Generalized Approval Strategy (Was Re: [EM] election's utility approach)

[Richard Moore]

Richard Moore wrote:
 >
 > delta-Pij = P2i - P1i = -sum_over_X(Fi(X)*Gj(X)*product_over_k(Bik(X)))
 >
 > where the product is for all k except i and j, and where
 >
 > Gj(X) = Bij(X) - Bij(X-1).
 >
 > The delta-Pij equation is for j != i.
 >
 > We also need the effect of voting for i on i's own probability:
 >
 > delta-Pii = -sum_over_j( delta-Pij ), for j != i.
 >
 > This equation for delta-Pij above is a general expression that applies
 > to any Approval election (large or small population, zero info or rich
 > info, correlated or uncorreslated voting patterns, etc.). We can make
 > various assumptions to reduce this equation to a form suited for a
 > specific type of situation.

I should note a correction to the formula, which is significant for the
small-population case.

Fi(X) is the probability that candidate i will receive X votes. Now, if
we have already decided whether or not we will approve candidate i, we
can assign a value Ai to this decision:

	Ai = 0 if we disapprove i
	Ai = 1 if we approve i

Now comes a philosophical question: What is our relationship to the
statistical model? Are we acting with free will, independent of that
model, or are we somehow tied up in the model so that the model takes
our decision into account? I'm going to assume the former. I will assume
the statistical model doesn't know we exist, as if we arrived on the
scene after the poll was taken but before the election was held. In that
case, Ai also is 0 if we haven't yet decided how to vote, because that
value agrees with our exclusion from the model. So, Fi(X) is the
probability of i getting X votes if we don't exist, or don't vote, or
if we do vote and disapprove i. Therefore, if we decide to approve i,
then the actual probability of i getting X votes inclusive of our vote
is the same as the probability of i getting X-1 votes exclusive of our
vote, or Fi(X-1). So the formula should use Fi(X-Ai) instead of Fi(X).

What about the other factors: Bik(X) (k != i, k != j) and Gj(X)?

Bik(X) is the probability, given X votes for i, that i will beat k. The
product of all Bik(X) where k != i (Note: this product includes k = j)
is the probability, given X votes for i, that i will win. Incidentally,
the Gj(X) factor results from finding the differential of that product
with respect to a one-vote change in favor of j. We are already using
Fi(X-Ai), which means our decision on i is accounted for. We also need
to account for our decision on k, which we do by noting that if we vote
for k (Ak = 1), i needs an extra vote to achieve the same probability of
beating k as it would be if we don't vote for k. That is, we need to use
Bik(X-Ak) instead of Bik(X).

Now since Gj(X) = Bij(X) - Bij(X-1), and we are substituting X-Ak for
all X in each Bik(X), we need to use Gj(X-Aj) = Bij(X-Aj) - Bij(X-Aj-1).

So my modified formula is

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

The sum is over all integer values of X for which Fi(X-Ai) is non-zero;
that could range from 0 to Xmax if Ai = 0, or from 1 to Xmax+1 if Ai = 1.
For convenience we could just express this as the sum over all integers
from 0 to Xmax+1. Note that j represents the candidate we are considering
whether or not to vote for, and i represents the candidate whose win
probability we want to determine the effect of our j vote upon. (Yeah, I
know, a preposition is a terrible thing to end a sentence with.)

Normally we would work through the calculations for each candidate in
sequence from our favorite down to our least favorite, or until we hit
the first disapproved candidate. Unless we have reason to believe there
is cause for vote-skipping, we will stop with the first disapproved
candidate. If we were interested in knowing the delta-Ps exactly, then
we would have to repeat all the calculations using the Ai, Aj, and Ak
values that were found in the first pass (these were originally assumed
to be zero for all but our favorite). But that isn't of any practical
use, for once we decide to approve candidate i, the decision to approve
a candidate we like less than i would not cause us to adjust our vote
on candidate i to a "no" vote -- except, again, in those rare cases
where vote-skipping is a possibility. So in the vast majority of cases,
the calculations will converge on the first pass.

I hope to have a small example worked out soon.

  -- Richard

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