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

[Abd ul-Rahman Lomax]

At 02:03 PM 7/23/2007, Chris Benham wrote:


>Abd ul-Rahman Lomax wrote:
>>
>>At 09:48 AM 7/23/2007, Kevin Venzke wrote:
>>
>>>
>>>Are you going to argue that it should make no difference to the voter
>>>how likely it is that he will be able to change the outcome given some
>>>way of voting?
>>>
>>
>>
>>No, for if it is so unlikely as to be impossible, rational voting
>>strategy is to not bother to vote.
>>
><snip>
>>
>>What I am arguing is that a voter should properly assume that many
>>other people will vote as he votes. If the voter knows that
>>assumption is true, then there is always a reasonable chance that the
>>voter's vote will shift the outcome, to the point where, if the voter
>>and those like him are in the majority, the voter's vote, depending
>>on the method, may be *certain* to affect the outcome.
>
>I find these two statements (before and after the cut) a bit contradictory.

If you do, then try to interpret them as having some common truth 
behind them, but possibly not stated well. It's generally a good policy.

I'm saying that it is a rational approach to voting to assume that 
one's voting power is not simply one vote, but is the amalgamated 
power of one and all others who are like one. That is, we should vote 
as if our vote *is* going to shift the outcome. If we are wrong, no 
harm was done. If we are right, then how we vote is important -- and 
strategic voting can backfire.

It is, in fact, also possible to vote in combination with others, 
effectively creating one big high-res Range voter, even if the method 
is Approval....

This is why I'm claiming that dealing with the two-voter case 
provides valuable information about the many-voter case. I just made 
one modification: I eliminated all scenarios that involved a 
three-way tie, because these are so vanishingly rare in the 
large-election case. What I was left with, given the three original 
candidates, was three pairwise races, and, as a zero-knowledge study, 
each is equally likely. There is then a limited set of possible votes 
leaving the voter able to affect the outcome, being, essentially, all 
the possible Range votes in those pairwise elections, all other vote 
totals leaving the voter unable to affect the outcome.



>>The case I'm studying is zero-knowledge, Range 2, i.e., CR3. Three
>>candidates, voter's preference is A>B>C, with midrange sincere rating for B.
>>
>>We now know that in the two-voter case, the optimal vote is sincere.
>
>No we don't. The "optimal vote" in your scenario is to max-rate A 
>and min-rate B and C.

Let's see where Benham gets this result.

>Comparing the two, the "truncated vote" beats the "sincere vote" in 
>3 opposing ballot situations.

This is correct.


>(1)B2 A0 C0: the truncated vote gives an AB tie while the sincere 
>vote elects B
>
>(2) B2 A1 C0: the truncated vote elects A while the sincere vote 
>gives an AB tie.
>
>(3) B2 C1 A0: the truncated vote gives an AB tie while the sincere 
>vote elects B.
>
>In all other cases the two give the same "voter satisfaction" score 
>(by giving the same result in all
>except one:)
>
>B2 C2 A0: the truncated vote gives an ABC tie while the sincere vote elects B.

In spite of what had been said by Ossipoff, and, indeed, by Benham, 
that it was equivalent to max rate B and to min rate B, the election 
function is not symmetrical. Utility is higher if B is min-rated.

I had not looked at the vote of 200 by our voter, because of my 
expectation that it was symmetrical. Before I read Benham's post I 
noted that I should go ahead and do that, but had not done it. I 
would have found that the maximum expectation was for the vote of 
only max-rating A, and min-rating the rest. It's 1.852 compared to 
1.722 expected through voting sincerely.

Remember, this is the two-voter situation, however, a very restricted 
special case. When we go to the many-voter situation, the 
configuration is different. When our Favorite A is moot and the 
pairwise election is between B and C, by voting min for both, we have 
abstained, so the prior results determine the outcome, and C wins 
one-third the time.

For our vote of 220, with many voters, the expected utility is 1.444, 
compared to the sincere 1.481. For the vote of 200, the optimal vote 
in the two-voter case, the expected utility is also 1.444. It's symmetrical!

So with many voters, by this analysis, sincere voting *is* optimal in 
the zero-knowledge case, and, indeed, the configuration is 
symmetrical, it is the same expected outcome if one max rates or min 
rates B. But utility is improved by mid-rating B.

The loss of utility from voting the extremes comes in the cases where 
the pairwise election is between B and C, so, by abstaining from that 
election with a vote of 200, the voter leaves the result to the other 
voters, and, by the initial conditions, it is equally likely for B or 
C to be elected, so the average expected utility is 0.5 in that 
pairwise election; whereas by voting sincerely, the expected utility 
in that election would be, I get, 0.77.

This is symmetrical with the loss involved in abstaining from the 
pairwise election between A and B with a vote of 220.

It's the loss of the third participant in the reduction of the 
election to three pairwise contests that results in this shift of 
strategy from exaggerated to sincere.

(I've state these initial cases abd our voter's vote as ABC, I don't 
know why Benham picked a different order.)

I'm sure there will be some discussion of the approach of only 
looking at non-moot votes; but, relatively speaking, no matter how 
you slice it, utility can't be based on truly moot actions!