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Non-linear strategies in a linear quadratic differential game

  • Colin Rowat

    (University of Birmingham)

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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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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  1. 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.
  2. 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.
  3. 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.
  4. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, June.
  5. 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.
  6. Mäler, K-G. & Xepapadeas, A. & de Zeeuw, A.J., 2003. "The economics of shallow lakes," Other publications TiSEM 368f83ad-bc2f-4ad4-b603-8, Tilburg University, School of Economics and Management.
  7. 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.
  8. Le Van, C. & Morhaim, L., 2000. "Optimal Growth Models with Bounded or Unbounded Returns : a Unifying Approach," Papiers d'Economie Mathématique et Applications 2000.64, Université Panthéon-Sorbonne (Paris 1).
  9. Dutta, Prajit K & Sundaram, Rangarajan K, 1993. "The Tragedy of the Commons?," Economic Theory, Springer, vol. 3(3), pages 413-26, July.
  10. Vencatachellum, Desire, 1998. "Endogenous growth with strategic interactions," Journal of Economic Dynamics and Control, Elsevier, vol. 23(2), pages 233-254, September.
  11. 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.
  12. 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.
  13. Colin Rowat, 2005. "Non-Linear Strategies in a Linear Quadratic Differential Game," Discussion Papers 05-05, Department of Economics, University of Birmingham.
  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. & 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.
  16. 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.
  17. 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.
  18. 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.
  19. 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.
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