delta-p Approval strategy

[Richard Moore]

MIKE OSSIPOFF wrote:
> But I was saying that Weber, like delta-p, can deal with n-member
> ties and near-ties. So I'm not saying that Weber has to only deal
> with 2-member ties. And when I say "...and near ties", that means
> situations where your ballot can change a decisive result into a tie.
> Change a decisive result that j wins into a tie that i has some
> probability of winning, and which has some utility to you different
> from j's utility, depending on who is in the tie.

OK, on my first reading I missed your reference to "near ties". I should
have read that message more carefully.

> What I'm saying is that Weber and delta-p could both use the assumption
> that all ties are 2-way, or they could take into account n-member
> ties & near-ties.

If you take those effects into account for both methods, or ignore those
effects for both methods, then the only difference is that pij requires
calculating the probability that there is any first-place tie. If you
remove that requirement (which as I pointed out is an unnecessary
complication since it is merely a scaling factor), then you end up with
exactly the same calculations: the same level of accuracy, and the same
difficulty. They are in fact identical methods.

Pij with the Pt scaling factor removed is Ptij, the probability that
i and j are in a tie for first place (with appropriate adjustments to
that definition depending on whether we are limiting it to 2-way ties
or not, and whether we are including near-ties or not).

Of course my original message, three or four postings back, was
comparing the exact delta-p method with the approximate pij method.
That was based on the definition of pij that had been stated prior
to that time. With the concession that pij is not restricted to the
approximations, then I would have said I was comparing the exact delta-P
(= Ptij =  Pt * Pij) method with the approximate delta-P (= Ptij =
Pt * Pij) method.

> And I emphasize that, when Weber takes n-member ties & near-ties
> into account, that doesn't mean that Weber is achieving delta-p's
> accuracy by borrowing something from delta-p. n-member ties & near-ties
> are part of the committee landscape, and are not the property of
> delta-p. Weber and delta-p are 2 approaches, either of which could
> take all the possibilities into account, for committee votes, or
> could just consider 2-member ties, for public elections.

The closer you look the more you will see that they are the same
approach. Only different terminology is used. The real variations
are in the approximations chosen.

> I noticed that when I reread the posting. It's a very different
> approach from Weber's.

Does this mean there's a different way to calculate pij than the
one I gave? I'm not aware of one. If there is one then that would
be a different approach, but so far I'm not convinced that they
are different.

> That's the way that sounds best to me, for Pij, or for the n-way
> ties & near-ties. But Hoffman is available too, for public elections
> or commmittees. Hoffman could sum the probability density in regions
> where n candidates have the same vote-total, & in regions where
> to vote totals differ only by one. The individual vote-total
> probability distribution approach sounds like it wouldn't get
> complicated in the way that Hoffman would, when more candidates are
> added. But I don't know how the probabilities would be gotten from
> the individual candidates' vote total probability distributions.
> If I remember correctly, that's what Crannor's method does.

 From my reading of your description of Hoffman's method, it sounds
like the method of calculation is to integrate probabilities over
the boundaries between the winning regions of candidates. That's
like taking the formula I gave and replacing the summmations with
integrals, and I presume the binomial distribution is replaced with
a continuous distribution (Gaussian?). So Hoffman's method doesn't
seem to be different in the calculation aspect. What is distinct
about it is the method of inference of the winning probabilities;
did you say that it was based on the outcome of a previous election?

> Exactly. If Weber doesn't make the simplifying assumptions that it
> uses for public elections, it's as accurate as delta-p. But that
> doesn't mean that Weber is then borrowing from delta-p. They're
> 2 different approaches, either of which could make or not make
> the many-voter simplifying assumptions.

Again, how are they different? And if they aren't, how could one
be borrowing from the other?

> Maybe, but it just seemed to me that Weber is more direct, and
> that delta-p is more computationally roundabout, calculating various
> intermediate quantities. But I only have experience with Weber
> with the many-voter assumptions, and so I can't say anything
> for sure about one being simpler than the other till I've used
> both in simplified & not-simplied form.

Unless you can present a simpler way of calculating Pij or Ptij, they
are computationally equivalent. For M candidates and N voters,
calculating a single Ptij or delta-Pij value requires M*N inferences
to determine the F(X), M*N calculations (of 1 or 2 additions, and
0 or 2 divide-by-two operations) to determine the cumulative
probabilities (Ck(X), which stand in for the Bik(X)), and N
subtractions to determine the Gj values. This is followed by N*(M-1)
multiplications, and the summation of the resulting N values.

If there's a simpler way (that isn't an estimate), I'd like to know.

> 
> You continued:
> 
> Also, if I'm just going to estimate the pij (as with the geometric
> mean or some other approximation), I could make a valid claim that
> this is also an estimate of the delta-Pij strategy, because, at
> least for large-scale elections, pij and delta-Pij are practically
> the same strategy.
> 
> I reply:
> 
> If they're both designed to maximize utility expectation, and
> they both are valid and don't make simplifying assumptions that
> the other doesn't make, and they both use the same inputs,
> then they're effectively the same method.

Exactly.

> Experience with both, with & without the many-voters simplifying
> assumptions, will tell which approach is easier.

It is the simplifying assumptions that make things easier, not the
choice of terminology.

> Again, I can't really say that delta-p is more computationally
> roundabout, till you post an example that makes the many-voters
> simiplifying assumptions that Weber made in his article, and until
> , for both methods, there are examples written in which
> the same kind of probability-info is available to both methods, and
> they both don't make the many-voters simplifying assumptions.

But if I posted such an example, it would calculate delta-Pij and
Pij in the exact same way, so what would we learn?

> But
> it _appears_, at this point, that delta-p is likely to be more
> computationally roundabout.

"Appears" based on what?

> Find, somehow, the probability of all the possible n-member ties
> & near-ties, before your vote is cast (last). For each of the 2^N ways that
> you could vote, a sum can be written, each of whose terms multiplies
> the probability of a tie or near-tie by the difference in the
> utility of what would happen if you didn't change that tie or near-tie
> and what would happen if you did. Find the way of voting that
> maximizes that sum.

If you calculate this based on your 2^N (or 2^M, since I use M to represent
number of candidates) ways of voting, you will have far more calculations
to do. Fortunately, this is approval, so you only have to determine
whether to vote for your second favorite assuming you've already
approved your favorite, and you third favorite based on whether you
approved your first two choices, and so on...that's M possible ways of
voting. And since you calculate your third choice after you know your
first two choices, this can accommodate the occassional candidate-skipping
phenomenon, too.

  -- Richard


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