American Mathematical Monthly, volume 117, number 1, january by Daniel J. Velleman

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 117, number 1, january 2010

Sample text

It is not (3, 4)agreeable, however, because among voters 1, 2, 4, 7 no three of them share a common platform. The same society, after restricting the spectrum to a set of candidates, is the linear society shown in Figure 3. It is not (2, 3)-agreeable, because among voters 2, 4, 7 there is no pair that can agree on a candidate (in fact, voter 7 does not approve any candidate). However, one may verify that this linear society is (2, 4)-agreeable. For a society S, the agreement number of a platform, a( p), is the number of voters in S who approve of platform p.

For a voter x, we use the symbols x and x to denote the left and right endpoints of A x , respectively. For example, the society from Figure 1 would have a string representation of B A C B A C . Of course, any string representation can be rotated cyclically without changing the society. We can disregard the platforms occurring at endpoints, because their support will coincide with the support of platforms either immediately to the left or right. Also, between two consecutive endpoints, any two platforms will have the same support, so we only need to consider one platform between two consecutive endpoints.

The Columbia Dictionary of Quotations, Columbia University Press, New York, 1993. 2. C. Berge, F¨arbung von Graphen deren s¨amtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung), Wissenschaftliche Zeitschrift, Martin Luther Universit¨at Halle-Wittenberg, MathematischNaturwissenschaftliche Reihe (1961) 114–115. 3. S. J. Brams and P. C. Fishburn, Going from theory to practice: The mixed success of approval voting, Social Choice and Welfare 25 (2005) 457–474. 1007/s00355-005-0013-y 4.

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