[EM] Multiwinner runoffs (Re: Proportional Representation Systems I'd Support)

Kristofer Munsterhjelm km-elmet 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 house-monotone
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 

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 
two-round method to return the same result as an automated version of 
the one-round 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 house-monotone 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) = x-1: 
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 
one-round version (just consider a maximal Condorcet cycle for proof - 
it has to be broken in *some* direction). However, house-monotone 
methods will come closer than those that are not.


Proportional orderings also have the Left-Right-Center 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 single-winner (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 single-winner 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 
Left-Right-Center 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 

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