Lomax says:
But then I noted that the optimum vote for B in the zero-knowledge case
(where we must assume equal probability of election of all the candidates)
was 50%
I reply:
Thats only because, in Approval it wouldnt make any difference,
expectation-wise, whether or not you vote for B. In Approval, flipping a
coin would be a good way to decide. Thats the only reason why an RV voter
wouldnt be voting suboptimally if s/he gives B middle rating instead of one
of the extremes. In RV, in your example, giving B top rating or bottom
rating would be just as good as giving hir a middle rating. You might
choose middle rating simply because its like flipping a coin in Approval,
and because, in this instance, it isnt suboptimal to rate sincerely.
Let me say it in more general terms: When a candidate is exactly at your
Approval cutoff, then, whether the method is Approval or RV, it makes no
difference what rating you give hir. In Approval the choices are top and
bottom. It doesnt matter which you give, but Id flip a coin.
In RV, when a candidate is at your Approval cutoff, so that it makes no
difference how you rate hir, obviously it doesnt matter whether you give
hir top rating, bottom rating, or anything in between. Of course that means
that you can rate hir sincerely if you want to.
So yes, a candidate exactly at your Approval cutoff is the one exception, in
a public election, to the statement that its always suboptimal to give an
intermediate rating in RV.
And, since it doesnt matter how you rate hir, you could give hir a sincere
rating. I would.
But Warren Schudes statement was correct: When a candidate is at your
Approval cutoff, so that it makes no difference how you rate hir, then there
is no such thing as an optimal (or sub optimal) rating. Therefore its
correct to say that, in public elections with RV, its never optimal to give
an intermediate rating.
Lomax says:
Now, to the core of Schudy's claim: He uses the word "never," which is an
absolute, making it trivial to refute his claim, all I have to show is a
simple counterexample, and I already did that.
I reply:
No. You didnt. As I said, in your example, B is at your Approval cutoff, so
that it makes no difference how you rate B. So theres no such thing as an
optimal way to rate B. You might want to then say that _all_ possible
ratings for B are optimal, but then the word pretty much loses its meaning.
Lomax continues:
And with Range, we are doing the same, counting all the votes, only every
voter can cast up to N votes for each candidate.) Now, in the Approval
example, we have a crisis point. If I am considering which candidate is the
frontrunner, between A and C, on which my choice turns, and as the relative
probability of the election of C increases, there is a point where my vote
for B suddenly flips from disapproval to approval. The discontinuity is a
sign that there is something amiss. If we look instead at this same context
for a Range election, there would be no discontinuity. As the relative
probability of the election of C increased, the optimum rating of B would
increase, until, at some point, the rating of B would become maximum.
I reply:
Thats incorrect. Its exactly the same in RV as in Approval. In your
example, with B at your Approval cutoff, it doesnt matter how you rate B.
But, when you make C more likely to outpoll A than vice-versa, that lowers
the Approval cutoff. Now B is above the Approval cutoff. There is then one
optimal way to vote: Give B a top rating. As soon as the Approval cutoff
goes below where B is, you abruptly must give B a top rating, for
optimality, whether the method is Approval or RV.
This is a good example of why a newcomer should speak not so assertively.
Of course, likewise, if the A were more likely to outpoll C than vice-versa,
raising the Approval cutoff, that would put B below the Approval cutoff, and
it would be necessary to bottom-rate B, for optimality.
Lomax repeats:
If A and C are equally likely to be elected, or we don't know which of them
is equally likely, so we must assume equality, and given the initial
condition that the preference strengths are balanced, it's clear that game
theory would optimize our vote as 50% for B.
I reply:
As I said, with B at your Approval cutoff, there is no such thing as an
optimal way to rate B, because it makes no difference how you rate B. Feel
free to rate hir sincerely, but dont call it optimal.
Mike Ossipoff
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