Dear Mike, you wrote (27 March 2005): > I told Markus that I was going to define majority > rule soon. My definition of majority wishes is > similar, and I guess that I'd better state that > definition now, instead of being vague about what > I mean by majority wishes and majority rule. > > If a majority prefer X to Y, that's a majority > pairwise preference (MPP). The strength of that > MPP is measured by the number of voters who prefer > X to Y. > > An MPP for X over Y is outdone if there is a sequence > of MPPs from Y to X, consisting of MPPs that are all > at least as strong as the MPP of X over Y. > > To violate majority wishes means to elect someone who > has an MPP against him that isn't outdone. > > Protecting majority wishes means avoiding a violation > of majority wishes. > > [end of definition of protecting majority wishes] > > Majority rule: > > X has a majority pairwise vote against Y if a majority > vote X over Y. > > Substituting majority pairwise vote for majority pairwise > preference in the definitions above leads to a definiltion > of majority rule instead of majority wishes. > > Well, it's better to say it explicitly: > > X has a majority pairwise vote (MPV) against Y if a > majority vote X over Y. > > An MPV's strength is measured by the number of people > who vote X over Y. > > An MPV for X over Y is outdone if there's a sequence of > MPVs from Y to X consisting of MPVs that are all at least > as strong as the one for X over Y. > > Violating majority rule means electing someone who has > an MPV against him that isn't outdone. > > [end of definition of violating majority rule]
Such criteria have already been proposed in the past. Suppose V is the number of voters. Suppose d[X,Y] is the number of voters who strictly prefer candidate X to candidate Y. Suppose p(z)[X,Y] is the strength of the strongest path from candidate X to candidate Y when the strength of a pairwise defeat is measured by "z" (e.g. "z" = "margins", "z" = "winning votes", "z" = "votes against"). Then I proposed the following criterion in 1997: If p(wv)[A,B] > V/2 and p(wv)[B,A] < V/2, then candidate B must be elected with zero probability. Steve Eppley proposed the following criterion in 2000: If d[A,B] > V/2 and p(wv)[B,A] < V/2, then candidate B must be elected with zero probability. Markus Schulze ---- Election-methods mailing list - see http://electorama.com/em for list info