A report on NISOCSol: An algorithm for approximating Markovian equilibria in dynamic games with coupled-constraints

Krawczyk, Jacek B. and Azzato, Jeffrey D. (2006): A report on NISOCSol: An algorithm for approximating Markovian equilibria in dynamic games with coupled-constraints.

Abstract

In this report, we outline a method for approximating a Markovian (or feedback-Nash) equilibrium of a dynamic game, possibly subject to coupled-constraints. We treat such a game as a "multiple" optimal control problem. A method for approximating a solution to a given optimal control problem via backward induction on Markov chains was developed in [Kra01]. A Markovian equilibrium may be obtained numerically by adapting this backward induction approach to a stage Nikaido-Isoda function (described in [KZ06]).

• Item Type: MPRA Paper
• Institution: Victoria University of Wellington
• Language: English
• Keywords: Computational techniques; Noncooperative games; Econometric software; Taxation; Water; Climate; Dynamic programming; Dynamic games; Applications of game theory; Environmental economics; Computational economics; Nikaido-Isoda function; Approximating Markov decision chains
• Subjects: C - Mathematical and Quantitative Methods > C8 - Data Collection and Data Estimation Methodology ; Computer Programs > C87 - Econometric Software; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling; Q - Agricultural and Natural Resource Economics ; Environmental and Ecological Economics > Q2 - Renewable Resources and Conservation > Q25 - Water; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; E - Macroeconomics and Monetary Economics > E6 - Macroeconomic Policy, Macroeconomic Aspects of Public Finance, and General Outlook > E62 - Fiscal Policy

References

[AD54] K. J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22(3):265–290, 1954.

[AK06] Jeffrey D. Azzato and Jacek B. Krawczyk. SOCSol4L: An improved MATLAB package for approximating the solution to a continuous-time stochastic optimal control problem. Working paper, School of Ecnomics and Finance, Victoria University of Wellington, Dec 2006. Available at http://mpra.ub.uni-muenchen.de/10015/ on 26/08/2008.

[BO82] T. Basar and G. J. Olsder. Dynamic Noncooperative Game Theory. Academic Press, New York, 1982.

[CKK04] Javier Contreras, Matthias Klusch, and Jacek B. Krawczyk. Numerical solutions to Nash–Cournot equilibria in coupled constraint electricity markets. IEEE T. Power Syst., 19(1):195–206, Feb 2004.

[HK97] Alain Haurie and Jacek B. Krawczyk. Optimal charges on river effluent from lumped and distributed sources. Environ. Model. Assess., 2(3):177–189, Oct 1997. DOI: 10.1023/A:1019049008557.

[HP07] Benjamin F. Hobbs and J.-S. Pang. Nash–Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints. 55(1):113–127, Jan/Feb 2007. DOI: 10.1287/opre.1060.0342.

[Kra01] Jacek B. Krawczyk. A Markovian approximated solution to a portfolio management problem. ITEM., 1(1), 2001. Available at http://www.item.woiz.polsl.pl/issue/journal1.htm on 26/08/2008.

[Kra05a] J. B. Krawczyk. Numerical solutions to lump-sum pension problems that can yield left skewed fund return distributions. In Christophe Deissenburg and Richard F. Hartl, editors, Optimal Control and Dynamic Games, number 7 in Advances in Computational Management Science, chapter 10, pages 155–176. Springer, New York, 2005.

[Kra05b] Jacek B. Krawczyk. Coupled constraint Nash equilibria in environmental games. Resource Energy Econ., 27(2):157–181, Jun 2005. DOI: 10.1016/j.reseneeco.2004.08.001.

[Kra06] Jacek B. Krawczyk. Coupled constraint Markovian equilibria in dynamic games of compliance. Seminar at the Kyoto University Institute of Economic Research, 30 Nov 2006.

[KT06] J. B. Krawczyk and M. Tidball. A discrete-time dynamic game of seasonal water allocation. J. Optim. Theory Appl., 128(2):411–429, Feb 2006. DOI: 10.1007/s10957-006-9020-0.

[KU00] Jacek B. Krawczyk and Stanislav Uryasev. Relaxation algorithms to find Nash equilibria with economic applications. Environ. Model. Assess., 5(1):63–73, Jan 2000. DOI: 10.1023/A:1019097208499.

[KZ06] Jacek B. Krawczyk and James Zuccollo. NIRA-3 An improved MATLAB package for finding Nash equilibria in infinite games. Working paper, School of Economics and Finance, Victoria University of Wellington, Dec 2006. Available at http://mpra.ub.uni-muenchen.de/1119/ on 26/08/2008.

[McK59] Lionel W. McKenzie. On the existence of general equilibrium for a competitive market. Econometrica, 27(1):54–71, Jan 1959.

[NI55] Hukukane Nikaidô and Kazuo Isoda. Note on noncooperative games. Pac. J. Math., 5(1):807–815, 1955.

[Ros65] J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, Jul 1965.