Smith criterion
The Smith criterion (sometimes the generalized Condorcet criterion) is a
The Smith criterion is also called the top
An alternative, stricter criterion is given by the Landau set.
Determination
The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ(n3) or Kosaraju's algorithm in time Θ(n2).
Example
When there is a
pro\con | A | B | C | D |
---|---|---|---|---|
A | — | 65 | 40 | 60 |
B | 35 | — | 75 | 60 |
C | 60 | 25 | — | 60 |
D | 40 | 40 | 40 | — |
max opp | 60 | 65 | 75 | 60 |
minimax | 60 | 60 |
In this example, under minimax, A and D tie; under Smith//Minimax, A wins.
In the example above, the three candidates in the Smith set are in a "rock/paper/scissors" majority cycle: A is ranked over B by a 65% majority, B is ranked over C by a 75% majority, and C is ranked over A by a 60% majority.
Other criteria
Any election method that complies with the Smith criterion also complies with the
The Smith set and Schwartz set are sometimes confused in the literature. Miller (1977, p. 775) lists as an alternate name for the Smith set, but it actually refers to the Schwartz set. The Schwartz set is actually a subset of the Smith set (and equal to it if there are no pairwise ties between members of the Smith set).
Complying methods
The Smith criterion is satisfied by
] Moreover, any voting method can be modified to satisfy the Smith criterion, by finding the Smith set and eliminating any candidates outside of it. For example, the voting method Smith//Minimax applies Minimax to the candidates in the Smith set. Another approach is to elect the member of the Smith set that is highest in the voting method's order of finish.Methods failing the Condorcet criterion also fail the Smith criterion. However, some Condorcet methods (such as
Examples
Minimax
The Smith criterion implies the mutual majority criterion, so Minimax's failure to satisfy the Mutual majority criterion is also a failure to satisfy the Smith criterion. Observe that the set S = {A, B, C} in the example is the Smith set and D is the Minimax winner.