The "problem" with circularity (was Re: Reply to Blake Cretney)

[Markus Schulze]

Dear Steve,

an election method meets the "Pairwise Majority
Criterion" (PMC) if & only if:

   Suppose, that there are only two candidates: X and Y.
   Suppose, that the number of voters, who strictly prefer
   candidate X to candidate Y, is strictly larger than the
   number of voters, who strictly prefer candidate Y to
   candidate X.
   Then: Candidate X must be elected.

Now consider the following example and suppose that the
used election method meets PMC:

    40 voters prefer A > B > C.
    35 voters prefer B > C > A.
    25 voters prefer C > A > B.
    Candidate A prefers candidate B to candidate C.
    Candidate B prefers candidate C to candidate A.
    Candidate C prefers candidate A to candidate B.

Case 1:

   Suppose, that candidate A is elected.
   Then candidate B is a spoiler, because he prefers
   candidate C to candidate A and because candidate C
   would have been elected if candidate B hadn't run.

Case 2:

   Suppose, that candidate B is elected.
   Then candidate C is a spoiler, because he prefers
   candidate A to candidate B and because candidate A
   would have been elected if candidate C hadn't run.

Case 3:

   Suppose, that candidate C is elected.
   Then candidate A is a spoiler, because he prefers
   candidate B to candidate C and because candidate B
   would have been elected if candidate A hadn't run.

Thus: _Independently of which election method is used_
there can always be spoilers (at least as long as the used
election method meets PMC).

I believe that the example above cannot be too complicated
because Donald understood the example although he
"is a turkey who almost never writes anything of value."

*****

You wrote (17 Mar 1999):
> The key phrase is "against his will." Markus hasn't
> identified which, if any, of the candidates in his
> example is forced to be a spoiler against his will.

Your comment is irrelevant because no election method
urges a person to run for a seat if this person fears
that he could be a spoiler.

You wrote (17 Mar 1999):
> > The following is Schulze's Example:
> >    40 voters prefer A > B > C.
> >    35 voters prefer B > C > A.
> >    25 voters prefer C > A > B.
> >    Candidate A prefers candidate B to candidate C.
> >    Candidate B prefers candidate C to candidate A.
> >    Candidate C prefers candidate A to candidate B.
> > 
> > Donald writes: I would call this a Circular Stand-Off. All
> > three candidates are still contenders - any one of them
> > could be the winner if only the "correct" candidate would
> > withdraw. 
>
> There is no "correct" candidate to withdraw (or to elect)
> when there is a sincere circular tie. See the discussion in
> this list regarding the merits of the Smith//Random method,
> which randomly elects one of the top cycle.

The problem is: If JITW//Condorcet violates the No-Spoiler
Criterion at least in all those cases in which any other
Condorcet Criterion method violates the No-Spoiler Criterion
(i.e. in all those cases with a circular tie) then there is
no reason why JITW//Condorcet should be less vulnerable by
spoilers than any other Condorcet Criterion method.

Markus Schulze