Three individuals are seated in a room. Each one of them is wearing a hat, which may be either red or white. Each of them sees the hats worn by the others, but cannot see his own hat (and in particular does not know its color). The true situation is that every person in the room is wearing a red hat.
(a) Depict this situation as a Harsanyi model of incomplete information, where a player's type is the color of his hat, and specify the vector of types corresponding to the true situation.
(b) Depict this situation as an Aumann model of incomplete information, and specify the state of the world corresponding to the true situation.
(c) A stranger enters the room, holding a bell. Once a minute, he rings the bell while saying "If you know that the color of the hat on your head is red, leave this room immediately." Does anyone leave the room after a few rings? Why?
(d) At a certain point in time, the announcer says, "At least one of you is wearing a red hat." He continues to ring the bell once a minute and requesting that those who know their hat to be red to leave. Use the Aumann model of incomplete information to prove that after the third ring, all three hat-wearers will leave the room.
(e) What information did the announcer add by saying that at least one person in the room was wearing a red hat, when this was known to everyone before the announcement was made?
(f) Generalize this result to n individuals (instead of 3).