[EM] Clone Defeats Revised Example
A revised example of clone defeats. 34 ABC 33 BCA 32 CAB 99 67 BC 32 66 AB 33 65 CA 34 AB, BC, CA circular tie Z, a 100 percent clone of B, is added after B 34 ABZC 33 BZCA 32 CABZ 99 99 BZ 0 67 BC 32 67 ZC 32 66 AB 33 66 AZ 33 65 CA 34 BZ, AZ, ZC AB, BC, CA (same as above) Another circular tie The total or partial clone in the worst defeat should be removed (whether there is 3, 30 or 300 in a circular tie) (???) Borda fans (if any)- do the Borda math with additional clones-- such as Z2 after Z, Z3 after Z2, etc.
Re: [EM] Method definitions (partial reply)
The translations of Condorcet's own words for his bottom-up iteration proposal have Plain Condorcet as their literal interpretation. Yes some of us, including me, believe that Mr. Condorcet meant more than Plain Condorcet, but what I call Plain Condorcet is the literal, simplest interpretation of that proposal, the name seems reasonable, to distinguish it from the more refined interpretations, that I call Cycle Condorcet interpretations, because, though they solve circular ties by dropping defeats, they won't drop a defeat unless it's the weakest defeat in some cycle. I fit Schulze into that category since Schulze, SSD, SD are equivalent when there are no pairwise ties or equal defeats. My understanding is that Condorcet only seriously considered the situation of three candidates and complete rankings. If so, we have the same problem as before. His words have been taken out of context to appear to advocate Minmax(winning-votes) when in the context he was using, margins was equal to winning-votes, and Tideman, Schulze and many others are equivalent to Minmax. Here's a translation: on page LXVIII of his "Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix" (Imprimerie Royale, Paris, 1785), Condorcet writes due to my own translation: From the considerations, we have just made, we get the general rule, that in all those situations, in which we have to choose, we have to take successively all those propositions that have a plurality, beginning with those that have the largest, to pronounce the result, that is created by those first propositions, as soon as they create one, without considering the following less probable propositions. On page 126, he writes due to my own translation: Create an opinion of those n*(n-1)/2 propositions, which win most of the votes. If this opinion is one of the n*(n-1)*...*2 possible, then consider as elected that subject, with which this opinion agrees with its preference. If this opinion is one of the (2^(n*(n-1)/2))-n*(n-1)*...*2 impossible opinions, then eliminate of this impossible opinion successively those propositions, that have a smaller plurality, accept the resulting opinion of the remaining propositions. This makes it clear that, in Condorcet's top-down proposal, and in his bottom-up proposal, he _isn't_ limiting his proposal to 3 candidates. As for assuming that he thinks everyone will rank all the candidates, does he actually say that? If not then we can't assume it. If he doesn't say one way or another, then we must assume that his proposals apply whether or not there's truncation. We can't read his mind, so we must go only by what he said. Mike (more tomorrow) __ Get Your Private, Free Email at http://www.hotmail.com
[EM] Method definitions
Dear participants, on page LXVIII of his "Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix" (Imprimerie Royale, Paris, 1785), Condorcet writes due to my own translation: From the considerations, we have just made, we get the general rule, that in all those situations, in which we have to choose, we have to take successively all those propositions that have a plurality, beginning with those that have the largest, to pronounce the result, that is created by those first propositions, as soon as they create one, without considering the following less probable propositions. On page 126, he writes due to my own translation: Create an opinion of those n*(n-1)/2 propositions, which win most of the votes. If this opinion is one of the n*(n-1)*...*2 possible, then consider as elected that subject, with which this opinion agrees with its preference. If this opinion is one of the (2^(n*(n-1)/2))-n*(n-1)*...*2 impossible opinions, then eliminate of this impossible opinion successively those propositions, that have a smaller plurality, accept the resulting opinion of the remaining propositions. ** I interpret the first quotation as follows: "Minimize the maximum pairwise defeat that has to be ignored to get a complete ranking of the candidates." I interpret the second quotation as follows: "Maximize the minimum pairwise defeat that has to be used to get a complete ranking of the candidates." Therefore I think that every election method, that minimizes the maximum pairwise defeat that has to be ignored and maximizes the minimum pairwise defeat that has to be used to get a complete ranking of the candidates, is a possible interpretation of Condorcet's words. Therefore I think that the Tideman method, the Schulze method and many other election methods are possible interpretations of Condorcet's words. ** Example (3 Feb 2000): 26 voters vote C A B D. 20 voters vote B D A C. 18 voters vote A D C B. 14 voters vote C B A D. 08 voters vote B D C A. 07 voters vote D A C B. 07 voters vote B D A = C. Then the matrix of pairwise defeats looks as follows: A:B=51:49 A:C=45:48 A:D=58:42 B:C=35:65 B:D=75:25 C:D=40:60 1. The "strongest cycle" is C B D C. The weakest pairwise defeat of this directed cycle is C:D=40:60. Therefore the ranking of the candidates should be compatible to all pairwise defeats that are stronger than 40:60. That means that the final ranking must include C B and B D. 2. If votes-against was used then A:C would be the weakest pairwise defeat. Therefore he would ignore A:C=45:48. Then he would ignore A:B=51:49. Then he would take A:D=58:42 into consideration because otherwise it wouldn't be possible anymore to get a complete ranking of the candidates. That means that the final ranking must include A D. Therefore I conclude that every election method that is compatible to Condorcet's words must result in a ranking that includes C B, B D and A D. Markus Schulze (this time without any virus)
