Minimax Condorcet method

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In voting systems, the Minimax Condorcet method (often referred to as "the Minimax method") is a single-winner ranked-choice voting method that always elects the majority (Condorcet) winner.^[1] Minimax compares all candidates against each other in a round-robin tournament, then ranks candidates by their worst election result (the result where they lose by the smallest margin). The candidate with the largest (maximum) margin of victory in their worst (minimum) matchup is declared the winner.

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.

Football analogy

Imagine politicians compete like football teams in a round-robin tournament, where every team plays against every other team once. In a matchup, candidate's score is equal to the number of votes they get, minus the number of votes their opponent gets.

Minimax finds each team's (or candidate's) worst game--the one where they got the smallest number of points (votes). Each team's tournament score is equal to the number of points they got in their worst game. The first place in the tournament goes to the team with the best tournament score.

This method is similar to Copeland's method, which is used more often in sports; in Copeland's method, the team with the most wins gets first place in the tournament (or election). However, in real elections, Copeland's method can often result in ties. Both methods are Condorcet methods, which means that if a team wins every one of their games, they are guaranteed to get first place.

Formal definition

Formally, let ${\displaystyle \operatorname {score} (X,Y)}$ denote the pairwise score for ${\displaystyle X}$ against ${\displaystyle Y}$. Then the candidate, ${\displaystyle W}$ selected by minimax (aka the winner) is given by:

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

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 ${\displaystyle d(X,Y)}$ be the number of voters ranking X over Y. The variants define the score ${\displaystyle \operatorname {score} (X,Y)}$ for candidate X against Y as:

1. The number of voters ranking X above Y minus the number of voters ranking Y above X. This variant is called margins, and is the most common variant.
   • ${\displaystyle \operatorname {score} (X,Y):=d(X,Y)-d(Y,X)}$
2. 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.
   • ${\displaystyle \operatorname {score} (X,Y):={\begin{cases}d(X,Y),&d(X,Y)>d(Y,X)\\0,&{\text{else}}\end{cases}}}$
3. 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.
   • ${\displaystyle \operatorname {score} (X,Y):=d(X,Y)}$

When one of the first two variants 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.

Satisfied and failed criteria

Minimax using winning votes or margins satisfies the Condorcet and the majority criterion, but not the Smith criterion, mutual majority criterion, or Condorcet loser criterion. When winning votes is used, minimax also satisfies the plurality criterion.

Minimax fails independence of irrelevant alternatives, independence of clones, local independence of irrelevant alternatives^{[broken anchor]}, and independence of Smith-dominated alternatives: 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^{[citation needed]}.

When the pairwise opposition variant 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.

Markus Schulze modified minimax to satisfy several of the criteria above. Nicolaus Tideman's ranked pairs method additionally satisfies clone independence and local independence of irrelevant alternatives.

Examples

Example with Condorcet winner

Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:

• Memphis, the largest city, but far from the others (42% of voters)
• Nashville, near the center of the state (26% of voters)
• Chattanooga, somewhat east (15% of voters)
• Knoxville, far to the northeast (17% of voters)

The preferences of each region's voters are:

The results of the pairwise scores would be tabulated as follows:

• [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [Y] 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:

• [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

Result: With the winning votes and margins alternatives, 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:

• [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [Y] 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 – main minimax article
• Wald's maximin model – Wald's maximin model
• Multiwinner voting - 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 library supporting multiple Condorcet methods, including the three variants of Minimax method.
• Electowiki: minmax

This page was last edited on 24 March 2024, at 12:15