NOTES
- See http:// www .brainyquote .com/ quotes/ authors/ o/ otto _von _bismarck .html. I’m
a big fan of quotes, and this site has a lot of great ones. - Allan Gibbard is a professor of philosophy at the University of Michigan, and
Mark Satterthwaite is a professor of strategic management and managerial eco-
nomics at Northwestern University. Despite the Midwestern locales of these two
universities, the Gibbard-Satterthwaite theorem was not hatched over a dinner
table while the two were discussing insincere voting. The original result is due
to Gibbard; the improvement to Satterthwaite, as the following papers indicate:
Allan Gibbard, “Manipulation of Voting Schemes: A General Result,” Economet-
rica 4 1 (4) (1973): pp. 587–601; Mark A. Satterthwaite, “Strategy-proofness and
Arrow’s Conditions: Existence and Correspondence Theorems for Voting Proce-
dures and Social Welfare Functions,” Journal of Economic Theory 10 (April 1975):
pp. 187–217. - See http:// en .wikipedia .org/ wiki/ United_States _presidential _election , _1876.
An interesting sidelight to the election is that there was a minor third party in
this election called the Greenback Party. Insert cynical remark here. - See http:// occawlonline .pearsoned .com/ bookbind/ pubbooks/ pirnot_awl/
chapter1/ custom3/deluxe-content.html#excel. This site has Excel spreadsheets
you can download for both the Alabama paradox and the Huntington-Hill ap-
portionment method. - See http:// www .cut -the -knot .org/ ctk/ Democracy .shtml. This site not only has
explanations of all the paradoxes, but nice Java applets that you can use to see
them in action. - M. L. Balinski and H. P. Young, Fair Representation, 2nd ed. (Washington, D. C.:
Brookings Institution, 2001). Unlike the authors of the Gibbard-Satterthwaite
theorem, who were separated by time and probably distance, Balinski and Young
were together at New York University for much of the period during which the
relevant ideas were formulated and the Balinski-Young theorem proved. - MathSciNet is a wonderful database, but you either have to belong to an institu-
tion that subscribes to it (many colleges and universities, as well as some re-
search-oriented businesses, are subscribers), or have a tidy chunk of change
burning a hole in your pocket. - See http:// rangevoting .org/ Apportion .html.
236 How Math Explains the World