[EM] Soliciting feedback for a modification of Allocated Score

[Andy Dienes]

Hi all,

I have recently come up with a small change to the way surplus handling is
performed on Allocated Score (AS). It is inspired by the way MES (which you
can read about here
https://proceedings.neurips.cc/paper/2021/hash/69f8ea31de0c00502b2ae571fbab1f95-Abstract.html)
operates on ranked ballots, which is also related to the Expanding
Approvals rule by Aziz.

Basically, the way it works in the reweighting step of AS is:
Set a threshold d such that the total ballot weight of voters who scored
the candidate >= d is at least one quota (using Hare for now, but other
choices are fine). Then, find the minimal amount of voting power that can
be subtracted equally from each ballot such that the total amount taken is
exactly one quota. Note that some ballots may have less than this amount
remaining, so they will be fully exhausted.

It is very similar to the original surplus handling, but rather than
exhausting fully all ballots with score > d and then fractionally ballots
with score = d, it chooses to subtract an equal amount of power from all
ballots above the threshold.

When all scores are 0,1 (i.e. approval ballots) it does not satisfy EJR (in
the same way that AS doesn't), but it does satisfy PJR.

I have already done some simulations and found favorable results, so what I
am mostly looking for is if there are any sneaky ways this can go very
wrong? Of course, every voting method has pathological examples so it's
never good to put too much stock in specific bad scenarios, but I've
already looked at this proposed modification from many other perspectives
so pathological examples are exactly what I'm after here :)

Best,
Andy Dienes
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