If we try to determine what the will of the people is with the result of the joint preferences of the three voters, this results in A > B > C > A > B > C > A. This is termed a Condorcet cycle and means that there is no winner to the election. We can summarize Condorcet’s paradox as such: even if the preferences of individual voters are transitive, this does not guarantee that the aggregate is transitive.

A manifestation of Condorcet’s Paradox is found within both FPFP and RCV since individuals’ preferences are not necessarily reflected in the group preferences. In other words, neither of these methods of voting guarantees a Condorcet Winner, i.e., a candidate in a head-to-head comparison is preferred by the voters over every other choice is the winner of the election.

There is a myriad of actual examples of this such as the 2009 mayoral election in Burlington, VT, but let’s look at the San Francisco Supervisor election for District 10 in 2010, which was decided by RCV. Of the 21 candidates, Malia Cohen received only about 11.77% of the total first-preference votes while Lynette Sweet had the highest percentage of first-preference votes (12.07%). In FPTP, Sweet would have won and Cohen would have placed third. But with RCV, there needed to be 20 iterations of the algorithm to declare a winner! Not only did Sweet have the highest percentage of first-preference votes, but she was also the top vote-getter of 13 of the 20 passes. However, she was eliminated in the 18th iteration, while Malia Cohen was finally declared the winner. …

Perhaps a mathematically superior voting method like Pairwise Comparison could win out, but until all states, territories, and districts of the United States of America are able to count all of the votes within a few hours of the polls closing like Florida, we should stick with first-past-the-post.