> Date: Thu, 16 Dec 2004 02:18:25 +0100 (CET) > From: Kevin Venzke > Subject: [EM] Defection, nomination disincentive, MMPO
> In "MinMax (Pairwise Opposition)" or "MMPO," a pairwise matrix > is formed as in a Condorcet method, but the winners of pairwise > contests are not determined. For each candidate, find how many > ballots favored each other candidate over him, and record the > largest number. Elect the candidate for whom this number is smallest. For the situation I am thinking of using a voting method on, at the moment I would use MMPO. This is because it is almost the same as MinMax (Winning votes) except for the fact that it is even easier to explain and present the results. There is no need to mention about head-to-head/pairwise results. You just need to mention how many voters ranked a candidate X above another candidate Y. > This method has flaws: It fails Condorcet, I initially thought MMPO did fail Condorcet. However, I thought I worked out why it did not. May be I am wrong. Can you give an example of this? > Majority, Plurality, and Clone Independence. May be MMPO can be patched by Forest's de-cloning method before applying MMPO. > But it does satisfy Later-no-Harm. I think that sacrificing Condorcet in order to gain Later-no-Harm is most probably a worthwhile sacrifice. For a while, I had been trying to think up of a reasonable pairwise method that fails Condorcet. It looks like MMPO is one such method. > 49 A, 24 B, 27 C>B or > 49 A, 24 B>C, 27 C>B: > A: score is 51 (number of B>A voters) > B: score is 49 (number of A>B voters) > C: score is 49 (number of A>C voters) > > In other words, a B-C tie, regardless of how many B or C voters > defect from the other, so long as over 49 voters rank the same > candidate above A. If at least 25 A voters pick a side between > B and C, then that would break the tie, also. > > If a tie still remains, I suggest breaking it with Random Ballot > or perhaps FPP, two methods which still satisfy Later-no-Harm. Let's say that Candidate A was John Andrews, Candidate B was George Brown and Candidate C was Tim Charles. The results of the above two scenarios could be presented as follows... 1st. Tim Charles - 49:JA, 24:GB 2nd. George Brown - 49:JA, 27:TC 3rd. John Andrews - 51:GB, 27:TC ...and... 1st. Tim Charles - 49:JA, 24:GB 2nd. George Brown - 49:JA, 27:TC 3rd. John Andrews - 51:GB, 51:TC The above are basically whole pairwise matrices except that each line (i.e. candidate), the number of opposing votes is ordered in decreasing numeric order. The lines are then ordered by number of opposing votes in the first "column" in increasing numeric order. For example, according to the first line, 49 ballots ranked JA (John Andrews) above Tim Charles and 24 ballots ranked GB (George Brown) above Tim Charles. As can be seen in the above, Tim Charles and George Brown are initially tied with 49 opposing votes. The most easily presentable tie-breaker I thought of using was to find the next highest opposing votes for each of the tied candidates. The candidate with the lowest number of these opposing votes is the winner. If there is still a tie here, then you go on to the next highest opposing votes for the tied candidates. And so on, if required. In the above situation, Tim Charles' next highest opposing votes is less than George Brown's. Therefore, Tim Charles is the winner. In this case, the tie-breaker also turns out to be a head-to-head contest between the Time Charles and George Brown, which Tim Charles wins. In this situation, the number of voters is large enough to say that this was an extremely close election. If George Brown had just got one less ballot that voted him below John Andrews, then George Brown would have won, regardless of the fact that George Brown loses the head-to-head against Tim Charles. It can be seen that for an N candidate election, there can be up to (N-2) tie-breakers. However, assuming that enough ballots are cast, it could be said any candidate who wins as a result of the tie-breaker is lucky. Therefore, it could be said that any tie-breaker could do. Thanks, Gervase. ---- Election-methods mailing list - see http://electorama.com/em for list info
