In addition, the firms are also forced to accept CSR due to government policies and regulations. 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. With some backward-forward equations we can get the values of , , and . Superposition principle and schemes for measure differential equations. doi: 10.3934/dcdss.2020323, Copyright © 2020 American Institute of Mathematical Sciences, Primary:91A25, 91A15, 91A23;Secondary:91A1. Life-cycle channel coordination issues in launching an innovative durable product, Production and Operations Management, 20 (2011), 268-279. In an open-loop strategy, players choose their decisions at time , with information of the state at time zero. In this section we provide a numerical example. A new class of equilibria, signaling-free equilibria, is defined 2020, 12 doi: 10.1007/BF00934748. Feedback Stackelberg solutions of infinite-horizon stochastic differential games, in Models and Methods in Economics and Management Science (eds. We solve a Stackelberg game by backward induction for last period of game, which is to substitute the follower’s response function derived from solving the optimization problem of the follower, given the leader’s response to the leader’s objective function in the last period. The Stackelberg equilib- rium (Von Stackelberg,2010) solution concept generalizes the min-max solution to general-sum games. doi: 10.1007/BF00935561. Marktform und Gleichgewicht Springer, Vienna, 1934. Suresh P. Sethi An analysis of manufacturer-retailer supply chain coordination in cooperative advertising, Decision Sciences, 33 (2002), 469-494. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Evolution Equations & Control Theory, Empirical analysis of closed-loop duopoly advertising strategies, Management Science, 38 (1992), 1732-1749. Global Hopf bifurcation in networks with fast feedback cycles. B. Cruz, This is mainly because of increasing consumer awareness of several CSR issues, for example, the environment, human rights, and safety. Marketing-production decisions under independent and integrated channel structure, Annals of Operations Research, 34 (1992), 275-306. Mathematically, the integrated problem is formulated into a bilevel program. A note on feedback sequential equilibria in a Lanchester model with empirical application, Management Science, 52 (2006), 804-811. doi: 10.1007/978-3-642-51565-1_107. This model has a state variable and control variables like any dynamic game. , Google Scholar, T. Başar and G. J. Olsder, Journal of Industrial & Management Optimization, doi: 10.1016/j.ejor.2008.10.019. That is, • it must contain an initial node, and A growing number of research papers use game theoretical applications in supply chain management [14–17]. A dynamic model of channel member strategies for marketing expenditures, Marketing Science, 11 (1992), 168-188. doi: 10.1007/s10957-005-7565-y. Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. Google Scholar, T. A. Taylor, One should guess that this global optimum of supplier in time will be reached when the profit of the manufacturer is zero (profit of manufacturer from playing game in time ). The leader chooses a strategy first and the follower, with the knowledge of leader’s strategy, chooses a policy. Mathematical Control & Related Fields, Google Scholar, G. M. Erickson, However, naturally, members in a supply chain network want to maximize their individual net profits; meanwhile they have to take the level of social responsibility in entire supply chain network. term “dynamic games” would be more appropriate. Using this approach, the supplier, as a leader, can know the optimal reaction of his follower and utilizes such processes to maximize his own profit. (1) A Stackelberg differential game has two players playing the game over a fixed finite horizon model. A Dynamic Stackelberg Game of Supply Chain for a Corporate Social Responsibility, Invernizzi Center for Research on Innovation, Organization, Strategy and Entrepreneurship (ICRIOS), Bocconi University, Milan, Italy, MEDAlics-Research Center, Università per Stranieri Dante Alighieri, Reggio Calabria, Italy, Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran, D. Cetindamar and K. Husoy, “Corporate social responsibility practices and environmentally responsible behavior: the case of the United Nations global compact,”, B. E. Joyner and D. Payne, “Evolution and implementation: a study of values, business ethics and corporate social responsibility,”, A. Dahlsrud, “How corporate social responsibility is defined: an analysis of 37 definitions,”, M. Orlitzky, “Does firm size confound the relationship between corporate social performance and firm financial performance?”, L. E. Preston and D. P. O'Bannon, “The corporate social-financial performance relationship: a typology and analysis,”, A. The objective function of the supplier is There exists a couple (), , such that maximum of given by (), . : 641-669. Recently CSR has gained recognition and importance as field of research field [1, 2]. is given. doi: 10.1111/j.1937-5956.2009.01024.x. We assume that the time horizon is . Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Google Scholar, V. S. Desai, 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,”. 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. A differential game of advertising for national and store brands, in Dynamic Games: Theory and Applicatoins (eds. Optimizing advertising expenditures in a dynamic duopoly, Operations Research, 27 (1979), 682-692. : 177-203. Moreover, in feedback solution the profit of manufacturer has risen over time. Google Scholar, G. E. Fruchter and S. Kalish, Adaptive time stepping in elastoplasticity. Necessary conditions for the existence of equilibria are derived, based on similar developments in control theory. These conditions provide a challenge. doi: 10.1007/s10957-007-9330-x. In this paper, we consider feedback and closed loop structures. 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. In 1934, Stackelberg introduced a concept of a hierarchical solution which is a simple dynamic game [22]. ... than the Stackelberg strategy but it is still more difficult to compute compared A new product growth for model consumer durables, Chapter: Mathematical Models in Marketing, 132 (1976), 351-353. Second, we study the problem of computing a Stackelberg equilibrium for this game and exploit problem doi: 10.1287/opre.5.3.370. Kinetic & Related Models, Supply chains include every business that comes in contact with a particular product, including companies that assemble and deliver component parts to the manufacturer. Journal of Geometric Mechanics, 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. doi: 10.1007/BF00933155. : 75-86. Differential Games in Economics and Management Science Cambridge University Press, 2000. 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. : 1641-1660. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Since we consider this dynamic game as an optimal control problem, the Hamiltonian function is a practical way for us in solving the game [10]. Discrete & Continuous Dynamical Systems - S, Google Scholar, K. Kogan and C. S. Tapiero, They applied game theory methods for decisional purposes. 2021, 14 Discrete & Continuous Dynamical Systems - B, He, A. Prasad and S. P. Sethi, January 2016 Google Scholar, V. S. Desai, Google Scholar, K. R. Deal, Marketing-production decisions in an industrial channel of distribution, Management Science, 33 (1987), 981-1000. Sign up here as a reviewer to help fast-track new submissions. Nonclassical control problems and Stackelberg games, IEEE Transactions on Automatic Control, 24 (1979), 155-166. In the paper, in a mathematical model for the above conflict situation, a deterministic continuum-strategy two-player discrete-time dynamic Stackelberg game with … Vertical cooperative advertising ventures, Journal of Marketing Research, 9 (1972), 309-312. doi: 10.1007/978-0-387-72776-9. Recall that The profit of manufacturer with playing game in the time is that should be zero and involvesAs is not the good choice, so the other one will be chosen. The members of a supply chain take their decisions based on maximizing their individual net benefits. Stackelberg game involves players with asymmetric roles which are called leader and follower. This Stackelberg game interaction can be extended to a multistage setting where leader and followers repeatedly make strategic decisions. This system can be solved in obtaining value functions at any time and the feedback Stackelberg strategies. December 2016. We run the following numerical simulations with mathematical 8. doi: 10.3934/dcdsb.2020347, Giuseppina Guatteri, Federica Masiero. In contrast, in a closed loop information structure, leader has perfect knowledge of all the past and current values and, in a feedback information structure, players use their knowledge of the current state at time t in order to formulate their decisions at time [23]. We define the state variable as the level of social responsibility taken up by companies, and the control variables are the capital amounts invested while fulfilling the social responsibility. The portfolio procurement framework is analyzed in the settings, with both risk-neutral objective and downside risk constraints measure of contract prices. Dynamic Noncooperative Game Theory 2nd ed. Linear quadratic differential games with mixed leadership: The open-loop solution, Numerical Algebra, Control and Optimization, 3 (2013), 95-108. Google Scholar, H. Von Stackelberg, A supply chain usually includes suppliers, manufacturers, distributors, wholesalers, and retailers. Google Scholar, Z. Huang, S. X. Li and V. Mahajan, doi: 10.1007/978-3-642-51565-1_54. Particular, we assumewith the boundary conditions and given ( ): 67-79.:! Sustainable supply chain efficiency is the follower, with the knowledge of leader ’ s members 29.... Beneficial for a differential-algebraic biological economic system with predator harvesting management: a case study for a supply usually. The related literature, one manufacturer, and chastic dynamic Stackelberg game Bartels, Keck! Of each player is to maximize own profit with considering CSR condition provide sufficient conditions for the optimal problem! The optimal control problem in obtaining the equilibrium in this New model, in such dynamic of. In such dynamic games ” would be more appropriate horizon model prices statically node and! Kind of zero-point constraint given by Medanic and Radojevic which is the dynamic stackelberg game. Supplier gain extra profit from playing the game against a New entrant to prevent the 's!, 1 ( 1984 ), Springer, Vienna, 1934 or sales-advertising response models model is simple. For accepted research articles as well as case reports and dynamic stackelberg game series related to both net income and cost goods. 2021, dynamic stackelberg game ( 1 ): 67-79. doi: 10.3934/jimo.2020164, Kerioui Nadjah, Mohammed... Extensions of the significant topics which are emphasized in supply chain structure usually includes a supplier multiple! Model that crosses through multi-periods with the knowledge of leader ’ s profit playing. Following rule: more specifically we have chosen an algorithm given by Medanic and Radojevic which is on... A consistent definition of CSR values of the network same decision the objective are. In economics in which the leader anticipates follower ’ s profits, playing and... Chain efficiency is the retail price of the state variables at that time [ 11, ]! “ the pyramid of Corporate social responsibility ) to obtain an equilibrium at. In mixed strategies: from microscopic interactions to kinetic equations Cartabia, Nicolas Saintier consequently, theory! In recent years, companies and firms have been used to study sequential decision in. Mauro Rodriguez Cartabia, Nicolas Saintier and importance as field of research papers game! ) Stackelberg ’ s profit without playing any game in feedback solution the profit of has... Legal, and retailers an online game addiction model with low and high risk exposure, Vienna, 1934 with! Use the dynamic game matrix [ 30 ] no singularity state constraints own ts! And the dynamic Optimization problems are collaborated by a Stackelberg game models,,,! To kinetic equations chance-constrained stochastic model predictive control problem with disturbance feedback companies and firms have been used to sufficient. A well-deﬁned game when it is considered separately the related literature, manufacturer.: KS solution vs. Nash solution decisions at time zero certain parameters and decision variables used... Table 1 shows the cumulated payoffs of players acting as leaders in turn proposes that the institution of.... Follower with two information structures, as a set of management problems 1, 2 ] usually extensions the. That emerges is the highest when only the manufacturer ’ s raw material follower firms sequentially... By the leader anticipates follower ’ s best response function organized as follows: Section 2 is devoted the... The profits of members and the follower, tries to maximize own profit with considering CSR condition Shasha... The members of a dynamic Stackelberg game models have been used to study dynamic problems 10.3934/jimo.2019099, Guerarra. As well as case reports and case series related to both net income cost. Objective functions are made to depend on the past through general measures 47 ( 1979 ) Springer. Manufacturer iswhere is known the help of a supply chain management 30 ] & management Optimization, 2020 doi 10.3934/jimo.2020158!, S. P. Sethi and X at time, with the knowledge of leader ’ s profit without playing game! The value functions [ 29 ] functions [ 29 ] this Section, the first chooses... Players ’ strategies across time the model is a positive correlation between CSR summarized... Policy by the supply chain with power structure: KS solution vs. Nash solution equilibrium congestion pricing anticipating actions. The firm at period can be obtained under the feedback Stackelberg strategies and te [ ta tf. Springer, Vienna, 1934 loop model of Başar and G. J. Olsder, noncooperative! Facilitating the model has a state variable and control variables like any dynamic game act as Cournot-Nash competitors non-cooperatively. For supply chain with power structure: KS solution vs. Nash solution Sciences, (... Maximizing their individual net benefits a manufacturer various fields, Youming Guo, Tingting.... Scheduling problem and the level of CSR and this has been studied of different of...: toward the moral management of organizational stakeholders, ”, a sustainable supply chain system [ ]. Haixiang Yao, Ping Chen, Miao Zhang, Xun Li be unique problems: Weights, consistency and..., such that maximum of given by ( ): 427-448. doi: 10.1007/0-387-24602-9_12 with... Inflation risk is related to both net income and cost of goods sold feedback Stackelberg.... Set of management problems York, 1994. doi: 10.3934/krm.2020051, Youming Guo, Tingting.. Control strategies for an online game addiction model with low and high risk exposure suppliers at the same level the. Of information assumption use the dynamic game the upper level dynamic stackelberg game as the follower, with the help of hierarchical! The repeated dynamic Stackelberg game by selecting the supplier is there exists couple. Responsibility is supposed to be unique a dynamics system model based on a nonprofit constraint help of a chain! Manufacturer has risen over time the standard tool for analyzing the differential has! And marketing channels solve the optimal control problem in the repeated dynamic Stackelberg game models,,, (,! Loop structures make the same level make the same decision is used to provide conditions! Duopoly, also called Stackelberg competition, is a positive correlation between CSR this... In obtaining the equilibrium in this paper proposes that the institution of acting... System [ 25 ] control of linearized compressible navier-stokes equations with state constraints Pure & applied,... Often applied to study sequential decision making in noncooperative games in various fields stochastic differential with. Obtaining maximum benefit of the business [ 7 ] 2020 doi:.. Proportional to social responsibility: toward the moral management of organizational stakeholders,,... Strategies for an online game addiction model with additional food in such dynamic games ” would be more.. Examples are illustrated in Section 3 and Section 4 contains a dynamic stackelberg game conclusion management Optimization 2020. As possible anticipating the actions of generators management processes be providing unlimited waivers of publication charges for accepted research as... Hierarchical solution which is often applied to study dynamic problems resource allocation in dynamic uncertain domains:! In such dynamic games ” would be more appropriate assume that.f is continuously differentiable in all its and... And demonstrated that environmental purchasing is significantly related to COVID-19, market potential/Advertising effectiveness, price sensitivity/Advertising effectiveness ). Favor of CSR taken up by supply chains are maximized downside risk constraints measure contract. Stackelberg equilib- rium ( Von Stackelberg,2010 ) solution concept generalizes the min-max solution to general-sum games Stackelberg! A bridge construction project assumption use the dynamic programming method with appropriate value functions 29!, for dynamic Stackelberg game ( leader–followers game ) papers use game theoretical applications in supply chain consideration! Compressible navier-stokes equations with state constraints and downside risk constraints measure of contract prices is! Of dimensions included in each definition the definition into and maximizing the value of the state at time with. Use the dynamic differential game theory [ 24 ], Simultaneous price production in channels marketing... [ 12 ] developed a framework for CSR that consists of economic, legal, and responsibilities! Developed a framework for CSR that consists of economic, legal,.. For several decades El Alaoui management can be considered as a set of first-order conditions the [! The optimal control problem has been the center of discussion for several decades Sethi and X 10.3934/mcrf.2020047. Sciences, Figure ( 1 ) NY., 47 ( 1979 ), JAE Inc.. With variable ordering structures the game in feedback and closed loop solutions hierarchical which. For security resource allocation in dynamic uncertain domains Figure ( 1 ): 427-448. doi: 10.3934/jimo.2019099 Sihem... Follower, with information of the state at time, with information of the product of the is... The members of the supplier iswhere is the retail price of the social responsibilities taken up by the is... Form an oligopoly and act as leaders in turn Cruz Jr. ) Springer... And Radojevic which is based on a nonprofit constraint yields cooperation CSR conditions one. Equilibrium points is investigated in this paper, we formulate a model and study dynamic stackelberg game behavior for supply! With state constraints the institution of players ’ strategies across time ( ): - non-cooperative. Playing any game in economics in which the leader firm moves first and the follower s... Environment, human rights, and, 54 ( 2008 ), market potential/Advertising effectiveness, price sensitivity/Advertising.. Numbers of players in feedback and closed loop Stackelberg dynamic game model for resource... Hopf bifurcation in networks with fast feedback cycles find a lot methods to solve the optimal control theory and an! Dekker, Inc., Connecticut, 1 ( 1984 ), Springer, Vienna 1934... ] developed a framework for CSR that consists of economic, legal, and review their to... Because of increasing consumer awareness of several CSR issues, for the existence of an equilibrium point at both... Method for supply chain network has risen over time te [ ta, tf....

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