[Paper Review] A Quantitative Arrow Theorem

Elchanan Mossel|ArXiv.org|Mar 14, 2009

Game Theory and Voting SystemsEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper establishes a quantitative version of Arrow's Impossibility Theorem, proving that any constitution satisfying Independence of Irrelevant Alternatives (IIA), minimal support for pairwise preferences, and bounded non-dictatorship must exhibit a non-transitive outcome with probability at least δ > 0, where δ depends only on ε. The result holds for any number of alternatives k ≥ 3 and generalizes to non-uniform, symmetric distributions over rankings, using inverse-hypercontractivity and non-linear invariance to bound paradox probabilities.

ABSTRACT

Arrow's Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider $n$ voters who vote independently at random, each following the uniform distribution over the 6 rankings of 3 alternatives. Arrow's theorem implies that any constitution which satisfies IIA and Unanimity and is not a dictator has a probability of at least $6^{-n}$ for a non-transitive outcome. When $n$ is large, $6^{-n}$ is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every $\eps > 0$, there exists a $δ= δ(\eps) > 0$, which depends on $\eps$ only, such that for all $n$, and all constitutions on 3 alternatives, if the constitution satisfies: The IIA condition. For every pair of alternatives $a,b$, the probability that the constitution ranks $a$ above $b$ is at least $\eps$. For every voter $i$, the probability that the social choice function agrees with a dictatorship on $i$ at most $1-\eps$. Then the probability of a non-transitive outcome is at least $δ$.

Motivation & Objective

• To provide a quantitative strengthening of Arrow’s Impossibility Theorem by bounding the probability of non-transitive outcomes in social choice functions.
• To generalize prior results by Kalai and Keller, which applied only to balanced constitutions and k=3 alternatives, to arbitrary k ≥ 3 and general symmetric distributions over rankings.
• To characterize all IIA-satisfying social choice functions that yield transitive outcomes with high probability, identifying structural constraints on pivotal voters.
• To extend the analysis beyond uniform distributions to any distribution where each ranking has probability at least α > 0, establishing robust quantitative bounds.

Proposed method

• Uses inverse-hypercontractivity to show that if transitivity holds with high probability, then no two distinct voters can be pivotal for multiple pairwise preferences with non-negligible probability.
• Applies non-linear invariance principles to lower-bound paradox probabilities in cases where individual voters have small influence (low total influence).
• Employs a decomposition of voter influence via Fourier analysis on the hypercube, focusing on low-degree Fourier coefficients to control transitivity failure.
• Establishes a connection between the probability of non-transitive outcomes and the influence structure of voters, using conditional expectation and noise stability bounds.
• Proves that for any ε > 0, there exists δ(ε) > 0 such that if the constitution is not close to a dictator and satisfies minimal pairwise support, then the paradox probability is at least δ.
• Generalizes results from uniform to symmetric distributions by adjusting constants in the exponent using the minimum ranking probability α.

Experimental results

Research questions

• RQ1 What is the minimal probability of a non-transitive outcome in a social choice function satisfying IIA, minimal pairwise support, and bounded non-dictatorship?
• RQ2 Can the quantitative Arrow theorem be extended beyond uniform distributions over rankings to any symmetric distribution with bounded minimum ranking probability?
• RQ3 How do influence structures of voters constrain the likelihood of transitive outcomes under IIA?
• RQ4 What is the precise relationship between voter influence and the probability of voting paradoxes in IIA-satisfying constitutions?
• RQ5 Can the techniques used here—especially inverse-hypercontractivity and non-linear invariance—be applied to other impossibility theorems in social choice, such as Gibbard-Satterthwaite?

Key findings

• For any ε > 0, there exists δ(ε) > 0 such that any IIA-satisfying constitution with pairwise preference probability ≥ ε and non-dictatorship probability ≤ 1−ε must have a non-transitive outcome with probability at least δ.
• The bound δ is independent of the number of voters n and depends only on ε, establishing a uniform lower bound on paradox probability.
• The result generalizes to k ≥ 3 alternatives and symmetric distributions where each ranking has probability at least α, with δ = exp(−C₁ / (α ε^{C₂(α)})) and C₂(α) = 3 + 1/(2α²).
• Constitutions that are transitive with high probability must have a very restricted structure: either one voter dominates on large subsets of alternatives, or influence is highly concentrated.
• The proof shows that if two voters are simultaneously pivotal for multiple pairwise comparisons with non-negligible probability, then paradoxes are unavoidable, even under weak assumptions.
• The use of non-linear invariance and inverse-hypercontractivity enables tight bounds in the low-influence regime, where individual voters have little impact on outcomes.