How should one select the winner of an election?
If there are only two candidates,
the answer is clear choose the
one who would win the most votes in a headtohead election.
But with three or more candidates, when
each voter has ranked his or her candidate preferences, the answer is less obvious.
Mathematically we can formalize the question in this way.
A social choice 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"?
Consider the following situation
with 3 voters and 3 candidates:
Suppose Voter 1 prefers A to B to C.
Suppose Voter 2 prefers B to C to A.
Suppose Voter 3 prefers C to A to B.
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.)
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.
Presentation Suggestions:
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 paradox to
N people.
The Math Behind the Fact:
The study of social choice functions and related questions
is called social choice theory, a subfield of game theory.
There are other famous impossibility results: most notably Arrow's Impossibility Theorem.
How to Cite this Page:
Su, Francis E., et al. "Social Choice and the Condorcet Paradox."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
