Summary: Condorcet’s original ideas of 1785 to 1788 are shown to lead naturally to Ranked Pairs, a well-known Condorcet-compatible method that is free of influence by clones. The computa- tional effort of solving for the rank order of the candidates under Ranked Pairs is shown for N candidates, whose least upper bound is O(N3) in worst case, to be empirically O(N1.8) when averaged over a large number of random elections, even for values of N up to 1024. Methods are demonstrated whereby an elections official or a common voter can prepare information that allows a voter who has no recourse to a computer to judge whether a Ranked Pairs rank-order has been computed correctly. As one method, the definition and implementation of the circ operation, which sorts all possible permutations of the candidates into an ordered list with the Ranked Pairs rank order at the head, are simplified so it is an O(N2) and not an O(N2 lnN) process, and one that runs ever faster the closer the Kendall-tau distance between two permutations becomes.

Summary: Beatpath, Split Cycle, and Ranked Pairs are the only known Condorcet-compatible election methods that are independent of Smith-dominated alternatives and also independent of influence by clones. Beatpath and Split Cycle output a partial order among the candidates; and only candidates who can be at the head of any linear extension of that partial order should have any chance to win. Often however a single winner is needed; and it has been unclear how to define the probability with which each undefeated candidate should win, so that introducing a set of clones for a candidate only divides his original winning probability among that candidate and his clones; and so that the sum over those divisions cannot be changed by introducing clones of other candidates. We demonstrate the one correct way the required probabilities can be derived from the partial order. We develop two alternate approaches, each equally applicable to both Beatpath and Split Cycle, and each of which is much easier to implement for a mass popular election and where the alternate probabilities, while slightly incorrect, are likewise strictly free of influence by clones. A parallel approach modifies Ranked Pairs so that when any number of sets of elements in an election’s victory matrix have equal magnitudes, the probability any candidate wins is similarly free of influence by clones.

Summary: We report for Beatpath and Ranked Pairs an exhaustive examination of how the winning candidate changes when one candidate is dropped, for initial numbers of candidates of 4 and 5, and report sampling results of what happens for initial numbers of candidates from 6 to 18. Consistent with all the searches is the observation that if under Beatpath there is a single rank order, and if when the winning candidate drops there is also a single rank order, then in the new rank order the candidate who formerly placed second must place above the candidate who formerly placed third; and therefore we add to the proof by M. Schulze that the candidate who placed second cannot become placed last, the observation that the candidate who placed third cannot become placed first. Other than those two excluded cases, for candidates numbering from 4 to 18 we find that a candidate who placed anywhere in the original rank order could be found to be placed anywhere in the new rank order; in particular the candidate who had placed last could come to be placed first, and the candidate who had placed second could come to be placed second-to-last. These results are compared to the known properties of Ranked Pairs, and their larger political significance discussed.