Circular Stand-Off

[Markus Schulze]

Mike Ossipoff wrote (19 Mar 1999):
> From: 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.
>
> That's a natural circular tie, with no right result, and no
> lesser-of-2-evils problem to avoid, and no really meaningful
> spoiler problem. When every candidate is equally a "spoiler",
> then it becomes just a matter of policy negotiation to
> determine which will withdraw via JITW. More later in the
> reply.
>
> > 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).
>
> It's important to emphasize that situations with a natural circular
> tie are situations where 1) One can't expect any kind of definite
> solution, and, as I said, it's merely a matter of policy negotiation
> to determine who withdraws via JITW; & 2) It doesn't really matter
> anyway, since there's no candidate for whom we could say that'
> it's wrong if he doesn't win, or if he does win. No solution, and
> it doesn't matter that there's no solution. A natural circular tie
> isn't the scenario for testing JITW or any method. 
>
> > 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.
>
> No, because, with JITW, not being a spoiler never requires not
> running. It's important not to discourage anyone from running.
> Creating "barriers to entry" preserves a sometimes undesirable
> status-quo. Similarly, it's important not to discourage voters
> from voting their favorite in 1st place.
>
> Condorcet(EM), Smith//Condorcet(EM), & Schulze's method have
> no requirement for defensive strategy unless offensive order-reversal
> is attempted. I believe that will be rare with those methods,
> meaning that those methods, for practical purposes, have no 
> strategy problem. But if offensive order-reversal _were_ used
> on a scale sufficient to change the outcome, then JITW neatly
> defeats the order-reversal. In an election between A, B, & C,
> where B is middle CW, and where the A voters order-reverse against
> B, so that C>B>A>C, and where any pairwise count method is used,
> any "Condorcet completion method", candidate C will withdraw, and
> thereby elect B. That outcome is stable, & will happen every
> time where A otherwise wins by order-reversal.
>
> JITW would similarly avoid the strategy problems of Plurality
> or IRO.
>
> > 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.
>
> But JITW _doesn't_ have that problem everywhere that other
> pairwise methods do. When there's a CW, and someone has won
> by order-reversal, the other methods leave it at that, but
> if JITW is an option, then that withdrawal option will be used
> to defeat the order-reversal & elect the CW. That's a certainty
> in a 3-candidate election, and I suspect that it's true no
> matter how many candidates there are. So, in the situations where
> there's a genuine right result, JITW brings about that result.
> The important thing is that it does so without any voter having
> to do other than rank his favorite alone in 1st place, and without
> requiring any candidate to not run.
>
> If the method is Condorcet(EM), or one of our other good
> votes-against methods, then JITW's withdrawal will only be needed
> if order-reversal succeeds. If the method is Margins, then JITW's
> withdrawal will also be needed in the much more common & plausible
> circular ties caused by truncation. But it will still accomplish
> the same thing, the election of the CW.
> 
> > Suppose that JITW//Plurality is used.
> >
> >   30 voters vote A > ...
> >   25 voters vote B > ...
> >   24 voters vote C > ...
> >   10 voters vote E > C > ...
> >   09 voters vote D > B > ...
> >   03 voters vote E > A > ...
> >   03 voters vote D > A > ...
> >   A, B, and C won't withdraw under
> >   any circumstances.
> >   Candidate D prefers B to C to E to A.
> >   Candidate E prefers C to B to D to A.
> >
> > ["..." means that it is irrelevant how these preferences
> > look like because -as candidate A, B, and C will never
> > withdraw- these preferences will never be counted if
> > JITW//Plurality is used.]
> >
> > What will happen if JITW//Plurality is used?
> > If nobody withdraws candidate A is elected.
> > If candidate D withdraws candidate B is elected.
> > If candidate E withdraws candidate C is elected.
> > But if both candidates (D and E) withdraw then
> > candidate A is elected.
> >
> > In other words: If JITW//Plurality is used then candidate
> > D will say: "I withdraw first." Candidate E will answer:
> > "No, I withdraw first." Candidate D will answer: "No!
> > I withdraw first." ......
> >
> > In other words: Both candidates (D and E) are spoilers
> > against their will because both candidates would prefer
> > that they never ran for that office.
>
> Why? D & E have the power to pass their votes on to their voters'
> next choice, A, regardless of whether they withdraw 1st, 2nd, or
> 3rd. Surely there's no rule saying that only one candidate may
> withdraw. If they can get a better result by withdrawing, then
> nothing's stopping them from doing so, and so they aren't spoilers
> against their will.
>
> Of course if E is the 2nd choice of the B & C voters, then
> those voters would insist that B & C withdraw, and would be unlikely
> to vote for them again if they refused to withdraw. Additionally,
> the election of A, when D & E withdraw, will be regretted by
> B & C if their preferences resemble those of their voters. They
> improve their outcome, in that case, if they withdraw too.

Markus Schulze