[EM] Threshold Approval Preference (Approval Margin Sort + Median rating score ideas)

[Ted Stern]

Approval Sorted Margins
<https://electowiki.org/wiki/Approval_Sorted_Margins> was first proposed
nearly 20 years ago by Forest Simmons, in a private correspondence between
Simmons, Chris Benham and myself. It's based on starting from an initial
ranking in descending order of some seed metric, determining the pairwise
preferences between successive candidates in the ranking, and then
reversing the pairwise-out-of-order pair with the smallest margin, relative
to the seed metric.

Approval sorted margins could thus be described as Pairwise Seed Ranking
Margin Sort (Approval), where Approval is the seed metric function, or
PSRMS(Approval), abbreviated.

This method gives a decent level of burial and defection resistance, but
forces the voter to either put less preferred candidates below a fixed
cutoff, or rate an explicit cutoff candidate.

Based on discussions a couple of months ago regarding an earlier Pairwise
Median Rating proposal with inputs from Chris Benham and Michael Ossipoff,
I thought about other ways to find the approval cutoff automatically.
Below is what I've come up with.

As in ER-Bucklin / Median Rating methods, a rating ballot is used. Any
candidate rated above 0 is considered Approved, with higher ratings
indicating stronger preference, and 0 is not approved.

To determine the Approval Preference threshold, find the maximum rating R
at which at least one candidate has a rating of R or above on more than
half the ballots.

For each candidate, determine their threshold approval preference by
counting ballots that score each candidate at or above rating R.

Then run PSRMS(Threshold Approval Preference) using the TAP as the seed for
the initial ranking.

This method satisfies Condorcet and Smith criteria like Approval Sorted
Margins, with additional limitations from Median Rating (aka ER-Bucklin
with gaps) methods.

As an example of burial/defection resistance, see Michael
Ossipoff's defection example from a few days ago (candidate A = Condorcet
Winner with no defection; Candidate B = Burier's Favorie; Candidate C =
non-contender with CW as second choice, used for burial by BF), with one
slight modification -- '>>' means there is a gap in the rating.

32: A >> B
34: B >> C
33: C > A

On a zero to 5 scale, this could mean 32: A5 >> B3; 34: B5 >> C3; 33: C5 >
A4.

In this case, the approval preference threshold is 4, because the C
faction's second choice passes 50% for rating 4 and above.

The seed ranking is therefore

A:64, B:34, C:33

A > B, B > C, so no pairs are out of order, and A wins.

If the A faction does not down-rate B, the burial will succeed. However,
unlike Approval Sorted Margins, the A faction does not have to put in an
explicit cutoff or rate B at 2. Instead, if a defection is feared,
down-rating by just one position should be sufficient. B voters likewise
have an incentive to down-rate C because if they don't, C might be at the
top of the TAP seed ranking, which could interfere with B's defection goal.
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