, this result seems to be the first of this kind. Indeed, the converse convergence result of [42] needs to be done with a slight extension of the notion of closed-loop Nash equilibria which they call S-closed loop Nash equilibria.…”

“…Without counting the fact that we take into account the empirical distribution of states and controls, even for the classical MFG framework, this result appears as new in the literature. As highlighted in [41] and later in [42], in the presence of multiple MFG equilibria, this kind of converse result is delicate to prove without extension of the notion of closed-loop Nash equilibrium (see [42]). To bypass this difficulty, the condition σ 0 is invertible turns out to be quite useful.…”

“…The convergence of Nash equilibria and converse convergence is provided. In particular, we are able to prove that any weak MFG is the limit of Nash equilibria without extending the notion of Nash equilibria as in [42].…”