[EM] Three rounds

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Sat, Nov 15, 2008 at 3:45 PM, Kristofer Munsterhjelm
> <km-elmet at broadpark.no> wrote:
>> I don't think so. Though I haven't investigated this method, I'm thinking
>> that since it uses a divisor method (Sainte-Laguë), there would be instances
>> where it breaks quota, just like ordinary Sainte-Laguë breaks quota, since
>> quota (no candidate or party should need more than a quota worth of votes to
>> get a seat, or get a seat with less than a quota's worth) is incompatible
>> with the two criteria Sainte-Laguë meets (population pair and house
>> monotonicity).
> 
> Well, I was thinking if the proposal was used with d'Hondt.

D'Hondt is also a divisor method, and since divisor methods meet the two 
monotonicity criteria, they are all incompatible with quota. To my 
knowledge, only divisor methods meet both monotonicity criteria. I'm 
unsure as to whether that is true for divisor methods on sets, like 
Ossipoff's Cycle Webster method, but it doesn't seem to be the case for 
Cycle Webster, at least.

>> Perhaps something like "if the method, when electing k winners,
>> returns the set X, and there is a way of partitioning the ballots into k
>> piles so that each pile has a CW, and each CW is in X, then the method
>> passes this criterion".
>> Or, is there something that is to the Droop proportionality criterion as the
>> Smith criterion is to mutual majority?
> 
> In the single winner case, Droop proportionality says that if a
> majority ranks a group of candidates above all other candidates, then
> one of those candidates will win.  All methods that meet the condorcet
> criterion would also meet the Droop proportionality criteron.
> However, all single winner methods that meet the Droop proportionality
> criterion don't necessarily meet the condorcet criterion.  IRV being
> an example that meets the Droop proportionality criterion but not meet
> the condorcet criterion.

The single-winner criterion corresponding to the DPC is the mutual 
majority criterion. Any method that's Smith also passes mutual majority, 
and since Condorcet is just the case of the Smith set being a singleton, 
any Condorcet method passes the criterion when there's a CW. When 
there's not, a method may pass or fail; it passes if it's Smith, and it 
may either if it's not. Minmax and Black both fail mutual majority, to 
my knowledge. While I'm not familiar with which well known Condorcet 
methods, if any, that pass mutual majority while not being Smith, it's 
easy to make one: "CW if there is one, else IRV", for instance.

> In that context, a multi-winner condorcet criterion would have to a
> stricter requirement than merely meeting the Droop criterion and any
> method that fails the Droop proportionality criterion would have to
> fail it.

It may pass it yet fail DPC if the multi-winner Condorcet "winner" 
(winner set?) is not present in all elections. If it elects the 
multiwinner Condorcet candidates (and in that case passes the DPC) when 
they exist, but fails DPC in all other cases, then it would fail DPC in 
general. But if it's like the Smith set, in that it's a subset of the 
Droop Proportionality set (mutual majority set in the case of 
single-winner), then what you say is true.

If it's a subset of the DP set, then we know that it can't always be a 
proper subset. Otherwise, there would be cases where there are no 
eliminations in STV, so that any method that passes DPC must elect the 
entire set; if the subset was a proper subset, it would the fail the 
DPC, which is not desirable. But that's not so surprising, since it's 
also the case with Smith (regarding the mutual majority set); just 
produce ballots that all vote A > B > C and then a cycle among the other 
candidates.

But what would this multi-winner Condorcet criterion be? That's the 
question. One may also ask whether it's a desirable criterion (like 
Condorcet), or if it's too strict (like Participation).