The Condorcet Paradox of Voting

Updated Jan 21, 2023

Most economies rely on democratic principles, such as majority rule to set government policies. That means people get to vote, and the majority gets its way. This works well in many cases. However, there are situations where democratic principles run into problems. In this article, we will take a closer look at one of those situations, namely the Condorcet Paradox of Voting, and learn why sometimes, the majority may be unable to reach the socially desirable outcome.

The Paradox

The Condorcet Paradox states that the majority rule sometimes fails to produce transitive preferences for society. That means we usually assume that for any three options A, B, and C, when A is preferred to B, and B is preferred to C, this implies that A must be preferred to C as well. According to this paradox, however, that does not always apply to democratic voting. Instead, the majority can choose different options in the same vote in some cases, depending on how the vote is structured.

Example of the Condorcet Paradox of Voting

The best way to explain the paradox is by looking at a simple example. Assume that a local government wants its citizens to vote on policies to fight poverty. There are three options to consider: (A) a minimum wage, (B) improved social security, and (C) a negative income tax.

The officials know that voters’ preferences vary across age groups. Starting from there, they have identified three groups of voters with different preferences: (1) young voters, who make up 35% of the electorate, (2) middle-aged voters, who account for 45% of all voters, and (3) old voters, who make up the remaining 20% of the electorate.

The different groups and their preferences are illustrated in the table below.

To ensure an absolute majority, one of the officials suggests that the government should try pairwise votes. That means the public could first choose between options A and B first, and then between the winner of the first vote and C. That way, there should be a clear winner at the end, right? Well, no. That’s a false assumption.

In this example, a vote between options A and B results in young and old voters choosing A, giving A the majority. In the second round (A vs. C), the winner is C, which is preferred by both middle-aged and old voters. By contrast, if A vs. C is the first vote, the majority still picks C in the first round. However, in the second round (C vs. B), the young and middle-aged voters both prefer B, which gives B the majority. Hence, the order in which the pairs are voted on affects the final outcome.

Implications

The fact that the order in which things are voted on can affect the outcome is an important implication to be aware of. After all, this suggests that whenever there are more than two options to vote on, the people responsible for setting the agenda may have considerable power over the final outcome of the vote. This, in turn, implies that a majority vote alone doesn’t necessarily accurately reflect the preferences of society as a whole.

Summary

The Condorcet Paradox of Voting states that the majority rule sometimes fails to produce transitive preferences for society. That means we usually assume that for any three options A, B, and C, when A is preferred to B, and B is preferred to C, this implies that A must be preferred to C as well. However, this is not always the case when it comes to majority votes. Instead, when there are more than two options to vote on, the order in which they are presented to the public can affect the final outcome. Thus, we can conclude that a majority vote alone doesn’t necessarily accurately reflect the preferences of society as a whole.