[Paper Review] Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach

Minyi Huang, Mengjie Zhou|arXiv (Cornell University)|Mar 20, 2019

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine50 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition for asymptotic solvability in linear-quadratic mean field games using a novel re-scaling technique that reduces the system to a low-dimensional non-symmetric Riccati ODE. It proves that asymptotic solvability implies feasibility of the fixed point approach, but not vice versa, and analyzes long-time behavior and non-uniqueness in the fixed point framework.

ABSTRACT

Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit. This is accomplished by developing a re-scaling method to derive a low dimensional ordinary differential equation (ODE) system, where a non-symmetric Riccati ODE has a central role. We next compare with the fixed point approach which determines a two point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem.

Motivation & Objective

• To formalize and analyze asymptotic solvability in the direct (bottom-up) approach to linear-quadratic mean field games.
• To establish a necessary and sufficient condition for asymptotic solvability using a re-scaling method that reduces the system to a low-dimensional non-symmetric Riccati ODE.
• To compare the direct approach with the fixed point (top-down) approach, clarifying their relationship and relative feasibility.
• To investigate the long-time behavior and non-uniqueness of solutions in the fixed point formulation.
• To provide a rigorous foundation for comparing the two dominant routes in mean field game theory within the LQ setting.

Proposed method

• Introduces an asymptotic solvability notion for large-population LQ games, requiring existence of feedback Nash strategies for all sufficiently large N.
• Applies a novel re-scaling technique to derive a low-dimensional non-symmetric Riccati ODE that captures the essential dynamics of the original high-dimensional coupled Riccati system.
• Uses dynamic programming to derive the coupled Riccati ODEs for the N-player game and applies re-scaling to extract the limiting behavior.
• Analyzes the fixed point approach by formalizing a two-point boundary value problem (TPBV) derived from mean field approximations.
• Compares the two approaches by showing that asymptotic solvability implies feasibility of the fixed point approach, but the converse does not hold.
• Employs stability and existence analysis on the non-symmetric Riccati ODE to study long-time behavior and solution uniqueness.

Experimental results

Research questions

• RQ1 What conditions ensure asymptotic solvability in the direct approach to linear-quadratic mean field games?
• RQ2 How does the re-scaling technique reduce the complexity of the N-player coupled Riccati system to a low-dimensional non-symmetric Riccati ODE?
• RQ3 Under what conditions is the fixed point approach feasible when the direct approach is asymptotically solvable?
• RQ4 Can the fixed point approach yield solutions when the direct approach fails, and what accounts for non-uniqueness in this case?
• RQ5 What is the long-time behavior of the non-symmetric Riccati ODE arising from the asymptotic solvability condition?

Key findings

• A necessary and sufficient condition for asymptotic solvability is derived in terms of the solvability of a low-dimensional non-symmetric Riccati ODE obtained via re-scaling.
• The solution of the asymptotically solvable system converges to a mean field limit as the population size N tends to infinity.
• Asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true, indicating a strict hierarchy between the two methods.
• Non-uniqueness can occur in the fixed point approach even when the direct approach is asymptotically solvable, highlighting a key difference in solution structure.
• The long-time behavior of the non-symmetric Riccati ODE is analyzed, with existence and stability conditions derived based on the eigenvalues of the associated characteristic equation.
• The re-scaling method successfully captures essential dynamics of high-dimensional systems, analogous to dimension reduction in statistical physics and mean field oscillator models.