Why fishes behave “better” than Condorcet

We see this problem in Sumpter et al. (2008), Consensus Decision Making by Fish, Curr Biol.

Condorcet Theorem just says the probability that, if we make N trials with probability p of getting the right answer in each trial, at least more than half of them get the right answer. Thus, each trial is completely independent. The solution to this problem is the same as the solution to problems of the kind “taking balls from a box”. Thus, if the number of fishes is odd, the probability of the majority taking the “right” decision is

If the number of fishes is even, we have the problem of ties. Assuming that ties are resolved by tossing a fair coin (so each option has 50% probability of being chosen), we have

On the other hand, as shown by Ward et al. (2008) Quorum decision-making facilitates… pnas, decisions of different fishes in the shoal are not independent. Once one or more fishes decide one option, the probability that other go for that option increases (in fact, Sumpter et al. 2008 also use this formula in other part of their paper). Therefore, it is to be expected that the probability increases faster for the case of fishes. It is as if p in the above formula was not constant, but increased.