How should one select the winner of an\u00a0election?<\/p>\n\n\n\n
If there are only two candidates, the answer is clear— choose the one who would win the most votes in a head-to-head election. But with three or more candidates, when each voter has ranked his or her candidate preferences, the answer is less obvious.<\/p>\n\n\n\n
Mathematically we can formalize the question in this way. A social choice<\/em> function is a function that takes lists of people’s ranked preferences and outputs a single alternative (the “winner” of the election). So the question becomes: is there a “good” social choice function that represents “the will of the people”?<\/p>\n\n\n\n Consider the following situation with 3 voters and 3 candidates:<\/p>\n\n\n\n Suppose Voter 1 prefers A to B to C. Notice that no matter who is selected as the “social choice” for this set of lists, then 2\/3 of the voters will be “unhappy” in the sense that those voters prefer another candidate to the one chosen by the social choice function! (For instance, if A is chosen as the winner, then Voters 2 and 3 will prefer C to A.)<\/p>\n\n\n\n This paradox, due to Maurice de Condorcet in 1785, shows that it is not always possible for a social choice function to pick a candidate that will beat all other candidates in pairwise comparisons. If there is a candidate that does, then that candidate is called a Condorcet winner<\/em>.<\/p>\n\n\n\n Presentation\u00a0Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> How to Cite this Page:<\/strong>
Suppose Voter 2 prefers B to C to A.
Suppose Voter 3 prefers C to A to B.
<\/p>\n\n\n\n
This may be a good starting point for students to ponder the role of third parties in American politics. You may also ask students to generalize this\u00a0paradox\u00a0to N people.<\/p>\n\n\n\n
The study of social choice functions and related questions is called\u00a0social choice theory<\/em>, a subfield of\u00a0game theory. There are other famous impossibility results: most notably\u00a0Arrow’s Impossibility Theorem.<\/p>\n\n\n\n
Su, Francis E., et al. “Social Choice and the Condorcet Paradox.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n