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Invoting systems, theMinimax Condorcet method is asingle-winnerranked-choice voting method that always elects themajority (Condorcet) winner.^[1] Minimax compares all candidates against each other in around-robin tournament, then ranks candidates by their worst election result (the result where they would receive the fewest votes). The candidate with thelargest (maximum) number of votes in theirworst (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 around-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${\displaystyle \mathrm{score}(X,Y)}$ denote the pairwise score for${\displaystyle X}$ against${\displaystyle Y}$. Then the candidate,${\displaystyle W}$ selected by minimax (aka the winner) is given by:

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

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${\displaystyle d(X,Y)}$ be the number of voters rankingX overY. The variants define the score${\displaystyle \mathrm{score}(X,Y)}$ for candidateX againstY as:

1. The number of voters rankingX aboveY, but only when this score exceeds the number of voters rankingY aboveX. If not, then the score forX againstY is zero. This variant is sometimes calledwinning votes is the most commonly used and preferred bysocial choice theorists.
   • 
2. The number of voters rankingX aboveY minus the number of voters rankingY aboveX. This variant is calledmargins, and is less used.
   • ${\displaystyle \mathrm{score}(X,Y):=d(X,Y)-d(Y,X)}$
3. The number of voters rankingX aboveY, regardless of whether more voters rankX aboveY or vice versa. This variant is calledpairwise opposition, and is also rarely used.
   • ${\displaystyle \mathrm{score}(X,Y):=d(X,Y)}$

When one of the first two variants is used, the method can be restated as: "Disregard the weakestpairwise defeat until one candidate is unbeaten." An "unbeaten" candidate possesses a maximum score against him which is zero or negative.

Minimax usingwinning votes ormargins satisfies theCondorcet and themajority criterion, but not theSmith criterion,mutual majority criterion, orCondorcet loser criterion. Whenwinning votes is used, minimax also satisfies theplurality criterion.

Minimax failsindependence of irrelevant alternatives,independence of clones,local independence of irrelevant alternatives, andindependence of Smith-dominated alternatives.^{[citation needed]}

With thepairwise opposition variant (sometimes called MMPO), minimax only satisfies the majority-strengthCondorcet criterion; a candidate with arelative majority over every other may not be elected. MMPO is alater-no-harm system and also satisfiessincere favorite criterion.

Nicolaus Tidemanmodified minimax to only drop edges that createCondorcet cycles, allowing his method to satisfy many of the above properties.Schulze's method similarly reduces to minimax when there are only three candidates.

SupposeTennessee is holding an election on the location of itscapital. The population is split between four cities, andall the voters want the capital to be as close to them as possible. The options 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 alternativesNashville has the lowest value and is elected winner.

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 winnerA 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.

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 choosesA as winner, since it has the lowest value of 35 votes for the winner in his biggest defeat;
• the margin alternative choosesB as winner, since it has the lowest difference of votes in his biggest defeat;
• and pairwise opposition chooses the Condorcet loserD as winner, since it has the lowest votes of the biggest opponent in all pairwise scores.

• Minimax – main minimax article
• Multiwinner voting – contains information on some multiwinner variants of Minimax Condorcet.

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

• Description of ranked ballot voting methods: Simpson by Rob LeGrand
• Condorcet ClassPHPlibrary supporting multiple Condorcet methods, including the three variants of Minimax method.
• Electowiki: minmax

• Monotonic Condorcet methods

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