- Smith set
In
voting system s, the Smith set is the smallest non-empty set of candidates in a particular election such that each member beats 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 theSmith criterion and are said to be "Smith-efficient".A set of candidates where every member of the set pair-wise beats every member outside of the set is also 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, non-empty, and the set of candidates is finite.
*The Smith set can have more than one candidate, either because of pair-wise ties or because of cycles, such as in
Condorcet's paradox .
*The Smith set contains, if they exist, the Condorcet winner or any weak Condorcet winners.
*The Condorcet winner exists if and only if the Smith Set has only one candidate.chwartz set comparison
The
Schwartz set is closely related to and is always asubset 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 pair-wise ties with candidates in the set,
* candidates that beat a candidate in the set. Note that candidates of the second type can only exist after candidates of the first type have be added.Alternative formulation
Any
binary relation R on a set A can generate a natural partial order on the R- cycleequivalence class es of set A, so that xRy implies [x] ≥ [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 pair-wise beats or ties y, then the resulting partial order is the beat-or-tie order which is a
total order . The Smith set is themaximal 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 ofKosaraju's algorithm in time Θ("n"2).References
*cite journal | author=Ward, Benjamin | title=Majority Rule and Allocation | journal=Journal of Conflict Resolution | year=1961 | volume=5 | pages=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.
*cite journal | author=Smith, J.H. | title=Aggregation of Preferences with Variable Electorates | journal=Econometrica | year=1973 | volume=41 | pages=1027–1041 | doi=10.2307/1914033 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.
*cite journal | author=Fishburn, Peter C. | title=Condorcet Social Choice Functions | journal=Siam Journal of Applied Mathematics | year=1977 | volume=33 | pages=469–489 | doi=10.1137/0133030 Narrows Smith's generalized Condorcet Criterion to the smallest dominating set and calls it Smith's Condorcet Principle.
*cite book | first=Thomas | last=Schwartz | year=1986 | title=The Logic of Collective Choice | publisher=Columbia University Press | location=New York Discusses the Smith set (named GETCHA) and the Schwartz set (named GOTCHA) as possible standards for optimal collective choice.
ee also
*
Smith criterion
*Schwartz set
*Condorcet Criterion
*Condorcet Method
*Preorder
*Partial order External links
* [http://wiki.electorama.com/wiki/Maximal_elements_algorithms Example algorithms to calculate the Smith set]
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