A Characterization Framework for Stable Sets and Their Variants

TL;DR

This paper extends classical stable set concepts to infinite alternatives and provides topological conditions for their existence, advancing the theoretical framework in social choice and game theory.

Contribution

It introduces an extension of stable set variants to infinite sets and offers a topological characterization for their existence.

Findings

Extended stable sets defined over infinite alternatives.

Topological conditions for the existence of stable sets.

Generalized solution frameworks for cyclic preferences.

Abstract

The theory of optimal choice sets offers a well-established solution framework in social choice and game theory. In social choice theory, decision-making is typically modeled as a maximization problem. However, when preferences are cyclic -- as can occur in economic processes -- the set of maximal elements may be empty, raising the key question of what should be considered a valid choice. To address this issue, several approaches -- collectively known as general solution theories -- have been proposed for constructing non-empty choice sets. Among the most prominent in the context of a finite set of alternatives are the Stable Set (also known as the Von Neumann-Morgenstern set) and its extensions, such as the Extended Stable Set, the socially stable set, and the $m$-, and $w$-stable sets. In this paper, we extend the classical concept of the stable set and its major variants - specifically, the extended stable set, the socially stable set, and the $m$- and $w$-stable sets - within the framework of irreflexive binary relations over infinite sets of alternatives. Additionally, we provide a topological characterization for the existence of such general solutions.