Characterizing Von Neumann-Morgenstern Stable Sets in Infinite Sets

The theory of optimal choice sets provides a well-established framework in social choice and game theory. When preferences are cyclic, as often occurs in complex economic environments, the set of maximal elements may be empty, thereby motivating alternative solution concepts such as the von Neumann--Morgenstern (vNM) stable set. In this paper, we study binary relations on infinite sets of alternatives within an order-theoretic and topological framework. Our main result yields a topological characterization of von Neumann--Morgenstern stable maximality: for consistent abstract decision problems satisfying Upper MacNeille Informational Monotonicity, the set of maximal elements is non-empty and stable if and only if there exists a compact topology on \(X\) with respect to which \(R\) is Nachbin closed and upper semicontinuous.