Smith set


In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as Generalized Top-Choice Assumption, is the smallest non-empty set of candidates in a particular election such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be "Smith-efficient".
A set of candidates where every member of the set pairwise defeats every member outside of the set is known as a dominating set.

Properties

The Schwartz set, known as the Generalized Optimal-Choice Axiom or GOCHA, is closely related to and is always a subset of the Smith set. The Smith set is larger if and only if a candidate in the Schwartz set has a pair-wise tie with a candidate that is not in the Schwartz set.
The Smith set can be constructed from the Schwartz set by repeatedly adding two types of candidates until no more such candidates exist outside the set:
Note that candidates of the second type can only exist after candidates of the first type have been added.

Alternative formulation

Any binary relation on a set can generate a natural partial order on the -cycle equivalence classes of set, so that implies.
When is the Beats-or-Ties binary relation on the set of candidates defined by Beats-or-Ties if and only if pairwise defeats or ties then the resulting partial order is the beat-or-tie order which is a total order. The Smith set is the maximal element of the beat-or-tie order.

Algorithms

The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ. It can also be calculated using a version of Kosaraju's algorithm or Tarjan's algorithm in time Θ.
It can also be found by creating a pairwise comparison matrix with the candidates ranked by their number of pairwise victories minus pairwise defeats, and then looking for the smallest top-left-most square of cells that can be covered such that all cells to the right of these cells show pairwise victories. All candidates named to the left of these cells are in the Smith set.
Example: Suppose candidates A, B, and C are in the Smith set, each pairwise beating one of the others, but all 3 pairwise beat D and E. A, B, and C would be placed in the top 3 rows of the pairwise comparison table, and then it would be seen that by covering all cells from "A beats A" to "C beats C", all cells to the right would show pairwise victories, whereas no smaller set of cells could do so, so A, B, and C would be in the Smith set.
Example using the Copeland ranking:
ABCDEFG
A---WinLoseWinWinWinWin
BLose---WinWinWinWinWin
CWinLose---LoseWinWinWin
DLoseLoseWin---TieWinWin
ELoseLoseLoseTie---WinWin
FLoseLoseLoseLoseLose---Win
GLoseLoseLoseLoseLoseLose---

A loses to C, so all candidates from A through C are confirmed to be in the Smith set. There is one matchup where a candidate already confirmed to be in the Smith set loses or ties someone not yet confirmed to be in the Smith set: C loses to D; so D is confirmed to be in the Smith set. Now there is another such matchup: D ties with E, so E is added into the Smith set. Because all of A through E beat all candidates not yet confirmed to be in the Smith set, the Smith set is now confirmed to be A through E.