Skip to main content
Log in

Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

The intrinsic comparative dynamics of locally differentiable feedback Stackelberg equilibria are derived for the ubiquitous class of autonomous and exponentially discounted infinite horizon differential games. It is shown that the follower’s intrinsic comparative dynamics agree in their form and qualitative properties with those of every player in a feedback Nash equilibrium, while those of the leader differ in form. The difference allows, in principle, an empirical test of the leader-follower role in a differential game. Separability conditions are identified on the instantaneous payoff and transition functions under which the intrinsic comparative dynamics of feedback Nash equilibria, feedback Stackelberg equilibria, and those in the corresponding optimal control problem are qualitatively identical.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amrouche N, Martin-Herran G, Zaccour G (2008) Feedback Stackelberg equilibrium strategies when the private label competes with the national brand. Ann Oper Res 164:79–95

    Article  MATH  MathSciNet  Google Scholar 

  2. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. Society for Industrial and Applied Mathematics, Philadelphia

    MATH  Google Scholar 

  3. Breton M, Jarrar R, Zaccour G (2006) A note on feedback sequential equilibria in a Lanchester model with empirical application. Manag Sci 52:804–811

    Article  MATH  Google Scholar 

  4. Caputo MR (1998) A dual vista of the Stackelberg duopoly reveals its fundamental qualitative structure. Int J Ind Organ 16:333–352

    Article  Google Scholar 

  5. Caputo MR (2003) The comparative dynamics of closed-loop controls for discounted infinite horizon optimal control problems. J Econ Dyn control 27:1335–1365

    Article  MATH  MathSciNet  Google Scholar 

  6. Caputo MR, Ling C (2013) The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games. J Econ Dyn Control 37:1982–1994

    Article  MathSciNet  Google Scholar 

  7. Dockner EJ, Jørgensen S, Long NV, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  8. Fujiwara K, Long NV (2011) Welfare implications of leadership in a resource market under bilateral monopoly. Dyn Games Appl 1:479–497

    Article  MATH  MathSciNet  Google Scholar 

  9. Groot F, Withagen C, de Zeeuw A (2003) Strong time-consistency in the cartel-versus-fringe model. J Econ Dyn Control 28:287–306

    Article  MATH  Google Scholar 

  10. 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 16:385–413

    Article  Google Scholar 

  11. Long NV (2010) A survey of dynamic games in economics. World Scientific, Singapore

    Book  Google Scholar 

  12. Long NV (2011) Dynamic games in the economics of natural resources: a survey. Dyn Games Appl 1:115–148

    Article  MATH  MathSciNet  Google Scholar 

  13. Long NV, Sorger G (2010) A dynamic principal-agent problem as a feedback Stackelberg differential game. Cent Eur J Oper Res 18:491–509

    Article  MATH  MathSciNet  Google Scholar 

  14. Mehlmann A (1988) Applied differential games. Plenum Press, New York

    Book  MATH  Google Scholar 

  15. Novak AJ, Feichtinger G, Leitmann G (2010) A differential game related to terrorism: Nash and Stackelberg strategies. J Optim Theory Appl 144:535–555

    Article  MathSciNet  Google Scholar 

  16. Partovi MH, Caputo MR (2006) A complete theory of comparative statics for differentiable optimization problems. Metroeconomica 57:31–67

    Article  MATH  Google Scholar 

  17. Partovi MH, Caputo MR (2007) Erratum: a complete theory of comparative statics for differentiable optimization problems. Metroeconomica 58:360

    Article  Google Scholar 

  18. Rubio SJ, Escriche L (2001) Strategic pigouvian taxation, stock externalities and polluting non-renewable resources. J Public Econ 79:297–313

    Article  Google Scholar 

  19. Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, Cambridge

    MATH  Google Scholar 

  20. Shimomura K, Xie D (2008) Advances on Stackelberg open-loop and feedback strategies. Int J Econ Theory 4:115–133

    Article  Google Scholar 

  21. Tahvonen O (1996) Trade with polluting nonrenewable resources. J Environ Econ Manag 30:1–17

    Article  MATH  Google Scholar 

  22. Wirl F (1994) Pigouvian taxation of energy for flow and stock externalities and strategic, noncompetitive energy pricing. J Environ Econ Manag 26:1–18

    Article  MATH  Google Scholar 

  23. Wirl F (2012) Global warming: prices versus quantities from a strategic point of view. J Environ Econ Manag 64:217–229

    Article  Google Scholar 

  24. Wirl F, Dockner E (1995) Leviathan governments and carbon taxes: costs and potential benefits. Eur Econ Rev 39:1215–1236

    Article  Google Scholar 

  25. Xie J, Ai S (2006) A note on “cooperative advertising, game theory and manufacturer retailer supply chains”. Omega 34:501–504

    Article  Google Scholar 

Download references

Acknowledgments

We would like to express our gratitude to two referees, whose comments and concerns have led us to make changes in our paper for the better. Chen Ling acknowledges the financial support from the National Natural Science Foundation of China (No. 71401127).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chen Ling.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caputo, M.R., Ling, C. Intrinsic Comparative Dynamics of Locally Differentiable Feedback Stackelberg Equilibria. Dyn Games Appl 5, 1–25 (2015). https://doi.org/10.1007/s13235-014-0121-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-014-0121-3

Keywords

JEL Classification

Navigation