Minimax Condorcet method

In voting systems, the Minimax Condorcet method (often referred to as "the Minimax method" and sometimes as "minmax" or "min-max") is one of several Condorcet methods used for tabulating votes and determining a winner when using ranked voting in a single-winner election. It is sometimes referred to as the Simpson–Kramer method, and the successive reversal method.

Minimax selects as the winner the candidate whose greatest pairwise defeat is smaller than the greatest pairwise defeat of any other candidate: or, put another way, "the only candidate whose support never drops below [N] percent" in any pairwise contest.

Variants

Minmax or Minimax method, also referred to as the Simpson-Kramer method, is the name of a class of election methods based on electing the candidate with the most consistently high performance in pairwise contests with other candidates. It is sometimes also called the least reversal or successive reversal method, although this term is ambiguous.

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.

Minmax may be indecisive, particularly for small electorates, because it only considers a single pairwise contest for each candidate. This tendency to tie can be broken by a leximax tiebreaker: if two or more candidates have the same strength greatest defeat, then elect the one that has the least next-greatest defeat. Repeat as needed.

Minmax with a leximax tiebreaker is sometimes called Ext-Minmax on the election-methods mailing list. It is the standard tiebreaker for MMPO as defined by Mike Ossipoff.

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. Given this election (called the "bad-example" on EM):

• x: A>B=C
• 1: A=C>B
• 1: B=C>A
• x: B>A=C

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

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 with three or fewer candidates. Because of this, defeat-dropping methods that pass ISDA are equivalent to Smith//Minimax when the cycle involves only 3 candidates.

Since the defeat-droppers are equivalent to Minimax when three or fewer candidates run, they all fail dominant mutual third burial resistance. This follows from the equivalence and the three-candidate Minimax DMTBR failure example given above.