# [EM] Reverse Symmetry Criterion

Forest Simmons fsimmons at pcc.edu
Sat Mar 24 16:34:56 PST 2001

```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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