[EM] Logic/Jargon question

[Daniel Bishop]

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.