[EM] Reverse Symmetry Criterion

[Martin Harper]

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