Nonsmooth Nash Equilibria on Smooth Manifolds

Abstract

Motivated by Nash equilibrium problems on ‘curved’ strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced by using nonsmooth analysis on Riemannian manifolds. Fixed point characterizations and existence of Nash-Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifolds, 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 nonsmooth calculus on Riemannian manifolds. Several examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices.