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

[Abd ul-Rahman Lomax]

At 01:12 AM 7/24/2007, Chris Benham wrote:

>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 had been told, explicitly, that there was no strategic difference 
between the vote of 210 and 200, that Candidate B, was at the 
"approval cutoff," exactly. Since this accorded with a certain -- 
apparently incorrect -- intuition, I didn't check it, though I knew 
it was a detail that required follow-up. Benham, ironically, was one 
who confirmed such a statement. In fact, if there is a strategic 
advantage to zero-rating the midrange candidate, in small elections, 
the Approval Cutoff would not be 50%, it would be some higher number. 
Slightly higher. With large elections, *very* slightly higher. This 
is only with Approval.

I should study the effect of shifting the original utilities up on 
the middle candidate. Theoretically, there is an actual Approval 
Cutoff, that is not 50% even though the utility is at 50%.

But it's a case of academic interest only. In real elections, I will 
note, there is effectively a very large Range resolution (created by 
blocks voting in a coordinated way). Some have noted this, in fact, 
by suggesting that Approval can be improved if voters use coordinated 
voting patterns to increase resolution, thus converting Approval, in 
effect, to Range.

Repeating, in large elections, sincere votes appear to have a slight 
edge over voting Approval Style, for the midrange candidate. I have 
not studied the effect of higher resolution Range and other than 
midrange candidates, and I expect to see larger effects there.


>>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.

Right. Because our half votes full rating for A. That's the 
protection. That protection disappears in the *last voter at the end 
of a 'voter can turn the outcome' election* case, because A can be, 
possibly, excluded. And that case is the only case where the voter's 
vote actually matters!


>>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

There are, in the two-voter case, 27 possible ballots, being all the 
base-3 numbers from 000 to 222. All are considered equally likely, 
which simplifies the calculations by assigning the same probability 
to each. In the many-voter, ballot-could-shift-result election,  the 
virtual ballots of 000 and 222 are, in fact and in real elections 
equally probable with the others. At least that's how I see it, I can 
easily make mistakes. I depend on feedback, or further thought, to find them.

There is no reason to assume that the voter fully normalizes. In the 
many-voter case, our virtual voter is voting as the resultant sum of 
many different voting patterns, and 000 is just as probable as, say, 
010 or 002, or even 222. To list all the equally possible ballots, 
each one having the same probability, we must include those.

This explains some of the deviation between Chris's results and mine. 
I was explicit that I considered *all* the possible first-voter 
votes, and people can and do cast blank or vote-for-all ballots, they 
turn up in real elections.

The blank ballot, for example, 000 is specially important. It gives 
all power to the subject voter, and this results in an imbalance 
between voting 220 and 200; in the former case it creates -- it *is* 
-- a tie between A and B and so the utility is average 1.5, and this 
is also true of 222. These two cases are ones where, contrary to what 
was asserted before, voting Approval style results in a loss of utility.

In the many voter situation, 222 is not possible in practice, because 
it would represent an exact three-way tie, likewise 000 is not 
possible for the same reason. But what we have in the many voter 
situation is x00, 0x0, and 00x, which I express in the table as -3, 
0, 0, etc. The exact negative number doesn't matter, but the 
resulting vote when our voter's vote is added should not reach 0, or 
it could participate in a tie. Note that a tie between A, B, and C 
has an expectation of 1.0, whereas a tie between B and C only has an 
expectation of 0.5.

>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.

Well, I tried to make sense out of it, but I found it *far* harder to 
read than the notation I used. The spreadsheet is at 
http://beyondpolitics.org/OptimalRangeVote.xls

In my notation, a vote of 210 is A=2, B=1, C=0. This notation allows 
each vote to be expressed as a three-digit base-3 number, and then 
all the possible votes can be listed simply by incrementing. What you 
did was to change the order depending on preference, I'd suggest that 
is what makes it really hard to read. I know, I know, election 
methods experts are used to A>B>C, and to ranked ballots that list 
preferences in order (though, in fact, the actual ballots have the 
candidates in fixed sequence, or sometimes in jumbled sequence to 
avoid the first-listed effect. But it's much simpler to see at a 
glance, if the votes are just treated as numbers with a fixed meaning 
for each place.

I keep finding interesting little details that fall out of this 
study.... more coming.