[EM] Peak Approval Sorted Margins

[Ted Stern]

Comments below:

On Tue, May 10, 2022 at 10:41 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> This is an aggregate method that can be taken apart and reassembled to fit
> the circumstances.
>
> The longest path from start to finish begins with ranked choice ballots.
>
> Step 1 converts the ballots to 3-slot ballots. This step can be
> accomplished in two main ways (detailed presently) or bypassed entirely by
> getting direct 3-slot approval input as in Ted Stern's Approval Sorted
> Margins, for example.
>

I would credit you as the originator of Approval Sorted Margins. My
modification was Preference Approval Sorted Margins, with 3 approval slots
of Preferred, Approved, Rejected.


>
> The first main way is to distinguish Top, Bottom, and Middle positions on
> the ranked ballots.
>
> The second main way is to give Top slot status to every candidate X on
> ballot B for which there is some candidate Y outranked by X, that defeats
> every candidate that outranks X.
>
> Bottom slot status goes to X if it is outranked by some Y that defeats
> every candidate that X outranks.
>
> Middle slot status goes to ranked candidates not assigned Top or Bottom
> status by the above rules.
>

I'm not quite clear on the TMB slots. Could you give an example of this in
a simple election, then apply the positions on an example ballot?

The unclear thing to me is whether the 3 slot approval is per-ballot or
based on the overall pairwise array.


>
> Step 2 is converting 3-slot Top, Middle, and Bottom tallies into Robert
> Bristow-Johnson's  "peak approval" scores.
>
> Let t, m, & b be the respective slot values. Then the peak approval score
> is ...
> (t-b)/(2-t-b) or (t-b)/(1+m)
>

To clarify, are t, b, and m the totals of individual ballot t/b/m scores
from each ballot?


>
> with the David Gale value t-b as the tie breaker.
>
> Step 3. Do peak approval sorted margins, as in any other Sorted margins
> method.
>
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