Stackelberg Leader-follower Game Research Articles

Existence of equilibrium in a dynamic supply chain game with vertical coordination, horizontal competition, and complementary goods

We consider supply chain competition and vertical coordination in a linear–quadratic differential game setting. In this setting, supply chains produce complementary goods and each of them includes a single manufacturer and a single retailer who coordinate their decisions through a revenue-sharing contract with a wholesale price and a fixed sales revenue share. We study a multiple leader-follower Stackelberg game where the manufacturers are the leaders and the retailers are the followers. Competition occurs at both levels of the supply chains. Retailers play Nash and compete in price; manufacturers also play Nash but they compete in choosing their production capacities by exploiting the equilibrium price decisions made by the retailers. We show that open-loop Nash equilibria exist when the manufacturers only receive a wholesale price (there are no longer exploiting the equilibrium price decision made by the retailers, however). When the manufacturers receive both a wholesale-price and a share of the retailers’ sales revenues, equilibria generally no longer exist. The non-existence of an equilibrium stems from the fact that the manufacturers’ instant profits are discontinuous functions of their production capacities. This discontinuity leads to a major technical difficulty in that one cannot apply standard optimal control approaches to study the equilibria of the dynamic game. Our results illustrate the possibility that competition between supply chains might not be sustainable when they sell complementary products and rely on a revenue-sharing agreement.

Research on time-of-use compensation pricing strategies for load aggregators based on user demand response

As the electric power market reform deepens, the strategic role of load aggregators in demand-side response becomes increasingly important. The inherent variability of distributed renewable energy generation and user demand response often leads to a mismatch between the purchased electricity and the market bid volume, resulting in punitive costs for companies. To address this issue, this study treats demand-side controllable loads as dispatchable resources and proposes a tiered pricing strategy to adjust power distribution. By establishing a Stackelberg leader-follower game model, the study promotes a mutually beneficial relationship between load aggregators and controllable load users. Through case studies, this paper examines the operational profits of load aggregators and the power adjustment behaviors of controllable load users under tiered and fixed compensation pricing schemes. The results indicate that tiered compensation pricing significantly reduces punitive costs and enhances user participation in demand response.

Knowledge sharing decision-making under stochastic factors in platform ecosystems: the diversified participants' perspective

PurposeKnowledge sharing is critical to creating value in platform ecosystems. However, participants refrain from sharing knowledge and even engage in free-riding behavior, thereby causing the value co-destruction of the platform ecosystems. To encourage knowledge sharing among participants, it is essential to analyze the influencing factors and decision-making mechanisms of knowledge sharing in the platform ecosystems.Design/methodology/approachThe study investigated the issue of knowledge sharing among participants in platform ecosystems, based on the stochastic differential game model. Considering the uncertain factors, the Nash non-cooperative game, Stackelberg leader-follower game, and cooperative game models were proposed. By utilizing system dynamics and numerical simulations, the key influencing factors and mechanisms of knowledge sharing were deeply explored, consequently providing game solutions to achieve the Pareto optimality of the ecosystem.FindingsParticipants' innovation capability and the marginal benefits of knowledge-sharing positively impact knowledge-sharing decisions, while the environmental knowledge decay rate has a negative influence. The platform subsidy mode enhances the knowledge-sharing effect, and the collaborative cooperation mode can realize the Pareto optimization of the system.Practical implicationsThe research findings will provide theoretical support for fostering knowledge innovation and sustainable development of platform ecosystems. Managers should cultivate an innovative environment, establish fair reward mechanisms, and utilize subsidies to promote knowledge sharing, leading to higher value creation.Originality/valueUtilizing the stochastic differential game model, the study proposed various game-theoretic frameworks to analyze participants' knowledge-sharing strategies. The integration of system dynamics and numerical simulations provides a practical approach to understanding the key influencing factors and decision-making processes.

A game theoretic approach to contract-based enviro-economic coordination of wood pellet supply chains for bioenergy production

Wood pellets have gained global attention due to their economic availability and increasing demand for bioenergy as part of sustainable energy solutions. Management of the wood pellet supply chains, from feedstock harvesting to bioenergy conversion, is critical to ensure competitiveness in the energy markets. In this regard, wood pellets supply chain coordination can play a strategic role in enhancing the efficiency and reliability of bioenergy generation. This study proposes a contract-based coordination mechanism for wood pellet supply chains and compares its performance in alternative centralized and decentralized decision-making structures. A bi-level nonlinear game-theoretic approach with two economic and environmental objective functions is developed. It utilizes the concept of life cycle assessment in a Stackelberg leader–follower game to obtain the bioenergy equilibrium solutions. Further, this study examines the case of wood pellet supply chains in three remote Canadian communities. The aim is to showcase the practicality and significance of the proposed approach and interpret the findings. By focusing on these communities, the crucial role of supply chain coordination in fostering sustainable development, particularly, in the context of bioenergy generation is emphasized. The study colludes by advocating a number of avenues for future research.

Credit rating- and credit score-based carbon emission quota trading model of city dwellers

Introduction: The reduction of electricity-related carbon emissions by city dwellers (CDs) is important for China to achieve low-carbon development and sustainable energy transformation. Due to the lack of incentives for reduction, electricity-related carbon emissions from CDs are increasing year by year. To this end, this paper proposes an electricity-related carbon emission quota trading model that integrates a credit rating and credit score system, particularly for motivating CDs to actively participate in carbon emission reduction.Methods: With the history of electricity bill payment data, the density-based spatial clustering of applications with noise (DBSCAN) algorithm is used to cluster CDs, forming different clusters of CDs with different sensitivity levels to carbon emission quota prices. Thereafter, based on the total carbon emission quota and tiered electricity prices from the power company, incentive rules according to the classification result and credit scores of CDs are formulated. Under certain conditions, a leader–follower Stackelberg game between CDs and the power company is built to determine the base price of the carbon emission quota, and thereby, referring to the credit scores of CDs, floating carbon emission quota prices are offered to them in the final settlement.Results: The simulation results for an actual community in a city in China show that the proposed method can considerably reduce the carbon emissions.Discussion: The proposed credit rating and credit score system outperforms the asymmetric Nash negotiation method in terms of promoting carbon emission reduction.

Strategic incentives for comparative advertising investments in non-zero-sum competition and economic consequences

Persuasive advertising is viewed as an effective tool to affect willingness-to-pay of customers for a firm’s product. In their use of persuasive advertising, firms typically face the choice of whether to tout the benefits of their own products without mentioning rival products (self-promotional advertising) or make references to rival products while promoting their own products (comparative advertising). A concomitant (and intrinsic) issue in the use of comparative advertising is whether to portray rival products negatively or positively and the level of comparative intensity (the valence of comparative advertising). Although it is often criticized and disapproved of by business decision makers and its effectiveness has been a subject of contentious debate, comparative advertising is used prevalently by firms in consumer goods markets. In this paper, we examine the strategic incentives of competing firms to abandon their self-promotional advertising in favor of comparative advertising, the ways in which they choose a positive or negative valence in their comparative advertising, and the economic consequences of these decisions. We consider a simultaneous (and a leader-follower Stackelberg) game between two asymmetric firms vying for customers in a market that is not characterized by zero-sum competition. We also study the possibility that comparative advertising elicits positive or negative market externalities on the advertised product. We seek market- and channel-based explanations for why firms may opt to undertake comparative advertising (in lieu of self-promotional advertising) and use its constructive or combative form. We show that the use of comparative advertising can be an effective tactic in attaining certain objectives and when carefully executed in certain ways, and detrimental to not only the advertising firm but also the product category in other ways.

Cooperative Jamming-Assisted Untrusted Relaying Based on Game Theory for Next-Generation Communication Systems

In this contribution, we investigate the performance of an untrusted relaying system when Cooperative Jammers (CJs) are available. We propose two scenarios, Untrusted Relaying-aided Multiple Cooperative-Jammers-based Simultaneous Transmission (UR-MCJST) and an Untrusted Relaying-aided Multiple Cooperative Jammers-based Time-Division Transmission (UR-MCJTDT). The performances of both UR-MCJST and UR-MCJTDT schemes are investigated. The source node is the primary user (PU) that has access to a transmission bandwidth. An untrusted relay is employed for improving the reliability of the PU transmission, while CJs are invoked for improving the secrecy rate of the PU transmission. The transmission strategy for the scheme proposed in this paper is split into three unique phases. The first phase is called the broadcast-and-jamming phase, while the second phase is called the relaying phase, and the final phase is called CJ’s secondary phase, which is used for transmitting the secondary user message. More explicitly, CJs transmit noise for impairing the untrusted relay node’s ability to decode the message during the broadcast-and-jamming phase. In compensation for their help, CJs are allocated part of the available spectrum by the PU for their transmission during the secondary transmission phase. A Stackelberg leader–follower game was considered between the PU and the CJs, while a power control game that is based on the well-known Nash equilibrium is employed by the CJs. Furthermore, an investigation into the effect of invoking simultaneous transmission and time-division-based transmission scenarios during the CJs’ secondary transmission phase was carried out. Finally, we evaluated the achievable secrecy rate of the PU and the utility rate of the CJs in our proposed scheme.

Endogenous Risk Management of Prosumers by Distributionally Robust Chance-Constrained Optimization

Distributed renewable energy sources owned by prosumers have been considered an effective way to fortify grid resilience, and enhance sustainability. However, prosumers serve their own interests, which are affected by their risk preferences. Moreover, their objectives are unlikely to align with those of the society. In contrast to the conventional assumption of constant and exogenous risk preference, we propose an alternative modeling framework in which prosumers endogenously determine their risk attitudes through optimization in the power sector with a day-ahead market and real-time imbalance settlement. The decisions made by prosumers and other participants in the wholesale power market are expressed as a Stackelberg leader-follower game, resulting in a mathematical program with equilibrium constraints. The problem is formulated in a distributionally robust chance-constrained framework to account for renewable generation uncertainty. We show that the effect of uncertainty can be mitigated in the market <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">if</i> prosumers are allowed to optimally determine their risk attitudes. We also examine the case of price-taking prosumers, in which consistent implications are obtained. Therefore, our work highlights the fact that endogenizing risk preferences can be a useful tool for managing risk in the power market.

Data-driven Game-theoretic Model Based on Blockchain for Managing Resource Allocation and Vehicle Routing in Modular Integrated Construction

ABSTRACT In modular integrated construction (MiC), the resource allocation problem (RAP) adopted by construction sites and vehicle routing problem (VRP) adopted by logistics companies are highly interdependent. However, this interdependence has been overlooked in the literature. Thus, the plans determined by each problem cannot be achieved practically. Moreover, there is no existing VRP model suitable for the MiC application. This study aims to propose a VRP model suitable for MiC, investigate the interdependence between the RAP and proposed VRP model, and develop a blockchain system to secure information sharing between the RAP and VRP. The first two objectives are achieved by developing a coordinated system formulated as a leader-follower Stackelberg game model (LFSGM). This model is presented as a bi-level optimisation model and solved using a nested ant colony optimisation-based algorithm. The effectiveness of the LFSGM is validated using a real case study. The results show that using the traditional approach (i.e. a separate RAP and proposed VRP) succeeds in providing a plan for the construction sites but not for the logistics company. In contrast, the LFSGM offers applicable plans for both the construction sites and logistics company. Lastly, some managerial implications are identified and discussed.

Leader–follower Stackelberg game oriented adaptive robust constraint-following control design for fuzzy exoskeleton robot systems

To comprehensively enhance the robust tracking performance, a leader–follower Stackelberg game oriented adaptive robust constraint-following control scheme has been proposed for a class of uncertain mechanical systems. In the proposed scheme, a fuzzy set-theoretic description represents the uncertainties (possibly fast time-varying), and an adaptive robust constraint-following control ensures the robust stability. With the fuzzy uncertainty description and control performance analysis, a leader–follower game theory is employed to obtain multiple optimal gains for the proposed control scheme. Further, the existence of these optimal parameters can be verified. The proposed approach is successfully applied to a lower limb exoskeleton (LLE) robot system for rehabilitation training.

Fairness concerns and extended producer responsibility transmission in a circular supply chain

Extended producer responsibility (EPR) is an important policy tool that aims to make manufacturers financially and/or physically responsible for recycling and remanufacturing their end-of-life products. To transfer these responsibilities, a number of manufacturers require their retailers to co-finance the additional investment on EPR through wholesale price markup, possibly leading to retailers' fairness concern issues. Along this line, we consider a two-echelon circular supply chain where a manufacturer transfers its recycling and remanufacturing responsibility to a retailer by determining the wholesale price. Thus, the retailer perceives unfairness and makes the corresponding pricing decisions to resist the manufacturer. To analyze the relationship between fairness concerns and the pricing-based responsibility transmission, we establish a leader–follower Stackelberg game model for this supply chain and compare cases with and without fairness concerns. The equilibrium results indicate that the two agents can achieve a win–win situation if the manufacturer considers retailer's fairness concerns when determining the wholesale price and if the retailer sets a relatively lower price to undertake part of upfront investment to fulfill EPR. We further conduct a survey and in-depth interviews as empirical evidences to support these results derived from our models.

How does regulatory monitoring of cause marketing affect firm behavior and donations to charity?

Cause marketing (CM) typically involves for-profit firms donating part of their sales revenue to a charity, with the hope that this will increase their revenue. We argue that it is important for a regulator to monitor firms’ CM activities, and to assess how differences in the enforcement of CM laws impact the CM practice by firms. Our analytical model uses a Stackelberg leader–follower game that endogenizes the regulator’s decision to enforce CM. The firm then decides whether to truthfully declare or overstate the amount it contributes to charity (and if overstate: by how much). We find the following results in equilibrium under different conditions: (i) CM campaigns are a win–win–win situation – they increase profit for the firm while being truthful, generate larger donations for the charity, and generate a cause marketing surplus for the regulator, resulting in doing well while doing good, (ii) the best response of the firm is to be strategic, even when the regulator is strict with monitoring, (iii) the regulator itself decides not to monitor CM, even though it knows that this results in untruthful behavior by firms.When we endogenize the extent of overstatement, we find that the firm tends to be strategic by overstating donation percentage, whether the regulator is strict or not. As the proportion of unsophisticated consumers (who believe a firm’s claims, whether truthful or not) increases, the donation proportion decreases in general, and the overstatement level increases when the regulator is lenient and decreases when the regulator is strict. In equilibrium, the regulator is strict if the market size is large, and lenient otherwise. A survey with consumers supports key modeling assumptions regarding consumers' lack of knowledge of CM laws.

Reformulation of bilevel linear fractional/linear programming problem into a mixed integer programming problem via complementarity problem

The bilevel programming problem is a static version of the Stackelberg's leader follower game in which Stackelberg strategy is used by the higher level decision maker called the leader given the rational reaction of the lower decision maker called the follower. The bilevel programming problem (BLPP) is a two-level hierarchical optimisation problem and is non-convex. This paper deals with finding links between the bilevel linear fractional/linear programming problem (BF/LP), the generalised linear fractional complementarity problem (GFCP) and mixed integer linear fractional programming problem (MIFP). The (BF/LP) is reformulated as a (GFCP) which in turn is reformulated as an (MIFP). The method is supported with the help of a numerical example.

A game-theoretic analysis of the impact of government subsidy on optimal product greening and pricing decisions in a duopolistic market

In compliance with environmental and emerging international imperatives, promoting sustainable consumption and production is far due and pivotal for greening supply chains. Rampant demand for environmentally-friendly products and regulatory changes have pressured manufacturing companies to reassess their products and processes. Yet, the cost of greening and its allocation remain challenging tasks. This study investigates government subsidy strategies to encourage firms to transition to green production strategies and improve environmental quality when heterogeneous consumers are sensitive to sustainability. We consider a leader-follower Stackelberg game between two profit-maximizing firms with different green technologies (the followers) and a government (the leader). Those two competing firms sell two differentiated products to a price- and pollution-sensitive market. We first discuss the target level of greenness that can improve the environmental quality and then design the appropriate subsidy rate. We show that the government subsidy can decrease the selling price, increase the market share and the profit from greener products, and positively affect EQ. Contrary to some findings in the literature, we find that a higher subsidy rate may not always simultaneously benefit the environment, social welfare, and social surplus. Finally, we validate our structural results with various numerical examples and sensitivity analyses.

Stackelberg game based co-firing biomass with coal under carbon cap-and-trade regulation

Coal-fired power industry is under enormous pressure to accomplish carbon emission reduction targets. This paper proposes a bi-level multi-objective model for co-firing biomass with coal under carbon cap-and-trade regulation which considers a leader-follower Stackelberg game between the authority and the coal-fired power plants. The upper level regards social welfare maximization and allocation satisfaction maximization as its multiple objectives, while the lower level attempts to maximize the profits of each coal-fired power plant. The inherent uncertainty prompts the motivation for employing fuzzy set theory to characterize the uncertain parameters and determine their exact values. A case study from Shandong Province, China is provided to demonstrate the practicality and efficiency of the optimization model. [Formula: see text]-constraint method and interactive algorithm are used to solve the model, and furthermore the solutions associated with different free carbon emission quota levels and minimal allocation satisfactions have been generated to examine the influences. Based on the analysis and discussion, the methodology can meet the carbon emission reduction goals and transit to a lower-carbon power generation. It also assists the decision makers to develop desired quota allocation strategy in accordance with their attitudes and actual conditions.

Asymptotic Optimality in Byzantine Distributed Quickest Change Detection

The Byzantine distributed quickest change detection (BDQCD) is studied, where a fusion center monitors the occurrence of an abrupt event through a bunch of distributed sensors that may be compromised. We first consider the binary hypothesis case where there is only one post-change hypothesis and prove a novel converse to the first-order asymptotic detection delay in the large mean time to a false alarm regime. This converse is tight in that it coincides with the currently best achievability shown by Fellouris et al.; hence, the optimal asymptotic performance of binary BDQCD is characterized. An important implication of this result is that, even with compromised sensors, a 1-bit link between each sensor and the fusion center suffices to achieve asymptotic optimality. To accommodate multiple post-change hypotheses, we then formulate the multi-hypothesis BDQCD problem and again investigate the optimal first-order performance under different bandwidth constraints. A converse is first obtained by extending our converse from binary to multi-hypothesis BDQCD. Two families of stopping rules, namely the simultaneous d-th alarm and the multi-shot d-th alarm, are then proposed. Under sufficient link bandwidth, the simultaneous d-th alarm, with d being set to the number of honest sensors, can achieve the asymptotic performance that coincides with the derived converse bound; hence, the asymptotically optimal performance of multi-hypothesis BDQCD is again characterized. Moreover, although being shown to be asymptotically optimal only for some special cases, the multi-shot d-th alarm is much more bandwidth-efficient and energy-efficient than the simultaneous d-th alarm. Built upon the above success in characterizing the asymptotic optimality of the BDQCD, a corresponding leader-follower Stackelberg game is formulated and its solution is found.

Balancing Large-Scale Wildlife Protection and Forest Management Goals with a Game-Theoretic Approach

When adopted, wildlife protection policies in Canadian forests typically cover large areas and affect multiple economic agents working in these landscapes. Such measures are likely to increase the costs of timber for forestry companies operating in the area, which may hinder their acceptance of the policies unless harvesting remains profitable. We propose a bi-level wildlife protection problem that accounts for the profit-maximizing behavior of forestry companies operating in an area subject to protection. We consider the regulator with a wildlife protection mandate and forestry companies licensed to harvest public forest lands. We depict the relationship between the regulator and forestry companies as a leader-follower Stackelberg game. The leader sets the protected area target for each license area and the followers adjust their strategies to maximize payoffs while meeting the protection target set by the leader. The leader’s objective is to maximize the area-wide protection of spatially contiguous habitat while accounting for the followers’ profit-maximizing behavior. We apply the approach to investigate habitat protection policies for woodland caribou in the Churchill range, Ontario, Canada. We compare the game-theoretic solutions with solutions that do not consider the forest companies’ objectives and also with solutions equalizing the revenue losses among the companies.

Supplier development under cooperative and non-cooperative investment structures

PurposeThis study analyzes supplier development investment decisions under a triadic setting (two buyers and a common supplier). In a triadic setting, the supplier development investment decision of one buyer can have a spillover effect of the benefits on other buyer. Therefore, it is utmost important for the investing buyer to understand the impact of benefit spillover on other competing buyers'. Therefore, one of the purposes of this study to analyze the supplier development investment decision of buyers under two scenarios. First, under cooperative development structure where both buyers jointly invest in supplier and share equal benefits. Second, non-cooperative investment structure where both buyers individually invest in supplier development and share unequal benefits.Design/methodology/approachIn order to assess the impact of supplier development investment decisions on the profitability of buyers and the common supplier, the authors used game-theoretic approach. The authors design a Stackelberg leader-follower game where the supplier acts as Stackelberg leader and buyers follow the supplier's pricing decision to maximize their profit level. Additionally, both buyers decide either to cooperate or non-cooperate while investing in supplier development.FindingsThe results show that the cooperative investment is always an optimal strategy for buyers and supplier. Interestingly, the efficient buyer's share of investment level is lower under non-cooperative investment structure and he is better-off due to its capability of taking advantage from the other buyer's investment. However, the inefficient buyer, on the other hand, is worse-off under non-cooperative investment. Furthermore, comparative analysis between the two shows that initially, the buyer who extracts more profit because of the other buyers' development investment tends to prefer the non-cooperative development investment set up. However, after a certain point, the same buyer is better-off under cooperative development investment through cooperation, and sharing equal benefit of the supplier's development, as the supplier in turn, starts charging a higher wholesale price under non-cooperative investment case.Originality/valueTo the best of authors’ knowledge, extant literature on supplier development has mostly focused on. One supplier-one buyer; thus, the learning spillover effect has almost been unexplored. In real-life, different buyers often purchase from the shared supplier. Therefore, it is important to analyze the spillover of supplier development benefits due to investment of one buyer on other buyer and deriving the condition under which buyers would be incentivized to invest jointly or individually.

Optimal scheduling of integrated demand response-enabled integrated energy systems with uncertain renewable generations: A Stackelberg game approach

In order to balance the interests of integrated energy operator (IEO) and users, a novel Stackelberg game-based optimization framework is proposed for the optimal scheduling of integrated demand response (IDR)-enabled integrated energy systems with uncertain renewable generations, where the IEO acts as the leader who pursues the maximization of his profits by setting energy prices, while the users are the follower who adjusts energy consumption plans to minimize their energy costs. Taking into account the inherent uncertainty of renewable generations, the probabilistic spinning reserve is written in the form of a chance constraint; in addition, a district heating network model is built considering the characteristics of time delay and thermal attenuation by fully exploiting its potential, and the flexible thermal comfort requirements of users in IDR are considered by introducing a predicted mean vote (PMV) index. To solve the raised model, sequence operation theory is introduced to convert the chance constraint into its deterministic equivalent form, and thereby, the leader–follower Stackelberg game is tackled into a mixed-integer quadratic programming formulation through Karush-Kuhn-Tucker optimality conditions and is finally solved by the CPLEX optimizer. The results of two case studies demonstrate that the proposed Stackelberg game-based approach manages to achieve the Stackelberg equilibrium between IEO and users by the coordination of renewable generations and IDR. Furthermore, the study on a real integrated energy system in China verifies the applicability of the proposed approach for real-world applications.

Adversarial Kendall’s Model Towards Containment of Distributed Cyber-Threats

This work examines propagation of cyber-threats over networks under an adversarial formulation. Exploiting Kendall's birth-death-immigration model, we propose an analytical framework to describe the stochastic dynamics of cyber-threat propagation in a collection of heterogeneous sub-networks characterized by different attributes. We propose two formalisations of the problem as zero-sum games involving two adversaries: an attacker, who launches cyber-threats across the distinct sub-networks; and a defender, who tries to mitigate the threats by delivering suitable countermeasures. According to the first formalisation, the interplay between the defender and the attacker is modelled as a Stackelberg leader-follower game, while the second formalisation considers a strategic game wherein the two contenders play simultaneously without knowing the choice of the other player. We derive the equilibrium strategies for both versions of the game, and discuss a number of insightful interplays and ramifications of the different equilibrium points for the problem at hand. The equilibrium strategies depend on three fundamental attributes: i) the available resource budget of the attacker and the defender; ii) the capacity of the legitimate nodes to (unintentionally) forward the threat across the network, after they have been compromised during the propagation of the threat; iii) the intrinsic characteristics of the sub-networks, namely, their immunity to the attacks, their inertia in responding to the countermeasures, and the importance of the individual sub-networks. The relevance of the proposed solution is illustrated through a series of examples and numerical simulations.