Minimax Condorcet

In voting systems, 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 also known 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

Minimax selects the candidate for whom the greatest pairwise score for another candidate against him is the least such score among all candidates.

Formally, let $score(X,Y)$ denote the pairwise score for $X$ against $Y$. Then the candidate, $W$ selected by minimax (aka the winner) is given by:

$W = \arg \min_X ( \max_Y score(Y, X))$

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 $d(X, Y)$ be the number of voters ranking X over Y. The variants define the score $score(X, Y)$ for candidate X against Y as:

1. 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.
   • $score(X,Y) := \begin{cases} d(X, Y) &, d(X, Y) > d(Y, X) \\ 0 &, else\end{cases}$
2. The number of voters ranking X above Y minus the number of voters ranking Y above X. This variant is called using margins.
   • $score(X,Y) := d(X, Y) - d(Y, X)$
3. 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.
   • $score(X,Y) := d(X, Y)$

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, Independence of clones criterion, or Condorcet loser criterion. When winning votes is used, Minimax also satisfies the Plurality criterion.

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.

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)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

The results of the pairwise scores would be tabulated as follows:

Pairwise election results

		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 (won-tied-lost):		0-0-3	3-0-0	2-0-1	1-0-2
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:

Pairwise election results

		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 (won-tied-lost):		2-0-0	0-0-2	1-0-1
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:

Pairwise election results

		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 (won-tied-lost):		2-0-1	2-0-1	2-0-1	0-0-3
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 choose A as winner, since it has the lowest value of 35 votes for the winner in his biggest defeat;
• the margin alternative choose B as winner, since it has the lowest difference of votes in his biggest defeat;
• and pairwise opposition choose the Condorcet looser 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

References

• Levin, Jonathan, and Barry Nalebuff. 1995. "An Introduction to Vote-Counting 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