[EM] MJ SFR (preliminary). Score vs Approval, based on considerations discussed.

Ted Stern araucaria.araucana at gmail.com
Tue Sep 11 12:25:29 PDT 2012

On 11 Sep 2012 08:39:48 -0700, Michael Ossipoff wrote:
> Two topics:
> 1. Brief preliminary comments in reply to Jameson's MJ SFR posting.
> 2. Score vs Approval based on considerations that have been discussed.
> 1.
> At first I said here that MJ doesn't have SFT. But later I said that
> it does have SFR, but that it's SFR is more complicated than that of
> Score/Approval. Jameson's recent MJ SFR posting seems to confirm
> that conclusion.
> Brevity and clarity can be mutually incompatible,and so it is with
> the descriptions of MJ's tiebreaking bylaws. As Jameson briefly
> described the tiebreaking bylaws of MJ and CMJ, I didn't understand
> them.
> I think we can agree that MJ's and CMJ's tiebreaking bylaws, when
> described completely would add a lot of words to the methods'
> definitions. I think we can agree that those tiebreaking bylaws
> spoil any simplicity or brevity that those method would otherwise
> have had.

Majority Judgment (MJ) and Continuous Majority Judgment (CMJ) are both
Median Ratings methods.  As is ER-Bucklin(whole).  You're probably
most familiar with the latter, so let me start there.  I will put
ER-Bucklin into the same formulation as MJ and CMJ so you understand
the terms, then define the MJ and CMJ rankings.

With a ratings ballot, you can find the totals for each candidate at
each ratings level.  Let's denote the number of ballots cast at level
M for candidate C as


Let's denote the number of ballots cast for C at levels higher than M as


Let's denote the number of ballots cast for C at levels lower than M


Let's say we successively add level totals, starting at the maximum
rating, to each candidate's approval until at least one candidate's
approval strictly exceeds 50%.  That threshold is the highest median
rating, M.  So the ER-Bucklin Approval for such a candidate is
equivalent, in our notation, to

     V_C(L>M) + V_C(M)

To find the median rating for other candidates, keep dropping the
threshold until either that candidate's total approval at that
threshold strictly exceeds 50%, or the lowest rating is reached.  The
threshold you find is their median rating.

In Majority Judgment, there are actually two different tie-breakers,
but they are nearly identical.  The following method is the "fast
computational method" described in Balinski and Laraki's book.

Using the process above, we can determine the median rating for each
candidate.  We say that candidate C has a score of M+ if V_C(L>M) is
greater than V_C(L<M).  Similarly, C has a score of M- if V_C(L<M) is
greater than V_C(L>M).  All the candidates can be ranked by their M
plus-or-minus scores, and if two candidates both have a score of M+,
the one with the larger V_C(L>M) is ranked higher.

In Majority Judgment: among the same set of candidates whose
ER-Bucklin total approval score and and above threshold M exceeds 50%,
the candidate with the highest MJ-adjusted M+ score wins.

In Continuous Majority Judgment, the scoring is similar.  Instead of
finding M+ or M-, the score is

       M  +  (V(L>M) - V(L<M)) / (2 * V(M))

Among the same set of candidates who achieve a >50% ER-Bucklin
approval at threshold M, the candidate with the highest CMJ score

As a specific example, consider a candidate whose total ratings above
M is 47%, total below M is 45%, and total at M is 8%, then that
candidate has an MJ score of M+ (47%), and a CMJ score of

  CMJ(C) =  M + (47 - 45) / (2 * 8) 

         = M + 0.125

In many cases, MJ, CMJ and ER-Bucklin will choose the same winner.

Note that the method Jameson has been describing for MJ is the more
general method, in which equal numbers of V(M) votes are removed from
each candidate's score until the median rating for one of them
changes.  However, this is roughly equivalent to the fast
computational method I described above.

Does that help?

> 2.
> Though Score's SFR can probably be approximated by intuitive
> subjective judgement, something indistinguishable from sincere rating,
> there is probably a tendency to not rate at extremes when one should.
> I used to criticize Score because people who inbetween-rate can be had
> by strategizing voters.
> But if someone would inbetween-rate a candidate whom s/he should
> bottom-rate, in Score--Might s/he not approve the same candidate in
> Approval? Which would be worse? Voters tend to overcompromise. Score's
> flexible fractional-rating capability lets people rate where they most
> feel like, without Approval's stark choice, and would probably lessen,
> not increase, the overcompromise problem.
> But, as has already been discussed, of course Score-like middle-rating
> can be effectively achieved probabilistically in Approval, in public
> elections where there are many voters.
> It isn't difficult. 0-10 Score can be simulated by drawing one of ten
> numbers from a bag. 0-100 Score could be simulated by twice drawing
> (with replacement) one of ten numbers from a bag.
> For 0-10, if you want to give a candidate 7/10, and the pieces of
> paper in the bag are numbered from 0 to 9, then approve hir if the
> number that you draw is less than 7.  If the pieces of paper are
> numbered from 1 to 10, then approve hir if the number you draw isn't
> more than 10--if the number is from 1 to 7.
> For 0-100, you should number the pieces of paper from 0 to 9. Draw a
> number. Write it down, Replace the piece of paper in the bag. Draw
> another number. Right it down to the right of where you wrote the
> first number. That gives you a two digit number. If you wanted to
> probabilistically give the candidate .87, then approve hir if the
> two-digit number you wrote is less than 87.
> A big advantage of Approval over Score is the much less
> labor-intensive count. Though Score is less computation-intensive than
> the rank methods, Approval is a lot easier and less laborious to count
> than Score is. As we all agree, count-labor = count-fraud-opportunity.
> Better to give the voter a little more to do, for fractional rating,
> than to increase the count-labor.
> There's nothing wrong with letting the voter be more directly involved
> with making the fractional rating, when s/he wants to fractionally
> rate.
> On another topic:
> By the way, Chris only suggests ICT as a 3-slot method. To me, because
> I don't recommend rank methods for official public elections, the only
> value of ICT or Symmetrical ICT is for informational polling. For
> that, it's desirable to allow unlimited rankings, because it's
> desirable to get as much preference information as possible. Chris
> said that more criteria are met by ICT when it's s 3-slot. Maybe, but
> I'm impressed and satisfied with ICT's and Symmetrical ICT's
> properties even with unlimited ranking.
> But I wonder how much less the count labor of Symmetrical ICT would be
> if it were only 3-slot. Would it start being competitive with Score in
> regards to count labor? Probably not, but I just thought I'd mention
> the question.
> I don't suppose that the instance-tallying of Score and MJ would work
> with a pairwise-count method. With N candidates, there'd still be
> N*(N-1) pairwise vote-totals to add up, even though finding which of 2
> candidates a ballot ranks higher would be easier on a 3-slot ballot.
> Mike Ossipoff
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info

My only comment about this is that, since your quoting style is
non-standard, I really wish you'd provide a glossary of abbreviations
somewhere in your message, either inline, using standard
first-reference style, or at the end of your message.

For example, which Chris are you referring to (Benham?), what does ICT
stand for, and where is ICT defined?

By the way, you needn't answer any of my comments; I'm just providing
them as clarification for other readers.

araucaria dot araucana at gmail dot com

Majority Judgment:


Continuous Majority Judgment:




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