Acyclic versions of the von Neumann and Nash equilibrium theorems (Q1963882)

The author rewrites earlier results of his [J. Korean Math. Soc. 35, No. 4, 803-829 (1998; Zbl 0923.47034)] for the case of compositions of acyclic-valued maps: For \(i=1,\dots,n\) let \(E_i\) be a topological vector space, \(X_i\subset E_i\) a convex set, \(K_i\subset X_i\) a nonempty compact set and \(T_i\) an upper semicontinuous map from \(X:=\prod_{j=1}^n X_j\) into the compact acyclic subsets of \(K_i\). If \(X\) is admissible in \(\prod_{i=1}^n E_i\) (i.e., if for every compact set \(K\subset X\) and each neighbourhood \(U\) of zero there is a finite dimensional map \(h:K\to X\) such that \(x-h(x)\in U\) for all \(x\in K\)) then there is an \(x\in\prod_{i=1}^nK_i\) such that \(x_i\in T_i(x)\) for \(i=1,\dots,n\). Standard arguments then yield the usual equilibrium theorems.