[EM] Reverse Symmetry Criterion

Forest Simmons fsimmons at pcc.edu
Mon Mar 26 12:13:54 PST 2001

```I like your idea.  But I still think there is no stigma attached to
failing the reverse symmetry criterion when there is no (unique) Condorcet
winner, so I would suggest that you throw those out of the count.

Forest

On Sun, 25 Mar 2001, Martin Harper wrote:

> I'm not that fond, in general, of past/fail criterion. Surely it would be
> better to know the /extent/ to which a method fails, if it fails.
>
> Qualitative Reverse Symmetry Test
> --
> Generate a large number of sets of votes, according to some appropriate model
> of the electorate. For each of these, determine if the method elects the same
> person for those votes, and those votes reversed. The proportion for which it
> does is the key - the method which has the lowest proportion is the best
> according to QRST.
>
> For comparing two methods, it makes sense to give them both the same set of
> votes to work with, to minimise the random factor.
> --
>
> Martin
>
> Forest Simmons wrote:
>
> > Consider the following summary of 90 preference ballots:
> >
> > 40 C > A > B
> > 20 A > B > C
> > 30 B > C > A
> >
> > IRV gives the win to B.  Reverse all of the preferences and IRV still
> > gives the win to B.  However, we cannot fault IRV in this case because
> > the candidates form a Condorcet cycle: C beats A beats B beats C, and are
> > therefore in some sense tied.
> >
> > Here's another reason to not fault IRV in this case. It is possible to
> > assign utilities to the candidates consistent with the preferences in such
> > a way that the average utility of each candidate is the same number:
> >
> > Suppose the first and last candidate in each faction have utilities of
> > 100% and zero, respectively. And suppose the middle candidates of the
> > respective factions have average utilities (within the respective
> > factions) of 65%, 80%, and 20%, respectively.  Then each of the three
> > candidates has an average utility (over the entire population of 90
> > voters) of 46/90 or about 51% .  And as far as this particular population
> > is concerned no candidate is better than any other.
> >
> > So in this case, who could fault IRV or Condorcet for giving some kind of
> > tie to the candidates on the basis of preference ballots? (Well, I can
> > probably guess a few characters that could.)
> >
> > Now let's consider a beefed up version of the example:
> >
> > 40 C > X > A > Y > B
> > 20 X > A > B > C > Y
> > 30 B > X > C > Y > A
> >
> > Both forward and reverse IRV still give the win to B.  But this time there
> > is a definite Condorcet Winner, so IRV cannot get off the hook with the
> > Condorcet cycle excuse.
> >
> > Can IRV get off the hook the other way (by appealing to some possible
> > assignment of ratings that would give all of the candidates the same
> > average rating)?
> >
> > No. The proof is simple. Candidate X is strictly preferred over candidate
> > Y by 100% of the voters, therefore candidate X's average rating must be
> > strictly greater than candidate Y's average rating.
> >
> > As someone pointed out recently, most methods based on preference ballots
> > fail the Reverse Symmetry Criterion. But how many of them still fail it
> > when there is a Condorcet Winner? And how many of then still fail it when
> > there is no possible assignment of ratings (consistent with the
> > preferences) that would give the least and greatest candidate (hence all
> > the candidates) the same average rating?
> >
> > It seems to me that these modified (weaker) Reverse Symmetry Criteria
> > would be more useful in distinguishing good methods from bad (more useful
> > than the version that most methods based on preference ballots fail
> > anyway).
> >
> > If your favorite method fails the ordinary Reverse Symmetry Criterion,
> > then you can say, "Big deal, so does just about every other method."  But
> > if your method fails these weaker versions, then you will have to make up
> > a lot of ridiculous excuses or find a better method to support.
> >
> > So here they are:
> >
> > First version:
> >
> > If a preference list based method picks the same winner forward and
> > backwards when there is a unique Condorcet winner, then it fails this
> > modified reverse symmetry criterion.
> >
> > Second version:
> >
> > If a method based on preference ballots picks the same winner with
> > preferences reversed when there doesn't exist any possible assignment of
> > ratings (consistent with the preferences) that would give all of the
> > candidates the same average rating, then the method fails this test.
> >
> > We have just seen that IRV fails both of these modified criteria.
> >
> > Forest
>
>

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