4 Fairness Criteria
The Majority Criterion
The Condorcet Criterion
The Monotonicity Criterion
The Independence of Irrelevant Alternatives Criterion.

For many years, mathematicians and others interested in voting theory searched for a preferential voting procedure that would satisfy a reasonable set of fairness criteria (such as the set of 4 above). Arrow's work in 1952 (mentioned previously) abruptly ended these efforts when he showed that such a search was in vain. In terms of the 4 fairness criteria above, Arrow's result means that there is NO consisent preferential voting method that can satisfy all four of them.

Thus, for example, a preferential procedure designed to satisfy the Majority Criterion will always violate at least one of the other criteria. This, in turn, means that there is no 'perfect' preferential voting procedure and the decision about the procedure to be used is, by necessity, subjective. The best one can hope for is to be able to objectively analyze the strengths and weaknesses of various preferential procedures and to apply that knowledge to the task of selecting a 'good' one for a particular situation.

The Majority Criterion
Any candidate receiving a majority of first place votes should be the winner.

In other words, it would seem unfair to most people if Candidate A got 51 first place votes, Candidate B got 40 first place votes, and Candidate C got 9 first place votes (100 votes in all) but Candidate B was declared the winner of the election. Such an outcome would violate the Majority Criterion.

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The Condorcet Criterion
(named after the Marquis de Condorcet)
A candidate who wins head-to-head matchups with all other candidates should be the winner.

Suppose 4 candidates A, B, C, and D run for mayor of a small town (a very small town!). There are 20 registered voters. The local newspaper performed a post-election survey of each of the 20 registered voters. Among other things, the survey asked the voters who they preferred in a two-way race between candidate C (the one endorsed by the paper's editorial staff) and each of the other candidates. Here are the results:
11 voters preferred candidate C over candidate A
11 voters preferred candidate C over candidate B
17 voters preferred candidate C over candidate D
So, in head-to-head competition, candidate C won against each of the other candidates.
Wouldn't it seem unfair if candidate C was not declared the winner?
When the actual votes were tabulated, candidate A got 9 first place votes, candidate B got no first place votes, candidate C got 8 first place votes, and candidate D got 3 first place votes. If candidate C is not declared the winner, this would be a violation of the Condorcet Criterion.

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The Monotonicity Criterion
If an election is held and a winner is declared, this winning candidate should remain the winner in any revote in which all preference changes are in favor of the winner of the original election.

Three students, Al, Bob, and Carrie are running for class president. The class will vote in rounds. The student with the fewest votes in the first round will drop out and a new vote will be taken between the two remaining candidates. The student with the most votes in this final round will be declared the winner of the election. In the first round, Al gets 11 first place votes, Bob gets 8 first place votes, and Carrie gets 10 first place votes. Bob drops out (since he had the fewest votes in the first round). In the final round, Al gets 11 first place votes and Carrie gets 18 first place votes. Carrie wins the election!
But wait! The chairman of the election oversight committee destroyed the ballots before the results had been officially certified by the administration. You guessed it! The election had to be repeated. In the first round of the repeated election everyone voted exactly as in the first round of the original election except for 4 voters who decided to jump on the bandwagon and vote for Carrie instead of Al. As a result, Al gets 7 first place votes, Bob gets 8 first place votes, and Carrie gets 14 first place votes. This causes Al to drop out instead of Bob so that the final round of the repeated election is between Bob and Carrie. BUT the 7 students who originally voted for Al prefer Bob over Carrie. So, all 7 of them cast their votes for Bob in the final round. This gives Bob 15 votes and Carrie 14 votes. Bob wins the repeated election EVEN THOUGH THE ONLY CHANGES IN VOTER PREFERENCE WERE THE 4 VOTES THAT CHANGED FROM AL TO CARRIE (THE ORIGINAL WINNER). This illustrates a violation of the Monotonicity Criterion.

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The Independence of Irrelevant Alternatives Criterion
If an election is held and a winner is declared, this winning candidate should remain the winner in any recalculation of votes as a result of one or more of the losing candidates dropping out..

As a publicity stunt for its soon-to-be-published cookbook, the Culinary Club of Smallville decided to have a 'Best Pie' contest. The entries were narrowed down to three for the final round of the contest. In this final round, each club member ranked the 3 pies. Each first place vote is worth 3 points, each second place vote is worth 2 points, and each third place vote is worth 1 point. Here is a summary of the results:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second, and Al's Apple Pie third;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third.
Based on the rules above,
Al's Apple Pie got 27 first place votes at 3 points each (81 points) and 26 third place votes at 1 point each (26 points) for a grand total of 107 points.
Chris' Cream Pie got 2 first place votes at 3 points each (6 points) and 51 second place votes at 2 points each (102 points) for a grand total of 108 points.
Pat's Peach Pie got 24 first place votes at 3 points each (72 points), 2 second place votes at 2 points each (4 points), and 27 third place votes at 1 point each (27 points) for a grand total of 103 points
Thus, Chris' Cream Pie gets first place (108 points), Al's Apple Pie gets second place (107 points), and Pat's Peach Pie gets third place (103 points).
Before the results can be publicized, Pat (who is upset about a third place finish) demands that her Peach Pie entry be retroactively withdrawn from the contest. Bowing to her wishes, the club removes the Peach Pie and recalculates points (2 points for each first place, 1 point for each second place).
Since Pat's Peach Pie is now out, the rankings are:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second first, and Al's Apple Pie third second;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third second.
The new results are,
Al's Apple Pie gets 27 first place votes at 2 points each (54 points) and 26 second place votes at 1 point each (26 points) for a grand total of 80 points;
Chris' Cream Pie gets 26 first place votes at 2 points each (52 points) and 27 second place votes at 1 point each (27 points) for a grand total of 79 points.
Thus, Al's Apple Pie gets first place (80 points), and Chris' Cream Pie gets second place (79 points).
Notice what happened. Because a loser (Pat) dropped out, the winner changed (from Chris to Al). This is a violation of the the Independence of Irrelevant Alternatives Criterion.

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