Fixed point theorems and applications in theory of games (Q2925775)

Let \(X\) be a convex set in a vector space and \(Y\) be a nonempty subset of a topological vector space. A correspondence \(T:X \longrightarrow 2^Y\) is said to be: (i) weakly naturally quasi-concave if for each finite set \(\{x_0,x_1, \dots, x_n\}\subseteq X\) there exist \(y_i\in T(x_i)\) and a continuous function \(g = (g_0,g_1, \dots, g_n): \Delta_n \rightarrow \Delta_n\) (here, \(\Delta_n\) is the standard \(n\)-simplex) with \(g_i(1) = 1\), \(g_i(0)=0\) such that \(\sum\limits_{i=0}^n g_i(\lambda_i)y_i\in T(\sum\limits_{i=0}^n x_i)\) for every \((\lambda_0,\lambda_1,\dots,\lambda_n)\in \Delta_n\); (ii) weakly *-concave if for each finite set \(\{x_0,x_1, \dots, x_n\}\subseteq X\) there exists \(y_i\in T(x_i)\) such that for every \((\lambda_0,\lambda_1,\dots,\lambda_n)\in \Delta_n\), \(\sum\limits_{i=0}^n y_i \in T(x)\) for all \(x\in X\). The author obtains first selection theorems for weakly naturally quasi-concave, respectively weakly *-concave correspondences defined on a simplex. Then, she uses these selection theorems to establish the existence of equilibrium for a noncompact abstract economy without continuity assumptions for the constraint and preference correspondences.