[EM] Tolerating: MJ, HMJ, GMJ, SARA, or MAJORITY SCORE

[Kristofer Munsterhjelm]

On 10/30/2016 04:00 PM, steve bosworth wrote:
> Tolerating: MJ, HMJ, GMJ, SARA, MAJORITY SCORE
> 
> To Jameson and everyone,
> 
> Highest Majority Judgment (HMJ)
> 
> Jameson seems to suggest that his SARA or Majority Score methods
> guarantees the election of the candidate with the ‘deepest and widest’
> support from the voters.  I am not yet clear that this claim could be
> sustained when compared with my refined version of Balinski &Laraki’s MJ
> which I will call Highest Majority Judgment (HMJ).
> 
> There is only one difference between HMJ and B&L’s MJ.  Both HMJ and MJ
> start by discovering all the candidates who have receive the highest
> median grade, i.e. ‘majority-grade’.  If only one candidate has received
> this grade, she wins.  If 2 or more candidates have received this grade,
> the winner in a large election will most probably be discovered simply
> by going on to compare their different ‘majority-guages’ (see below). 
> However, if the winner is not obvious from these comparisons, HMJ next
> uses a different tie-breaker than used by B&L.  HMJ discovers which has
> received the highest average evaluative score.  For example, using
> Excellent (5), Very Good (4), Good (3), Acceptable (2), Poor (1), or
> Reject (0), it divides the sum of the score equivalents of all the
> median and higher grades received by each candidate by the total number
> of these supporting median and higher than median grades.
> 
> In contrast, MJ’s most precise if laborious way of discovery the winner
> is to go on to calculate which candidate has received the highest
> ‘majority-value’.   B&L see the complete ‘majority-value’ of each
> candidate as expressed by ‘the sequence of his (first) majority-grade,
> his second majority-grade, his third majority-grade, down to his /n/th
> majority-grade (if there are/n/ [voters])’ (B&L, Majority Judgment,
> p.6).  Thus, the number expressing this value starts with the score
> equivalent to their common ‘majority-grade’, followed after the decimal
> point by the sequence of each of all the scores of all the remaining
> majority-grades that would result, one by one, by removing each new
> majority-grade once it is found.

I'd like to mention, from a theory purist's perspective, that one of the
good things about MJ is that it doesn't make any assumptions about how
far away Excellent is from Very Good or Very Good from Poor. That is, if
you ran MJ with the following assignments:

Excellent = 5
Very Good = 4
Good = 3
Acceptable = 2
Poor = 1
Reject = 0

you'd get the same result as if you ran it with these:

Excellent = 1000
Very Good = 800
Good = 500
Acceptable = 100
Poor = 20
Reject = 0

All that's required for MJ to work is that the grading standard is
reasonably close to the same for every voter.

If you introduce averaging into the mix, this property is (obviously) lost.

Since you only do that as a last tiebreaker, the compromise would
probably not be that great in practice, but in theory, it changes the
perspective of the method: no longer do only the grades matter, but it
also matters what relative levels of satisfaction each grade corresponds to.