Editing Smith set (section)

==The Smith criterion==
The Smith criterion is a [[Voting system criteria|voting system criterion]] that formalizes a stronger idea of [[majority rule]] than the [[Condorcet winner criterion|Condorcet criterion]]. A [[voting system]] satisfies the Smith criterion if it always picks a candidate from the Smith set.

Though less common, the term ''Smith-efficient'' has also been used for methods that elect from the Smith set.<ref name="y421">{{cite | last=Boudet | first=Samuel | title=Bipartisan/Range Voting in Two Rounds Reaches a Promising Balance between Efficiency and Strategy-Resistance | publisher=MDPI AG | date=2023-09-06 | doi=10.20944/preprints202309.0388.v1 | doi-access=free | page=}}</ref>

Here is an example of an electorate in which there is no Condorcet winner: There are four candidates: A, B, C and D. 40% of the voters rank D>A>B>C. 35% of the voters rank B>C>A>D. 25% of the voters rank C>A>B>D. The Smith set is {A,B,C}. All three candidates in the Smith set are majority-preferred over D (since 60% rank each of them over D). The Smith set is not {A,B,C,D} because the definition calls for the ''smallest'' subset that meets the other conditions. The Smith set is not {B,C} because B is not majority-preferred over A; 65% rank A over B. (Etc.)
{| class="wikitable"
!{{diagonal split header|pro|con}}
!A
!B
!C
!D
|-
!A
|—
|65
|40
|60
|-
!B
|35
|—
|75
|60
|-
!C
|60
|25
|—
|60
|-
!D
|40
|40
|40
|—
|-
!max opp
|60
|65
|75
|60
|-
!minimax
|60
|
|
|60
|}
In this example, under minimax, A and D tie; under Smith//Minimax, A wins.

In the example above, the three candidates in the Smith set are in a "rock/paper/scissors" ''majority cycle'': A is ranked over B by a 65% majority, B is ranked over C by a 75% majority, and C is ranked over A by a 60% majority.

=== Other criteria ===
Any election method that complies with the Smith criterion also complies with the [[Condorcet criterion|Condorcet winner criterion]], since if there is a Condorcet winner, then it is the only candidate in the Smith set. Smith methods also comply with the [[Condorcet loser criterion]], because a Condorcet loser will never fall in the Smith set. It also implies the [[mutual majority criterion]], since the Smith set is a subset of the MMC set.<ref name="Brandt" /> Conversely, any method that fails any of those three majoritarian criteria (Mutual majority, Condorcet loser or Condorcet winner) will also fail the Smith criterion.

=== Complying methods ===
The Smith criterion is satisfied by [[ranked pairs]], [[Schulze method|Schulze's method]], [[Nanson's method]], and several other methods. Moreover, any voting method can be modified to satisfy the Smith criterion, by finding the Smith set and eliminating any candidates outside of it. 

For example, the voting method Smith//Minimax applies Minimax to the candidates in the Smith set. Another example is the [[Tideman alternative method|Tideman alternative]] method, which alternates between eliminating candidates outside of the Smith set, and eliminating the candidate who was the plurality loser (similar to [[Instant-runoff voting|instant-runoff]]), until a Condorcet winner is found. A different approach is to elect the member of the Smith set that is highest in the voting method's order of finish. 

Methods failing the Condorcet criterion also fail the Smith criterion. However, some Condorcet methods (such as [[Minimax Condorcet|Minimax]]) can fail the Smith criterion.