From
Wikipedia:
In voting systems, the Smith set is the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. Ideally, this set consists of only one candidate, the Condorcet winner. However, when the electorate is conflicted (as in Condorcet's paradox), the set has at least one cycle of candidates for whom A beats B, B beats C, and C beats A. See also Schwartz set.
If there are N
candidates, how can the size of the Smith set be smaller than N-1 if it is not
exactly 1 (i.e. there is a Condorcet winner)?
If there's no CW,
then disregarding ties there can be only one candidate who pairwise-loses to all
of the others, so candidates for the Smith set are all who pairwise defeat that
one.
Suppose there are
two members of the Smith set's complement. Then one would have pairwise-beaten
the other, and therefore would not have pairwise-lost to anybody outside of
the Smith set, which would make it a part of the Smith set.
I can see how there
can be "a Smith partition", but THE Smith set just seems to be by definition a
partitioning that excludes either N-1 candidates (when there is a Condorcet
winner) or 1 candidate (when there isn't a CW).
Or is there
something I'm missing?
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