Mathematical utility expectation maximization in Approval

[Richard Moore]

MIKE OSSIPOFF wrote:

> >> > Pij is the probability that i & j will be the 2 frontrunners.
> >
> >
> >Slight correction/clarification:  The precise meaning of Pij is usually
> >taken to mean the probability, given a tie exists for first place, that
> >i and j will be involved in that tie.
>
> 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. It's easy to think they'd be proportional, but keep
in mind Bayes' Theorem for conditional probabilities.

Ignoring three-way (and higher) ties...

Let Pt|ij be the probability that there is a tie given that i and j are the
front runners.
Let Pij be the probability that i and j are front runners (Mike's
definition).
Let Ptij be the probability of a tie between i and j.
Let Pij|t be the probability that, given a tie, i and j are the front
runners (Bart's).
Let Pt be the probability of any tie.

Two conditional probability equations can be written:

Ptij = Pt|ij * Pij
Ptij = Pij|t * Pt

Setting the two expressions for Ptij equal to each other, and rearranging
the terms, you get:

Pij = (Pt * Pij|t) / Pt|ij

or

Mike = Bart * (Pt / Pt|ij)

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.

While Mike has matched the correct definition to the notation he chose,
I think Bart's conditional probability is better for determining strategy.
The probability that A and B are front runners isn't the probability that
your vote will decide between them. I suspect Mike's strategy as
modified by Bart is mathematically equivalent to mine, which I defined
in terms of differential probabilities. I didn't give a formal equation in
my earlier post but what I have in mind is something like

Ui = sum over j of (dPj|i * uj)

where Ui is the incremental utility of voting for i, uj is the utility of
candidate j, and dPj|i is the change in probability of j winning caused
by my vote for i (for monotonic voting methods, dPj|i will be
non-negative when i = j).

For zero-info voting with N candidates, we can add that

dPj|i = -dPi|i / (N-1)

for all j not equal to i, which simplifies the solution, as I showed in
my earlier post.

Hopefully my differential method and Bart's method will prove
equivalent. I wonder which is harder to calculate for the non-zero
info case, Pt|ij or dPj|i?

 -- Richard