On the Game of Googol

In the classical secretary problem the decision maker can only observe the relative ranks of the items presented. Recently, Ferguson building on ideas of Stewart showed that, in a game theoretic sense, 'there is no advantage if the actual values of the random variables underlying the relative ranks can be observed (game of googol). We extend this to the case where the number of items is unknown with a known upper bound. Corollary 3 extends one of the main results in [HK] to all randomized stopping times. We also include a modified, some­ what more formal argument for Ferguson's result.