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Noncooperative Differential Games

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Abstract

This paper, having a tutorial character, is intended to provide an introduction to the theory of noncooperative differential games.

Section 2 reviews the theory of static games. Different concepts of solution are discussed, including Pareto optima, Nash and Stackelberg equilibria, and the co-co (cooperative-competitive) solutions.

Section 3 introduces the basic framework of differential games for two players. Open-loop solutions, where the controls implemented by the players depend only on time, are considered in Section 4. These solutions can be computed by solving a two-point boundary value problem for a system of ODEs, derived from the Pontryagin maximum principle. Section 5 deals with solutions in feedback form, where the controls are allowed to depend on time and also on the current state of the system. In this case, the search for Nash equilibrium solutions leads to a highly nonlinear system of Hamilton-Jacobi PDEs. In dimension higher than one, we show that this system is generically not hyperbolic and the Cauchy problem is thus ill posed. Due to this instability, feedback solutions are mainly considered only in the special case with linear dynamics and quadratic costs.

In Section 6, a game in continuous time is approximated by a finite sequence of static games, by a time discretization. Depending of the type of solution adopted in each static game, one obtains different concepts of solutions for the original differential game.

Section 7 deals with differential games in infinite time horizon, with exponentially discounted payoffs. Section 8 contains a simple example of a game with infinitely many players. This is intended to convey a flavor of the newly emerging theory of mean field games.

Modeling issues, and directions of current research, are briefly discussed in Section 9. Finally, the Appendix collects background material on multivalued functions, selections and fixed point theorems, optimal control theory, and hyperbolic PDEs.

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References

  1. Aubin J.P.: Optima and equilibria. Second edition. Springer-Verlag, Berlin (1998)

    Google Scholar 

  2. Aubin J.P., Cellina A.: Differential inclusions. Set-valued maps and viability theory. Springer-Verlag, Berlin (1984)

    MATH  Google Scholar 

  3. Aumann R.: Rationality and bounded rationality. Games Econom. Behav. 21, 214 (1997)

    Article  MathSciNet  Google Scholar 

  4. R. J. Aumann, Game engineering. In: Mathematical programming and game theory for decision making, 279-285, World Scientific Publ., Hackensack, NJ, 2008.

  5. Aumann R., Brandenburger A.: Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakushinsky A., Goncharsky A.: Ill-posed problems: theory and applications. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  7. Bardi M., Capuzzo Dolcetta I., (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser

  8. Benzoni-Gavage S., Serre D.: Multidimensional hyperbolic partial differential equations. First-order systems and applications. Oxford University Press, Oxford (2007)

    Google Scholar 

  9. Basar T., Olsder G.J.: Dynamic Noncooperative Game Theory. Reprint of the second edition. SIAM, Philadelphia (1999)

    Google Scholar 

  10. Bressan A.: Bifurcation analysis of a noncooperative differential game with one weak player. J. Differential Equations 248, 1297–1314 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series in Applied Mathematics, Springfield Mo. 2007.

  12. Bressan A., Priuli F.: Infinite horizon noncooperative differential games. J. Differential Equations 227, 230–257 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bressan A., Shen W.: Small BV solutions of hyperbolic noncooperative differential games. SIAM J. Control Optim. 43, 104–215 (2004)

    Article  MathSciNet  Google Scholar 

  14. Bressan A., Shen W.: Semi-cooperative strategies for differential games. Intern. J. Game Theory 32, 561–593 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Cardaliaguet P., Rainer C.: On a continuous-time game with incomplete information. Math. Oper. Res. 34, 769–794 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cellina A.: Approximation of set valued functions and fixed point theorems. Ann. Mat. Pura Appl. 82, 17–24 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Cesari, Optimization Theory and Applications, Springer-Verlag, 1983.

  18. Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R.: Nonsmooth Analysis and Control Theory. Springer-Verlag, New York (1998)

    MATH  Google Scholar 

  19. A. Cournot, Recherches sur les principes mathématiques de la théorie des richesses, L. Hachette, Paris, 1838.

  20. E. J. Dockner, S. Jorgensen, N. V. Long, and G. Sorger, Differential games in economics and management science. Cambridge University Press, 2000.

  21. Evans L.C.: Partial Differential Equations Second edition. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  22. Fleming W.H., Rishel R.W.: Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975)

    MATH  Google Scholar 

  23. Friedman A.: Differential Games. Wiley, New York (1971)

    MATH  Google Scholar 

  24. Friedman A.: Stochastic differential games. J. Differential Equations 11, 79–108 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  25. O. Guéant, J. M. Lasry and P. L. Lions, Mean field games and applications. in: Paris-Princeton Lectures on Mathematical Finance 2010, Springer Lecture Notes in Mathematics, Vol. 2003, pp. 205-266.

  26. Isaacs R.: Differential Games. Wiley, New York (1965)

    MATH  Google Scholar 

  27. A. T. Kalai and E. Kalai, A cooperative value for Bayesian games, preprint, 2010.

  28. Krasovskii N.N., Subbotin A.I.: Game-Theoretical Control Problems. Springer-Verlag, Berlin (1988)

    Book  MATH  Google Scholar 

  29. Lasry J.M., Lions P.L.: Mean field games. Japanese J. Math. 2, 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mirica S.: Verification theorems for optimal feedback strategies in differential games. Int. Game Theory Rev. 5, 167–189 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nash J.: Non-cooperative games. Annals of Math. 2, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  32. Priuli F.S.: Infinite horizon noncooperative differential games with nonsmooth costs. J. Math. Anal. Appl. 336, 156–170 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shiryaev A.N.: Probability, Second edition. Springer-Verlag, New York (1996)

    Google Scholar 

  34. Souganidis P.E.: Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 57, 1–43 (1985)

    Article  MathSciNet  Google Scholar 

  35. Srikant R., Basar T.: Iterative computation of noncooperative equilibria in nonzerosum differential games with weakly coupled players. J. Optim. Th. Appl. 71, 137–168 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. von Stackelberg H., (1952) The Theory of the Market Economy. Oxford Univ Press

  37. Vinter R.B.: Optimal Control. Birkhaüser, Boston (2000)

    MATH  Google Scholar 

  38. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Third edition. Princeton University Press, 1980.

  39. J. Wang, The Theory of Games. Oxford University Press, (1988)

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Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, Pa., 16802, U.S.A.

    Alberto Bressan

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Correspondence to Alberto Bressan.

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Lecture held on February 25, 2010, by Alberto Bressan, recipient of the Luigi and Wanda Amerio gold medal awarded by the Istituto Lombardo Accademia di Scienze e Lettere.

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Bressan, A. Noncooperative Differential Games. Milan J. Math. 79, 357–427 (2011). https://doi.org/10.1007/s00032-011-0163-6

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  • DOI: https://doi.org/10.1007/s00032-011-0163-6

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