delta-p Approval strategy

[MIKE OSSIPOFF]

Richard:

I'd said:

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.

You replied:

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.

I reply:

Ok, I didn't know that. Though I figured they'd give the same results,
the wording is so different that they sound like different approaches.
If the actual calculations are the same, so both approaches are equally
difficult, that's good, because it simplifies the choice of strategy
method.

I'd said:

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

You replied:

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.

I reply:

When I said that, I assumed that they were completely different
approaches, completely different calculations that get the same
result, because their wording is so different. That patticular
statement of mine, copied above, was in reply to when you pointed
out that the delta-p approach doesn't use tie or near-tie probabilities.
What I was saying, then, was that I hadn't realized that the
approaches were that different. But of course it's possible for
the same calculations to be done, under 2 very different wordings,
one of which speaks of tie probabilities and one of which doesn't,
and which can appear to be completely different approaches.

I'd said:

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.

You replied:

>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?).

I reply:

Yes. The probability density at a point in outcome space is
related to that point's distance from the most likely outcome-point
position, by the Gaussian distribution.

You continued:

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?

I reply:

Yes. Or the previous iteration in Crannor's DSV.

You continued:

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

I reply:

I'd just assumed that they're different because they sound so different,
in their wording. I'm not still claiming that they're different now.


I'd said:

Maybe, but it just seemed to me that Weber is more direct, and
that delta-p is more computationally roundabout, calculating various
intermediate quantities.

You replied:

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.

I reply:

I haven't brought the Weber method, without the simplifying assumptions, any 
farther than the rough definition that I gave. If I'd actually
pursued both approaches, then I'd have noticed that they're the same.
The reason why I said that Weber seemed more direct was because
it only looks at tie & near-tie probabilities, and the difference in
utilities depending on whether you change those ties & near-ties.

The delta-p approach _appeared_ more computionally roundabout because
of the intermediate quantities. But, not having actually worked-out
either, I of course wasn't in a position to say for sure that
one approach is roundabout. And, when I hear that the calculations
are the same, despite the very different sound of the wording, I have
no reason to doubt that, since I haven't gone into those calculations.
Appearances can be deceptive, and, though the methods seemed different, I'm 
not now claiming that they're different.

I'd said:

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.

You replied:

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...

I reply:

...and isn't it also assumed that you're not voting for any candidate
whom you like less than the one currently being considered?

At first my concern was that if you later decide to vote for someone
lower in your sincere ranking, that could reduce the utility of a
tie that had made it worthwhile to vote for your 2nd choice, making
it no longer desirable to vote for the 2nd choice. But then I realized
that that's skipping, something that's been said to be called for only
situations so implausible that they can be ignored. So one could
safely use the stepwise procedure that you describe, if it can
be assumed that skipping doesn't pay.

You continued:

, and so on...that's M possible ways of
voting.

I reply:

Yes, much easier. Considering all 2^M ways of voting would be
good to avoid if possible.

But if it were felt possible that skipping conditions could obtain,
wouldn't it then be necessary to look at all 2^M ways of voting?

But I'm not saying that skipping conditions are likely enough to
warrant considering them.

Mike






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