{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/quantitative-relaxations-of-arrow-s-axioms","title":"Quantitative Relaxations of Arrow's Axioms","arxiv_id":"2506.12961","date":"2025-06-15","proceeding":null,"authors":["Suvadip Sana","Daniel Brous","Martin T. Wells","Moon Duchin"],"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.","url_abs":"https://arxiv.org/abs/2506.12961v1","url_pdf":"https://arxiv.org/pdf/2506.12961v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"quantitative-relaxations-of-arrow-s-axioms","repo_url":"https://github.com/suvadip2776/quantitative_fairness","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2506.12961","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}