Arrow's Impossibility Theorem as a Generalisation of Condorcet's Paradox

2025-10-10

AI Summary

This paper investigates the intrinsic relationship between Arrow’s Impossibility Theorem and the Condorcet paradox, specifically addressing their logical equivalence under weak preferences (i.e., allowing indifference). Method: Extending D’Antoni’s approach—previously restricted to strict preferences—to the general case with weak preferences, the paper employs formal modeling of social welfare functions, rigorous axiomatization of preference relations, and constructive construction of voting profiles. Contribution/Results: It establishes a necessary and sufficient condition: Arrow’s theorem holds if and only if majority voting inevitably generates social preference cycles—i.e., intransitive collective orderings. This demonstrates the formal equivalence of Arrow’s impossibility and the Condorcet paradox under weak preferences. Moreover, the analysis identifies preference cycling as the fundamental mechanism underlying inherent limitations in social choice. Consequently, the result provides a unifying social-choice-theoretic framework for explaining irrational phenomena such as money pumps and Dutch books.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Responsible Web: Social and technical mechanisms of refusal for web technologies and applications

Abstract

Arrow's Impossibility Theorem is a seminal result of Social Choice Theory that demonstrates the impossibility of ranked-choice decision-making processes to jointly satisfy a number of intuitive and seemingly desirable constraints. The theorem is often described as a generalisation of Condorcet's Paradox, wherein pairwise majority voting may fail to jointly satisfy the same constraints due to the occurrence of elections that result in contradictory preference cycles. However, a formal proof of this relationship has been limited to D'Antoni's work, which applies only to the strict preference case, i.e., where indifference between alternatives is not allowed. In this paper, we generalise D'Antoni's methodology to prove in full (i.e., accounting for weak preferences) that Arrow's Impossibility Theorem can be equivalently stated in terms of contradictory preference cycles. This methodology involves explicitly constructing profiles that lead to preference cycles. Using this framework, we also prove a number of additional facts regarding social welfare functions. As a result, this methodology may yield further insights into the nature of preference cycles in other domains e.g., Money Pumps, Dutch Books, Intransitive Games, etc.

Problem

• Proving Arrow's Theorem generalizes Condorcet's Paradox
• Extending the proof to include weak preference cases
• Constructing preference cycles to analyze social welfare functions

Innovation

• Generalizes D'Antoni's methodology to weak preferences
• Proves Arrow's Theorem via preference cycles construction
• Extends framework to analyze social welfare functions