[EM] The D in DMC

[Forest Simmons]

We vacillated between D for Democratic, Definite, or Definitive.

Definitive was in the running because one of our first formulations was to
start by eliminating all of the definitively defeated candidates, i.e. all
of those beaten in both approval and pairwise, that is if X is approved by
more ballots than Y is, and X outranks Y on more ballots than not, then
these two criteria agree and reinforce each other, and therefore should
definitively determine that Y be disqualified.

A nice way to do this is to list the candidates in approval order, too to
bottom. Then draw arrows from the pairwise winners to their respective
losers. Any candidate on the point end of a downward pointing arrow is
crossed off.

The remaining set we called P for some reason or other.

The members of P form a chain that is totally ordered by defeat.
Furthermore, this defeat order is precisely the opposite of the approval
order restricted  to P. The higher on the approval list the lower in the
defeat chain.

So which do you go by? You could say they are tied. But if you want the
method to be Condorcet efficient, you have to defer to the defeat order.
Elect the head of the defeat chain.
More to the point, we trust voters' ordinal judgments more than their
approval cutoff decisions, because they are easier to make.
If the defeat chain P is maximal, as in the usual case, then the DMC
winner  is a Banks candidate.
If you want to make sure the chain is maximal, incorporate as many
additional candidates as possible into the total pairwise order of the
chain, trying the highest approval candidates first.

Only once in a trillion will this change the winner to someone outside of
P, that is to someone that was definitively eliminated ...not worth
mentioning in the context of generic public elections.

This formulation is not as simple as the Benham formulation, but it sheds
light on the simplest possible formulation of DMC:

Elect the lowest approval candidate that defeats every candidate with
greater approval.

-Forest
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