[EM] delta-p Approval strategy

[Richard Moore]

Mike,

I think there may be far less of a difference here than meets the eye.
Consider the equations for the two methods:

delta-Pij = -sum_over_X(Fi(X-Ai)*Gj(X)*product_over_k(Bik(X-Ak)))

pij = -sum_over_X(Fi(X-Ai)*Fj(X)*product_over_k(Wik(X-Ak))/pt)

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. For large populations,
more-than-two-way ties are much less likely than two-way ties,
and so can be ignored; this means Wik is very nearly equal to Bik
for large populations.

Of course, you could accommodate multiway ties by simply replacing
the Wik with Bik.

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). pt can vary slightly with changes in the Ai/Ak
values. However, pt will be constant for each pij for a given j,
so all the pij for that j will scale by the same amount, so the
resulting approval decision for candidate j is not affected.
Therefore, we can ignore pt altogether.

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.

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

The only area where I expect delta-Pij to perform significantly better
is in the small-population cases. There are two effects that are
significant for small elections: multi-way ties, and the effect
of Gj vs. Fj.

More comments below.

MIKE OSSIPOFF wrote:
 >
 > When I said that tie probabilities are more fundamental than delta-p,
 > I just meant that calculating the delta-p values depends on the
 > probabilities of the revelevant ties & near-ties, whereas you don't
 > need the delta-p values to calculate estimate the tie probabilities.

My equations don't use the tie probabilities. They use the score
probabilities, Fi(X); all other variables are derived from the Fi(X).
(In the case of correlated score probabilities you would need additional
information not found in the Fi(X).) But at any rate you would need
Fi(X) to calculate pij, as well as to calculate delta-Pij.

 > But, just as the delta-p approach could take n-way ties into account,
 > for committee voting, or could assume that all ties are 2-way, for
 > public voting, that's also true of Weber's method. Weber can take
 > into account n-way ties & near-ties.

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.

 > 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.

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

 > For public elections, where only 2-way ties are considered, can
 > the delta-p method match the great simplicity of Weber's method?

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

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.

 > If delta-p is more complicated, more calculation-intensive, then
 > , more likely than not, it has more roundoff error than Weber would,
 > and is slightly (maybe not significantly) less accurate. I'm not saying
 > that delta-p isn't accurate enough, only that it's probably less
 > accurate if it's considerably more calculation-intensive.

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?

 > Until Weber & delta-p are completely worked-out for taking into account
 > n-way ties & near-ties, for small committees, I don't know which is
 > simpler. Of course the 2-way tie experience suggests that Weber will be
 > simpler, but we won't know about that till both methods are worked out
 > for small committees. So I'm not now claiming that Weber will be
 > simpler for small committees.

Weber's method would be *slightly* simpler, since we can always leave pt
out of the equation.

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 don't want to seem too partisan about Approval strategy methods,
 > or Condorcet(wv) strategies. If I've seemed so, I'm not someone who'd
 > fight about those issues. I wanted to just state the case for
 > truncation in Condorcet(wv), and, in the case of the Approval strategy
 > methods, I objected to Richard's implication that delta-p is more
 > accurate than Weber, because it's more accurate when we look at
 > n-way ties & near-ties with delta-p, but not with Weber.

It's not just n-way ties. I don't see that Weber takes into account
the other small-population effect. If you put the small-population
adjustments (Bik and Gj) into Weber then it becomes the exact
equivalent of the delta method (pt has no effect on the outcome).
Which is why I say there's less of a difference than meets the eye;
both methods are formulaically very similar, and ultimately rely on
the same data: the F(X) values, which are inferred from polls or
other available data through a statistical model.

I'm not out to politicize this issue either; I just hope to set the
record straight where I think some inaccuracies have crept in. And if
I've made any erroneous statements then corrections are welcome,
because I certainly don't want to contribute any new inaccuracies.

   -- Richard

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