There is a famous class of problems concerning public announcements. There is a philosophical (sociological) appeal in these problems. Can a public announcement change the behaviour of the citizens? Of course, if it is for example a news article about dangerous epidemic. But what if it is a news article about something everybody already knew? Can such a news article affect the behaviour of the citizens? Analogously, can our economics theories affect our economics once published? The following mathematical problem and its generalization show that even when the announcement seems to be completely uninformative, it might affect the behaviour of the citizens in a radical way.
Since I want to continue the Prisoners’ Problem series, I reformulate this problem similarly to the previous ones. In the following we assume that the prisoners are very smart which is essential for this argument to work — but clearly such problems may be formulated for populations with limited deductive power and (little) irrationality; such problems would be of much higher complexity of course.
Consider a prison with 60 prisoners. One morning the prisoners are collected to the yard and every one is given a hat. Half of them (30) gets a black and half a white hat, but the prisoners do not know how many there is of a kind. The prisoners see each other’s hats but nobody sees her own hat and they cannot communicate. Every 10 minutes each prisoner can guess her own hat’s colour if she wants. If a guess is correct, the prisoner is freed from the prison and she leaves immediately. We assume that the prisoners never guess unless they are completely sure that they know the correct answer (because they are hanged otherwise). So no one guesses anything… But then the
chairman of the prison gives an announcement: “at least one of you has a white hat!”
What happens? Why?
a more elaborate version:
the chairman announces something non-trivial about the number of the white hats. (Non-trivial means: it is true, but it could be false taking into account the total number of prisoners)
What happens? Why?