Condorcet's jury theorem

Condorcet's jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de Condorcet.

It states that where the average chance of a member of a voting group making a correct decision is greater than fifty percent the chance of the group as a whole making the correct decision will increase with the addition of more members to the group.

Example

 * An innocent man is accused of murder on the border between England and Scotland. All other considerations aside, he would be wiser to hand himself over in Scotland than England.  This is because, given exactly the same evidence, a jury of fifteen persons is more likely to reach a true verdict than a jury of twelve.

The logic behind the claim is based on probability.

Imagine throwing a weighted coin, with a 51% chance of landing heads. If you threw it three times, and it landed tails three times, you'd not be too surprised. If you threw it ten times and it game up tails all those times, knowing it had a 51% chance of coming up heads, you'd raise an eyebrow. If it came up tails 100 times, you'd be safe in concluding that it did not indeed have a 51% chance of coming up heads, but a 100% chance of landing tails.

The more iterations of a probabilistic outcome there are, the more likely that the distribution of outcomes will conform with the base probability.

In a voting group, therefore, where members have an average chance of reaching a true decision above fifty percent, the more times votes occur, the more likely it will be that an overall majority will come to that decision. Individual probabilities turn into a group probability that is greater.

This raises two linked questions - is there a correct decision to be had? and what is a fifty one percent chance of finding it?

The latter is easier to answer. A fifty percent chance of reaching an answer to any question, where there are two choices, is a blind guess. Any answer based on true information correctly processed must have a greater than fifty percent chance of being correct. The only way to go below fifty percent chance is to have either incorrect information or badly processed information.

As to the claims that an answer is true, that will vary. Whether a person killed another is reducible to a true or false dichotomy. Whether higher or lower taxes are right for the economy is a much more complex question; it might depend on goals for the economy, what various group members consider acceptable levels of income inequality, preferences for work or leisure, and dozens of other normative factors.

Criticisms and corrections
There have been a variety of corrections and criticisms offered about the Jury Theorem over the years. At the heart of jury theorem is basically the law of large numbers and a very simplistic probabilistic math that assumes that each vote is independent. David Estlund pointed out that there is little reason to believe that there will only be two choices. We need to investigate how the condition holds when there are multiple choices and perhaps even along multiple dimensions. Considering these scenarios can give rise to what is known as Condorcet's paradox - one will get cycling (among choices) as number of choices becomes greater than two and will lead to at the minimum requirements for larger n to provide 'correct decision'.

Another common criticism is that there is little reason to believe that a voter has a greater than 50% chance of getting the right answer. If this is the case then majority rule would actually result in zero probability of producing the correct result. However this criticism is simply countered when one recalls that these voters are honest and there are only two issues at stake. Voters who get the wrong answer more often than not would soon realize that just flipping a coin would yield better results, and so he would decide to do that.

This is all based on simplistic mathematics and simplistic assumption of independence. In reality coin flips or decisions of people are not independent. In that case there can be strategic voting or a variety of other malaise that can suck away the power of producing correct decisions.