[EM] Fragmented Condorcet doesn't imply DPC
Kristofer Munsterhjelm
kmelmet at broadpark.no
Mon Jan 12 11:41:36 PST 2009
From offlist conversation, I discovered an example of that my
tentative multiwinner criterion, Fragmented Condorcet, doesn't imply DPC.
Consider this bulletvoting situation:
400: A
400: B
400: C
300: D
Three to be elected.
The Droop quota is 375. So, according to DPC, A, B, and C should be elected.
There's at least one way of splitting these votes into three bundles so
that "the right" candidates (A, B, and C) get elected, and so that each
contains 500 ballots (1500/3). For instance,
first bundle: 400 A, 100 D
second bundle: 400 B, 100 D
third bundle: 400 C, 100 D
but there's also a way that isn't proportional:
first bundle: 300 A, 100 B, 100 C A beats B and C, A wins
second bundle: 300 C, 100 B, 100 A C beats A and B, C wins
third bundle: 300 D, 200 B D beats B, D wins
The parallels to packing and cracking are obvious. I suppose I shouldn't
be surprised, since Condorcet doesn't imply mutual majority, either, but
this allows for the possibility that Fragmented Condorcet contradicts
the DPC.
If it does, the construction would probably be something like: arrange a
setup so that there's only one way of arranging Condorcet winners in
each bundle, all others causing cycles in at least one bundle. Then
modify this arrangement so that the only CWpermitting partitioning
contradicts the DPC.
I don't know if that's possible, though, and it would have to use full
preference votes (e.g not just bullet votes).

In general, I think my surprise at this confirms what I've suggested
before: that it's not enough to technically satisfy criteria, one must
also gracefully fail towards them. Clone independence isn't worth much
if the system is "remove clones then run Borda". Similarly, if it's
possible to pass both FC and the DPC, then the method, for a ballot set
where DPC and FC provide no constraints, must elect results that are in
some fashion "close" to a ballot set where they would provide constraints.
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