[EM] Winners or representation?

[Kristofer Munsterhjelm]

On 25.03.2022 20:50, Richard Lung wrote:
> 
> I can't help but think that this group is absorbed in determining
> election winners, rather than representatives of the people, as in a
> democracy.

For me at least - I don't know if that's why others are focusing on
single-winner - it's because multi-winner is so much harder, it's
difficult to see even where to begin to prove anything.

Take Droop proportionality, for instance. It is known that party list
methods that are based around fulfilling a quota criterion must fail
what's called population pair monotonicity (or the Alabama paradox); and
Droop proportionality implies a sort of one-sided quota criterion, call
it a lower quota.

But is then Droop proportionality compatible with population pair
monotonicity? Is it only so for methods that reduce to D'Hondt (which
passes a lower quota property)? I don't know. Or perhaps population pair
monotonicity is the analog of the participation criterion, which almost
all voting methods fail anyway, and thus is not something we need to pay
attention to.

There are many questions like this. Another one is (strong
seat-independent) summability: we know of a bunch of methods that are
summable for single-winner. But is it even possible to pass both Droop
proportionality and summability for a general ranked voting method?
Again, very difficult to prove *or* disprove. I have been playing with
this on and off, and I suspect it is impossible. But I also suspected
that the combination (for single-winner) of monotonicity and DMTBR was
impossible, and I was at least partially shown wrong by my optimal
strategy solver.

The cardinal voting camp has probably been more successful, but they
have the benefit of being able to treat ballots numerically. Trying to
do so with ordinal ballots usually leads to clone dependence, Condorcet
failure or similar problems.

There's also an argument that the best representatives of the people are
a representative sample of the people, i.e. that one should not elect at
all, but just populate the representative body with an unbiased
selection of all kinds of people who make up society. If so, then
large-scale PR is "solved" by removing the need to solve it by elections
in the first place. On the other hand, this might also make the question
of a good single-winner method moot as well, since parliamentary
procedure is usually very simple.

That's not to say I've been completely uninterested in multi-winner. My
first post here showed tradeoffs between proportionality (in a simple
yes-no model) and total satisfaction: to some degree, if you please each
faction more, you end up pleasing all of society less as the faction
representatives become more polarized.
https://munsterhjelm.no/km/elections/multiwinner_tradeoffs/ Which might
be an obvious result in retrospect, but the gap between social optimum
and known methods shows that there's potentially a lot more improvement
to be had.

I also devised MCAB, which is a more strategy resistant variant of EAR
or the Bucklin transferable vote:
https://electowiki.org/wiki/Maximum_Constrained_Approval_Bucklin

-km