Alexander Strang, Peter Thomas
With so many candidates running for the White House, who will come out on top?
With 24 declared candidates for the Democratic Party’s presidential nomination (and counting), many Americans are likely wondering how the party will ultimately make up its mind and settle on the best candidate.
Asmathematicians, we wondered whether there might not even be a best candidate. In fact, this is an established mathematical paradox. The more candidates there are, the greater the chance there is no clear favorite.
Here’s what we mean.
Suppose there were only two candidates for some office, and that each voter preferred one or the other. Barring a perfect tie, one candidate will end up with the most votes. Ignoring complications like the Electoral College or voter turnout, the election process provides a way to measure the “will of the people.”
Now imagine there were three candidates instead of just two.
Three friends and a pollster walk into a bar and discuss the upcoming election. The first friend thinks that candidate A is better than B, and that C is the worst of all. The next agrees that B is better than C, but she thinks that B and C are both better than A. The final friend partially agrees with both of them: He thinks C is the best candidate, followed by A, and then B.
The pollster cannot say which is the best candidate, since, for these voters, there is no best candidate! Their ranked preferences are inconsistent with each other.
This situation is an example of Condorcet’s paradox. It was named for the French Enlightenment philosopher and mathematician Marie Jean Antoine Nicolas de Caritat, marquis of Condorcet, an advocate of democratic reforms who perished in 1794, a victim of the French Revolution.
To Condorcet, a winner is a candidate who would win a one-on-one election against any other candidate. But, a paradox arises when there is no candidate who wins head-to-head against all opponents – which implies that voters’ ranked preferences contradict one another.
How likely is a situation like Condorcet’s paradox to arise in practice? It depends on how many candidates there are, and how evenly distributed the voters’ preferences are.