Slow and Steady Wins the Race: Approximating Nash Equilibria in Nonlinear Quadratic Tracking Games Steter Tropfen höhlt den Stein: Approximation von Nash Gleichgewichten in Nicht-linearen Dynamischen Spielen

Ivan Savin 1 , Dmitri Blueschke 2 ,  and Viktoria Blueschke-Nikolaeva 3
  • 1 Chair for Economic Policy, Ural Federal University, Russian Federation, Karlsruhe, Germany
  • 2 Alpen-Adria-Universität Klagenfurt, Klagenfurt am Wörthersee, Klagenfurt, Austria
  • 3 Alpen-Adria-Universität Klagenfurt, Klagenfurt am Wörthersee, Klagenfurt, Austria
Ivan Savin
  • Corresponding author
  • Chair for Economic Policy, Ural Federal University, Russian Federation, Karlsruhe, Germany
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, Dmitri Blueschke and Viktoria Blueschke-Nikolaeva

Abstract

We propose a new method for solving nonlinear dynamic tracking games using a meta-heuristic approach. In contrast to ‘traditional’ methods based on linear-quadratic (LQ) techniques, this derivative-free method is very flexible with regard to the objective function specification. The proposed method is applied to a three-player dynamic game and tested versus a derivative-dependent method in approximating solutions of different game specifications. In particular, we consider a dynamic game between fiscal (played by national governments) and monetary policy (played by a central bank) in a monetary union. Apart from replicating results of the LQ-based techniques in a standard setting, we solve two ‘non-standard’ extensions of this game (dealing with an inequality constraint in a control variable and introducing asymmetry in penalties of the objective function), identifying both a cooperative Pareto and a non-cooperative open-loop Nash equilibria, where the traditional methods are not applicable. We, thus, demonstrate that the proposed method allows one to study more realistic problems and gain better insights for economic policy.

  • Assenmacher-Wesche, K. (2006), Estimating Central Banks’ Preferences from a Time-varying Empirical Reaction Function. European Economic Review 50 (8): 1951–1974.

  • Başar, T., Olsder G.J. (1998), Dynamic Noncooperative Game Theory. Philadelphia, SIAM.

  • Bednar-Friedl, B., Behrens D., Getzner M. (2012), Optimal Dynamic Control of Visitors and Endangered Species in a National Park. Environmental and Resource Economics 52 (1): 1–22.

  • Bellman, R. (1957), Dynamic Programming. Princeton University Press, Princeton.

  • Blueschke, D., Neck R. (2011), “Core” and “Periphery” in a Monetary Union: A Macroeconomic Policy Game. International Advances in Economic Research 17: 334–346.

    • Crossref
    • Export Citation
  • Blueschke, D., Savin I. (2017), No Such Thing Like Perfect Hammer: Comparing Different Objective Function Specifications for Optimal Control. Central European Journal of Operations Research 25: 377–392.

    • Crossref
    • PubMed
    • Export Citation
  • Blueschke, D., Blueschke-Nikolaeva V., Savin I. (2013a), New Insights into Optimal Control of Nonlinear Dynamic Econometric Models: Application of a Heuristic Approach. Journal of Economic Dynamics and Control 37 (4): 821–837.

    • Crossref
    • Export Citation
  • Blueschke, D., Neck R., Behrens D.A. (2013b), OPTGAME3: A Dynamic Game Solver and an Economic Example. 29–51 in: V. Krivan, G. Zaccour (eds.), Advances in Games Dynamic. Theory, Applications, and Numerical Methods, Birkhäuser Verlag. Basel, Switzerland.

  • Boryczka, U., Juszczuk P. (2013), Differential Evolution as a New Method of Computing Nash Equilibria. in: Nguyen Ngoc Thanh (ed.), Transactions on Computational Collective Intelligence IX. Lecture Notes in Science Computer, vol 7770, Springer. Craiova; Romania.

  • Brumm, J., Scheidegger S. (2017), Using Adaptive Sparse Grids to Solve High-dimensional Dynamic Models. Econometrica 85 (5): 1575–1612.

  • Cruz, I., Willigenburg L., Straten G. (2003), Efficient Differential Evolution Algorithms for Multimodal Optimal Control Problems. Applied Soft Computing 3: 97–122.

    • Crossref
    • Export Citation
  • Cukierman, A. (2002), Are Contemporary Central Banks Transparent about Economic Models and Objectives and What Difference Does it Make? The Federal Reserve Bank of St Louis, pp. 15–35.

  • Das, S., Suganthan N. (2011), Differential Evolution: A Survey of the State-of-the-art. IEEE Transactions on Evolutionary Computation 15 (1): 4–31.

  • Dennis, R. (2004), Inferring Policy Objectives from Economic Outcomes. Oxford Bulletin of Economics and Statistics 66: 735–764.

    • Crossref
    • Export Citation
  • Dockner, E.J., Long N.V. (1993), International Pollution Control: Cooperative versus Noncooperative Strategies. Journal of Environmental Economics and Management 25 (1): 13–29.

  • Eggertsson, G., Woodford M. (2003), Zero Bound on Interest Rates and Optimal Monetary Policy. Brookings Papers on Economic Activity, pp. 139–233.

  • Engwerda, J., van Aarle B., Plasmans J. (2002), Cooperative and Non-cooperative Fiscal Stabilization Policies in the EMU. Journal of Economic Dynamics and Control 26 (3): 451–481.

  • Friedman, B.M. (1972), Optimal Economic Stabilization Policy: An Extended Framework. Journal of Political Economy 80: 1002–1022.

    • Crossref
    • Export Citation
  • Gilli, M., Schumann E. (2011), Optimal Enough? Journal of Heuristics 17 (4): 373–387.

    • Crossref
    • Export Citation
  • Gilli, M., Schumann E. (2014), Optimization Cultures. WIREs Computational Statistics 6 (5): 352–358.

    • Crossref
    • Export Citation
  • Gilli, M., Winker P. (2009), Heuristic Optimization Methods in Econometrics. 81–119 in: D. Belsley, E. Kontoghiorghes (eds.), Handbook of Computational Econometrics. Wiley, Chichester.

  • Herrmann, J., Savin I. (2017), Optimal Policy Identification: Insights from the German Electricity Market. Technological Forecasting and Social Change 122: 71–90.

    • Crossref
    • Export Citation
  • Krause, MU., Moyen S. (2016), Public Debt and Changing Inflation Targets. American Economic Journal: Macroeconomics 8 (4): 142–176.

  • Leitmann, G. (1974), Cooperative and Non-Cooperative Many Player Differential Games. Springer, Vienna.

  • Leon, M., Xiong N. (2016), Adapting Differential Evolution Algorithms for Continuous Optimization via Greedy Adjustment of Control Parameters. Journal of Artificial Intelligence and Soft Computing Research 6 (2): 103–118.

  • Mandler, M. (2012), Decomposing Federal Funds Rate Forecast Uncertainty Using Time-varying Taylor Rules and Real-time Data. The North American Journal of Economics and Finance 23 (2): 228–245.

  • Modares, H., Sistani M. (2011), Solving Nonlinear Optimal Control Problems Using a Hybrid IPSO-SQP Algorithm. Engineering Applications of Artificial Intelligence 24: 476–484.

    • Crossref
    • Export Citation
  • Nakov, A. (2008), Optimal and Simple Monetary Policy Rules with Zero Floor on the Nominal Interest Rate. International Journal of Central Banking 4 (2): 73–127.

  • Neck, R., Blueschke D. (2014), “Haircuts” for the EMU Periphery: Virtue or Vice? Empirica 41 (2): 153–175.

  • Nobay, A.R., Peel D.A. (2003), Optimal Discretionary Monetary Policy in a Model of Asymmetric Central Bank Preferences. Economic Journal 113: 657–665.

    • Crossref
    • Export Citation
  • Oliveira, H. (2016), Nash Equilibria of Finite Strategic Games and Fuzzy ASA. 77–92. in: Optimization Evolutionary Global, Manifolds and Applications, Studies in Systems, Decision and Control, Heidelberg vol 43.

    • Crossref
    • Export Citation
  • Oliveira, H., Petraglia A. (2014), Establishing Nash Equilibria of Strategic Games: A Multistart Fuzzy Adaptive Simulated Annealing Approach. Applied Soft Computing 19: 188–197.

    • Crossref
    • Export Citation
  • Pontryagin, L., Boltyanskii V., Gamkrelidze R.V., Mishchenko E.F. (1962), The Mathematical Theory of Optimal Processes. English Translation: Interscience, New York.

  • Savin, I., Blueschke D. (2016), Lost in Translation: Explicitly Solving Nonlinear Stochastic Optimal Control Problems Using the Median Objective Value. Computational Economics 48 (2): 317–338.

  • Storn, R., Price K. (1997), Differential Evolution: A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces. Journal of Global Optimization 11: 341–359.

    • Crossref
    • Export Citation
  • Varian, H. (1974), A Bayesian Approach to Real Estate Assessment. in: Feinberg S., Zellner A. (eds.), Studies in Bayesian Economics in Honour of L.J. Savage, North-Holland, Amsterdam.

  • Zellner, A. (1986), Bayesian Estimation and Prediction Using Asymmetric Loss Functions. Journal of the American Statistical Association 81: 446–451.

    • Crossref
    • Export Citation
  • Zhuang, Y., Huang H. (2014), Time-optimal Trajectory Planning for Underactuated Spacecraft Using a Hybrid Particle Swarm Optimization Algorithm. Acta Astronautica 94 (2): 690–698.

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The Journal of Economics and Statistics is a scientific journal published in Germany since 1863. The Journal publishes papers in all fields of economics and applied statistics. A specific focus is on papers combining theory with empirical analyses.

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