A gambler who plays a sequence of games in n different rooms of a gamblehouse recieves as the expected reward the maximum of the optimal stopping values (according to so called threshold stopping rules) of the n sequences. In the case that the games in each room are represented as sequences of independent uniformly bounded random variables, this reward is compared with the expected gain of a ‘prophet’, i.e. the expected supremum of all games. The main theorems obtained here yield known results of Hill, Hill and Kennedy, and Samuel–Cahn as immediate corollaries