As a part of my thinking about generalizing STV, which is based on a
weighted positional method (namely, Plurality), I got to think about
what one may say about a loser elimination method's criteria based on
the criteria of the method it's built on top of.
For instance, STV methods are interesting in that they pass the Droop
proportionality criterion. If they don't, they're not very proportional,
and thus wouldn't be of much interest as PR methods either. It is
reasonable (although I haven't proven this) that if any "elect and
punish" STV generalization that reduces to loser elimination in the
single case is to work, it must pass mutual majority in the single
winner case, since that's Droop proportionality for a single winner.
Therefore, it's useful to know what election methods one can combine
with loser elimination so that the result passes mutual majority. Now,
it might be that my intuition is wrong here and you can get a good
multiwinner method out of something that doesn't pass mutual majority,
but I don't quite see how; it probably won't be much like STV.
From this, one may ask, when does a loser elimination method pass
mutual majority? To simplify things, let's call a method LE-X (for
LoserElimination-X) when it is a loser elimination method based on X.
Then LE-X passes mutual majority if X passes (simple) majority. To see
how this works, consider the case when all but one member of the mutual
majority set has been eliminated. By the definition of mutual majority,
this remaining candidate is now preferred to all other candidates by a
majority of the voters. Therefore, that candidate has a (simple)
majority, and so won't be eliminated if the method passes Majority.
The general proof above might be generalized: if there is a set, so that
eliminating a random candidate doesn't make other candidates drop out of
the set (unless that candidate was in the set), and method X elects the
candidate in that set if there's a single member in that set, then LE-X
always elects from that set.
To give a concrete example: LE-[Condorcet method] should be in Smith,
since a Condorcet method elects the CW if it exists. When all but one
Smith set member has been eliminated, the remaining member is the CW.
The reasoning above is sufficient, but it's not necessary: Borda fails
Majority, but LE-Borda (Baldwin) passes mutual majority. Still, because
of the above, it would be interesting to know when a weighted positional
method passes Majority, since weighted positional methods can be easily
inserted into STV (if I can find a way of doing the reweighting correctly).
However, here's the other thing I discovered. Unless I'm wrong, all
weighted positional systems that pass Majority are like Plurality, but
with differing tiebreakers. How do we see that? Consider the worst case
scenario for a weighted positional system with regards to majority. It is:
(x+1): A > B > [rest]
x : B > [rest] > A
Assuming a somewhat reasonable method where placing someone nth place
gives him higher score than or equal score to placing him (n-1)th place,
this is the worst case, since B gets all the second place points from
the (x+1) majority, and all the first place points from the x minority,
whereas A gets only enough to fall within the Majority criterion.
Call the score given to first place s_0, and that given to second place,
s_1, where these scores are normalized so that last place gets zero
points. Then, for the method to satisfy Majority, s_0 * (x+1) > s_0 * x
+ s_1 * (x+1) for any and all positive finite x. There are two ways of
doing this. Either s_1 = 0, or (the more general case) s_0 > s_1 * x. In
the first case, we have Plurality; but in the second, it's obvious that
being in second place versus not being in second place on a ballot only
makes a difference if you're tied for first, because (as was the
design), no amount of second place votes can count as a single first
place vote.
Therefore, among weighted positional methods, only Plurality and methods
that work like Plurality but with some implied tiebreak, can satisfy
Majority.
The example above did not regard modified weighted positional methods
where it's possible to have equal preferences, since even for those, the
ballots above are valid and so the argument holds; you only have to fail
Majority once to fail it entirely, so the only thing it can do is to
make Plurality fail Majority.
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Finally, I'll mention another observation that seems correct: If X meets
Condorcet Loser and Reversal Symmetry, then LE-X meets Condorcet. The
logic goes like this: For LE-X not to pass Condorcet, it must at some
point eliminate the CW. So there must be a CW that's placed last
according to X on a subset of all candidates. But reverse all the
ballots. Now the CW is the Condorcet loser. Since the method meets
Reversal Symmetry, and the CW was placed last before we reversed the
ballots, the Condorcet loser must be placed first; but that's impossible
because X meets Condorcet Loser.
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As usual, there may be errors in the above. If you find any, let me know.
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