[EM] Truncation, defeat strength, Landau
Markus Schulze
markus.schulze at alumni.tu-berlin.de
Sat Sep 6 05:01:01 PDT 2003
Dear participants,
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.
******
A "Landau winner" (aka an "uncovered candidate") is a
candidate A such that for each other candidate B at least
one of the following two statements is valid:
1. d[A,B] >= d[B,A].
2. There is a candidate C with
d[A,C] >= d[C,A] and d[C,B] >= d[B,C].
******
The "Landau set" (aka "uncovered set", aka "Fishburn set")
is the set of all Landau winners.
Markus Schulze
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