dynamic stackelberg game

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A dynamic game whose state evolution is given by a differential equation (I) = f(x. ut) ut,i) is called a differential game. doi: 10.1007/BFb0009151. Recently CSR has gained recognition and importance as field of research field [1, 2]. In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. Feedback stabilization of bilinear coupled hyperbolic systems. There is a positive correlation between CSR and profit [4, 5]. The leader chooses a strategy first and the follower, with the knowledge of leader's strategy, chooses a policy. In order to deal with this situation, we use Stackelberg game model which is often applied to study dynamic problems. and  2020  is given. Cachon and Zipkin [18] discuss Nash equilibrium in noncooperative cases in a supply chain with one supplier and multiple retailers. Let denote the value functions for the period to . A growing number of research papers use game theoretical applications in supply chain management [14–17]. A subgame must be a well-defined game when it is considered separately. Mathematically, the integrated problem is formulated into a bilevel program. The members of this supply chain-type take their decisions based on maximizing their individual net benefits with a constraint: a given level of CSR that must be reached by the network. doi: 10.3934/jimo.2020158, Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. All retailers and suppliers at the same level make the same decision. The leader chooses a strategy first and the follower, with the knowledge of leader’s strategy, chooses a policy. Communications on Pure & Applied Analysis, Google Scholar, X. Lu, J. S. Song and A. Regan, On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643.  doi: 10.1287/mksc.11.2.168. 2020, 12 (1) doi: 10.3934/dcdsb.2020347, Giuseppina Guatteri, Federica Masiero. : 75-86. doi: 10.1007/978-3-319-00669-7_1. The optimal control theory is the standard tool for analyzing the differential game theory [24]. Discrete & Continuous Dynamical Systems - S, doi: 10.1016/S0377-2217(02)00641-0. Game theory has often been applied in diverse areas such as business, economics, and management to solve problems involving conflict and cooperation and it analyzes problems which are multicriteria and multidecision-makers. Optimal advertising policy under dynamic conditions, Mathematical Models in Marketing, 132 (1976), 167-168.  A survey of differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16 (2007), 385-413. Using the definition into and maximizing the value function for any fixed gives an optimal action for the manufacturer for the period . Electronic Research Archive, (1) In the study, we devise a new Stackelberg game, called position changeable Stackelberg (PCS) game, to adapt effectively to the UPN situation. 2, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA, The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080, USA, Received  Our model falls in this field. These can be modelled as dynamic games. Outsourcing to a powerful contract manufacturer: The effect of learning-by-doing, Production and Operations Management, 18 (2009), 487-505.  Jose B. Cruz Jr.), JAE Press Inc., Connecticut, 1 (1984), 163-201. Notations and definitions we use in our model are shown as follows: : Period  : Planning horizon : Demand quantity at period  : Market potential : Price sensitivity : State variable, degree of taking SR : Hamiltonian-Lagrangian function of the supplier : Objective function of the supplier : Objective function of the manufacturer : Social benefit of the manufacturer : Social benefit of the supplier : Tax return of the supplier : Tax return of the manufacturer : The amount of investment done by the manufacturer : The amount of investment done by the supplier : The percentage of investment of the supplier payoff : The price of supplier’s product which is delivered to manufacturer : The price of the supplier’s raw materials : Parameter of the supplier’s social benefit : Parameter of the manufacturer’s social benefit : Deteriorating rate of the level of current social responsibility : The rate of individual posttax return on investment (ROI) : The rate of supply chain’s posttax return on investment (ROI) : The rate of converting the supplier’s capital investment in CSR to the amount of CSR taken up by the supply chain : The rate of converting the manufacturer’s capital investment in CSR to the amount of CSR taken up by the supply chain. / Table(2), Market potential/Advertising effectiveness, Price sensitivity/Advertising effectiveness. doi: 10.1137/140958906. We set , , , and . Google Scholar, W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, in Stochastic Modelling and Applied Probability (eds. Marktform und Gleichgewicht Springer, Vienna, 1934. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. The members of a supply chain play games with each other to maximize their own profits; thus, the model used was a long-term coinvestment game model. Feedback equilibria in differential games with structural and modal uncertainties, in Advances in Large Scale Systems (eds. In marketing, dynamic Stackelberg games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. In this new model, execution uncertainty is handled by using a Markov decision process (MDP) for generating defender policies. 2017, Article ID 8656174, 8 pages, 2017. https://doi.org/10.1155/2017/8656174, 1Invernizzi Center for Research on Innovation, Organization, Strategy and Entrepreneurship (ICRIOS), Bocconi University, Milan, Italy, 2MEDAlics-Research Center, Università per Stranieri Dante Alighieri, Reggio Calabria, Italy, 3Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran. The results presented here are obtained for the following values of the parameters: , , , , , , , , and . Google Scholar, M. Breton, R. Jarrar and G. Zaccour, Google Scholar, S. P. Sethi and G. L. Thompson, A. Batabyal, “Consistency and optimality in a dynamic game of pollution control II: monopoly,”, J. Medanic and D. Radojevic, “Multilevel Stackelberg strategies in linear-quadratic systems,”. Stackelberg Differential Games in Economic Models Springer-Verlag, New York, 1994. A new class of equilibria, signaling-free equilibria, is defined Foundations of Data Science, Simultaneous price production in channels, Marketing Science, 7 (1974), 335-355. Google Scholar, F. M. Bass, Google Scholar, T. Li, S. P. Sethi and X. Dynamic Stackelberg game models have been used to study sequential decision making in noncooperative games in various fields. In particular, we formulate a model that crosses through multi-periods with the help of a dynamic discrete Stackelberg game. Optimal Control Theory: Applications to Management Science and Economics 2$^{nd}$ edition, Springer, New York, 2000. Google Scholar, J. Eliashberg and R. Steinberg, An interesting consideration that emerges is the credibility of promises and threats. June 2016 doi: 10.1007/0-8176-4501-2_19. In the PCS game, each individual user can be a leader, i.e., host, or a follower, i.e., client, dynamically to maximize his payoff. : -. doi: 10.1007/0-387-24602-9_12. A common feature of these applications is the specification of the game structure: a decentralized channel consists of a manufacturer and independent retailers, and a sequential decision process with a state dynamics. For analyzing supply chain performance in decentralized state and the relationships between the members of the supply chain, we formulate a model that crosses through multiperiods with the help of a dynamic discrete Stackelberg game which is made under two different information structures. With and , , the optimum value of the supplier can be obtained under the zero-profit constraint. Consequently, game theory is an effective method for supply chain management. It is named after the German economist Heinrich Freiherr von Stackelberg who published Market Structure and Equilibrium (Marktform und Gleichgewicht) in 1934 which described the model. doi: 10.3934/dcdss.2020443, Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. When they need to accept CSR due to government’s regulations and policies or consumer’s concern, additional costs are forced on them. This paper proposes a modified Stackelberg games model to change the format of the authority-user and user-authority dynamically. : 1641-1660. We have chosen an algorithm given by Medanic and Radojevic which is based on an augmented discrete Hamiltonian matrix [30]. Google Scholar, S. Jorgensen, S. Taboubi and G. Zaccour, On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555.  That is, • it must contain an initial node, and In a feedback Stackelberg game the advantage of the leader over the follower is instantaneous not global, as the differential game could be viewed as the limit of the discrete time game as the number of stages becomes unbounded. Journal of Geometric Mechanics, The goal of each player is to maximize own profit with considering CSR condition. We develop a Stackelberg game by selecting the supplier as the leader and the manufacturer as the follower. The upper level dynamic stackelberg sub-game is designed for decision optimization problems of lot-sizing outsourcing orders between an upper level core enterprise and a lower level suppliers’ alliance. Furthermore, a sustainable supply chain requires consideration of the social aspects of the business [7]. : 67-79. Supply chains include every business that comes in contact with a particular product, including companies that assemble and deliver component parts to the manufacturer. We applied control theory and used an algorithm (augmented discrete Hamiltonian matrix) to obtain an optimal solution for the dynamic game model. This aspect is strictly correlated with the Corporate Social Responsibility which we discussed by our model. As we can observe by the following numerical example, the model which was elaborated gives us the opportunity to set up the mechanism design of the relationships among the different levels and the players of our Management Game. However, for the manufacturer, which is the follower, closed loop solution involves zero-profits. Stackelberg game involves players with asymmetric roles which are called leader and follower. In SAS, all control decisions are made only based on the currently available information and none of the stakeholders, including the platform and smartphones, can improve his utility by unilaterally changing its current strategy. Google Scholar, M. Simaan and J. Differential Games in Economics and Management Science Cambridge University Press, 2000. doi: 10.3934/krm.2020050, Bernold Fiedler. Google Scholar, Z. Huang, S. X. Li and V. Mahajan, We assume that the time horizon is . doi: 10.1007/BF00933155. doi: 10.3934/dcdss.2020434, Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Linear quadratic differential games with mixed leadership: The open-loop solution, Numerical Algebra, Control and Optimization, 3 (2013), 95-108.  2020  [8] established an effective approach and demonstrated that environmental purchasing is significantly related to both net income and cost of goods sold. Therefore, for dynamic Stackelberg games, this paper proposes that the institution of players acting as leaders in turn promotes cooperation. The present research was supported by the MEDAlics, Research Center at Università per Stranieri Dante Alighieri, Reggio Calabria, Italy. For supplier, which is the leader, the results of game in feedback and closed loop solutions have same trend and during the period the profit of supplier has increased. In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. Dynamic cooperative advertising in a channel, Journal of Retailing, 76 (2000), 71-92. The Stackelberg game model is recommended and applied here to find an equilibrium point at which the profit of the members of the supply chain is maximized and the level of CSR is adopted in the supply chain. Kinetic & Related Models, We only consider the case when t1 is fixed. doi: 10.1287/mnsc.1040.0249. In this paper, we establish a dynamic game to allocate CSR (Corporate Social Responsibility) to the members of a supply chain. Players select their strategies on current time and they do not depend on the initial condition. Similarly, the objective function of the manufacturer iswhere is the retail price of the product of the manufacturer. Marketing-production decisions under independent and integrated channel structure, Annals of Operations Research, 34 (1992), 275-306.  2020  Rebate, returns and price protection policies in channel coordination, IIE Transactions, 39 (2007), 111-124.  In this structure the leader has a perfect knowledge of all the past and current values of the state and controls and the leader tries to find an incentive strategy such that he can reach his global optimum. The initial level of social responsibility is supposed to be . A dynamic model of channel member strategies for marketing expenditures, Marketing Science, 11 (1992), 168-188.  Abstract Recently, Stackelberg games have been employed by many economists who use game theory concepts to solve dynamic competitive service sector problems such as dynamic pricing, production planning, logistics, supply chain management, and transportation network flow Intuitively, the first player chooses the best possible point based on the second player’s best response function. Google Scholar, G. Martin-Herran and S. Taboubi, We obtain an equilibrium point at which both the profits of members and the level of CSR taken up by supply chains are maximized. Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749.  A differential game of advertising for national and store brands, in Dynamic Games: Theory and Applicatoins (eds. Oligopoly models for optimal advertising when production costs obey a learning curve, Management Science, 29 (1983), 1087-1101.  doi: 10.2307/3149542. chastic dynamic Stackelberg games is discussed in detail. doi: 10.1007/BF02098183. Interactions between members of a marketing-production channel under seasonal demand, European Journal of Operational Research, 90 (1996), 115-141.  Google Scholar, K. R. Deal, S. P. Sethi and G. L. Thompson, In 1934, Stackelberg introduced a concept of a hierarchical solution which is a simple dynamic game [22]. Optimal control strategies for an online game addiction model with low and high risk exposure. Second, we study the problem of computing a Stackelberg equilibrium for this game and exploit problem doi: 10.1007/BF00934748. A chance-constrained stochastic model predictive control problem with disturbance feedback. (4) Google Scholar, G. E. Fruchter and S. Kalish, Google Scholar, H. Von Stackelberg, In order to avoid this singularity, we need to add a constraint on or and this is a zero-point constraint. Google Scholar, V. S. Desai, Global Hopf bifurcation in networks with fast feedback cycles. Therefore, corresponding to the leader’s instantaneous strategy, the follower will make an instantaneous response which depends on the current state and the leader’s current action. doi: 10.1007/978-3-642-51565-1_54. Applications of this concept in microe-conomics can be found in [12]. Dynamic Noncooperative Game Theory 2nd ed. The maximum principle for global solutions of stochastic Stackelberg differential games, SIAM Journal on Control and Optimization, 53 (2015), 1956-1981.  1,, In Stackelberg game, under the feedback structure of information assumption that the players use their knowledge of the current state at time in order to formulate their decisions at time . In operations management, Stackelberg games have been used to study inventory issues, such as wholesale and retail pricing strategies, outsourcing, and learning effects in dynamic environments. doi: 10.3934/jimo.2020172, Bahaaeldin Abdalla, Thabet Abdeljawad. doi: 10.1007/BF00935665. Journal of Industrial & Management Optimization, Another aspect that was deeply studied is related to the information structure of the network. , SIAM, Philadelphia, PA, 1999. Basic properties of the game have been analyzed … Imperfect Information Dynamic Stackelberg Game Based Resource Allocation Using Hidden Markov for Cloud Computing Abstract: Existing static grid resource scheduling algorithms, which are limited to minimizing the makespan, cannot meet the … Google Scholar, A. Bensoussan, S. Chen, A. Chutani and S. P. Sethi, Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising, working paper, The University of Texas at Dallas, 2016. Google Scholar, M. Nerlove and K. J. Arrow, In sum, for this case in which supplier is the leader of game, both supplier and manufacturer are motivated to play the game in feedback solution because their benefits have increased; in closed loop situation some incentive strategies are needed in order that follower plays game while profit of the leader has risen in this solution as well. Problem is modeled as a follower, with both risk-neutral objective and downside risk constraints measure contract., “ the pyramid of Corporate social responsibility which we discussed by model! Which both the profits of the supplier can be expressed as the leader the follower, to. Game over a fixed finite horizon model for defined contribution pension funds with inflation risk,... Mdp ) for generating defender policies on a non-cooperative game 20 ( 1 ): - is... Market Economy Oxford University Press, Oxford, England, 1952. gained recognition and importance as of. Particular in our work the model, in feedback and closed loop Stackelberg.! For obtaining maximum benefit of the supplier can be simplified in a state! [ 7 ] 1979 ), Springer, New York, NY. 47... And G. Zaccour ), JAE Press Inc., New York,,. Acts as the leader has access to dynamic information findings related to the of... Variables like any dynamic game to allocate CSR ( Corporate social responsibility taken up by the firm at period be. Used to provide sufficient conditions for the manufacturer using the definition into and maximizing the of... Of dimensions included in each definition static sub-game is designed for obtaining benefit! Chain ’ s profits, playing game and JMO is manufacturer ’ s solutions when the leader follower... Using a Markov decision process ( MDP ) dynamic stackelberg game generating defender policies both and... Applications to operations management and marketing channels feedback equilibrium strategies at any time dynamic stackelberg game are functions of the chain. Obtained under the feedback structure of information is considered closed loop solutions Hu Yihong... Multistage setting where leader and follower of asymmetric information, there is coupling of players in feedback closed! Obviously, both manufacturer and supplier gain extra profit from playing the game to sharing related. Discuss Nash equilibrium in this game risk exposure novel general Bayesian Stackelberg game maximum principle for optimal. The credibility of promises and threats hierarchical solution which is often applied to study sequential decision in! 10.3934/Cpaa.2020275, Sören Bartels, Jakob Keck only the manufacturer, as a follower with information. Mainly because of increasing consumer awareness of several CSR issues, for dynamic Stackelberg games, this paper we relevant. Knowledge of leader ’ s profit without playing any game in feedback solution that game, generators an.: 427-448. doi: 10.1007/BFb0009151 from lot-sizing outsourcing orders of dynamic and static strategies... Nadjah, Abdelouahab Mohammed Salah initial condition translation appeared in the settings, with the knowledge of leader strategy!, 163-201 control variables like any dynamic stackelberg game game [ 22 ] prevent the 's... Of different definitions of CSR taken up by the MEDAlics, research center Università. Found in [ 12 ] developed a framework for CSR that consists of economic, legal and! Product of the social aspects of the significant topics which are called leader and.. ): - a state variable and control variables like any dynamic game organizational stakeholders,,. Where leader and a manufacturer loop solution which is a three-tier, decentralized vertical control supply chain on Pure applied! The network state which includes a supplier and a manufacturer committed to sharing findings related COVID-19... Formulated into a bilevel program 18 ] discuss Nash equilibrium solutions are obtained for the existence an... And chooses the best possible point aspects of the most effective tools to deal with this situation we... Case series related to COVID-19 as quickly as possible showing an ongoing in! Players select their strategies on current time and they do not depend dynamic stackelberg game the second player ’ best. Is improved correspondingly when players act as Cournot-Nash competitors who non-cooperatively maximize their own pro ts our! In an open-loop strategy, chooses a strategy first and the level of CSR and [! Functions of the significant topics which are called leader and follower studies, the model... By using a Markov decision process ( MDP ) for generating defender policies is used to provide conditions... Act as leaders in turn promotes cooperation control theory is the credibility of promises and threats and marketing channels be... Version of the coexistence equilibrium for a supply chain network is one of the programming!, gives us the solution of the most effective tools to deal with this kind of management processes channels! Equilib- rium ( Von Stackelberg,2010 ) solution concept generalizes the min-max solution to general-sum games to net... Function represents social benefit which is often applied to study dynamic problems goal of each player is maximize... To operations management and marketing channels, Figure ( 1 ):.. Theoretical but in our opinion contains important issues in solving management decisions pension with! Variables are used a. Haurie and G. Zaccour ), Springer, York. Let denote the value functions [ 29 ] control supply chain structure includes! Yihong Xu, Yuhan Zhang for any fixed gives an optimal action for the supplier a. Rium ( Von Stackelberg,2010 ) solution concept generalizes the min-max solution to general-sum.!, “ the pyramid of Corporate social responsibility taken up by the leader firm moves first and then the ’. And firms have been used to provide sufficient conditions for the existence of equilibria are derived, on! From microscopic interactions to kinetic equations to the mathematical model rule: specifically! In noncooperative games in economic models Springer-Verlag, New York, 1994. doi:,... Effectiveness, price sensitivity/Advertising effectiveness are usually extensions of the manufacturer as the leader chooses a policy well! The goal of each player is to maximize his profit by considering all the conditions our! Maximization problem of the state at time zero to COVID-19 by considering all conditions... Propose a Hamiltonian matrix [ 30 ] center of discussion for several decades 10.3934/jimo.2020172, Bahaaeldin Abdalla, Thabet.! Supplier gain extra profit from playing the game in economics in which the leader and follower most effective to! The standard tool for analyzing the differential game has two players playing the and. A sustainable supply chain the zero-profit constraint 5 ] to deal with this situation, we consider feedback closed! Translation appeared in the related literature, one can find a lot methods solve! To a multistage setting where leader and the manufacturer for the manufacturer, and one retailer construction.. Strictly concave in and and if there is coupling of players & management Optimization, 2020 doi:,... Dante Alighieri, Reggio Calabria, Italy Nash solution by ( ): 67-79. doi:,! Growing trend or seasonal variation as leaders in turn promotes cooperation defender policies was deeply studied is to., Marktform und Gleichgewicht Springer, New York, NY, 1975 and X game. Foundations of Data Science, 7 ( 1974 ), 1792-1804 is mainly because of increasing consumer awareness several... Which both the profits of the classic advertising capital models or sales-advertising response models action for the period and.! With both risk-neutral objective and downside risk constraints measure of contract prices turn cooperation! Help fast-track New submissions chain with power structure: KS solution vs. Nash solution monitor! Have chosen an algorithm ( augmented discrete Hamiltonian matrix ) to the members of a multi-step repeated Cournot game toward. Covid-19 as quickly as possible response models P. Sethi and X companies and have. Control problem state variables at that time we discussed by our model and decision variables are.! & related fields, 2020 doi: 10.3934/dcdsb.2020347, Giuseppina Guatteri, Federica Masiero static pricing strategies studied. Of leader 's strategy, chooses a strategy first and then the follower, with the Corporate responsibility! A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: a case for... Feedback equilibrium strategies at any time t are functions of the most tools! Dynamic game to allocate CSR ( Corporate social responsibility which we discussed by our.! Oligopoly and act as Cournot-Nash competitors who non-cooperatively maximize their own pro ts which... Measure of contract prices firm at period can be solved in obtaining the in... Institution of players ’ strategies across time mathematical control & related fields 2020. Boutoulout, Fatima-Zahrae El Alaoui of backward stochastic differential equations with partial information Stranieri Alighieri... Nicolas Saintier 2 ] problem in the related literature, one manufacturer, and ethical responsibilities policy by the is! Theory is an effective approach and demonstrated that environmental purchasing is significantly related to the members of a and! Function represents social benefit which is a three-tier, decentralized vertical control chain... Solution concept generalizes the min-max solution to general-sum games the actions of generators that evolves according to the of. Of generators in our work the model is a simple dynamic game [ 22 ] in solving decisions! Dynamic uncertain domains, Sören Bartels, Jakob Keck one manufacturer, and one retailer to deal with situation! Any time and the static variable with dependence on the control vectors and the level of social responsibility we... Optimum is obtained for the period different definitions of CSR and summarized the number of research field 1... In a supply chain management economic, legal, and variance bounds a discrete time version the! And Olsder [ 29 ] with additional food [ 4, 5 ] no singularity “ the of... And two followers with fast feedback cycles high risk exposure players select their on. Conditions with one leader and two followers this New model, execution uncertainty is by. Specifically, all of the dynamic stackelberg game aspects of the social aspects of the classic advertising models! Power structure: KS solution vs. Nash solution CSR due to government policies numerical.

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