Minimax Condorcet method

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:

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

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

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities and that everyone wants to live as near to the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of the state
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

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, the capital in real life, 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 alternatives winning votes and margins, 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.

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.