Multi Choice Min Prophet

Abstract

The prophet inequality is a fundamental problem in optimal stopping theory. Given n independent variables drawn from known distributions, a player observes values sequentially and must decide irrevocably whether to stop and accept the current value or continue. The goal is to select a single element while maximizing the ratio between the value chosen and that of the maximum value in the sequence. In this paper, we study the minimization counterpart, often termed the min prophet or cost prophet inequality. Unlike the maximization setting, where simple threshold algorithms achieve half of the prophet’s value, the minimization setting is significantly harder, with an exponential lower bound even for i.i.d. variables.
We study a multi-choice relaxation in which the algorithm may select multiple variables and gets to choose the best amongst them (the minimum amongst those selected). Our goal is to minimize the expected number of selections while achieving a constant competitive ratio. For adversarial order, we show that a constant competitive ratio requires a nearly linear number of choices in expectation, ergo, Ω(n/ln n). In contrast, we show that for the prophet secretary model (random order) one can attain constant competitiveness while requiring only an exponentially smaller expected number of choices i.e. O(ln n). We give a refined analysis and define M to be the ratio of the minimum expected value of any single variable to the expected minimum value of all variables (the prophet’s value) and present an algorithm that achieves a constant competitive ratio with O(min{ln ln M, ln n}) choices in expectation for the prophet secretary. We show that this is tight up to low order log factors even for the special case of the i.i.d. model. Specifically, the lower bound on the expected number of choices for any constant competitive algorithm is Ω(min{ln ln M/ln ln ln M, ln n/ln ln n}). We also show that if we insist on a deterministic bound on the number of choices then every constant competitive algorithm requires n choices. This holds even in the i.i.d. setting and shows that to achieve a constant competitive algorithm there is an exponential gap between the lower bound on the deterministic number of choices and the upper bound on the expected number of choices.
Finally, we consider a variant where both the algorithm and the adversary choose r values and pay their sum, this is the minimization multi unit version. We extend our techniques to the multi-unit variant for i.i.d. variables, achieving a constant competitive ratio with a small expected number of choices.