Here's an idea of a possible method inspired by my MCAB and IFPP:
Say that a set of candidates is eliminable if their combined first
preference count is less than or equal to a third of the total number of
voters, but adding any candidate would make the total exceed 1/3.
For one candidate: A's score is the number of voters.
For two candidates: A's score is A>B.
For three or more: A's score is the maximum score obtained by choosing
some eliminable set, eliminating it, and recursing. The set is chosen so
as to maximize A's score.
The idea here is that suppose A is a DMT candidate, but burial reversed
A>C into C>A. Then it's likely that A>C was weaker than A>B, so that A's
score was A>B. So A's score isn't affected by the burial. Suppose X is
not a DMT candidate. Then A must be retained because just adding A would
push the set over capacity. So X's score must be X>A which (given the
assumption above that C>A is weak) must be weak; either A beats X or X
is C and X just barely beats A pairwise.
That the set is maximizing A's score should (hopefully?) retain
monotonicity because shuffling candidates who are part of an eliminable
set around won't affect the outcome. Maybe there's a possibility for
nonmonotonicity involving some other candidate going out of or into the
eliminable set? But I can't quite see it.
So yeah, this is all kinda sloppy, but I thought the idea would be
interesting. The recursion would make it a pain to program, though;
maybe dynamic programming tricks can be used to make it polytime.
-km
Here's an idea of a possible method inspired by my MCAB and IFPP:
Say that a set of candidates is eliminable if their combined first
preference count is less than or equal to a third of the total number of
voters, but adding any candidate would make the total exceed 1/3.
For one candidate: A's score is the number of voters.
For two candidates: A's score is A>B.
For three or more: A's score is the maximum score obtained by choosing
some eliminable set, eliminating it, and recursing. The set is chosen so
as to maximize A's score.
The idea here is that suppose A is a DMT candidate, but burial reversed
A>C into C>A. Then it's likely that A>C was weaker than A>B, so that A's
score was A>B. So A's score isn't affected by the burial. Suppose X is
not a DMT candidate. Then A must be retained because just adding A would
push the set over capacity. So X's score must be X>A which (given the
assumption above that C>A is weak) must be weak; either A beats X or X
is C and X just barely beats A pairwise.
That the set is maximizing A's score should (hopefully?) retain
monotonicity because shuffling candidates who are part of an eliminable
set around won't affect the outcome. Maybe there's a possibility for
nonmonotonicity involving some other candidate going out of or into the
eliminable set? But I can't quite see it.
So yeah, this is all kinda sloppy, but I thought the idea would be
interesting. The recursion would make it a pain to program, though;
maybe dynamic programming tricks can be used to make it polytime.
-km
