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, a candidate's score is equal to the number of voters who support them over their opponent.
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.
Formally, let
\operatorname{score}(X,Y)
X
Y
W
W=\argminX\left(maxY\operatorname{score}(Y,X)\right)
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
d(X,Y)
\operatorname{score}(X,Y)
\operatorname{score}(X,Y):=\begin{cases}d(X,Y),&d(X,Y)>d(Y,X)\ 0,&else\end{cases}
\operatorname{score}(X,Y):=d(X,Y)-d(Y,X)
\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.
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.
With the pairwise opposition variant (sometimes called MMPO), minimax only satisfies the majority-strength Condorcet criterion; a candidate with a relative majority over every other may not be elected. MMPO is a later-no-harm system and also satisfies sincere favorite criterion.
Nicolaus Tideman modified minimax to only drop edges that create Condorcet cycles, allowing his method to satisfy many of the above properties. Schulze's method similarly reduces to minimax when there are only three candidates.
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 |
|---|---|---|---|
| 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% | ||
| 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 | |
|---|---|---|---|---|---|---|---|---|
| 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 | ||
| 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: