# Condorcet Jury Theorem

Imagine a jury of three people.

They each have an identical probability, p>0.5, that they make the right decision. The right decision in this case means:

• they convict a guilty person, or
• they acquit an innocent person,

and let’s say the decision rule is a simple majority i.e. if two or more people make the right decision, then the jury’s verdict will be right.

Condorcet’s jury theorem basically states that in such circumstances, the jury will more likely than not get it right, and in fact, as the group gets larger the probability of getting it right goes to 1. This theorem is often cited in favour of democracies and the way they make decisions. But is voting by simple majority always best?

The first thing to note about Condorcet’s theorem is that there are three HUGE caveats:

1. each individual must be ‘competent’ (i.e. individually more likely to get it right than wrong, or their ‘p’ has to be greater than 0.5)
2. each individual’s decision must be independent
3. the theorem only applies to questions of fact and can’t really address issues where there is no objectively right or wrong answer

This rules out a lot of situations to which Condorcet’s theorem can be applied. Is it reasonable to assume that people get it right all the time? Not always, according to Hans Rosling. Do people always make their decisions in a vacuum? Unlikely, and increasingly difficult to do so due to social media echochambers. Lastly, society’s most difficult, divisive problems are usually ones that involve a conflict between two value systems, rather than a factual disagreement.

In fact the assumption that everyone is competent is a very strong assumption and means that mathematically it always works out that the majority is right most of the time. This problem is analgous to the coin-flipping problem and every other problem governed by a binomial distribution. Note that the expected outcome is n*p, i.e. the number of people multiplied by the probability that an individual gets it right. If p is greater than 0.5, then the expected outcome will always be more than half the number of people getting it right i.e. the majority makes the correct decision.

I did some good old excel number-crunching just to show that if we lowered everyone’s competency (say, every individual only has a 0.48 chance of getting it right), then all the probabilities change:

 No. of people Threshold for majority Probability the majority votes for the right decision (probability individual getting it right = 0.48) 1 1 0.4800 11 6 0.4460 51 26 0.3870 101 51 0.3434 10001 5001 0.0000 1000001 500001 0.0000

Basically, a confederacy of dunces is not a good idea. But what if only some people were incompetent? Would they drag the majority down?

I decided to work it out using specific numbers first.

Let’s say we have a jury of 3 people, each with a 0.52 chance of getting it right. In this scenario, the majority will get it right 53% of the time. Now let’s say we add 2 incompetent people, each with a 0.48 chance of getting it right. What’s the probability that the majority will get it right?

We can break it down into three cases:

•  If the initial 3 all get it right, then it doesn’t matter how the additional 2 vote, as they won’t change how the majority votes
• If 2 of the initial 3 get it right, then for the majority to get it right, at least 1 of the additional 2 have to vote correctly
• If 1 of the initial 3 get it right, then for the majority to get it right, both the additional 2 have to vote correctly

This probability that our new majority gets it right is given by:

$P(\texttt{majority gets it right}) = 0.52^3 + \binom{3}{2}(0.52^2)(1-0.52)[1-(1-0.48)^2] +\binom{3}{1}(0.52)(1-0.52)^2(0.48^2)$

Where:

• the first term on the right hand side represents the initial three getting it right (in which case it doesn’t matter how the additional 2 people vote)
• the second term is the probability that 2 of the initial 3 vote correctly, and at least 1 of the additional 2 voting correctly (this is equal to 1 minus the probability that none of the additional 2 vote correctly)
• the third term is the proability that 1 of the initial 3 vote correctly and the remaining 2 both vote correctly

When calculted, this gives 0.5075, which is lower than 0.53 (the probability that the initial 3 get it right), but the new majority still gets it right more than half the time.

The mathematical intuition is that by adding incompetent people, we are actually weighting the second and third terms on the right hand side of the equation by a factor that is less than 1, so the sum will therefore decrease. In fact if we increase the incompetency of the additional people (say p = 0.32), this further decreases the weight. If we add more incompetent people then eventually they outweigh the competent voters.

The policy implication is that we should only let experts (i.e. the ‘competent’ people) vote (at least on matters where there is a clear right or wrong answer). This defeats the central tenet of democracy which is that each person’s opinion has equal dignity. Then again, Condorcet was only trying to examine whether simple majorities are the most efficient way to make decisions, not whether they are the fairest.