The jury theorem

Hi, I’m Geoff Pynn, and I teach philosophy at Elgin Community College. In this video, I’m going to talk about a famous argument for democracy based on a mathematical theorem. Plato argued that democracy led to tyranny. That’s because most people don’t have the intelligence, education, or interest in the matter to know what’s best for society as a whole. Plus, people are generally motivated by their own interests more than the common good. This makes it easy for a demagogue to hijack an election by manipulating people’s ignorance and self-interest for his own nefarious ends. Instead, Plato thought, the only way to ensure that a society flourished was to put an elite guardian class in charge. Through their natural intelligence, broad education, and rigorous training in virtue, Plato’s guardians would be motivated to do what’s best for society as a whole and they would know what that was. Rousseau disagreed. He thought that democratic elections were the best way to figure out what’s best for society as a whole. You ask the citizens whether a certain policy would promote the common good, and “Each man, in giving his vote, states his opinion on that point; and the general will is found by counting votes.” Just as Plato’s scheme assumes that the guardian class would rule in society’s best interests, and not for their own personal gain, Rousseau’s argument assumes that people vote from the point of view of the common good. He thought that the right kind of education and social institutions would ensure that democratic citizens would vote this way. “We can not doubt,” he wrote, that people brought up in his envisioned democratic society “will learn to cherish one another mutually as brothers, and to will nothing contrary to the will of society.” Well suppose we could somehow ensure that democratic citizens cast their votes for the general will, and not their particular ends. And, let’s also assume that they have some ability to discern what really is in our common interest. They still might not know as much about the needs of society as a brilliant, virtuous, very well-informed ruler would. So why would taking a vote be a more effective method for promoting society’s best interests than letting a wise ruler make the decisions for us? Nicolas de Condorcet was a French contemporary of Rousseau’s. He defended liberal ideas about voting, education, social reform, women’s rights, and more. He was a supporter of the French Revolution, until his views put him out of favor with the leaders of the Revolution, leading to his imprisonment and death. Today he is remembered for his optimistic defense of science and rationalism as the keys to human progress. But the idea that he’s best known for is a mathematical discovery called the Jury Theorem. Many thinkers have viewed the Jury Theorem as a vindication of Rousseau’s idea that democratic elections are the best way to discern what’s best for society. Imagine a twelve-person jury tasked with determining whether a suspect is guilty or not guilty. Each juror is intelligent, reasonable, and wants to make an accurate decision. But of course, each of them is limited. Like all of us, they bring their own assumptions and biases to the table, and none of them has access to all of the facts. Plus, everybody makes mistakes. Given their individual limitations, why think that letting the jury as a whole decide the question is a reliable way to arrive at the correct verdict? Condorcet provided a mathematical answer to this question. He showed that, as long as each member of the jury is more than 50% likely to be right, the jury’s collective decision is more likely to be correct than the average juror is. He also showed that the more individuals you add to the jury, the more likely it is to make a correct decision. All you need is for each added juror to be slightly more than 50% likely to get the answer right — and the bigger the jury, the more reliable it will be. The same principle applies to voters. If each voter is more than 50% likely to be correct about the answer to a question, taking a vote and going with the majority’s opinion is significantly more likely to lead to the right answer. Remarkably, Condorcet showed, if you have 10,000 voters, and each voter is only 51% likely to get the right answer, the answer that gets the most votes is almost certain to be correct. That's not just philosophical speculation. It’s a mathematical fact. So in a large enough electorate, if we assume that voters are deciding whether a certain policy is in our common interest, and that each of them is just slightly more likely to get the answer right than to get it wrong, the outcome of an election will almost certainly be correct about whether the policy would advance the common good. Contrary to Plato’s pessimism, Condorcet’s Jury Theorem suggests democracy really does have the best chance of leading to the best outcomes for society as a whole. Of course, it rests on some pretty strong assumptions. No matter how rigorously they’ve been trained in democratic values, it’s not clear how many people would vote for a policy they thought was best for society if they perceived that it wasn’t in their own interest. Rousseau thought that in his ideal society, people would identify the general will with their own, but that seems pretty optimistic. The deeper problem isn’t about people’s values, integrity, or attitude towards voting. It’s about their knowledge. It is very hard to figure out what’s truly in our common interest. Partly, this is because society is an immensely complicated system, and we can’t predict with any confidence everything that will happen as a result of a new law. This is what economists call the “law of unintended consequences”– that any large-scale action will have significant effects that were neither foreseen, nor intended. But it’s also hard for philosophical reasons. We don’t all agree on what it even means to benefit society. Disagreements about morality, values, and so on are pervasive, and these disagreements will lead us to different views about what’s really best. Given the empirical and philosophical complexities of discerning the common good, how can we be confident that a typical voter is more than 50% likely to know what’s in the common interest? It seems like we can’t. And if we can’t, not only does Condorcet’s theorem fail to provide an argument in favor of democracy – it threatens to provide one against it. That’s because another consequence of Condorcet’s theorem is that if all the voters are less than 50% likely to get the answer right, the outcome of their vote is even less likely to do so. The more ignorant voters there are, the less likely the election is to determine the right outcome. If 10,000 voters are each 49% likely to be right, the outcome of their election is almost certain to be wrong. So if the voters are even slightly more likely to be wrong than right, elections may inevitably lead us astray. Since Condorcet, many theorists have demonstrated versions of his Jury Theorem that model more accurately the conditions of actual group decision-making. But still, there’s no consensus that any actual election is sufficiently similar to the ideal conditions required for a Jury Theorem to apply. So it’s not clear whether we have mathematical reason to be confident about any actual democratic process. The jury’s still out. What do you think?