Price of Anarchy for Mean Field Games

TL;DR

This paper extends the concept of the price of anarchy to mean field games, analyzing its properties and behaviors in linear quadratic models with explicit calculations and numerical illustrations.

Contribution

It introduces and studies the price of anarchy in mean field games, providing explicit formulas and asymptotic analysis for linear quadratic cases.

Findings

Explicit computations of the price of anarchy in linear quadratic mean field games

Asymptotic behaviors of the price of anarchy under different model limits

Numerical simulations illustrating theoretical results

Abstract

The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are presented.