Mathematical utility expectation maximization in Approval

[Bart Ingles]

It should be pointed out that this entire discussion is more-or-less
academic unless you plan to use the formulae to model strategy in
simulations, based on randomly generated utilities or some such.  This
is because in the real world, utility is determined by the willingness
of a voter to vote for a candidate, and not the other way around.

If you are a voter attempting to assign utilities to candidates for the
purpose of devising a strategy, remember that these assigned 'utilities'
are really just ratings -- surrogates for real utilities.  Thus the
'neural net' approach is probably the more accurate (although
calculations involving ratings might help to get you into the ball
park).

So if, after calculating your intended strategy, you feel an urge to
make a couple of changes, you would probably be right to follow your
intuition.  That is if the election method in use is not
counter-intuitive.

Bart



Richard Moore wrote:
> 
> 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