The Prophet Dilemma: A Challenge to Induction
by Lin Zhong, August 2016
While deduction has been widely accepted as logically sound, induction has been an everlasting focus of debate amongst philosophers of science. The essence of induction goes like this: if we have observed A to be true, A will be true next time we observe. I would like to defer you to this nicely written book for further entertainment. Here, rather, I would like to talk about an understudied phenomenon that challenges induction in the same spirit of the famed Russell’s chicken.
Imagine there is a village of 256 people in the primordial time. The village lies in the valley of a large river that randomly floods (with 50% odds) every year during the summer. The lowland of the village is fertile but gets destroyed whenever the river floods; the highland is safe but has poor yield of crops. Therefore, to know if the river floods has great value to the villagers. As a result, each everyone of them tries to predict if the river will flood or not at the beginning of the year. Sadly, none of them have any magic power and therefore all predict randomly, with odds of being correct being 50%. What is the probability that at least one villager makes N consecutive predictions?
The answer is P=1-(1-1/2^N)^256. Perhaps to your surprise, P=0.9997 when N=5 and P=0.6328 when N=8. That means that it is almost guaranteed that at least one villager will be able to predict correctly five years in a row, despite that she/he does it completely randomly and possesses no magic power.
The question is: shall we induce that this villager somehow possesses a magic power for predicting the flood? Well, obviously not if you know the setup of the story. But what if you didn’t know that all the villagers make predictions randomly? I call this the Prophet Dilemma.
The essence of this phenomenon is a large number of random experiments conducted in parallel. It provides the mathematical explanation for Paul the Octopus and the notorious English football scam where half of the victims receive one prediction and another half the other. It is the parallel counterpart of the famous Infinite Monkey Theorem.
This Prophet dilemma perhaps can explain the origin of myth and religion. It is also at the heart of evolution and innovation: with a large number of individuals experimenting in parallel, a successful design is almost guaranteed to emerge.