[Election-Methods] Correction of false statements by Ossipff & Schudy about range voting.

[Abd ul-Rahman Lomax]

At 06:25 PM 7/31/2007, Peter Barath wrote:
> >>"Range voting is a generalisation of approval voting where you can
> >> give each candidate any score
> >>between 0 and 1. Optimal strategies never vote anything other than
> >> 0 or 1, so range voting
> >>complicates ballots and confuses voters for little or no gain."
> >>
> >>Ossipoff: Warren Schude's statement was correct
>
> >--CORRECTION: optimal strategies can vote other than 0 and 1, and
> >voting 0 or 1 can be suboptimal.
>
>Seems you were not so assiduous as to actually read the footnotes
>in Warren Schudy's paper, which in this particular case (footnote
>number 1) reads: "As long as the population is sufficiently big
>and uncertain."

The statement is not correct. With some patterns 
of utilities and probabilities, intermediate 
votes are not suboptimal at all. I'll give an 
obvious counterexample. I'm working on exact 
numbers, but it looks currently like the 
suboptimality of the "sincere" Range vote takes 
place when it is *not* fully sincere, because the 
resolution does not allow an exact sincere 
expression. The error can then make the Range 
vote suboptimal. And this is a work in progress, 
and I'm not aware of any formal study, only 
theories, which can be drastically incorrect unless rigorous.

At the time that Warren Smith wrote the above, he 
had not read, I think, Schudy's paper, he was 
responding to the comment as made, which was made 
here without the qualification.

The study I'm working on is the zero-knowledge 
case. It's obvious that if you have a 
two-candidate election, your sincere vote is 
Approval style, even if the election method 
allows more resolution, and the same is 
effectively the case when there are more 
candidates, but you judge them moot, even if they include your favorite.

With a large number of voters in the simple Range 
2 election, the sincere vote we postulated of 2, 
1, 0 had the same expected utility as 2, 2, 0 or 
2, 0, 0. But to come to this result we had to 
neglect three-way ties; the greatly increased 
possibility of this is the reason that the 
optimal Approval vote is not balanced in the small election case.

If the utility is the same, it cannot be said 
that voting sincerely is "suboptimal." Rather, 
what can be said about this case is that either 
strategy is "optimal," there is no cost to voting 
sincerely. And there is a cost in regret to 
having voted insincerely. If you vote sincerely 
and lose, you will think, "I did my best." If you 
exaggerate, and lose because of that, you will be 
kicking yourself. "Why didn't I just tell it like 
it was?" Hard to quantify this, but it is a 
reasonable assumption that there is a value to 
sincerity and accuracy, independent from actual 
strategy, such that if strategies are equally 
effective, the sincere one is better.

>"Suppose that the method is 0-10 RV...
>Now, suppose that you consider the points that you're awarding
>one-at-a-time, as if it were a series of 10 Approval elections...
>We're assuming that it's a public election, so that there are so many
>voters
>that your own votes have no significant effect on the probabilities.
>Your Approval strategy is based on two things: The candidates' utility
>to
>you, and the probabilities that you estimate...
>Your utilities don't change during that series of 10 Approval
>elections that
>you vote. The probability estimates don't change either...
>If you give to a candidate any points at all, you give hir 10 points."

Okay, counterexample to the principle. A group of 
people are going to place bets on the number of 
beans in a jar, to win a prize. They look at the 
jar, and use some method to estimate the number. 
If they guess it exactly, they get a much bigger 
prize than if they are merely the closest guess. 
Now, you are ten of these people, say you are all 
one family, there is no competition between you. 
Any one of you winning the prize is the same as 
any other one, your interests coincides. How 
should you guess? If you only had one chance, you 
would use your estimate, best shot. But if you 
have ten chances, you would spread them.

Further, there is a specific problem here. In 
order to prove that the optimal strategy is to 
vote Approval style, Ossipoff is *assuming* that 
the optimal strategy is approval style.... He 
merely does not give the individual voter a chance to vote any other way.

Now, if I look only at the situations where the 
voter can influence the outcome, in a Range 2 
3-candidate election with sincere utilities of 2, 
1, 0, zero knowledge, the sincere vote and the 
two possible Approval Votes all have the same 
expectation, 40% over not voting. If I do the 
same, making the election Approval, the expected 
utility is 33% over not voting.

There is an assumption in Ossipoff's "proof" that 
a series of decisions, optimized, must be the 
same as a set of individual decisions multiplied by the number in the series.

Suppose we were trying to make a course 
correction. We can do it with ten small moves or 
one large one. What is the optimum move to make? 
If we look at the small correction, we will see 
that it is binary, perhaps. We make a full move 
to swing our direction in one way. If we make a 
series of these identically, however, we 
overcorrect. If we are going to make a series, we 
will make some in one direction, and some in the 
other, with some net effect. If we do it all at 
once, we would want to use some intermediate force.

So I'm highly suspicious of the "proof," and 
attaching words like "linear programming" to it 
doesn't help, unless the specific principle and its application is examined.

>is informal, but fully correct. (Which doesn't mean I agree with
>him in everything else.) And those who refer to linear
>programming essentially say the same thing. If the field is big
>enough, a small part of it may be considered as having linear
>probability-distribution (or whatever) functions, so the optimum
>lies somewhere on the border. So if you started to go to a
>direction you have to stay on that course.

But this clearly doesn't work in control systems 
where some precision is involved.

>So, they say if the number of voters goes toward infinity, the
>probability of a case where Approval-style voting is suboptimal
>goes toward zero.

Well, this is a different case than saying, 
"Optimal strategies never vote anything other 
than 0 or 1, so range voting complicates ballots 
and confuses voters for little or no gain."

The first part is false, as stated. It can be 
equally optimal to vote "sincerely." (The 
language is loaded, I do not define an Approval 
style vote as "insincere," as long as preferences are not reversed.")

Secondly, the conclusion does not follow. It 
presumes that the only function of elections is 
to determine the winner. The performance of 
losers is also of great interest and effect. 
Intermediate ratings, we think, have a value in 
expressing much more accurately what the gap was 
between winner and loser. *How* much would a 
candidate have had to rise in rating to win*? If 
people vote Approval style, which they properly 
are free to do, this information is lost. There 
is a value to the information. Further, the zero 
knowledge case is a special one. I used it for my 
study because it is simpler. But it is not 
realistic. Most people have some idea who is 
likely to win the election, and especially they 
usually know if it is a certain pair, and sometimes there are three.

Approval style votes can make very much sense if 
you know a pairwise contest is the real one. It 
is obvious that if this knowledge is strong, your 
optimal vote is Approval style. For those two candidates. What about others?

To claim that there is no value to these other 
votes is utterly without basis. It is an 
*opinion* of the voter, not a fact, that this is 
the pairwise contest. And voting intermediate 
votes for candidates that the voter does not 
expect to win can have no negative impact on the 
expected utility for that voter.

And it has another value: individual voting 
strategy does not determine the overall benefit 
of the election to society. It may be true that 
voting Approval style is optimal strategy for the 
individual, but it does not follow that this 
optimizes the value to society of the election. 
We consider public methods and procedures based 
on overall public benefit, and it is clear that 
voting Approval style reduces the information 
available for doing this. It is like having a 
series of sensors for light. We could have analog 
sensors that give us a range of values for the 
light, or binary ones that tell us that the light 
is above a certain threshhold. Which one can give 
us better vision for making decisions? If we have 
the analog ones, and we want to, we can convert 
them to binary. But we can't convert the binary 
to analog, particularly if there is a systemic 
bias. Sometimes we can average together a lot of 
binary values and get a refined value, but as it 
was described above, if you take each example and 
make a determination on it, which way should it 
go, you can bias the outcome. When you are going 
to average together a lot of binary values, it 
helps to introduce noise, which causes the binary 
information to become far more accurate when 
averaged together. It is as if you are making a 
lot of binary measurements with different cutoff levels.

And this may explain why the expected outcome is 
better, under the conditions stated, for Approval 
Voting strategy, if the election is Range. Maybe. 
At this point this is a conjecture.

(To really know that the expected outcome is 
better, I need to determine the absolute utilities, I haven't done that yet.)



>The counterexamples? most of them have extremally small number
>of votes. And even which does not so, uses the less-then infinite,
>non-linear attributes or simply wrong.
>
>http://beyondpolitics.org/Range2Utility.htm
>
>when shifts from Range(0,1,2) to Approval, calculates like those
>"odd number" cases would simply vanish. And vanishing some good
>vote value, the average worsens when Approval becomes the method.
>But those cases don't vanish. The logical statistic assumption is
>that they evenly distribute themselves among the neighboring cases.
>And some of previously irrelevant cases become relevant cases, so
>the probability of decisive vote rises. This rise exactly compensates
>for the loss of utility rise. As for
>
>http://rangevoting.org/RVstrat5.html
>
>it's more reality, but only by using the three-candidate-tie event,
>attributed with a T probability. If T=0, the classic case happens:
>giving the in-between B candidate max or min is optimal, or all
>the same. And if the number of voters goes toward infinity, T
>goes toward zero.
>
>So, please, don't infer "which graduation is best for range voting"
>type statements from these calculations. We can go back to the
>consensus (used even in your simulations) that _essentially_
>a strategic Range vote is an Approval vote.
>
>Which doesn't decide which one is better. Valid arguments exists
>on both sides. Range voters can choose from more possibilities, but
>is this choice a pleasant one? I can be a "sucker" or a "cheater",
>maybe I would be more glad without it.
>
>I think they are so close that even their fans can be close and
>fight side by side. I'm looking for the future when TV-personalities
>as well as people at the coffee machine dispute about whether
>Approval or Range is better method.
>
>Peter Barath
>
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