[Election-Methods] Intermediate rating is never optimal

[Abd ul-Rahman Lomax]

At 07:03 AM 7/26/2007, Michael Ossipoff wrote:
>Lomax replies:
>
>In what I wrote, B was not at the voters "approval cutoff."
>
>I reply:
>
>Wrong. You said that the voter had no information about whom the pair-tie
>would be between if there were one. That's what is called a zero-information
>election. In such an election, for the expectation-maximizing voter, one's
>Approval cutoff is at the mean of all the candidates. In your example, B is
>at that mean. In your example, B is exactly at the Approval cutoff.
>
>And, as I said, that's why it doesn't matter how the voter rates B.

If Ossipoff had followed the ensuing discussion, he would know that 
what he just repeated in the face of my denial is incorrect.

A candidate being exactly at the approval cutoff would indicate that 
either vote produces the same expected return. I initially made this 
assumption, with Ossipoff. However, it only turns out to be true, the 
analysis, in the many voter election. And it still is not *exactly* 
true. The approval cutoff, for equally balanced utilities, is higher 
than midrange, so B is *below* the approval cutoff, so the vote of 
200 has higher utility than 220. The approval cutoff is higher than 
midrange by an amount which declines with increasing voters, because 
three-way close ties affect that cutoff, and we only force the 
utilities to be equal by eliminating them from consideration. They 
are vanishingly rare, however.

On the other hand, Ossipoff assumes that "public elections" are many 
voters. There are public elections with only a handful of voters. For 
what he is saying to be relatively correct we would have to be 
talking about "large public elections," not just public elections.

Voting methods are of interest and apply to small elections as well 
as large ones, so general discussions of voting theory should 
consider all cases. It's even important to consider a single voter 
election. They happen, though very, very rarely.

>Lomax continues:
>
>This is the situation described:
>
>The voter prefers A>B>C, with the preference strength between A and B
>being the same as the strength between B and C.
>
>I reply:
>
>No, that's not the situation described. You also said that the voter had no
>information about who would be in the pair-tie if there were one.

Ossipoff is remembering incorrectly. I've written a lot about this 
test. The initial conditions, however, were described repeatedly as 
what he quoted. I *defined* this condition, repeatedly, so it is 
beyond me how Ossipoff missed it. But he generates reams of these 
kinds of misunderstandings. And thus responding to him, if I address 
them, generates lots of words, even for me.

The initial condition was as described, it is the voter's utilities 
and has nothing to do with the other voters. Somehow Ossipoff has 
confused this with the zero-knowledge condition, which is that the 
voter has no knowledge of the preferences of the other voters.

As a result -- and as a condition -- we can assume that all 
preferences are equally likely. This does not necessarily match 
real-world conditions, it's a special condition. However, it makes 
the study enormously simpler. And it turns out that preference 
frequencies really aren't that important; they actually vanish 
because we are only looking at very delicate balance points, in large 
elections, and even a handful of voters voting wonky preferences, 
ones that we would not expect, like Nader>Bush>Gore, will fill out 
those possibilities and thus make the votes spread equally.

I'm describing my intuitions about the math. I'm not a mathematician, 
and there are very many ways to make mistakes about this. I *think* 
I've done it right. So far, nobody has pointed out a mistake, after 
the first example, the 2-voter case, where I reported results based 
on an error, an assumption that the vote of 220 would have the same 
utility as 200. It doesn't, contrary to what Ossipoff continues to 
claim. However, the utility of 220 and 200 approach each other as the 
number of voters increases. If we can neglect three-way ties, as I do 
in the many-voter study, the utility of 200 and 220 is balanced.

Nevertheless the nonzero probability of three way ties means that the 
approval cutoff is *not* at exact midrange in this election. It is 
above midrange, by an amount that declines with the number of voters. 
Meaning that the vote of 200 *always* has better utility than 220.

Interesting result, eh? Quite counter-intuitive, to me. But those are 
the numbers, and it looks solid to me.

>Lomax continues:
>
>There is nothing here about Approval cutoff, there is nothing that
>says that the voter does or does not "approve" of *any* candidate.
>
>I reply:
>
>As I said, for the voter maximizing hir expectation, in a 0-info election
>such as you describe, the strategy is to vote for the above-mean candidates.

Well, it turns out, in the many-voter case, that this is the best 
*approval* strategy, but only by a tiny amount. You could also vote 
for the candidate at the mean, with the tiniest of losses of utility. 
(The difference in utility is approximately 1/V, where V is the 
number of voters.

And this utility, of course, is amplified over what is normally 
stated because it only looks at votes when the voter is in a position 
to influence the outcome. Otherwise it really doesn't matter how the 
voter votes.... as far as determining the winner. There are other 
issues which will shift utilities in the real world. For example, 
people value sincerity. They would rather be sincere, if it does not 
cost them too much. This means that there is some nonzero utility 
assigned simply to being sincere!

But we are not considering that here.

>  In your example, B is at the mean, and so B is exactly at the Approval
>cutoff for the expectation-maximizing voter.

No, it turns out that the approval cutoff is slightly above the mean, 
which is why the candidate at the mean should receive, optimally, a zero vote.

Look, I'm being exact here, as exact as I can. I may make mistakes, 
but I'm not blowing smoke.

B with a midrange utility is *not* at "exactly" the Approval cutoff. 
He is below the Approval cutoff. If this were very high resolution 
Range, the Approval cutoff would be visible above midrange, such that 
a candidate might actually be above the mean and yet optimal Approval 
strategy might be to vote against the candidate.

>Lomax continues:
>
>Ossipoff confused the fact that the candidate was intermediate
>between A and C in sincere rating, i.e., being midrange, with being
>"at your Approval cutoff."
>
>
>I reply:
>
>In the 0-info situation that you described, for the expectation-maximizing
>voter, that mean position is indeed the Approval cutoff.

It is being argued that there is some "Approval cutoff" with a 
definite meaning. However, there are alternate strategies for 
determining Approval cutoff, and Ossipoff is referring to a form of 
Mean-based Approval strategy. Usually the definitions I've seen for 
this state that one should vote for candidates above the Approval 
cutoff, meaning above the mean. But it is clear from what I've found 
that the *optimal* Approval cutoff is above the mean, though only by 
an amount that declines with increasing vote count. It is very 
significantly above it when the election only has two voters.

Apparently Ossipoff hasn't noticed that result. In other words, he 
and I agreed initially, I got slapped for making an unwarranted 
assumption that the two cases would have equal utility (votes of 220 
and 200), when, in fact, that is only approximately true in the 
many-voter case. It is *never* exactly true, contrary to Ossipoff's 
use of the term, which I believe has a clear meaning. N and N + 1/M 
are never exactly equal for any positive value of M, no matter how large.

>Lomax continues:
>
>And, quite clearly, it *does* matter how
>you rate B in some scenarios
>
>I reply:
>
>...but not in your scenario. It should be obvious that I was referring to
>your example. Because I said so.

No, to the contrary, it matters in my first scenario, two voters. It 
does not matter in my later scenario, many voters. There were two 
examples before us, and if Ossipoff can misinterpret what I write, 
it's guaranteed that he will. It matters how one rates B in some 
scenarios. Not in others. And "my example" was two different 
examples: 2 voters and many voters.

>Please, don't anyone send me copies of Lomax's postings, because I don't
>want to read his rebuttals, because I don't intend to answer them, for the
>reasons that I've given.

This list sets up automatic copies of replies that are themselves 
copied. Because of Ossipoff's request, I'm taking him off the routine 
copies that are sent to the authors of posts when I reply to them. I 
don't do that deliberately, it is automatic, and apparently that is 
true for many others.

My suggestion to Ossipoff, though he may not read this, is that he 
simply not read email that he doesn't want. Deleting it is faster 
than complaining.

And I don't give a fig if Ossipoff answers, indeed, I prefer that he 
doesn't. However, I'd, in fact, appreciate it very much if he could 
find a flaw in the study. He's being presented, numerous times now, 
with a clear counterexample to what he is claiming, and apparently he 
isn't interested in that. He's interested, instead, in belaboring the 
arguments. Which is what he's famous for. I used to have a policy of 
not responding to Ossipoff, because it was such a huge time-waster.

However, this particular sequence, I'm learning new things, and 
having an irritating person making off-the-wall claims actually 
helps, it makes me think about the issues from many different perspectives.

>But Chris's reply to this Lomax posting was well-expressed, and my own reply
>mostly just repeats Chris's answers.

Ossipoff has completely missed that Chris is the one that pointed out 
the error that Ossipoff repeats above, the opinion that the utilities 
are balanced for the votes 220 and 200 in the subject case. That 
opinion is only *approximately* true in the many voter case, and the 
first study was explicitly two voters, yet, writing about that case 
-- I had not yet published the many voter data, I think -- Ossipoff 
started making the balanced utility claim.

The utilities are *never* "exactly* balanced, which is why it is 
always optimal to vote against the middle utility candidate, no 
matter how many voters. The approval cutoff is *above* midrange, by a 
nonzero amount, such that it is possible that one could have a 
utility above the mean and still have optimal Approval strategy be to 
vote against the candidate.

These are results falling out from an exhaustive mathematical study 
of the utilities, for all relevant vote combinations. They are not 
opinions, they are reports, generally. (I tend to tack on some 
opinions, but....)

It's math, and apparently Ossipoff thinks he can argue with math. The 
only *real* argument would be a discovery of error in the math, not 
reams of theoretical arguments which can be far fuzzier than we 
sometimes realize.