Smith set: Difference between revisions

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There are two definitions of the '''Smith set'''. One is general and holds for any voting method; the other can be used only when the votes permit a comparison of any pair of candidates. Thus, the Robert's Rules single elimination pairwise voting method (which is used for voting on motions and amendments) requires the more general definition because it doesn't elicit enough information to allow all pairs of alternatives to be compared. Most voting methods that use a single round of ~~voting--even~~ methods like Plurality Rule and ~~Approval--are~~ compatible with both definitions.

There are two definitions of the '''Smith set'''. One is general and holds for any voting method; the other can be used only when the votes permit a comparison of any pair of candidates. Thus, the Robert's Rules single elimination pairwise voting method (which is used for voting on motions and amendments) requires the more general definition because it doesn't elicit enough information to allow all pairs of alternatives to be compared. Most voting methods that use a single round of voting—even methods like Plurality Rule and Approval—are compatible with both definitions.

[Actually, Plurality and Approval always choose from the Smith set, when the definition referred to here as the simple definition is used. In that way, Plurality and Approval pass the Smith Criterion as it is often defined. But not when the Smith Criterion is defined as I define it two paragraphs below on this page]

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Also of note is the '''sincere Smith set''': the smallest non-empty set of candidates such that every candidate in the set is ''sincerely preferred'' by a majority of the voters over every candidate not in the set. (That isn't correct. It's only necessary that each candidate in the set be preferred to each candidate outside the set by more voters than vice-versa. It needn't be a majority Maybe some voters are indifferent between some candidate-pairs] Since the definition depends on voters' preferences, it does not matter what voting method is used, assuming voters' preferences do not depend on the voting method. However, that assumption is naive, because candidates who want to win choose positions on issues that they hope will help them win, and winning positions depend in part on the voting method, and candidates' positions affect voters' preferences. For example, Plurality Rule, Top Two Runoff, Instant Runoff and many other methods can cause candidates to avoid centrist positions due to the risk of being sandwiched between a candidate "on the left" and a candidate "on the right" whereas voting methods that satisfy the Smith criterion can cause candidates to take more centrist positions. In this important sense, the sincere Smith set (and the voted Smith set) depend on the voting method. However, this is beyond the scope of this article. (Unfortunately, many comparisons of voting methods naively oversimplify their analysis by assuming candidates' positions and voters' preferences are constant, and neglect the possibility that the most important criteria for comparing voting methods may involve the effects that voting methods have on candidates' positions.)

The Smith set can differ from the sincere Smith set because votes may misrepresent voters' preferences. (Voters sometimes have an incentive to strategically misrepresent their preferences, as the Gibbard-Satterthwaite "manipulability" theorem shows. Also, some voting methods such as "vote for one, plurality rule" and Approval Voting simply do not allow voters to accurately represent their preferences when there are more than two candidates.) When people say it is desirable that voting methods satisfy the Smith criterion, what they usually mean is that a candidate in the sincere Smith set should be elected. (Similarly, when people say it is desirable that voting methods satisfy the [[Condorcet criterion]] they mean it is desirable to elect the ''sincere'' Condorcet ~~winner--which~~ is defined according to voters' preferences rather than ~~votes--when~~ it exists.)

The Smith set can differ from the sincere Smith set because votes may misrepresent voters' preferences. (Voters sometimes have an incentive to strategically misrepresent their preferences, as the Gibbard-Satterthwaite "manipulability" theorem shows. Also, some voting methods such as "vote for one, plurality rule" and Approval Voting simply do not allow voters to accurately represent their preferences when there are more than two candidates.) When people say it is desirable that voting methods satisfy the Smith criterion, what they usually mean is that a candidate in the sincere Smith set should be elected. (Similarly, when people say it is desirable that voting methods satisfy the [[Condorcet criterion]] they mean it is desirable to elect the ''sincere'' Condorcet winner—which is defined according to voters' preferences rather than votes—when it exists.)

Satisfaction of the Smith criterion does not imply the winner is in the sincere Smith set. Therefore, satisfaction of additional criteria (for example, [[Truncation Resistance]] and [[Minimal Defense criterion]]) can help to elect a candidate in the sincere Smith set.

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==Properties==

The Smith set is the maximal cycle equivalence class of the [[Beatpath#Beat-or-~~tie_path~~ | beat-or-tie order]].

The Smith set is the maximal cycle equivalence class of the [[Beatpath#Beat-or-tie path|beat-or-tie order]].

The Smith set and the sincere Smith set always exist, regardless of the voting method. When a [[Condorcet Criterion |Condorcet winner]] exists, the Smith set contains only that candidate. This means every voting method that satisfies the Smith criterion also satisfies the [[Condorcet criterion]]. Similarly, when a sincere Condorcet winner exists, the sincere Smith set contains only that candidate.

The Smith set and the sincere Smith set always exist, regardless of the voting method. When a [[Condorcet Criterion|Condorcet winner]] exists, the Smith set contains only that candidate. This means every voting method that satisfies the Smith criterion also satisfies the [[Condorcet criterion]]. Similarly, when a sincere Condorcet winner exists, the sincere Smith set contains only that candidate.

If the Smith set contains more than one candidate, this is due to majority cycles such as in [[Condorcet's paradox]], or to pairwise ties.

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The [[Schwartz set]] is always a subset of the Smith set. The Schwartz set is equal to the Smith set except when there is a candidate in the Schwartz set that has a pairwise tie with a candidate outside of the Schwartz set.

The Uncovered set and the Banks set are also subsets of the Smith set. The Banks set is defined as the alternatives that can win given the Robert's Rules sequential pairwise voting method for at least one agenda order of candidates, assuming the voters are strategically sophisticated and know each ~~others~~' preferences. (An assumption more likely to hold when the voters are a professional legislature.)

The Uncovered set and the Banks set are also subsets of the Smith set. The Banks set is defined as the alternatives that can win given the Robert's Rules sequential pairwise voting method for at least one agenda order of candidates, assuming the voters are strategically sophisticated and know each other's preferences. (An assumption more likely to hold when the voters are a professional legislature.)

Some examples of the Smith set and Schwartz set are provided, some [[Beatpath examples 3 | with 3 candidates]] or with [[Beatpath example 12 | 12 candidates]].

Some examples of the Smith set and Schwartz set are provided, some [[Beatpath examples 3|with 3 candidates]] or with [[Beatpath example 12|12 candidates]].

The Smith set can be calculated using versions of either [http://en.wikipedia.org/wiki/Kosaraju%27s_algorithm Kosaraju's algorithm] or the [http://en.wikipedia.org/wiki/Floyd-Warshall_algorithm Floyd-Warshall algorithm]. Examples of both algorithms applied to calculating the Smith set and the Schwartz set are available [[maximal elements algorithms | here]].

The Smith set can be calculated using versions of either [http://en.wikipedia.org/wiki/Kosaraju%27s_algorithm Kosaraju's algorithm] or the [http://en.wikipedia.org/wiki/Floyd-Warshall_algorithm Floyd-Warshall algorithm]. Examples of both algorithms applied to calculating the Smith set and the Schwartz set are available [[maximal elements algorithms|here]].

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* [[Schwartz set]]

* [[Beatpath]]

* [[Beatpath examples 3 | Examples with 3 candidates]]

* [[Beatpath examples 3|Examples with 3 candidates]]

* [[Beatpath example 12 | Example 12 candidates]]

* [[Beatpath example 12|Example 12 candidates]]

* [[maximal elements algorithms | Algorithms to calculate the Smith set]]

* [[maximal elements algorithms|Algorithms to calculate the Smith set]]