Evaluating Fairness of Voting Systems: Simulating Violations of Arrow's Conditions

TL;DR

The paper addresses the empirical fairness of voting systems by simulating preference profiles under IC and IAC to evaluate Arrow's conditions for elections with 3–6 alternatives and up to 10,000 voters. It introduces a joint-violation metric to compare systems and extends prior work to larger alternative sets, revealing that Pairwise Majority typically has the lowest joint violation risk. As the number of alternatives grows, IIA violations rise sharply and joint violations tend toward near-ubiquitous levels for many systems, though Ranked Pairs and some transitive-rule subsets fare better. The work emphasizes the practical limitations of Arrow's theorem, highlights the ubiquity of IIA violations, and provides improved methods and open data/code for replication and further study.

Abstract

This paper builds upon the work of Dougherty and Heckelman (2020) by determining the frequency that 13 voting systems violate Arrow's social choice criteria with up to six alternatives. These results determine which of the 13 voting systems, is the fairest based on their probabilistic likelihood of violating Arrow's social choice criteria. The voting systems considered are: Plurality, Borda, Dowdall, Top Two, Hare, Coombs, Baldwin, Copeland, Anti-Plurality, Nanson, Ranked Pairs, Pairwise Majority, and Minimax. Elections with up to 10,000 voters and between three and six alternatives are simulated using both Impartial Culture and Impartial Anonymous Culture. These simulations show that Pairwise Majority is the least likely to jointly violate Arrow's criteria. As the number of alternatives increases, the joint-violation frequencies increase for each voting method. For all systems except Pairwise Majority, the joint-violation frequencies in elections with at least 30 voters and four alternatives are greater than 98%.