Mathematical utility expectation maximization in Approval

[MIKE OSSIPOFF]

> > > > 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.

Ok, then you can't support your claim that the probability
that i & j will be the 2 frontrunners in the election can be
different from the probability that, if there's a tie, it will
be between i & j.

Below, you give us an idea of you arrived at your error, and I'll
answer it when I get to that point in your letter.

>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.

But you're the only one who's talking about conditional &
unconditional probabilities. You're saying that my claim contains
an unspoken assumption about those. I'll reply to that at the
point where you talk about it below.


>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.

No, that's what I meant to say that Pij is. By "the 2 frontrunners",
I mean "the 2 frontrunners in the election's votecount". (as opposed
to merely frontrunners in a pre-election poll). By "the 2 frontrunners", I 
mean "the 2 candidates with the most votes".

>
>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.

I make no claims about what a conditional probability is the same as in 
general. You keep straying from the matter of those particular
definitions, and talking about generalities.

>
> > 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:

I'll try to tell you why you're mistaken. It will more likely
just take the form of telling you what's wrong with your
statement below:

>
>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.

Presumably you agree with both of these statements (If you don't
agree with one of them, show me an example where it isn't true):

1. If A & B are frontrunners, and if there's a 2-way tie for 1st
place, it will be between A & B.

2. If A & B are not frontrunners, and there's a 2-way tie for 1st
place, it won't be between A & B.

Let's summarize that by saying: If & only if A & B are frontrunners,
then it's true that if there's a 2-way tie for 1st place, it will be between 
A & B.

That means that if one of the following statements is true, then
the other is:

1. A & B are the 2 frontrunners.
2. If there's a 2-way tie for 1st place, it's between A & B.

If there's a point at which you've stopped agreeing with me, then
tell me which part you disagree with. If not,
then you agree that if one of the above 2 statements is true then
the other is.

If it's true that if one of the above 2 statements is true then
the oher is, then the probability that one of them is true is
the same as the probability that the other is true. It must be,
since they're either both true or both false.

If there's a claim in the above argument that you disagree with,
then tell me which claim it is.

Obviously you've read something about Bayes' work, and are eager
to apply it, and to show that you've read it. But Professor Bayes
would not be proud of your mistaken attempt to apply his ideas.

By the way, the Approval strategy that I posted here is Weber's
strategy. Weber was the initial proponent of Approval, and he
described the strategy for it. Most likely he was the first to
describe Approval's utility expectation maximization strategy.

So: Weber defined Approval & described its strategy. Other authors,
including Merrill wrote about it too. I've posted it here a number
of times. I posted it a few days ago. Bart then posted an alternative
wording for the definition of Pij. He posted a different wording
of the same definition. His definition defines the same probability
that my definition defines.

And, Richard, if my argument above still didn't convince you, then
I suggest that you post an example in which  the probability that
A & B will be  the 2 frontrunners in the election count is different from 
the probability that it's true that if there's a 2-way tie, it will be 
between A & B.

I'm not asking for more symbolic argument. I'm asking for an
example consisting of actual probabilities, whatever ones that you
consider relevant, like the probability that A & B will be frontrunners, the 
probability that if there's a 2-way tie it will
be betweent them, and the probability that there will be a 2-way tie.
I ask that in your example all the probabilities that you mention
be explicitly numerically stated.

Mike Ossipoff

_________________________________________________________________
Get your FREE download of MSN Explorer at http://explorer.msn.com