The Stationary Prophet Inequality Problem

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 Saritaç (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

On the Significance of Knowing the Arrival Order in Prophet Inequality

“…That is, rather than using the standard prophet benchmark, we use the arguably more realistic benchmark of the best order-aware online algorithm. This is part of a growing interest in alternative benchmarks to the prophet benchmark Kessel et al [2021], Niazadeh et al [2018], Papadimitriou et al [2021]. For example, Niazadeh et al [2018] quantify the loss due to single-threshold algorithms by the worst-case ratio between the best single-threshold algorithm and the best general online algorithm (single-threshold or not), both under a known order, and show that this 1/2 ratio is tight.…”