The probability of violating Arrow’s conditions

• Published in: European Journal of Political Economy 2020-12, Vol.65, p.101936, Article 101936
• Main Authors: Dougherty, Keith L., Heckelman, Jac C.
• Format: Article
• Language: English
• Subjects: Arrow impossibility theorem; Independence of irrelevant alternatives; Pareto; Political economy; Probability; Social choice; Transitivity
• ISSN: 0176-2680; 1873-5703
• DOI: 10.1016/j.ejpoleco.2020.101936

Arrow’s impossibility theorem shows that all preference aggregation rules (PARs) must violate a specific set of normative conditions (transitivity, Pareto, IIA, nondictatorship) over an unrestricted domain of preference profiles. However, the theorem does not address which PARs are more likely to violate those conditions across preference profiles. We compare the probabilities that thirteen PARs (anti-plurality, Hare, Nanson, plurality, plurality runoff, Simpson–Kramer, Baldwin, Borda, Coombs, Copeland, Dowdall, pairwise majority, and ranked pairs) violate Arrow’s conditions. We prove that Baldwin, Borda, Coombs, Copeland, Dowdall, and ranked pairs are less likely to violate IIA than the first six PARs, and they are less likely to violate Arrow’s conditions jointly. In contrast, pairwise majority never violates IIA but can violate transitivity. Simulations with three alternatives reveal that among the PARs studied, pairwise majority is the most likely to satisfy Arrow’s conditions jointly. Our results suggest pairwise majority violates transitivity with a small probability, while the other PARs violate IIA with much larger probabilities. •Compares the rate at which thirteen voting rules violate conditions of Arrow’s theorem.•Proves which rules always violate independence and which rules sometimes do not.•Estimates probability of joint and marginal violations using simulations.•Finds pairwise majority rule is the most likely to satisfy Arrow’s conditions jointly.