Summary:

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.