The simple process of voting leads to surprisingly counterintuitive paradoxes. For example, if three people vote for three candidates, giving the rankings A, B, C; B, C, A; and C, A, B. A majority prefers A to B, B to C, but also C to A (Gardner 1984, p. 25)! It is also possible to conduct a secret ballot even if the votes are sent in to a central polling station (Lipton and Widgerson, Honsberger 1985).
See alsoArrow's Paradox, Ballot Problem, Cake Cutting, May's Theorem, Quota System, Social Choice Theory
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ReferencesBlack, D. Theory of Committees and Elections. Cambridge, England: Cambridge University Press, 1958.Black, D. A Mathematical Approach to Proportional Representation: Duncan Black on Lewis Carroll. Boston, MA: Kluwer, 1995.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 25, 1984.Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Springer-Verlag, pp. 317-330, 1997.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 157-162, 1985.Huntington, E. V. "A Paradox in the Scoring of Completing Teams." Science 88, 287-288, 1938.Lipton, R. G.; and Widgerson, A. "Multi-Party Cryptographic Protocols." Unpublished manuscript. May 1982.Niemi, R. G. and Riker, W. H. "The Choice of Voting Systems." Sci. Amer. 234, 21-27, Jun. 1976.Riker, W. H. "Voting and the Summation of Preferences." Amer. Political Sci. Rev., Dec. 1961.Saari, D. G. Math. Intell. 10, 32, 1988.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 72-74, 1999.Trott, M. "The Mathematica Guidebooks Additional Material: EU Voting Scheme." http://www.mathematicaguidebooks.org/additions.shtml#N_1_12.
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Weisstein, Eric W. "Voting Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VotingParadoxes.html