At 01:27 PM 3/24/2008, Abd ul-Rahman Lomax wrote: >So we eliminate, for the individual voter, the exact match candidate. >We now have a choice of candidates who agree with the voter on three >out of four issues. The problem, you will note, isn't solved on that >web page. We could make a nice neat assumption that somehow these >issues have been arranged in a universal sequence of importance, so >that agreement in the first position is more important than that in >second, and so on. The effect is that the fourth issue is dropped, >there are now eight unique positions, each held by two candidates; >one of these, however, was in first rank, so the other one is now the >second choice. And, by recursion, we can determine the vote of each >voter. I haven't done it. Once upon a time, I would have. Life moves on.
I decided to construct the rankings. For each row in the table, if we make the assumption above about issue importance (not the general case, to be sure, but a reasonable example), we can determine the sincere vote. the table (monospaced) shows the order of preference for each voter profile. Each voter votes first preference for the candidate with the exact same profile as the voter. Then the voters rank the candidates according to two considerations: the voters always prefer a candidate which agrees with them on more issues. The table shows, at the top, the number of agreements (the first column total of 4 is not shown). Then, within the set of candidates with the same number of agreements, the issues are considered to be presented in order of importance, which would necessarily be minor compared to the preference between, say, a candidate with three agreements rather than two. If a ranked ballot allowed equal preferences to be expressed, then these columns with the same number at the top would be equated, but then we'd need to deal with a tie-breaking method even more. voter count agree: 3 3 3 3 2 2 2 2 2 2 1 1 1 1 (0) 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 0 YYYY YYYN YYNY YNYY NYYY YYNN YNYN YNNY NYYN NYNY NNYY YNNN NYNN NNYN NNNY NNNN 4 YYYN YYYY YYNN YNYN NYYN YYNY YNYY YNNN NYYY NYNN NNYN YNNY NYNY NNYY NNNN NNNY 4 YYNY YYNN YYYY YNNY NYNY YYYN YNNN YNYY NYNN NYYY NNNY YNYN NYYN NNNN NNYY NNYN 1 YYNN YYNY YYYN YNNN NYNN YYYY YNNY YNYN NYNY NYYN NNNN YNYY NYYY NNNY NNYN NNYY 4 YNYY YNYN YNNY YYYY NNYY YNNN YYYN YYNY NNYN NNNY NYYY YYNN NNNN NYYN NYNY NYNN 1 YNYN YNYY YNNN YYYN NNYN YNNY YYYY YYNN NNYY NNNN NYYN YYNY NNNY NYYY NYNN NYNY 1 YNNY YNNN YNYY YYNY NNNY YNYN YYNN YYYY NNNN NNYY NYNY YYYN NNYN NYNN NYYY NYYN 1 YNNN YNNY YNYN YYNN NNNN YNYY YYNY YYYN NNNY NNYN NYNN YYYY NNYY NYNY NYYN NYYY 4 NYYY NYYN NYNY NNYY YYYY NYNN NNYN NNNY YYYN YYNY YNYY NNNN YYNN YNYN YNNY YNNN 1 NYYN NYYY NYNN NNYN YYYN NYNY NNYY NNNN YYYY YYNN YNYN NNNY YYNY YNYY YNNN YNNY 1 NYNY NYNN NYYY NNNY YYNY NYYN NNNN NNYY YYNN YYYY YNNY NNYN YYYN YNNN YNYY YNYN 1 NYNN NYNY NYYN NNNN YYNN NYYY NNNY NNYN YYNY YYYN YNNN NNYY YYYY YNNY YNYN YNYY 1 NNYY NNYN NNNY NYYY YNYY NNNN NYYN NYNY YNYN YNNY YYYY NYNN YNNN YYYN YYNY YYNN 1 NNYN NNYY NNNN NYYN YNYN NNNY NYYY NYNN YNYY YNNN YYYN NYNY YNNY YYYY YYNN YYNY 1 NNNY NNNN NNYY NYNY YNNY NNYN NYNN NYYY YNNN YNYY YYNY NYYN YNYN YYNN YYYY YYYN 5 NNNN NNNY NNYN NYNN YNNN NNYY NYNY NYYN YNNY YNYN YYNN NYYY YNYY YYNY YYYN YYYY Is this the correct presentation of voter rankings, given the assumptions? (I did cross-check it by looking at the binary patterns.) Now, to try to use IRV with this, we immediately run into a rather nasty problem. Massive ties. Most of the tie-breaking methods don't work; I will eliminate the first candidate on the list of tied candidates. So this is the way the rounds go: YYYY 0 - YYYN 4 5 5 5 5 5 5 5 5 5 5 5 - YYNY 4 4 5 5 5 5 5 5 5 5 5 5 10 15 20 wins YYNN 1 1 - YNYY 4 4 4 5 5 5 5 5 5 5 5 5 5 - YNYN 1 1 1 - YNNY 1 1 1 1 - YNNN 1 1 1 1 2 2 2 2 2 - NYYY 4 4 4 4 4 5 5 5 5 5 5 5 5 5 - NYYN 1 1 1 1 1 - NYNY 1 1 1 1 1 1 - NYNN 1 1 1 1 1 1 2 2 2 2 - NNYY 1 1 1 1 1 1 1 - NNYN 1 1 1 1 1 1 1 2 2 2 2 - NNNY 1 1 1 1 1 1 1 1 - NNNN 5 5 5 5 5 5 5 5 6 8 10 11 11 11 11 YYNY wins. IRV performs better than Plurality, which elects NNNN. Now, this used a full set of rankings. One could look at this with, say, RCV (three ranks), or using the five ranks that allow ranking all the three or four issue matches. I thought it would be interesting to do this with Bucklin. Allow to proceed until majority or to the fifth round (which is the last round in which voters are voting for candidates they match on three issues out of four.) In Duluth Bucklin, perhaps, these votes would be third round votes, which allowed multiple candidates to be chosen. Notice that if every voter votes "strategically," they get NNNN. Prisoner's dilemma, isn't that? YYYY 0 4 4 4 4 = 16, majority YYYN 4 0 1 1 1 = 7 YYNY 4 1 0 1 1 = 7 YYNN 1 4 4 1 1 = 11 YNYY 4 1 1 0 1 = 7 YNYN 1 4 1 4 1 = 11 YNNY 1 1 4 4 1 = 11 YNNN 1 1 1 1 5 = 9 NYYY 4 1 1 1 0 = 7 NYYN 1 1 1 1 4 = 8 NYNY 1 1 4 1 4 = 11 NYNN 1 1 1 5 1 = 9 NNYY 1 1 1 4 4 = 11 NNYN 1 1 5 1 1 = 9 NNNY 1 5 1 1 1 = 9 NNNN 5 1 1 1 1 = 9 Bucklin does detect the issue-majority winner, the candidate who agrees with the majority on all four issues. These studies could be replete with errors. Handle with care. The full ranking table is probably correct, I would not bet much on the IRV counting. Remind me not to count an IRV election with many candidates. If someone wants to create the Condorcet matrix.... ---- Election-Methods mailing list - see http://electorama.com/em for list info