Mathematical utility expectation maximization in Approval

[Richard Moore]

MIKE OSSIPOFF wrote:

> I'd said:
>
> > > Different wording, same thing. If i & j are the 2 highest votegetters,
> > > and there's a tie for 1st place, then it's between them. If they
> > > aren't the 2 highest votegeters, and there's a tie for 1st place, then
> > > it isn't between them.
> >
> >They aren't the same.
>
> (Ignoring ties with more than 2 candidates)
>
> Would you post an example in which 2 candidates are the 2 biggest
> votegetters in the election, and there's a tie for 1st place, and
> those 2 candidates aren't the ones that are tied for 1st place?

No.

> Or would you post an example in which 2 candidates are not the 2
> biggests votegetters in the election, and there's a tie for 1st
> place, and those 2 candidates are the ones that are tied for 1st
> place?

No.

> >It's easy to think they'd be proportional
>
> ...that what's proportional? I'm saying that iff 2 candidates are
> the 2 biggest votegetters in the election, then, if there's a 2-
> candidate tie, it's between those 2 candidates.

Nobody has tried to contradict that if there's a two-way tie for
first place it's between the two front-runners. In fact I think we
are all in agreement about that.

What's disputed here is the difference between conditional and
unconditional probabilities.

> The fact that you thought I was saying something was proportional
> suggests that you misunderstood what I said. My statement in the
> paragraph before this one is what I'm saying.

You were saying (regarding Bart's post) that he was using different
wording but describing the same thing. I'm saying that Bart was
not describing the same thing at all. Sorry if the proportionality
statement was misleading; I was merely describing my own initial
reaction to the difference between what Bart was saying and what
you were saying. After some thought I realized that not only were
they not equal, they weren't even related by a constant of
proportionality (they couldn't be -- or you would have a set of
probabilities that don't add up to one). So I was actually correcting
myself. Feel free to disregard the "proportional" statements, because
I think that's where we got off on the wrong foot.

> >but keep
> >in mind Bayes' Theorem for conditional probabilities.
>
> Ok, so then you're saying that Bayes' Theorem means that
> 2 candidates can be the biggest votegetters in an election, and
> not be the ones who are in a tie, when there's a tie? Or that 2
> candidates can not be the biggest votegetters in an election, and
> still be the two who are tied when there's a tie.

No. I suspect you didn't read what I said very carefully at all.

> Instead of wading through the definitions and arguments that follow,
> for the purpose of finding the part that expresses the misunderstanding,
> I'll just repeat my request for the examples that I asked you for
> above.

And again I'll deny that request. It just isn't relevant since I never
claimed anything of the sort.

> [definitions & arguments deleted]
> >Since Pt is constant for all candidate pairs, and Pij|t is different for
> >each
> >pair of candidates, then the relationship between Mike's and Bart's
> >definition isn't a proportional one.
>
> I didn't say that the relationship between Bart's definition and my
> definition was a proportional one. I said that both definitions
> define the same probability. They're 2 wordings of the same
> definition.

If they are the same then they are proportional.

It's possible that you only misstated when you said, in the message
forwarded by Martin Harper,

> Pij is the probability that i & j will be the 2 frontrunners.

Bart modified this to mean the probability that, IF A TIE EXISTS,
i & j will be the 2 frontrunners. Then you said these two definitions
are the same. But a conditional probability is not, in general, the
same as the probability of the outcome. Bayes Theorem gives the
relationship between the two.

> Both Pij definitions are wordings for the same probabilities, Richard.
> Unless you can post the examples that I asked for above.

If they are meant to be the same then one of them was misstated. If
you do not believe you misstated the definition, can you then prove
the definitions as stated are the same? I would like to see a proof
that:

The probability of two candidates A and B being the front runners
equals the conditional probability that, given a two-way tie for first
place exists, it is between A and B.

This is not the same as saying that:

If A and B are front runners, and a two-way tie for first place
exists, it is between A and B.

The latter statement is incontrovertible, but it says nothing about
probabilities.

> Perhaps you should have a formal equation, a clearly-stated procedure
> such as the one that I posted, and perhaps you should have a better
> idea of what your procedure is, before finding fault with my
> definitions. If you don't know what your procedure is, are you sure
> that you couldn't be a little premature & presumptuous when finding
> fault with my procedure and its definitions?

I'm only challenging one of the definitions. The rest of the procedure is
valid. And I already know what my procedure is, but I'm not using my
procedure to critique the other; in fact I was making a conjecture that
they are equivalent (provided we get the terms defined correctly), i.e.,
that they will produce the same results under the same circumstances.

If you like I'll expand on my procedure, though I thought my post
already described the basic mathematics. I won't attempt the math
of deriving the differential probabilities needed in step 2, which is
an academic exercise similar to deriving the probabilities in Merrill's
procedure. For the ZI strategy it's simple as I pointed out. Someday
I may try to derive it for a strategy based on a poll in which the only
source of error is assumed to be sampling error, but today is not that
day. We don't need that detail here, just as we can discuss Merrill's
procedure without an equation for Pij. Here is the step-by-step
procedure:

1. Always vote for your favorite.
2. For the candidate with the next-highest utility, determine the
(positive) change in probability of winning that a vote for that
candidate will produce, and the (negative) changes in probability
for all other candidates that such a vote will produce.
3. Calculate the dot product of these probabilities with the
utilities of the candidates.
4. If the result of 3 is positive, vote for the candidate in question
in addition to any previously selected candidates, then go back
to step 2. If the result of 3 is zero or negative, then stop and do
not include this candidate in your vote.

 -- Richard