[EM] One Way to Respect "Pairwise Loser Should Not Win"

[Forest Simmons]

Elimination methods generally proceed by eliminating the "worst" remaining
candidate at each step.

"Worst" is defined in various ways ... the simpler the better, all else
being equal.

No gauge of "worst" is infallible ... especially when defined in one sound
bite.

Robert's dictum, can be thought of as a corrective... all else being equal,
do not elect a pairwise beaten candidate ... especially if she is
considered to be "worse" than the other remaining candidates.

Think of BTR IRV. Here "worst" means fewest transferred top votes ... an
appealing criterion ... but not infallible.

The tentative judgment based on "fewest top votes" is not a reliable
standard of worst when center squeeze is a real possibility.

Here's a suggestion for incorporating Robert's dictum in conjunction with
any notion of "worst" ... whether fewest top votes, most bottom, least
pairwise support, most pairwise opposition, or any thing else:

While there remains at least one un-eliminated candidate ...
eliminate the "worst" remaining one ...
AFTER
eliminating any (and every) candidate pairwise defeated by it.
EndWhile

[Then elect the last candidate to be eliminated.]

Rationale: If there exists a candidate X pairwise defeated by the candidate
Y considered to be "worst" by some tentative criterion, then evidently a
majority of the participating voters consider X to be even worse than Y ...
not withstanding the tentative judgment of of Y being "worst."

In other words, whatever the criterion for "worst" may be ... that judgment
is only tentative until confirmed by a democratic majority of the
participating voters ... hence it can and should be overridden when the
voters (according to their ranked preference ballots) prefer keeping Y over
X.

Any elimination method following this template will be Condorcet efficient.
Beyond that ... absent a Condorcet Winner, it will still elect an uncovered
candidate ... important insurance against loser complaints ... insurance
that no extant public methods offer.

So fill in the template wih your favorite standard for "worst" ... least
GPA, most disapproval, worst majority judgment, fewest total yards gained,
fewest free throws completed, fewest technical errors, etc... use your
imagination ... but remember, for public election proposals ... the simpler
the better ..."defeat by the strongest majority of participating voters"
might barely pass.

In general, elimination methods (like IRV) fail monotonicity ... except a
few them when they are based on a fixed, monotonically generated agenda.

For example,  when "worse" means  worst according to a monotonically
generated agenda, this method (like Sequential Pairwise Elimination) is
monotone ... otherwise probably not.

-Forest
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