Nash-type equilibria on Riemannian manifolds: a variational approach

TL;DR

This paper introduces Nash-Stampacchia equilibrium points on Riemannian manifolds, characterizing their existence and stability using variational inequalities, and highlights Hadamard manifolds as the ideal setting for these equilibrium problems.

Contribution

It develops a variational inequality framework for Nash equilibria on curved strategy sets, specifically on Hadamard manifolds, and characterizes the geometric conditions for their existence and stability.

Findings

Nash-Stampacchia equilibria characterized via variational inequalities.

Existence and stability results depend on the non-positivity of sectional curvature.

Hadamard manifolds are optimal for Nash equilibrium problems on curved spaces.

Abstract

Motivated by Nash equilibrium problems on 'curved' strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash-Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifoleds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the {non-positivity} of the sectional curvature of complete and simply connected Riemannian spaces, delimiting the Hadamard manifolds as the optimal geometrical framework of Nash-Stampacchia equilibrium problems. Our analytical approach exploits various elements from set-valued and variational analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincar\'e upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.