Smith set

In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as GETCHA, is the smallest non-empty set of candidates in a particular election such that each member defeats every other 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 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.

Schwartz set comparison

The Schwartz set 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 $R$ on a set $A$ can generate a natural partial order on the $R$ -cycle equivalence classes of set $A$ , so that $xRy$ implies $[x]\geq [y]$ .

When $R$ is the Beats-or-Ties binary relation on the set of candidates defined by $x$ Beats-or-Ties $y$ if and only if $x$ pairwise defeats or ties $y$ 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 $(n^{3})$ . It can also be calculated using a version of Kosaraju's algorithm or Tarjan's algorithm in time Θ $(n^{2})$ .

References

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 of 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.

External links

Example algorithms to calculate the Smith set

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