Paul Kislanko wrote:
From Wikipedia:
In voting systems <http://en.wikipedia.org/wiki/Voting_system>, 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 <http://en.wikipedia.org/wiki/Condorcet_winner>. However, when the electorate is conflicted (as in Condorcet's paradox <http://en.wikipedia.org/wiki/Voting_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 <http://en.wikipedia.org/wiki/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.
There can be a Condorcet loser *after* you've eliminated the Condorcet loser, and that candidate will not be in the Smith Set either.
For example: With the ballots
A>B>C>D>E B>C>A>D>E C>A>B>D>E
The Smith Set is {A, B, C}, which excludes the Condorcet loser E and the secondary Condorcet loser D.
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