Hello, I have been trying to design a method that satisfies Minimal Defense and as much Later-no-harm as possible. I believe the method I'll describe in this message is as close as I can get.
A method with similar properties is MinMax (Pairwise Opposition), which strictly satisfies Later-no-harm, but fails Minimal Defense when there are more than three candidates. My new method also satisfies both criteria when there are only three candidates. First let me define some terms: Candidate A "dominates" candidate B if more than half of the voters rank A above B. Candidate A has a "majority-strength beatpath" to candidate B if A dominates a candidate who dominates B, or who dominates someone else who dominates B, etc. "Minimal Defense" says that if more than half of the voters rank A above B, and don't rank B above anybody, then B must be elected with 0% probability. This implies Mike Ossipoff's SDSC. When only pairwise contests are considered, we have to strengthen Minimal Defense to say that if some A dominates B, and B dominates nobody, then B must be elected with 0% probability. "Later-no-harm" says that adding another preference to a ballot (i.e., modifying a ballot so that a candidate previously ranked above no one is now ranked above the other candidates who had been ranked above no one, but still strictly lower than the same candidates as before) must not decrease the probability of winning of any candidate ranked above this new preference. When only pairwise contests are considered, it's impossible to determine the preference order of individual ballots, so we have to strengthen Later-no-harm to say that increasing the number of votes for A over B must not reduce the probability of election of any candidate except for B. Now I'll define the method. First, determine a ranking of the candidates using a method that satisfies Later-no-harm. You could use Random Candidate, Random Ballot, FPP, IRV, Woodall's DSC, or the MMPO method mentioned above. DSC probably has the best properties (alone, it satisfies Clone Independence, Mono- raise, Mono-add-top, and Participation) but is hard to count. MMPO only needs the pairwise matrix, but it can be indecisive. The simplest way to see the properties of the method would be Random Candidate or Random Ballot, though. Elect the highest-ranked candidate who has a majority-strength beatpath to any and all candidates which dominate him. The set of all such candidates is the interesting part of the method. (By the way, you shouldn't eliminate the candidates not in the set and then try to use a LNHarm ranking method, since that will probably needlessly fail Mono-raise. You don't want the candidate ranking to be influenced by which candidates are in the choice set.) When the winner always comes from this set, then Minimal Defense is guaranteed, since a "Minimal Defense loser" will be dominated by somebody, but won't possess any majority-strength beatpaths to anyone. When there are only 3 candidates, Later-no-harm is guaranteed: Suppose your preference order is A>B>C and you alter your vote from just A to A>B>C. The only way this can alter the set is if you cause B to dominate C. Only two changes could result: C moves out of the choice set; or A and B both move into the choice set. With more than 3 candidates, Later-no-harm can't be guaranteed anymore, since new preferences could give other candidates (liked less than A or B) majority-strength beatpaths which move them back into the choice set. Here is an example, where ">>" means "dominates": Situation 1: C>>D, D>>B. A isn't involved in any dominations. The members of my set are {a,c}. Situation 2: You change your vote from A to A>B>C>D. The domination B>>C is created. Now the members of my set are {a,b,c,d}. So adding B>>C has potentially hurt A. Here's an example showing why I don't think Minimal Defense and Later-no-harm can be satisfied simultaneously with 4+ candidates: Situation 1: A>>B, B>>C, C>>A, A>>D The members of my set are {a,b,c}. D is a Minimal Defense loser. Situation 2: Add in D>>B. Now the set members are {a,b,c,d}. But potentially this has reduced the odds of winning of candidates other than B, so Later-no-harm is failed. I want to emphasize that a LNHarm failure can only occur when there are majority-strength cycles. Cycles created by truncated ballots can't create these. There have to actually be cyclic preferences in the electorate. My hope is that this degree of LNHarm failure is small enough to permit the voters to ignore the possibility that additional preferences could harm earlier preferences. Minimal Defense essentially guarantees that a majority of the voters won't have to rank their compromise choice insincerely high in order to defeat someone they prefer him to. In other words, if the race is essentially between two strong candidates, then the entrance of weaker candidates, and the willingness of voters to rank these weaker candidates high, shouldn't confuse the method into electing the wrong one of the two strong candidates. I feel these two criteria together make for an impressive pairing of guarantees. They should reduce voters' incentive to truncate and uprank compromises, in particular, and as a result this should reduce disincentive for candidates to enter the race. I appreciate any thoughts or criticism. Kevin Venzke Découvrez le nouveau Yahoo! Mail : 250 Mo d'espace de stockage pour vos mails ! Créez votre Yahoo! Mail sur http://fr.mail.yahoo.com/ ---- Election-methods mailing list - see http://electorama.com/em for list info