Minimax Condorcet
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voting system An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections m ...
s, the Minimax Condorcet method (often referred to as "the Minimax method") is one of several
Condorcet method A Condorcet method (; ) is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any others, whenever ...
s used for tabulating votes and determining a winner when using
ranked voting The term ranked voting (also known as preferential voting or ranked choice voting) refers to any voting system in which voters rank their candidates (or options) in a sequence of first or second (or third, etc.) on their respective ballots. Ra ...
in a single-winner election. It is sometimes referred to as the Simpson–Kramer method, and the successive reversal method. Minimax selects as the winner the candidate whose greatest pairwise defeat is smaller than the greatest pairwise defeat of any other candidate: or, put another way, "the only candidate whose support never drops below percent" in any pairwise contest.


Description of the method

The Minimax Condorcet method selects the candidate for whom the greatest pairwise score for another candidate against him or her is the least such score among all candidates.


Formal definition

Formally, let \operatorname(X,Y) denote the pairwise score for X against Y. Then the candidate, W selected by minimax (aka the winner) is given by: : W = \arg \min_X \left( \max_Y \operatorname(Y, X)\right)


Variants of the pairwise score

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 variants are equivalent. Let d(X, Y) be the number of voters ranking ''X'' over ''Y''. The variants define the score \operatorname(X, Y) for candidate ''X'' against ''Y'' 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 variant is sometimes called winning votes. #*\operatorname(X,Y) := \begin d(X, Y), & d(X, Y) > d(Y, X) \\ 0, & \text \end #The number of voters ranking ''X'' above ''Y'' minus the number of voters ranking ''Y'' above ''X''. This variant is called using margins. #*\operatorname(X,Y) := d(X, Y) - d(Y, X) #The number of voters ranking ''X'' above ''Y'', regardless of whether more voters rank ''X'' above ''Y'' or vice versa. This variant is sometimes called pairwise opposition. #*\operatorname(X,Y) := d(X, Y) When one of the first two variants is used, the method can be restated as: "Disregard the weakest
pairwise Pairwise generally means "occurring in pairs" or "two at a time." Pairwise may also refer to: * Pairwise disjoint * Pairwise independence of random variables * Pairwise comparison, the process of comparing two entities to determine which is prefer ...
defeat until one candidate is unbeaten." An "unbeaten" candidate possesses a maximum score against him which is zero or negative.


Satisfied and failed criteria

Minimax using ''winning votes'' or ''margins'' satisfies the
Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher and mathematician. His ideas, including support for a liberal economy, free and equal p ...
and the majority criterion, but not the
Smith criterion The Smith criterion (sometimes generalized Condorcet criterion, but this can have other meanings) is a voting systems criterion defined such that it's satisfied when a voting system always elects a candidate that is in the Smith set, which is the ...
,
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. The criterion states that if there is a subset S of the candidate ...
, or
Condorcet loser criterion In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion. A voting system complying wi ...
. When ''winning votes'' is used, minimax also satisfies the
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: :If the number of ballots ranking A as the first preference is greater than the number of b ...
. Minimax cannot satisfy the
independence of clones criterion In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the ...
because clones will have narrow win margins between them; this implies Minimax cannot satisfy
local independence of irrelevant alternatives The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it a ...
because three clones may form a cycle of narrow defeats as the first-, second-, and third-place winners, and removing the second-place winner may cause the third-place winner to be elected. When the ''pairwise opposition'' variant is used, minimax also does not satisfy the
Condorcet criterion An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...
. 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 The later-no-harm criterion is a voting system criterion formulated by Douglas Woodall. Woodall defined the criterion as " ding a later preference to a ballot should not harm any candidate already listed." For example, a ranked voting method in w ...
criterion, which means that by listing additional, lower preferences in one's ranking, one cannot cause a preferred candidate to lose. When constrained to the Smith set, as Smith/Minimax, minimax satisfies the Smith criterion and, by implication, the mutual majority, independence of Smith-dominated alternatives, and Condorcet loser criterion. Markus Schulze modified minimax to satisfy several of the criteria above. Compared to Smith/Minimax, Nicolaus Tideman's
ranked pairs Ranked pairs (sometimes abbreviated "RP") or the Tideman method is an electoral system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. The ranked-pairs procedure can also be used to create ...
method additionally satisfies clone independence and local independence of irrelevant alternatives.


Examples


Example with Condorcet winner

The results of the pairwise scores would be tabulated as follows: * indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption * indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption Result: In all three alternatives Nashville has the lowest value and is elected winner.


Example with Condorcet winner that is not elected winner (for pairwise opposition)

Assume three candidates A, B and C and voters with the following preferences: The results would be tabulated as follows: * indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption * indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption Result: With the alternatives winning votes and margins, the Condorcet winner A is declared Minimax winner. However, using the pairwise opposition alternative, C is declared winner, since less voters strongly oppose him in his worst pairwise score against A than A is opposed by in his worst pairwise score against B.


Example without Condorcet winner

Assume four candidates A, B, C and D. Voters are allowed to not consider some candidates (denoting an n/a in the table), so that their ballots are not taken into account for pairwise scores of that candidates. The results would be tabulated as follows: * indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption * indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption Result: Each of the three alternatives gives another winner: * the winning votes alternative chooses A as winner, since it has the lowest value of 35 votes for the winner in his biggest defeat; * the margin alternative chooses B as winner, since it has the lowest difference of votes in his biggest defeat; * and pairwise opposition chooses the Condorcet loser D as winner, since it has the lowest votes of the biggest opponent in all pairwise scores.


See also

*
Minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When ...
– main minimax article *
Wald's maximin model In decision theory and game theory, Wald's maximin model is a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad worst outco ...
– Wald's maximin model *
Multiwinner voting Multiwinner voting, also called multiple-winner elections or committee voting or committee elections, is an electoral system in which multiple candidates are elected. The number of elected candidates is usually fixed in advance. For example, it can ...
- contains information on some multiwinner variants of Minimax Condorcet.


References

*Levin, Jonathan, and Barry Nalebuff. 1995. "An Introduction to Vote-Counting Schemes." Journal of Economic Perspectives, 9(1): 3–26.


External links


Description of ranked ballot voting methods: Simpson
by Rob LeGrand
Condorcet Class
PHP PHP is a general-purpose scripting language geared toward web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by The PHP Group ...
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supporting multiple Condorcet methods, including the three variants of Minimax method.
Electowiki: minmax
{{voting systems Monotonic Condorcet methods