[EM] majority judgement question

[Jameson Quinn]

Great suggestion. I've been thinking along those lines, but I hadn't
expressed it as clearly.

And now that Ross has given me this idea, I can make it even simpler.
 Ross's suggested process is of course equivalent to, and harder to explain
than, using (number above median grade)-(number below median grade) as a
score. The only disadvantage of my version is that it could give negative
numbers. But almost all people over the age of 10 (and a lot of people under
that age) can handle negative numbers just fine, so I think that's OK.

This tiebreaker process is good. It will also tend to agree with the MJ one,
as long as the tied candidates have approximately the same number of votes
at the median grade - which will generally be true for two candidates whose
strengths are similar enough to tie the median grade in the first place.

Here's another "tiebreaker" which I've developed. The advantage is that it
gives a single real-number grade to each candidate, thus avoiding the issue
of "ties" in the first place. I call it "Continuous Majority Judgment" or
CMJ.

Rc= Median rating for candidate c (expressed numerically; thus, letter
grades would be converted to grade-point-average numbers, etc.)
Mc= Number of median ratings for candidate c
Ac= Number of ratings above median for candidate c
Bc= Number of ratings below median for candidate c
|x| = standard notation for absolute value of x

CMJ rating for c = Rc + ((Ac - Bc)/(Mc + |Ac - Bc|))

For approval (that is, binary ratings), the CMJ rating works out to be equal
to the fraction of 1s, as you'd expect. Note that the adjustment factor is
always in the range of -0.5 to 0.5, because the difference |Ac - Bc| can
never be greater than Mc or it wouldn't be the median.

I prefer either of these methods to the MJ method - not for results, but for
simplicity. (Ac - Bc) is simplest to explain, while CMJ is simplest to
compare candidates / post results. All three of them should give the same
results in almost all cases. But Balinski and Laraki preferred the
remove-one-median-rating-at-a-time method because they could prove more
theorems about it, and they wrote the MJ book, so until I write my own book
about it I'm fine with promoting their method.

JQ


2011/10/19 Ross Hyman <rahyman at sbcglobal.net>

> It seems to me that there is a simpler way to compare candidates with the
> same median grade in majority judgement voting than the method described in
> the Wikipedia page for majority judgement.  Why isn't this simpler way used?
>
>
> Every voter grades every candidate.  Elect the candidate with the highest
> median grade (the highest grade for which more than 50% of voters grade the
> candidate equal to or higher than that grade.)  If there are two or more
> candidates with the same highest median grade, elect the candidate with the
> highest score of those with the highest median grade.  A candidate's score
> is equal to the the number of voters that grade the candidate higher than
> the median grade plus the number of voters that grade to candidate equal to
> or higher than the median grade.  This is equivalent to giving one point to
> each candidate for each voter who grades the candidate its median grade and
> two points for each voter who grades the candidate higher than its median
> grade.  Motivation:  voters who vote median grade instead of something lower
> should increase the score for the candidate by the same amount as voters who
> vote above the median grade instead of equal to the median grade.  With this
> scoring, going from less than median to median increases the candidate score
> by one point and going from median to higher than median also increases the
> candidate score by one point.
>
> Example using same example from Wikipedia's majority judgement entry:
> 26% of voters grade Nashville as Excellent and 42% of voters grade
> Nashville as Good.  Nashville's median grade is Good and its score is
> 26+26+42 = 94
> 15% of voters grade Chattanooga as Excellent and 43% of voters grade
> Chattanooga as Good.  Chattanooga's median grade is Good and its score is
> 15+15+43 = 73.
> Nashville wins.
>
>
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>
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