Nash-stampacchia Equilibrium Points on Manifolds

Motivated by Nash equilibrium problems on ’curved’ strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced for a finite family of non-smooth functions defined on geodesic convex sets of certain Riemannian manifolds. Characterization, existence, and stability of NashStampacchia equilibria are studied when the strategy sets are compact/noncompact subsets of certain Hadamard manifolds, exploiting two well-known geometrical features of these spaces both involving the metric projection operator. These two 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 analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds developed by Yu. S. Ledyaev and Q. J. Zhu [Trans. Amer. Math. Soc. 359 (2007), 3687-3732].