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Electoral systems 


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 singlewinner election. It is sometimes referred to as the Simpson–Kramer method, and the successive reversal method.^{[citation needed]}
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.
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 denote the pairwise score for against . Then the candidate, 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 be the number of voters ranking X over Y. The variants define the score for candidate X against Y as:
 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.
 The number of voters ranking X above Y minus the number of voters ranking Y above X. This variant is called using margins.
 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.
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 thirdplace winners, and removing the secondplace winner may cause the thirdplace winner to be elected.
When the pairwise opposition variant is used, minimax also does not satisfy the Condorcet criterion. However, when equalranking is permitted, there is never an incentive to put one's firstchoice candidate below another one on one's ranking. It also satisfies the laternoharm 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 Smithdominated 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
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:
42% of voters (close to Memphis) 
26% of voters (close to Nashville) 
15% of voters (close to Chattanooga) 
17% of voters (close to Knoxville) 





The results of the pairwise scores would be tabulated as follows:
X  
Memphis  Nashville  Chattanooga  Knoxville  
Y  Memphis  [X] 58% [Y] 42% 
[X] 58% [Y] 42% 
[X] 58% [Y] 42%  
Nashville  [X] 42% [Y] 58% 
[X] 32% [Y] 68% 
[X] 32% [Y] 68%  
Chattanooga  [X] 42% [Y] 58% 
[X] 68% [Y] 32% 
[X] 17% [Y] 83%  
Knoxville  [X] 42% [Y] 58% 
[X] 68% [Y] 32% 
[X] 83% [Y] 17%  
Pairwise election results (wontiedlost):  003  300  201  102  
worst pairwise defeat (winning votes):  58%  0%  68%  83%  
worst pairwise defeat (margins):  16%  −16%  36%  66%  
worst pairwise opposition:  58%  42%  68%  83% 
 [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.
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:
4% of voters  47% of voters  43% of voters  6% of voters 

1. A and C  1. A  1. C  1. B 
2. C  2. B  2. A and C  
3. B  3. B  3. A 
The results would be tabulated as follows:
X  
A  B  C  
Y  A  [X] 49% [Y] 51% 
[X] 43% [Y] 47%  
B  [X] 51% [Y] 49% 
[X] 94% [Y] 6%  
C  [X] 47% [Y] 43% 
[X] 6% [Y] 94% 

Pairwise election results (wontiedlost):  200  002  101  
worst pairwise defeat (winning votes):  0%  94%  47%  
worst pairwise defeat (margins):  −2%  88%  4%  
worst pairwise opposition:  49%  94%  47% 
 [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.
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.
30 voters  15 voters  14 voters  6 voters  4 voters  16 voters  14 voters  3 voters 

1. A  1. D  1. D  1. B  1. D  1. C  1. B  1. C 
2. C  2. B  2. B  2. C  2. C  2. A and B  2. C  2. A 
3. B  3. A  3. C  3. A  3. A and B  
4. D  4. C  4. A  4. D  
n/a D  n/a A and D  n/a B and D 
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 35 [Y] 30 
[X] 43 [Y] 45 
[X] 33 [Y] 36  
B  [X] 30 [Y] 35 
[X] 50 [Y] 49 
[X] 33 [Y] 36  
C  [X] 45 [Y] 43 
[X] 49 [Y] 50 
[X] 33 [Y] 36  
D  [X] 36 [Y] 33 
[X] 36 [Y] 33 
[X] 36 [Y] 33 

Pairwise election results (wontiedlost):  201  201  201  003  
worst pairwise defeat (winning votes):  35  50  45  36  
worst pairwise defeat (margins):  5  1  2  3  
worst pairwise opposition:  43  50  49  36 
 [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.
See also
 Minimax – main minimax article
 Wald's maximin model – Wald's maximin model
 Multiwinner voting  contains information on some multiwinner variants of Minimax Condorcet.
References
 Levin, Jonathan, and Barry Nalebuff. 1995. "An Introduction to VoteCounting Schemes." Journal of Economic Perspectives, 9(1): 3–26.
External links
 Description of ranked ballot voting methods: Simpson by Rob LeGrand
 Condorcet Class PHP library supporting multiple Condorcet methods, including the three variants of Minimax method.
 Electowiki: minmax