[EM] Consistency in PR methods

[Forest Simmons]

Recently, in his "grand compromise" proposal, Jobst suggested k-consistency as 
a valuable criterion.

In the multiwinner context,
            a method is k-consistent
                      iff
            a candidate set S winning
        in each of its k candidate supersets
                implies that
             S is the winning set.
For example, Proportional Approval Voting (PAV) is k-consistent for all k 
greater than or equal to twice the number of seats.

This follows from the fact that under PAV the winning set S is the set 
that gets the over all highest "sum of discounted redundant layers of 
representation" from the ballots, and that this kind of sum depends only 
on which candidates are in the set, not on how many are excluded. Here we 
assume (as usual) that the approval ballots are not altered for scoring 
different subsets.

It isn't possible in general for a PR method to be k-consistent when k is 
between the number of seats and twice that number, as the following two 
seat example shows:

25 A=C
25 A=D
25 B=D
24 B=C
1  B
Any decent PR method must seat the set {A,B}.
However the set S={C,D} will come out winner in any of its three 
candidate supersets, since it doesn't have to compete directly with {A,B}.

Of course {A,B} will also win in any of its three candidate supersets.
PAV satisfies the stronger property that the winning set of candidates 
always wins in any of its supersets.

Does anybody know of any other PR method that satisfies k-consistency for any k 
greater than twice the number of seats? Perhaps some version of List PR 
would meet this criterion.

STV doesn't satisfy k-consistency for even one value of k greater than the 
number of seats, not even in the case of single winner elections.

I would be extremely surprised if CPO-STV satisfied k-consistency for some 
value of k greater than the number of seats when that number is greater than 
one.

Also PAV is the only method of which I am aware that satisfies the other 
kind of consistency (relative to subsets of voters rather than subsets of 
candidates):  if the ballot set is partitioned into subsets, and the 
candidate set S wins according to each of those subsets of ballots, then S will 
win according to the entire ballot set.

Perhaps some list PR method also satisfies this kind of consistency.
How about a version of Candidate Proxy that satisfies both?
Forest
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