Dear Mike Rouse, you wrote (6 Oct 2004): > I was just wondering if there were examples of three-way circular ties > in that "unwind" the wrong way in the Kemeny order. For example, if the > cycle is A>B>C>A, it would put the order as A>C>B. (There is probably > a relatively trivial example somewhere on the net if there is, I just > don't remember seeing it).
Suppose N is the number of candidates. Suppose d[X,Y] is the number of voters who strictly prefer candidate X to candidate Y. Suppose K(1),...,K(N) is the Kemeny ranking of the candidates. Then d[K(i),K(i+1)] >= d[K(i+1),K(i)] for all i = 1,...,(N-1). Otherwise, suppose that there is a j with d[K(j),K(j+1)] < d[K(j+1),K(j)]. Then you can find a ranking with a better Kemeny score simply by switching K(j) and K(j+1). Therefore, it is not possible to find an example where the Kemeny-Young method "unwinds" a cycle in its opposite direction. (The same is true for Tideman's ranked pairs method.) Markus Schulze ---- Election-methods mailing list - see http://electorama.com/em for list info
