My conjectures turned out to be true: Lemma: If range values are limited to k levels, and alternative X beats alternative Y with a margin ratio greater than (k-1)/1, then alternative X has a greater range score than alternative Y.
Proof: Without loss in generality assume that the k possible ratings on each ballot are 0, 1, ...(k-1). If there are x ballots on which X is rated above Y for every y ballots on which Y is rated above X, then the least the difference in the respective range scores could be is d = 1*x - (k-1)*y , since the least possible difference in ratings on any single ballot is one, and the greatest possible difference in ratings on any ballot is (k-1). But when the margin ratio x/y is greater than (k-1)/1, the value of d is positive. Therefore X has a greater total range score than Y. Corollary 1. If range values are limited to k levels, then there can be no beat cycle where all of the defeats have margin ratios greater than (k-1)/1. Corollary 2. If range values are limited to k levels, then no beatpath with margin ratio strength greater than (k-1)/1 can be longer than k times the number of ballots, no matter how many alternatives are rated on the ballots. Corollary 3. In the case of ordinal ballots, if no ballot ranks candidates at more than (k-1) levels, then the conclusions of Corollaries 2 and 3 still hold. Corollary 4. If there are only k candidates , then the conclusions of Corollaries 2 and 3 still hold. How can we put this information to good use? Suppose that we are dealing with 3 slot ballots as in MCA, APV, MAFP, etc. It may not be too common for one candidate to have a wv score against another candidate consisting of more than two thirds of the vote. But that is not needed here, only a margin ratio greater than two to one is needed. In other words, if eleven percent of the voters prefer X over Y but only five percent of the voters prefer Y over X, then we have a margin ratio that cannot be sustained indefinitely in a beatpath, and (more to the point) cannot sustain any cycle no matter how long or short. So the losers in all such defeats can be eliminated without fear of eliminating all of the candidates. Doing so would automatically eliminate all of the Pareto dominated candidates, too, and make the method independent from Pareto dominated candidates. Any other ideas on how to put these facts to use? Forest ---- Election-Methods mailing list - see http://electorama.com/em for list info