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Non-linear strategies in a linear quadratic differential game

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  • Colin Rowat

    (University of Birmingham)

Abstract

We study non-linear Markov perfect equilibria in a two agent linear quadratic differential game. In contrast to the literature owing to Tsutsui and Mino (1990), we do not associate endogenous subsets of the state space with candidate solutions. Instead, we address the problem of unbounded-below value functions over infinite horizons by use of the `catching up optimality' criterion. We present sufficiency conditions for existence based on results in Dockner, Jorgenson, Long and Sorger (2000). Applying these to our model yields the familiar linear solution as well as a condition under which a continuum of non-linear solutions exist. As this condition is relaxed when agents are more patient, and allows more efficient steady states, it resembles a Folk Theorem for differential games. The model presented here is one of atmospheric pollution; the results apply to differential games more generally.

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Bibliographic Info

Paper provided by EconWPA in its series GE, Growth, Math methods with number 0502001.

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Length: 31 pages
Date of creation: 01 Feb 2005
Date of revision:
Handle: RePEc:wpa:wuwpge:0502001

Note: Type of Document - pdf; pages: 31. University of Birmingham Department of Economics Discussion Paper 05-05
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Web page: http://128.118.178.162

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Keywords: differential game; non-linear strategies; catching up optimal; Folk Theorem;

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  1. Karp, L., 1992. "Depreciation erodes the coase conjecture," Discussion Paper Series In Economics And Econometrics 9210, Economics Division, School of Social Sciences, University of Southampton.
  2. Rubio, Santiago J. & Casino, Begona, 2002. "A note on cooperative versus non-cooperative strategies in international pollution control," Resource and Energy Economics, Elsevier, vol. 24(3), pages 251-261, June.
  3. Le Van, Cuong & Morhaim, Lisa, 2002. "Optimal Growth Models with Bounded or Unbounded Returns: A Unifying Approach," Journal of Economic Theory, Elsevier, vol. 105(1), pages 158-187, July.
  4. Vencatachellum, Desire, 1998. "Endogenous growth with strategic interactions," Journal of Economic Dynamics and Control, Elsevier, vol. 23(2), pages 233-254, September.
  5. Dutta, Prajit K & Sundaram, Rangarajan K, 1993. "The Tragedy of the Commons?," Economic Theory, Springer, vol. 3(3), pages 413-26, July.
  6. Karl-Göran Mäler & Anastasios Xepapadeas & Aart de Zeeuw, 2003. "The Economics of Shallow Lakes," Environmental & Resource Economics, European Association of Environmental and Resource Economists, vol. 26(4), pages 603-624, December.
  7. Tsutsui, Shunichi & Mino, Kazuo, 1990. "Nonlinear strategies in dynamic duopolistic competition with sticky prices," Journal of Economic Theory, Elsevier, vol. 52(1), pages 136-161, October.
  8. Lockwood, Ben, 1996. "Uniqueness of Markov-perfect equilibrium in infinite-time affine-quadratic differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 20(5), pages 751-765, May.
  9. Dockner Engelbert J. & Van Long Ngo, 1993. "International Pollution Control: Cooperative versus Noncooperative Strategies," Journal of Environmental Economics and Management, Elsevier, vol. 25(1), pages 13-29, July.
  10. Wirl, Franz & Dockner, Engelbert, 1995. "Leviathan governments and carbon taxes: Costs and potential benefits," European Economic Review, Elsevier, vol. 39(6), pages 1215-1236, June.
  11. Maria Luisa Gota & Luigi Montrucchio, 1999. "On Lipschitz continuity of policy functions in continuous-time optimal growth models," Economic Theory, Springer, vol. 14(2), pages 479-488.
  12. Colin Rowat, 2005. "Non-linear strategies in a linear quadratic differential game," GE, Growth, Math methods, EconWPA 0502001, EconWPA.
  13. Feichtinger, Gustav & Wirl, Franz, 1993. "A Dynamic Variant of the Battle of the Sexes," International Journal of Game Theory, Springer, vol. 22(4), pages 359-80.
  14. Wirl, Franz, 2007. "Do multiple Nash equilibria in Markov strategies mitigate the tragedy of the commons?," Journal of Economic Dynamics and Control, Elsevier, vol. 31(11), pages 3723-3740, November.
  15. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, 9.
  16. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, December.
  17. Seierstad, Atle & Sydsaeter, Knut, 1977. "Sufficient Conditions in Optimal Control Theory," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(2), pages 367-91, June.
  18. Dockner, Engelbert J. & Sorger, Gerhard, 1996. "Existence and Properties of Equilibria for a Dynamic Game on Productive Assets," Journal of Economic Theory, Elsevier, vol. 71(1), pages 209-227, October.
  19. Tsutsui, Shunichi O., 1996. "Capacity constraints and voluntary output cutback in dynamic Cournot competition," Journal of Economic Dynamics and Control, Elsevier, vol. 20(9-10), pages 1683-1708.
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Citations

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Cited by:
  1. Siciliani, Luigi & Rune Straume, Odd & Cellini, Roberto, 2013. "Quality competition with motivated providers and sluggish demand," Journal of Economic Dynamics and Control, Elsevier, vol. 37(10), pages 2041-2061.
  2. Fujiwara, Kenji, 2008. "Duopoly can be more anti-competitive than monopoly," Economics Letters, Elsevier, vol. 101(3), pages 217-219, December.
  3. Dockner, E.J. & Wagener, F.O.O., 2013. "Markov–Perfect Nash Equilibria in Models With a Single Capital Stock (Revised version, March 2013)," CeNDEF Working Papers 13-03, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
  4. Rémy Dullieux & Lionel Ragot & Katheline Schubert, 2011. "Carbon tax and OPEC's rents under a ceiling constraint," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00976591, HAL.
  5. Colin Rowat, 2005. "Non-Linear Strategies in a Linear Quadratic Differential Game," Discussion Papers, Department of Economics, University of Birmingham 05-05, Department of Economics, University of Birmingham.
  6. Hans Gersbach & Noemi Hummel & Ralph Winkler, 2011. "Sustainable Climate Treaties," Diskussionsschriften, Universitaet Bern, Departement Volkswirtschaft dp1105, Universitaet Bern, Departement Volkswirtschaft.
  7. Franz Wirl, 2009. "Non-cooperative investment in partnerships and their termination," Central European Journal of Operations Research, Springer, vol. 17(4), pages 479-494, December.
  8. Wang, Hefei, 2012. "Costly information transmission in continuous time with implications for credit rating announcements," Journal of Economic Dynamics and Control, Elsevier, vol. 36(9), pages 1402-1413.
  9. Dina Tasneem & Jim Engle-Warnick & Hassan Benchekroun, 2014. "An Experimental Study of a Common Property Renewable Resource Game in Continuous Time," CIRANO Working Papers 2014s-09, CIRANO.

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