[EM] Smith set

[MIKE OSSIPOFF]

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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