On 25.03.2022 20:50, Richard Lung wrote:

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

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

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.

It’s putting oneself at an unfair disadvantage to try to prove something
that doesn’t exist. The term “winners” assumes some preordained election
results, that one has to try to discover. The statistical assumption, as
distinct from the determinist assumption, is that there are only best
estimates of representation. Some candidates win beyond reasonable
doubt. Other contests may leave the voters indifferent, with no
candidates a clear winner. This is just a fact of life that defies
mathematical certainty.

Laplacehad a metaphysical belief in determinism. (That is an interesting
hypothesis, as Bonaparte commented.) Yet his adjudication between
Condorcet and Borda appeared in his treatise on probability. Elections
are a statistic. (I follow the account by JFS Ross, in Elections and
Electors.)

Laplacehad the first and perhaps the last word on Condorcet pairing or
Round Robin elections: they don’t take into account the relative
importance of preferences.

Over a century later, in the 1940s, this message was reinforced, by SS
Stevens on scales of measurement. Judging winners solely by binary
elections, staying at the least powerful, nominal scale of measurement,
leaves out a comparison of the whole range of candidates, at once, that
is to say, an ordinal scale of measurement, which preference voting is
the next, more powerful scale. It is the difference between absolute,
all or nothing choice, of a single order of preference, and the relative
choice of a whole range of preferences.

Condorcet pairing may occasionally contradict preference voting. But
this cross-check never departs from minimal choices, which do not
justify a veto over the more general relative choice of preference voting.

Moreover, a case in about 135, like Burlingtonis not statistically
significant enough, to inaugurate Condorcet pairing as an arbiter in
(preferential) elections, considered as statistical estimates.

After the nominal and ordinal scales, Stevens progresses to the
successively more powerful interval scale and ratio scale. These are
both proportional measurements. The only difference is that the interval
scale lacks a real zero. An example is the conventional temperature
scales like Fahrenheit and centigrade. Their different zero points on
the scale are a matter of convention, such as making zero, when water
freezes (either in salt water or fresh-water).

The quota is an example of a ratio scale. Party lists, however are a
particular vote in a general election. Likewise, the local constituency
vote in a national election. The Andrae or Hare system are the more
logical general vote in a general election. The interval scale appears
in voting method, in surplus or deficit of a quota, which sets its
conventional level of zero. A candidate whose votes exactly equal the
quota has a zero surplus and a zero deficit of votes.

The transferable vote possesses all four scales of measurement; a rare
quality in voting methods, which I have long since argued qualifies it
as something like scientific method of elections.

Regards,

Richard Lung.

On 02/04/2022 10:51, Kristofer Munsterhjelm wrote:

On 03.04.2022 11:54, Richard Lung wrote:

An estimator can itself be deterministic without making the process of
estimation deterministic. And properties can be proven about these
estimators (e.g. whether they're biased or not); it's not like these
properties don't exist.

So too with voting methods: you may from a statistical perspective
consider voting methods to be estimators of some parameter (this analogy
is particularly direct for Kemeny).[1] That the methods respond in
predictable ways given predictable data does not make them deterministic
when applied to random variables any more than it does the sample mean.

Perhaps we could replace the term "winner" or "winning set" with
"estimated best fit". But this would not affect whether voting method
criteria exist.

-km

[1] Strictly speaking, Kemeny is an estimator connected to a function
that processes the estimate. The estimate is the consensus ordering and
the function returns the candidate on top of that estimate ordering as
the best fit to the electorate.