A Qualitative Characterization of Symmetric Open-Loop Nash Equilibria in Discounted Infinite Horizon Differential Games

  • C. Ling
  • M. R. Caputo


The local stability, steady state comparative statics, and local comparative dynamics of symmetric open-loop Nash equilibria for the ubiquitous class of discounted infinite horizon differential games are investigated. It is shown that the functional forms and values of the parameters specified in a differential game are crucial in determining the local stability of a steady state and, in turn, the steady state comparative statics and local comparative dynamics. A simple sufficient condition for a steady state to be a local saddle point is provided. The power and reach of the results are demonstrated by applying them to two well-known differential games.


Symmetric open-loop Nash equilibria Local stability Steady state comparative statics Local comparative dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Caputo, M.R.: The qualitative structure of a class of infinite horizon optimal control problems. Optim. Control Appl. Methods 18, 195–215 (1997) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Dockner, E., Feichtinger, G., Jørgensen, S.: Tractable classes of nonzero-sum open-loop Nash differential games: theory and examples. J. Optim. Theory Appl. 45, 179–197 (1985) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fershtman, C., Muller, E.: Capital accumulation games of infinite duration. J. Econ. Theory 33, 322–339 (1984) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Reynolds, S.S.: Capacity investment, preemption, and commitment in an infinite horizon model. Intermt. Econ. Rev. 28, 69–88 (1987) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Fershtman, C., Kamien, M.I.: Dynamic duopolistic competition with sticky prices. Econometrica 55, 1151–1164 (1987) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cellini, R.: Lambertini. L.: Dynamic oligopoly with sticky prices: closed-loop, feedback, and open-loop solutions. J. Dyn. Control Syst. 10, 303–314 (2004) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Lambertini, L.: Oligopoly with hyperbolic demand: a differential game approach. J. Optim. Theory Appl. 145, 109–119 (2010) CrossRefGoogle Scholar
  8. 8.
    Negri, D.H.: The common property aquifer as a differential game. Water Resour. Res. 25, 9–15 (1988) CrossRefGoogle Scholar
  9. 9.
    Plourde, C., Yeung, D.: Harvesting of a transboundary replenishable fish stock: a noncooperative game solution. Mar. Resour. Econ. 6, 59–70 (1989) Google Scholar
  10. 10.
    Arnason, R.: Minimum information management in fisheries. Can. J. Econ. 23, 630–653 (1990) CrossRefGoogle Scholar
  11. 11.
    Caputo, M.R., Lueck, D.: Natural resource exploitation under common property rights. Nat. Resour. Model. 16, 39–67 (2003) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Long, N.V.: Pollution control: a differential game approach. Ann. Oper. Res. 37, 283–296 (1992) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dockner, E.J., Long, V.N.: International pollution control: cooperative versus noncooperative strategies. J. Environ. Econ. Manag. 24, 13–29 (1993) CrossRefGoogle Scholar
  14. 14.
    Mehlmann, A.: Applied Differential Games. Plenum, New York (1988) MATHGoogle Scholar
  15. 15.
    Dockner, E.J., Jørgensen, S., Long, N.V., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  16. 16.
    Engwerda, J.C.: LQ Dynamic Optimization and Differential Games. Wiley, Chichester (2005) Google Scholar
  17. 17.
    Cellini, R., Lambertini, L.: A differential oligopoly game with differential goods and sticky prices. Eur. J. Oper. Res. 176, 1131–1144 (2007) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Brock, W.A.: Differential games with active and passive variables. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory: Essays in Honor of Oskar Morgenstern, pp. 34–52. Springer, Berlin (1977) Google Scholar
  19. 19.
    Dockner, E.J., Takahashi, H.: On the saddle-point stability for a class of dynamic games. J. Optim. Theory Appl. 67, 247–258 (1990) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory, 2nd edn. Academic Press, San Diego (1995) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of EconomicsSouthwestern University of Finance and EconomicsChengduChina
  2. 2.Department of EconomicsUniversity of Central FloridaOrlandoUSA

Personalised recommendations