39 16 voter C101 C102 C103 C104 C105 C106 C107 C108 C109 C110 C111 C112 C113 C114 C115 C116 V1 7 9 1 5 6 8 2 9 9 7 5 6 9 5 0 0 V2 8 9 8 9 2 7 5 2 4 8 8 3 9 2 9 9 V3 7 9 0 4 9 9 3 9 3 8 0 9 9 0 0 4 V4 8 7 8 7 8 9 3 6 6 7 4 4 5 5 0 5 V5 8 8 9 9 1 8 9 9 8 7 9 4 2 6 0 9 V6 9 0 0 9 0 9 9 9 9 0 0 0 9 0 0 0 V7 8 9 9 7 8 9 7 X 8 8 X 2 0 7 5 6 V8 6 7 9 2 9 9 6 4 9 9 7 3 2 7 8 6 V9 9 7 5 6 9 9 7 5 9 8 6 0 3 7 9 7 V10 9 9 9 X 1 9 8 5 9 9 9 6 1 9 1 1 V11 9 2 7 9 9 5 7 7 9 9 X 5 7 7 7 X V12 9 9 9 6 0 9 9 7 5 5 X 2 0 5 5 9 V13 9 8 8 9 6 8 8 9 5 5 5 4 5 5 9 4 V14 X 9 X X 0 0 0 9 0 9 X 0 0 X 0 X V15 8 6 9 7 7 8 3 4 3 6 7 0 3 7 3 5 V16 9 6 8 5 5 9 4 6 8 8 2 4 4 5 1 7 V17 8 6 5 2 9 5 4 1 3 0 X 0 0 2 9 0 V18 7 7 5 6 0 8 9 9 9 9 7 1 1 9 9 9 V19 9 9 8 8 9 9 6 9 9 9 9 8 7 6 7 8 V20 5 8 9 3 9 9 3 7 9 6 9 4 4 2 4 4 V21 9 8 9 5 8 8 7 7 8 9 7 5 3 5 4 7 V22 0 8 9 5 5 7 6 4 6 6 7 8 4 0 7 7 V23 8 9 7 9 0 9 0 9 0 9 6 6 4 0 0 0 V24 9 9 X X X 9 X X X 9 X X X X X X V25 7 7 8 4 8 8 7 6 9 6 8 6 4 4 5 6 V26 9 9 0 5 0 9 9 5 9 0 0 0 0 0 0 9 V27 8 9 8 7 9 9 2 7 8 9 3 6 6 2 0 5 V28 2 7 8 5 0 6 9 9 9 6 9 4 6 6 0 8 V29 9 9 6 9 9 6 3 5 5 9 X 2 6 5 3 6 V30 7 7 9 0 2 9 5 9 9 2 2 0 0 0 0 9 V31 9 9 5 9 9 5 9 9 9 5 9 5 9 5 5 5 V32 9 9 9 7 9 9 7 8 9 9 7 4 3 4 2 7 V33 X 5 X X 3 5 5 6 5 5 X 4 X 7 X 5 V34 7 9 3 6 5 X X X X X X 5 2 9 5 3 V35 9 9 X X 9 9 X X 9 X X 9 X X 9 X V36 6 8 8 X 0 9 X 7 7 6 6 3 0 6 5 7 V37 4 6 9 3 6 9 7 8 9 4 9 7 5 5 9 1 V38 9 9 0 5 5 9 3 0 0 2 3 0 9 5 0 9 V39 8 8 7 4 5 8 6 8 8 7 4 4 5 5 6 6 voter C101 C102 C103 C104 C105 C106 C107 C108 C109 C110 C111 C112 C113 C114 C115 C116 -- Vote data used as INPUT for program. (View widescreen.) ABOUT THIS ELECTION: This election had 45 potential voters, of whom 39 actually voted (denoted V1...V39), and 16 candidates (denoted C101..C116). The election's purpose was to elect 9 winners. Roughly speaking, the stakes were equivalent to jobs for few years for the winners. The voters each were supposed to rate each candidate on a scale from 0=terrible to 9=great. However, evidently some voters left some candidates unscored (X), perhaps due to apathy, laziness, or self-perceived ignorance. Q. Why it is that some voters did not score some candidates? A. Voters were instructed "PLEASE RATE ALL CANDIDATES BELOW unless you feel you do not know someone well enough to evaluate him/her. You may register a vote of 'no opinion' by leaving a candidate's rating blank which will neither help nor hurt someone's score." While most people in the 45 know each other, there was a possibility that some qualified candidates would be disadvantaged by the fact that some voters didn't have the opportunity to work closely with them. (Some candidates had the advantage of having worked with all voters for several years, while other candidates - based on factors beyond their control - had less time to work with all voters.) Q. Is it ok if I reveal that you're a prestigious musical group and the vote-scores are presumably largely based on hearing performances by the candidates? A. Sure, but allow me to correct one misconception here. The primary criteria was NOT based on performance ability - there were no playing auditions for this vote. The goal of this election was focused on choosing a "leadership team" from amongst the 45 players. The 9 members of the leadership team will be charged with setting the artistic/administrative direction of the organization. Playing ability was likely one factor among many that voters considered when evaluating the candidates. Other factors that were (hopefully!) considered were leadership ability, organizational ability, etc. In addition to the body of work that each candidate has produced over the past 1-3 years, all candidates were required to submit a statement describing their vision/qualifications for serving on the leadership team. The specific instructions to voters were; "Based on your informed opinion and a close reading of the candidates' statements, please rate each candidate's fitness for a spot on the 'Leadership Team'." So this election was not just about hiring people (or making their jobs more permanent) it has a political aspect as well. FULL RANKING: I was asked actually to rank the 16 candidates in order so that if some of the 9 winners could not serve, they could go further down to list to hire "ALTERNATES." The RRV program did so, but there is a CRITICISM of that, pointed out by Aaron Hamlin -- it violates the PR (proportional representation) philosophy. To explain that vis-a-vis "Democrats" vs "Republicans," PR assures that Republicans (if a fraction-F minority) will win about fraction F of the seats (as opposed to naive voting methods which would yield 100% Democrat winners). RRV does so by continually "reweighting" the votes. In that situation the Dems would win seat #1, but then the voters who rated winner#1 high, would get their votes deweighted. With enough Dem-winners and hence Dem-deweighting, the Republican voters would obtain the majority of the weight and be able to elect a Republican winner (but then they'd be deweighted). The weighting formulas are devised to assure PR. Now Aaron Hamlin's point is this. If among the 9 winners, the guy who drops out is a Republican, then that ought to bias things so that the 1st alternate is Republican. Or ditto with "Democrat." Point is, who should be the first alternate, DEPENDS on which of the first 9 winners dropped out -- it isn't just one fixed ordering. If two of the 9 drop out, it would depend on them both and could be even more complicated. Fortunately there does not appear to have been a lot of factionalism in this election (at least compared to USA politics) so these issues probably do not matter hugely. RRV VOTING METHOD described here: http://RangeVoting.org/RRV.html It produces an ordering from best to worst, such that the first K candidates in the ordering are the "best" K winners in a PR sense (for any & every value of K). FACTIONS: The strategic voters seem a lot different to naive clustering algorithms, e.g. {V24, V35} appear to form "one faction," with everybody else in the "other faction." That may just indicate that naive clustering algorithms are not a good way to identify factions. OUTPUT OF PROGRAM RRVvoting.c: 39 voters. 16 candidates. The 16 candidates are: C101 C102 C103 C104 C105 C106 C107 C108 C109 C110 C111 C112 C113 C114 C115 C116 Am assuming single-digit scores {0,1,2,3,4,5,6,7,8,9} and "X" for "no score." The 39 votes were: voter V1: 7 9 1 5 6 8 2 9 9 7 5 0 voter V2: 8 9 8 9 2 7 5 2 4 8 8 9 voter V3: 7 9 0 4 9 9 3 9 3 8 0 4 voter V4: 8 7 8 7 8 9 3 6 6 7 4 5 voter V5: 8 8 9 9 1 8 9 9 8 7 9 9 voter V6: 9 0 0 9 0 9 9 9 9 0 0 0 voter V7: 8 9 9 7 8 9 7 X 8 8 X 6 voter V8: 6 7 9 2 9 9 6 4 9 9 7 6 voter V9: 9 7 5 6 9 9 7 5 9 8 6 7 voter V10: 9 9 9 X 1 9 8 5 9 9 9 1 voter V11: 9 2 7 9 9 5 7 7 9 9 X X voter V12: 9 9 9 6 0 9 9 7 5 5 X 9 voter V13: 9 8 8 9 6 8 8 9 5 5 5 4 voter V14: X 9 X X 0 0 0 9 0 9 X X voter V15: 8 6 9 7 7 8 3 4 3 6 7 5 voter V16: 9 6 8 5 5 9 4 6 8 8 2 7 voter V17: 8 6 5 2 9 5 4 1 3 0 X 0 voter V18: 7 7 5 6 0 8 9 9 9 9 7 9 voter V19: 9 9 8 8 9 9 6 9 9 9 9 8 voter V20: 5 8 9 3 9 9 3 7 9 6 9 4 voter V21: 9 8 9 5 8 8 7 7 8 9 7 7 voter V22: 0 8 9 5 5 7 6 4 6 6 7 7 voter V23: 8 9 7 9 0 9 0 9 0 9 6 0 voter V24: 9 9 X X X 9 X X X 9 X X voter V25: 7 7 8 4 8 8 7 6 9 6 8 6 voter V26: 9 9 0 5 0 9 9 5 9 0 0 9 voter V27: 8 9 8 7 9 9 2 7 8 9 3 5 voter V28: 2 7 8 5 0 6 9 9 9 6 9 8 voter V29: 9 9 6 9 9 6 3 5 5 9 X 6 voter V30: 7 7 9 0 2 9 5 9 9 2 2 9 voter V31: 9 9 5 9 9 5 9 9 9 5 9 5 voter V32: 9 9 9 7 9 9 7 8 9 9 7 7 voter V33: X 5 X X 3 5 5 6 5 5 X 5 voter V34: 7 9 3 6 5 X X X X X X 3 voter V35: 9 9 X X 9 9 X X 9 X X X voter V36: 6 8 8 X 0 9 X 7 7 6 6 7 voter V37: 4 6 9 3 6 9 7 8 9 4 9 1 voter V38: 9 9 0 5 5 9 3 0 0 2 3 9 voter V39: 8 8 7 4 5 8 6 8 8 7 4 6 No errors found in input. Done reading input. Approval-style strategic voters: V6 V14 V24 V35 (4 total) Did not employ full score range: V2 V8 V10 V11 V13 V16 V19 V20 V21 V24 V25 V29 V31 V32 V33 V34 V35 V37 V39 (19 total) Left 1 or more candidates unscored: V7 V10 V11 V12 V14 V17 V24 V29 V33 V34 V35 V36 (12 total) Left all candidates unscored: (0 total) Done inspecting input. Average Scores for the 16 candidates: C101=7.595 C102=7.641 C103=6.600 C104=5.939 C105=5.237 C106=7.868 C107=5.629 C108=6.657 C109=6.838 C110=6.486 C111=5.759 C112=3.763 C113=4.056 C114=4.556 C115=3.946 C116=5.514 This would naively induce the following order (top to bottom) for the candidates: C106, C102, C101, C109, C108, C103, C110, C104, C111, C107, C116, C105, C114, C113, C115, C112 That order just is for perceived quality, without any attempt at "proportional representation" (PR). [With a majority-Democratic electorate, the naive order would yield 100% Democratic winners and 0% Republican winners. But PR would yield about fraction F Republican winners if fraction F of voters Republican.] Now for proportional representation... Embarking on RRV process using K=0.500000: Winner[1] is C106. Winner[2] is C102. Winner[3] is C101. Winner[4] is C108. Winner[5] is C109. Winner[6] is C110. Winner[7] is C103. Winner[8] is C104. Winner[9] is C107. Winner[10] is C116. Winner[11] is C105. Winner[12] is C111. Winner[13] is C114. Winner[14] is C113. Winner[15] is C115. Winner[16] is C112. RRV process(using K=0.500000) complete. Embarking on RRV process using K=1.000000: Winner[1] is C106. Winner[2] is C102. Winner[3] is C101. Winner[4] is C108. Winner[5] is C109. Winner[6] is C110. Winner[7] is C103. Winner[8] is C104. Winner[9] is C107. Winner[10] is C116. Winner[11] is C105. Winner[12] is C111. Winner[13] is C114. Winner[14] is C113. Winner[15] is C115. Winner[16] is C112. RRV process(using K=1.000000) complete. All done. -- MORE COMMENTS AND OTHER ELECTION METHODS... Note the K=0.5 and K=1 RRV processes gave the exact same order, which is fortunate because there is some debate about the best value of K -- but for this election, K made no difference. The naive and RRV orders were the same except for swapping C108,C109 and C103,C110 and then further changes from the 9th place onward. The 9th placer actually matters since the 9th and last winner (since they wanted 9 winners only) changes using RRV versus naive; RRV elects C107, naive elects C111. Unfortunately due to a bug in the program's handling of "X" votes (now fixed) they may have gone with C111. Jameson Quinn says in this election he prefers K=1 but they returned the same winner set so it does not matter. I (Warren Smith) don't have any clear preference on K=1/2 versus K=1, there is a small discussion of that on http://RangeVoting.org/RRV.html which doesn't really help me decide for your case. Jameson Quinn informs me meanwhile that he has programmed a voting method of his own invention "AT-TV method" and it yielded this ordering and top 9: [C106, C102, C109, C101, C103, C108, C105, C110, C104] 7 candidates elected in round 1 (cutoff=9), 1 in round 2 (cutoff=8), and one in round 5 (cutoff=5). Continuing onward to order the remaining 7 candidates, Quinn finds... C112, C107, C116, C114, C115, C113, C111. Quinn writes: The biggest discrepancy with the RRV results is that C105 is much higher on my list. That is apparently because C105 had a strong core of supporters who rated him/her at 9, but did not have broad support (relatively low average rating). I suspect that is also because my method penalizes the C106 voters much less than RRV, and the voters for lower elected candidates (i.e, C108) more. So perhaps my method chooses C105 to "balance out" C108. That's just a superficial guess, though. Later: Ted Stern has run a method he calls PR-CTV, and Kristofer Munsterhjelm has run several more methods "birational," "range PAV," and "STV" (using certain random tie-breaks) and all four of these methods produced the same top 9 as AT-TV. KM also ran "QPQ" and "Meek STV" both of which produced these top 9 (not in order): {C102 C103 C104 C105 C106 C107 C109 C110 C116} note C101 did not win while "Schulze STV" produced these top 9 (not in order) {C102 C103 C104 C105 C106 C107 C109 C114 C116} note C101 did not win. Kristofer Munsterhjelm converted the 39 ballots to rank-order form (i.e. ignoring the numerical scores aside from their ordering) finding the following: 1: C113=C109=C108=C102>C106>C110=C101>C112=C105>C114=C111=C104>C107>C103>C116=C115 1: C116=C115=C113=C104=C102>C111=C110=C103=C101>C106>C107>C109>C112>C114=C108=C105 1: C113=C112=C108=C106=C105=C102>C110>C101>C116=C104>C109=C107>C115=C114=C111=C103 1: C106>C105=C103=C101>C110=C104=C102>C109=C108>C116=C114=C113>C112=C111>C107>C115 1: C116=C111=C108=C107=C104=C103>C109=C106=C102=C101>C110>C114>C112>C113>C105>C115 1: C113=C109=C108=C107=C106=C104=C101>C116=C115=C114=C112=C111=C110=C105=C103=C102 1: C106=C103=C102>C110=C109=C105=C101>C114=C107=C104>C116>C115>C112>C113 1: C110=C109=C106=C105=C103>C115>C114=C111=C102>C116=C107=C101>C108>C112>C113=C104 1: C115=C109=C106=C105=C101>C110>C116=C114=C107=C102>C111=C104>C108=C103>C113>C112 1: C114=C111=C110=C109=C106=C103=C102=C101>C107>C112>C108>C116=C115=C113=C105 1: C110=C109=C105=C104=C101>C115=C114=C113=C108=C107=C103>C112=C106>C102 1: C116=C107=C106=C103=C102=C101>C108>C104>C115=C114=C110=C109>C112>C113=C105 1: C115=C108=C104=C101>C107=C106=C103=C102>C105>C114=C113=C111=C110=C109>C116=C112 1: C110=C108=C102>C115=C113=C112=C109=C107=C106=C105 1: C103>C106=C101>C114=C111=C105=C104>C110=C102>C116>C108>C115=C113=C109=C107>C112 1: C106=C101>C110=C109=C103>C116>C108=C102>C114=C105=C104>C113=C112=C107>C111>C115 1: C115=C105>C101>C102>C106=C103>C107>C109>C114=C104>C108>C116=C113=C112=C110 1: C116=C115=C114=C110=C109=C108=C107>C106>C111=C102=C101>C104>C103>C113=C112>C105 1: C111=C110=C109=C108=C106=C105=C102=C101>C116=C112=C104=C103>C115=C113>C114=C107 1: C111=C109=C106=C105=C103>C102>C108>C110>C101>C116=C115=C113=C112>C107=C104>C114 1: C110=C103=C101>C109=C106=C105=C102>C116=C111=C108=C107>C114=C112=C104>C115>C113 1: C103>C112=C102>C116=C115=C111=C106>C110=C109=C107>C105=C104>C113=C108>C114=C101 1: C110=C108=C106=C104=C102>C101>C103>C112=C111>C113>C116=C115=C114=C109=C107=C105 1: C110=C106=C102=C101 1: C109>C111=C106=C105=C103>C107=C102=C101>C116=C112=C110=C108>C115>C114=C113=C104 1: C116=C109=C107=C106=C102=C101>C108=C104>C115=C114=C113=C112=C111=C110=C105=C103 1: C110=C106=C105=C102>C109=C103=C101>C108=C104>C113=C112>C116>C111>C114=C107>C115 1: C111=C109=C108=C107>C116=C103>C102>C114=C113=C110=C106>C104>C112>C101>C115=C105 1: C110=C105=C104=C102=C101>C116=C113=C106=C103>C114=C109=C108>C115=C107>C112 1: C116=C109=C108=C106=C103>C102=C101>C107>C111=C110=C105>C115=C114=C113=C112=C104 1: C113=C111=C109=C108=C107=C105=C104=C102=C101>C116=C115=C114=C112=C110=C106=C103 1: C110=C109=C106=C105=C103=C102=C101>C108>C116=C111=C107=C104>C114=C112>C113>C115 1: C114>C108>C116=C110=C109=C107=C106=C102>C112>C105 1: C114=C102>C101>C104>C115=C112=C105>C116=C103>C113 1: C115=C112=C109=C106=C105=C102=C101 1: C106>C103=C102>C116=C109=C108>C114=C111=C110=C101>C115>C112>C113=C105 1: C115=C111=C109=C106=C103>C108>C112=C107>C105=C102>C114=C113>C110=C101>C104>C116 1: C116=C113=C106=C102=C101>C114=C105=C104>C111=C107>C110>C115=C112=C109=C108=C103 1: C109=C108=C106=C102=C101>C110=C103>C116=C115=C107>C114=C113=C105>C112=C111=C104 Based on this we find that C106 is not only the average-score winner, he/she also is the Condorcet (beats all rivals pairwise) winner and the IRV (instant runoff) winner (according to Eric Gorr, where note I believe his program regards unranked candidates as ranked coequal bottom). Gorr's program indeed finds that there is an unambiguous notion of the top 5 candidates in order: C106 > C102 > C101 > C109 > C103 where each ">" means "preferred pairwise by a voter majority versus all candidates to the right (if unranked candidates regarded as being ranked coequal bottom)" -- although the 6th-placer is unclear since there is a CONDORCET CYCLE C105 = C108 = C110 > C105. 17:17 16:16 16:13 (6th placer could be C108 or C110... C105 would be the most likely IRV winner with the top 5 removed, "most likely" meaning with random tiebreaks.) Note that this order agrees with RRV's election order (regardless of K=0.5 or K=1) for these first 5, and RRV also agrees to take two out of three of the members of the Condorcet cycle (C108 and C110) as its next two winners. RRV however does not elect C105, presumably because of deweighting. C105 had a high correlation with C110 and C101, both of whom had already been elected, but it seems as though this is not enough to explain why C105 was not elected while C104 and C111 or C107 were. It is not clear to me why RRV does not like C105; apparently PAIRwise correlations are not enough to explain it and higher order correlations (triples...) would be needed? Centered Correlation Coefficients between candidates: ...... C101 C102 C103 C104 C105 C106 C107 C108 C109 C110 C111 C112 C113 C114 C115 C116 C101 1.000 0.029 -0.223 0.415 0.140 0.134 0.003 -0.037 -0.094 0.066 -0.299 -0.267 0.037 0.154 -0.088 0.008 C102 0.029 1.000 0.090 -0.032 0.059 0.110 -0.231 -0.031 -0.204 0.341 0.253 0.244 -0.153 0.013 -0.076 0.287 C103 -0.223 0.090 1.000 -0.029 0.093 0.009 0.084 0.067 0.198 0.441 0.635 0.192 -0.396 0.234 0.280 0.213 C104 0.415 -0.032 -0.029 1.000 -0.088 -0.209 0.121 0.184 -0.153 0.314 0.145 0.081 0.378 0.168 -0.037 -0.004 C105 0.140 0.059 0.093 -0.088 1.000 0.094 -0.202 -0.193 0.178 0.243 0.123 0.326 0.299 0.072 0.321 -0.148 C106 0.134 0.110 0.009 -0.209 0.094 1.000 0.189 -0.011 0.354 -0.038 -0.427 0.187 0.012 -0.133 -0.036 0.061 C107 0.003 -0.231 0.084 0.121 -0.202 0.189 1.000 0.167 0.623 -0.223 0.238 -0.051 -0.147 0.306 0.328 0.329 C108 -0.037 -0.031 0.067 0.184 -0.193 -0.011 0.167 1.000 0.334 0.231 0.014 0.324 0.054 -0.014 -0.230 -0.114 C109 -0.094 -0.204 0.198 -0.153 0.178 0.354 0.623 0.334 1.000 0.052 0.206 0.217 -0.040 0.252 0.183 0.119 C110 0.066 0.341 0.441 0.314 0.243 -0.038 -0.223 0.231 0.052 1.000 0.447 0.438 0.038 0.437 0.078 0.119 C111 -0.299 0.253 0.635 0.145 0.123 -0.427 0.238 0.014 0.206 0.447 1.000 0.305 -0.113 0.530 0.470 0.036 C112 -0.267 0.244 0.192 0.081 0.326 0.187 -0.051 0.324 0.217 0.438 0.305 1.000 0.386 -0.039 0.109 -0.254 C113 0.037 -0.153 -0.396 0.378 0.299 0.012 -0.147 0.054 -0.040 0.038 -0.113 0.386 1.000 -0.207 -0.096 -0.187 C114 0.154 0.013 0.234 0.168 0.072 -0.133 0.306 -0.014 0.252 0.437 0.530 -0.039 -0.207 1.000 0.310 0.109 C115 -0.088 -0.076 0.280 -0.037 0.321 -0.036 0.328 -0.230 0.183 0.078 0.470 0.109 -0.096 0.310 1.000 0.057 C116 0.008 0.287 0.213 -0.004 -0.148 0.061 0.329 -0.114 0.119 0.119 0.036 -0.254 -0.187 0.109 0.057 1.000 WARNING: The below may be buggy, needs to be recomputed: One way to assess the impact of the 4 strategic approval-style voters V6 V14 V24 V35 is to re-count the election with them removed. We would then have these Average Scores for the 16 candidates: C101=7.471 C102=7.743 C103=6.794 C104=5.844 C105=5.429 C106=8.000 C107=5.697 C108=6.515 C109=6.912 C110=6.529 C111=5.964 C112=3.829 C113=4.029 C114=4.686 C115=4.029 C116=5.676 This would naively induce the following order (top to bottom) for the candidates: C106, C102, C101, C109, C103, C110, C108, C111, C104, C107, C116, C105, C114, C115, C113, C112 as opposed to the naive order with the strategic voters left in, which was: C106, C102, C101, C109, C108, C103, C110, C104, C111, C107, C116, C105, C114, C113, C115, C112 which note was the same top 9 (indeed the same top 1,2,3,4,7,9,10, and 11). So in that naive sense, the strategic voting has zero effect. The RRV(K=0.5 and K=1.0) order with the strategists removed was: C106, C102, C101, C109, C103, C110, C108, C111, C116, C104, C107, C105, C113, C114, C115, C112 which compares with this RRV order with strategists included: C106, C102, C101, C109, C103, C108, C110, C104, C107, C111, C116, C105, C113, C114, C115, C112 .................................A swap B....AAAAAAAA swap BBBBBBB........... Thus in some sense the strategists had little effect, but they were able to change RRV winners #8 & #9.