[EM] Inferring Approval Strategy From Ranked Ballots

[Forest Simmons]

As Mike, Rob LeGrand, Richard Moore, Kevin, Bart, and others have pointed
out both recently and anciently, it is not always possible to infer
optimal Approval strategy from ranked ballots alone.

In fact the more interesting the election, the more relevant the strategic
values, which depend on information not contained in the mere ranked
ballots.

How good can we do with a simple rule of conversion of ranked ballots to
approval ballots?

Various ideas have been suggested in the past depending on what
supplemental information we are supposed to be privy to.

It would be great if we had winning probabilities, but they seem to be an
illusive "Will O' Wisp."

In any case, here's an idea that's as good as any I've ever seen along
these lines:

Given a set of preference ballots, let n1, n2, ... be the numbers of top
rank votes for candidates c1, c2, ... , respectively. (For now assume that
every ballot fully ranks the candidates.)

Use these numbers n1, n2, ... as weights in the weighted median method for
determining the approval cutoff for each ballot.

That's it, except for a review of the "weighted median method for
determining approval cutoffs."

Suppose the weights are w1, w2, ... .  In the present case the w's are the
n's .  In Joe Weinstein's original proposal, the w's were the respective
winning probabilities.

A typical example should clarify the method:

Suppose that there are five candidates (A, B, C, D, and E) and 100
ballots, and that the candidates' respective weights are 13, 25, 8, 36,
and 18.

Here's how we determine the approval cutoff for the the ranked ballot

                      C>B>A>E>D .

Replicate each candidate name in proportion to the weights, and find the
halfway mark:

        CCCCCCCCBB...BBAAAAAAAAAAAAAEEEE|EEEEEEEEEEEEEEDD...DD

Since the halfway mark is less than halfway through the E's, we put the
approval cutoff between A and E, i.e. candidates C, B, and A are approved,
but E and D are not.

If the halfway bar "|" appears among copies of the name of the top ranked
candidate (or bottom ranked candidate, resp.), then it is adjusted to the
right (left, resp.) no matter which side it leans towards.

Various options exist for adapting this procedure to ballots that do not
fully rank the candidates.  We won't worry about that now.

It is interesting to contemplate the action of the method in issue space
in the present case where the weights are the top rank supports n1, n2,
etc.

The upshot is that you tend to approve the favorites of the half of the
voters that are closest to you in issue space.

To be slightly more precise, before the halfway marker "|" is adjusted to
the right or left to become the approval cutoff, the only names to the
left of the marker are those that represent candidates who are favorites
among the half of the voters that are closest to you in issue space.

This statement is precise if we measure our closeness to another voter by
how high we rank that voter's favorite. Other measures of closeness could
be used to derive other similar methods.

In the case of a one dimensional issue space, it follows that the name of
the median voter candidate (i.e. the CW) will be represented to the left
(i.e. high ranking side) of the halfway bar.

In the case where the CW's name is represented on both sides of the bar,
whether or not the CW is actually approved (on the ballot in question)
depends on which way the bar gets adjusted.

Thus we see that it is highly likely, though not necessarily certain, that
most of the ballots will approve the CW when there is one.

Comments? Suggestions?

Forest