9.2 Majority Rule and Condorcet's Method
When a choice is being made between two candidates, the first type of voting system to suggest itself is majority rule. Each voter indicates a preference for one of the two, and the one with the most votes wins. There are three good things about majority rule.
1) All voters are treated equally. (Everyone's vote counts the same.)
2) Both candidates are treated equally. (If a new election were held and every voter reversed his or her vote, the outcome would be reversed as well.)
3) It is monotone. (If candidate X is the winner and a new election is held in which the only change made is for one more person to vote for X, then X will remain the winner.)
These things seem obvious to us, but consider situations in which each of these properties would not hold. For example, condition 1) is not satisfied in a dictatorship (such as in Cuba), where the voters are not treated equally. The ballots of all voters except the dictator are ignored. Condition 2) is not satisfied in a situation of imposed rule (such as in a small­town good­old­
boy network) where the candidates are not treated equally. It may be a matter of "who you know" rather than "what you know." And condition 3) is not satisfied in minority rule (such as in a golf game, where the lowest score wins), because if the person with the lowest score (the winner) got another point (vote), they may not still be the winner.
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May's Theorem (proved in 1952 by Kenneth May) tells us that among all two­candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is monotone.
But what if there are three or more candidates? Is there a way to build on the strengths of majority rule? It turns out that there is a system that does just that, and it is called Condorcet's method. Our description of Condorcet's method begins with the observation that if we have a sequence of preference list ballots, then, for each pair of candidates, we can determine who the winner would have been had the election involved only these two in a one­on­one contest using majority rule. Consider the following example involving candidates A, B, and C:
Rank
First
Second
Third
Number of Voters (3)
A
B
C
B
C
A
C
A
B
The 1st voter ranked A first, B second, and C third, the 2nd voter ranked B first, C second, and A third, and the 3rd voter ranked C first, A second, and B third.
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To help us better understand a preference list ballot, let's do one of our own. We can rank holidays, ice cream flavors, or college courses, to name a few. 3
Description of Condorcet's Method:
With the voting system known as Condorcet's method, a candidate is a winner precisely when he or she would, on the basis of the ballots cast, defeat every other candidate in a one­on­one contest using majority rule. (Historically, the voting system attributed to the Marquis de Condorcet in the 18th century was actually developed by Ramon Llull in the 13th century.)
Example: Determine the winner in each preference list ballot below using Condorcet's method.
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Condorcet’s voting paradox can occur with three or more candidates in an election where Condorcet’s method yields no winners. For example, in a three­candidate race, two­thirds of voters could favor A over B, two­thirds of voters could favor B over C, and two­thirds of voters could favor C over A. This is the example given in the text. With three or more candidates, there are elections in which Condorcet’s method yields no winners.
Does Condorcet’s voting paradox occur in the following tables?
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