[EM] Schwartz//Tideman for meetings & small committees

[Markus Schulze]

Dear Mike,

you wrote (2 Aug 2000):
> Speaking of demonstrations of nonmonotonicity, your 7-candidate
> SD nonmonotonicity example, like Blake's 9-candidate example,
> doesn't specify all of the defeats. You showed why it isn't
> necessary to specify the rankings, but I don't believe that you
> ever showed that it isn't necessary to show all the defeats.
> If you're assuming that all of the defeats you didn't specify
> are smaller, and will be dropped first, I remind you that
> leastness isn't enough to drop a defeat in SD: The defeat must
> also be in a cycle, not only initially, but also at the time
> when it is being considered for dropping.
>
> Norm is the only one who has posted an SD nonmonotonicity
> example, with no pair-ties or equal defeats, which specifies all
> of the defeats. But Norm, did you actually run that example with
> your SD count program? Because it seemed to me that Blake's
> posting about his example contained a count error.

If you need concrete numbers then you can use the following example:

Act I:

   AB 21
   BC 17
   CD 15
   DE 22
   EF 18
   FG 19
   GA 14
   DB 16
   GE 20

   AC 1
   AD 2
   AE 3
   AF 4
   BE 5
   BF 6
   BG 7
   CE 8
   CF 9
   CG 10
   DF 11
   DG 12

   SD chooses candidate A.

Act II:

   If "DE 22" is changed to "DE 13" then SD chooses
   candidate D. Therefore SD violates monotonicity
   even for 7 candidates.

But it is only necessary to specify the first 9 defeats
because as there is a cycle A > B > C > D > E > F > G > A
it is clear that every additional defeat will lock in a
cycle.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de