Minmax

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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.

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.

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.

Criterion compliances

Minmax with winning votes or margins passes monotonicity, the Condorcet criterion, but fails clone independence, the mutual majority criterion, the Condorcet loser criterion, and the favorite betrayal criterion.

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

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 combined with a tiebreaking rule. However, in return 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.

None of the Minmax methods pass the Smith criterion. Minmax also fails dominant mutual third burial resistance:

```5: A>B>C
4: B>A>C
2: C>B>A
```

B is the Condorcet winner and DMT candidate. But if the A-first group buries B under C:

```5: A>C>B
4: B>A>C
2: C>B>A
```

then that produces a cycle and the Minmax winner becomes A, which this group prefers.

Criticism

MMPO has been criticized for its counter-intuitive behavior on some elections.[1] Given this election (called the "bad-example" on EM):

• x: A
• 1: A=C
• 1: B=C
• x: B

MMPO elects C even if x is made arbitrarily large (say, 3.95 billion voters).

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).

Since the defeat-droppers are equivalent to either Minmax or Smith//Minmax when three or fewer candidates run, they all fail dominant mutual third burial resistance.

References

1. Benham, C. (2016-09-21). "Re: MMPO objections (hopefully better posted)". Election-methods mailing list archives.