[EM] FBC, Clone-Winner, and pairwise components seem incompatible
stepjak at yahoo.fr
Sat Jun 25 09:42:21 PDT 2005
I spent some amount of time trying to determine whether a method could
simultaneously satisfy FBC, SFC, SDSC, and Clone-Winner. I'm thinking not.
SDSC is the easy one, since you can satisfy it using an approval measure,
whereby ranked candidates are approved and the others are disapproved. Seen
this way, already ranked-ballot Approval could be said to satisfy it.
Ranked-ballot Approval would also already satisfy Clone-Winner, assuming a
clone is always ranked or unranked based on whether the original candidate
was. Also, Approval satisfies FBC. So an obvious question is: "Is there some
minimal modification to (ranked-ballot) Approval that would add some (any)
pairwise comparison, without losing any of the other properties?"
Schulze and the other cloneproof pairwise methods satisfy all of the
properties except for FBC. I've written before (I believe) that I don't
think beatpaths (i.e., the notion of the contest A:B affecting other candidates)
are compatible with FBC, since increasing the strength of A>B (imagining that
both are favorites) typically *could* make A win at the expense of some
candidate other than B.
However, Majority Defeat Disqualification Approval ("MDDA," a good name even
without the punctuation) satisfies SFC, which at first glance seems to
require a beatpath notion: In practice, it seems to say that if there is a
(majority) defeat A>B but no defeat anyone>A, then B can't win, while if I
add a win C>A then B is no longer barred from winning. But MDDA "goes beyond
the call of duty" by making all of the defeats damning, and if all candidates
are damned, then adding C>A can only make A lose, no one else. So MDDA
satisfies FBC and SFC without the use of beatpaths, although it does use
a pairwise component.
I can't think of any way to satisfy SFC without using a pairwise component.
Summary so far:
"beatpath methods": SFC, Clone-Winner, beatpaths.
"ranked" Approval: FBC, Clone-Winner.
MDDA and MMPO: FBC, SFC, pairwise component.
I considered some other substitutes for SFC; for example, you could try to
add Majority Favorite to ranked-ballot Approval. But then Clone-Winner is lost,
unless you also satisfy Majority (for solid coalitions). There are a couple
of ways to do that: You could directly measure the size of solid coalitions,
as in DSC, but then FBC is lost. You could use IRV eliminations somehow, but
surely this would break FBC and SDSC (as well as monotonicity). Finally, you
could use pairwise contests and ensure the winner comes from Woodall's CGTT
(the minimal set of candidates with majority-strength wins over all the other
candidates), but as I've written earlier, I don't believe Smith (etc.) can
even be *modified* to satisfy FBC.
So suppose that FBC, Clone-Winner, and pairwise components (including even
Majority Favorite) are incompatible. It seems doubtful that one could do much
better than Schulze when satisfying the latter two. It seems also that ranked-
ballot Approval is the best possible using the first two. So the interesting
question is how far we can get satisfying FBC and criteria like Majority
Favorite or SFC. Could we even add something *stronger* than SFC?
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