Forest Simmons wrote (Thursday, January 20, 2005) > Here's a quick way to find the Condorcet Winner if there is one: > > Use Rob LeGrand's ballot by ballot approval idea, but instead > of ballot by > ballot, use voter by voter. > > For fairness, either randomize the order of polling the > voters or else > poll them twice, once from left to right, and once from right > to left, so > that each voter gets to vote twice.
This is an example of how terminology can confuse me. In what way is "voter by voter" different from "ballot by ballot"? Until I read this I would have considered those phrases synonymous. But, Mike Ossippoff was kind enough to provide me the other day a very efficient way to find the CW if there is one, and more generally how to find the Smith Set which will have one member if there's a CW. 1. For each ballot, for each pair of candidates {X, Y} count defeat.X = defeat.X if the voter ranked Y better than X, defeat.Y = defeat.Y +1 if the voter ranked X better than Y. If equal rankings are allowed, leave both defeat.X and defeat.Y at their current values if on this ballot the voter specified X=Y. 2. Sort the candidates by defeat.candidate in ascending order. The candidate with the lowest number of these pairwise-by-voter defeats are in the Smith Set, and if no alternative pairwise-defeats (according to the pairwise-matrix SUM over all ballots) then the that identifies the CW. If there's not a CW you can continue to construct the Smith Set by the simple rule "all X with defeat.X equal to or higher than the most recent addition to the Smith Set and have a pairwise-win over the most recent addition to the Smith Set are also in the Smith Set", and iterate until no remaining candidate has a pairwise win over any of the previously identified candidates. This is very fast, as it can almost be done as a byproduct of constructing the pairwise matrix. The key is you have to "do over ballots" to count pairwise defeats. You cannot get that information from the pairwise matrix. ---- Election-methods mailing list - see http://electorama.com/em for list info