We use Ehrhart polynomials to estimate the likelihood of each three-candidate social ranking produced by pairwise majority rule assuming an even number of voters and the Impartial Anonymous Culture condition. We then calculate the probability the ranking is transitive and the probability of a weak Condorcet winner. Finally, we determine the weak Condorcet efficiency of various voting rules. We prove that Baldwin, Nanson, Copeland, and ranked pairs are weak Condorcet efficient, as is Borda unless there are no ties. Simulations show that among the rest, Dowdall is typically the most efficient rule for small voter groups and anti-plurality the least.
Diss, M., Dougherty, K., & Heckelman, J. C. (2023). When ties are possible: Weak Condorcet winners and Arrovian rationality. Mathematical Social Sciences, 123, 128–136. https://doi.org/10.1016/j.mathsocsci.2023.03.004
