Control of McKean-Vlasov Dynamics versus Mean Field Games

TL;DR

This paper compares two approaches to analyze large-player stochastic differential games: mean-field games and McKean-Vlasov control, focusing on their mathematical differences and solutions.

Contribution

It clarifies the distinctions between the two methods and analyzes their associated forward-backward stochastic differential equations, including linear-quadratic examples.

Findings

Different solution structures for the two approaches.

Explicit analysis of linear-quadratic cases.

General theoretical results on the two methods.

Abstract

We discuss and compare two methods of investigations for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which optimization and passage to the limit are performed. When optimizing first, the asymptotic problem is usually referred to as a mean-field game. Otherwise, it reads as an optimization problem over controlled dynamics of McKean-Vlasov type. Both problems lead to the analysis of forward-backward stochastic differential equations, the coefficients of which depend on the marginal distributions of the solutions. We explain the difference between the nature and solutions to the two approaches by investigating the corresponding forward-backward systems. General results are stated and specific examples are treated, especially when cost functionals are of linear-quadratic type.