[EM] Unifying DMTC and DMTCBR, and method X criterion

[Kristofer Munsterhjelm]

Here's a way to unify DMTC (elects a DMT candidate) and burial resistance:

Suppose that candidate A has more than 1/3 of the first preferences, and 
beats B pairwise. Then B must not win.

This works because B>A voters can't influence A's first preferences, nor 
can they influence whether A beats B pairwise. A neat trick is also that 
this can never lead to a cycle, because there can't be more than two 
candidates with more than 1/3 first preferences, and A can't both beat B 
and B beat A at the same time.

In that spirit, here's a weak criterion that's passed by method X:

Let an election restricted to set S be the election after everybody but 
the candidates in S are eliminated. Then if, in every restricted 
election involving A and B, A has more than 1/|S| of the first 
preferences, and A also beats B pairwise, then B must not be elected. 
Here |S| is the cardinality of set S, i.e. the number of non-eliminated 
candidates in the restricted election in question.

Method X passes this criterion because when B's finding a maximizing 
elimination path, no matter who's eliminated, A always has more than 
1/|S| first preferences. Therefore A can never be eliminated, so B is 
either forced to be eliminated (in which case he obviously can't be 
elected) or his final matchup is against A, who beats him pairwise. On 
the other hand, A can do no worse than following the same path, leading 
to a matchup against B, which he wins.

It's weak because we need the criterion to hold for all restricted 
elections, not just one of them. So the DMT set has to be 
well-distributed: all its candidates must have about the same support. 
But in return, if the criterion bars B from being elected, voters may 
flip pairwise preferences between DMT members and non-DMT members all 
they want; it'll never help. Furthermore, a voter lowering his last 
ranked DMT set member under a bunch of non-DMT candidates also never helps.

Weak as it is, it could be useful in method design, just like clone 
independence. I'll take every handhold scaling this cliff :-)

-km