[Election-Methods] [EM] Is "sincere" voting in Range suboptimal?

[Chris Benham]

Abd ul-Rahman Lomax wrote:

>>> We also want to see how the results fall if the voter votes 
>>> "Approval style." There are two possible Approval style votes in 
>>> this example, being 220 and 200. Because of the symmetry of the 
>>> scenario, however, I expect that the expected satisfaction is the 
>>> same between the votes of 220 and 200, and I want to keep things 
>>> simple, so, if someone considers it necessary, I can say that the 
>>> voter decides that he "slightly" prefers B additionally such that he 
>>> will vote that way, but this does not significantly affect the 
>>> utilities.
>>
>> "Keep things simple"? The scenario is so simple that it is easy to 
>> work out everything exactly.
>
>
> Yes. Which is what I did. Because I kept it simple!


No, you just assumed "because of the symmetry of the scenario" that the 
2 possible approval votes (220 and 200) must give the same "expected 
satisfaction"
and therefore you would only bother comparing 220 with 210  (and ignore 
200).

> I get different results. First of all, A2, B0, C0 should be the same 
> as A2, B2, C0, because of the symmetry in preference strength. Now, if 
> that isn't true, I'd wonder why, but I'd certainly be open to having 
> made some mistake here. However, given that B is exact midrange in 
> utility, there is no reason for preferring the vote of B 0 to B 2, 
> they both deviate the exact same amount from true utility, but in 
> opposite directions, so the loss of utility, if any, or the gain, if 
> any, would be the same in either case, matching what was said by 
> others, including Benham, that it was equally advantageous to 0 rate B 
> as to max rate B. 

It is a special situation being half  the voters, so there is no danger 
that C can win decisively.

> To compare Benham's results with mine, I think he would have to more 
> exactly specify what he did

Only considering votes that  both max-rate at least one and min-rate at 
least one, there are only 12 possible opposition ballots
(3 that max-rate 2, 3 that min-rate 2, 6 that mid-rate 1). 

Opp. Ballot          A2 B0 C0        A2 B1 C0       A2 B2 C0 
B2 A0 C0            A=B  1.5          B     1             B     1
B2 A1 C0            A       2            A=B 1.5          B     1
B2 C1 B0            A=B   1.5         B     1             B      1

These are the 3 where A2 B0 C0  does better than  A2 B1 C0.  The numbers 
represent  voter satisfaction with the result, assuming that
our voter's sincere ratings are A2,B1,C0 and that ties are broken by 
random candidate.

Opp. Ballot          A2 B0 C0              A2 B1 C0       A2 B2 C0 
A2 B2 C0            A           2              A     2             
A=B      1.5
B2 C2 A0            A=B=C 1               B     1             B          1
C2 A0 B0            A=C      1              A=C 1            A=B=C 1
C2 B1 A0            A=C      1              A=C 1            B           
1       

The other 5 possible opposition  ballots all give the same results.

A2 C2 B0          A          2
A2 B0 C0          A          2
A2 B1 C0          A          2
C2 A1 B0          A          2
A2 C1 B0          A          2

So  A2 B0 C0 dominates A2 B1 C0 which in turn dominates  A2 B2 C0.  
Averaged over the 12 possible opposition ballots  I score
A2 B0 C0 as  20/12 = 1.666,  A2 B1 C0 as  18.5/12 = 1.54166, and  A2 B2 
C0 as  17.5/12 =  1.45833.

I hope my table is still lined up when it arrives.

Chris Benham