Minimax Condorcet method

Minimax is often considered to be the simplest of the Condorcet methods. 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 $score(X,Y)$ denote the pairwise score for $X$ against $Y$. Then the candidate, $W$ selected by minimax (aka the winner) is given by:

$W=\arg \min _{X}(\max _{Y}score(Y,X))$

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:

1. 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.
2. The number of voters ranking x above y minus the number of voters ranking y above x. This is called using margins.
3. 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 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.

See also

• Minimax - main minimax article
• Wald's maximin model - Wald's maximin model

External resources

• Description of ranked ballot voting methods: Simpson by Rob LeGrand
• Electowiki: minmax