[EM] Proportionality vs utility: Droop quota and feasible points

[Closed Limelike Curves]

On Wed, Sep 11, 2024 at 1:45 PM Kristofer Munsterhjelm <
km-elmet at munsterhjelm.no> wrote:

> By using the set of every outcome compatible with the Droop
> proportionality criterion, the region shows what results a method that
> passes the DPC can get. And at least for this model, it's surprisingly
> large: the Droop proportionality criterion doesn't seem to be a very
> strong constraint.
>
Doesn't surprise me :) I've been pointing out that PSC is an extremely weak
condition for a while now. It's only slightly stronger than
color-proportionality is.

I'd also point out the space in the top-right, which shows either our
definition of "proportionality" or "utility" is wrong, or Droop-PSC is
wrong, because there are Pareto improvements on this graph ruled out by
Droop-PSC. I'd probably go with both.

Droop-PSC guarantees *dis*proportional representation overall, because as
I've mentioned before, the Droop quota is the most-biased possible quota
(it is maximally friendly to large parties; any rule more friendly to them
would no longer count as proportional). I'm not 100% sure, but IIRC Webster
satisfies the *k-1* quota rule (which has no common name that I know of).
This makes me think that if we *did* want to enforce PSC, (k-1)-PSC would
be a more sensible rule of proportionality. It also lines up quite well
with the "proportionality up to one object" constraint that's common for
fair division.

On the topic: I think if we ever find ourselves tempted to make a tradeoff
between something we're calling "utility" and some other quantity, we've
defined utility incorrectly. Specifically, I doubt voters' utilities are
equal to the sum of scores they assign to each candidate. Unlike in the
single-winner case, I'm not sure how we can interpret the ratings voters
assign to each candidate. If scores were actually linear and additive, that
would imply colored voters care just as much about going from 49% of seats
for their party up to 51% as they do about going from 97% to 99%.
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