<HTML><FONT FACE=arial,helvetica><FONT SIZE=2 FAMILY="SANSSERIF" FACE="Arial" LANG="0">Hello everybody<BR>
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My favourite type of example to post on this list is the following:<BR>
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45 A>B>C <BR>
6 B>A>C<BR>
5 B>C>A<BR>
44 C>B>A<BR>
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I argue that B shouldn't win because he/she is very likely to be a low utility compromise- the least worst.<BR>
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Many people on this list disagree with me and feel that B should win. They argue that he/she is a genuine compromise and the most generally preferred candidate.<BR>
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Actually neither myself nor those who disagree with me can be certain as to whether B is really a low utility turkey ( the least worst) or a popular compromise ( the most best). This is because ranked ballots just tell us that the first choice is preferred to the second choice not how much the first choice is preferred to the second choice.<BR>
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I've been trying to think of a method that overcomes this problem and I've come up with something that I've snappily called " cardinal rating Condorcet loser elimination".<BR>
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<B><U>How people vote using this method<BR>
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</B></U>Voters rate candidates on a scale from 1 (lowest utility) to 100 (highest utility). Voters may rate as few or as many candidates as they wish. Unrated candidates are considered to have a utility of 0. An example of the rating process is given below for 5 ballot papers in a 4 candidate (A,B,C,D) election:<BR>
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1 2 3 4 5<BR>
A 100 90 55 ---- -----<BR>
B 75 60 42 ----- 52<BR>
C 30 20 8 ---- 30<BR>
D ----- 10 ---- 95 71<BR>
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The cardinal rating ballots are then " standardised ". The standardisation involves dividing the ratings of all candidates rated on a ballot paper by the rating of the highest rated candidate on that ballot. Below are the standardised ballots for the 5 votes shown above:<BR>
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1 2 3 4 5<BR>
A 1.00 1.00 1.00 ----- -----<BR>
B 0.75 0.67 0.76 ----- 0.73<BR>
C 0.30 0.22 0.15 ----- 0.42<BR>
D ------ 0.11 ------ 1.00 1.00<BR>
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<B><U>How the ballots are counted<BR>
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</B></U>1/ If any candidate is rated first on a majority of ballot papers he/she is elected.<BR>
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2/ If no candidate is rated first on a majority of ballots the condorcet loser is eliminated.In pairwise comparisons the utility values of each candidate are used in determining the Condorcet loser. For example:<BR>
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45 A 1.00> B 0.70<BR>
8 B 1.00> C 0.50<BR>
47 C 1.00> B 0.80<BR>
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A versus B 45 v 45.6<BR>
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A = 45 @ 1.00 B = 8 @ 1.00 + 47 @ 0.80<BR>
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A versus C 45 v 51<BR>
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A = 45 @ 1.00 C = 47 @ 1.00 + 8 @ 0.50<BR>
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B versus C 39.5 v 47<BR>
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B = 45 @ 0.70 + 8 @ 1.00 C = 47 @ 1.00<BR>
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A is defeated pairwise by both B and C and is eliminated as the Condorcet loser.<BR>
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3/ If the candidate rated highest on a ballot paper is eliminated the utility value of the second highest rated candidate is reset to 1.00.<BR>
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<B><U>An example election with 2 low utility centrists<BR>
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</B></U>41 A1.00 > B 0.4 > C 0.3<BR>
11 B1.00 > C 0.9 > A 0.8<BR>
9 C1.00 > B 0.9 > D 0.8<BR>
39 D1.00 > C 0.4 > B 0.3<BR>
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Pairwise comparisons:<BR>
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A v B 41 v 30.8<BR>
A v C 41 v 34.5<BR>
A v D 49.8 v 46.2<BR>
B v C 27.4 v 24.6<BR>
B v D 35.5 v 39<BR>
C v D 31.2 v 39<BR>
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C is defeated by A, B and D and is eliminated as Condorcet loser.<BR>
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The ballots are now :<BR>
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41 A1.00 > B 0.4<BR>
11 B1.00 > A 0.8<BR>
<B>9 B1.00 > D 0.8</B><BR>
39 D1.00 > B 0.3<BR>
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Note that on the 9 ballots that rated C highest B ( the second rated candidate) has had his/her utility reset to 1.00.<BR>
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Second round pairwise comparison:<BR>
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A v B 41 v 37<BR>
A v D 49.8 v 46.2<BR>
B v D 36.4 v 39<BR>
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B is defeated by A and D and is eliminated as the Condorcet loser.<BR>
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The ballots are now<BR>
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41 A 1.00<BR>
<B>11 A 1.00<BR>
9 D 1.00</B><BR>
39 D 1.00<BR>
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The 11 ballots that rated B first have been reset to give A as the highest rated non-eliminated candidate a utility of 1.00.<BR>
The 9 ballots that rated C first and B second have had their values reset to give D as the highest remaining non-eliminated candidate a utility of 1.00.<BR>
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In summary the result is:<BR>
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1st 2nd 3rd<BR>
A 41 41 52<BR>
B 11 20 ----<BR>
C 9 ---- ----<BR>
D 39 39 48<BR>
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In this example with low utility centrists ( B and C ) the result is that A wins ( the same as in IRV).<BR>
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<B>An example with two high utility centrists<BR>
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</B>41 A1.00 > B 0.9 > C 0.8<BR>
11 B1.00 > C 0.9 > A 0.8<BR>
9 C1.00 > B 0.9 > D 0.8<BR>
39 D1.00 > C0.9 > B 0.8<BR>
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This is becoming a very long post so I'll skip the detailed workings:<BR>
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In summary the result is :<BR>
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1st 2nd 3rd<BR>
A 41 41 -----<BR>
B 11 11 52<BR>
C 9 48 48<BR>
D 39 ------ -----<BR>
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On the first count D is eliminated as the Condorcet loser. On the second count A is eliminated as the Condorcet loser. On the third count B beats C. With these high utility Compromise candidates B is elected as in Condorcet.<BR>
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This method appears to have the ability to elect high utility compromise candidates with low first preference votes ( like Condorcet ) but defeat low utility compromise candidates ( like IRV ). <BR>
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Comments, suggestions, criticisms!!!!! <BR>
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David Gamble<BR>
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