[EM] majority judgement question

[Jameson Quinn]

2011/10/19 Andy Jennings <elections at jenningsstory.com>

> Good points, Ross and Jameson.
>
> Section 4.3 of my dissertation (http://ajennings.net/dissertation.pdf)
> talks about this very thing.
>
> The Ac-Bc rule was proposed by David Gale (
> http://en.wikipedia.org/wiki/David_Gale) before his passing, to Balinski
> and Laraki directly.  I proposed a rule very similar to Jameson's (Ac -
> Bc)/(Mc + |Ac - Bc|).  Mine was (Ac - Bc)/(2*Mc).  Both are continuous
> everywhere (assuming fractional voters), even at the junction where the
> median changes from one grade to another.  (See graphs in the pdf.)
>

"Andy's" rule is simpler to state. "My" rule makes straighter lines, reduces
to something more sensible in the case of approval, and is a bit better for
stating the error on a poll. The two rules always give the same winner, so
it doesn't really matter.


> Also in my dissertation, I gave examples for the other two rules where a
> small change in the voter profile could cause a candidate to fall multiple
> rankings.
>
> On the other hand, Balinski and Laraki's rule is constant with respect to
> either Ac or Bc almost everywhere.  I think this might make it a little more
> resistant to strategic voting.
>

I've explored this a bit - not rigorously, just enough to sharpen my
intuition - and it does not seem to be true.


>  Plus, the remove-one-median-rating-at-a-time method has a certain
> simplicity and elegance to it, especially for very small electorates, even
> if it gets a little convoluted for large ones.
>
> ~ Andy
>
>
>
> On Wed, Oct 19, 2011 at 9:12 AM, Jameson Quinn <jameson.quinn at gmail.com>wrote:
>
>> 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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