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

[Abd ul-Rahman Lomax]

This is actually getting interesting. In short, I've discovered that, 
in the many-voter case studied, sincere voting in Range 2 has 
slightly higher utility for the voter than Approval. Further, 
reducing the method from Range 2 to Approval (Range 1), lowers the 
voter's expected utility. The simulations referred to below are 
probably Range 999. This leads to an inference that there is a N such 
that Range N is not only better than Approval, as a method and/or as 
a strategy, it is better than higher-resolution range, it is an 
optimal resolution. This is, to my knowledge, a new finding, and unexpected.

So it becomes important to ferret out my errors, if any remain!

At 12:55 AM 7/25/2007, Chris Benham wrote:
>Abd ul-Rahman Lomax wrote:
>>At 09:51 AM 7/24/2007, Kevin Venzke wrote:
>>>
>>>Warren implemented his own version afterwards; I suggest his results if
>>>you're really interested. 
>>><http://rangevoting.org/RVstrat3.html>http://rangevoting.org/RVstrat3.html
>>>
>>Yes, I'm familiar with the page.
>I had seen it but forgotten about it.  Apparently Abd doesn't 
>understand this table with which he is so
>"familiar".

That's correct. I can be familiar with something and not understand 
every aspect of it. In particular, that page does not state what the 
numbers, the results for each case, mean. They are stated as 
"improvement over not voting," but the scale is not stated. I'd guess 
that it is relative to the utility for not voting at all, but then 
other aspects of the report remain mysterious.

>>These simulations are looking that the return to the voter from
>>various strategies, a direct answer to the issue posed by those who
>>claim that voting Approval style is optimal. It turns out that it's
>>optimal in some limited cases:
>
>No.  Only when there are no other voters does any other strategy do 
>as well. Of the ten strategies
>considered in the table, seven of them are versions of  "voting 
>Approval style".

Yes. The no-other-voter case is a special case, but I think I've 
stumbled onto something. When there are many voters, there are still 
cases which are equivalent to the no-other-voter case, and which also 
cover every possible vote when there is only one other voter. These 
cases are rare with many voters, so they do not affect the 
simulations within the precision involved, but they are the only 
cases where the voter's vote actually counts, so they loom large.

But there are only a few vote combinations where the voter's vote 
actually matters. And in the example shown, Range 2 -- which is not 
studied in Warren's simulations, he is working with higher resolution 
range - there are vote patterns where the Approval strategy produces 
a worse result than the sincere strategy. (I don't like using that 
word, it's loaded; I'm just using it to mean that the voter votes the 
straight utility as best expressed within the limits of the method, 
and in this case the method, Range 2, expresses it exactly.)

There are 27 vote patterns where the voter's vote counts, neglecting 
the patterns that require the race to be near a three-way tie, near 
enough for the voter to be able to affect the win of more than two 
candidate, such as an exact three-way tie. In a very large election, 
this probability is so low compared to the probability of a two-way 
tie that we can neglect it. (There is more to be said on this, I suspect.)

Of the 27 vote patterns, 3 result in loss of utility from the sincere 
vote compared to approval. These occur in the cases where the sincere 
vote causes or eliminates a tie. In each of these cases there is a 
loss of 0.5 in utility (using the sincere Range scores as the measure 
of utility, and they were defined as accurate). Total loss: 1.5.

However, there are 5 other vote patterns where the voter gains 
utility compared to approval. These are all cases where the voter's 
favorite cannot win, so the vote for the midrange candidate becomes 
important, and under Approval, with midrange being rated zero, a loss 
for B by one point becomes a tie, average gain of 0.5, and a tie 
becomes a victory; again, the average gain is 0.5. Total gain: 2.5.

Net gain: 1.0. 40/27 is the predicted average utility with the 
sincere vote and 39/27 is the predicted average utility with the approval vote.

One point difference vanishes in the noise when we are looking at 
very large elections and are considering absolute utility over all 
possible prior votes. Obviously, one point should still shift the 
utility positively, but the utilities have not been stated with 
sufficient precision to see it, and it's been thought that such a low 
change in utility would be of no significance at all, which is why it 
is said that it doesn't matter.

However, the reasoning is false. Suppose I were considering taking a 
divot of grass from the public commons. The divot is tiny, the effect 
on the public at large so small that it would not have a value of 
even one cent. Should we allow people to steal a penny?

Problem is, if everyone does it, and there are billions of people.

As many have noting, the immediate value of voting is about nil. If I 
can't vote for some reason, it would not be rational for me to be 
seriously upset about it. It did *almost* no harm. There has never 
been a Presidential election, I suspect, where one vote changed the outcome.

However, if I deliberately don't vote, I must consider the 
consequences of *thinking* like this. If many people think this way, 
it can shift the outcome in a direction I think harmful. So I vote.

Now, in considering *how* to vote, I only need consider the cases 
where my vote counts. That *vastly* simplifies the problem! I don't 
need to worry about the vast number of vote combinations where my 
vote is moot. How I vote in those cases has no effect on the outcome.

I only need to consider the few cases where my vote counts. And if 
everyone does this, we all benefit! (That is, more benefit than 
lose). This isn't "altruistic," it is, indeed, selfish, the only 
"altruistic" part of it is that I vote at all! By voting, I'm 
performing a public service, at substantial cost, my individual cost 
is considerable, while my gain is nearly zero.

I really think this is a new kind of vote analysis, here, though I 
think I heard some hints that it has been mentioned by others.

If Warren were to take his simulations and eliminate all the cases 
where the voter's vote is moot (that is, no possible vote by the 
voter could improve the outcome for the voter), he would have much 
larger utilities to report, they would be commensurable.

As they say, the devil is in the details.

1 point difference is, in Range 2, one-half vote. This improvement of 
utility, 1/27 in value compared to the maximum possible utility for 
this voter; if we reference it to the value of not voting, 1.0, then 
we estimate the value of voting at 48.15% for sincere and 44.44% for 
approval. (This is 40/27 or 39/27 compared to 1.0.)

In the many-voter case, the utility is the same whether the voter 
votes 0 or 2 for the middle candidate, as others stated. But they 
didn't state it, I will carp, as being for all elections, they did 
not specify that it was many-voter case. Now, it should have been 
obvious, but, unfortunately, something isn't obvious to me until the 
reason occurs to me.....

So there are a series remaining problems for me in studying voting 
strategy in the 3-candidate case. Warren's simulations, reported 
above, are for high-resolution Range. If the best strategy is 
Approval, why both providing people with an inferior strategy? There 
is an answer for this, but it is also a very good question. These 
simulations, however, compare high-resolution Range, (vote in 0-1 
range, precision not stated, but probably substantially higher than 
is being considered for public elections. Warren prefers no limit on 
range resolution, but he also would have wanted to speed up the 
calculations, I don't know what choice he made.

But what I'm studying is the effect, it turns out, of adding only one 
new rating level. What I find is that Range 2 is *slightly* better 
than Approval. By the way, the results get even worse if Approval is 
the *method*, rather than merely the strategy. Turns out, it appears, 
that *enforcing* an approval strategy does harm the voter. (See the 
spreadsheet, the Approval Method case is pulled out on the ManyVoter tabs.

So, even if the analysis on the page cited is correct (that is, even 
if the precision *is* sufficent), what is being shown is that 
Approval is better than, perhaps, Range 999.

But if Range 2 is better than Approval, we may speculate that there 
is an N such that Range N is the best method. And it should now be 
relatively easy to find that number.

Don't you think that's interesting?