[Election-Methods] Test message/resending "Using range ballots as an extension of ranked ballot voting"

[Michael Rouse]

Hi, everyone. I was just sent a note letting me know my last message was in an unreadable format (a copy of it here: http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20080308/a1502ffe/attachment.html), so I thought I'd resend it using Thunderbird instead of the web-based email program I used before. If someone can let me know if the formatting is garbled (I know the content is often garbled, but that's a given :) ) then please let me know. Thanks!

(I wouldn't re-send it, but I'd like it to appear in the archive correctly. Not sure what happened last time, though). The text between the underlines ("________") is what I wrote before.

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Snipping the message:

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On Mar 3, 2008, at 1:45 , <mrouse1 at mrouse.com> <mrouse1 at mrouse.com> wrote:


 > juho4880 at yahoo.co.uk:
 >

 > >>Can you also clarify a bit how step 3 is counted when some candidate X is beaten by two other candidates (Y and Z). 
 > >>I find the proposed method interesting since it seems to aim at electing good winners (using a function minimizes the problems caused to the voters, from one point of view).
 >
 > I'd be happy to try. Do you have an example election for me to play with? I'm assuming you mean where I said
 >
 >
 > 3. If there is no Condorcet winner, find the shortest distance (sum of individual ranges) necessary to produce a Condorcet winner.
 >

Sorry for some delay in replying. Here's one quick example.

1: A=10 B=2 C=1 D=0
1: A=10 C=7 B=6 D=0
1: B=10 C=6 A=5 D=0
3: C=10 D=5 A=1 B=0
3: D=10 B=4 A=3 C=0

C is now beaten by both A and B, and C has to win them both in order to become a Condorcet winner. What is the "shortest distance (sum of individual ranges)" for C in this example and how do you count it? 

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Okay, here's how I did it by hand (sorry if it's a bit cryptic).

Given the following:

1: A=10 B=2 C=1 D=0
1: A=10 C=7 B=6 D=0
1: B=10 C=6 A=5 D=0
3: C=10 D=5 A=1 B=0
3: D=10 B=4 A=3 C=0

The question was: "C is now beaten by both A and B, and C has to win them both in order to become a Condorcet winner. What is the "shortest distance (sum of individual ranges)" for C in this example and how do you count it?" 

Here are the number of pairs each way:
A>B (1+1+3)
A>C (1+1+3)
A>D (1+1+1)
B>A (1+3)
B>C (1+1+3)
B>D (1+1+1)
C>A (1+3)
C>B (1+3)
C>D (1+1+1+3)
D>A (3+3)
D>B (3+3)
D>C (3)

Simplifying (numbers in parenthesis indicate surplus votes) and showing the pair relations:
A>B (1)
A>C (1)
B<C (1)
C>D (3)
D>A (3)
D>B (3)

To remove the relation A>B, it would take 1 vote, the smallest total distance of which is 1 (1-0).
To remove the relation A>C, it would take 1 vote, the smallest total distance of which is 3 [(10-7) or (3-0)]
To remove the relation B>C, it would take 1 vote, the smallest total distance of which is 1 (2-1)
To remove the relation C>D, it would take 3 votes, the smallest total distance of which is 11 [(1-0)+(10-5)+(10-5)] 
To remove the relation D>A, it would take 3 votes, the smallest total distance of which is 12 [(5-1)+(5-1)+(5-1)]
To remove the relation D>B, it would take 3 votes, the smallest total distance of which is 15[(5-0)+(5-0)+(5-0)]

To make A the weak Condorcet winner (A>=B,C,D), removing the relation D>A is sufficient. The total distance is 12.
To make B the weak Condorcet winner (B>=A,C,D), removing the relation A>B and D>B is sufficient. The total distance is 16 (1+15)
To make C the weak Condorcet winner (C>=A,B,D), removing the relation A>C and B>C is sufficient. The total distance is 4 (3+1)
To make D the weak Condorcet winner (D>=A,B,C), removing the relation C>D is sufficient. The total distance is 11.

Using this method, C would be the winner, since 4 is the shortest distance. The complete order is C>D>A>B.

(I use the weak Condorcet criterion, because an infinitesimal amount added to either candidate in a tie is sufficient to create a winner.)

Let me know if anything is unclear, and I'll try to give a better explanation (grin).


I might play around with the same election and see what removing the lowest order of preferences (and not just the closest preferences) would yield.

Michael Rouse.

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Anyway, if it's garbled but readable enough to see this line, let me know. Or if it came through fine, that would be nice to know, and I might try a test with the webmail program again. Thanks!

Mike