# [EM] Cyclic Ambiguities = misinformed voters? (was: The True Majority Ghost)

Mon Jul 1 07:48:48 PDT 2002

```Alex wrote:

> If all candidates fit on a one-dimensional spectrum then
> this  is certainly true.  The person whose preference
> order is  Bush>Nader>Buchanan>Gore needs to start reading
> some newspapers.  The person whose  preference order is
> Buchanan>Gore>Nader>Bush probably lives in Florida ;)
> But what if we have two dimensional candidates?
> [example follows]

I'd like to generalize Alex's example by giving a simple geometric argument
why a sincere cyclic tie can occur in the electorate.  For simplicity, I
will assume there are only three candidates.  These candidates have opinions
on a wide range of issues, so they each have a point on the "issue space"
that corresponds to their stances.  Their three points in the issue space
define a plane, so we can effectively ignore any differences outside that
plane.  Voters will find the point on that plane that is nearest to them,
and prefer candidates who are closer to that point.

So without loss of generality, we have reduced the situation to a
two-dimensional issue space.  The candidates are represented by points in
this space.  Now draw three lines; each line is a perpendicular bisector of
the line connecting two of the candidates.  On one side of the line, all the
voters prefer A to B, and on the other side all the voters prefer B to A,
and so on.  The three lines (A/B, B/C, and C/A) will meet in one point and
divide the plane into six sectors.  This is guaranteed by plane geometry as
long as the three candidates are not perfectly collinear (which is very
unlikely).  These six sectors correspond to the six permutations of
preference in a three way election; ABC, ACB, BAC, BCA, CAB, and CBA.

So I have clearly shown that, given merely a two-dimensional issue space
(the most common example is economic issues and social issues), reasonable
and informed voters could have any permutation of preferences between the
candidates, as long as their opinions were in a certain position relative to
the three candidates.  So if there were, say, more voters in the ABC, BCA,
and CAB sectors than the other three sectors, then a three way cyclic
ambiguity would frequently result.

It's not a flaw in the voting system, or a case of confused voters.  It's a
real conflict of majority preferences that can exist in certain situations.

> The CW is the one who says  "Come one, come all!  None
> can defeat me!"  It seems quite reasonable to elect him.
> If, however, there is no such champion, it seems plausible
> (for political reasons) to say "Well, whoever has the
> fewest fans can leave now."  Of course, I will have to
> think a little about strategic aspects of this method, but
> it seems reasonable.

This has been proposed before, yes?  The problem, or course, is that any
problem that presents itself in IRV can present itself in this method.  The
Condorcet front end provides a pretty good shield from IRV's flaws most of
the time, but that doesn't justify IRV's use on the back end.  The ranked
pairs and beatpath methods of resolving cyclic ambiguities are far more
reliable.

> > Donald:  Why should any method be required to elect the
> > so called `Condorcet Wiener Winner'.  Your bogus remark
> > can be turned around.  It can also be said that it is
> > true that Condorcet can fail to elect the IRVing winner
> > and fail to elect the Approval winner and fail to elect
> > the Bucklin winner, etc, etc.

Donald is fond of this retort, but it is spurious for a very simple reason.
As Donald is fond of pointing out himself, Condorcet is not a method in and
of itself.  It is, simply, a standard to use when comparing methods.

There are a whole host of methods which can be explained by saying, "We will
always match this particular standard, which we call the Condorcet
Criterion.  That alone will uniquely determine the winner of the election
most of the time.  If that criterion does not determine the winner straight
out, then we will do (XXXXX)."  We call these methods Condorcet methods, but
that doesn't make them all one method.  They're just methods that all match
one particular standard.

Now if you like, you could argue that the Condorcet Criterion is a bad
standard, or that an "IRV standard" would be a good one.  But there's
something undeniably intuitive about a standard that says, "the candidate
that would win any head-to-head runoff against any other candidate should
win the election".  On the other hand, I don't see anything intuitive about
"the candidate who would win a majority of the remaining votes after the
candidate with the least votes is successively eliminated, and votes for
that candidate are transferred to the remaining candidate who is each
voter's next favorite choice, should win the election".  In other words,
there's a very good reason why there's a Condorcet Criterion but there's no
IRV criterion.