[EM] Defeat Strength Demystified

[Forest Simmons]

When I first joined the EM List twenty years ago the main topic of debate
was margins vs winning votes for measuring defeat strength. It was all very
mysterious to me  ... to a mathematician the symmetry of margins was
appealing ... and as a  a sympathizer of underdog minorities to me it
seemed callous to totally disregard the losing votes when they might help
resolve a Condorcet cycle. On the other hand, there was the point of view
that when there are competing majorities, the proposition supported by the
greatest majority is the one most likely to be true. However a cynic might
question this altruistic truth seeking assumption and assert that it's not
so much a question of right or wrong but of who can get their way.

Which brings us to game theory, which looks at elections as multiplayer
games with the players (voters or voter factions) strategically trying to
optimize their expected personal or factional "utilities" given the rules
of the game as well as the information they have about the preferences,
desires, or "utilities" of the other players.

Once I became aware of this point of view, I saw the futility of Borda's
assumption of honest voters, and the irrelevance of Saari's appeal to
geometric symmetry in Borda's defense. Also it made it more obvious why the
standard use of Cardinal Ratings/Score/Range/Grade ballots might just as
well be replaced by simple Approval, since they all have the same optimal
strategy ... only the naive voter would vote strictly between the extremes.

[Of course there are some extremely sophisticated voters who might factor
in an externality that we could call the "ultimate utility of supporting
eternal truth" ... not part of the limited scope of the voting game proper
... perhaps something more along the lines of Pascal's wager.]

After this point of view soaked in ... the defeat strength debate started
to make more sense. In fact, a paramount ranked voting strategy problem is
the insincere "burial" of a second choice to give added support to a first
choice. This problem is especially evident in pairwise methods like Borda
and Condorcet. But this kind of attack against a sincere Condorcet
candidate is easier to defend against when defeat strength is measured by
winning votes.

Once this fact soaked in to my newbie psyche, I saw the wisdom of the
Ossipoff camp with its impressive array of defense criteria based on
winning votes.

Eventually Mike O. went on to bigger and better things but a few years ago
he made a brief, but passionate, return to the EM List when the
Possibilities of Hope seemed to include a real possibility of election
reform.  As we weighed the merits of various methods it suddenly became
apparent that we didn't have a Condorcet method that was immune to both
burial abd "Chicken," a ploy that had not concerned us much in the past but
now loomed larger.

O course IRV came up as a method that was immune to both Burial and
Chicken, but at the expense of the Condorcet Criterion.  A flurry of
activity on the EM list searched for a hybrid between IRV and Condorcet
similar to what we have seen since the resurrection of IRV as RCV.

BTR-IRV and Benham were the leading contenders, but neither of these
inspired the fire in anybody from the glory days of the List. A few of us
toyed with a hybrid between Condorcet and Approval called Approval Sorted
Margins (ASM) that gave a way of defending against both Burial and Chicken,
but nobody took it seriously because unlike the automatic defense under IRV
it required awareness of the problem to know when to lower the approval of
a potential chicken defector ... furthermore the addition of approval into
the mix went against the Universal Domain purity ethos in the form of
ranked ballots only.

It was soon pointed out that margins automatically defends against chicken
ploys, and it was already well known that with minimal precaution wv
defends against burial, neither requiring any approval lever ... but nobody
quite managed to combine them into one holy grail ... because, as I
recently (last week) showed, the limitation to Universal Domain makes it
impossible. However, the method under Universal Domain requiring the least
vigilance to defend against both of these kinds of attacks is Fractional
Approval Sorted Margins. [Here the margins referred to are approval
differences, not pairwise defeat margins.] The defensive maneuvers required
are truncations or raising to equal top in the respective cases of Chicken
or Burial.

Now here is a suggestion for a minimal departure from Universal Domain that
makes both Burial and Chicken gambits too risky to be practical with added
benefit of potentially settling the wv versus margins debate once and for
all!

[But probably not before the debate about round or flat earth:-)]

To each ranked preference ballot append a check box labeled "symmetric
completion?"

Here we are making use of the equivalence of margins and wv under symmetric
completion of the ballots.

If more than half of the voters check the optional box, then defeat
strength will be according to margins ... otherwise wv is used.

Another more elegant way to finesse this thang is to symmetrically complete
those ballots having checked boxes, and then tally all of the resulting
ballots (checked or not) by wv rules.

The symmetric completion of a ballot takes place operationally at the
pairwise matrix stage ... if candidates i and j are ranked equally on a
ballot, then that ballot normally contributes nothing to row i or row j of
the pairwise matrix. But under symmetric completion it contributes 1/2 to
both the (i, j) and the (j, i) entries of the pairwise matrix.

I hope this has been interesting and stimulating to the imagination of
possibilities ... not the end of all debate... which would be more of a
tragedy than a triumph!
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