[EM] thoughts on recent messages

[Forest Simmons]

Tied rankings:

I know the context for this was Ranked Pairs, Beatpath, and River,
primarily but in the context of MaxMin, Chris Benham's suggestion gives an
FBC compliant method similar to MinMaxPO with power truncation. Neither of
these methods is Condorcet Compliant, but they are about as close as you
can get to Condorcet Compliance without sacrificing the FBC.

Here's a version of MMPO(with symmetric completion in non-top ranks, and
power truncation at bottom) which is related to both the MaxMin method and
the MinMax method mentioned above:

Construct a matrix whose entry in row i, and column j is the number of
ballots on which candidate i is ranked strictly above candidate j plus one
half the number of ballots on which i and j are ranked equally between top
and bottom, plus the number of ballots on which both i and j are unranked
(truncated).

The MinMax winner for this matrix is the candidate whose column max is
minimal.

If we subtract each element of this pairwise matrix from the total number
of tallied ballots and then take the transpose we get the matrix whose
entry in row n and column n is the number of ballots on which candidate n
is ranked strictly above candidate m, plus the number of ballots on which n
and m are ranked equal top, plus half the number of ballots on which n and
m are ranked equal between top and bottom.

If we elect the candidate whose row min is maximal we get the MaxMin winner
for the matrix.

Which of these two methods is better.  Neither (or both) since they are the
same method in disguise.

So counting equal top as a full defeat in the MaxMin method is the same as
counting equal top as no defeat in the MinMax method. So these respective
conventions in MaxMin and MinMax, respectively are sufficient to ensure the
FBC for those methods.

So we can now begin to understand Chris Benham's cryptic suggestion about
measuring defeat strength with losing votes and counting equal top as a
full ballot defeat for both candidates.

My suggested treatment of middle rank ties is different from Chris's, but
only for the sake of symmetry, so that there is a simpler duality between
the MinMax and the MaxMin matrices, analogous to the fact that the
transpose of the margins matrix is the same as its opposite.

Knowing Chris, I'm sure he has a practical reason for his deviation from my
symmetric version (not to suggest that he knew that I was thinking along
these lines)

A ratings version of some of my my thoughts along these lines can be found
under the title "MinMax Pairwise Approval" in the electowiki list of
election methods at ...

https://electowiki.org/wiki/Minmax_pairwise_approval

Somehere in these archives is a description of a variant of MDDA, namely
MDDA(symmetric completion) which also makes use of the same idea to
preserve the FBC while avoiding a Plurality failure and also avoiding the
Chicken Dilemma in a Burial resistant form.

Avoiding Chicken and Burial at the same time is a challenge.  Different
sincere ballots can lead to the same set of insincere voted ballots, one
from a chicken challenge and the other as a burial challenge.  Since the
election method cannot read minds the winner selection has to punish both
kinds of defections from sincerity at the same time!  In MDDA(sc) it is
assumed that the voters know which kind of threat has to be countered, and
the only difference in defense strategy is whether or not to approve ranks
between top and bottom for threatened faction of voters. (Approve in the
case of burial threat, but not in case of chicken threat).

That's all for now!

Forest
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