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

[Abd ul-Rahman Lomax]

At 02:42 PM 7/23/2007, Stephane Rouillon wrote:
>For more voters, it is not simple. For an optimal strategy, one that 
>would optimize the result form the voter point of view,
>the question depends of the probability of the rest of the ballots. 
>Saying it's a no-information election is not sufficient.
>Are non-extreme position like B2 A1 C1 or B1 A0 C0 equiprobable to others?
>Some would say they have a 0 probility but I doubt even if those 
>cases do not seem optimals (thus logicals) ...
>(Other) voters are not always logicals...

Well, by "zero information," I do mean that the election of three 
candidates must be considered equally likely. It's

>
>
>But for the two-voters case, I have to agree with Chris analysis: 
>the sincere ballot is not the optimal strategy.
>The truncated strategy is even dominant in the sense that there is 
>no case where the results is expected to be worse
>according to the mean social utility (averaging ties).

I came to the same conclusion myself, after looking at the 200 vote, 
whereas I had only looked at the 220 vote, being misled by assertions 
that the situation was symmetrical, one could equally vote for B or 
not for B. That turns out to be true in the case of many voters, but 
not with only two.

Not what I expected. Indeed, I was expecting that with midrange 
utility for the middle candidate, it might be a wash. There are 
aspects of this that I still don't understand....

But, then, the situation changes in the many-voter case, where I 
relied upon an assumption that three-way ties would be so rare as to 
be utterly moot. The vast majority of presenting vote situations 
where the voter's vote can make a difference have only two possible 
winners, we just don't know which two. And, yes, this should be 
symmetrical, and it is. The neglected situations, where there is a 
possibility of shifts involving a third candidate simultaneously, 
cannot significantly change the expected utilities.

>With more voters it would not be the case because truncation could 
>lead to C wins, something that cannot happen with
>two voters as A is always at least tied with C. Thus probabilities 
>would matter.

Right. That's exactly it. C wins sometimes with the Approval case, a 
large drop in utility, with three initial vote patterns causing this. 
With the sincere vote, only one pattern allows C to win, x02. (x 
means that the vote for A was below the threshold of relevance, our 
vote can't bring it up to victory or a tie).

>Could you (both Chris and M. Lomax) provide the other voters ballot 
>repartition you use according to your own no-information understanding?

I will. Chris has so far reported only the two-voter case, which is 
far less interesting, of course, than the many-voter case, though it 
has some surprises in it (such as the lack of symmetry).

My spreadsheet shows, for the two-voter case, all possible vote 
patterns for the unknown voter, being 000 to 222. Those extremes are 
important, for they are cases where the sincere vote with double 
approval results in a tie and a loss of utility over the sincere 
vote, whereas only one approval has no such effect.

Then, with the many-voter case, there are three sets of nine vote 
patterns, the nine patterns being the votes of 0 to 2 for two 
candidates. I listed this by putting a negative number in the 
"absent" candidate's position, so the sum would never reach a tie.

Each of these patterns, with the initial assumptions, would occur 
with equal frequency, so I don't need to worry about relative 
probabilities. The restriction to studying only the effect of 
functional votes drastically simplified the problem.