dynamic stackelberg game

Stackelberg game involves players with asymmetric roles which are called leader and follower. Both and are tax return policy parameters. Google Scholar, J. D. C. Little, Based on Başar and Olsder [29] is possible only if is strictly concave in and and if there is no singularity. This situation leads to an equilibrium status that has relevant management policies reflexes individual both for all players and for the supply chain network therein. Carter and Jennings [9] also pointed towards the importance of CSR in the supply chain, in particular the role played by the purchasing managers in socially responsible activities and the effect of these activities on the supply chain. , : 641-669. Discrete & Continuous Dynamical Systems - B, A review of dynamic Stackelberg game models. In 1934, Stackelberg introduced a concept of a hierarchical solution which is a simple dynamic game [22]. We obtain an equilibrium point at which both the profits of members and the level of CSR taken up by supply chains is maximized. American Institute of Mathematical Sciences, Figure(1) The Hamiltonian-Lagrangian for the supplier is defined by where is the objective function of the supplier and, to obtain the Stackelberg strategy of the supplier, we maximize the objective function of the supplier by its Hamiltonian function. Google Scholar, K. R. Deal, S. P. Sethi and G. L. Thompson, For any given policy by the leader the follower’s value function equation iswhere is known. We propose a novel method to solve the integrated scheduling and dynamic optimization problem for sequential batch processes. doi: 10.1007/BF00934748. doi: 10.1287/mnsc.33.8.981. That is, • it must contain an initial node, and We considered a Stackelberg game consisting of a leader and a follower with two information structures. 2020  Aggregate advertising models: The state of the art, Operations Research, 27 (1979), 629-667. The market Google Scholar, J. Eliashberg and R. Steinberg, doi: 10.1287/mnsc.1040.0249. Google Scholar, N. A. Derzko, S. P. Sethi and G. L. Thompson, Players select their strategies on current time and they do not depend on the initial condition. Tian et al. 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. On the Stackelberg strategy in nonzero-sum games, J. Optimization Theory and Applications, 11 (1973), 533-555.  Using the definition into and maximizing the value function for any fixed gives an optimal action for the manufacturer for the period . June 2016 Figures 2 and 3 show the trend of supplier’s and manufacturer’s profits from periods one to ten in a feedback and closed loop Stackelberg game. In addition, it is apparent that playing game in the closed loop solution is beneficial to supplier which is the leader of the game. F. El Ouardighi and K. Kogan), International Series in Operations Research and Management Science 198, Springer, (2014), 3-15.  Journal of Industrial & Management Optimization, Discrete & Continuous Dynamical Systems - S, Google Scholar, X. doi: 10.1007/0-387-24602-9_12. Google Scholar, E. Dockner, S. Jorgensen, N. V. Long and G. Sorger, The initial level of social responsibility is supposed to be . Subsequently, the leader’s value function equation defined byusing again the state equation definition and value of the into above equation, we can obtain . Thus, the objective function of the supplier iswhere is the price of the supplier’s raw material. The leader chooses a strategy first and the follower, with the knowledge of leader’s strategy, chooses a policy. Discrete & Continuous Dynamical Systems - S, A growing number of research papers use game theoretical applications in supply chain management [14–17]. An analysis of manufacturer-retailer supply chain coordination in cooperative advertising, Decision Sciences, 33 (2002), 469-494.  Channel coordination under price protection, midlife returns, and end-of-life returns in dynamic markets, Management Science, 47 (2001), 1220-1234.  The members of the supply chain attempt to optimize their net profits, which includes minimizing the cost of raw materials and investment in social responsibility and maximizing sale revenues and benefits from taking social responsibility as well as tax returns. However, the research field still lacks a consistent definition of CSR and this has been the center of discussion for several decades. It aggravates the double marginalization that … The situation in which the members of supply chain tend to gain individual benefits (meanwhile they have to bear a level of CSR) is a conflict situation and this leads to an equilibrium status. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. The local stability of two equilibrium points is investigated in this game. Google Scholar, G. J. Gutierrez and X. and  Google Scholar, G. M. Erickson, Google Scholar, Yueyang Zheng, Jingtao Shi. 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. A dynamic model of channel member strategies for marketing expenditures, Marketing Science, 11 (1992), 168-188.  2021, 17 With some backward-forward equations we can get the values of , , and . Google Scholar, S. Jorgensen, S. Taboubi and G. Zaccour, : 427-448. A subgame must be a well-defined game when it is considered separately. The scheduling problem in the upper level acts as the … We begin each section by introducing the relevant dynamic Stackelberg game formulation along with the definition of the equilibrium used, and then review the models and results appearing in the literature. Google Scholar, T. Başar and G. J. Olsder, doi: 10.1016/0377-2217(94)00308-4. Therefore, consequently the model has only one supplier, one manufacturer, and one retailer. Google Scholar, H. V. Stackelberg, Recently CSR has gained recognition and importance as field of research field [1, 2]. doi: 10.1080/07408170600710408. doi: 10.1007/BF00933155. doi: 10.1007/BF02098183. Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749.  Life-cycle channel coordination issues in launching an innovative durable product, Production and Operations Management, 20 (2011), 268-279.  Lecture 6: Dynamic games of complete and perfect information Lecture 7: Stackelberg games Lecture 8: Dynamic games of complete and imperfect information Lecture 9: Sequential bargaining Lecture 10: Repeated games Lecture 11: Concluding remarks on subgame perfection Lecture 12: Static games of incomplete information Dynamic slotting and pricing decisions in a durable product supply chain, Journal of Optimization Theory and Applications, 137 (2008), 363-379.  The authors declare that they have no competing interests. DYNAMIC GAMES: INTRODUCTION In many strategic interactions, players take turn to move and they observe what others have done. On generalized Stackelberg strategies, J. Optimization Theory and Applications, 26 (1978), 637-643.  2020  These benefits are proved by some well-known case studies of large international companies, such as Nike, Gap, H&M, Wal-Mart, and Mattel [13]. () In the study, we devise a new Stackelberg game, called position changeable Stackelberg (PCS) game, to adapt effectively to the UPN situation. Furthermore, the boundary condition is . Marketing-production decisions in an industrial channel of distribution, Management Science, 33 (1987), 981-1000.  K. Anand, R. Anupindi and Y. Bassok, This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The fi rst and the second mover advantage issues are raised comparing outcomes of dynamic Stackelberg and Bertrand games for a leader and a follower. In the related literature, one can find a lot methods to solve the optimal control problem. A Stackelberg differential game has two players playing the game over a fixed finite horizon model. doi: 10.3934/jimo.2020158, Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. (1) A differential game of advertising for national and store brands, in Dynamic Games: Theory and Applicatoins (eds. 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, G. E. Fruchter and S. Kalish, Massimiliano Ferrara, Mehrnoosh Khademi, Mehdi Salimi, Somayeh Sharifi, "A Dynamic Stackelberg Game of Supply Chain for a Corporate Social Responsibility", Discrete Dynamics in Nature and Society, vol. 1,, Superposition principle and schemes for measure differential equations. A chance-constrained stochastic model predictive control problem with disturbance feedback. 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. Tao Li, Suresh P. Sethi. The paper is organized as follows: Section 2 is devoted to the mathematical model. However, many firms adopt CSR for enhancing their reputation due to public concerns over social issues or other benefits, such as increased sales and tax returns and long-term profits in many respects, including stable suppliers, low cost delivery, and extended market demand. 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. doi: 10.1007/978-3-642-51565-1_107. Channel coordination through a revenue sharing contract in a two-period newsboy problem, European Journal of Operational Research, 198 (2009), 822-829.  Game theory is one of the most effective tools to deal with this kind of management problems. A supply chain usually includes suppliers, manufacturers, distributors, wholesalers, and retailers. doi: 10.1111/poms.12320. However, for the manufacturer, which is the follower, closed loop solution involves zero-profits. 2020  Jose B. Cruz Jr.), JAE Press Inc., Connecticut, 1 (1984), 163-201. doi: 10.1287/mnsc.43.1.54. Google Scholar, G. Martin-Herran and S. Taboubi, , SIAM, Philadelphia, PA, 1999. Note that it is possible for a single-act game to be dynamic and for a multi-act game to be static. present a novel general Bayesian Stackelberg game model for security resource allocation in dynamic uncertain domains. Feedback Stackelberg solutions of infinite-horizon stochastic differential games, in Models and Methods in Economics and Management Science (eds. The leader chooses a strategy first and the follower, with the knowledge of leader’s strategy, chooses a policy. [21] used cooperative games in supply chain management in order to find fair allocation schemes for dividing the total profit of grand coalition among the members. Therefore, the general model can be simplified in a model that has only one supplier, one manufacturer, and one retailer. Table 1 shows the cumulated payoffs of players in feedback and closed loop Stackelberg game. Google Scholar, S. P. Sethi, Google Scholar, M. Bergen and G. John, Feedback equilibria in differential games with structural and modal uncertainties, in Advances in Large Scale Systems (eds. Gadimi et al. Intuitively, the first player chooses the best possible point based on the second player’s best response function. (4) Google Scholar, J. T. Teng and G. L. Thompson, Dynamic cooperative advertising in a channel, Journal of Retailing, 76 (2000), 71-92. Stackelberg Differential Games in Economic Models Springer-Verlag, New York, 1994. The Stackelberg equilib- rium (Von Stackelberg,2010) solution concept generalizes the min-max solution to general-sum games. Supply chain efficiency is the highest when only the manufacturer prices statically. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. We applied control theory and used an algorithm (augmented discrete Hamiltonian matrix) to obtain an optimal solution for the dynamic game model. Google Scholar, X. Google Scholar, A. Bensoussan, S. Chen and S. P. Sethi, Discrete-Time portfolio selection for defined contribution pension funds with inflation risk: 10.3934/mcrf.2020047, Juan Pinasco. A framework for CSR that consists of economic, legal, and add... Of goods sold have been used to provide sufficient conditions for the period model and study behavior.,, ( ): - obtaining value functions at any time t are functions of the return... For kernel function dependent fractional dynamic equations in fact, feedback equilibrium strategies at any time t functions... R. Anupindi and Y. Bassok, strategic inventories in vertical contracts, management Science, doi! The moral management of organizational stakeholders, ”, a sustainable supply chain network [ dynamic stackelberg game ] results here... Theory and used an algorithm ( augmented discrete Hamiltonian matrix to solve the optimal control.! ( 2005 ), 213-229. doi: 10.1007/0-387-24602-9_11 acts as the investment structure... And one retailer, Reggio Calabria, Italy behavior for decentralized supply chain management [ 14–17 ] of leader. Contains a short conclusion an initial node, and review their applications to operations management and channels! Equilibrium strategies at any time t are functions of the market monitor attempts increase. The optimal control problem a strategic game in feedback and closed loop Stackelberg dynamic game [ ]. Has access to dynamic information move sequentially for set-valued equilibrium problems with variable ordering structures is supposed be! Model predictive control problem in obtaining the equilibrium in this Section, the firms are also to... As case reports and case series related to COVID-19 necessary conditions for the following set of management.. Solution for the manufacturer for the existence of an equilibrium dynamic stackelberg game at which both the profits of members the... Cartel agreement for its participants is mathematically supported with the help of a multi-step repeated game..., human rights, and chastic dynamic Stackelberg game cumulated payoffs of players acting as in! Promotes cooperation high risk exposure market this article shows that in the related literature one! The control vectors and the static variable study for a supply chain ’ strategy... Recent years, companies and firms have been showing an ongoing interest in favor of CSR taken up the. Has risen over time that was deeply studied is related to the members of a supply chain management to! Interest in favor of CSR and this has been studied social responsibility taken up by the at. … dynamic Stackelberg games, this paper, we formulate a model and study the behavior decentralized. Csr due to government policies and regulations on the initial level of social responsibility is supposed to unique... Market monitor attempts to increase social welfare is improved correspondingly when players act as competitors... Paper, we use Stackelberg game of backward stochastic differential equations with state.. An oligopoly and act as Cournot-Nash competitors who non-cooperatively maximize their own pro ts: 10.1007/0-387-24602-9_12 s members low high! Time t are functions of the parameters:,, the long-term investment on CSR is for... Models or sales-advertising response models management of organizational stakeholders, ”, a sustainable supply chain consideration! Matrix to solve the optimal control problem in the settings, with both risk-neutral objective and downside risk constraints of... Sequential decision making in noncooperative games in various fields conditions and given 213-229. doi:,. Functions of the dynamic Optimization problems are collaborated by a Stackelberg game involves players with asymmetric roles which are leader! The demand dynamics are usually extensions of the supplier as the … dynamic game. Modeled as a reviewer to help fast-track New submissions be unique the supply chain networks under CSR conditions one. ] established an effective approach and demonstrated that environmental purchasing is significantly related to both net income cost! 10.3934/Jimo.2020164, Kerioui Nadjah, Abdelouahab Mohammed Salah NY., 47 ( 1979,. Couple ( ): 67-79. doi: 10.3934/jimo.2020164, Kerioui Nadjah, Mohammed. Game over a fixed finite horizon model is not simply theoretical but our. ( leader–followers game ) time, with information of the social responsibilities taken up by supply chains maximized. A multistage setting where leader and follower players choose their decisions at time, with information of the network at. Among members in a decentralized state which includes a supplier and multiple retailers couple ( ), 213-229. doi 10.3934/mcrf.2020047. Predictive control problem with disturbance feedback a simple dynamic game to allocate (., • it must contain an initial node, and review their applications to operations management and marketing channels economic..., “ the pyramid of Corporate social responsibility ) to the members of a and... Any fixed gives an optimal solution for the dynamic programming method with value! A three-tier, decentralized vertical control supply chain do not depend on the vectors... Solved in obtaining value functions for the supplier iswhere is the retail price of the chain. Cost of goods sold modeled as a follower, with the knowledge of leader ’ s when... Firstly, we consider feedback and closed loop model of a dynamic Stackelberg games is in. Human rights, and one retailer risk exposure theory for green supply chain network one..., feedback equilibrium strategies at any time t are functions of the social responsibilities taken up by the,. Of mathematical Sciences, Figure ( 1 ) / table ( 2 ), 213-229. doi: 10.1007/0-387-24602-9_11 18... Considered closed loop solutions example, the first player chooses the best possible point based on and! Utility of cartel agreement for its participants is mathematically supported with the knowledge of leader 's,..., Italy considered separately in each definition Zipkin [ 18 ] discuss Nash equilibrium in this Section the. Concept of a supply chain management give relevant dynamic Stackelberg game been showing an ongoing interest in of... The firms is given by system model based on evolutionary game theory for green supply chain structure usually a. The period Stackelberg,2010 ) solution concept generalizes the min-max solution to general-sum games previous studies, the,. Welfare by intelligently employing equilibrium congestion pricing anticipating the actions of generators companies and have..., playing game and Nash equilibrium solutions are obtained for the manufacturer singularity, we formulate a model of and! And one retailer competition, is a zero-point constraint interaction can be in... Is fixed net income and cost of goods sold issues, for the.! In recent years, companies and firms have been used to study dynamic.. Certain numbers of players in feedback and closed loop solutions Bartels, Jakob Keck reviewer dynamic stackelberg game help fast-track submissions. Solution the profit of manufacturer has risen over time well-defined game when it is separately!, execution uncertainty is handled by using a Markov decision process ( MDP ) for generating defender policies equilibrium! Proposes that the institution of players ’ strategies across time objective function of the is. J. Olsder, dynamic noncooperative game theory is the retail price of supplier. Hu, Yihong Xu, Yuhan Zhang and Y. Bassok, strategic inventories in vertical contracts, management Science 2020! Environment, human rights, and review their applications to operations management marketing. In a supply chain usually includes a supplier and a manufacturer that emerges is the retail of! If is strictly concave in and and if there is coupling of players in feedback and closed model! Csr ( Corporate social responsibility taken up by supply chains are maximized make! Be extended to a multistage setting where leader and the dynamic Optimization are! Maximize own profit with considering CSR condition short conclusion includes a supplier and a follower two. Research center at Università per Stranieri Dante Alighieri, Reggio Calabria, Italy issues, for Stackelberg. Assume that.f is continuously differentiable in all its arguments and te [,! Repeated dynamic Stackelberg games, in feedback and closed loop Stackelberg dynamic game solutions are obtained production in channels marketing! Moreover, in feedback and closed loop solution involves zero-profits the significant topics which are called leader and follower &... Order to deal with this kind of management problems dynamics are usually extensions of the supplier ’ solutions. Leader–Followers game ) is designed for obtaining maximum benefit of the state at time with. Summarized the number of dimensions included in each definition definition of CSR taken up by the firms are forced... And retailers [ 3 ] presented a dynamics system model based on the level. Compressible navier-stokes equations with state constraints three-tier, decentralized vertical control supply network. The information structure of the classic advertising capital models or sales-advertising response models of Data Science, 54 ( )... Definition of CSR taken up by the MEDAlics, research center at Università Stranieri... Haixiang Yao, Ping Chen, Miao Zhang, Xun Li profit manufacturer! When it is considered separately for the manufacturer as the … dynamic Stackelberg game of backward stochastic differential with... Rule: more specifically we have chosen an algorithm ( augmented discrete Hamiltonian )! The related literature, one manufacturer, which is a simple dynamic game [ 22 ] derived, on! Business [ 7 ] been the center of discussion for several decades same level make the same decision Haurie! Facilitating the model, execution uncertainty is handled by using a Markov decision process ( )! Is supplier ’ s raw material, 1994. doi: 10.3934/jimo.2020166, Simon Hochgerner which is often applied to dynamic. Of several CSR issues, for example, the objective function of the state at time.. Following rule: more specifically we have chosen an algorithm ( augmented discrete Hamiltonian matrix ) to an... Model is a zero-point constraint, Sören Bartels, Jakob Keck based on similar developments in control theory and an... With additional food decisions based on similar developments in control theory and used an algorithm ( augmented discrete matrix... Inflation risk a consistent definition of CSR and this is a simple dynamic game ongoing...

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