[EM] FWD: Borda Count by Paul Dumais

[Bart Ingles]

Blake Cretney wrote:
> 
> Bart Ingles wrote:
> > > > I used Average Ratings as a means of showing overall support (as in
> > > > strength times breadth), and not as a "gold standard" for election
> > > > methods.  I had considered including Median Ratings as an additional
> way
> > > > of quantifying support, but didn't want to unnecessarily clutter the
> > > > examples.  It wouldn't have made much difference in these examples
> > > > anyway.
> > >
> > > > Nobody is suggesting that the voters "self-rate the certainty of their
> > > > opinions".  They are rating the suitability of the candidates to hold
> > > > office.  If a voter's assessment that "candidate B is much worse than
> A,
> > > > but only slightly better than C" cannot be trusted, then neither can a
> > > > ranking that shows B somewhere between A and C.
> > >
> > > This is the result of a fundamental difference between the way you
> > > and I are approaching the issue.  You seem to be approaching the
> > > problem as one of
> > >
> > > "Which candidate creates the greatest over-all satisfaction?"
> > >
> > > where I usually ask
> > >
> > > "Which candidate is most likely the best?"
> > >
> > > If you look at the issue from my perspective, the problem is that
> > > people often will make wrong decisions, but are at least slightly more
> > > likely to make right ones.  So, when a person says that
> > > A>B
> > > you can use this as evidence that A is in fact better than B.  Note
> > > that the assumption here is that people are attempting to find the
> > > best candidate from a global perspective, but may get the answer
> > > wrong, due to self-interest or ignorance.  From this perspective, if
> > > you give one person's
> > > A>B
> > > equal weight to ten persons'
> > > B>A
> > > this is only justified if you place ten times as much certainty on
> > > the A>B.  Of course, since in Average Ratings this is caused by the
> > > A>B person giving ten times the difference between A and B as the B>A
> > > people, I give the interpretation that people are in effect
> > > self-rating their certainty level.
> >
> > I guess it depends what you mean by "certainty level".  A person might
> > rate A as 100% suitable to hold a given office, B as 10% suitable, and C
> > totally unsuitable, and still be absolutely certain that B is better
> > than C.
> 
> True, but then, what you want to know is not "how much is B preferred
> to C," but "is B better than C".
> I was trying to interpret the ratings in a way that would make them
> relevant to this question.


In the following ratings example:

                  Candidates
group votes	A     B     C
  I    50      100    95    0
 II    50        5   100    0

By arguing that ratings are irrelevant, you are saying that the above
example is an exact tie between A and B.  To me this is not even a close
contest.

Suppose groups I and II each have a list of 20 issues that they consider
equally important, and that candidates A and B are otherwise equal in
education, honesty, etc.  Group I agrees with candidate A on all 20
issues, and with B on only 19 of the 20.  Group II only agrees with
candidate A on one of the 20.  It should be clear that we are not
measuring levels of certainty, but levels of agreement.

If you want to inject levels of uncertainty into this argument, you
could suppose that the voters were potentially wrong, or wrong about the
candidates, on (say) two of the 20 issues.  Then the ratings would be
accurate to within +/- 10 points.  If you clip the ratings at 100, that
would mean that the correct group I score for A could be as low as 90,
and for B as high as 100.   In this were to happen, then 50% of the
votes would be absolutely wrong under a ranked system.  A rated system
could treat group I's A/B vote as less significant, thereby minimizing
this error.

If you don't believe in ratings, then presumably you wouldn't allow
equal rankings (an oxymoron?), since you would in effect be allowing
voters to make ratings decisions (i.e. how high does B have to be rated
to deserve equal ranking with A?  67 points?  75?)

If you want to argue that group I's rating for B cannot be accurate to
within 10 points, and that we can only say that it is between 0 and 100,
then how could the voters possibly handle a ranking situation with 50
candidates?  In order to do so, they would have to resolve differences
between candidates to an equivalent of 5 points, on average.

 
> > I don't see how you can improve accuracy by discarding that
> > information.
> >
> > On the other hand, I am not that caught up with Average Ratings.  Median
> > Ratings would answer your objection, and still make my point.  Of
> > course, I don't advocate either as election methods, but only use them
> > as tools to quantify sincere-rated examples.
> 
> Average ratings matches well with our concepts of finding the
> candidate with the most support and finding the candidate to maximize
> [normalized] utility.  So, I understood why you would view it as a
> standard, and I felt it was necessary to criticize it.
> 
> Median Ratings, however, is just another method.  Why should I care
> if one method is closer to it than another?

Medians are a natural way of evaluating rated examples, since a
candidate with the highest median rating is by definition the candidate
rated higher than all others *by a majority of voters*.  It would be an
good method on its own, if not for a serious strategy weakness.  Since
strategy is not an issue with sincere examples, I believe it is useful
as a standard, especially when used as a cross-check along Average
ratings.


> > > The "satisfaction maximization" method by which you are interpreting
> > > the results is as follows.  Each person states their satisfaction with
> > > each potential result on a 0 to 100 scale.  From this point of view
> > > the only incorrect vote is one that does not accurately describe the
> > > voters perceived self-interest.  This perspective assumes people will
> > > not vote altruistically.
> >
> > Not at all.  A voter is just as likely to have the good of the community
> > in mind when rating candidates as when ranking.  I suppose an outcome
> > beneficial to the community would provide most voters some satisfaction,
> > though, even if not based on self-interest.
> >
> > Just to be clear, my scale wasn't based on personal satisfaction, but on
> > the individual's assessment of suitability for the office, where 100 is
> > completely adequate, and 0 is completely useless.
> 
> Consider the following situation.  Candidates A, B, and C are running
> for office.  Their ratings, based purely on self-interest are
> group  votes     A    B    C
> I      60      100   90    0
> II     40       90  100    0
> 
> A 96
> B 94
> C 0
> 
> So, the winner is A.  From the Average-Ratings point of view this is
> also the candidate that provides the greatest average satisfaction.
> 
> Now, if the group I voters decide to vote based on community
> interests instead of self-interest, they would use these average
> ratings as their individual ratings, normalizing them of course.  So
> we get
> 
> A 96 * 100/96=100
> B 94 * 100/96=97.9 approx 98
> C 0 * 100/96=0
> 
> So, if group I is voting with community interest the votes are
> 
> group  votes     A    B    C
> I      60      100   98    0
> II     40       90  100    0
> 
> A 96
> B 98.8
> C 0
> 
> So, B is the winner.  So, group I not only hurt themselves, but hurt
> the community as a whole by their community interest.  I think it's
> clear from this that Ratings assumes people vote for their
> self-interest, or it won't work properly.

I think you're off on several counts here.  First, you cannot
mathematically transorm selfish votes into altruistic votes simply by
averaging them together.  Second, even if an altruistic vote were the
same as an average vote, your example would not prove that the original
votes were selfish -- the original votes could have all been altruistic
and yielded the same result.

The rated votes in my examples all reflect the same inseparable
combination of self- and community-interest as any real vote, whether in
a ranked, approval, or plurality setting.  After all, the whole purpose
of using rated examples was to be able to set up scenarios that would
allow comparison of other methods.


> --snip--
> > > > > In particular, if 10 people say that A is better than B, do you
> > > > > really think that should be equaled by one person saying that B is
> > > > > better than A, even if that person feels 10 times as strongly about
> > > > > it?
> > > >
> > > > Again, the scales do not show "strength of feeling", but "assessment of
> > > > fitness to hold office".  These are not open-ended scales, so extreme
> > > > views would be clipped or compressed to fit the 0-100 scale.  The only
> > > > way 91% of the voters would rank two candidates only ten points apart
> > > > would be if there is at least one other candidate who is either much
> > > > better or much worse than the first two.
> > >
> > > If both candidates are reasonable then there will probably be lots
> > > that are a lot worse.
> >
> > That is why I am more interested in eliminating candidates that are much
> > worse, than in worrying about which of two good candidates win.
> >
> 
> I think this statement refers to an earlier posting where you made
> the claim that Approval keeps out bad candidates, while other methods,
> if enough people use misguided strategy, can elect them.  I think it
> is unfair, however, to give Approval examples that assume everyone
> uses perfect strategy, and Borda examples that assume the opposite.

I wasn't showing misguided strategy on the part of other methods, unless
you think that burying is always misguided.  In my last Borda example,
repeated here:

       Rating:
       100    80    60    40    20    0
votes  ----------------------------------
45      A              C              B
15      B                        C    A
40      C              A              B

I showed voters starting off by using a "semi-burying" strategy, by
truncating, as:

45  A
15  BC
40  C

This is perfectly reasonable, and in this situation poses little risk of
electing B.  In fact, the identical strategy would be used in Approval,
in this election.

The problem with the ranked methods is that the voters need not stop at
truncation; some of them may decide to actually use reverse ranking
(under Prof. Saari's verson of Borda, truncation is penalized, so voters
may be more likely to use reversal).  This is also reasonable, so long
as they leave enough safety margin that B is unlikely to win
accidentally.  But if the three groups are not homogenous, and some of
the A and C voters rate each others' candidates closer to B, they will
view B as less of a threat and be more willing to risk letting B win if
this also gives their own candidate a better chance.

I don't view this as misguided on the part of the people doing the
voting.  In fact, it appears that the voters are faced with the same
decisions and trade-offs that they would have under Approval, but with
more opportunities for mischief.

Bart