Characterizing Von Neumann-Morgenstern Stable Sets in Infinite Sets

2026-07-29

AI Summary

This study addresses the nonexistence of traditional maximal elements when preference relations exhibit cycles and the set of alternatives is infinite. Within a unified order-theoretic and topological framework, the paper establishes the first topological characterization of von Neumann–Morgenstern (vNM) stable maximality by integrating Upper MacNeille information monotonicity, compact topology, Nachbin closedness, and upper semicontinuity. Leveraging tools from order theory and topology, the authors prove that under consistency and information monotonicity conditions, there exists a specific compact topology ensuring that the set of maximal elements is nonempty and vNM stable. This result yields necessary and sufficient conditions for the existence of such stable solutions.

Abstract

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.

Problem

• von Neumann-Morgenstern stable set
• infinite sets
• cyclic preferences
• maximal elements
• social choice

Innovation

• von Neumann-Morgenstern stable set
• infinite alternatives
• topological characterization
• upper semicontinuity
• Nachbin closed