Schulze and Ranked Pairs Under Stress: A Computational Comparison of Two Condorcet Voting Methods

Abstract

Schulze and Ranked Pairs are two of the most theoretically attractive Condorcet voting methods. Both satisfy a long list of axiomatic criteria that older methods like Borda count fail, and both have been adopted by real organizations, including the Pirate Party, Wikimedia, and Debian. Despite their popularity in voting theory, direct empirical comparisons between the two are uncommon, and the question of how often they actually disagree on a winner is not well documented. This paper reports a Monte Carlo study comparing Schulze, Ranked Pairs, and Borda count across four voter preference models (Impartial Culture, Mallows, Spatial 1D, and Spatial 2D), with the number of candidates varying from 3 to 5 and the number of voters varying from 25 to 101. Three metrics are used: Condorcet efficiency, single-voter manipulability, and a new dropout stability metric measuring how often the winner survives the removal of a random ballot. The main finding is that under structured preference models, Schulze and Ranked Pairs are nearly indistinguishable, agreeing on the winner in more than 99% of elections and showing near-identical manipulability. Disagreement is only visible under Impartial Culture, where the two methods select different winners in 1 to 7% of elections depending on the number of candidates. Manipulability of the two Condorcet methods is consistently lower than Borda count, by a factor ranging from roughly 1.6 to about 30 across the conditions tested. On dropout stability, however, Borda performs comparably to and sometimes slightly better than the two Condorcet methods, which suggests that small-margin Condorcet decisions can be more fragile than the score-summing logic of Borda.