The Existence and Uniqueness of a Nash Equilibrium in Mean Field Game Theory

TL;DR

This paper broadens the conditions for the existence and uniqueness of Nash equilibria in Mean Field Game Theory by showing that convexity of individual parts is unnecessary if their combination is convex, aligning better with real-world scenarios.

Contribution

It demonstrates that only an appropriate combination of action components needs to be convex, relaxing previous convexity assumptions and enhancing the model's applicability.

Findings

Generalizes conditions for Nash equilibrium existence and uniqueness

Relaxes convexity assumptions on individual action components

Aligns theoretical models more closely with real-world applications

Abstract

In recent and past works, convexity is usually assumed on each individual part of the action functional in order to demonstrate the existence and uniqueness of a Nash equilibrium on some interval [0, T] (this meant that each hessian was assumed to be nonnegative). Particularly, a certain assumption was imposed in order to quantify the smallness of T. The contribution of this project is to expand on this with the key insight being that one does not need the convexity of each part of the action, but rather just an appropriate combination of them, which will essentially "compensate" for the other two terms to yield convexity in the action. This is meaningful in both the pure and applied settings as it generalizes the existence and uniqueness of a Nash equilibrium slightly more, but maybe more importantly matches real world application slightly closer, as in reality there are many settings in which not each part of the action have convexity. Thus, it is more accurate for modern application of Mean Field Game Theory.