The Wisdom Of Crowds Reduction

Recently I attended a talk where Hannah Fry mentioned The Wisdom of Crowds in the following context: she twitted a picture of a jar with sweets and offered her followers to guess how many sweets are in the jar. The average of guesses turned out to be almost spot-on, somewhat like 117.9, whilst the true number is 117.

Call the quantity to be estimated \( \mu \). If we were to model each guess as an independent sample from some random variable \( X \)1, then we know that the average has to converge to the expectation of \( X \), call it \( E[X] \), by the Strong Law Of Large Numbers.

In the context of Hannah’s talk, the phenomena was mentioned to highlight that aggregately people are predictable. And the stress is on the fact that \( E[X] \approx \mu \).

However, I fail to see the significance of the Wisdom of the Crowds presented unless the conditions when \( E[X] \approx \mu \) are mentioned. Obviously it is not hard to make sure that public gets it really wrong. Try integer factorisation2 or any NP-complete (i.e. hard) problem3. You don’t have to go all the way to NP-complete spectrum of problems, there is most definitely plenty of easy problems where the average guess is really bad.

Therefore I really need to know when the public has a fair chance to get it right. This is a simple matter of Making Beliefs Pay Rent (in Anticipated Experience). The phenomena, as presented, has no predictive power and does not generate testable hypotheses.

So it wouldn’t have changed anything had the average guess for the sweets been within a reasonable margin rather than spot-on.

In conclusion, I insist that people who talk about The Wisdom shift focus from the fact that the crowds can be wise to when they are wise.

  1. Here do not treat \( X \) as a random variable assigned to each person rather it is a RV which picks a person at random and each person represents a guess so that sampling comes from identically distributed random variables. 

  2. If only I could break RSA by asking people around… 

  3. For the same reason the idea that the public could self-organise so that the society is operating at some kind of stationary level (like in pure capitalism) makes me cringe because that is short of implying N=NP