Kevin Venzke [EMAIL PROTECTED] 3/5/03 I said in my first post that I was designing an Approval variant aimed at more often producing the Condorcet winner. Here it is. I hope it's of some interest, that it's novel, and that I don't come across as too pretentious.
"MEDIAN" ELECTION METHOD This is a proposal for a single-winner election method I call "Median." Votes are cast as under Approval, and a method, justified by a geometry analogy, is used to determine the "median candidate." More precisely, this is the candidate most likely to be supported by the "median voter" as drawn on a political spectrum. The number of dimensions of the spectrum is unimportant, and the method will not be broken if there is in fact no spectrum behind the voters' choices. METHOD The voter votes for as many candidates as he wishes. The total number of voters "approving" each candidate is counted, as under Approval. However, a two-dimensional table has also to be maintained, recording the number of voters that approved each pair of candidates. For instance, if the ballots read: 12: AB 15: BC 10: AD 9: D After counting the votes, we would have counted not just that A, B, C, and D's total votes received were 22, 27, 15, and 19 respectively. We would also have maintained the following table of the "overlap" of support between each pair of candidates: A B C D A . B 12 . C 0 15 . D 10 0 0 . Next, for each of the candidates, we determine which of the other candidates had the most supporters who did not also vote for the candidate we're talking about. This is why the overlap information is needed: If we are determining A's supporters' disapproval for B, for example, we must subtract the A-B overlap from A's vote count. For the above scenario, we could make the following table. Each row is the disapproval for a candidate, while each column is the disapproval expressed by a candidate's supporters. A B C D greatest A 0 15 15 9 15 (from B or C) B 10 0 0 19 19 (from D) C 22 12 0 19 22 (from A) D 12 27 15 0 27 (from B) The "greatest" column is what matters. It tells us the "greatest constructive disapproval" for each candidate. In other words, for candidate A, there is no group of voters larger than 15 people who did not vote for A and who were agreed on a different candidate. Since 15 is the smallest "GCD," A is judged to be the median candidate and would, by this method, be the winner. (A tie could be broken simply by number of votes received.) You can also imagine the method this way: If you removed all of the ballots approving candidate X, how many votes would the top-scoring candidate then have? Elect X if the answer is the smallest. (You wouldn't have to use this method alone. You could elect the candidate who maximizes (votes received / GCD). Approval elects B, of course. If you do this division, A gets 1.467 while B gets 1.42. A would still win, but it's much closer.) I won't consider the question of whether the "median candidate," however defined, should be the winner. I will only use a geometrical analogy to argue that the candidate with the smallest GCD is probably the median candidate. Consider the following very simple scenario: 26: A 25: AB 25: BC 27: C Clearly we have a left, right, and center candidate. At least, it's possible to draw it that way. The Approval scores are 51 A, 50 B, 52 C, with C winning. Under Median, the scores are 52 A, 27 B, 51 C, with B winning. The spectrum basically looks like this, with B obviously being the median: AAAAAAAA BBBBBBBB CCCCCCCC Here is the analogy: Think of the candidates as points on a line or plane, and the "constructive disapproval" of (for instance) candidate A's supporters towards candidate B, as the distance (measured in voters) that B would need to "travel" in order to win over A's supporters. (Note that the distance is not reciprocal. If it were, then two-candidate elections would always be ties.) We can think of the "greatest constructive disapproval" for a given candidate as the greatest distance he would have to travel in one direction to obtain the approval of all the voters. Candidates A and C above would each have to win over 50 voters in a single direction, while B is only 27 voters away from the furthest candidate. It's unnecessary to be able to draw the candidates on a line or plane precisely. It's intuitive to suppose that the point with the least greatest distance to another point, is the one which is closest to the center. STRATEGY I think that it would be difficult to vote strategically under Median. This is because votes aren't counted directly. For instance, in order to "hurt" a certain candidate, it's not sufficient to not vote for him. You need to vote for a candidate who is likely to provide the GCD against him. In other words, if you want to "hurt" your second-favorite who is similar to your favorite, you're probably out of luck since the latter's supporters are probably not the largest bloc of constructive disapproval against the candidate you don't like. A candidate's fate is essentially decided by the voters who dislike him sincerely. Unlike in Approval, in Median there is reason for centrist voters to vote for second-favorite, off-center candidates in addition to their favorites. So doing can hurt a least favored candidate. It can never cause the favorite to lose, unless the favorite is otherwise the "most distant" candidate from the second-favorite. If anyone is interested, I have a scenario where if some centrist B voters had also approved off-center C, then B would have been judged the median instead of opposite off-center A. FLUKES Consider this: 3: A 1: AB 2: BC 3: C Approval scores are A 4, B 3, C 5. Median scores are A 5, B 3, C 4. B is the winner even though C has majority approval. Defensible? Is it necessary to claim that B was everyone's second choice, at worst? Or is "median voter's candidate" a valid criterion on its own? Two of the five C voters said B was acceptable. If C wins, 4 B supporters are unhappy. If B wins, 6 people are unhappy, but they're divided 3 and 3 on who would've been better. You could say that if B wins, no more than 3 people will attend the same protest. You could say that only 3 people are alienated by B in the same way. Possibly one could claim that it's impractical to give power to whichever majority manages to assemble itself for a given election. And consider this disaster: 1000: A 1000: B 1: AC 1: BC Approval scores are A 1001, B 1001, C 2. Median scores are A 1001, B 1001, C 1000. C wins with only two votes. Of course, this is not very likely; A and B have to be exactly tied for C to win like this. Specifically, for every A-only voter we add, C needs one more voter overlap with A in order to still win. Additional overlap with B will not help C in this case. Example: 1050: A 1: AC 1001: BC Median scores are A 1001, B 1051, C 1050. A is deemed to be the pick of the median voter. I'm interested in comments and criticism, especially regarding philosophy, voter strategy, and election method criteria. I suspect that Median meets similar criteria to Approval, although Median doesn't meet Independence from Irrelevant Alternatives. It is very possible for a non-winning candidate to be "kingmaker." I would actually propose that, assuming perfect information, the more candidates there are, the "better" the result will be. Thanks to any who may read this. Kevin Venzke Stepjak ___________________________________________________________ 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