Location of Nash equilibria: A Riemannian geometrical approach

Abstract

Abstract. Existence and location of Nash equilibrium points are studied for a large class of a finite family of payoff functions whose domains are not necessarily convex in the usual sense. The geometric idea is to embed these non-convex domains into suitable Riemannian manifolds regaining certain geodesic convexity properties of them. By using recent non-smooth analysis on Riemannian manifolds and a variational inequality for acyclic sets, an efficient location result of Nash equilibrium points is given. Some examples show the applicability of our results.

Stackelberg equilibria via variational inequalities and projections

“…Moreover, since card(R SV (x 1 )) = 1 for every x 1 ∈ K 1 , and the map x → R SV (x) is of class C 1 , then one has that S SV = ∅, see Remark 3.1 For the same functions and sets as in Example 3.1, we state that the set of Nash equilibrium points is empty. This fact can be seen by following the arguments from the paper of Kristály [5], where a very general framework is discussed for non-smooth function on finite-dimensional Riemannian manifolds; see also the monograph of Kristaly, Radulescu and Varga [6]. More precisely, the first step is to determine the Nash critical points, i.e., the solutions for the system…”

“…Now, the set of Nash equilibrium points is a subset of N SV , due to [5]. Note however that none of the above points fulfil the system for Nash equilibria, i.e.,…”

A Nash type solution for hemivariational inequality systems

“…The following result is an existence result for Nash generalized derivative points and is an infinite-dimensional version of a result from the paper [13]. Therefore, if in Theorem 3.1 we choose F i = 0, i ∈ {1, .…”

“…We consider the function f i : K 1 × · · · × D i × · · · K n → R which are continuous and locally Lipschitz in the ith variable. The next notion was introduced recently by A. Kristály [13] and is a little bit different form for functions defined on Riemannian manifolds. .…”

“…In the last years many papers have been dedicated to the of study the existence and multiplicity of solutions for hemivariational inequality systems or differential inclusion systems defined on bounded or unbounded domain, see [2], [3], [11], [12], [13], [16]. In these papers the authors use critical points theory for locally Lipschitz functions, combined with the Principle of Symmetric Criticality and different topological methods.…”