Optimal Stopping Rule Research Articles

Comparison of real data and simulated data analysis of a stopping rule based on the standard error of measurement in computerized adaptive testing for medical examinations in Korea: a psychometric study.

This study aimed to compare and evaluate the efficiency and accuracy of computerized adaptive testing (CAT) under 2 stopping rules (standard error of measurement [SEM]=0.3 and 0.25) using both real and simulated data in medical examinations in Korea. This study employed post-hoc simulation and real data analysis to explore the optimal stopping rule for CAT in medical examinations. The real data were obtained from the responses of 3rd-year medical students during examinations in 2020 at Hallym University College of Medicine. Simulated data were generated using estimated parameters from a real item bank in R. Outcome variables included the number of examinees’ passing or failing with SEM values of 0.25 and 0.30, the number of items administered, and the correlation. The consistency of real CAT result was evaluated by examining consistency of pass or fail based on a cut score of 0.0. The efficiency of all CAT designs was assessed by comparing the average number of items administered under both stopping rules. Both SEM 0.25 and SEM 0.30 provided a good balance between accuracy and efficiency in CAT. The real data showed minimal differences in pass/ fail outcomes between the 2 SEM conditions, with a high correlation (r=0.99) between ability estimates. The simulation results confirmed these findings, indicating similar average item numbers between real and simulated data. The findings suggest that both SEM 0.25 and 0.30 are effective termination criteria in the context of the Rasch model, balancing accuracy and efficiency in CAT.

Monitoring a sequence of Bernoulli random variables subject to gradual changes in the success rates where the success rates are unknown

We look at a sequence of Bernoulli random variables where the success rates change from θ 1 to θ 2. We will assume that both the success rates before and after the change are unknown but increasing. We assume that this change does not happen abruptly but gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop late.

Optimizing Sequential Decisions: Enhancements to the Brickman Principle with Cumulative Punishment and Probability Adjustments

Introduction Determining the optimal stopping point in sequential decision-making scenarios is crucial for maximizing rewards and minimizing costs. Traditional models like the original Brickman Principle often simplify this process by assuming fixed critical values and equal probabilities at each decision stage. These assumptions may not accurately reflect real-world complexities, where costs can be cumulative and probabilities variable. Objective This work seeks to enhance the Brickman Principle by including cumulative punishment elements and non-uniform probability distributions, therefore improving its capacity to accurately represent the intricacies of real-world decision-making. Methods Through a rigorous experimental study, we evaluate the impact of these modifications on optimal stopping rules and expected profits. Results In line with Prospect Theory's emphasis on loss aversion, the results reveal a distinct pattern of risk-averse behavior, with most participants choosing to stop sooner in the sequence to avoid growing fines. Furthermore, we saw substantial variation in both the termination points and anticipated earnings across participants, suggesting that individual disparities in risk tolerance and decision-making approaches are crucial in influencing results

Machine learning of surrender: Optimality and humanity

AbstractWe develop a novel machine learning (ML) framework to estimate a surrender charge for variable annuities (VAs) with the balance between human behavior and rational optimality. Optimality accounts for insurers' potential losses from strategic surrenders by policyholders who attempt to take advantage of the market situation. However, policyholders sometimes need to surrender a VA because of sudden personal financial distress or a terminal illness. The literature contains contributions for these two surrender decisions separately, but we consider them simultaneously using ML. The ML framework is a Bayesian mixture of a deep optimal stopping rule based on potentially high‐dimensional financial variables and a statistical model with historical data. This framework can help insurers and pension funds to set surrender charges and perform stress testing in ways that balance profits and social responsibility by incorporating policyholders' behavioral data.

Optimal Stopping Rules for Preventing Overloading of Multicomponent Systems

When random-strength components work as an interconnected parallel system, then its carrying capacity is random as well. In a case where such a multicomponent system is a subject of the stepwise-growing workload, some of its components fail and their loads are taken over by the ones that are intact. When the loading process is continued, the additional loads trigger consecutive failures that degrade the system, eventually leading to a complete failure. If the goal of the system is to carry as much load as possible, then the loading process should be continued, but no longer than until the loading capacity of the whole system is reached. On the other hand, with every additional load step, a failure of the system becomes more probable, as the carrying capacity is random and known solely through its probability distribution. In such cases, the decision on when to cease the loading process is not obvious. We introduce and analyse a minimal model of failure spreading in an array of progressively loaded pillars controlled by a decision-maker who stops the process when a required load is attained. We show how to construct an optimal stopping rule. Under some additional assumptions regarding the adopted loss function, it is argued that the optimal stopping rule is of the threshold type and it significantly depends on the shape of the load-step probability distribution.

Costly multi-unit search

We formulate and solve a costly multi-unit search problem for the optimal selling of a stock of goods. Our showcase application is an inventory liquidation problem with holding costs, such as warehousing, salaries or floor planning. A seller faces a stream of buyers periodically arriving with random capped demands. At each decision point, he decides how to price each unit and also whether to stop searching or not. We set this as a dynamic programming problem and solve it inductively by characterizing optimal search rules and reservation prices. Our inductive logic does not rely in the use of the first order conditions for optimal pricing and optimal search decisions.We show that combining multiple units with a fixed per period search cost might translate into non-monotone selling costs and reservation prices. This lack of monotonicity naturally leads to discontinuities of the pricing strategy. In particular, the seller optimally employs strategies such as bundling, and more sophisticated ones that endogenously combine purchase premiums, when inventory is large, with clearance sales and discounts, when inventory is low. Instead, a variable search cost always yields an increasing marginal selling cost and thus can only make sense of pricing strategies that employ purchase premiums.Our model extends search theory by explicitly accounting for the effects of search costs on optimal multi-unit pricing strategies, pushing it into a richer class of problems and offering solutions that extend beyond optimal stopping rules.

Optimal Stopping and Loading Rules Considering Multiple Attempts and Task Success Criteria

Numerous engineering systems gradually deteriorate due to internal stress caused by the working load. The system deterioration process is directly related to the workload, providing opportunities for decision-makers to manage system deterioration by modifying the workload. As one of the most effective ways to control system malfunction risk, mission stopping has been extensively studied. Most existing research on mission stopping ignores the effect of working loads on the internal deterioration of safety-critical systems. The purpose of this work is to examine the optimal joint loading and stopping rules for systems subject to internal degradation under two types of mission success requirements (MSR). The problem is formulated using the recursive algorithm to minimize the expected cost over the mission. Mission reliability and system safety are assessed, and the optimal loading and stopping rules are investigated. The established models are illustrated by practical examples, and comprehensive policy comparison and parameter sensitivity analysis on the allowable mission time, mission duration and the number of mission tries are conducted. Our findings indicate that dynamic load level modification has a substantial effect on system deterioration and predicted long-term costs. For the purpose of decision-making, several managerial implications for the joint development of load adjustment and abort implementation are obtained.

The Extended David-Yechiali Rule for Kidney Allocation

The First Come First Served (FCFS) queuing policy is routinely assumed to be the benchmark policy for “fairness” in waiting-time performance. In this article, we propose a slight modification of the FCFS policy based on a natural extension of the well-established David and Yechiali (DY) rule and analyze it in the context of managing a waiting list for kidney transplants. In the proposed policy, the queuing agents are sequentially offered a stochastically arriving organ on a “first come, first served” basis while applying the individually optimal DY stopping rule. Through a realistic simulation, we show that the proposed policy, which we term Extended David and Yechiali (EDY), favorably compares to the FCFS policy in terms of medical efficiency while maintaining a comparable level of equity (i.e., fairness). Possible implications and practical aspects of the EDY are discussed.

Exactly Optimal Quickest Change Detection of Markov Chains

This paper establishes that an exactly optimal rule for Bayesian Quickest Change Detection (QCD) of Markov chains is an optimal stopping rule in the form of a threshold test on the no change posterior. We also provide a computationally efficient scalar filter for the no change posterior whose effort is independent of the dimension of the chains. We establish that an (undesirable) weak practical super-martingale phenomenon can be exhibited by the no change posterior when the before and after chains are too close in a relative entropy rate sense. The proposed detector is examined in simulation studies.

Monitoring a Bernoulli process subject to gradual changes in the success rates of a sequence of Bernoulli random variables

We look at a sequence of Bernoulli random variables where the success rates change from θ 1 to θ 2. We will assume that both the success rates before and after the change are known. We also assume that this change does not happen abruptly but gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early.

Computation Offloading and Service Caching for Mobile Edge Computing Under Personalized Service Preference

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mobile edge computing</i> (MEC) has emerged as an attractive solution by executing computation-intensive services at a powerful edge server instead of mobiles. Two types of data are necessary to this end. One is user-specific data acquired from mobiles, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">computation offloading</i> (CO). The other is service-specific data downloaded from a central cloud, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">service caching</i> (SC). It is noteworthy that CO and SC decisions are coupled when each user’s <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">service preference</i> (SP) is personalized. Specifically, noting that the optimal SC is to cache services likely to be requested more frequently, the resultant SC tends to be biased to the SP of the user whose offloading rate is high. On the other hand, such an SC decision causes longer computing latency of users with a relatively low offloading rate, which ultimately limits a CO decision for agile MEC services. This work tackles this issue from a sum-utility maximization perspective under radio-resource and computation-latency constraints. The average computation latency is first derived in closed-form by modeling a computation as a stochastic process following a hyper-exponential distribution. Based on it, we first consider the case for homogeneous SP where CO and SC decisions are decoupled. Thus, SC can be deterministically controlled using the homogeneous SP, while CO decision is independently determined, lying between water-filling and channel-inversion allocations. Next, we design a joint CO-and-SC policy for heterogeneous SP. CO and SC decisions are iteratively optimized with the other fixed by leveraging the homogeneous SP’s result. The optimal stopping rules are derived, guaranteeing the sum-utility enhancement. The proposed algorithm’s effectiveness is verified by simulations that the proposed CO-and-SC design for heterogenous SP always outperforms that for homogeneous SP.

The access time of random walks on trees with given partition

Denote by T(s,t) the set of trees, whose vertex set can be partitioned into two independent sets of sizes s and t respectively. Given a tree T with stationary distribution π and a vertex v∈T, the access time HT(π,v) is the expected length of optimal stopping rules from π to v. In this paper, we get a sharp upper bound for maxv∈THT(π,v) and a sharp lower bound for minv∈THT(π,v) among T(s,t), respectively. The corresponding extremal graphs are also obtained. As a byproduct, it is proved that the path Pn maximizes maxv∈THT(π,v) and the star K1,n−1 minimizes minv∈THT(π,v) among all trees on n vertices.

Cautious stochastic choice, optimal stopping and deliberate randomization

We study cautious stochastic choice (CSC) agents facing optimal timing decisions in a dynamic setting. In an expected utility setting, the optimal strategy is always a threshold strategy—to stop/sell the first time the price process exits an interval. In contrast, we show that in the CSC setting, where the agent has a family of utility functions and is concerned with the worst case certainty equivalent, the optimal strategy may be of non-threshold form and may involve randomization. We provide some carefully constructed examples, including one where we can solve explicitly for the optimal stopping rule and show it is a non-trivial mixture of threshold strategies. Our model is consistent with recent experimental evidence in dynamic setups whereby individuals do not play cut-off or threshold strategies.

An entrepreneur's dilemma: An optimal stopping rule in pivoting

Though pivoting is widely practiced by entrepreneurs, an optimal stopping rule in pivoting remains unexplored. Proposing an optimal foraging model and accounting for competition in exploiting an opportunity, we identify an optimal stopping rule of 30% gain‐to‐loss ratio. That is, additional gains are less feasible when 70% of the pivoting across future plausible versions of opportunities are traversed to. The findings have practical implications for entrepreneurs.

A sequential estimation problem with control and discretionary stopping

&lt;p style='text-indent:20px;'&gt;We show that “full-bang” control is optimal in a problem which combines features of (i) sequential least-squares &lt;i&gt;estimation&lt;/i&gt; with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded &lt;i&gt;control&lt;/i&gt; of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary &lt;i&gt;stopping&lt;/i&gt;. We develop also the optimal filtering and stopping rules in this context.&lt;/p&gt;

Smart Energy Management in Virtual Power Plant Paradigm With a New Improved Multilevel Optimization Based Approach

A virtual power plant (VPP) is a cloud based distributed power plant that aggregates the capacities of diverse distributed energy resources (DERs) for the purpose of enhancing power generation as well as trading or selling power on the electricity market. The main issue faced while working on VPPs is energy management. Smart energy management of a VPP is a complex problem due to the coordinated operation of diverse energy resources and their associated uncertainties. This research paper proposes a real-time (RT) smart energy management model for a VPP using a multi-objective, multi-level optimization-based approach. The VPP consists of a solar, wind and thermal power unit, along with an energy storage unit and some flexible demands. The term multi-level refers to three different energy levels depicted as three homes comprising of different amounts of loads. RT operation of a VPP is enabled by exploiting the bidirectional communication infrastructure. Multi-objective RT smart energy management is implemented on a community-based dwelling system using three alternative algorithms i.e., hybrid optimal stopping rule (H-OSR), hybrid particle swarm optimization (H-PSO) and advanced multi-objective grey wolf optimization (AMO-GWO). The proposed technique focuses on achieving the objectives of optimal load scheduling, real-time pricing, efficient energy consumption, emission reduction, cost minimization and maximization of customer comfort altogether. A comparative analysis is performed among the three algorithms in which the calculated real-time prices are compared with each other. It is observed that on average H-PSO performs 7.86 % better than H-OSR whereas AMO-GWO performs 10.49% better than H-OSR and 5.7% better than H-P-SO. This paper concludes that AMO-GWO is the briskest, most economical, and efficient optimization algorithm for RT smart energy management of a VPP.

A Class of Recursive Optimal Stopping Problems with Applications to Stock Trading

In this paper, we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multidimensional Markovian setting, we show that the problem is well posed in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues, and we determine the optimal stopping rule in that case.

Optimal stopping of random walk with exponential utility function

The article provides solutions for optimal stopping problems for finite and infinite horizons in the case of random walk with exponential utility function. For these problems we found: i) explicit solution of the corresponding Bellman equation; ii) solution of free boundary problem; iii) optimal stopping rule.

Monitoring a Poisson process subject to gradual changes in the arrival rates where the arrival rates are unknown

We look at a Poisson process where the arrival rates change from λ 1 to λ 2. We will assume that the arrival rates both before and after the change are unknown. We also assume that this change does not happen abruptly but gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early.

A deep learning approach for computations of exposure profiles for high-dimensional Bermudan options

In this paper, we propose a neural network-based method for approximating expected exposures and potential future exposures of Bermudan options. In a first phase, the method relies on the Deep Optimal Stopping algorithm (DOS) proposed by Becker, Cheridito, and Jentzen (2019), which learns the optimal stopping rule from Monte-Carlo samples of the underlying risk factors. Cashflow paths are then created by applying the learned stopping strategy on a new set of realizations of the risk factors. Furthermore, in a second phase the cashflow paths are projected onto the risk factors to obtain approximations of pathwise option values. The regression step is carried out by ordinary least squares as well as neural networks, and it is shown that the latter results in more accurate approximations.The expected exposure is formulated, both in terms of the cashflow paths and in terms of the pathwise option values and it is shown that a simple Monte-Carlo average yields accurate approximations in both cases. The potential future exposure is estimated by the empirical α-percentile.Finally, it is shown that the expected exposures, as well as the potential future exposures can be computed under either, the risk neutral measure, or the real world measure, without having to re-train the neural networks.