[EM] Usual probability definition not for elections?

[MIKE OSSIPOFF]

I've just posted 2 not-valid definitions of probability. That
doesn't mean that I understand it less well than other members of
this list, however. I'm familiar with the usual frequency definition of 
probability, which goes like this:

If a trial is repeated many times, and the fraction of the trials resulting 
in a certain outcome
can be made as close as desired to some number P, by doing enough trials,
then P is the probability of that outcome of that trial.

[end of definition]

The trouble is that I don't know how that can mean anything for
elections. An election can't be repeated like a roulette trial
or coin flip. Literally, if November's election were repeated
many times, under the same conditions, we'd get the same outcome
each time.

That's why I (unsuccessfully) tried to write probability definitions
that would mean something for elections. If I try again today,
this should be my last try at it. Anything further that I post
about it will be copied from some reference source.

One thing that occurred to me was to say:

Say T is a series of trials, events with uncertain ourcome. They
needn't have anything in common other than that. Say you bet on
some of those trials.

The trials are numbered so that Ti is the ith trial. For each trial
Ti that you bet on, you bet with a payoff ratio of Ri. P is a series of 
numbers, each between 0 & 1. Those numbers in that series P are
numbered so that Pi is the ith number in that series.

The Pi aren't the probabilities of the outcomes that you bet for
in the Ti if the following isn't true:

If each Ri is more than (1/Pi)-1, then you can make your gain as positively 
large as you want by betting on a sufficient number of
trials. If each Ri is less than (1/Pi)-1, then you can make your loss
as great as you want to by betting on a sufficient number of trials.

That's just half of a definition, though, and I don't know if it's
useful. Maybe it can be made into a complete definition.

Maybe the easiest way to find a complete definition would be to
base it more closely on expectation-maximization, so that the
definition is just a sort of re-statement of the expectation-maximization 
goal, said in such a way that it defines
probability in terms of the other numbers.

I'm doing this because someone said that I don't know what probability is. 
As I said, I'm familiar with the usual frequency definition, but
I don't think it's adequate for elections, and so that means that
if I don't have a definition more useful for elections, then I
_don't_ know what probability means for elections. That's why I
felt that I should try to write my own definitions. But I may have
to give up & look it up.

Say we just use Weber's method for maximizing expectation in Approval.

Say we do many Approval elections, numbered from 1 to k.
The Pijk used by Weber's method in those elections are the probabilities 
that the candidates pairs (i,j) will be  the 2 frontrunners in election k if 
& only if it's certain that, if we do enough Approval elections,
a person who used certain Pijk for his strategy determinations can be made
to get a higher overall total for his utilities for the winnners
than another person who has the same utilities but who used different
Pijk.

[end of definition]

I won't post about this again unless I'm _sure_ that I have
a definition that is valid, or if I quote one that I found from
some reference source.

Mike Ossipoff









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