On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

Pensavalle, C. A. ; Pieri, G.

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 337-348

Résumé

Consider a M-player game in strategic form $G = (X_{1},\cdots,X_{M},g_{1},\cdots,g_{M})$ where the set $X_{i}$ is a closed interval of real numbers and the payoff function $g_{i}$ is concave and differentiable with respect to the variable $x_{i} \in X_{i}$, for any $i = 1,\cdots,M$. The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence, which is the well-posedness of an appropriate minimum problem.