[Paper Review] Arrow's Impossibility Theorem Without Unanimity

Elchanan Mossel|ArXiv.org|Jan 29, 2009

Game Theory and Voting Systems4 references3 citations

TL;DR

This paper presents a new, concise proof of Arrow's Impossibility Theorem under relaxed conditions, showing that any constitution satisfying Transitivity and Independence of Irrelevant Alternatives (IIA) is a dictatorship if it satisfies Weak Non-Imposition (WNI), even without the Unanimity condition. The proof uses a local pivotal voter argument on three alternatives and fully characterizes all such functions, enabling quantitative extensions in follow-up work.

ABSTRACT

Arrow's Impossibility Theorem states that any constitution which satisfies Transitivity, Independence of Irrelevant Alternatives (IIA) and Unanimity is a dictatorship. Wilson derived properties of constitutions satisfying Transitivity and IIA for unrestricted domains where ties are allowed. In this paper we consider the case where only strict preferences are allowed. In this case we derive a new short proof of Arrow theorem and further obtain a new and complete characterization of all functions satisfying Transitivity and IIA.

Motivation & Objective

• To re-derive Arrow’s Impossibility Theorem under weaker conditions than Unanimity, specifically using Weak Non-Imposition (WNI) instead.
• To provide a complete characterization of all constitutions satisfying Transitivity and IIA when only strict preferences are allowed.
• To develop a local, pivotal-voter-based proof technique that facilitates quantitative extensions in social choice theory.
• To overcome limitations of prior work by avoiding reliance on the Unanimity condition, which is hard to quantify in probabilistic settings.

Proposed method

• Uses a local argument based on 2 voters and 3 alternatives to establish the existence of a pivotal voter under IIA and Transitivity.
• Applies a variant of Barbera’s pivotal voter method to trace how social preferences propagate across alternative sets.
• Employs induction on the number of alternatives, building up from smaller sets where the constitution is known to be a dictator or non-constant.
• Leverages Lemma 3.2 and Lemma 3.3 to show that if a subset of alternatives induces a dictator or non-constant function, adding a new alternative preserves the structure.
• Uses contradiction arguments to rule out non-dictatorial outcomes when WNI and IIA hold, particularly by analyzing social rankings across triplets.
• Relies on the structure of preference profiles and the behavior of social choice functions under permutations to derive global dictatorship.

Experimental results

Research questions

• RQ1 Can Arrow’s Impossibility Theorem be proven without the Unanimity condition, under the weaker WNI assumption?
• RQ2 What is the complete set of constitutions satisfying Transitivity and IIA when only strict preferences are allowed?
• RQ3 How can a local pivotal voter argument be used to derive global dictatorship results in social choice?
• RQ4 Can the proof technique be adapted to derive quantitative versions of Arrow’s Theorem?
• RQ5 What is the structural difference between constitutions satisfying WNI versus Unanimity in the context of IIA and Transitivity?

Key findings

• Any constitution satisfying Transitivity, IIA, and WNI on three or more alternatives must be a dictatorship, even without Unanimity.
• The paper provides a complete characterization of all such functions: there are exactly 2n possible dictatorships (identity or reversal) for n individuals.
• The proof establishes that if a function is a dictator on a subset of size ≥3 or non-constant on a pair, then adding any new alternative preserves the dictator structure.
• The result implies that WNI is sufficient for dictatorship, even when Unanimity fails, and that the standard definition of dictatorship must be generalized to include reversal mappings.
• The method enables a quantitative Arrow theorem in follow-up work, showing that any IIA function with high transitivity probability is ε-close to a dictator function.
• The proof avoids reliance on quantifiable unanimity, making it suitable for probabilistic and robustness analyses in large societies.