delta-p Approval strategy

MIKE OSSIPOFF nkklrp at hotmail.com
Sat Apr 20 21:31:13 PDT 2002


You wrote:

In the pij equation, Wik represents the probability that i will
defeat k strictly on votes (i.e, they will not tie), compared to
Bik which includes the possibility that i beats k in a tiebreaker.
Wik is used here since the definition of pij as I understand it
only includes two-way tie possibilities.

I reply:

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.

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.

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.

You continued:

Also, pt is the probability that there is a tie for first place.
Dividing by pt converts pij to a conditional probability. pij is
the probability, given a tie for first place, that it is between
i and j (i and j both score some value X, and all other candidates
score less than X).

I reply:

Yes, that's what I've meant by Pij.

You continued:

The distinction between Gj(X) and Fj(X) needs a little more
consideration. If we only consider the probability of j being
involved in a first-place tie (2-way or multi-way), so that we
want to calculate the probability that our vote will convert
j from a possible tie-break winner to a decisive (by one vote)
winner, we can use Fj(X). But we should also consider the probability
that our vote converts j from a loser by one vote to a tie-breaker
participant. Gj(X) accounts for both possibilities. Gj(X) is
different from Fj(X) only if Fj(X-1) is different from Fj(X).
When the population is large, this is the case within a very small
margin of error. When the population is small, the more accurate
Gj(X) should be used.

I reply:

That's what I meant when I said n-way ties _& near-ties_. The near-ties
are situations where your ballot can change a decisive win into a
tie, not necessarily 2-member.

You continued:

So the differences between the two methods all become insignificant
for large populations.

I reply:

But the methods don't have that difference, when I'm referring to
Weber that takes into account n-way ties & near-ties.

You continued:

My equations don't use the tie probabilities. They use the score
probabilities, Fi(X); all other variables are derived from the Fi(X).

I reply:

I noticed that when I reread the posting. It's a very different
approach from Weber's. If Weber's method is ignoring something,
that that would make it less accurate, but, when talking of
committee Weber, I didn't intend for it to ignore n-way ties or
near-ties.

You continued:

But at any rate you would need
Fi(X) to calculate pij, as well as to calculate delta-Pij.

I reply:

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 continued:

Obviously you could take the pij equation above and replace all the
Wik with Bik, and that would account for n-way ties. That would make
Weber nearly as accurate as delta-Pij.

I reply:

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.

I'd said:

>So, let's not say that Weber's method is less accurate because, in
>that article, it doesn't take into account n-way ties & near-ties.

You replied:

No, it's less accurate because it uses Fj rather than Gj.

I reply:

If that means ignoring the possibility that your ballot could change
a decisive win into a tie, Weber needn't make that assumption.
That's why I've been referring to Weber with n-member ties & near-ties.

You continued:

The added complexity is actually pretty trivial, though I wouldn't
want to calculate either one by hand.

I reply:

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.

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.

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

You continued:

The only added calculations are in the Bik and Gj terms. Are you
suggesting that if I use calculated Bik or Gj values then it will be
less exact than if I just throw in the Wik and Fj terms in their place?

I reply:

Not at all. But if delta-p gets its results in a more computationally
roundabout way, and the methods are otherwise doing the same thing,
then that could make delta-p very slightly less accurate, due to
roundoff error.

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

I haven't said exactly what I mean by Weber without the many-voters
simplifying assumptions. Here's all I mean. This isn't intended
as an instruction, just a rough definition:

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.

That's what I mean by Webster's method. Easier said than done, of
course, for committees.

For public elections, Weber does that in a very simple way, with the
simplifying assumptions that are possible under those conditions.

You continued:

But if it's simplicity you're after, then the grosser approximation
of the geometric mean could be used even for small groups. Just include
an adjustment to the winning probability for each candidate you've
already determined you will vote for.

I reply:

Sure, the assumption that ties are 2-way isn't really so bad even
in committees. 3-way ties & near-ties aren't vanishingly unlikely
in committees, but they're still significantly less likely than
2-member ties & near-ties. Most likely, one's guesses about candidates'
utilities, and whatever probability-input is used, aren't going to
be reliable enough so that there'd be enough precision to be spoiled
by the 2-way assumption.

So, right now, since I haven't worked out n-way strategy, if I
had to do that calculation in a committee, I'd just use many-voter
Weber.

In our polls last year, my Approval strategy was the one based
on unacceptable candidates. But otherwise, my Approval strategy
would have been the one that votes down to the likely CW, and no
farther, or to the better of the 2 strongest contenders. Approval
was the expected CW (and of course that estimate turned out to be
right). I'd have probably just voted down to the CW & no farther,
rather than try to guess the other strongest contender. In any
case, though, I'm sure that I like Approval better than whatever
would have been the other strongest contender, and so that strategy
too would have led me to vote down to Approval and no farther.

Mike




_________________________________________________________________
Join the world’s largest e-mail service with MSN Hotmail. 
http://www.hotmail.com

----
For more information about this list (subscribe, unsubscribe, FAQ, etc), 
please see http://www.eskimo.com/~robla/em



More information about the Election-Methods mailing list