# Minmax

**Minmax** or Minimax (Simpson-Kramer method) is the name of several election methods based on electing the candidate with the lowest score, based on votes received in pairwise contests with other candidates.

**Minmax(winning votes)** elects the candidate whose greatest pairwise loss to another candidate is the least, when the strength of a pairwise loss is measured as the number of voters who voted for the winning side.

**Minmax(margins)** is the same, except that the strength of a pairwise loss is measured as the number of votes for the winning side *minus* the number of votes for the losing side.

Criteria passed by both methods: Condorcet criterion, majority criterion

Criteria failed by both methods: Smith criterion, mutual majority criterion, Condorcet loser criterion.

Minmax(winning votes) also satisfies the Plurality criterion. In the three-candidate case, Minmax(margins) satisfies the Participation criterion.

**Minmax(pairwise opposition)** or **MMPO** elects the candidate whose greatest *opposition* from another candidate is minimal. Pairwise wins or losses are not considered; all that matters is the number of votes for one candidate over another.

Pairwise opposition is defined for a pair of candidates. For X and Y, X's pairwise opposition in that pair is the number of ballots ranking Y over X. MMPO elects the candidate whose greatest pairwise opposition is the least.

Minmax(pairwise opposition) does not strictly satisfy the Condorcet criterion or Smith criterion. It also fails the Plurality criterion, and is more indecisive than the other Minmax methods (unless it's used with a tiebreaking rule such as the simple one described below). However, it satisfies the Later-no-harm criterion, the Favorite Betrayal criterion, and in the three-candidate case, the Participation criterion, and the Chicken Dilemma Criterion.

MMPO's choice rule can be regarded as a kind of social optimization: The election of the candidate to whom fewest people prefer another. That choice rule can be offered as a standard in and of itself.

MMPO's simple tiebreaker:

If two or more candidates have the same greatest pairwise opposition, then elect the one that has the lowest next-greatest pairwise opposition. Repeat as needed.

## Notes

The Minimax method can be thought of as "Until there is a candidate or group of candidates with no pairwise losses, repeatedly drop (turn into a pairwise tie) the weakest pairwise defeat."

This contrasts with Schulze, which alternates between eliminating all candidates not in the Schwartz set and dropping defeats.

All defeat-dropping Condorcet methods become equivalent to Minimax when there are 3 or fewer candidates with no pairwise ties between them. Because of this, defeat-droppers that pass ISDA are equivalent to Smith//Minimax when the above conditions hold for the Smith set. Example:

25 A>B>C

40 B>C>A

35 C>A>B

The pairwise victories are 60 A>B, 65 B>C, 75 C>A. The A>B defeat is weakest by winning votes, so dropping it results in B being undefeated (alternatively, B's win can be explained as them being on the losing end of this defeat, and this defeat being their strongest defeat, since it's their only defeat).