[EM] Soliciting feedback for a modification of Allocated Score

[Kristofer Munsterhjelm]

On 07.04.2022 01:56, Andy Dienes wrote:
> 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
> <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 :)

I don't think there should be a problem as long as you make sure to
never elect more than one candidate in one go. E.g. if both A and B are
above the quota, apply the tiebreaker of your choice (say it picks A),
then elect A, deweight the voters, and *then* check if there's a quota
for B. Otherwise a majority might get too many candidates elected.

Your change might also reduce free-riding. Let's say my honest vote is
A: 5, B: 4. I know some other voters are also voting A:5, so it's
tempting for me to vote A: 4 instead so that my voting weight is only
exhausted fractionally. With the original reweighting (as you describe
it), I may get exhausted fully with A: 5 but only partially with A: 4.
But with your change, I would get exhausted equally.

In general, as long as you reasonably fairly reweight the votes, then
just how you distribute the surpluses shouldn't matter much. You'll get
quota proportionality anyway, as long as you reweight everybody who
contributes to getting a candidate elected, in such a way that their
total weight afterwards is equal to the surplus.

For instance, I constructed a more strategy resistant variant of EAR,
https://electowiki.org/wiki/Maximum_Constrained_Approval_Bucklin, which
maximally delays the decision of just what ballot weights to attenuate.
So although it's very hard to reason about just whose votes will count
less, it still passes Droop proportionality, because it does the
reweighting as soon as a candidate is backed by a quota.

-km