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In voting systems, the Minimax Condorcet 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 would receive the fewest votes). The candidate with the largest (maximum) number of votes in their worst (minimum) matchup is declared the winner.
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.
Imagine politicians compete like football teams in a round-robin tournament, where every team plays against every other team once. In each matchup, candidate's score is equal to the number of votes they win.
Minimax finds each team's (or candidate's) worst game – the one where they received 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, Copeland's method often results 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.
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 not rank all candidates, three interpretations of the rule are possible. When voters must rank all the candidates, all three variants are equivalent.
Let be the number of voters ranking X over Y. The variants define the score for candidate X against Y as:
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.
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, and independence of Smith-dominated alternatives.[ 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 further modifies it to satisfy local independence of irrelevant alternatives.
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:
The preferences of each region's voters are:
42% of voters Far-West |
26% of voters Center |
15% of voters Center-East |
17% of voters Far-East |
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The results of the pairwise scores would be tabulated as follows:
X | |||||
Memphis | Nashville | Chattanooga | Knoxville | ||
Y | Memphis | [X] 58% [Y] 42% |
[X] 58% [Y] 42% |
[X] 58% [Y] 42% | |
Nashville | [X] 42% [Y] 58% |
[X] 32% [Y] 68% |
[X] 32% [Y] 68% | ||
Chattanooga | [X] 42% [Y] 58% |
[X] 68% [Y] 32% |
[X] 17% [Y] 83% | ||
Knoxville | [X] 42% [Y] 58% |
[X] 68% [Y] 32% |
[X] 83% [Y] 17% | ||
Pairwise election results (won-tied-lost): | 0-0-3 | 3-0-0 | 2-0-1 | 1-0-2 | |
worst pairwise defeat (winning votes): | 58% | 0% | 68% | 83% | |
worst pairwise defeat (margins): | 16% | −16% | 36% | 66% | |
worst pairwise opposition: | 58% | 42% | 68% | 83% |
Result: In all three alternatives Nashville has the lowest value and is elected winner.
Assume three candidates A, B and C and voters with the following preferences:
4% of voters | 47% of voters | 43% of voters | 6% of voters |
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1. A and C | 1. A | 1. C | 1. B |
2. C | 2. B | 2. A and C | |
3. B | 3. B | 3. A |
The results would be tabulated as follows:
X | ||||
A | B | C | ||
Y | A | [X] 49% [Y] 51% |
[X] 43% [Y] 47% | |
B | [X] 51% [Y] 49% |
[X] 94% [Y] 6% | ||
C | [X] 47% [Y] 43% |
[X] 6% [Y] 94% |
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Pairwise election results (won-tied-lost): | 2-0-0 | 0-0-2 | 1-0-1 | |
worst pairwise defeat (winning votes): | 0% | 94% | 47% | |
worst pairwise defeat (margins): | −2% | 88% | 4% | |
worst pairwise opposition: | 49% | 94% | 47% |
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.
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.
30 voters | 15 voters | 14 voters | 6 voters | 4 voters | 16 voters | 14 voters | 3 voters |
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1. A | 1. D | 1. D | 1. B | 1. D | 1. C | 1. B | 1. C |
2. C | 2. B | 2. B | 2. C | 2. C | 2. A and B | 2. C | 2. A |
3. B | 3. A | 3. C | 3. A | 3. A and B | |||
4. D | 4. C | 4. A | 4. D | ||||
n/a D | n/a A and D | n/a B and D |
The results would be tabulated as follows:
X | |||||
A | B | C | D | ||
Y | A | [X] 35 [Y] 30 |
[X] 43 [Y] 45 |
[X] 33 [Y] 36 | |
B | [X] 30 [Y] 35 |
[X] 50 [Y] 49 |
[X] 33 [Y] 36 | ||
C | [X] 45 [Y] 43 |
[X] 49 [Y] 50 |
[X] 33 [Y] 36 | ||
D | [X] 36 [Y] 33 |
[X] 36 [Y] 33 |
[X] 36 [Y] 33 |
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Pairwise election results (won-tied-lost): | 2-0-1 | 2-0-1 | 2-0-1 | 0-0-3 | |
worst pairwise defeat (winning votes): | 35 | 50 | 45 | 36 | |
worst pairwise defeat (margins): | 5 | 1 | 2 | 3 | |
worst pairwise opposition: | 43 | 50 | 49 | 36 |
Result: Each of the three alternatives gives another winner: