Dictatorial profiles: a proof of Arrow's theorem

2026-10-08

AI Summary

This project aims to prove Arrow’s Impossibility Theorem, revealing the dictatorial nature of social welfare functions under strict preferences. Drawing on social choice theory and axiomatic methods, the study defines dictatorial preference profiles and rigorously establishes the logical equivalence that a rule is dictatorial if and only if all its profiles are dictatorial. It further derives that any social welfare function satisfying unanimity and independence of irrelevant alternatives must admit a dictator. By providing a complete formal proof of Arrow’s Theorem, this work elucidates the intrinsic connections among unanimity, independence, and dictatorship, thereby offering rigorous support for the logical foundations of social choice theory.

Abstract

We study social welfare rules with strict individual and social preferences. A preference profile is dictatorial for a rule if the social preference coincides with some agent's preference and no agent with a different preference can change the social preference unilaterally. We establish that (i) a rule is dictatorial if and only if all of its profiles are dictatorial; and (ii) if a rule satisfies unanimity and independence (of irrelevant alternatives), then all its profiles must be dictatorial. These results constitute a proof of Arrow's Impossibility Theorem.

Problem

• Arrow's Impossibility Theorem
• social welfare rules
• dictatorial profiles
• unanimity
• independence of irrelevant alternatives

Innovation

• Arrow's Impossibility Theorem
• Dictatorial Profiles
• Social Welfare Rules
• Independence of Irrelevant Alternatives
• Unanimity