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Electoral systems 


In voting systems, the Minimax Condorcet method is a singlewinner rankedchoice voting method that always elects the majority (Condorcet) winner.^{[1]} Minimax compares all candidates against each other in a roundrobin tournament, then ranks candidates by their worst election result (the result where they would receive the fewest votes). The candidate with the largest (maximum) number of votes in their worst (minimum) matchup is declared the winner.
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Voting Theory: Plurality Method and Condorcet Criterion

The Condorcet Method (Voting System)

Simulating alternate voting systems

DEFINITE INTEGRALS using SIMPSON'S 1/3rd RULE  C PROGRAM [TUTORIAL]

Voting Paradoxes and Combinatorics  Noga Alon
Transcription
 WELCOME TO A LESSON ON THE PLURALITY VOTING METHOD. IN THIS LESSON WE'LL DEFINE THE PLURALITY VOTING METHOD, DETERMINE WINNERS OF ELECTIONS USING THE PLURALITY METHOD, DEFINE THE CONDORCET FAIRNESS CRITERION AND ALSO FIND A CONDORCET WINNER. THE PLURALITY VOTING METHOD IS PROBABLY THE METHOD YOU'RE MOST FAMILIAR WITH, WHERE THE CHOICE WITH THE MOST FIRST PREFERENCE VOTES IS DECLARED THE WINNER. TIES ARE POSSIBLE AND WOULD HAVE TO BE SETTLED THROUGH SOME SORT OF RUN OFF. THIS METHOD IS SOMETIMES MISTAKENLY CALLED THE MAJORITY METHOD, OR MAJORITY RULES, BUT IT IS NOT NECESSARY FOR A CHOICE TO HAVE GAINED A MAJORITY OF VOTES TO WIN, WHERE A MAJORITY IS OVER 50% OF THE VOTES. SO IT IS POSSIBLE FOR A WINNER TO HAVE A PLURALITY WITHOUT HAVING A MAJORITY. LET'S TAKE A LOOK AT A COUPLE OF EXAMPLES. THE SURVEY ASKED TO RANK WHICH WEST COAST STATES PEOPLE PREFER TO LIVE. THE RESULTS ARE BELOW. USE THE PLURALITY METHOD TO SELECT THE WINNER. WE'RE LOOKING AT THE PREFERENCE TABLE HERE, C = CALIFORNIA, O = OREGON, AND W = WASHINGTON. NOTICE THAT WE FIND THE SUM OF THESE VALUES HERE, WE CAN DETERMINE THE TOTAL VOTES IS 300. TO DETERMINE THE PLURALITY WINNER WE'LL DETERMINE HOW MANY FIRST CHOICE VOTES CALIFORNIA RECEIVED, THEN HOW MANY FIRST CHOICE VOTES OREGON RECEIVED, AND THEN HOW MANY FIRST CHOICE VOTES WASHINGTON RECEIVED. WELL, CALIFORNIA RECEIVED 75 + 94 FIRST CHOICE VOTES, WHERE 75 + 94 = 169. OREGON RECEIVED 51 + 12 FIRST CHOICE VOTES, WHICH IS 63 FIRST CHOICE VOTES. AND FINALLY, WASHINGTON RECEIVED 43 + 25 OR 68 FIRST CHOICE VOTES. SO IN THIS CASE, NOTICE THAT CALIFORNIA RECEIVED THE MOST FIRST CHOICE VOTES. AND THEREFORE, CALIFORNIA IS THE PLURALITY WINNER. NOTICE HOW IN THIS CASE CALIFORNIA RECEIVED 169 FIRST CHOICE VOTES OUT OF 300, WHICH IS APPROXIMATELY 56.3%, WHICH IS MORE THAN 50%. AND THEREFORE, CALIFORNIA WOULD ALSO BE THE MAJORITY WINNER. REMEMBER, A WINNER DOES NOT HAVE TO BE A MAJORITY WINNER TO BE THE PLURALITY WINNER. LET'S TAKE A LOOK AT A SECOND EXAMPLE, WHERE HERE A SMALL GROUP OF COLLEGE STUDENTS RANK THE BEST DESTINATION FOR SPRING BREAK WHERE S = SAN DIEGO, L = LAKE HAVASU, AND R = ROCKY POINT. AGAIN, BY FINDING THE SUM OF THESE VALUES HERE WE CAN DETERMINE THERE ARE A TOTAL OF 17 VOTES. NOTICE, SAN DIEGO RECEIVED A TOTAL OF 4 + 4, OR 8, FIRST PLACE VOTES. LAKE HAVASU RECEIVED A TOTAL OF TWO FIRST PLACE VOTES. AND ROCKY POINT RECEIVED A TOTAL OF 5 + 2, OR 7, FIRST PLACE VOTES. AND SINCE SAN DIEGO RECEIVED THE MOST FIRST CHOICE VOTES, OR FIRST PLACE VOTES, SAN DIEGO IS THE WINNER. NOTICE IN THIS CASE, SAN DIEGO RECEIVED A TOTAL OF 8 FIRST PLACE VOTES OUT OF 17, WHICH IS APPROXIMATELY 47.1%. SO NOTICE HOW HERE EVEN THOUGH SAN DIEGO IS NOT THE MAJORITY WINNER, IT STILL IS THE WINNER USING THE PLURALITY METHOD. THIS LEADS US TO A DISCUSSION ABOUT WHAT CAN BE WRONG ABOUT THE PLURALITY VOTING METHOD. IF THERE ARE THREE OR MORE CHOICES IT IS POSSIBLE THAT A CHOICE COULD LOSE, BUT WHEN COMPARED IN A ONE TO ONE COMPARISON IT COULD BE PREFERRED OVER THE PLURALITY WINNER. AND THIS VIOLATES WHAT'S CALLED A FAIRNESS CRITERION WHERE THE FAIRNESS CRITERIA ARE STATEMENTS THAT SEEM LIKE THEY SHOULD BE TRUE IN A FAIR ELECTION. THE FIRST FAIRNESS CRITERION WE'LL CONSIDER IS CALLED THE CONDORCET CRITERION WHERE IF THERE IS A CHOICE, IT IS PREFERRED IN EVERY ONE TO ONE COMPARISON WITH THE OTHER CHOICES. THAT CHOICE SHOULD BE THE WINNER AND WE CALL THIS WINNER THE CONDORCET WINNER OR CONDORCET CANDIDATE. LET'S LOOK AT TWO MORE EXAMPLES. THIS IS THE EXAMPLE THAT WE SAW BEFORE WHERE WE KNOW THE PLURALITY WINNER WAS SAN DIEGO WITH A TOTAL OF 8 VOTES, BUT NOW WE WANT TO FIND THE CONDORCET WINNER. SO TO FIND THE CONDORCET WINNER WE'LL DO A ONE TO ONE COMPARISON WITH OUR THREE OPTIONS. SO WE'LL COMPARE SAN DIEGO VERSUS LAKE HAVASU. WE'LL COMPARE SAN DIEGO VERSUS ROCKY POINT. AND WE'LL COMPARE LAKE HAVASU VERSUS ROCKY POINT. TO DO THE ONE TO ONE COMPARISON WITH SAN DIEGO AND LAKE HAVASU WE WOULD IGNORE ROCKY POINT. SO WE'LL IGNORE ROCKY POINT HERE, HERE, HERE, HERE, AND HERE. REMEMBER, WE HAVE A TOTAL OF 17 VOTES. SO OF THE 17, SAN DIEGO IS PREFERRED OVER LAKE HAVASU 4 + 4 + 5 TIMES, SO THAT WOULD BE 8 + 5 = 13. SO SAN DIEGO WINS OVER LAKE HAVASU 13 TO 4. NOW WE'LL COMPARE SAN DIEGO TO ROCKY POINT SO WE'LL IGNORE LAKE HAVASU. SO NOTICE SAN DIEGO BEATS ROCKY POINT HERE AND HERE, BUT NOTICE HOW ROCKY POINT WINS HERE, HERE, AND HERE. AND THEREFORE, FOR SAN DIEGO VERSUS ROCKY POINT THE VOTE IS 8 TO 9. NOTICE IN THIS ONE TO ONE COMPARISON ROCKY POINT WINS. AND THEN FINALLY, WE WANT TO CONSIDER LAKE HAVASU VERSUS ROCKY POINT. SO NOW WE'LL IGNORE SAN DIEGO. SO LAKE HAVASU'S PREFERRED OVER ROCKY POINT HERE AND HERE AND THEREFORE, LAKE HAVASU VERSUS ROCKY POINT WOULD BE 6 TO 11. NOW, LOOKING AT THESE ONE TO ONE COMPARISONS NOTICE HOW ROCKY POINT BEATS LAKE HAVASU HERE AND ROCKY POINT ALSO BEATS SAN DIEGO HERE. THEREFORE ROCKY POINT ALWAYS WINS IN A ONE TO ONE COMPARISON. AND THEREFORE, ROCKY POINT IS THE CONDORCET WINNER. SO EVEN THOUGH SAN DIEGO WAS THE PLURALITY WINNER, UNDER THE CONDORCET FAIRNESS CRITERION ROCKY POINT SHOULD BE THE WINNER. LET'S TAKE A LOOK AT ONE MORE EXAMPLE. WE WANT TO FIND THE CONDORCET WINNER, OR CONDORCET CANDIDATE, IF THERE IS ONE. SO THE CANDIDATES ARE "A," B, AND C SO WE'LL DO A ONE TO ONE COMPARISON. WE'LL HAVE "A" VERSUS B, "A" VERSUS C, AND B VERSUS C. NOTICE THE PLURALITY WINNER WOULD BE C WITH A TOTAL OF 16 FIRST CHOICE VOTES. SO FOR "A" VERSUS B WE'LL IGNORE C. SO "A" WOULD WIN OVER B HERE AND HERE. SO "A" VERSUS B WOULD BE 31 TO 10. NEXT, FOR "A" VERSUS C WE'LL IGNORE B. NOTICE, "A" WINS ONLY HERE SO "A" VERSUS C WOULD BE 15 TO 26. AND THEN FOR B VERSUS C WE'LL IGNORE "A". NOTICE HOW B WINS HERE AND C WINS HERE AND HERE. SO B VERSUS C WOULD BE 10 TO 31. SO AGAIN, LOOKING AT THESE TWO HERE NOTICE C WINS OVER B AND HERE C ALSO WINS OVER "A" AND THEREFORE CANDIDATE C IS THE CONDORCET WINNER, BUT NOTICE HOW C IS ALSO THE PLURALITY WINNER HERE. I HOPE YOU FOUND THIS HELPFUL.
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.
Football analogy
Imagine politicians compete like football teams in a roundrobin tournament, where every team plays against every other team once. In each matchup, a candidate's score is equal to the number of voters who support them over their opponent.
Minimax finds each team's (or candidate's) worst game – the one where they received the smallest number of points (votes). Each team's tournament score is equal to the number of points they got in their worst game. The first place in the tournament goes to the team with the best tournament score.
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 not rank all 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 is the most commonly used and preferred by social choice theorists.
 The number of voters ranking X above Y minus the number of voters ranking Y above X. This variant is called margins, and is less used.
 The number of voters ranking X above Y, regardless of whether more voters rank X above Y or vice versa. This variant is called pairwise opposition, and is also rarely used.
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 fails independence of irrelevant alternatives, independence of clones, local independence of irrelevant alternatives, and independence of Smithdominated alternatives.^{[citation needed]}
With the pairwise opposition variant (sometimes called MMPO), minimax only satisfies the majoritystrength Condorcet criterion; a candidate with a relative majority over every other may not be elected. MMPO is a laternoharm system and also satisfies sincere favorite criterion.
Nicolaus Tideman modified minimax to only drop edges that create Condorcet cycles, allowing his method to satisfy many of the above properties. Schulze's method similarly reduces to minimax when there are only three candidates.
Examples
Example with Condorcet winner
Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:
 Memphis, the largest city, but far from the others (42% of voters)
 Nashville, near the center of the state (26% of voters)
 Chattanooga, somewhat east (15% of voters)
 Knoxville, far to the northeast (17% of voters)
The preferences of each region's voters are:
42% of voters FarWest 
26% of voters Center 
15% of voters CenterEast 
17% of voters FarEast 





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 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 winning votes and margins alternatives, 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
 Multiwinner voting – contains information on some multiwinner variants of Minimax Condorcet.
References
 ^ "[EM] the name of the rose". lists.electorama.com. Retrieved 20240212.
 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