Optimal Shopping When the Sales Are on—a Markovian Full-Information Best-Choice Problem

Abstract

We study a full-information best-choice problem viewed in a shopping context. A certain commodity can be found at certain random times with stochastically fluctuating prices. While the prices may have a tendency to decrease, the instants at which items are offered become less frequent and it is possible that the item currently found will be the last one. The prospective customer's objective is to buy at the right time so as to minimize the expected price of the acquired item. We propose a two-dimensional Markov chain model with a rather general continuous-time point process structure and dependence of the random prices on the availability times of the items. The value function v of the associated optimal stopping problem is characterized as the smallest solution of a two-dimensional integral equation; this allows us to find the optimal policy under certain conditions. In particular, we consider a nonhomogeneous Poisson model for which more specific results can be obtained. We derive a differential equation of which v is the uniformly smallest nonnegative solution. This way v is determined up to a boundary condition at infinity. We provide criteria for identifying a solution as the value function and also for the natural stopping rule to be optimal. Several examples are given.

Keywords:

• Best-choice problem
• Boundary condition at infinity
• Full information
• Integral equation
• Nonhomogeneous Poisson process
• Two-dimensional continuous-time Markov chain
• Value function

Mathematics Subject Classification:

• Primary 60G40
• Secondary 60J20

ACKNOWLEDGMENT

We would like to thank two referees for their careful reading of our manuscript and for several valuable suggestions that led to considerable improvements.