[EM] Standard MJ Reformulation

[Forest Simmons]

Symmetric MJ when there are an even number of categories ...

Temporarily augment with a virtual empty category midway between the two
extreme categories, i.e. having the same number of categories to either
side of it.

Now use the odd category version to find the Majority Judgment.

The only problem with this is the possibility of the MJ category turning
out to be the virtual category.

Even this is not a problem unless two candidates end up with the same
virtual  MJ ... the standard MJ tie breaker can be thrown into an endless
loop by this case.

We need a new tie breaker, and the easiest approach is to build it into a
new more sensitive procedure for finding the MJ ... and this new procedure
requires the introduction of one new virtual judgment category between each
pair of the original categories.

For example, if the four originals are A, B, C, and D, the virtual judgment
categories might be called ab, bc, and cd ... so now we have seven
categories with the implied order A>ab>B>bc>C>cd>D.
The virtual categories start out empty ... they are not ballot categories
... only accessories to facilitate the tally, including the tie breaker
information.

Note that midway between any two "real" categories is a member of the
extended set of categories, and that this extended set is totally ordered.

For now we concentrate on one candidate X.  We build up a string S(X) of
category codes (including code names of virtual categories) for X as
follows...

Initialize S(X) as a blank string ... then ..

WHILE there remains more than one judgment vote for X, append to the string
S(X) the code for the judgment category midway between X's two outer most
remaining judgment votes before discarding them.
ENDWHILE

If there still remains one undiscarded vote, append its code name to the
end of X's  string S(X) before discarding it also.

Now repeat until you have for each candidate X, a code string S(X).

Finally, reverse each string S(X) to rS(X), and put these reversed strings
into lexicographical (alphabetical) order (assuming the order of the
judgment categories is consistent with the alphabetical order of their
respective code words)

The lexicographical order of this set of reversed strings {rS(X)} gives the
order of finish of the set of candidates {X}.

More than two categories, and more  than a dozen voters make ties highly
unlikely ... the built in tie mechanism takes care of everything short of
two candidates with identical vote distributions or both having all of
their votes symmetrically distributed about the same judgment category ....
extremely rare.

However a tie of any two tied symmetrical (but non-identical) distributions
where the common center of symmetry is not the middle category may may be
resolved by collapsing inward to the middle category.

For example ....
x1, _ , y1, * , y2, vmid , x2, _ , _ , _ , _
based on six real categories plus five virtual ones.

The X and Y votes are both distributed symmetrically around *, but not
around the virtual middle, vmid (midway between y2 and x2).

If we collapse towards the virtual middle, then y1 and y2 end up in the
current y2 position, while x1 and x2 end up in the virtual middle... so
even symmetrical ties can be resolved if the common center of symmetry is
not the middle category (whether real or virtual makes no difference).

Central symmetry ties about the (real or virtual middle) cannot be resolved
even in principle by a y method that satisfies Reverse Symmetry.

In conclusion, it appears that all ties can be resolved by these
considerations except the ones that cannot be resolved in principle because
of Reverse Symmetry or identical distributions of voter judgments.

It would be interesting to see by simulation how frequently these identical
or centrally symmetrical distributions are likely to arise by chance or
even by extreme intentional coordination among factions.

FWS


El lun., 25 de oct. de 2021 5:16 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> To better understand MJ's relation to Symmetrical MJ, let's look at two
> equivalent descriptions of MJ ...
>
> If there are an odd number of judgments (votes) for candidate X, then the
> Majority Judgment of X is the median judgment.
>
> If there are an even number of judgments, add one additional vote to the
> bottom judgment category ... the median of this augmented set of judgments
> is X's MJ.
>
> For Symmetric MJ add the extra vote to the middle category.
>
> The other equivalent procedure for MJ is to start at the top category and
> collapse downward until a majority of votes is reached.
>
> The symmetrical version collapses from the outside towards the middle
> category.
>
> FWS
>
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