What happens in the Condorcet jury theorem, when p<0.5?

I think we are living it.

As it turns out, my little thought already occurred to Condorcet, and the hunch is trivially true. Thw wisdom of crowds and the madness of crowds are what happens for different values of p, the probability the the average person arrives at the correct answer on a binary choice. As the number of people participating gets larger, even slight differences from .5 rapidly converge to 100% correct or 0% correct in the aggregate.

Sigh. Thanks for the link, John. I had come across the concept in a book on data modeling, and typed it in on a Kindle just to get the thing posted.

Here, for those unfamiliar with it, is an introduction to Condorcet’s jury theorem.

By definition they are more likely to choose incorrectly.

So there is a mathematical formula for calculating a hung jury? Maybe calculating the probability of a hung jury?

Maybe, but that’s not this. This is just an observation that in a population composed of individuals each slightly better than 50% at binary choices, increasing the number of people voting on a single answer rapidly goes to high likelihoods of success.

Unfortunately, the converse is true for slightly stupid populations.

Condorcet’s theorem isn’t really about hung juries but the likelihood of reaching the “correct” result (when that can be defined) by adding together the votes of individuals with a wide dispersion of opinions. As long as slightly more than 50% of voters will choose the “correct” option, then increasing the number of voters increases the probability that the electorate as a whole will arrive at the correct outcome.

The flip side of this is that if less than 50% of voters will select the correct outcome, then increasing voter participation and the size of the electorate doesn’t help—they’ll still get the “wrong” answer.

If you look at it though a sufficiently twisted telescope, this can be seen in much the same light as Claire Berlinski’s ongoing screeds which I wrote about on 2019-07-17 in “Is Democracy Doomed?”. You see, those rubes are just too stupid and uninformed to vote for what’s good for them (which their betters know, and up until recently have been able to lead them toward by controlling their access to information and presenting them with a set of candidates and policies from which it didn’t really matter much which they chose).