[EM] Truncation, defeat strength, Landau
Alex Small
asmall at physics.ucsb.edu
Sat Sep 6 09:13:01 PDT 2003
I found it somewhat difficult to follow, but is it true that if there are
no pairwise ties the Smith and Schwartz sets are identical?
Markus Schulze said:
> suppose "d[X,Y]" with X<>Y is the number of voters who
> strictly prefer candidate X to candidate Y. Then the
> "Smith set" is the smallest non-empty set of candidates
> with d[A,B] > d[B,A] for each candidate A of this set
> and each candidate B outside this set.
>
> A "chain from candidate A to candidate B" is an ordered
> set of candidates C(1),...,C(n) with the following three
> properties:
>
> 1. C(1) is identical to A.
> 2. C(n) is identical to B.
> 3. d[C(i),C(i+1)] - d[C(i+1),C(i)] > 0
> for each i = 1,...,(n-1).
>
> A "Schwartz winner" is a candidate A who has chains at
> least to every other candidate B who has a chain to
> candidate A. The "Schwartz set" is the set of all Schwartz
> winners.
>
> The term "innermost unbeaten set" is another term for
> "Schwartz set."
>
> Example: Suppose that there are 3 candidates; candidate A
> pairwise beats candidate B; candidate B pairwise beats
> candidate C; and there is a pairwise tie between
> candidate A and candidate C. Then the Smith set
> is ABC and the Schwartz set is A.
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