Technology Category
Game Theory and Economic Paradigms: Social Choice / VotingKnowledge Representation and Reasoning: PreferencesReasoning under Uncertainty: Decision/Utility Theory
A Characterization Framework for Stable Sets and Their Variants
🤖 AI Summary
When preferences exhibit cycles, the set of maximal elements may be empty, necessitating well-defined choice solutions. This paper systematically extends the notion of stable sets—and their principal variants: extended stable sets, socially stable sets, m-stable sets, and w-stable sets—to infinite option spaces, modeling preferences via irreflexive binary relations. Drawing on order theory, set theory, and topology, we characterize the essential conditions for their existence. Our main contribution is a unified existence framework, establishing necessary and sufficient topological conditions for the existence of these solution concepts in general topological spaces: specifically, the underlying relation must satisfy certain closure and compactness properties. This constitutes the first general theoretical foundation for rational choice under cyclic preferences applicable to infinite domains, with direct implications for social choice theory and game theory.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Extends stable set theory to infinite alternative sets
Addresses cyclic preferences in social choice theory
Provides topological characterization for solution existence
Innovation
Methods, ideas, or system contributions that make the work stand out.
Extends stable sets to infinite alternatives
Topological characterization of solution existence
Generalizes Von Neumann-Morgenstern set variants
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