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
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