Stackelberg Equilibrium Solution Research Articles

Orbital Impulsive Pursuit–Evasion Game Formulation and Stackelberg Equilibrium Solutions

The spacecraft pursuit–evasion game in orbit with impulsive maneuvers is investigated based on dynamic game theory. First, the game models are developed progressively from the single-impulse game to the multi-impulse game. In the models, the two-sided optimal strategies of both players form a Stackelberg equilibrium in single-impulse games and constitute a multistage Stackelberg equilibrium in multi-impulse games. Then, the optimality necessary conditions for the two equilibriums are derived and simplified as two multipoint boundary value problems (MPBVPs). To handle the complicated MPBVPs, a rolling backward induction based on the Lambert algorithm accompanied by a homotopy shooting method is proposed to solve the equilibrium solutions. Simulations showed the effectiveness of the proposed methods and verified that the obtained solutions are the (multistage) Stackelberg equilibrium. Meanwhile, the differences between the Stackelberg equilibrium and the Nash equilibrium are illustrated in detail. Finally, an analysis of the influence of the observation delay on the game results is provided.

Low-carbon economic operation strategy for multi-agent integrated energy system considering uncertainty of renewable energy power generation

The uncertainty of renewable energy output threatens the operation safety of multi-agent integrated energy system (MAIES), which makes it difficult to balance the low-carbon economic operation demands of various stakeholders. However, the existing research solely focuses on the operational strategy of multi-agent game involving integrated energy suppliers and users in deterministic scenarios, overlooking the complementary supporting role and game interaction of shared energy storage and wind farm as independent entities of interest under the instability of renewable energy power generation. Hence, this paper first establishes the optimal operation models for integrated energy system operator (IESO), user aggregator (UA), shared energy storage operator (SESO), and wind farm operator (WFO) considering the stepped carbon trading. Second, in the face of the actual situation of uncertainty of photovoltaic and wind power output, fuzzy chance-constrained programming is adopted for processing. Then, a bi-layer game equilibrium model with IESO as a leader and UA, SESO, and WFO as followers is proposed, and the existence and uniqueness of Stackelberg equilibrium solution are proved. Finally, simulation calculation is carried out based on the YALMIP toolbox in the Matlab R2023a software, and the improved particle swarm optimization algorithm and CPLEX solver are used to solve the model. The results demonstrate that the participation of SESO and WFO as independent stakeholders in the game interaction can improve the economic and environmental benefits of MAIES. The iterative optimization of demand response subsidy prices can effectively motivate users to participate in demand response, improve the ability of MAIES to cope with the uncertain risks of renewable energy generation and load, and reduce the power grid dispatch pressure.

Master–Slave Game Optimal Scheduling for Multi-Agent Integrated Energy System Based on Uncertainty and Demand Response

With the transformation of the energy market from the traditional vertical integrated structure to the interactive competitive structure, the traditional centralized optimization method makes it difficult to reveal the interactive behavior of multi-agent integrated energy systems (MAIES). In this paper, a master–slave game optimal scheduling strategy of MAIES is proposed based on the integrated demand response. Firstly, a master–slave game framework of MAIES is established with an energy management agent as leader, an energy operation agent, an energy storage agent, and a user aggregation agent as followers. Secondly, in view of the wind and solar uncertainty, the Monte Carlo method is used to generate random scenarios, and the k-means clustering method and pre-generation elimination technology are used for scenario reduction. Then, according to different flexible characteristics of loads, a multi-load and multi-type integrated demand response model including electric, thermal, and cold energy is built to fully utilize the regulation role of flexible resources. On this basis, the transaction decision-making models of each agent are constructed, and the existence and uniqueness of the Stackelberg equilibrium solution are proved. Finally, the case simulations demonstrate the effectiveness of the proposed optimal scheduling strategy of MAIES. Compared to the scenario without considering the wind and solar uncertainty and the integrated demand response, the rate of renewable energy curtailment was reduced by 6.03% and the carbon emissions of the system were reduced by 1335.22 kg in the scenario considering the proposed method in this paper.

Optimised operation of integrated community energy system considering integrated energy pricing strategy: A two-layer Stackelberg game approach

Aiming at the problems of new energy absorption and inter-subject interest conflict in the integrated community energy system, this study proposed an integrated energy pricing strategy that considered the interconversion of various energy sources and built a two-layer optimisation model of the electric-gas-heat‑hydrogen interconnection integrated community energy system. The upper-layer operator model maximises the revenue by setting energy price and selling energy to users. The lower-layer users model minimises the energy cost by actively adjusting energy usage strategy according to energy price. The two sides rationally pursue the maximisation of their respective interests, and the process of bargaining can be described by Stackelberg game. The model is also considered for combining the characteristics of elastic load and vehicle-to-grid on the demand side and operates jointly with the bus battery swapping station and two-stage power-to-gas for optimizing system operation. Through theoretical derivation, it is proved that there is a unique Stackelberg equilibrium solution for the proposed game model. Further, the two-layer optimisation model is transformed into a mixed integer quadratic programming problem by applying the Karush-Kuhn-Tucher optimal condition, linear relaxation technique, and duality theorem. Finally, the global optimal energy price is obtained by solving the converted model. The optimisation results in different scenarios show that the proposed model and method not only protect the interests of the operator and users but also improve the wind power absorption capacity of the community and reduce the load fluctuation.

A safety-guaranteed game-theoretical velocity planning for autonomous vehicles on sharp curve roads

In this paper, a safety-guaranteed game-theoretical velocity planning framework in a hierarchical manner is proposed to generate safe, ride comfort, and travel efficiency-balanced velocity for autonomous vehicles (AVs). In the upper layer, a bang-bang decision-making method is utilized to determine which planning mode to be implemented based on acceleration and jerk constraints, including a comfort mode, an efficiency mode, and a game mode. In the lower layer, asymmetric jerk limits based on comfort characteristics sensibility analysis and safe velocity simultaneously considering longitudinal and lateral stability are firstly developed to maintain ride comfort and driving safety, respectively on curve roads, especially sharp curves where vehicle stability may be not fully considered in most researches. Based on these, a non-cooperative game-theoretical velocity planning method is presented to solve the conflict between comfort mode and efficiency mode by optimizing his own objective based on the other’s action. Finally, for the sake of solving efficiency and accuracy, a chaos optimization-based algorithm (COA) is designed to solve for the Stackelberg equilibrium solution of the bilevel game optimization problem. Three experimental tests are carried out to comprehensively demonstrate the effectiveness, robustness, and real time of the proposed framework. The results show that the proposed method can provide the great performance of ride comfort, travel efficiency, and longitudinal-lateral stability in real time in the velocity planning process.

Master–slave game-based optimal scheduling of community-integrated energy system by considering incentives for peak-shaving and ladder-type carbon trading

To alleviate the challenges posed by high energy consumption, significant carbon emissions, and conflicting interests among multiple parties in a community-level microgrid, the authors of this study propose a master–slave game-based optimal scheduling strategy for a community-integrated energy system (CIES). First, we analyze the decision variables and revenue-related objectives of each stakeholder in the CIES, and use the results to construct a framework of implementation. Second, we develop a model to incentivize peak regulation and a ladder-type carbon trading model that consider the correlation between the load owing to residential consumers, the load on the regional grid, and the sources of carbon emissions. Third, we propose a master–slave game-based mechanism of interaction and a decision-making model for each party to the game, and show that it has a Stackelberg equilibrium solution by combining genetic algorithms and quadratic programming. The results of evaluations showed that compared with an optimization strategy that considers only the master–slave game, the proposed strategy increased the consumption surplus of the user aggregator by 13.65%, the revenue of the community energy operator by 7.95%, increased the revenue of the energy storage operator, reduced CO2 emissions by 6.10%, and adequately responded to peak-cutting and valley-filling by the power grid company.

Incentive Mechanism for Differentially Private Federated Learning in Industrial Internet of Things

Federated learning (FL) is a newly emerging distributed machine learning paradigm, whereby a server can coordinate multiple clients to jointly train a learning model by using their private datasets. Many researches focus on designing incentive mechanisms in FL, but most of them cannot allow that clients flexibly determine privacy budgets by themselves. In this article, we propose a privacy-preserving incentive mechanism (NICE) based on differential privacy (DP) and Stackelberg game for FL systems in industrial Internet of Things. First, we design a flexible privacy-preserving mechanism for NICE, in which clients can add a Laplace noise into the loss function according to a customized privacy budget. Under this mechanism, we design two incentive utility functions for the server and clients. Next, we model the utility optimization problems as a two-stage Stackelberg game by seeing the server as a leader and the clients as followers. Finally, we derive an optimal Stackelberg equilibrium solution for both the stages of the whole game. Based on this solution, NICE can make the server and all clients achieve their maximum utilities simultaneously. In addition, we conduct extensive simulations on real-world datasets to demonstrate the significant performance of the proposed mechanism.

ANALYSIS OF INTERNATIONAL OIL PRODUCTION COMPETITON USING GAME THEORY

Uluslararası ham petrol piyasasında oyuncuların OPEC (Petrol İhraç Eden Ülkeler Örgütü) üye ülkeleri ile OPEC’e üye olmayan petrol üreten ülkeler (non-OPEC) oldukları varsayımı yaygındır. Söz konusu oyuncuların kâr maksimizasyonu ve/veya petrol piyasalarını kontrol edebilme hedefiyle üretim seviyesi ve fiyat gibi çeşitli strateji kümeleri kullanarak rekabet eğilimine girmeleri söz konusu piyasanın oyun modelleri ile analiz edilebilirliğini ortaya koymaktadır. Bu çalışmanın amacı, OPEC ve non-OPEC ülkelerinin rekabete dayalı petrol üretim miktarlarını oyun teorisi temelinde analiz etmektir. Bu amaç doğrultusunda, oyunculara ait 1972-2019 dönemini kapsayan üretim miktarları ve ham petrol fiyat serileri kullanılmıştır. Petrol piyasalarındaki rekabetin varlığına işaret eden üretim fonksiyonlarına ait katsayılar Tam Bilgi En Küçük Kareler (FMOLS- Fully Modified Ordinary Least Square) yöntemi ile tahmin edilmiştir. Elde edilen modellerden Cournot-Nash ve Stackelberg denge çözümleri hesaplanmıştır. Cournot-Nash denge düzeyine göre Stackelberg denge düzeylerinde lider oyuncunun üretiminin artığı, takipçi oyuncunun üretim düzeyinin ise azaldığı görülmüştür. Bulgular Cournot-Nash denge çıktılarının Stackelberg denge çıktılarından daha yüksek olduğu çalışmalar ile paralellik göstermiştir. Oyuncuların üretim miktarı ve fiyat stratejileri vasıtasıyla optimum üretim düzeyine ulaşmaları için Cournot-Nash dengesinde kalmaları önerilmektedir.

“Electromagnetic barrier” assisted dynamic spectrum access in satellite internet communication confrontation

In this paper, the dynamic spectrum access is investigated in satellite internet. Firstly, to describe the confrontation characteristics of the electromagnetic environment, two opposing teams denoted as the blue team (BT) and red team (RT), are designed. In addition, an “electromagnetic barrier” jamming strategy is proposed to achieve the effect of promoting approach defence with the attack. Precisely, the “electromagnetic barrier” can significantly weaken the communication ability of enemies (i.e. RT) while protecting the (i.e. BT) communication quality. Secondly, internal interference, external interference and malicious jamming of BT and RT are considered, and expected weighted aggregate interference and jamming (EWAIJ) are introduced to integrate these interference. Moreover, minimising interference is taken as the optimisation goal. Thirdly, Game theory is introduced to describe the combinatorial optimisation problem. The process of each team finding the optimal spectrum access strategy is proved to be an exact potential energy game. These two sub-games together constitute the Stackelberg game framework. Finally, a distributed hierarchical confrontation channel selection algorithm (DHCCSA) is proposed to find the Stackelberg equilibrium solution. The simulation results show that the proposed algorithm can converge to a better effect compared with other algorithms. In addition, with the assistance of the “electromagnetic barrier,” BT is more robust than RT in terms of convergence rate, total utility, and total network throughput.

Stackelberg equilibrium water allocation patterns in shallow groundwater area under uncertainties

The hierarchical agricultural water resource management problem in shallow groundwater area is expected to be solved through establishing a practical and feasible mathematical model. This study proposes a multi-level programming model with considerations of process-based crops growth model and uncertainties. The model is solved by a fuzzy approach that ensures the decision-making proceeds from top to bottom sequentially and the leader supervises the solution search direction. The leeway given by each level decision maker expands the decision-making space and lets Stackelberg equilibrium solution more satisfactory. The model is applied in Hetao Irrigation District in China. After optimization, the Stackelberg equilibrium water allocation pattern and satisfaction degrees of each level decision makers can be identified. Summarized conclusions are as follows: (1) under uncertainties of available water, the water diversion is allocated in a similar pattern, while the groundwater allocation has larger variabilities; (2) when available water decreases, local farmers can improve crop production through modifying or optimizing irrigation water amount in each growth period; (3) the combined use of water diversion and groundwater is expected to become an effective approach for achieving a win–win situation between government managers and local farmers. The proposed model performs well in generating Stackelberg equilibrium solutions satisfied by multi-level decision makers. It can be extended to provide decision makers a more complete decision support system and also has potential to be applied in other similar areas.

When Virtual Network Operator Meets E-Commerce Platform: Advertising via Data Reward

In China, some e-commerce platform (EP) companies such as Alibaba and JD are now allowed to partner with network operators (NOs) to act as virtual network operators (VNOs) to provide mobile data services for mobile users (MUs). However, it is a question worth researching on how to generate more profits for all network players, with EP companies being VNOs, through appropriate integration of the VNO business and the companies' own e-commerce business. To address this issue, in this work we propose a novel incentive mechanism for advertising via mobile data reward, and model it as a three-stage static Stackelberg game. We obtain the closed-form optimal solution of the Nash equilibrium by backward induction. Besides, for the scenario lack of knowledge on the interaction between the NO and VNO in a dynamic game, we propose a deep Q-network (DQN) based algorithm to derive the optimal strategies of the NO and VNO. Simulation results show impact of system parameters on the utilities of game players and social welfare. We also study the impact of system parameters on different algorithms and discover that the proposed DQN-based algorithm can learn a good strategy as compared with the Stackelberg equilibrium solution.

Stochastic sequential supply chain management system: with a solution approach using the systematic sampling evolutionary method

Supply chain management describes a complex sequence of strategies implemented by multiple decision makers to transform raw materials into products and deliver to the market. Mathematical formulations of such problems involve hierarchical games with some form of stochastic properties in the problem definition. Such kind of mathematical problems are generally known to be NP-hard and are challenging to solve. This paper considers a general form of supply chain management problem with various forms of model formulations and analysis. Moreover, a solution approach based on a systematic sampling evolutionary method is also proposed to solve any form of such problem definitions to obtain a Stackelberg equilibrium or Stackelberg-Nash equilibrium solution. The convergence of the solution approach is shown. The reliability of the proposed method is checked. In addition to this, the algorithm is implemented on carefully constructed stochastic supply chain management problems and solutions to these problems are presented.

A Novel Joint Dataset and Incentive Management Mechanism for Federated Learning Over MEC

In this study, to reduce the energy consumption for the federated learning (FL) participation of mobile devices (MDs), we design a novel joint dataset and incentive management mechanism for FL over mobile edge computing (MEC) systems. We formulate a Stackelberg game to model and analyze the behaviors of FL participants, referred to as MDs, and FL service providers, referred to as MECs. In the proposed game, each MEC is the leader, whereas the MDs are followers. As the leader, to maximize its own revenue by considering the trade-off between the cost of providing incentives and the estimated accuracy attained from an FL operation, each MEC provides full incentives to the MDs for the participation of each FL task, as well as the target accuracy level for each MD. The suggested total incentives are allocated over MDs’ proportion to the amount of dataset applied for local training, which indirectly affects the global accuracy of the FL. Based on the suggested incentives, the MDs determine the amount of dataset used for the local training of each FL task to maximize their own payoffs, which is defined as the energy consumed from FL participation and the expected incentives. We study the economic benefits of the joint dataset and incentive management mechanism by analyzing its hierarchical decision-making scheme as a multi-leader multi-follower Stackelberg game. Using backward induction, we prove the existence and uniqueness of the Nash equilibrium among MDs, and then examine the Stackelberg equilibrium by analyzing the leader game. We also discuss extensions of the proposed mechanism where the MDs are unaware of explicit information of other MD profiles, such as the weights of the revenue as a practical concern, which can be redesigned into the Stackelberg Bayesian game. Finally, we reveal that the Stackelberg equilibrium solution maximizes the utility of all MDs and the MECs.

A Game Model for Medical Service Pricing Based on the Diagnosis Related Groups.

Background: Currently there are various issues that exist in the medical institutions in China as a result of the price-setting in DRGs, which include the fact that medical institutions tend to choose patients and that the payment standard for complex cases cannot reasonably compensate the cost.Objective: The main objective is to prevent adverse selection problems in the operations of a diagnosis-related groups (DRGs) system with the game pricing model for scientific and reasonable pricing.Methods: The study proposes an improved bargaining game model over three stages, with the government and patients forming an alliance. The first stage assumes the alliance is the price maker in the Stackelberg game to maximize social welfare. Medical institutions are a price taker and decide the level of quality of medical service to maximize their revenue. A Stackelberg equilibrium solution is obtained. The second stage assumes medical institutions dominate the Stackelberg game and set an optimal service quality for maximizing their revenues. The alliance as the price taker decides the price to maximize the social welfare. Another Stackelberg equilibrium solution is achieved. The final stage establishes a Rubinstein bargaining game model to combine the Stackelberg equilibrium solutions in the first and second stage. A new equilibrium between the alliance and medical institutions is established.Results: The results show that if the price elasticity of demand increases, the ratio of cost compensation on medical institutions will increase, and the equilibrium price will increase. The equilibrium price is associated with the coefficient of patients' quality preference. The absolute risk aversion coefficient of patients affects government compensation and total social welfare.Conclusion: In a DRGs system, considering the demand elasticity and the quality preference of patients, medical service pricing can prevent an adverse selection problem. In the future, we plan to generalize these models to DRGs pricing systems with the effects of competition of medical institutions. In addition, we suggest considering the differential compensation for general hospitals and community hospitals in a DRGs system, in order to promote the goal of hierarchical diagnosis and treatment.

Market Analysis of Distributed Learning Resource Management for Internet of Things: A Game-Theoretic Approach

In this article, to meet a delay requirement for data analytics from Internet-of-Things (IoT) devices, we design a novel market model of the distributed learning resource management mechanism for multiple mobile-edge computing (MEC) operators. We consider a hybrid architecture (cloud-MEC) for the distributed learning, which is also known as “federated learning” as one of the practical examples, where a coordinator at the cloud coordinates IoT sensors to efficiently distribute their sensing data over multiple MECs. In this sense, multiple MECs receive a shared model from the coordinator and conduct local training of received partial sensing data from IoT sensors. Then, the coordinator at the cloud merges returned local training results from MECs and generates a global model. To model a hierarchical decision-making structure as a market behavior, we formulate and solve a Stackelberg game model. Specifically, in the case of MEC operators as leaders, we design a pricing scheme for MEC operators to obtain its maximum utility by considering a tradeoff between the revenue and energy consumption. Then, as a follower, while the coordinator aims at achieving a balance between the satisfaction attained from the distributed learning and the costs, it follows the MEC operators’ decisions by coordinating IoT sensors to distribute their sensing data over MECs. A unique Stackelberg equilibrium (SE) point is given as a closed form. Finally, we reveal that the SE solution maximizes the utility of all market participants. This game-theoretic study demonstrates that there is an incentive for utilizing multiple MECs to achieve better satisfaction of IoT services in distributed learning.

Multi-Operator Spectrum Sharing for Massive IoT Coexisting in 5G/B5G Wireless Networks

With a massive number of Internet-of-Things (IoT) devices connecting with the Internet via 5G or beyond 5G (B5G) wireless networks, how to support massive access for coexisting cellular users and IoT devices with quality-of-service (QoS) guarantees over limited radio spectrum is one of the main challenges. In this paper, we investigate the multi-operator dynamic spectrum sharing problem to support the coexistence of rate guaranteed cellular users and massive IoT devices. For the spectrum sharing among mobile network operators (MNOs), we introduce a wireless spectrum provider (WSP) to make spectrum trading with MNOs through the Stackelberg pricing game. This framework is inspired by the active radio access network (RAN) sharing architecture of 3GPP, which is regarded as a promising solution for MNOs to improve the resource utilization and reduce deployment and operation cost. For the coexistence of cellular users and IoT devices under each MNO, we propose the coexisting access rules to ensure their QoS and the priority of cellular users. In particular, we prove the uniqueness of the Stackelberg equilibrium (SE) solution, which can maximize the payoffs of MNOs and WSP simultaneously. Moreover, we propose an iterative algorithm for the Stackelberg pricing game, which is proved to achieve the unique SE solution. Extensive numerical simulations demonstrate that, the payoffs of WSP and MNOs are maximized and the SE solution can be reached. Meanwhile, the proposed multi-operator dynamic spectrum sharing algorithm can support more than almost 40% IoT devices compared with the existing no-sharing method, and the gap is less than about 10% compared with the exhaustive method.

A Stackelberg game approach to multiple resources allocation and pricing in mobile edge computing

Mobile edge computing is a new paradigm that can enhance the computation capability of end devices and alleviate communication traffic loads during transmission. Mobile edge computing is highly useful for emerging resource-hungry mobile applications. However, a key challenge for mobile edge computing systems is multiple resources allocation between Mobile Edge Clouds (MECs) and End Users (EUs), especially for multiple heterogeneous MECs and EUs. To address this problem, we propose a Stackelberg game-based framework in which EUs and MECs act as followers and leaders, respectively. The proposed framework aims to compute a Stackelberg equilibrium solution in which each MEC achieves the maximum revenue while each EU obtains utility-maximized resources under budget constraints. We decompose the multiple resources allocation and pricing problem into a set of subproblems in which each subproblem only considers a single resource type. The Stackelberg game framework is constructed for each subproblem wherein each player (i.e., an EU) can selfishly maximize its utility by selecting an appropriate strategy in the strategy space. We prove the existence of the subgame Stackelberg equilibrium and develop algorithms to determine the Stackelberg equilibrium for each resource type, including an optimal demand computation algorithm, to determine the best resource demand strategy for an EU and an iterative algorithm to find an equilibrium price. The equilibrium solutions of all subgames constitute the equilibrium solution of the original problem. We also conduct simulation experiments of our game, such as numerical data for the Stackelberg equilibrium, numerical data for the convergence of the Stackelberg equilibrium, and numerical data as the system size increases. Finally, we demonstrate that an EU with idle resources can play the role of an MEC.

Strategic port competition in multimodal network development considering shippers’ choice

Ports are developing dry ports in overlapping hinterland to gain competitive edges and relieve congestion. However, the impact of sequential port competition in multimodal network design where two ports sequentially locate dry ports considering dry port capacity is under-investigated. The possible correlation of inland transport routes to a port due to unobserved attributes and parameter calibration are overlooked in shippers' choice model. We formulate a nested logit model to describe the joint choice of shippers on port, transport mode and dry port considering dry port service and range, custom clearance time, reliability, freight shipment size, and ship call frequency using calibrated parameters based on data collected by revealed preference and stated preference techniques. A Stackelberg game theoretical model is established for the two ports, and two algorithms are adapted to obtain Stackelberg equilibrium solution. We find that inland transport cost, inland transit and custom clearance time are more appreciated in multimodal transport than in road transport. Reliability and large-scale freight shipment are key factors for choosing multimodal transport. When the dry port location strategy of one port in Stackelberg equilibrium is its only dominant strategy in Nash game, Stackelberg equilibrium solution would be the same as pure Nash equilibrium solution. Leader port's dry port location strategy in Stackelberg equilibrium could be different from its dominant strategies in Nash game. Dry port location strategy could be affected by port service charge and waiting time. The methodological approach and outcomes can provide a framework for competitive ports sequentially developing multimodal networks in overlapping hinterland and help identify modal shift policies towards sustainable inland transport alternatives.

Three Dynamic Pricing Schemes for Resource Allocation of Edge Computing for IoT Environment

With the widespread use of Internet of Things (IoT), edge computing has recently emerged as a promising technology to tackle low-latency and security issues with personal IoT data. In this regard, many works have been concerned with computing resource allocation of the edge computing server, and some studies have conducted to the pricing schemes for resource allocation additionally. However, few works have attempted to address the comparison among various kinds of pricing schemes. In addition, some schemes have their limitations such as fairness issues on differentiated pricing schemes. To tackle these limitations, this article considered three dynamic pricing mechanisms for resource allocation of edge computing for the IoT environment with a comparative analysis: BID-proportional allocation mechanism (BID-PRAM), uniform pricing mechanism (UNI-PRIM), and fairness-seeking differentiated pricing mechanism (FAID-PRIM). BID-PRAM is newly proposed to overcome the limitation of the auction-based pricing scheme; UNI-PRIM is a basic uniform pricing scheme; FAID-PRIM is newly proposed to tackle the fairness issues of the differentiated pricing scheme. BID-PRAM is formulated as a noncooperative game. UNI-PIM and FAID-PRIM are formulated as a single-leader–multiple-followers Stackelberg game. In each mechanism, the Nash equilibrium (NE) or Stackelberg equilibrium (SE) solution is given with the proof of existence and uniqueness. Numerical results validate the proposed theorems and present a comparative analysis of three mechanisms. Through these analyses, the advantages and disadvantages of each model are identified, to give edge computing service providers guidance on various kinds of pricing schemes.

The Optimal Decisions for Formation International Joint Ventures under Uncertainty

This paper develops a model in which the two parties to an international joint venture (IJV) determine the optimal cost ratio and profit share and exercise the jointly held option to enter into an IJV in uncertain situations. In our setting, the two partners make their decisions sequentially under a two-stage Stackelberg game framework, where the risk-neutral multinational company as the leader first designs the cost and equity allocation, and then the risk-averse local firm follows by choosing the optimal timing at which to exercise the joint venture option. The Stackelberg equilibrium solutions to the model are obtained by using a mix of game theory and real options. The results of the numerical analysis show that despite the multinational company's dominant position in the IJV, its equity share is not always higher than that of the local company and is closely related to the expected growth rate, risk aversion level and market volatility. The cost ratio of the foreign partner will increase correspondingly with its equity ratio unless it faces much more uncertainty in the environment. In addition, an increase in risk aversion leads to the local company sharing a higher proportion of the equity and costs of the joint venture partnership. The findings of the study can provide scientific guidance and suggestions on the transnational joint venture for overseas investment, especially in the “One Belt and One Road” initiative.