[EM] Uncovered MMPO based Approval

[Forest Simmons]

After putting together some ideas from recent conversations on approval
based chain building and MMPO based approval ... here's a burial resistant
method that popped out:

Let CO be the candidate whose max pairwise opposition is smaller than any
other candidate's MPO.

[Think CO for Cut Off as in approval cutoff candidate]

If this MMPO candidate is undefeated, elect it, otherwise ...

Initialize a chain with the candidate X with the greatest pairwise
opposition to CO, that is with the most winning votes against the cutoff CO.

[This X has the greatest.MMPO based approval]

If X is uncovered, add to the chain the most approved candidate that covers
X.

[That is, the candidate covering X with the greatest pairwise opposition to
CO]

In general, as long as the (head of) the chain is uncovered, keep
augmenting the chain with the most approved candidate that covers the (head
of) the chain.

Finally, elect the last candidate that was added to the chain ... the one
that covers all of the other members of the chain.

In the examples below all three candidates are uncovered, because in a
cycle of three candidates each has a short beatpath to each of the others.

Example 1.

46 A>B
44 B>C(Sincere B>A)
5. C>A
5  C>B

Candidate B is the MMPO approval cutoff CO. The chain is initialized with
candidate A, the candidate with the greatest pairwise opposition to B.

As the only member of the chain, candidate A wins.

Example 2.

49 A
26 B>C
25 C(sincere C>B)

Candidates B and C are tied for MMPO candidate ... but the tie is broken in
favor of B, because B beats C pairwise.

So B is the approval cutoff candidate.

Candidate A is the approval winner, that is the one with the most PO
against the cutoff B.

And A is uncovered, so A wins.

Example 3.

a A>B
b B>C
c C>A

Where c is the smallest faction size, but large enough to keep both a and b
below half of the total number of ballots.

The respective max PO's of A, B, and C are b+c, a+c, and a+b, which can be
written respectively as n-a, n-b, and n-c, where n is the total number of
ballots.

The largest of these MPO's is n-c, since C is the smallest faction size, so
the MinMaxPO candidate must be either A or B, depending which of A or B has
the larger faction.

Case 1: a>b>c
Candidate A is the cutoff. Then C has the greatest approval since C's PO
against A is  c+b which is greater than b, which is B's PO against A. So C
wins ... which means that if the cycle was created by the largest faction
burying the smallest faction, the buried sincere CW won anyway.

Case 2. b>a>c.
Candidate B is the cutoff. Candidate A has the most winning votes against
this approval cutoff, so A wins as the one and only member of the approval
chain.
If the cycle was created by the largest faction (B) burying its sincere
second choice C below its last choice A, then its ploy backfired since its
anti-favorite A got elected.

Note that IRV agrees with this method except on example 3 in the case where
a>b>c.  In that case IRV rewards the burial of C by the A faction.

fws
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