[EM] Cycles and Stubborn (but rational) Voters

[Dave Ketchum]

Perhaps I miss something but suppose:

x, y, and z are a set of clones, including absolute inflexibility as to 
moderating their preferences as to color, but differing as to which color 
is preferred (red, green, and blue each preferred by one candidate).

Now assume 1/3 of the voters prefer each candidate, and do bullet voting 
since they do not care what happens if their preferred candidate fails to win.

My intent is a 3-way tie with no way out other than a random selection of 
a winner.  No way to get to a 2-way runoff and, even if you fake your way 
into that, this collection of voters will give you a tie to resolve by 
flipping a coin.

Dave Ketchum

On Thu, 9 Oct 2003 15:24:23 -0700 (PDT) Alex Small wrote:

> 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

-- 
  davek at clarityconnect.com    people.clarityconnect.com/webpages3/davek
  Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
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