Backward Induction Research Articles

A Subsidization Scheme for Maximizing Social Welfare in Mobile Communications Markets

In contemporary mobile communications markets, various agents or players interact to pursue welfare. Regulatory policies enacted by governments in certain markets aim to maximize social welfare. However, some countries, both least developed and developing, often adopt successful models from developed nations without local market validation. Therefore, network economics serves as a pertinent framework for analyzing such policies. This paper introduces a novel scheme based on constrained optimization problems, where the constraints represent multilevel economic game equilibria within a system model involving three agents: the central planner, the mobile network operator, and the mobile data users. These agents strategically optimize their payoff functions by considering benefit factors and decision variables such as the subsidization factor, pricing, and data consumption. To this end, a three-stage dynamic game is proposed to model the players’ interactions, employing the backward induction method to ascertain the subgame perfect equilibrium from the Nash equilibrium. A case study is presented, demonstrating a 31.16% increase in social welfare between scenarios involving no adoption of the subsidization factor and its adoption at the optimal value when the central planner enacts it to other players in the game, even if they do not necessarily attain maximum payoff values. In countries aligning with this proposed model, social welfare is maximized through a subsidization scheme.

Decisions of dual-channel fresh agricultural product supply chains based on information sharing

PurposeThis paper studies the optimal decision-making and coordination problem of a dual-channel fresh agricultural product (FAP) supply chain. The purpose is to analyze the impact of information sharing on optimal decisions and propose a coordination mechanism to encourage supply chain members to share information.Design/methodology/approachThe two-echelon dual-channel FAP supply chain includes a manufacturer and a retailer. By using the Stackelberg game theory and the backward induction method, the optimal decisions are obtained under information symmetry and asymmetry and the coordination contract is designed.FindingsThe results show that supply chain members should comprehensively evaluate the specific situation of product attributes, coefficient of freshness-keeping cost and network operating costs to make decisions. Asymmetric information can exacerbate the deviation of optimal decisions among supply chain members and information sharing is always beneficial to manufacturers but not to retailers. The improved revenue-sharing and cost-sharing contract is an effective coordination mechanism.Practical implicationsThe conclusions can provide theoretical guidance for supply chain managers to deal with information asymmetry and improve the competitiveness of the supply chain.Originality/valueThis paper combines the three characteristics that are most closely related to the reality of supply chains, including horizontal and vertical competition of different channels, the perishable characteristics of FAPs and the uncertainty generated by asymmetric demand information.

Stochastic Resource Optimization for Wireless Powered Hybrid Coded Edge Computing Networks

With the ubiquitous sensing enabled by the Internet-of-Things (IoT), massive amount of data is generated every second, transforming the way we interact with the world. To manage big data and enable analytics at the edge of the network, large amount of computation power is required to perform the computation intensive tasks. However, the energy-constrained IoT devices are not able to perform the computation tasks without compromising the quality-of-service of the applications. In this paper, we propose a hybrid network in which users can fully offload their computation tasks to edge servers through coded edge offloading or perform local computation with the wireless power transfer derived from coalitions of unmanned aerial vehicles (UAVs) serving as mobile charging stations. To minimize the cost of the network, we consider a two-level optimization approach. At the lower level, an optimal UAV coalition that minimizes the network cost is formed. At the upper level, the computation approach for each user is decided while taking into account the stochastic nature of wireless charging efficiency. Given the interrelation between the two levels, we adopt the backward induction optimization strategy to jointly derive the optimal coalition structure and computation approach for each user. In the performance evaluation, we provide extensive sensitivity analyses to study the performance of the cost minimization approaches amid varying network parameters. Moreover, we show that our approach outperforms deterministic optimization approaches in network cost minimization.

Joint optimization of inspection and condition-based maintenance for a deteriorating product under extended warranty

Condition-based maintenance (CBM) has been widely applied in engineering practice, but it has been rarely integrated into extended warranty policies. When the inspection cost is high, it becomes important to schedule the inspection plan when making CBM decisions. This paper jointly optimizes inspection and CBM for a deteriorating product under extended warranty. The deterioration of the product is characterized as a non-stationary gamma process. At each inspection epoch, the decision maker needs to determine whether to repair the product and when to perform the next inspection. The objective is to simultaneously optimize the inspection policy and the maintenance policy that minimize the warranty servicing cost. The joint optimization problem is formulated by stochastic dynamic programming. The optimal policy is proven to be of a control-limit form. A backward induction algorithm integrating the obtained structural properties is proposed to find the optimization results. A numerical example is conducted to illustrate the effectiveness of the proposed approach. A graphical representation of the obtained policy is provided to support practical maintenance implementation. The comparison with a popular CBM policy confirms the superiority of the proposed policy.

Estimation of optimal treatment regimes with electronic medical record data using the residual life value estimator.

Clinicians and patients must make treatment decisions at a series of key decision points throughout disease progression. A dynamic treatment regime is a set of sequential decision rules that return treatment decisions based on accumulating patient information, like that commonly found in electronic medical record (EMR) data. When applied to a patient population, an optimal treatment regime leads to the most favorable outcome on average. Identifying optimal treatment regimes that maximize residual life is especially desirable for patients with life-threatening diseases such as sepsis, a complex medical condition that involves severe infections with organ dysfunction. We introduce the residual life value estimator (ReLiVE), an estimator for the expected value of cumulative restricted residual life under a fixed treatment regime. Building on ReLiVE, we present a method for estimating an optimal treatment regime that maximizes expected cumulative restricted residual life. Our proposed method, ReLiVE-Q, conducts estimation via the backward induction algorithm Q-learning. We illustrate the utility of ReLiVE-Q in simulation studies, and we apply ReLiVE-Q to estimate an optimal treatment regime for septic patients in the intensive care unit using EMR data from the Multiparameter Intelligent Monitoring Intensive Care database. Ultimately, we demonstrate that ReLiVE-Q leverages accumulating patient information to estimate personalized treatment regimes that optimize a clinically meaningful function of residual life.

Benefit distribution of guaranteed water saving management contract based on principal-agent theory

Aiming at benefit distribution of guaranteed water saving management contract (WSMC) projects, principal-agent theory is used to propose benefit distribution scheme. Firstly, principal-agent models are established in two cases: (1) the water user and WSCO are risk-neutral; (2) the water user is risk-neutral, while WSCO is risk-averse. Then the models are solved by the backward induction method under symmetric information and asymmetric information. The models' results are analyzed and the results show that (1) WSCO's benefit distribution proportion is negatively correlated with effort cost coefficient, risk aversion degree and project uncertainty; (2) the water user's expected utility is positively correlated with project investment scale and WSCO's water saving capacity; (3) WSCO's risk aversion will reduce the water user's expected utility; (4) incentives are only required when WSCO's effort is unobservable. Finally, numerical examples are used to show the above results' rationality and suggestions are given for the water user and WSCO.

Allocation of Eavesdropping Attacks for Multi-System Remote State Estimation

In recent years, the problem of cyber–physical systems’ remote state estimations under eavesdropping attacks have been a source of concern. Aiming at the existence of eavesdroppers in multi-system CPSs, the optimal attack energy allocation problem based on a SINR (signal-to-noise ratio) remote state estimation is studied. Assume that there are N sensors, and these sensors use a shared wireless communication channel to send their state measurements to the remote estimator. Due to the limited power, eavesdroppers can only attack M channels out of N channels at most. Our goal is to use the Markov decision processes (MDP) method to maximize the eavesdropper’s state estimation error, so as to determine the eavesdropper’s optimal attack allocation. We propose a backward induction algorithm which uses MDP to obtain the optimal attack energy allocation strategy. Compared with the traditional induction algorithm, this algorithm has lower computational cost. Finally, the numerical simulation results verify the correctness of the theoretical analysis.

Research on distributed energy sharing strategies of industrial clusters under consumers’ low-carbon preferences

Distributed energy sharing is an effective means for enterprises to improve energy utilization efficiency and reduce carbon emissions. With the continuous improvement of consumers' low-carbon awareness, low-carbon preference has an important impact on distributed energy sharing strategies. In this context, we introduce the energy saving benefit sharing contract and use the backward induction method to analyze the optimal equilibrium strategies of distributed energy sharing in industrial clusters. Then we conduct a comparative analysis of the equilibrium results. Finally, the above results are analyzed and verified by numerical simulation. The results show that when the ratio of energy low-carbon cost coefficient is within a certain range, energy sharing with integrated energy service providers is the optimal choice for enterprises in industrial clusters, and the centralized decision is better than the decentralized decision; The higher the consumer's low-carbon preference, the higher the product's low-carbon level and total system profitability; As the energy saving benefit sharing ratio increases, the low carbon level of the product and the profit of the integrated energy service provider also increases, and the total profit of the enterprises first increases and then decreases.

Equilibrium strategies of sharing CSR in a supply chain with manufacturer encroachment

Purpose Motivated by the real-world practice of the thriving e-commerce, manufacturers are transcending traditional boundaries of merely producing and selling directly by implementing encroachment. Concurrently, supply chain (SC) members who overlook the corporate social responsibility (CSR) sharing will be left behind, which is closely linked to their profits. This paper aims to investigate a better way to share CSR under the scenarios of manufacturer encroachment and no-encroachment.Design/methodology/approach This paper constructs game-theoretic models in a SC consisting of a manufacturer (M) and a retailer (R), where the manufacturer can sell products by retailing, and may sell directly by implicating encroachment. The manufacturer and retailer jointly consider whether to share CSR and the proportion of it by taking consumer surplus into account. Furthermore, equilibriums for each model are derived using backward induction. Then, the authors analyse the impact of CSR sharing proportion and compare the equilibrium outcomes under different scenarios. Finally, the numerical analyses are presented to verify the results.Findings Several interesting results are found in this paper. First, the retailer shares more CSR can benefit SC members and social welfare when the manufacturer does not implement encroachment. However, the results may change which is decided by the unit cost of encroachment when the manufacturer does so. Second, the proportion of CSR shared by manufacturer and the unit cost of encroachment has an interactive impact on equilibrium outcomes. Finally, both manufacturer encroachment and SC members share CSR may be the best for the perspective of SC members and consumers.Practical implications Based on the analytical results, this paper provides novel managerial implications to assist manufacturer and retailer in determining the optimal strategies for CSR sharing and encroachment. Furthermore, the appropriate proportion of CSR shared by manufacturer and the unit cost of encroachment may let manufacturer, retailer and consumer surplus achieve a win-win-win situation.Originality/value To the best of the authors’ knowledge, this paper is the first attempt to explore the strategy of CSR sharing under the scenarios of manufacturer encroachment and no-encroachment.

Blockchain adoption and contract coordination of poverty alleviation supply chain considering altruistic preference

This study discusses blockchain adoption strategy and coordination of a poverty alleviation supply chain (PASC) consisting of one retailer with altruistic preference and one farmer cooperative. Four decentralized models are considered, including scenario AN (with altruistic preference and without blockchain), scenario AB (with altruistic preference and with blockchain), scenario DN (without altruistic preference and without blockchain), and scenario DB (without altruistic preference and with blockchain). We solve the game models in these four scenarios based on backward induction and obtain the optimal solutions. Additionally, two centralized models, namely scenario CN (without blockchain) and scenario CB (with blockchain), are used as benchmarks for coordinating the PASC. Through comparative analysis, we find that (1) the retailer does not necessarily benefit from blockchain adoption. The retailer has an incentive to adopt blockchain when the fixed and unit operating costs of blockchain adoption are low. Interestingly, even when the unit operating cost of blockchain adoption is high, the retailer can still increase its profits by adopting blockchain when consumers’ acceptance level of agricultural products is low. (2) Under certain conditions, blockchain adoption can lead to a win–win situation for the retailer and the farmer cooperative. (3) The retailer’s altruistic preference increases the profits of the farmer cooperative and the PASC, but at the cost of reducing its own profits. (4) The two proposed contracts, the altruistic preference joint two-part tariff contract and the altruistic preference joint profit-sharing contract, can effectively coordinate the PASC. (5) Three extensions illustrate that the findings regarding the retailer’s blockchain adoption strategy remain robust under strict conditions.

A closed-loop supply chain operation decision under life cycle: Ecological design, service design and recycling effort perspectives

The commercial, environmental and social value of closed-loop supply chain (CLSC) has been widely recognised in the literature. Enterprises are increasingly aware of the importance of product life cycle management; that is, enterprises conduct CLSC management of the production, use and recycling of products. Existing research lacks research on CLSC operation decision under product life cycle. This paper introduces ecological design, service design and recycling into a CLSC with the manufacturer, retailer and recycler. Stackelberg models considering different leadership structures and service design providers are constructed using game theory, and we use backward induction to solve the game in different cases. Results show that from the system-operating efficiency and total profits maximisation perspectives, the recycler-led is always the most effective. However, for the selection of the optimal service design provider, when the manufacturer (retailer) is the leader, the retailer (manufacturer) providing the service design should be the dominant strategy. When the recycler is the leader, the effect is the same no matter who provides the service design. Finally, we find an interesting conclusion that in the case of retailer provides service design, supply chain members do not always achieve the highest profits under their own leadership structure. The numerical analysis shows that when the consumer’s preference for service design is greater than 6.3, the manufacturer can obtain greater profits under the recycler-led structure, and when consumer’s preference for ecological design is greater than 7.1, the retailer can gain greater profits under the recycler-led structure.

Multi-energy pricing strategy for port integrated energy systems based on contract mechanism

This article studies a pricing scheme to increase the profit of ports and gas utility company (GUC), and improve social welfare. Firstly, a system model of the port integrated energy system (PIES) is established and the government green subsidy on biomass gas is considered. Secondly, a multi-stage game is formulated to characterize the interaction between the port, GUC and the government. Besides, the contract mechanism is used to improve the operation performance of the energy supply chain. Then, the equilibrium solutions of the game are obtained by the backward induction. Finally, the theoretical analysis and simulation show that the proposed pricing strategy can improve the profit of GUC, ports and government, and the proposed revenue sharing contract mechanism can coordinate the energy supply chain.

A XIVth century approach to the points problem

The application of expected value as the first decision-theoretical criterion for evaluating risky lotteries is traditionally attributed to the 1654 correspondence between Pierre de Fermat and Blaise Pascal. However, an anonymous author of the XVth century Italian manuscript stored in the Vatican Apostolic Library calculated expected value by backward induction in a similar way to Pascal’s approach. This note argues that an even earlier anonymous XIVth century Italian manuscript calculates expected value by a different method resembling the direct rule (regula recta) described in 1202 by Leonardo of Pisa (Fibonacci) in his Liber Abaci as borrowed from the Arabs.

The coordination strategies of coal supply chain considering carbon emissions reduction

The coordination strategies of a two-echelon coal supply chain consisting of a coal enterprise and a downstream enterprise are proposed considering market low-carbon preference and carbon emissions reduction. The cleaning and matching degree (CMD) is introduced to measure the quality of commercial coal which have influence on the production cost and energy consumption of the downstream enterprise. Stackelberg models are established to investigate the coordination strategies using cost-sharing contracts. We use backward induction to solve the models and analyse the optimal decisions in four scenarios. It is revealed that promoting CMD level can extend the market portion and realise more profit. The cost-sharing contract can coordinate the supply chain members, obtain profit Pareto improvement in certain condition of cost-sharing ratio and establish a win-win situation. Additionally, the emission reduction is greater than that under decentralised decision-making.

Pricing strategy of first-enjoy-after-pay service offered by two-sided media platforms

PurposeThis study highlights the impact of mental accounts on a user's decision-making regarding payment schemes and aims to determine the pricing strategy for the first-enjoy-after-pay service offered by the two-sided media platforms.Design/methodology/approachThis study establishes a game-theoretic model and utilizes backward induction to derive the equilibrium price by maximizing the monopolist's profit.FindingsThe findings indicate that the conditions for a two-sided media platform to offer the first-enjoy-after-pay service depend on the trade-off between pleasure attenuation and pain buffering and the effect of time discounts. Moreover, the authors found that the time discount is a critical factor in determining pricing strategies under various payment schemes offered by the platform.Research limitations/implicationsThis work adopts a uniform pricing strategy for users who opt for either immediate or post-payment schemes. Nevertheless, it is important to note that this approach has limitations in terms of offering discriminatory pricing for those who choose both payment schemes.Practical implicationsThis analytical work provides valuable insights for two-sided media platforms to optimize their payment scheme strategies and pricing considering the influence of a user's mental account.Originality/valueIn a two-sided media platform, the authors provide applicable conditions for the platform to offer first-enjoy-after-pay service considering the effect of mental accounts. Further, the authors show the optimal pricing strategy under different payment schemes provided by the platform.

Energy Trading and Power Allocation Strategies for Relay-Assisted Smart Grid Communications: A Three-Stage Game Approach

In smart grid, serious packet loss often occurs in the process of information interaction between the Utilities and customers, which results in supply-demand deviation and further increases the cost of the Utilities. In order to improve the information transmission performance of communication networks, the Utilities purchase relay service from telecom operator to help data aggregator units (DAU) transfer information to gateway (GW), so as to improve communication quality and reduce the cost of the Utilities. Secondly, in order to solve the problem of telecom operators' energy reduction and reduce the cost of purchasing energy, we utilize energy supply point (ESP) to collect the surplus energy of retail customers for energy supply, and telecom operator pays a certain amount of remuneration to ESP in exchange for ESP to continuously supply energy to telecom operator. Then, we establish a three-stage game method and system model between the Utilities, telecom operator and ESP, and propose the relay power allocation and energy transaction pricing strategy. Due to the real-time change of energy demand, we consider two situations of energy oversupply and conservative supply, and use the backward induction method and iterative algorithm to obtain the equilibrium solution of the Stackelberg game, including the unit energy price of ESP, total power of telecom operator, the proportion of transmission power allocated to the relay service, and payment scheme of the Utilities. Simulation results show that the proposed algorithm can quickly and accurately converge to the optimal solution of the problem, and the method can improve the stability of demand-side regulation, reduce the cost of the Utilities and increase the profit of telecom operator.

Risk-neutral valuation of GLWB riders in variable annuities

In this paper we propose a model for pricing GLWB variable annuities under a stochastic mortality framework. Our set-up is very general and only requires the Markovian property for the mortality intensity and the asset price processes. The contract value is defined through an optimization problem which is solved by using dynamic programming. We prove, by backward induction, the validity of the bang-bang condition for the set of discrete withdrawal strategies of the model. This result is particularly remarkable as in the insurance literature either the existence of optimal bang-bang controls is assumed or it requires suitable conditions. We assume constant interest rates, although our results still hold in the case of a Markovian interest rate process. We present extensive numerical examples, modelling the mortality intensity as a non mean reverting square root process and the asset price as an exponential Lévy process, and compare the results obtained for different parameters and policyholder behaviours.

Insider Trading with Semi-Informed Traders and Information Sharing: The Stackelberg Game

This paper presents a financial Stackelberg game model with two partially informed risk neutral insiders. Each insider receives a private signal about the stock value and competes with the other insider under a Stackelberg setting. Linear strategies for the game’s participants are considered and normal distributions for the fundamentals are assumed. Based on the Stackelberg game and the Backward Induction theory, the unique linear equilibrium is characterized. The findings reveal that the level of partial information might increase/decrease the insiders’ profits as well as the market parameter in the Stackelberg setting relative to the Cournot setting. Additionally, this paper considers the information sharing scenario between the two insiders competing in this Stackelberg game. The results show that multiple equilibria exist in contrast to the information sharing scenario in the Cournot game where the Nash equilibrium is unique.

Subsidization of public transit service under double moral hazard

This paper aims to explore incentive issues and characteristics of optimal contract schemes for subsidizing public transit services under double moral hazard, which refers to situations where both the regulator and the operator face incentives to engage in opportunistic behavior due to information asymmetry in the provision of public transit service. This specific aspect of the regulator-operator interrelationship has not been thoroughly investigated in the existing literature. To address this gap, we utilize principal-agent theory and develop a non-cooperative game-theoretic model to analyze the relationship between the regulator (the principal) and the operator (the agent). Through backward induction, we examine the first-best equilibrium (without moral hazard) and second-best equilibrium (under double moral hazard) solutions. Numerical analysis based on a linear contract reveals some key characteristics of the optimal subsidy contract arrangements. Our findings reveal that under double moral hazard, both the regulator and the operator only exert their second-best efforts. The result analysis shows that while higher incentives can reduce the effort required by the regulator, they can also induce the operator to work harder. The marginal effort utilities of the two parties have distinct impacts on determining the optimal sharing rule. Furthermore, we illustrate how the interaction dynamics of the efforts exerted by the two involved parties vary with the sharing rule. Our study extends the current literature by shedding light on the optimal incentive arrangements and contractual requirements for public transport subsidy contracts when considering the double moral hazard issue.

Dynamic supervision of counterfeit products based on blockchain technology: A differential game on goodwill accumulation.

Counterfeiting is a serious problem in many industries, and the annual profit and tax losses caused by counterfeit products in China have reached over 60 billion yuan. The focus of this paper is to study the effect of blockchain technology in terms of supervising counterfeit products as well as product quality and service decisions under blockchain from a dynamic optimization perspective. We consider the role of blockchain when disclosing product quality information and develop a dynamic optimization model before and after blockchain adoption using differential game theory. Meanwhile, we solve the model using the Hamilton-Jacobi-Bellman (HJB) equation and backward induction and conduct parametric, comparative, and numerical analyses of the equilibrium solutions. The main findings are as follows: First, we find that in the absence of blockchain, counterfeit products can reduce brand goodwill and decrease the quality and service level of genuine products; however, supervising counterfeit product sales through blockchain can improve this problem under the critical conditions determined in this work. Second, in the direct sales mode, we find that if the unit operating cost and fixed cost of blockchain are small, the brand owner's adoption of blockchain can not only combat the counterfeiter's profit but also increase its own profit. Third, in the wholesale sales mode, we find that the best conditions in which a brand owner can establish blockchain are only related to the related costs of blockchain, while retailers also need to increase the selling price of genuine products when establishing blockchain. Fourth, we find that supply chain performance and blockchain supervisory effects are sensitive to key parameters through numerical analysis in a quantitative form. These findings have important implications for genuine enterprises in terms of determining the conditions for establishing blockchain to combat counterfeiting and for optimizing product quality and service decisions when using blockchain technology.