A preliminary definition:
X is publicly preferred to Y if more people prefer X to Y than vice-versa.
X beats Y if more people rank X over Y than Y over X.
The _sincere_ Smith set is smallest set of candidates such that they're all publicly preferred to everyone outside the set.
[end of sincere Smith set definition]
The Smith set is the smallest set of candidates such that they all pairwise-beat everyone outside that set.
[end of Smith set definition]
Someone wrote:
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 reply:
Not losing to anyone outside the Smith set doesn't make him a member of the Smith set. Yes, that nonmember who beats the other nonmember is in a set such that every member of that set beats everyone outside that set. But the Smith set is the _smallest_ such set. And that nonmember isn't in the smallest set of candidates who beat everyone outside that set.
Mike Ossipoff
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