Minimax Condorcet method

In voting systems, the Minimax Condorcet method (often referred to as "the Minimax method") 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.
Description of the method.
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.
Formal definition.
Formally, let formula_1 denote the pairwise score for formula_2 against formula_3. Then the candidate, formula_4 selected by minimax (aka the winner) is given by:
Variants of the pairwise score.
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 formula_6 be the number of voters ranking "X" over "Y". The variants define the score formula_7 for candidate "X" against "Y" as:
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.
Satisfied and failed criteria.
Minimax using "winning votes" or "margins" satisfies the Condorcet and the majority criterion, but not the Smith criterion, mutual majority criterion, or Condorcet loser criterion. When "winning votes" is used, minimax also satisfies the Plurality criterion. Minimax cannot satisfy the independence of clones criterion because clones will have narrow win margins between them; this implies Minimax cannot satisfy local independence of irrelevant alternatives because three clones may form a cycle of narrow defeats as the first-, second-, and third-place winners, and removing the second-place winner may cause the third-place winner to be elected.
When the "pairwise opposition" variant is used, minimax also does not satisfy the Condorcet criterion. However, when equal-ranking is permitted, there is never an incentive to put one's first-choice candidate below another one on one's ranking. It also satisfies the later-no-harm criterion, which means that by listing additional, lower preferences in one's ranking, one cannot cause a preferred candidate to lose.
When constrained to the Smith set, as Smith/Minimax, minimax satisfies the Smith criterion and, by implication, the mutual majority, independence of Smith-dominated alternatives, and Condorcet loser criterion.
Markus Schulze modified minimax to satisfy several of the criteria above. Compared to Smith/Minimax, Nicolaus Tideman's ranked pairs method additionally satisfies clone independence and local independence of irrelevant alternatives.
Examples.
Example with Condorcet winner.
The results of the pairwise scores would be tabulated as follows:
Result: In all three alternatives Nashville, the capital in real life, has the lowest value and is elected winner.
Example with Condorcet winner that is not elected winner (for pairwise opposition).
Assume three candidates A, B and C and voters with the following preferences:
The results would be tabulated as follows:
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.
Example without Condorcet winner.
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:
Result:
Each of the three alternatives gives another winner: