Abstract
This chapter provides an overview of the theory of nonzero-sum differential games, describing the general framework for their formulation, the importance of information structures, and noncooperative solution concepts. Several special structures of such games are identified, which lead to closed-form solutions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See, e.g., the classical textbook on optimal control by Lee and Markus (1972) for examples of synthesis of state-feedback control laws.
- 2.
This property will be discussed later in the chapter; in the context of static games, “strategic equivalence” has been discussed in Chap. 1.
- 3.
- 4.
With a slight abuse of notation, we have included here also the pair (x0, t0) as an argument of \(\bar {J}_{j}\), since under the SF information γ does not have (x0, t0) as an argument for t > t0.
- 5.
We use \(\mathcal {T}\) instead of T because, in a general setting, T may be endogenously defined as the time when the target is reached.
- 6.
We use the convention that λj(t)f(⋅, ⋅, ⋅) is the scalar product of two n dimensional vectors λj(t) and f(⋯ ).
- 7.
This is a standard result in optimal control, which can be found in any standard text, such as Bryson et al. (1975).
- 8.
The existence of a conjugate point in [0, T) implies that there exists a sequence of policies by the maximizer which can drive the value of the game arbitrarily large, that is, the upper value of the game is infinite.
- 9.
The setup can be easily extended to the case of several followers. A standard assumption is then that the followers play a (Nash) simultaneous-move game vis-a-vis each other, and a sequential game vis-a-vis the leader (Başar and Olsder 1999).
- 10.
This assumption allows us to use the strong-optimality concept and avoid introducing additional technicalities.
- 11.
Here, and in the balance of this section, we depart from our earlier convention of state-time ordering (x, t), and use the reverse ordering (t, x).
- 12.
In a linear-state differential game, the objective functional, the salvage value and the dynamics are linear in the state variables. For such games, it holds that a feedback strategy is constant, i.e., independent of the state and hence open-loop and state-feedback Nash equilibria coincide.
References
Başar T (1974) A counter example in linear-quadratic games: existence of non-linear Nash solutions. J Optim Theory Appl 14(4):425–430
Başar T (1975) Nash strategies for M-person differential games with mixed information structures. Automatica 11(5):547–551
Başar T (1976) On the uniqueness of the Nash solution in linear-quadratic differential games. Int J Game Theory 5(2/3):65–90
Başar T (1977) Informationally nonunique equilibrium solutions in differential games. SIAM J Control 15(4):636–660
Başar T (1979) Information structures and equilibria in dynamic games. In: Aoki M, Marzollo A (eds) New trends in dynamic system theory and economics. Academic, New York, pp 3–55
Başar T (1982) A general theory for Stackelberg games with partial state information. Large Scale Syst 3(1):47–56
Başar T (1985) Dynamic games and incentives. In: Bagchi A, Jongen HTh (eds) Systems and optimization. Volume 66 of Lecture Notes in Control and Information Sciences. Springer, Berlin, pp 1–13
Başar T (1989) Time consistency and robustness of equilibria in noncooperative dynamic games. In: Van der Ploeg F, de Zeeuw A (eds) Dynamic policy games in economics, North Holland, Amsterdam/New York, pp 9–54
Başar T, Bernhard P (1995) H∞-optimal control and related minimax design problems: a dynamic game approach, 2nd edn. Birkhäuser, Boston
Başar T, Haurie A (1984) Feedback equilibria in differential games with structural and modal uncertainties. In: Cruz JB Jr (ed) Advances in large scale systems, Chapter 1. JAE Press Inc., Connecticut, pp 163–201
Başar T, Olsder GJ (1980) Team-optimal closed-loop Stackelberg strategies in hierarchical control problems. Automatica 16(4):409–414
Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. SIAM series in Classics in Applied Mathematics. SIAM, Philadelphia
Başar T, Selbuz H (1979) Closed-loop Stackelberg strategies with applications in the optimal control of multilevel systems. IEEE Trans Autom Control AC-24(2):166–179
Bellman R (1957) Dynamic programming. Princeton University Press, Princeton
Blaquière A, Gérard F, Leitmann G (1969) Quantitative and qualitative games. Academic, New York/London
Bryson AE Jr, Ho YC (1975) Applied optimal control. Hemisphere, Washington, DC
Carlson DA, Haurie A, Leizarowitz A (1991) Infinite horizon optimal control: deterministic and stochastic systems, vol 332. Springer, Berlin/New York
Case J (1969) Toward a theory of many player differential games. SIAM J Control 7(2):179–197
Dockner E, Jørgensen S, Long NV, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge
Engwerda J (2005) Linear-quadratic dynamic optimization and differential games. Wiley, New York
Friedman A (1971) Differential games. Wiley-Interscience, New York
Hämäläinen RP, Kaitala V, Haurie A (1984) Bargaining on whales: a differential game model with Pareto optimal equilibria. Oper Res Lett 3(1):5–11
Haurie A, Pohjola M (1987) Efficient equilibria in a differential game of capitalism. J Econ Dyn Control 11:65–78
Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games. World scientific now series in business. World Scientific, Singapore/Hackensack
Isaacs R (1965) Differential games. Wiley, New York
Jørgensen S, Zaccour G (2004) Differential games in marketing. Kluwer, Boston
Krassovski NN, Subbotin AI (1977) Jeux differentiels. Mir, Moscow
Lee B, Markus L (1972) Foundations of optimal control theory. Wiley, New York
Leitmann G (1974) Cooperative and non-cooperative many players differential games. Springer, New York
Mehlmann A (1988) Applied differential games. Springer, New York
Selten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55
Sethi SP, Thompson GL (2006)Optimal control theory: applications to management science and economics. Springer, New York. First edition 1981
Starr AW, Ho YC (1969) Nonzero-sum differential games, Part I. J Optim Theory Appl 3(3): 184–206
Starr AW, Ho YC (1969) Nonzero-sum differential games, Part II. J Optim Theory Appl 3(4): 207–219
Tolwinski B, Haurie A, Leitmann G (1986) Cooperative equilibria in differential games. J Math Anal Appl 119:182–202
Varaiya P, Lin J (1963) Existence of saddle points in differential games. SIAM J Control Optim 7:142–157
von Stackelberg H (1934) Marktform und Gleischgewicht. Springer, Vienna [An English translation appeared in 1952 entitled The theory of the market economy. Oxford University Press, Oxford]
Yeung DWK, Petrosjan L (2005) Cooperative stochastic differential games. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this entry
Cite this entry
Başar, T., Haurie, A., Zaccour, G. (2018). Nonzero-Sum Differential Games. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-44374-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44373-7
Online ISBN: 978-3-319-44374-4
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering