Smith set

Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results The Smith set,[note 1] sometimes called the top-cycle or Condorcet winning set,[1] generalizes the idea of a Condorcet winner to cases where no such winner exists.

It does so by allowing cycles of candidates to be treated jointly, as if they were a single Condorcet winner.

The Smith set provides one standard of optimal choice for an election outcome.

Proof: Suppose on the contrary that there exist two dominating sets, D and E, neither of which is a subset of the other.

Though less common, the term Smith-efficient has also been used for methods that elect from the Smith set.

The Smith set is not {A,B,C,D} because the definition calls for the smallest subset that meets the other conditions.

For example, the voting method Smith//Minimax applies Minimax to the candidates in the Smith set.

Another example is the Tideman alternative method, which alternates between eliminating candidates outside of the Smith set, and eliminating the candidate who was the plurality loser (similar to instant-runoff), until a Condorcet winner is found.

A different approach is to elect the member of the Smith set that is highest in the voting method's order of finish.

These certainly belong to the Smith set, and any candidates whom they do not defeat will need to be added.

Now we look at any new cells which need to be considered, which are those below the top-left square containing {A,D,G}, but excluding those in the first two columns which we have already accounted for.

We repeat the operation for the new cells below the four members which are known to belong to the Smith set.

The cells which come into consideration are shaded pale green, and since all their entries are zero we do not need to add any new candidates to the set, which is therefore fixed as {A,D,G,C,F}.