[EM] Multiwinner runoffs (Re: Proportional Representation Systems I'd Support)
Kristofer Munsterhjelm
kmelmet at broadpark.no
Sat Mar 27 02:30:18 PDT 2010
Raph Frank wrote:
> This is less complex, but not as fair as transferring the surpluses.
Your first method is more proportional than your second. If what you
want is a majoritarian/centrist outcome, use the second.
In any case, I think it can be generalized if you have a housemonotone
method (i.e. one that generates a proportional ordering). Say you want
to elect to a council of K seats:
1. People vote and thus produce a set of ballots. Run this ballot
through the prop. ordering. Pick the first q, q > K, for the second round.
2. People vote on these q for the second round, and the outcome gives K
winners.
The limitation of this approach is that you have to use a proportional
ordering. However, this limit, or one close to it, is fundamental to the
runoff process. In a runoff, to the extent possible, you'd want the
tworound method to return the same result as an automated version of
the oneround method if everybody voted equally: a top two runoff should
return the same result as the contingent vote if everybody submitted the
exact same ballots for the first and second round.
However, if the outcome for q winners doesn't include the outcome for K
winners, that is impossible. The method can't produce the outcome for K
winners when limited to the outcome for q winners even with the same
ballots  because not all the candidates are available!
Thus the method should either be housemonotone outright  a
proportional ordering  or have two numbers as input (a and b), so that
a < b, the outcome for council size a is a subset of the outcome for
council size b, which it will provide. The latter is a more relaxed kind
of house monotonicity, but note that it also requires a function that
maps larger numbers to smaller ones, because the smaller council (of
size a) must also obey an (a', a)house proportionality criterion, and
we don't specify a' explicitly.
For ordinary house monotonicity, this function is merely f(x) = x1:
each outcome contains the council that's one seat smaller, and thus that
which is two seats, three seats, etc.
I'll note that IIA failure means you can't always have the property that
people submitting the same ballots will get the same result as the
oneround version (just consider a maximal Condorcet cycle for proof 
it has to be broken in *some* direction). However, housemonotone
methods will come closer than those that are not.

Proportional orderings also have the LeftRightCenter problem. Just to
repeat it: consider an electorate split between Left and Right, but both
preferring Center to the other pole. A proportional ordering either
picks Center for singlewinner (as should be) but produces an unbalanced
council for council size 2, or produces a balanced (Left, Right) council
for size 1 but fails to elect the centrist in the singlewinner case.
For a runoff situation, it's less important that the larger sized
council is balanced than that the smaller one is, because the larger one
isn't going to be elected anyway. However, the distortion can still
appear, but one is free to pick what q (larger size) one pleases.
Perhaps there are some values of q that produce more accurate initial
councils than others? If the electorate was perfectly polarized, a
highly composite value for q would make sense  but probably not in the
LeftRightCenter case.

An even nicer thing to have would be a proportional ordering based on
Approval: then voters could pick acceptable candidates out of a large
number, and later consider these more carefully. Once, I proposed using
ordinary Approval for that purpose, but Benham (I think?) showed that
that would not be cloneproof: a "rich party" could simply flood the
initial selection with its candidates, and as long as the party got a
plurality, all those candidates would be elected and so crowd out the
others.
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