I haven't seen a description of beatpath that is really easy to visualize, so I pondered the concept a while and came up with my own version. It's very similar to the Dodgson-like method I describe in my last post (RE Don't ignore the margins), except instead of eliminating by row or column total, margin entries are eliminated in order of increasing size. This elimination method is at least in the spirit of beatpath, and may be functionally equivalent to it, apart from my use of run-offs. Start with a matrix of winning margins. Eliminate Condorcet winners and losers until none are left. Winners accumulate in top-to-bottom ranking order; losers accumulate separately, bottom-to-top. Keep a copy of the original matrix to use in run-offs (below). Eliminate all occurrences of the smallest margin from the working matrix. This eliminates the weakest paths: candidates who do not win any contests by a larger margin are beatpath losers, because there are no paths from them to anywhere else that are stronger than that margin. Similarly, those who do not lose any contest by a larger margin are beatpath winners. During this step, if you like, you can say, "You are the weakest link. G'bye!" Collect any losers (those with no remaining victories) and, if there are more than one, conduct a run-off among them, using numbers from the original matrix, to determine the order in which they will be added to the losers. Collect any winners and do likewise. Remove winners and losers from the working matrix. Repeat Elimination and Collection until the working matrix is empty.
