[EM] Winners or representation?

[Forest Simmons]

El sáb., 2 de abr. de 2022 2:51 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> 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.
>

Yes ...Cardinal/Score ballots are essentially vectors ... so all of the
linear algebra vector space tools are readily available.  The biggest of
these advantages from my point of view is that the familiar vector space
visualizations help to guide our intuitive understanding of potential
election methods ... whether single or multiwinner.

In particular, vector norms provide standard metrics on vector spaces,
while, as near as I know, (up until  three months ago) only the clone
dependent Kendall-tau metric has been available as a topologically faithful
distance function. But now we have a clone free analog of Kendall-tau
called "swap cost" that is faithful to
 the natural Universal Domain ordinal topology.

In particular any geographical method of apportionment based on distance
considerations can be faithfully mimicked in ballot space.

>
> 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
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>
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