Linear quadratic differential games with mixed leadership: The open-loop solution, Numerical Algebra, Control and Optimization, 3 (2013), 95-108.  Competition among members in a supply chain network is one of the significant topics which are emphasized in supply chain management. Mathematically, the integrated problem is formulated into a bilevel program. 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. For fixed , the reaction function of the manufacturer is directly given by , that is:The objective function of supplier in period issubstituting the value of given by (4) to (5) the maximum of is obtained when After some algebra we get, for the last period, resolution to the problem at the period . The function represents social benefit which is proportional to social responsibility taken up by the supply chain system [25]. We propose a Hamiltonian matrix to solve the optimal control problem in obtaining the equilibrium in this game. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. 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 … doi: 10.1287/mnsc.1050.0475. Google Scholar, G. E. Fruchter and S. Kalish, 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. Since strategy of a Google Scholar, X. 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. : 177-203. As it is well-known, business logistics management refers to the production and distribution process within the company, while supply chain management includes suppliers, manufacturers, and retailers that distribute the product to the end customer. Google Scholar, T. A. Taylor, Google Scholar, X. The paper is organized as follows: Section 2 is devoted to the mathematical model. 2021, 11 Solving a dynamic stochastic Stackelberg game when both the attacker and the defender have a private Markovian states is computationally challenging. doi: 10.1007/BFb0009151. Tao Li Tian et al. 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. Google Scholar, A. Bensoussan, S. Chen and S. P. Sethi, doi: 10.3934/naco.2013.3.95. Google Scholar, T. Li, S. P. Sethi and X. discrete-time dynamic games was first treated in [9]. doi: 10.1007/978-0-387-72776-9. 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. doi: 10.1287/mnsc.43.1.54. Second, we study the problem of computing a Stackelberg equilibrium for this game and exploit problem Google Scholar, G. M. Erickson, That is our closed loop solution which is based on a nonprofit constraint. Journal of Geometric Mechanics, 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. The Theory of the Market Economy translated by Peacock A. T. , William Hodge and Co. , London, 1952. Incentives for retailer promotion in a marketing channel, Annals of the International Society of dynamic Games, 8 (2006), 365-378.  Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. In a vision which includes the social mission of the firms but looking for reaching profits which are fundamental for the economical sustainability of the business, we find a model by which we capture the existing correlations between these two issues. In the simplest formulation, one player acts as the leader who is endowed with the power to select an action knowing the other player (follower) plays a best-response. An operations research study of sales response to advertising, Operations Research, 5 (1957), 370-381.  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. Dynamic Stackelberg game models have been used to study sequential decision making in noncooperative games in various fields. Google Scholar, A. Bagchi, Review articles are excluded from this waiver policy. December 2016. In particular, we formulate a model that crosses through multi-periods with the help of a dynamic discrete Stackelberg game. A systematic process for evaluating structured perfect Bayesian equilibria in dynamic games with asymmetric information D Vasal, A Sinha, A Anastasopoulos IEEE Transactions on Automatic Control 64 … Markov perfect equilibrium advertising strategies of Lanchester duopoly model: A technical note, Management Science, 50 (2004), 995-1000.  An analysis of manufacturer-retailer supply chain coordination in cooperative advertising, Decision Sciences, 33 (2002), 469-494.  A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. doi: 10.1007/978-3-642-51565-1_107. doi: 10.3934/mcrf.2020048, Shasha Hu, Yihong Xu, Yuhan Zhang. June 2016 A bilinear-quadratic differential game in advertising, in Control Theory in Mathematical Economics (eds. Firstly, we assumewith the boundary conditions and given. 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. Google Scholar, H. L. Lee, V. Padmanabhan, T. A. Taylor and S. Whang, A solution (equilibrium) of the game consists of such strategy choices of both players, with which each player maximizes her/his total payoff (over the whole time horizon of the game) provided that the other player also maximizes her/his own payoff. doi: 10.1007/BF00933155. Google Scholar, S. Jorgensen, S. Taboubi and G. Zaccour, We assume that the time horizon is . In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. Feedback Stackelberg solutions of infinite-horizon stochastic differential games, in Models and Methods in Economics and Management Science (eds. He, doi: 10.3934/krm.2020050, Bernold Fiedler. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Players select their strategies on current time and they do not depend on the initial condition. Dahlsrud [3] presented an overview of different definitions of CSR and summarized the number of dimensions included in each definition. doi: 10.1287/opre.27.4.682. Evolution Equations & Control Theory, 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. Google Scholar, S. P. Sethi and G. L. Thompson, With and , , the optimum value of the supplier can be obtained under the zero-profit constraint. doi: 10.3934/jimo.2020172, Bahaaeldin Abdalla, Thabet Abdeljawad. Google Scholar, P. D. Berger, In this paper, we establish a dynamic game to allocate CSR (Corporate Social Responsibility) to the members of a supply chain. Discrete & Continuous Dynamical Systems - S, Hennet and Arda [19] presented a paper to evaluate the efficiency of different types of contracts between the industrial partners of a supply chain. Google Scholar, W. H. Fleming and R. W. Rishel, Deterministic and stochastic optimal control, in Stochastic Modelling and Applied Probability (eds. Google Scholar, S. P. Sethi, However, for the manufacturer, which is the follower, closed loop solution involves zero-profits. 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,”. When they need to accept CSR due to government’s regulations and policies or consumer’s concern, additional costs are forced on them. American Institute of Mathematical Sciences, Figure(1) The demand dynamics are usually extensions of the classic advertising capital models or sales-advertising response models. Google Scholar, J. V. Gray, B. Tomlin and A. V. Roth, Adaptive time stepping in elastoplasticity. 2020  Numerical Algebra, Control & Optimization, doi: 10.3934/dcdss.2020443, Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. Another aspect that was deeply studied is related to the information structure of the network. Figures 4 and 5 compare the cumulated profits of the supply chain’s members. Google Scholar, S. Jorgensen, S. P. Sigue and G. Zaccour, Pub Date: June 2015 arXiv: Therefore, for dynamic Stackelberg games, this paper proposes that the institution of players acting as leaders in turn promotes cooperation. Using the definition into and maximizing the value function for any fixed gives an optimal action for the manufacturer for the period . doi: 10.1287/mnsc.1040.0249. notion to general dynamic games: Definition 11.1 A Nash equilibrium is said to be subgame perfect if an only if it is a Nash equilibrium in every subgame of the game. Publication: arXiv e-prints. As we can see by the numerical example which follows, if we choose feedback and closed loop information structures among the players, the global environment of the supply chain promotes the interaction between all the members of the network that are naturally oriented in playing the game, that is, strengthening the immaterial but productive structure represented by the supply chain. Moreover, in feedback solution the profit of manufacturer has risen over time. Carter et al. / Table(2), Market potential/Advertising effectiveness, Price sensitivity/Advertising effectiveness. Cachon and Zipkin [18] discuss Nash equilibrium in noncooperative cases in a supply chain with one supplier and multiple retailers. doi: 10.3934/naco.2020016, Chao Xing, Jiaojiao Pan, Hong Luo. 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. 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]. Google Scholar, E. Dockner, S. Jorgensen, N. V. Long and G. Sorger, 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. Google Scholar, A. Bensoussan, S. Chen and S. P. Sethi, doi: 10.3934/cpaa.2020275, Sören Bartels, Jakob Keck. We also have Consequently, we can obtain the unique optimal response of the players as follows: And we have After some algebra, we obtainSince we use closed loop information, the structure variables depend on the current time variable and the initial state variables. A Stackelberg game between one manufacturer and one online retailer is built. Both and are tax return policy parameters. This output is not simply theoretical but in our opinion contains important issues in solving Management Decisions. Obviously, both manufacturer and supplier gain extra profit from playing the game in feedback solution. Imagine an incumbent firm threatens to launch a price war against a new entrant to prevent the latter's entry. Google Scholar, H. Von Stackelberg, Recently, based on many interesting social phenomena, Nie proposed a dynamic Stackelberg game with alternating leaders and analysed the properties of the game [2–4]. This is the derivation of (global) Stackelberg’s solutions when the leader has access to dynamic information. The idea of declaring a reward (or punishment) for a decision-maker according to his particular choice of action in order include a certain desired behavior on the part of decision-maker is known as an incentive (or in case of the punishment, as a threat) [29]. 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 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. B. Cruz, A subgame must be a well-defined game when it is considered separately. The optimal control theory is the standard tool for analyzing the differential game theory [24]. Optimal Control Theory: Applications to Management Science and Economics 2$^{nd}$ edition, Springer, New York, 2000. The results presented here are obtained for the following values of the parameters: , , , , , , , , and . Tao Li, Suresh P. Sethi. doi: 10.3934/eect.2020110, Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. Strategic inventories in vertical contracts, Management Science, 54 (2008), 1792-1804. 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  A discrete time version of the dynamic differential game has been studied. Interactions between members of a marketing-production channel under seasonal demand, European Journal of Operational Research, 90 (1996), 115-141.  Channel coordination under price protection, midlife returns, and end-of-life returns in dynamic markets, Management Science, 47 (2001), 1220-1234.  It aggravates the double marginalization that … Google Scholar, T. Başar and A. Haurie, The portfolio procurement framework is analyzed in the settings, with both risk-neutral objective and downside risk constraints measure of contract prices. Google Scholar, G. Leitmann, Google Scholar, J. T. Teng and G. L. Thompson, The scheduling problem and the dynamic optimization problems are collaborated by a Stackelberg game (leader–followers game). Google Scholar, F. M. Bass, Recently CSR has gained recognition and importance as field of research field [1, 2]. DYNAMIC GAMES: INTRODUCTION In many strategic interactions, players take turn to move and they observe what others have done. The maximization problem of the supplier, over and , gives us the solution of the following set of first-order conditions. In that game, generators form an oligopoly and act as Cournot-Nash competitors who non-cooperatively maximize their own pro ts. doi: 10.1287/mnsc.33.8.981. doi: 10.3934/krm.2020051, Youming Guo, Tingting Li. Google Scholar, X. 2021, 17 Stackelberg game involves players with asymmetric roles which are called leader and follower. 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 . Google Scholar, M. Simaan and J. doi: 10.1080/07408170600710408. Moreover, social welfare is improved correspondingly when players act as leaders in turn. Sutinen ), Springer, New York, NY, 10 ( 2005 ) 1792-1804! Bassok, strategic inventories in vertical contracts, management Science, 7 1974... 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Responsibility which we discussed by our model hierarchical solution which is the of... In Section 3 and Section 4 contains a short conclusion: 10.3934/mcrf.2020048, Hu... General-Sum games with disturbance feedback Stackelberg differential games in economic models Springer-Verlag, New,! A multi-step repeated Cournot game with appropriate value functions [ 29 ] a strategic game in feedback.... Our opinion contains important issues in solving management decisions and chastic dynamic Stackelberg games is discussed in detail dynamic ”! - B, 2020 doi: 10.1007/0-387-24602-9_11 to operations management and marketing channels the existence of an equilibrium solution feedback. Intelligently employing equilibrium congestion pricing anticipating the actions of generators results of the product of the supplier ’ profits. Setting where leader and follower Scholar, D. Pekelman, Simultaneous price production in channels, marketing,...