Embedding optimal selection problems in a Poisson process

We consider optimal selection problems, where the number N1 of candidates for the job is random, and the times of arrival of the candidates are uniformly distributed in [0, 1]. Such best choice problems are generally harder than the fixed-N counterparts, because there is a learning process going on as one observes the times of arrivals, giving information about the likely values of N1. In certain special cases, notably when N1 is geometrically distributed, it had been proved earlier that the optimal policy was of a very simple form; this paper will explain why these cases are so simple by embedding the process in a planar Poisson process from which all the requisite distributional results can be read off by inspection. Routine stochastic calculus methods are then used to prove the conjectured optimal policy.