Abstract

Mean field games of controls (MFGC) are mean field games in which players interact with each other through their controls, in addition to their states. MFGC provide a natural modelling framework in economics, where players (households, firms, etc.) often interact with each other through some macroeconomic variable (price, interest rate, etc.) which is determined by the distribution of the players’ controls (consumption vs. saving, production rate, etc.). Mathematically, the main novelty is the appearance of a fixed point equation on the space of measures, which is coupled with the usual equations of MFG theory. In this talk, I will discuss an ongoing joint work with Alpár Mészáros, in which we study the convergence problem for MFGC, i.e. the problem of rigorously justifying the connection between MFGC and the corresponding sequence of finite player games. Our main results provide convergence rates for the equilibria of the finite player games towards their MFGC limit as the number of players goes to infinity. Rather than analyzing the master equation (which seems to be quite challenging in the setting of MFGC), we first prove a convergence result for open-loop equilibria via “forward-backward propagation of chaos”, and then transfer this estimate to closed-loop equilibria by using some uniform in N estimates on the N-player Nash system.

Seminars in Probability and Finance

Seminari di Equazioni Differenziali e Applicazioni