Abstract
When there are more than a single decision-maker, each having one’s own objective function that each is trying to maximize, subject to a set of differential equations, then we require an extension of the optimal control theory referred to as the theory of differential games. While representing a generalization of optimal control problems in cases where there is more than one controller or player, differential games are conceptually far more complex than optimal control problems in the sense that it is no longer obvious what constitutes a solution. Indeed, there are different types of solutions such as minimax, Nash, and Stackelberg.
We discuss minimax solutions for two-person zero-sum differential games in Sect. 13.1, where one player maximizes his objective function and the other minimizes the same function. Section 13.2 considers nonzero-sum games where all players make simultaneous moves over and each player aims to maximize his own objective function. These are formulated as Nash differential games and their solutions in terms of open-loop and feedback equilibria are discussed. We also apply the theory to a common-property fishery resources game. In Sect. 13.3, we solve a feedback Nash stochastic differential game in advertising. In Sect. 13.4, we discuss a Stackelberg stochastic differential game in which two players make their decisions hierarchically. The player having the right to move first is called the leader and the other player is called the follower. The game is one of cooperative advertising between a manufacturer as the leader deciding on a percentage of the advertising expenditure that he will contribute toward the advertising expenditure of the retailer as the follower. The equilibrium feedback solution that maximizes the objective function of each player is obtained. There are many exercises at the end of the chapter.
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References
Basar T , Bensoussan A, Sethi SP (2010) Differential games with mixed leadership: the open-loop solution. Appl Math Comput 217:972–979
Basar T, Olsder GJ (1999) Dynamic noncooperative game theory, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia
Bensoussan A , Bultez AV , Naert PA (1978) Leader’s dynamic marketing behavior in oligopoly. In: Bensoussan A et al (eds) TIMS studies in the management sciences, vol 9. North-Holland, Amsterdam, pp 123–145
Bensoussan A , Chen S , Chutani A , Sethi SP , Siu CC , Yam SCP (2019) Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising. SIAM J Control Optim 57(5):3413–3444
Bensoussan A , Chen S , Sethi SP (2014) Feedback Stackelberg solutions of infinite-horizon stochastic differential games. In: Ouardighi FE, Kogan K (eds) Models and methods in economics and management science, essays in honor of Charles S. Tapiero, Series 6161, vol 198. Springer International Publishing, Cham, pp 3–15
Bensoussan A , Chen S , Sethi SP (2015a) The maximum principle for global solutions of stochastic Stackelberg differential games . SIAM J Control Optim 53(4):1956–1981
Berkovitz LD (1994) A theory of differential games. In: Basar T, Haurie A (eds) Advances in dynamic games and applications. Birkhäuser, Boston, pp 3–22
Breton M, Jarrar R, Zaccour G (2006) A note on feedback Stackelberg equilibria in a Lanchester model with empirical application. Manag Sci 52(5):804–811
Case JH (1979) Economics and the competitive process. New York University Press, New York
Chintagunta PK, Jain D (1994) A study of manufacturer-retailer marketing strategies: a differential game approach. Lecture notes in control and information sciences. Springer, New York
Chintagunta PK, Jain D (1995) Dynamic duopoly models of advertising competition: estimation and a specification tests. J Econ Manag Strateg 4(1):109–131
Chintagunta PK, Vilcassim NJ (1992) An empirical investigation of advertising strategies in a dynamic duopoly. Manag Sci 38(9):1230–1244
Clark CW (1976) Mathematical bioeconomics: the optimal management of renewal resources. Wiley, New York
Deal KR (1979) Optimizing advertising expenditures in a dynamic duopoly. Oper Res 27(4):682–692
Deal KR , Sethi SP, Thompson GL (1979) A bilinear-quadratic differential game in advertising. In: Liu PT, Sutinen JG (eds) Control theory in mathematical economics. Marcel Dek̇ker, New York, pp 91–109
Dockner EJ, Jørgensen S (1986) Dynamic advertising and pricing in an oligopoly: a Nash equilibrium approach. In: Basar T (ed) Dynamic games and applications in economics. Springer, Berlin, pp 238–251
Dockner EJ, Jørgensen S (1992) New product advertising in dynamic oligopolies. Z Oper Res 36(5):459–473
Dockner EJ , Jørgensen S, Long NV, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge
Erickson GM (2003) Dynamic models of advertising competition. Springer, Boston
Friedman A (1971) Differential games. Wiley, New York
Fruchter GE (1999a) The many-player advertising game. Manag Sci 45(11):1609–1611
Hämäläinen RP , Haurie A, Kaitala VT (1984) Bargaining on whales: a differential game model with Pareto optimal equilibria. Oper Res Lett 3(1):5–11
Hämäläinen RP , Haurie A, Kaitala VT (1985) Equilibria and threats in a fishery management game. Optimal Control Appl Methods 6:315–333
Hämäläinen RP , Ruusunen J, Kaitala VT (1986) Myopic Stackelberg equilibria and social coordination in a share contract fishery. Mar Resour Econ 3(3):209–235
Hämäläinen RP , Ruusunen J, Kaitala VT (1990) Cartels and dynamic contracts in sharefishing. J Environ Econ Manag 19:175–192
Haurie A , Tolwinski B, Leitmann G (1983) Cooperative equilibria in differential games. In: Proceedings ACC, San Francisco
He X , Prasad A , Sethi SP, Gutierrez GJ (2007) A survey of Stackelberg differential game models in supply and marketing channels. J Syst Sci Syst Eng 16(4):385–413. Erratum (2008) 17(2):255
He X , Prasad A, Sethi SP (2009) Cooperative advertising and pricing in a stochastic supply chain: feedback Stackelberg strategies. Prod Oper Manag 18(1):78–94
He X, Sethi SP (2008) Dynamic slotting and pricing decisions in a durable product supply chain. J Optim Theory Appl 137(2):363–379
Ho YC (1970) Differential games, dynamic optimization and generalized control theory. J Optim Theory Appl 6:179–209
Isaacs R (1965) Differential games. Wiley, New York
Jarrar R , Martín-Herrán G, Zaccour G (2004) Markov perfect equilibrium advertising strategies of Lanchester duopoly model: a technical note. Manag Sci 50(7):995–1000
Jørgensen S (1982a) A survey of some differential games in advertising. J Econ Dyn Control 4:341–369
Jørgensen S , Kort PM, Zaccour G (2009) Optimal pricing and advertising policies for an entertainment event. J Econ Dyn Control 33(3):583–596
Jørgensen S, Zaccour G (2004) Differential games in marketing. International series in quantitative marketing. Kluwer Academic Publishers, Boston
Jørgensen S, Zaccour G (2007) Developments in differential game theory and numerical methods: economic and management applications. Comput Manag Sci 4(2):159–182
Leitmann G (1974) Cooperative and non-cooperative many players differential games. Springer, Wien
Martín-Herrán G , Taboubi S, Zaccour G (2005) A time-consistent open loop Stackelberg equilibrium of shelf-space allocation. Automatica 41:971–982
Mehlmann A (1988) Applied differential games. Plenum, New York
Naik PA , Prasad A, Sethi SP (2008) Building brand awareness in dynamic oligopoly markets. Manag Sci 54(1):129–138
Prasad A, Sethi SP (2004) Competitive advertising under uncertainty: stochastic differential game approach. J Optim Theory Appl 123(1):163–185
Prasad A, Sethi SP (2009) Integrated marketing communications in markets with uncertainty and competition. Automatica 45(3):601–610
Rao RC (1984) Advertising decisions in oligopoly: an industry equilibrium analysis. Optimal Control Appl Methods 5(4):331–344
Rao RC (1990) Impact of competition on strategic marketing decisions. In: Day G , Weitz B, Wensley R (eds) Interface of marketing and strategy. JAI Press, Greenwich
Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55
Spiegel MR , Lipschutz S, Liu J (2008) Schaum’s outline of mathematical handbook of formulas and tables, 3rd edn (Schaum’s Outline Series). McGraw-Hill, New York
Starr AW, Ho YC (1969) Nonzero-sum differential games. J Optim Theory Appl 3:184–206
Tolwinski B (1982) A concept of cooperative equilibrium for dynamic games. Automatica 18:431–447
Varaiya PP (1970) N-person nonzero-sum differential games with linear dynamics. SIAM J Control 8:441–449
Zaccour G (2008a) On the coordination of dynamic marketing channels and two-part tariffs. Automatica 44:1233–1239
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Sethi, S.P. (2021). Differential Games. In: Optimal Control Theory. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-91745-6_13
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