Prophet Inequalities for Cost Minimization

Abstract

Prophet inequalities for rewards maximization are fundamental results from optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality due to Krengel, Sucheston, and Garling [KS77], where one is facing a sequence of costs X 1 , X 2 , . . . , X n in an online manner and must "stop" at some point and take the last cost seen. 1 Given that the X i 's are independent random variables drawn from known distributions, the goal is to devise a stopping strategy S (online algorithm) that minimizes the expected cost. The best cost possible is E [min i X i ] (offline optimum), achievable only by a prophet who can see the realizations of all X i 's upfront. We say that strategy S is an α-approximation to the prophet (α ≥ 1) ifWe first observe that if the X i 's are not identically distributed, then no strategy can achieve a bounded approximation, no matter if the arrival order is adversarial or random, even when restricted to n = 2 and distributions with support size at most two. 2 This leads us to consider the case where the X i 's are independent and identically distributed (I.I.D.). For the I.I.D. case, we give a complete characterization of the optimal stopping strategy. We show that it achieves a (distribution-dependent) constant-factor approximation to the prophet's cost for almost all distributions and that this constant is tight. In particular, for distributions for which the integral of the hazard rate 3 is a polynomial H(x) = k i=1 a i x di , where d 1 < • • • < d k , the approximation factor is λ(d 1 ), a decreasing function of d 1 , and is the best possible for H(x) = x d1 . Furthermore, when the hazard rate is monotonically increasing (i.e. the distribution is MHR), we show that this constant is at most 2, and this again is the best possible for the MHR distributions.For the classical prophet inequality for reward maximization, single-threshold strategies have been powerful enough to achieve the best possible approximation factor. Motivated by this, we analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a O (polylog n)-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound.We believe that our results may be of independent interest for analyzing approximately optimal (posted price-style) mechanisms for procuring items.