[EM] Cardinal Rating Condorcet Loser Elimination

[Dgamble997 at aol.com]

Hello everybody

My favourite type of example to post on this list is the following:

45 A>B>C 
6 B>A>C
5 B>C>A
44 C>B>A

I argue that B shouldn't win because he/she is very likely to be a low 
utility compromise- the least worst.

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.

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.

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".

How people vote using this method

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:

     1         2        3       4        5
A  100      90      55    ----       -----
B  75        60      42    -----      52
C  30        20       8     ----       30
D  -----      10       ----    95       71

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:

      1         2        3         4       5
A   1.00     1.00    1.00    -----    -----
B    0.75     0.67    0.76   -----   0.73
C    0.30     0.22    0.15   -----   0.42
D    ------     0.11    ------   1.00  1.00

How the ballots are counted

1/ If any candidate is rated first on a majority of ballot papers he/she is 
elected.

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:

45  A 1.00> B 0.70
8    B 1.00>  C 0.50
47  C 1.00>  B 0.80

A versus B    45 v  45.6

A = 45 @ 1.00        B = 8 @ 1.00 + 47 @ 0.80

A versus C    45 v 51

A = 45 @ 1.00         C = 47 @ 1.00 + 8 @ 0.50

B versus C    39.5 v 47

B = 45 @ 0.70 + 8 @ 1.00          C = 47 @ 1.00

A is defeated pairwise by both B and C and is eliminated as the Condorcet 
loser.

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.

An example election with 2 low utility centrists

41  A1.00 > B 0.4 > C 0.3
11  B1.00 > C 0.9 > A 0.8
9    C1.00 > B 0.9 > D 0.8
39  D1.00 > C 0.4 > B 0.3

Pairwise comparisons:

A v B    41 v  30.8
A v C    41 v  34.5
A v D    49.8 v  46.2
B v C    27.4 v  24.6
B v D    35.5 v  39
C v D    31.2 v  39

C is defeated by A, B and D and is eliminated as Condorcet loser.

The ballots are now :

41 A1.00 > B 0.4
11 B1.00 > A 0.8
9   B1.00 > D 0.8
39 D1.00 > B 0.3

Note that on the 9 ballots that rated C highest B ( the second rated 
candidate) has had his/her utility reset to 1.00.

Second round pairwise comparison:

A v B        41 v 37
A v D        49.8 v  46.2
B v D        36.4 v   39

B is defeated by A and D and is eliminated as the Condorcet loser.

The ballots are now

41 A 1.00
11 A  1.00
9   D  1.00
39  D  1.00

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.
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.

In summary the result is:

       1st     2nd    3rd
A     41      41      52
B     11      20      ----
C      9       ----     ----
D      39     39       48

In this example with low utility centrists ( B and C ) the result is that A 
wins ( the same as in IRV).

An example with two high utility centrists

41  A1.00  > B 0.9  > C 0.8
11  B1.00  > C 0.9  > A 0.8
9    C1.00  > B 0.9  > D 0.8
39  D1.00  >  C0.9  >  B 0.8

This is becoming a very long post so I'll skip the detailed workings:

In summary the result is :

       1st    2nd    3rd
A     41     41     -----
B     11     11      52
C      9      48      48
D     39     ------   -----

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.

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 ). 

Comments, suggestions, criticisms!!!!!  
  

David Gamble

















 



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