Forest, Thanks much for your reply. I had to read it slowly and I learned some math terms.
First, below is the drawing of my first example. Although B is Beats-All, I think an A victory is defensible. Under Condorcet, A could easily win, provided that the D supporters prefer A to B, and 5 AB voters prefer A. The practical explanation for why A beats B under Median is that B was more disagreeable to D supporters than A was to CB voters, measured of course in number of voters. ...And of course if Beats-All had to be elected, people would use Approval strategy when voting. DDDDDDDDDDDDDDDDDDD AAAAAAAAAAAAAAAAAAAAAA BBBBBBBBBBBBBBBBBBBBBBBBBBB CCCCCCCCCCCCCCC Looking at some of your proposals for picking the winner: >(1) Take the team that gave up the fewest points per game on >average (i.e. had the fewest points scored against them per >game on average). This might be considered the team with the >best defense. >(5) Take the team that scored the highest point average per game. >(6) Take the team with the maximal minimum number of points per game. >In other words, if team A scored at least 25 points in every game, and >every other team scored fewer than 25 points in at least one of their >games, then A should be the winner according to this criterion. etc. Using these would influence the number of candidates. Under (1), it's best for candidates to run alongside many clones. Under (5) or (6), candidates are punished for having similar support bases, like under Plurality. >(3) Take the team for which the maximum margin against them was >minimal. This is analogous to MinMax (margins). If there is a beats >all team, this method will pick it. >(4) Take the team for which the maximum score against them was >minimal. This is analogous to MinMax (winning votes), and also will >detect a beats all winner. All of these methods pick teams based on >measurements of defensive prowess. But what about offensive prowess? I'm a little confused. Will these pick different winners from Approval? >This combination method (8) would take care of the anomalous results >at the end of your message. Note for example, that disaster associated >with 1000: A, 1000: B, 1: AC, and 1: BC is avoided by taking into >account the offensive prowess of candidates A and B in comparison to C >who is relatively strong defensively but weak offensively. So you're saying that you'd compute the same (pairwise, final) table: A B C A 0 1001 1 B 1001 0 1 C 1000 1000 0 And the scores become (1001 - 1000) or 1 for A and B each, but (1001 - 1) or 1000 for C. So this is an A-B tie. But, there is something artificial in that in finding the "least disapproval expressed" down each column, we must ignore the reflexive 0 value. And that leads to why candidates are punished for being similar to others. Imagine the modified scenario: 1000: A 1: AC 1: BC 500: B 500 BD. (I.e., half of B's supporters also like new candidate D.) A B C D A 0 1001 1 500 B 1001 0 1 0 C 1000 1000 0 500 D 1001 501 2 0 Median still gives it to C, unaffected by D's entrance. But using method (8) as you describe, the scores are A 1, B 500, C 999, D 501. Yes, C is now crushed, but so is B, unjustifiably. B is literally punished for not having the exclusive support of the BD voters. But what do you think about "factoring in" the raw Approval score, as the measurement of offensive strength? I suggested doing (Approval score / Median score) and electing the candidate with the greatest result. These scores are ugly, though, and I usually don't like the results as much. There is more incentive to use Approval-style strategy in voting. A nice thing about Median's final scores is that you can imagine what they represent: "the largest group of voters who would oppose the given candidate, and who are agreed on an alternative." > Also, I believe that your median method and the >modified median method both satisfy the FBC. Considering what I wrote above, it seems to me that the "modified median" method (8) wouldn't satisfy FBC. I could misunderstand the issue, though. The BD voters get an inferior result than if they had all just voted B. (Well, if you remove the AC voter, let's say, to eliminate the tie.) Thanks much for your thoughts! Kevin Venzke [EMAIL PROTECTED] ___________________________________________________________ Do You Yahoo!? -- Une adresse @yahoo.fr gratuite et en français ! Yahoo! Mail : http://fr.mail.yahoo.com _______________________________________________ Election-methods mailing list [EMAIL PROTECTED] http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com