# [EM] MJ Symmetrized ... special cases

Forest Simmons forest.simmons21 at gmail.com
Sat Oct 23 21:59:08 PDT 2021

Another valuable special case is five categories because of the popular A,
B, C, D, F scale.

IF either the A or the F category (by itself) holds a majority of the voter
judgments for candidate X,  then we're done ...
ELSE fold the A and F categories into the B and D categories,
respectively...
AND ... use the three slot version to find the MJ.

This symmetrical 5-slot version of MJ is way simpler than 6-slot MJ.

Why not stick with 3, 5, or 7 slots (categories) for most single winner
elections ? Ordinary 6 slot MJ lacks the  simplicity and symmetry of these
nearby odd slot versions.

FWS

El sáb., 23 de oct. de 2021 9:34 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> One other noteworthy case is for 3, 9, 27, etc ...  3^k judgment
> categories...
>
> WHILE 3,9,27, or 81 categories remain ...
> IF more than half of the remaining ballot judgments are in the upper third
> or in the lower third of the categories ...
> THEN ...
> move the judgments from the minority and middle third categories to the
> majority category they are closest to ...
> AND apply this method recursively to the remaining third of categories
> (from the majority side)
> ELSE move the outer two thirds of categories to their nearest neighbor
> category in the remaining middle third of caategories.
> END END [IF and While]
>
> Tbe one category remaining (after this procedure) holds the Majority
> Judgment.
>
> Remark: This version of MJ goes well with Ranked Rankings when there are
> lots of candidates ... typical ballot:
>
> A>B>C>>D>E>F>>G>H>I>>> ...>>>, etc.
>
> ... a version of Ranked Rankings built up recursively on the popular
> 3-slot approval motif that we used to play around with ... but I don't
> think we ever actually came up with an MJ version of 3-slot in all of our
> brain storms ... did we Kevin?
>
> We probably shied away from it instinctively because of the high
> probability of ties ... not realizing how handily the gradual removal of
> (well defiined) mj's can resolve those ties.
>
> Three slot approval seemed so much easier to vote sincerely than ordinary
> two slot Approval ... that middle category softens the high stakes Approval
> decisions ... just like the possibility of giving half credit makes it
> easier to grade math tests.
>
> Three-slot MJ fits into both the 3^k and 2^j-1 patterns.
>
> FWS
>
> El sáb., 23 de oct. de 2021 8:28 p. m., Forest Simmons <
> forest.simmons21 at gmail.com> escribió:
>
>> The special cases of 3, 7, 15, 31, or 2^k-1 categories can be handled
>> much more succintly:
>>
>> WHILE more than one category remains ... [i.e. while 3, 7, ... or 2^k-1
>> categories remain]...
>> IF more than half of the judgments are better than the middle remaining
>> category or more than half are worse than the middle remaining category..
>> THEN ...
>>      ... move the remaining middle category judgments and those from the
>> minority side of the remaining middle category to the majority side
>> category that they are nearest to....
>> AND  apply this method recursively to the remaining categoties.
>> ELSE ...move all  judgments to the middle category among those remaining.
>> EndWhile
>>
>> After execution of the above procedure the one remaining category is the
>> MJ category.
>>
>>
>>
>>
>> El sáb., 23 de oct. de 2021 6:35 p. m., Forest Simmons <
>> forest.simmons21 at gmail.com> escribió:
>>
>>> It is easier to bring MJ into compliance with Reverse Symmetry if there
>>> are an odd number of categories 3, 5, 7, ...
>>>
>>> Let's call the middle category "neutral" or Greek "nu" for short because
>>> it is the only category without either a positive or negative connotation.
>>>
>>> A "majority judgment" for X will be one of the judgment categories (of
>>> which there are an odd number) and which also make up the possible voter
>>> judgments of the candidates.
>>>
>>> To find the voters' Majority Judgment for candidate X, list the
>>> induvidual voter judgments of X  while respecting the natural judgment
>>> order from left to right, say. [no slight intended, lefties!]
>>>
>>> If there is a median member of the list (because of an odd number of
>>> judgments of X), then that median judgment is X's Majority Judgment (MJ).
>>>
>>> Or (in the even case) if the two middle judgments are the same, then
>>> that common judgment is the MJ.
>>>
>>> Or if the two middle judgments are in categories situated symmetrically
>>> about neutral, then neutral is the majority judgment.
>>>
>>> Otherwise, the outside one (the one furthest away from neutral in the
>>> category order) is the MJ.
>>>
>>> It seems to me that these conventions respect Reverse Summetry: if all
>>> judgments are reversed (i.e. replaced with their symmetrical opposites),
>>> then the MJ will also move to its symmetrical opposite.
>>>