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Voting in a democratic society does not always yield conclusive results. Consider a club with 55 voting members. Five of the members (Alice, Bill, Carol, Don, and Herkimer) decide to run for club president. Members are asked to rank the five candidates from 1 to 5, with 1 being the first choice, 2 being the second choice, and so on. The results of the ranking are shown below.

	1	2	3	4	5
18 members	ALICE	BILL	CAROL	DON	HERKIMER
12 members	HERKIMER	CAROL	BILL	DON	ALICE
10 members	DON	HERKIMER	CAROL	BILL	ALICE
9 members	BILL	DON	CAROL	HERKIMER	ALICE
4 members	CAROL	HERKIMER	BILL	DON	ALICE
2 members	CAROL	DON	BILL	HERKIMER	ALICE

OK, who wins the election if

a) The winner is the candidate with the most first-place votes?
b) There is a run-off between the two candidates receiving the most first-place votes?
c) Five points are given for a first-place ranking, four points for a second-place ranking, three points for a third-place ranking, two points for a fourth-place ranking, and one point for a fifth-place ranking?
d) The winner is the person who beats each candidate in a two-person contest?

If you take the time to answer the questions, you'll see that the choice of voting process is not a trivial decision.

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Here's another voting situation. Suppose there are three candidates, Alice, Bill, and Carol.

PREMISE: It is known that 2/3 of the voters prefer Alice to Bill, and 2/3 of the voters prefer Bill to Carol.

QUESTION: Can we conclude that the voters prefer Alice to Carol?

If you think the answer is YES, look at the table below. The PREMISE is true (check it out).

Preference--->	1	2	3
1/3	ALICE	BILL	CAROL
1/3	BILL	CAROL	ALICE
1/3	CAROL	ALICE	BILL

What portion of the voters prefer Carol to Alice?

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Related Readings:

Davis, Kenneth C. Don't Know Much About History. New York: Avon Books, 1990.

Paulos, John Allen. A Mathematician Reads the Newspaper. New York: Anchor Books, 1995.

Paulos, John Allen. Beyond Numeracy: Ruminations of a Numbers Man. New York: Alfred A. Knopf, 1991.

Smith, Sanderson. Agnesi to Zeno: Over 100 Vignettes from the History of Math. Berkeley, CA: Key Curriculum Press, 1996.

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