Continuous time approximation of Nash equilibria in monotone games

We consider the problem of approximating Nash equilibria of N functions f(1), ... , f(N) of N variables. In particular, we show systems of the formu(j)(t) = -?(xj)f(j)(u(t))(j = 1, ... , N) are well-posed and the large time limits of their solutions u(t) = (u1(t), ... , uN(t)) are Nash equilibria for f(1), ... , f(N) provided that these functions sat-isfy an appropriate monotonicity condition. To this end, we will invoke the theory of maximal monotone operators on a Hilbert space. 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.

Automated summary

In this paper, we discuss the problem of approximating the Nash equilibria of N functions f(1), ..., f(N) of N variables. Specifically, we prove that the systems of the form u(j)(t) = -λ(xj)f(j)(u(t))(j = 1, ..., N) are well-posed and the large time limits of their solutions u(t) = (u1(t), ..., uN(t)) are Nash equilibria for f(1), ..., f(N), under the condition that these functions satisfy an appropriate monotonicity condition. For this purpose, we employ the theory of maximal monotone operators on a Hilbert space. We also explore the application of these ideas in game theory and provide a method to approximate equilibria in certain game theoretic problems in function spaces.