Continuous time approximation of Nash equilibria

TL;DR

This paper introduces a continuous-time dynamical system approach to approximate Nash equilibria in multi-variable functions, utilizing maximal monotone operator theory to analyze well-posedness and convergence.

Contribution

It establishes conditions for the well-posedness of a continuous-time approximation scheme and demonstrates its application in approximating Nash equilibria in game theory.

Findings

Conditions for well-posedness of the dynamical system

Convergence of solutions to Nash equilibria

Application to game theoretic problems in function spaces

Abstract

We consider the problem of approximating Nash equilibria of $N$ functions $f_1,\dots, f_N$ of $N$ variables. In particular, we deduce conditions under which systems of the form $$ \dot u_j(t)=-\nabla_{x_j}f_j(u(t)) $$ $(j=1,\dots, N)$ are well posed and in which the large time limits of their solutions $u(t)=(u_1(t),\dots, u_N(t))$ are Nash equilibria for $f_1,\dots, f_N$. To this end, we will invoke the theory of maximal monotone operators. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces.