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 (GETCHA), 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 Smith set always exists and is well-defined. There is only one smallest dominating set since dominating sets are nested and non-empty and the set of candidates is finite.
- The Smith set can have more than one candidate, either because of pairwise ties or because of cycles, such as in Condorcet's paradox.
- The Condorcet winner, if one exists, is the sole member of the Smith set. If weak Condorcet winners exist then they are in the Smith set.
- The Smith set is always a subset of the mutual majority-preferred set of candidates, if one exists.[1]
Schwartz set comparison
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:
- candidates that have pairwise ties with candidates in the set,
- candidates that defeat a candidate in 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 (a Copeland method ranking), 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. (This works because Copeland ranks the candidates such that the Smith set candidates have more points than the non-Smith set candidates[2])
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 (suppose that they're placed in that order for this example) of the pairwise comparison table, and then it would be seen that by covering all cells from "A beats A" (the cell comparing A to themselves) to "C beats C", all cells to the right (the cells comparing A, B, and C to D and E) 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:
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| A | --- | Win | Lose | Win | Win | Win | Win |
| B | Lose | --- | Win | Win | Win | Win | Win |
| C | Win | Lose | --- | Lose | Win | Win | Win |
| D | Lose | Lose | Win | --- | Tie | Win | Win |
| E | Lose | Lose | Lose | Tie | --- | Win | Win |
| F | Lose | Lose | Lose | Lose | Lose | --- | Win |
| G | Lose | Lose | Lose | Lose | Lose | Lose | --- |
A loses to C, so all candidates from A through C (A, B, and 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.
References
- http://dss.in.tum.de/files/brandt-research/dodgson.pdf
- This is because every candidate in the Smith set can only be pairwise beaten or tied by other Smith set members, while every candidate not in the Smith set is beaten by every Smith set member.
- Ward, Benjamin (1961). "Majority Rule and Allocation". Journal of Conflict Resolution. 5 (4): 379–389. doi:10.1177/002200276100500405. In an analysis of serial decision making based on majority rule, describes the Smith set and the Schwartz set.
- Smith, J.H. (1973). "Aggregation of Preferences with Variable Electorates". Econometrica. The Econometric Society. 41 (6): 1027–1041. doi:10.2307/1914033. JSTOR 1914033. Introduces a version of a generalized Condorcet Criterion that is satisfied when pairwise elections are based on simple majority choice, and for any dominating set, any candidate in the set is collectively preferred to any candidate not in the set. But Smith does not discuss the idea of a smallest dominating set.
- Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. Narrows Smith's generalized Condorcet Criterion to the smallest dominating set and calls it Smith's Condorcet Principle.
- Schwartz, Thomas (1986). The Logic of Collective Choice. New York: Columbia University Press. Discusses the Smith set (named GETCHA) and the Schwartz set (named GOTCHA) as possible standards for optimal collective choice.