Nash-type equilibria on Riemannian manifolds: A variational approach

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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 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 non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.

Original languageEnglish
Pages (from-to)660-688
Number of pages29
JournalJournal des Mathematiques Pures et Appliquees
Volume101
Issue number5
DOIs
Publication statusPublished - May 2014

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Keywords

  • Metric projection
  • Nash-Stampacchia equilibrium point
  • Non-positive curvature
  • Non-smooth analysis
  • Riemannian manifold

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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