Minimax Condorcet
Encyclopedia
In voting system
s, the Minimax method is one of several Condorcet method
s used for tabulating votes and determining a winner when using preferential voting
in a single-winner election. It is also known as the Simpson-Kramer method, and the successive reversal method.
Minimax selects the candidate for whom the greatest pairwise
score for another candidate against him is the least such score among all candidates.
Formally, let
denote the pairwise score for 
against 
.
Then the candidate,
selected by minimax (aka the winner) is given by:
When it is permitted to rank candidates equally, or to not rank all the candidates, three interpretations of the rule are possible. When voters must rank all the candidates, all three rules are equivalent.
The score for candidate x against y can be defined as:
When one of the first two interpretations is used, the method can be restated as: "Disregard the weakest pairwise
defeat until one candidate is unbeaten." An "unbeaten" candidate possesses a maximum score against him which is zero or negative.
Minimax using winning votes or margins satisfies Condorcet
and the majority criterion
, but not the Smith criterion
, mutual majority criterion
, independence of clones criterion
, or Condorcet loser criterion
. When winning votes is used, Minimax also satisfies the Plurality criterion
.
When the pairwise opposition interpretation is used, minimax also does not satisfy the Condorcet criterion
. However, when equal-ranking is permitted, there is never an incentive to put one's first-choice candidate below another one on one's ranking. It also satisfies the Later-no-harm criterion, which means that by listing additional, lower preferences in one's ranking, one cannot cause a preferred candidate to lose.
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
s, the Minimax method is one of several Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
s used for tabulating votes and determining a winner when using preferential voting
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...
in a single-winner election. It is also known as the Simpson-Kramer method, and the successive reversal method.
Minimax selects the candidate for whom the greatest pairwise
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
score for another candidate against him is the least such score among all candidates.
Formally, let
denote the pairwise score for against .Then the candidate,
selected by minimax (aka the winner) is given by:
When it is permitted to rank candidates equally, or to not rank all the candidates, three interpretations of the rule are possible. When voters must rank all the candidates, all three rules are equivalent.
The score for candidate x against y can be defined as:
- The number of voters ranking x above y, but only when this score exceeds the number of voters ranking y above x. If not, then the score for x against y is zero. This is sometimes called winning votes.
- The number of voters ranking x above y minus the number of voters ranking y above x. This is called using margins.
- The number of voters ranking x above y, regardless of whether more voters rank x above y or vice versa. This interpretation is sometimes called pairwise opposition.
When one of the first two interpretations is used, the method can be restated as: "Disregard the weakest pairwise
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...
defeat until one candidate is unbeaten." An "unbeaten" candidate possesses a maximum score against him which is zero or negative.
Minimax using winning votes or margins satisfies Condorcet
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...
and the majority criterion
Majority criterion
The majority criterion is a single-winner voting system criterion, used to compare such systems. The criterion states that "if one candidate is preferred by a majority of voters, then that candidate must win"....
, but not the Smith criterion
Smith set
In voting systems, the Smith set, named after John H. Smith, 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...
, mutual majority criterion
Mutual majority criterion
The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion...
, independence of clones criterion
Independence of clones criterion
In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman first formulated the criterion, which states that the addition of a candidate identical to one already present in an election will not cause the winner...
, or Condorcet loser criterion
Condorcet loser criterion
In single-winner voting system theory, the Condorcet loser criterion is a measure for differentiating voting systems. It implies the majority loser criterion....
. When winning votes is used, Minimax also satisfies the Plurality criterion
Plurality criterion
Plurality criterion is a voting system criterion devised by Douglas R. Woodall for ranked voting methods with incomplete ballots. It is stated as follows:...
.
When the pairwise opposition interpretation is used, minimax also does not satisfy the Condorcet criterion
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...
. However, when equal-ranking is permitted, there is never an incentive to put one's first-choice candidate below another one on one's ranking. It also satisfies the Later-no-harm criterion, which means that by listing additional, lower preferences in one's ranking, one cannot cause a preferred candidate to lose.