[EM] Old Topics ... New Insights

[Forest Simmons]

The geometry of MinMax has come to the fore in our attempt to decide which
two candidates should participate in the sincere runoff.

The candidate C whose max opposition is least .. the MMPO candidate ... can
be visualized as being surrounded by the other candidates with voters
spread out among them ... the voters that prefer Y over C all on the Y side
of C.

Let V(Y) be the set of voters that prefer Y over C. If the distribution of
voters is lopsided around C, bulging out in the direction of Y, then the
number #V(Y) will be larger than average.

If, on the other hand, #V(Y) is relatively small, there will be no
prominent bulge in the voter distribution in the direction of Y.

So adjusting the position (or choice of) C to minimize the largest value of
#V(Y) tends to minimize bulges, and put C close to the center of the voter
distribution (center of symmetry in the ideal case as in a Yee diagram).

This minimization puts C in a central position relative to the voters, but
is no guarantee that they are huddled in close to C; they might be spread
out in a crater shaped distribution.

Nevertheless, C is a valuable reference position; any candidate preferred
pairwise over C should have some credibility.

If you use C as the approval cutoff candidate, the candidate with the
greatest pairwise opposition to C is the approval winner ... corresponding
to the "greatest bulge candidate" [or the greatest threat to the defensive
champ, depending on your POV].

>From another point of view, the candidates that prefer Y over C are the
ones that will be disappointed if C is elected instead of Y ... so electing
the candidate C that minimizes max #V(Y),( is a way of minimizing the
number of disappointed voters.

>From a Round Robin tournament point of view, the MMPO team/ player is the
defensive champion of the tournament ... the one limiting their opponents
to the fewest points.

The other finalist team (the "bulge" team) is the one scoring the most
points against the defensive champ.

Taking into account all of these considerations, we suggest electing the
sincere winner between the MMPO candidate C, and the candidate Y whose
supporters form the greatest bulge in the inevitibly asymmetric
distribution ... i.e. the candidate that, if not elected, would result in
the most disappointment... in other words, the candidate that outranks C on
the greatest number of ballots.

Example 1.

46 A
2 C=A
4 C
4 C=B
44 B

Candidate C has a max PO of 46 (from opposition by A)
candidate A has a max PO of 48 by B
Candidate B has a max PO of 48 by A

The MMPO candidate is C, since 46<48.

The max opposition to C comes from A ... so A is the bulge/ main opposition
candidate ... which makesthe sincere runoff is between C and A.

Who will win?

That depends on how many of the B voters (if any) sincerely prefer C over A.

Note that the MMPO candidate C could also be characterized as the Majority
Judgment Candidate, because it is the choice of the median voter.

The ballot Condorcet winner is A, beating C 46 to 8, with pairwise tie
between A and B broken in favor of A by asymmetry considerations.

So the Condorcet winner is not always the median voter choice, after all!

Is it always true that for approval ballots the MJ winner is the same as
the MMPO winner?

Who should the sincere finalists be with cardinal ratings ballots?

In particular, who makes a better approval cutoff candidate ...??? ... the
MMPO candidate? ... the MJ candidate? ... or the Bucklin candidate?

In light of example 1 above, should we always elect the ballot CW when
there is one? Or should we wait and see if the sincere runoff between the
MMPO candidate and its greatest pairwise challenger detects the ballot CW
to be a compromise candidate ... perhaps via favorite betrayal?

[Relevant here is the fact that with only two judgment categories (say good
vs bad), the candidate approved on the most ballots is undefeated pairwise.]

Lots of food for thought!

Practical takeaway/ public proposal suggestion:

Lacking a ballot CW, elect the winner of a sincere runoff between the
ballot MMPO candidate C and the candidate ranked ahead of C on the greatest
number of (original, possibly insincere) ballots.

fws
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