Unobservable Case Research Articles

Technology licensing contract design considering royalty transparency and demand information asymmetry with downstream co-opetition

This paper studies how technology suppliers can optimally design licensing contracts when there is incomplete information on the licensee’s demand potential for the product in a high-technology supply chain. A licensing contract typically contains an up-front payment and royalties on product sales. The licensor decides to license one technology to a licensee, which designs products based on the supplier’s technology and outsources the product manufacturing to a competitive downstream manufacturer. We apply principal-agent models to formulate the licensor’s contracting problem while considering the transparency of the licensing royalty and find that under demand information asymmetry, the optimal contract structure changes with the demand potential. More specifically, both the separating and pooling strategies can be the optimal strategies for the licensor under the observable contract case. Conversely, under the unobservable case, only the separating strategy is the optimal licensing strategy. And the licensor can always charge zero royalty to a high-type design firm regardless of the menu strategy or the single contract. Furthermore, we examine the effect of licensing royalty transparency on licensing decisions and firms’ profits and highlight the effect of downstream competition on the licensor’s profit. Our results show that the licensor can be better off in the observable contract case. By contrast, the design firm may be better off in the unobservable contract case. We provide a rationale for the licensing contract design under demand information asymmetry and further shed light on the choice between public and secret licensing terms.

Modeling strategic walk-in patients in appointment systems: Equilibrium behavior and capacity allocation

We consider an outpatient clinic with strategic patients who choose between making an appointment with an indirect wait cost (advance patients) and walking in with an inconvenience cost that includes the risk of being rejected and waiting in the clinic (walk-ins). Patients have different indirect waiting costs and show up with some probability. The clinic allocates slots to advance and walk-in patients to minimize the expected blockage of walk-in patients. We characterize the equilibrium behavior of patients and investigate the optimal capacity allocation, for unobservable (patients know the expected waiting time) and observable (patients know their exact waiting time) cases. For the unobservable case, one of the three options is optimal: allocating all slots to advance patients, allocating all slots to walk-ins, or allocating a certain number of slots to advance patients so that only urgent patients would choose the walk-in option. In contrast, for the observable case, no such structure exists. We investigate the value of information numerically. Finally, we develop a simulation platform to examine the effects of model assumptions. We find the optimal capacity allocation for the simulation model to benchmark the performance of the theoretical models and two simple policies. These analyses verify that our models work well in realistic simulations, offering a useful tool in practice. In contrast to the common practice of allocating some slots to walk-ins, our results suggest that the clinics should prefer a system that allocates all slots to advance patients in certain environments due to the strategic behavior of patients.

The Value of Observing the Buyers’ Arrival Time in Dynamic Pricing

We consider a dynamic pricing problem in which a firm sells one item to a single buyer to maximize expected revenue. The firm commits to a price function over an infinite horizon. The buyer arrives at some random time with a private value for the item. He is more impatient than the seller and strategizes over the timing of the purchase in order to maximize his expected utility, which implies either buying immediately, waiting to benefit from a lower price, or not buying. We study the value of the seller’s ability to observe the buyer’s arrival time in terms of her expected revenue. When the seller can observe the buyer’s arrival, she can make the price function contingent on the buyer’s arrival time. On the contrary, when the seller can’t, her price function is fixed at time zero for the whole horizon. The value of observability (VO) is defined as the worst-case ratio between the expected revenue of the seller when she observes the buyer’s arrival and that when she does not. First, we show that, for the particular case in which the buyer’s valuation follows a monotone hazard rate distribution, the upper bound of VO is [Formula: see text]. Next, we show our main result: in a setting very general on valuation and arrival time distributions: VO is at most 4.911. To obtain this bound, we fully characterize the solution to the observable arrival problem and use this solution to construct a random and periodic price function for the unobservable case. Finally, we show by solving a particular example to optimality that VO has a lower bound of 1.136. This paper was accepted by Itai Ashlagi, revenue management and market analytics. Funding: This work was partially supported by the Agencia Nacional de Investigación y Desarrollo Chile [Grants ACT210005 and FONDECYT 1220054], and by the Center for Mathematical Modeling at the University of Chile [Grant FB210005]. The work of D. Pizarro has benefited from the Artificial and Natural Intelligence Toulouse Institute, which is funded by the French “Investing for the Future – PIA3” program [Grant ANR-19-P3IA-0004]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.4794 .

Observability Analysis for Orbit Determination Using Spaceborne Gradiometer

The application of a spaceborne gradiometer to satellite orbit determination has been proposed in recent years. The advantages lie in its full autonomy and its immunity against spoofing attacks. The objective of this paper is to study the observability of the spaceborne gradiometer-based orbit determination (SGOD) system. To address the problem of unknown biases in spaceborne gradiometer measurements, the joint estimation of satellite orbital states and gradiometer biases is adopted. Then, the system observability matrices are formulated based on the theory of nonlinear observability. The local weak observability is assessed numerically by singular values and condition numbers of the observability matrices. A special unobservable case is demonstrated, which is the circular orbit in the Keplerian two-body model. Different orbital parameters, attitude control modes, and gravity models are furtherly employed to analyze their effects on system observability. The observability analysis shows that low altitudes, large eccentricities, large inclinations, attitude maneuverings, and the J2 perturbation can improve the system observability. Furthermore, a numerical simulation using the extended Kalman filter (EKF) is developed to obtain accuracies and convergence rates of the SGOD system. Effects of orbital parameters, attitude control modes, and the J2 perturbation on orbit determination results are also studied. It is shown that the simulation results agree with the observability analysis. Using a spaceborne gradiometer with 0.1 E (1 E=10−9 s−2) accuracy, position accuracy of tens of meters is achieved for a low Earth orbit.

The Effect of a Service Experience Cost on a Queueing System

In this work, the authors consider the effect of a service experience cost (SE cost) on customer behaviour in the M/M/1 queueing system. Based on customer individual equilibrium strategy, social welfare is also analyzed in unobservable and observable cases. The SE cost decreases the equilibrium joining probability and social welfare in an unobservable case. However, there might exist multiple individual equilibrium thresholds in an observable case. Furthermore, numerical results show that the SE cost can be used as a feasible policy to make an incentive for customers and regulate the system for improved social welfare in some scenarios.

How to minimally modify a dynamical system when constructing flat inputs?

AbstractIn this paper, we study the problem of constructing flat inputs for multi‐output dynamical systems, in particular, we address the issue of the minimal modification of the initial dynamical system (the measure of modification being the number of equations that have to be changed by adding flat inputs). We show that in the observable case, control vector fields that distort m equations only (where m is the number of measurements) can always be constructed (and this is the minimal possible number of equations that have to be modified by adding flat inputs), while in the unobservable case, the best that we can hope for is that equations only are modified such that they involve flat inputs. We discuss when the original output is a flat output of minimal possible differential weight for the minimally modified control system (where by the differential weight, we mean the minimal number of derivatives of components of a flat output needed to express all states and controls). We propose a solution for constructing flat inputs leading to a minimally modified control system consistent with the minimal differential weight and, moreover, for which the observable part is affected by the minimal possible number of controls (this last property being important in applications). We show that, in that case, at least equations have to be affected by the flat inputs.

On observability analysis and observer design for a class of nonlinear uncertain systems with general elastic vibration dynamics

AbstractThis paper focuses on the observability analysis for a general class of rigid‐flexible coupling systems constituted of rigid‐body mode and elastic vibration modes with uncertain dynamics and disturbances. And the measured output is a linear combination of rigid‐body mode and elastic vibration modes. Firstly, the necessary and sufficient condition of the observability for such system is given. Secondly, the general extended state observer design is proposed to estimate the states of both rigid‐body mode and elastic vibration modes as well as “total disturbance.” Next, the convergence of estimation errors of the general extended state observer is discussed. Then, under unobservable case, a compressed extended state observer is designed to estimate the states and the “total disturbance” acting on rigid‐body mode, and the corresponding estimation errors are studied. In addition, the theoretical results are demonstrated via the simulation on a rigid‐flexible coupling system.

The effect of information on queue-scalping service systems

We consider an M/M/1 queueing system with a single queue scalper who makes profit by selling his position in the queue. The equilibrium purchasing strategies of customers under observable case (with double-threshold strategy) and unobservable case (with mixed strategy) are derived. The profits of the queue scalper are obtained under two information levels. Numerical experiments suggest that the queue scalper is better off with unobservable case when the system is too congested or the operating cost is too low.

Joint pricing and inventory control in a make-to-stock queue with delay-sensitive customers

We consider price and inventory decisions in a make-to-stock system from the perspectives of a profit-maximizing server and a social planner in which customers are delay-sensitive. Under two information cases (i.e. an almost unobservable case and a fully observable case), customers decide whether to purchase the product based on their utility. The utility is closely related to whether the inventory of the product is available, the product’s price, and the waiting time. We first investigate the optimal decision in the almost unobservable case. In this case, the server’s inventory strategy depends on the relationship between the unit holding cost of the product and the unit waiting cost of the customer. Additionally, the system utilization factor also affects the inventory strategy. Then, we analyse the corresponding results in the fully observable case. The results illustrate that, in both cases, profit maximization and welfare maximization lead to different pricing and inventory strategies. Finally, the numerical results show that the optimal price is sensitive to delay information, whereas the optimal stock level is not.

Knowledge, congestion, and economics: Parameter uncertainty in Naor’s model

This paper studies an extension of Naor’s model in which there is parameter uncertainty. In particular, the arrival rate is known, to customers and system managers, only through its distribution. For the observable case, the relationship between the optimal individual threshold and the thresholds for a social optimizer or revenue maximizer does not change from the classical model with a known arrival rate. However, in the unobservable case, it is shown that the decisions of the social optimizer and revenue maximizer no longer coincide. Furthermore, in the unobservable case with arrival rate uncertainty, the social optimizer induces a lower expected arrival rate than the revenue maximizer. This stands in contrast to the observable case, in which the social optimizer prefers a more congested system.

Strategic Behavior and Optimization in an Unobservable Constant Retrial Queue with Balking and Set-Up Time

AbstractAn M/M/1 constant retrial queue with balking customers and set-up time is considered. Once the system becomes empty, the server will be turned down to reduce operating costs, and it will be activated only when there is a customers arrives. In this paper, the almost unobservable case is studied, in which the information of the queue length is unavailable, whereas the state of the server can be obtained. Firstly, the steady state solutions are derived and the individual equilibrium strategies are analyzed. In addition, social optimization problems, including cost analysis and social welfare maximization are investigated by using the PSO algorithm. Finally, by appropriate numerical examples, the sensitivity of some main system parameters is shown.

Strategic joining in a single-server retrial queue with batch service

<p style='text-indent:20px;'>We consider a single-server retrial queue with batch service where potential customers arrive according to a Poisson process. The service process has two stages: busy period and admission period, which are corresponding to whether the server is in service or not, respectively. In such an alternate renewal process, if arrivals find busy period, they make join-or-balk decisions. Those joining ones will stay in the orbit and attempt to get into server at a constant rate. While, if arrivals find the server is in an admission period, they get into the server directly. At the end of each admission period, all customers in the server will be served together regardless of the size of the batch. The reward of each arrival after completion of service depends on the service size. Furthermore, customers in the orbit fail to get into the server before the end of each service cycle will be forced to leave the system. We study an observable scenario that customers are informed about the service period upon arrivals and an unobservable case without this information. The Nash equilibrium joining strategies are identified and the social- and profit- maximization problems are obtained, respectively. Finally, the optimal joining strategies in the observable queue and the comparison of social welfare between the two queues are illustrated by numerical examples.</p>

Equilibrium balking strategies in the single-server retrial queue with constant retrial rate and catastrophes

ABSTRACT We study an M/M/1 retrial queueing system with constant retrial rate and Poisson generated catastrophes. When a catastrophe arrives to the system that the server is working, it deletes all customers in system and breaks the server down. The failed server is repaired immediately and the repair time is exponentially distributed. We investigate a joining/balking dilemma of customers with a natural reward-cost structure under two information levels, i.e. the unobservable case where customers can only observe server’s state, and the observable case where customers know both the queue length in orbit and the status of the server. Individual equilibrium joining probabilities and socially optimal strategies are obtained in these two cases. Finally, numerical experiments are carried out to illustrate our theoretic findings and the impact of information levels on social welfare. Interestingly, by comparing the corresponding results in the standard system without catastrophes, we find that customers in our system incur a lower waiting cost, which results in the fact that customers will receive more benefit and they are more inclined to join the system with catastrophes.

The Value of Observability in Dynamic Pricing

We consider a dynamic pricing problem where a firm sells one item to a single buyer in order to maximize expected revenues. The firm commits to a price function over an infinite horizon. The buyer arrives at some random time with a private value for the item. He is more impatient than the seller and strategizes the time of his purchase in order to maximize his expected utility, which implies either buying immediately or waiting to benefit from a lower price. We study how important is to observe the buyer arrival time in terms of the seller's expected revenue. When the seller can observe the arrival of the buyer, she can make the price function contingent on his arrival time. On the contrary, when the seller cannot observe the arrival, her price function is fixed at time zero for the whole horizon. The value of observability (VO) is defined as the worst case ratio between the expected revenue of the seller when she observes the buyer's arrival and that when she does not. First, we show that for the particular case where the buyer's valuation follows a monotone hazard rate distribution, the upper bound is e, and it is tight. Next, we show our main result: In a very general setting about valuation and arrival time distributions, the value of observability is at most 4.911. To obtain this bound we fully characterize the observable arrival setting and use this solution to construct a random and periodic price function for the unobservable case. Finally, we show by solving a particular example to optimality that VO has a lower bound of 1.017.

On Equilibrium in a Constant Retrial Queuing System with Reserved Time and Vacations

This paper investigates an M/M/1 constant retrial queue with reserved time and vacations. A new arriving customer will take up the server and accept service immediately if the server is idle. Otherwise, if the server is busy or on vacation, customers have to join a retrial orbit and wait for retry. Once a service is completed, the server will reserve a random time to seek a customer from the orbit at a constant retrial rate. If there is no arrivals (from the orbit or outside) during the idle period, to save energy, the server will take a vacation. This paper studies the fully unobservable case. First, the steady-state condition of the system is analyzed by using the Foster’s criterion, and the customers’ expected waiting time is obtained based on the generating function technique. And then, by introducing an appropriate revenue structure, the equilibrium strategies of customers and the socially optimal strategy are all derived. Furthermore, a comparison between them is made and the effect of some main system parameters is studied.

Modeling Strategic Walk-in Patients in Appointment Systems: Equilibrium Behavior and Capacity Allocation

We consider an outpatient clinic with strategic patients. Based on the level of available information, patients either take an appointment and wait until the appointment day or they select walk-in and eliminate the indirect waiting cost. Those patients who prefer walking-in face the risk of not receiving service if there are more walk-in patients than the available slots for walk-ins. Patients make their decision by taking other patients' choice into consideration. Considering the equilibrium behavior of patients, the clinic should determine the optimal number of slots reserved for walk-ins. We conduct our analysis under two different levels of information: un-observable (patients know expected waiting time only) and observable (patients know their exact waiting time). In each case, we investigate the optimal capacity reserved for advanced patients that minimizes the expected daily cost for the society, considering strategic customer case and comparing it to the centralized decision maker case. We show that for the un-observable case either all-appointment or all walk-in policy is optimal, while for the observable case no such structure exists. We also analyze value of information and coordination via numerical examples and show that providing waiting time information is most valuable for hybrid systems sharing capacity between walk-ins and advanced patients. We also show that not accounting for strategic behavior may lead to sub-optimal capacity allocation policy and increase costs.

Equilibrium Joining Strategies in the Geo/GeoK/1 Queueing System

We study strategic behavior in the G e o / G e o K / 1 queueing system under both fully observable case and fully unobservable case. Furthermore, equilibrium and socially optimal strategies are obtained according to the available information and the linear utility function. We compare the impact of system parameters on the equilibrium strategies and socially optimal strategies. At the same time, we illustrate the effects of parameters on the obtained equilibrium social benefit. Finally, some numerical examples are presented.

Optimal inventory threshold for a dynamic service make‐to‐stock system with strategic customers

AbstractWe consider a make‐to‐stock production system with one product type, dynamic service policy, and delay‐sensitive customers. To balance the waiting cost of customers and holding cost of products, a dynamic production policy is adopted. If there is no customer waiting in the system, instead of shutting down, the system operates at a low production rate until a certain threshold of inventory is reached. If the inventory is empty and a new customer emerges, the system switches to a high production rate where the switching time is assumed to be exponentially distributed. Potential customers arrive according to the Poisson process. They are strategic in the sense that they make decisions on whether to stay for product or leave without purchase on the basis of on their utility value and the system information on whether the number of products is observable to customers or not. The strategic behavior is explored, and a Stackelberg game between production manager and customers is formulated where the former is the game leader. We find that the optimal inventory threshold minimizing the cost function can be obtained by a search algorithm. Numerical results demonstrate that the expected cost function in an observable case is not greater than that in an unobservable case. If a customer's delay sensitivity is relatively small, these two cases are entirely identical. With increasing of delay sensitivity, the optimal inventory threshold might be positive or zero, and hence, a demarcation line is depicted to determine when a make‐to‐stock policy is advantageous to the manager.

EQUILIBRIUM BALKING STRATEGIES IN THE REPAIRABLE M/M/1 G-RETRIAL QUEUE WITH COMPLETE REMOVALS

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

Block Trading: Building up a Stock Position Under a Regime Switching Model

This work is intended to systematically characterize the problem of block trading. The author characterizes it as a constrained optimal purchasing (or control) problem, on behalf of the vendee. The purpose of this work is to investigate the optimal purchasing strategies and give quantitative reference information on the pricing of block trade. Noting that large block of stock trading may result in the price fluctuation and different market environment may lead to distinct investing strategies, the controlled diffusion model with regime switching is adopted, which can be used to represent distinct environment of financial market, and the model is modified to allow the price impact, by allowing the drift depend on the purchase rate. Where the switching regimes can be completely observable or unobservable (hidden). Which is a significant difference in contrast to most market models in the existing literature. The objective is to maximize the total discounted shares of the underlying stock, subject to the expected fund for building up corresponding position with an upper bound. This work mainly focuses on the completely observable case, as to the case with hidden regimes, this work just briefly talk about how to reduce the unobservable model to a completely observable one. In the completely observable case, to solve this constrained control problem, the Lagrange multiplier methods and the dynamic programming approaches are used. Though optimality is obtained, what is still lacking is an explicit solution to the optimality equation. To overcome this difficulty, the two loop approximation scheme is given. For fixed Lagrange multiplier, the inner loop is to obtain optimal strategies, which is based mainly on the finite difference approximation. While, the outer loop is a stochastic recursive approximation algorithm to get the optimal Lagrange multiplier. Convergence results are obtained for both the inner and the outer approximations. The quantitative pricing information is related to the threshold levels of the optimal purchasing strategy. Numerical examples are also provided to illustrate our results. Finally, as to the unobservable case, by using the well known Wonham filter approach, one can reduce the unobservable case to a completely observable one.