The Stationary Prophet Inequality Problem

TL;DR

This paper introduces the stationary prophet inequality problem, analyzing pricing policies that achieve optimal or near-optimal revenue approximations in a continuous-time, infinite-horizon setting with Poisson arrivals.

Contribution

It provides the first optimal prophet inequality for a stationary problem and improves bounds on online policy approximations, advancing theoretical understanding.

Findings

Achieves a 1/2-approximation of the optimal offline policy.

Achieves better than (1-1/e)-approximation of the optimal online policy.

Improves upon previous bounds by Collina et al. and Aouad and Saritaç.

Abstract

We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller. Our main results are pricing-based policies which (i) achieve a $1/2$-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than $(1-1/e)$-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a $1-1/e$ bound implied by recent work of Aouad and Sarita\c{c} (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.