[EM] Three rounds

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Sat, Nov 15, 2008 at 8:36 PM, Kristofer Munsterhjelm
> <km-elmet at broadpark.no> wrote:
>> 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.
> 
> Mutual majority looks the same as the Droop criterion, but for single
> winner cases.
> 
> I wouldn't think much of a condorcet method that doesn't meet Smith,
> but the two criteria aren't the same.

Yes. Smith is a subset of mutual majority. The Condorcet winner is 
always in Smith, so when there's a CW, it's in the mutual majority set.

>> 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).
> 
> If the objective is to find a multi-winner equivalent of the condorcet
> criterion rather the Smith criterion, I am not so sure how useful that
> is.
> 
> It would be a criterion that covers less cases than the Droop criterion.
> 
> Maybe
> 
> An outcome is not a valid outcome if there is any non-elected
> candidate who is preferred to all the winning candidates by a Droop
> quota of the voters.  No invalid outcome may be used unless there are
> no valid outcomes.
> 
> This would be similar to re-defining the condorcet criterion as
> 
> A candidate shall be deemed an invalid winner if a majority prefer any
> other candidate to that candidate.  An invalid candidate may not be
> declared the winner unless there are no valid candidates.

That rule would admit more sets than the DPC. Call the candidates that a 
Droop quota supports above the others, "Droop CWs". Your criterion 
basically says "if you're picking k winners, and there are at least k 
Droop CWs, all the winners have to be Droop CWs; if there are less than 
k Droop CWs, those have to be included in the winning set".

If there are Droop CWs, and also there's a subset that has to be 
included as the winners, then those winners will be Droop CWs (similar 
to how the Condorcet winner, when there is one, is in the Mutual 
Majority set). However, if there's a single winner CW for the election 
in question, that winner will also be a Droop CW. Similarly, if there's 
a candidate that x voters prefer to all others, where x is larger than 
the Droop quota, that candidate will also be a Droop CW.

I guess that shouldn't surprise us; since Condorcet doesn't imply Mutual 
Majority, a multiwinner Condorcet criterion wouldn't imply the DPC 
either. However, the failure mode is different. Condorcet fails MM only 
when there's no CW (and the Condorcet criterion can't say which 
candidate you should elect); however, this fails even when there are 
Droop CWs (since we know Condorcet and the DPC is incompatible, and that 
a Condorcet winner must also be a Droop CW).

So we may need a Smith set, and that set would have to be defined so 
that electing from it implies DPC. I have no idea how it would actually 
be defined, though.