The Condorcet paradox
When majority votes can’t solve the problem.
Three people come together to try to make a decision — perhaps something as simple as what to have for dinner. The first suggestion made is spaghetti, but it turns out that two of them would rather have pizza. Then one person suggests lasagna would be better than pizza, and is quickly seconded.
But, adds the second, they would rather simply have spaghetti than lasagna. The third agrees. The first person groans in frustration and picks up a Chinese takeout menu. How is it possible for spaghetti to win 2–1 over lasagna when lasagna won 2–1 over pizza, and pizza won 2–1 over spaghetti?
It’s simple. The first person prefers lasagna to pizza to spaghetti; the second prefers spaghetti to lasagna to pizza; the third prefers pizza to spaghetti to lasagna. The first and second will vote for lasagna over pizza; the second and third will vote for spaghetti over lasagna; and the first and third for pizza over spaghetti. When examining the preferences of the voters side by side, the cyclic component can be seen in the diagonal stripes.
This paradox is not limited to trios of people deciding between three different kinds of Italian food, but can occur with any type of decision. The more options and the more voters, the more extreme the cycle can be.
Whenever there are three or more options, majority voting between pairs of options can produce a cycle — if the preferences of voters have a strong enough cyclic component. The Marquis de Condorcet described this paradox in 1785, and so we refer to it as a Condorcet paradox.
How common is the Condorcet paradox in everyday life? The answer to this depends on the structure of how preferences are distributed. If there is only a single salient dimension of difference between options (such as a simple spectrum from political left to political right), then the Condorcet paradox simply will not occur.
Within the academic literature, the principle criterion identified as simple enough to exclude majority cycles is single-peakedness, introduced by Duncan Black. If voters’ preferences are single-peaked, there will be no majority vote cycles.
In cases where preferences are distributed in symmetric ways across the spectrum of possible preferences, my own research shows that the probability of a Condorcet paradox ranges from 6.25% (1 in 16) to 8.8% (slightly under 1 in 11). For various reasons, it’s difficult to measure how likely Condorcet paradoxes are “in the wild,” because it requires information that is usually not collected during a plurality election.
Relationship to the Condorcet criteria and methods
Several other closely related concepts are similarly named after Condorcet. If there is no Condorcet paradox in play, then any sequence of pairwise majority voting can leave the majority satisfied.
In cases where pairwise majority votes reveal an unambiguous winner, one who wins every contest, that candidate is called a Condorcet winner. Similarly, in cases where pairwise majority votes reveal an unambiguous loser, one who loses every contest, that candidate is called a Condorcet loser.
These two concepts, in turn, leads to two related criteria for voting systems: Condorcet losers should not win elections elections, and Condorcet winners should win elections. The relatively few voting systems that satisfy these criteria are called Condorcet methods. Another voting-related result introduced in M.J.A.N. de Caritat’s 1785 paper and named with his title is Condorcet’s jury theorem, which leverages probability to show that a jury of independent decision-makers is likelier to vote for the correct verdict.
