[EM] MMPO(IA>MPO) (was IA/MMPO)

[Forest Simmons]

Kevin,

In the first step of the variant method  MMPO[IA >= MPO] (which, as the
name suggests, elects the MMPO candidate from among those having at least
as much Implicit Approval as Max Pairwise Opposition) all candidates with
greater MPO than IA are eliminated.

I have already shown that this step does not eliminate the IA winner.  Now
I show that this step does not eliminate the Smith\\IA winner either:

Let X be the Smith candidate with max Implicit Approval, IA(X), and let Y
be a candidate that is ranked above X on MPO(X) ballots.  There are two
cases to consider (i) Y is also a member of Smith, and (ii) Y is not a
member of Smith.

In both cases we have MPO(X) is no greater than IA(Y), because Y is ranked
on every ballot expressing opposition of Y over X.

Additionally in the first case IA(Y) is no greater than IA(X) because X is
the Smith\\IA winner.  So in this case MPO(X) is no greater than IA(X) by
the transitive property of "no greater than."

In the second case, X beats Y pairwise since X is in Smith but Y is not.
This entails that X is ranked above Y on more ballots than Y is ranked
above X.  In other words, X is ranked on more ballots than MPO(X).
Therefore IA(X) > MPO(X),

In sum, in neither case is the Smith\\IA winner X eliminated by the first
step in the method MMPO[IA>=MPO].

We see as a corollary that step one never eliminates a (ballot) Condorcet
Winner.  In particular, it does not eliminate a (ballot) majority winner.
And since MMPO always elects a ballot majority unshared first place winner
when there is one, and MMPO is the second and final step of the method
under consideration, this method satisfies the Majority Criterion.

Also worth pointing out is this: since step one eliminates neither the IA
winner nor the Smith\\IA winner, if there is only one candidate that
survives the first step, then the IA winner is a member of Smith, and the
method elects this candidate.

Also in view of this result, I suggest a strengthening of the Plurality
Criterion as a standard required of any method worthy of public proposal.

A method (involving rankings or ratings) satisfies the Minimum Ranking
Requirement MRR if it never elects a candidate whose max pairwise
opposition is greater than the number of ballots on which it is rated above
MinRange or (in the case of ordinal ballots) ranked above at least one
other candidate.

What do you think?

Also we need a nice name for the set of candidates that is not eliminated
by step one.

Any suggestions?

Forest
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