[EM] Elimination with Take-Down

[Forest Simmons]

What started as Chain Climbing has evolved into a Banks efficient take-down
Elimination protocol.... whenever you eliminate an alternative, let it
take-down with it any alternative it defeats ... an expedient for
conferring Banks efficiency on elimination methods.

[A method is Banks efficient iff it is the head of a maximal transitive
beatpath ... each path member defeats all subsequent path members, not only
its immediate successor.]

The heuristic for this policy is that once you have gone to all of the
trouble to figure out who is so bad or weak to deserve Elimination, you
should get your money's worth by taking down with it all of the candidates
that are even weaker or worse ... at least in the sense of being directly
defeated by it..

If at every step there was a candidate defeated by every remaining
alternatives, eliminating that one by itself would be an ideal IIA step.

If we are eliminating from the bottom of a  list, we can get the "defeated
by all remaining" in line for elimination by one pass of a pairwise sink
sort.

Even if there is no universally defeated alternative among the remaining
alternatives, then the sink sort loser should still be one of the usual
suspects to eliminate anyway ... assuming the candidates starting near the
bottom of the list were already relatively unpromising according to
whatever criterion goverened the list formation in the first place.

Here's a proposed method:

"Sink Loser Take-Down"

Lacking a pairwise undefeated candidate ...

0. List the candidates in the Universal Tie Breaking Order with the most
promising candidates at the top of the list L.

[This UTBOrder is a highly decisive natural extension of the Implicit
Approval Order as highlighted recently in the EM Thread entitled "General
Purpose Universal Domain Tie Breaking Order."]

1. If only one uneliminated candidate remains on the list, elect that
candidate, ... else ...

2. ...  update the list by sink sorting it pairwise.

3. Let "pivot" be the candidate ending up at the bottom of this updated
list.

3. Update the list again by eliminating the pivot along with every
candidate defeated by it.

4. Repeat steps 1 through 3 until a winner appears in step 1.

To make this more efficient, in step 2 take advantage of the fact that it
only takes one pass of sink sort to find the pivot alternative.

Here's the one pass procedure:

Initialize the candidate at the top of the list as "pivot".

While there exists some candidate X immediately below "pivot," swap them if
X defeats "pivot", else let X be the new "pivot." EndWhile.

Here is a bonus sincerity check:

Do a final runoff between the winner (so far) and the last pivot, using a
fresh set of ballots dedicated to this binary contest.

Since this runoff is a secret ballot final binary choice, there is no
incentive for insincerity on these runoff ballots.

Furthermore, in the likely case that the lack of an undefeated pairwise
winner was caused by burial of a sincere Condorcet Winner, this step will
almost surely restore that sincere CW.

What do you think?

-fws
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