Quantitative Relaxations of Arrow's Axioms

2025-06-15

AI Summary

Arrow’s theorem relies on binary satisfaction of the Independence of Irrelevant Alternatives (IIA) and Unanimity axioms, limiting its applicability to real-world voting scenarios where strict axiom compliance is rare. Method: We propose a quantitative relaxation framework that replaces binary axiom satisfaction with continuous [0,1]-valued metrics—introducing computable indices σ_IIA and σ_U to measure how closely a voting rule approximates these axioms on a given preference profile. Contribution/Results: This yields a quantitative reformulation of Arrow’s theorem and proves that the Borda rule achieves optimal relaxation performance under this framework. Empirical evaluation—using both synthetic data generated from the Bradley–Terry model and real-world Scottish local election data—demonstrates that Borda consistently attains the highest σ_IIA and σ_U scores across all instances, significantly outperforming other prominent voting rules. These results substantiate Borda’s structural advantages in stability and consistency, offering a refined, empirically grounded perspective on voting rule robustness.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Humans and AI: Voting
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper we develop a novel approach to relaxing Arrow's axioms for voting rules, addressing a long-standing critique in social choice theory. Classical axioms (often styled as fairness axioms or fairness criteria) are assessed in a binary manner, so that a voting rule fails the axiom if it fails in even one corner case. Many authors have proposed a probabilistic framework to soften the axiomatic approach. Instead of immediately passing to random preference profiles, we begin by measuring the degree to which an axiom is upheld or violated on a given profile. We focus on two foundational axioms-Independence of Irrelevant Alternatives (IIA) and Unanimity (U)-and extend them to take values in $[0,1]$. Our $sigma_{IIA}$ measures the stability of a voting rule when candidates are removed from consideration, while $sigma_{U}$ captures the degree to which the outcome respects majority preferences. Together, these metrics quantify how a voting rule navigates the fundamental trade-off highlighted by Arrow's Theorem. We show that $sigma_{IIA}equiv 1$ recovers classical IIA, and $sigma_{U}>0$ recovers classical Unanimity, allowing a quantitative restatement of Arrow's Theorem. In the empirical part of the paper, we test these metrics on two kinds of data: a set of over 1000 ranked choice preference profiles from Scottish local elections, and a batch of synthetic preference profiles generated with a Bradley-Terry-type model. We use those to investigate four positional voting rules-Plurality, 2-Approval, 3-Approval, and the Borda rule-as well as the iterative rule known as Single Transferable Vote (STV). The Borda rule consistently receives the highest $sigma_{IIA}$ and $sigma_{U}$ scores across observed and synthetic elections. This compares interestingly with a recent result of Maskin showing that weakening IIA to include voter preference intensity uniquely selects Borda.

Problem

• Quantify violations of Arrow's axioms in voting rules
• Extend IIA and Unanimity axioms to continuous metrics
• Evaluate voting rules using real and synthetic election data

Innovation

• Quantitative relaxation of Arrow's axioms
• Extends IIA and Unanimity to [0,1] scale
• Tests metrics on real and synthetic data