Political Analysis: Condorcet Jury Theorem

Last post I introduced the concept of aggregation in political analysis, and how you can’t always make inferences about members of a group based on the character of the group as a whole—or visa versa. This post will go further into that, and also why democracy works.

In college, it’s easy to forget how much smarter you are than everyone, because you’re surrounded by people as smart or smarter than you—but really, a lot of people are highly ignorant. [NOTE FROM THE NOTE-TAKER: This is how Professor Dion introduced the topic, not me just throwing in my own color. But it is kind of true.]

For example, only 74% of Americans believe that the Earth orbits around the sun. As for politics, in 2010 only 54% knew the controlling party of the House of Representatives.

So, is democracy doomed? The idea with democracy is that an individual is the best judge of their own interests, and will elect a good representative for themself. But if people are ignorant, will they really?

Never fear, aggregation is here! The rock pile method would suggest that a group, if composed of people who each have only a 51% chance of being correct, will have only a 51% of electing a good representative. But the rock pile method is dumb, so we’re going to use instead Condorcet Jury Theorem.

The Marquis de Condorcet came up with a jury model, as juries are one of a few historical examples of democracy in the 18th century. The model makes four assumptions. First, that there is a correct and an incorrect decision. Second, that people are fallible, and will make correct decisions with a probability p (p = 1 means someone is always correct, p = 0 means they’re never correct.) Third, that there is statistical independence—each person’s decision does not influence the others. Finally, that the decision is made by majority rule.

It turns out the majority amplifies the abilities of its individual members—for example, let’s say there are three voters, each with a 2/3 chance to be correct (so p = 0.67.) To figure out the chance of the majority to be correct, we have to look at every possible result. Here is a table of that, with each row representing a different combination of votes cast. C = correct, I = incorrect.

Voter 1 Voter 2 Voter 3
C C C
C C I
C I C
C I I
I C C
I C I
I I C
I I I

Next we look at the permutations in which the majority is correct, which I have underlined above. We only need to look at these results, because once we know the  probability of the majority being correct, we just subtract it from 100% and can find the probability of them being incorrect.

So next we find the probability of each of those results, by multiplying the probability of each individual vote. Here’s that:

Voter 1 Voter 2 Voter 3 Equation Result
C (2/3) C (2/3) C (2/3) 2/3*2/3*2/3 = 8/27
C (2/3) C (2/3) I (1/3) 2/3*2/3*1/3 = 4/27
C (2/3) I (1/3) C (2/3) 2/3*1/3*2/3 = 4/27
I (1/3) C (2/3) C (2/3) 1/3*2/3*2/3 = 4/27

Then just add up the results (8/27 + 4/27 + 4/27 + 4/27) and we get 20/27, which equals ~0.741. To be clear then, when there are three voters, each with a 67% chance of voting correctly, the majority will have a 74% of voting correctly. The group is greater than the average of its parts.

So the Condorcet jury theorem states that if a group has an average p value greater than .5, the probability of the majority being correct approaches 1 as the population approaches infinity. And the reverse is true—if we take three voters with a less than .5 probability of being correct, the majority will be dumber than the individuals. But on average, people should have a greater p value than .5, right? On average they should be smarter than a coin toss, yeah? You’d hope?

A practical example:

If you’re in a relationship and the majority of people you talk to think it’s a bad relationship and you disagree, you may think—hey, I should know best, my p value is like .8, because I’m actually in the relationship. What do they know? They’re a bunch of .6s. But you’d be wrong, because as a group, all those .6s will have a better chance of being correct than .8.

If the CJT is true, then any country whose population’s p value is on average greater than 0.5 would be best off being run by majority rule. If it’s below 0.5, they’d do better with any random individual being chosen to run the country—being run monarchically.

Isn’t that great? Society isn’t doomed after all. At least for this post. Next time we’ll talk about the prisoner’s dilemma.

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