With or without replacement? Sampling uncertainty in Shepp's urn scheme

TL;DR

This paper investigates a variant of Shepp's urn problem where the sampling method (with or without replacement) is unknown to the stopper, providing bounds and explicit solutions in certain cases.

Contribution

It introduces a new variant of Shepp's urn problem and derives bounds and explicit solutions, revealing that the optimal strategy remains unchanged in balanced urns.

Findings

Optimal strategy in balanced urns matches classical problem

Bounds on the value function are established

Explicit solution for the balanced urn case

Abstract

We introduce a variant of Shepp's classical urn problem in which the optimal stopper does not know whether sampling from the urn is done with or without replacement. By considering the problem's continuous-time analog, we provide bounds on the value function and in the case of a balanced urn (with an equal number of each ball type) an explicit solution is found. Surprisingly, the optimal strategy for the balanced urn is the same as in the classical urn problem.