Back in December 2008 I criticised Marcus Schulze's "beatpath
Generalized Majority Criterion" (which says in effect that if any
candidate X has a majority-strength beatpath to candidate Y then Y can't
win unless Y has a majority-strength beatpath back to X) in part
because the concept is vulnerable to Mon-add-Plump. That is, extra
ballots that plump for candidate A can cause A to fall out of the set of
candidates that the criterion specifies are qualified to win.
Then it was pointed out to me that to some extent the Mutual Majority
(aka Majority for Solid Coalitions) criterion has the same problem. A
candidate X can be in the set of candidates that are qualified to win
and then some extra ballots that plump for X are added and then the set
of candidates the criterion specifies are qualified to win expands to
include one or more new candidates. X doesn't actually fall out of the
set (as with beatpath GMC), but nonetheless according to the criterion
X's case has been weakened by the new ballots that plump for X.
I propose a replacement for Mutual Majority which addresses this problem
and also unites it with Majority Favourite.
Preliminary definitions:
A "solid coalition" of candidates of size N is a set S of (one or more)
candidates that on N number of ballots have all been voted strictly
above all outside-S candidates.
Any given solid coalition A's "rival solid coalitions" are only those
that contain a candidate not in A.
Statement of criterion:
*If one exists, the winner must come from the smallest solid coalition
of candidates that is bigger than the sum of all its rivals.*
[end criterion definition]
This wording could perhaps be polished, and I haven't yet thought of a
name for this criterion and resulting set. (Any suggestions?)
It might be possible to use the set as part of an ok voting method.
Chris Benham
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