How to Rig a Majority Vote

Diana Thomas,

Release Date
May 28, 2013


Democracy and Voting

Do you think that a majority vote is always the fairest way to reach a consensus? Think again! In this Learn Liberty video, Professor Diana Thomas illustrates a paradoxical outcome that arises when people vote on three or more items – known as Condorcet’s Paradox – and proves that it is quite easy to manipulate the voting process in this scenario.
Condorcet’s paradox occurs when a vote is taken on a set of three options that nobody ranks in the same order. Even though a vote of two of the options may yield a consistent winner, it’s impossible to achieve a consistent outcome between all three choices. Usually, a majority vote is taken on only two options, so whoever gets to choose which two options are on the table (known as the agenda setter) has the power to dictate the winner of the vote.

The Impossibility of Democracy: Condorcet’s Paradox [blog post]: A written explanation of Condorcet’s paradox and how it poses problems for the efficacy of democracy
What’s So Good about Democracy? [article]: A Freeman article critiquing democratic systems from a definitional, logical, and social perspective
“Public Choice” [encyclopedia entry]: The Concise Encyclopedia of Economics entry on public choice theory
Arrow’s Theorem Proves No Voting System Is Perfect [article]: An explanation of Arrow’s Impossibility Theorem, a theory that states that no consistent or fair voting system can lead to sensible results
The Science of Voting [article]: Another clear explanation of Arrow’s Impossibility Theorem and proposed solutions to voting paradoxes

How to Rig a Majority Vote
Voting is supposed to be about getting what the majority wants. But that is not always the way it works. Let’s imagine that you and your two friends Jim and John are on your way home from a party. The three of you want to get something to eat before you head home. All three of you immediately call out a different preference, so there is no clear majority in favor of any option. But all of you want equal say in the decision about where to eat. So you propose voting on two options at a time to figure out what the group’s preferences are. So you first say, “Tacos versus burgers – which do you guys want?” Personally, you want pizza more than anything, but you’d also be alright with tacos. Burgers sound awful right now.
So you vote tacos. Jim he votes burgers. John votes tacos. So tacos are the clear winner. But since pizza wasn’t even on the table for the last vote you ask for another round of voting to see how pizza ranks against the current winner. Your friends agree. In this new matchup tacos versus pizza you obviously vote for pizza. Jim also votes for pizza. John is left as the only person who would prefer tacos to pizza. So pizza appears to be the new champion. You’re thrilled.
But now Jim is not happy. He says, “That is not fair. We decided that tacos win out of burgers and pizza wins out over tacos, but how do you know that there is not a majority in favor of burgers over pizza. We never voted on that.” You counter, “That doesn’t matter. We already voted on burgers.”
But Jim doesn’t budge, so to make him happy you agree to have one last vote – burgers versus pizza. You of course vote for pizza. Jim votes for burgers like you expected, but now John also raises his hand in favor of burgers. So burger are declared the winner. What happened?
Well, let’s look at everyone’s preferences again. As you can see, the problem is no one ranks any of the options in the same order, so even though a vote between any two options yields a winner, between all three choices it is impossible to achieve a consistent outcome. This is called Condorcet’s Paradox.
In this scenario voting will result in what we called a cycle, so after voting on two pairings you may seem to have a clear answer, but if you change the order in which you voted on the pairings you would get a completely different result. None of the three options is preferred by a majority of the voters, and voting cannot resolve the problem. If you are surprised by this let me take you one step further.
The fact that any outcome may be possible implies that whoever gets to decide the order of the options is really the one who picks the outcome. This person is called the agenda setter. If the agenda setter is savvy and if he has any inkling of the relative preferences of the other voters, he can change the order of voting to achieve his preferred outcome. If you had been savvy you would have made the last vote tacos versus pizza and you would have gotten what you wanted. But would that have been fair?
The Condorcet Paradox shows that taking a vote will not always select what the majority prefers. In fact, when an agenda setter manipulates the voting process he is the one who will decide what the group does.