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

[Abd ul-Rahman Lomax]

At 09:28 AM 7/27/2007, Chris Benham wrote:
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.

With the correction of the missing votes, the spreadsheet now 
contains all the possible vote combinations where the voter has a 
possible gain (or loss) in utility from how the voter acts. Benham's 
analysis seems correct to me, based on the new results, which are, 
indeed, that with the voter utilities of 2, 1, 0, the expected return 
is almost exactly equal for the two possible approval votes. I'm not 
sure about one thing he says, which I have not verified, yet, which 
is what happens with three-way ties. Setting that aside, the two 
possible approval votes also have the same expected utility as the 
sincere range vote.

However, there is something very interesting question that remains. 
Take my spreadsheet, which covers all possible vote combinations 
where the voter's actino has a say -- unless there is some other 
error, which is getting less likely, and restrict the votes to only 
Approval votes. I.e., take only the vote totals which are divisible 
by two. What happens to the outcome utility?

Instead of publishing it now, I'll ask for predictions. Doe it 
increase, stay the same, or decrease. In other words, for the 
*voter's* expected utility with optimal strategy, but balanced 
utilities, which method should the voter rationally prefer, Range 2 
or Approval?

I think I may have written something about this before, but the 
foundation data, the list of votes, was in error. Having corrected 
that, what results can be seen for changing the election from Range 2 
to Approval?



>>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).

That claim may be true. However, I ask a very pertinent question 
above. Which voting method, Approval or Range 2, provides the best 
expected outcome for the voter, whether the voter votes sincerely or 
approval style?

I'm now in a position to answer that definitively, if I haven't made 
any more errors.

No, the table did not line up, and it is presenting the data in a 
different way and does not state the conclusions, the actual expected 
utilities. Since different ways of presenting the data should still 
produce the same result, here are my results:

Both the sincere vote and the approval vote have a utility exactly 
40% above that of not voting, given the initial conditions.

And the remaining question is .... what if we exclude all votes that 
are not divisible by 2, leaving us with only Approval style votes, 
which is exactly equivalent to not allowing any intermediate votes?

Does the utility improve, stay the same, or decrease? And by how much?

We can now answer exactly and easily, if my list of votes and 
consequences is correct. And it is a far shorter list now, only 15 
possible votes. (I had missed combinations before, but many of the 
combinations were equivalent, my selection of preceding vote patterns 
was far more complex than it needed to be, and simplifying it appears 
to have removed the error.)

And we can then look at Range 3, and higher. It's not as difficult as 
I thought it might be. If there are P pairwise elections and the 
election is Range N, the number of combinations is P*(2*N+1), I 
think. It's not exponential, that was an error.

I'll put up the revised spreadsheet and post the URL when I can. And 
it does now have an explanation in it. If there are shortcomings in 
the explanation, I'd like to know. It should be extremely easy to 
understand and review, it's very straightforward now.