Computational Economics

, Volume 45, Issue 3, pp 407–433

Approximating Solutions for Nonlinear Dynamic Tracking Games

Article

Abstract

This paper presents the OPTGAME algorithm developed to iteratively approximate equilibrium solutions of ‘tracking games’, i.e. discrete-time nonzero-sum dynamic games with a finite number of players who face quadratic objective functions. Such a tracking game describes the behavior of decision makers who act upon a nonlinear discrete-time dynamical system, and who aim at minimizing the deviations from individually desirable paths of multiple states over a joint finite planning horizon. Among the noncooperative solution concepts, the OPTGAME algorithm approximates feedback Nash and Stackelberg equilibrium solutions, and the open-loop Nash solution, and the cooperative Pareto-optimal solution.

Keywords

Dynamic game theory Algorithm Noncooperative game Cooperative game Tracking problem Nash equilibrium Markov-perfect equilibrium Stackelberg equilibrium Pareto optimum 

References

  1. Basar, T., & Olsder, G. J. (1999). Dynamic noncooperative game theory (2nd ed.). Philadelphia: SIAM.Google Scholar
  2. Dockner, E., Jorgensen, S., Long, N. V., & Sorger, G. (2000). Differential games in economics and management science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  3. Doraszelski, U., & Satterthwaite, M. (2010). Computable Markov-perfect industry dynamics. RAND Journal of Economics, 41(2), 215–243.CrossRefGoogle Scholar
  4. Fershtman, C., & Pakes, A. (2012). Dynamic games with asymmetric information: A framework for empirical work. Quarterly Journal of Economics, 127(4), 1611–1661.CrossRefGoogle Scholar
  5. Doraszelski, U., & Markovich, S. (2007). Advertising dynamics and competitive advantage. RAND Journal of Economics, 38(3), 557–592.CrossRefGoogle Scholar
  6. Jørgensen, S., & Zaccour, G. (2004). Differential games in marketing. Boston: Kluwer.CrossRefGoogle Scholar
  7. Acoccella, N., Di Bartolomeo, G., & Hughes Hallett, A. (2013). The theory of economic policy in a strategic context. Cambridge: Cambridge University Press.Google Scholar
  8. Blueschke, D., & Neck, R. (2011). “Core” and “periphery” in a monetary union: A macroeconomic policy game. International Advances in Economic Research, 17(3), 334–346.CrossRefGoogle Scholar
  9. Neck, R., & Behrens, D. A. (2004). Macroeconomic policies in a monetary union: A dynamic game. Central European Journal of Operations Research, 12(2), 171–186.Google Scholar
  10. Neck, R., & Behrens, D. A. (2009). A macroeconomic policy game for a monetary union with adaptive expectations. Atlantic Economic Journal, 37(4), 335–349.CrossRefGoogle Scholar
  11. Petit, M. L. (1990). Control theory and dynamic games in economic policy analysis. Cambridge: Cambridge University Press.Google Scholar
  12. Plasmans, J., Engwerda, J., van Aarle, B., Di Bartolomeo, G., & Michalak, T. (2006). Dynamic modeling of monetary and fiscal cooperation among nations. Dynamic modeling and econometrics in economics and finance (Vol. 8). Dordrecht: Springer.Google Scholar
  13. Van Long, N. (2011). Dynamic games in the economics of natural resources: A survey. Dynamic Games and Applications, 1(1), 115–148.CrossRefGoogle Scholar
  14. Behrens, D. A., Caulkins, J. P., Feichtinger, G., & Tragler, G. (2007). Incentive Stackelberg strategies for a dynamic game on terrorism. In S. Jørgensen, M. Quincampoix, & V. L. Thomas (Eds.), Advances in dynamic game theory. Annals of the international society of dynamic games (Vol. 9, pp. 459–486). Boston: Birkhäuser.Google Scholar
  15. Grass, D., Caulkins, J. P., Feichtinger, G., Tragler, G., & Behrens, D. A. (2008). Optimal control of nonlinear processes. With applications in drugs, corruption, and terror. Berlin/Heidelberg: Springer.Google Scholar
  16. Novak, A. J., Feichtinger, G., & Leitmann, G. (2010). A differential game related to terrorism: Nash and Stackelberg strategies. Journal of Optimization Theory and Applications, 144(3), 533–555.CrossRefGoogle Scholar
  17. Engwerda, J. C. (2005). LQ dynamic optimization and differential games. Chichester: Wiley.Google Scholar
  18. Pachter, M., & Pham, K. D. (2010). Discrete-time linear-quadratic dynamic games. Journal of Optimization Theory and Applications, 146(1), 151–179.CrossRefGoogle Scholar
  19. W. J. McKibbin (1987). Solving rational expectations models with and without strategic behavior. Reserve Bank of Australia Research, Discussion Paper 8706.Google Scholar
  20. Michalak, T., Engwerda, J., & Plasmans, J. (2011). A numerical toolbox to solve n-player affine LQ open-loop differential games. Computational Economics, 37, 375–410.CrossRefGoogle Scholar
  21. A. Pakes, G. Gowrisankaran, and P. McGuire (1993). Implementing the Pakes-McGuire algorithm for computing Markov-perfect equilibria in GAUSS. Working Paper, Department of Economics, Yale University.Google Scholar
  22. Pakes, A., & McGuire, P. (1994). Computing Markov-perfect Nash equilibria: Numerical implications of a dynamic differentiated product model. RAND Journal of Economics, 25, 555–589.CrossRefGoogle Scholar
  23. Pakes, A., & McGuire, P. (2001). Stochastic algorithms, symmetric Markov perfect equilibrium, and the ‘curse’ of dimensionality. Econometrica, 69, 1261–1281.CrossRefGoogle Scholar
  24. Ericson, R., & Pakes, A. (1995). Markov-perfect industry dynamics: A framework for empirical work. Review of Economic Studies, 62(1), 53–82.CrossRefGoogle Scholar
  25. Farias, V., Saure, D., & Weintraub, G. Y. (2012). An approximate dynamic programming approach to solving dynamic oligopoly models. RAND Journal of Economics, 43(2), 253–282.CrossRefGoogle Scholar
  26. Anderson, B. D. O., & Moore, J. B. (1998). Optimal control: linear-quadratic methods. New York: Prentice Hall.Google Scholar
  27. Pachter, M. (2009). Revisit of linear-quadratic optimal control. Journal of Optimization Theory and Applications, 140(2), 301–314.CrossRefGoogle Scholar
  28. de Zeeuw, A. J., & van der Ploeg, F. (1991). Difference games and policy optimization: A conceptional framework. Oxford Economic Papers, 43(4), 612–636.Google Scholar
  29. Lucas, R. E. (1983). Econometric policy evaluation: A critique. In K. Brunner & A. H. Meltzer (Eds.), Theory policy, institutions: Papers from the Carnegie-Rochester conference series on public policy (pp. 257–284). Amsterdam: North-Holland.Google Scholar
  30. Behrens, D. A., Hager, M., & Neck, R. (2003). OPTGAME 1.0: A numerical algorithm to determine solutions for two-person difference games. In R. Neck (Ed.), Modelling and Control of Economic Systems 2001. A Proceedings volume from the 10th IFAC Symposium, Klagenfurt, Austria, 6–8 September 2001 (pp. 47–58). Oxford: Elsevier.Google Scholar
  31. Hager, M., Neck, R., & Behrens, D. A. (2001). Solving dynamic macroeconomic policy games using the algorithm OPTGAME 1.0. Optimal Control Applications and Methods, 22(5–6), 301–332.CrossRefGoogle Scholar
  32. Chow, G. C. (1975). Analysis and control of dynamic economic systems. New York: Wiley.Google Scholar
  33. Chow, G. C. (1981). Econometric analysis by control methods. New York: John Wiley & Sons.Google Scholar
  34. Reinganum, J. F., & Stokey, N. S. (1985). Oligopoly extraction of a common property natural resource: The importance of the period of commitment in dynamic games. International Economic Review, 26(1), 161–173.CrossRefGoogle Scholar
  35. Matulka, J., & Neck, R. (1992). OPTCON: An algorithm for the optimal control of nonlinear stochastic models. Annals of Operations Research, 37, 375–401.CrossRefGoogle Scholar
  36. S. P. Boyd (2005). Linear dynamical systems. Linear quadratic regulator: Discrete-time finite horizon. See website http://www.stanford.edu/class/ee363.
  37. Hamada, K., & Kawai, M. (1997). International economic policy coordination: Theory and policy implications. In M. U. Fratianni, D. Salvatore, & J. von Hagen (Eds.), Macroeconomic policy in open economies (pp. 87–147). Westport, CT: Greenwood Press.Google Scholar
  38. R. Neck and D. Blueschke. Policy interactions in a monetary union: An application of the OPTGAME algorithm. In J. Haunschmied, V. M. Veliov, and S. Wrzaczek, editors, Dynamic Games in Economics. Springer, Berlin, forthcoming 2014.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Controlling and Strategic ManagementAlpen-Adria Universität KlagenfurtKlagenfurt/WörtherseeAustria
  2. 2.Lakeside Labs GmbHKlagenfurt/WörtherseeAustria
  3. 3.Department of Mathematical Methods in EconomicsVienna University of TechnologyViennaAustria
  4. 4.Department of EconomicsAlpen-Adria Universität KlagenfurtKlagenfurt/WörtherseeAustria

Personalised recommendations