Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

Finding a Home: Stopping Theory and Its Application to Home Range Establishment in a Novel Environment

Familiarity with the landscape increases foraging efficiency and safety. Thus, when animals are confronted with a novel environment, either by natural dispersal or translocation, establishing a home range becomes a priority. While the search for a home range carries a cost of functioning in an unfamiliar environment, ceasing the search carries a cost of missed opportunities. Thus, when to establish a home range is essentially a weighted sum of a two-criteria cost-minimization problem. The process is predominantly heuristic, where the animal must decide how to study the environment and, consequently, when to stop searching and establish a home range in a manner that will reduce the cost and maximize or at least satisfice its fitness. These issues fall within the framework of optimal stopping theory. In this paper we review stopping theory and three stopping rules relevant to home range establishment: the best-of-n rule, the threshold rule, and the comparative Bayes rule. We then describe how these rules can be distinguished from movement data, hypothesize when each rule should be practiced, and speculate what and how environmental factors and animal attributes affect the stopping time. We provide a set of stopping-theory-related predictions that are testable within the context of translocation projects and discuss some management implications.

Proactive Mitigation of the Revocation Risk for Preemptible Virtualized Resources

<div>The provision of resources at the Cloud follows two generic models. The first model guarantees the provided resources for the requested time while the second involves unreliable resources with lower price compared to the former scheme but with no guarantees concerning an unexpected revocation due to a high demand. In this paper, we focus on the latter model and propose a scheme that monitors the course of execution of tasks placed at unreliable resources and decides when to store their current progress avoid jeopardizing intermediate outcomes in unexpected revocations. We rely on the principles of Optimal Stopping Theory (OST) to manage multiple tasks and decide for which task and when we have to save its current status. The outcome is a novel checkpointing mechanism fully aligned with the needs of the dynamics of an unreliable environment. The proposed model builds upon the heterogeneity of the available services in the Cloud and concludes a proactive mitigation approach of the revocation risk for unreliable virtualized resources. We present the theoretical basis of our mechanism and describe the solution of the identified problem. The pros and cons of our approach are evaluated through extensive simulations and a set of performance metrics.</div>

Proactive Mitigation of the Revocation Risk for Preemptible Virtualized Resources

<div>The provision of resources at the Cloud follows two generic models. The first model guarantees the provided resources for the requested time while the second involves unreliable resources with lower price compared to the former scheme but with no guarantees concerning an unexpected revocation due to a high demand. In this paper, we focus on the latter model and propose a scheme that monitors the course of execution of tasks placed at unreliable resources and decides when to store their current progress avoid jeopardizing intermediate outcomes in unexpected revocations. We rely on the principles of Optimal Stopping Theory (OST) to manage multiple tasks and decide for which task and when we have to save its current status. The outcome is a novel checkpointing mechanism fully aligned with the needs of the dynamics of an unreliable environment. The proposed model builds upon the heterogeneity of the available services in the Cloud and concludes a proactive mitigation approach of the revocation risk for unreliable virtualized resources. We present the theoretical basis of our mechanism and describe the solution of the identified problem. The pros and cons of our approach are evaluated through extensive simulations and a set of performance metrics.</div>

Reflected BSDEs when the obstacle is predictable and nonlinear optimal stopping problem

In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable [Formula: see text] expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of [Formula: see text]-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

Dynamic Pricing (and Assortment) Under a Static Calendar

This work is motivated by our collaboration with a large consumer packaged goods (CPG) company. We have found that whereas the company appreciates the advantages of dynamic pricing, they deem it operationally much easier to plan out a static price calendar in advance. We investigate the efficacy of static control policies for revenue management problems whose optimal solution is inherently dynamic. In these problems, a firm has limited inventory to sell over a finite time horizon, over which heterogeneous customers stochastically arrive. We consider both pricing and assortment controls, and derive simple static policies in the form of a price calendar or a planned sequence of assortments, respectively. In the assortment planning problem, we also differentiate between the static vs. dynamic substitution models of customer demand. We show that our policies are within 1-1/e (approximately 0.63) of the optimum under stationary demand, and 1/2 of the optimum under nonstationary demand, with both guarantees approaching 1 if the starting inventories are large. We adapt the technique of prophet inequalities from optimal stopping theory to pricing and assortment problems, and our results are relative to the linear programming relaxation. Under the special case of stationary demand single-item pricing, our results improve the understanding of irregular and discrete demand curves, by showing that a static calendar can be (1-1/e)-approximate if the prices are sorted high-to-low. Finally, we demonstrate on both data from the CPG company and synthetic data from the literature that our simple price and assortment calendars are effective. This paper was accepted by Hamid Nazerzadeh, big data analytics.

OEDDBOS: An Efficient Data Distributing Strategy with Energy Saving in Sensor-Cloud Systems

Sensor-cloud is a developing technology and popular paradigm for various applications. It integrates wireless sensor into a cloud computing environment. On the one hand, the cloud offers extensive data storage and analytical and processing capabilities not available in sensor nodes. On the other hand, data distribution (such as time synchronization and configuration files) is always an important topic in such sensor-cloud systems, which leads to a rapid increase in energy consumption by sensors. In this paper, we aim to reduce the energy consumption of data dissemination in sensor-cloud systems and study the optimization of energy consumption with time-varying channel quality when multiple nodes use the same channel to transmit data. Suppose that there is a certain probability that the nodes send data for competing channel. And then, they decide to distribute data in terms of channel quality for saving energy after getting the channel successfully whether or not. Firstly, we construct the maximization problem of average energy efficiency for distributing data with delay demand. Then, this maximization problem transferred an optimal stopping problem which generates the optimal stopping rule. At last, the thresholds of the optimal transmission rate in each period are solved by using the optimal stopping theory, and the optimal energy efficiency for data distribution is achieved. Simulation results indicate that the strategy proposed in this paper can to some extent improve average energy efficiency and delivery ratio and enhance energy optimization effect and network performance compared with other strategies.

An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers

An asset selling problem is one of well-known problems in the decision making literature. The problem assumes a stream of bidders who would like to buy one or several identical objects (assets). Offers placed by the bidders once rejected cannot be recalled. The seller is interested in an optimal selling strategy that maximizes the total expected revenue. In this paper, we consider a multi-asset selling problem when the seller wants to sell several identical assets over a finite time horizon with a variable number of offers per time period and no recall of past offers. We consider the problem within the framework of the optimal stopping theory. Using the method of backward induction, we find an optimal sequential procedure which maximizes the total expected revenue in the selling problem with independent observations.