[Election-Methods] Study Data, Personal Utility with Range 2 election

[Chris Benham]

Abd ul-Rahman Lomax wrote:

>This election is a counterexample to the claim that optimal voting in 
>Range is never the sincere vote.
>
>The election is many voters, Range 2 (3 Cardinal Ratings), three 
>candidates, zero knowledge.
>
>The voter has utilities of 2, 1, 0.
>
>The *relative* utility for voting sincerely is 40/27, for voting 
>Approval Style, either 220 or 200, it is 39/27.
>
>Numerous writers have informed me that this is impossible, but have 
>given me theoretical arguments which do not address the conditions of 
>this study. Most notably, this is a Range 2 election, and it is 
>totally zero knowledge, and the utilities are exactly balanced for 
>the middle candidate.
>
Abd,
I couldn't understand your spreadsheet "proof", but  after analysing the 
problem afresh I can
assure you that your conclusion that the voter does better by voting 
A2,B1,C0 than by voting
A2,B0,C0 is wrong.

I've looked at it from the point of view that all the other votes have 
been cast and it is possible
that our vote can change the result. Since our 3 possible ballots differ 
only in how they rate B,
the only situations that interest us (for the time being barring the 
3-way tie) are B tied with A
or C, B leading A by 1 or 2 points, and B in second place trailing C by 
1 or 2 points.

Looking at these six situations and for each one measuring the equity 
gain from voting 200,
210 or 220 versus not voting we see that the total gain for each is the 
same (+2).

So assuming that without our voter's vote there is never an exact 3-way 
tie, it doesn't make any difference
in this election what rating our (sincere A2,B1,C0) voter gives B. But 
including that very small chance means
that 200 and 210 are slightly better than 220.

>This election is a counterexample to the claim that optimal voting in 
>Range is never the sincere vote.
>

The "claim" you refer to was made by no-one. What was pointed out was 
that the "sincere" vote is never
better than some approval-style vote (giving no intermediate ratings).

Prevote result.    Equity  of  not  voting.     Equity  gain  by  
voting  200.      Equity  gain  by  voting  210.     Equity  gain  by  
voting  220. 
B>A (m2)              1                                    0.5          
                                      0                                 
              0                           
B>A (m1)              1                                     
1                                                  
0.5                                            0
B=A (m0)              1.5                                 
0.5                                                
0.5                                            0

C>B (m2)              0                                    0             
                                       0                                 
             0.5                          
C>B(m1)               0                                    0 
                                                 
0.5                                             1
 C=B(m0)              0.5                                 0 
                                                 
0.5                                            0.5

A=B=C                 1                                     
1                                                  
1                                              0.5


I hope my table lines up, and that my explanation is clear.


Chris Benham



>http://www.beyondpolitics.org/OptimalRangeVote.htm
>
>Look at the last sheet. If your browser has trouble with the 
>formatting, there is a simple page at
>http://www.beyondpolitics.org/OptimalRangeVote_files/sheet006.htm
>
>That page and the previous tab, sheet005, are reformatted for 
>clarity, the other pages are my original 2-voter study and the first 
>arrangement of the many-voter study.
>
>The election is many voters, so many that three-way ties can be 
>neglected. The study looks only at conditions where the voter's vote 
>can affect the outcome; conditions other than these cannot affect the 
>voter's relative utilities for choices which *can* affect the 
>outcome. (The utility for all those other conditions is 1.0 in this election.)
>
>This election is a counterexample to the claim that optimal voting in 
>Range is never the sincere vote.
>
>The election is many voters, Range 2 (3 Cardinal Ratings), three 
>candidates, zero knowledge.
>
>The voter has utilities of 2, 1, 0.
>
>The *relative* utility for voting sincerely is 40/27, for voting 
>Approval Style, either 220 or 200, it is 39/27.
>
>Numerous writers have informed me that this is impossible, but have 
>given me theoretical arguments which do not address the conditions of 
>this study. Most notably, this is a Range 2 election, and it is 
>totally zero knowledge, and the utilities are exactly balanced for 
>the middle candidate.
>
>
>
>
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>
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