# Linear-Quadratic McKean-Vlasov Stochastic Differential Games

## TL;DR

This paper studies multi-player stochastic differential games with linear McKean-Vlasov dynamics and quadratic costs, providing a direct solution approach and characterizing Nash equilibria via Riccati and backward stochastic differential equations.

## Contribution

It introduces a simple direct method using weak martingale optimality and fixed point arguments to solve mean-field games with linear McKean-Vlasov dynamics.

## Key findings

- Characterization of Nash equilibria through Riccati ODEs and mean-field BSDEs.
- Existence and uniqueness conditions for the equilibrium systems.
- Application to a toy example demonstrating the theoretical results.

## Abstract

We consider a multi-player stochastic differential game with linear McKean-Vlasov dynamics and quadratic cost functional depending on the variance and mean of the state and control actions of the players in open-loop form. Finite and infinite horizon problems with possibly some random coefficients as well as common noise are addressed. We propose a simple direct approach based on weak martingale optimality principle together with a fixed point argument in the space of controls for solving this game problem. The Nash equilibria are characterized in terms of systems of Riccati ordinary differential equations and linear mean-field backward stochastic differential equations: existence and uniqueness conditions are provided for such systems. Finally, we illustrate our results on a toy example.