[EM] Cycles and Stubborn (but rational) Voters

[Alex Small]

OK, let's imagine a simple situation:  3 voters, named A, B, and C.  3
candidates, named x, y, and z.  Is it possible that, no matter what
stances the 3 candidates take, the voters will always form a cycle so that
A thinks x>y>z, B thinks y>z>x, and C thinks z>x>y?

Let's say A, B, and C each have well-defined stances on a variety of
issues, and all voters have some function F_i(x,y,z) (i=A, B, or C) where
x, y, and z are vectors data specifying each candidate's stances on the
issues.  F_i(x,y,z) returns a transitive preference order for each voter,
dividing the space of candidate positions into 6 regions (neglecting, for
the moment, the possibility of equal rankings).

Now, we could simply specify F_A(x,y,z) however we want, and then
construct F_B and F_C in such a manner that a cycle is always guaranteed. 
So, if we define three voters to have irreconcilable positions then a
cycle is inevitable.

But that seems contrary to what intuition would suggest.  Intuition would
suggest that there must be some sort of compromise that will please at
least 2 of the voters so that the cycle can be broken.

How to address this?  Well, first we should impose some conditions on
F_i(x,y,z).  The first is symmetry.  If two candidates swap their stances
on the issues then the voter swaps those candidates in his rankings. 
Pretty obvious.

Of course, we'll need to impose other conditions on the voter preferences.
 I toss out this question to the list:  Can anybody think of the most
minimal condition necessary so that it is possible for a candidate to
become the CW if he positions himself correctly?  I'm not trying to banish
the possibility of cycles.  Cycles are an inevitable possibility.  I'm
trying to work out whether it is feasible to have an electorate with such
irreconcilable differences that a cycle will occur no matter how the
candidates position themselves.

One obvious thing to do is to have all 3 candidates adopt the same
stances, at which point the voters become indifferent and they all have
the same transitive ranking:  x=y=z.  But that cheapens the question.  So
let's assume that the candidates are always distinct from one another.

Any thoughts?



Alex