Approximative solutions of best choice problems

Abstract

We consider the full information best choice problem from a sequence $X_1,\dots, X_n$ of independent random variables. Under the basic assumption of convergence of the corresponding imbedded point processes in the plane to a Poisson process we establish that the optimal choice problem can be approximated by the optimal choice problem in the limiting Poisson process. This allows to derive approximations to the optimal choice probability and also to determine approximatively optimal stopping times. An extension of this result to the best $m$-choice problem is also given.