Abstract
A new approach to the approximate solution of matrix games is proposed. It is based on the reduction of the original problem to a variational inequality of a special form. In particular, this makes it possible to design preconditioned iterative methods, which proved to be effective as a tool for the numerical solution of large and ill-conditioned systems of linear algebraic equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Karmarkar, “A New Polynomial-Time Algorithm for Linear Programming,” Combinatorica 4, 373–395 (1984).
M. H. Wright, “The Interior-Point Revolution in Optimization: History, Recent Developments, and Lasting Consequences,” Bull. Amer. Math. Soc. 42(1), 39–56 (2004).
A. S. Nemirovski and M. J. Todd, “Interior-Point Methods for Optimization,” Acta Numerica 17, 191–234 (2008).
F. P. Vasil’ev and A. Yu. Ivanitskii, Linear Programming (Faktorial, Moscow, 1998) [in Russian].
L. G. Khachiyan, Complexity of Linear Programming Problems (Znanie, Moscow, 1987) [in Russian].
V. Z. Belen’kii, V. A. Volkonskii, S. A. Ivankov, et al., Iterative Methods in Game Theory and Programming (Nauka, Moscow, 1974) [in Russian].
E. G. D’yakonov, “On the Use of Spectrally Equivalent Operators for Solving Difference Analogs of Strongly Elliptic Systems,” Dokl. Akad. Nauk SSSR 163, 1314–1317 (1965).
E. G. D’yakonov, “The Construction of Iterative Methods Based on the Use of Spectrally Equivalent Operators,” Zh. Vychisl. Mat. Mat. Fiz. 6, 12–34 (1966).
E. V. Chizhonkov, Relaxation Methods for Saddle-Point Problems (IVM RAN, Moscow, 2002) [in Russian].
I. V. Konnov, Methods for Solving Finite-Dimensional Variational Inequalities: A Course of Lectures (DAS, Kazan, 1998) [in Russian].
A. V. Lapin, Iterative Methods for Solving Network Variational Inequalities (Izd-vo Kazanskogo Gos. Univ., Kazan, 2008) [in Russian].
J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton, 1953; Nauka, Moscow 1970).
N. N. Vorob’ev, Game Theory for Ecomomists and Cyberneticians (Nauka, Moscow, 1985) [in Russian].
Matrix Games, Ed. by Vorob’ev N. N. (Gosfizmatlit, Moscow, 1961) [in Russian].
D. Gale, H. W. Kuhn, and A. W. Tucker, “On Symmetric Games,” in Contributions to the Theory of Games Ann. Math. Study 1 24 (Princeton Univ. Press, Princeton, New York, 1950), Vol. 1, pp. 81–87.
S. I. Gaas, P. M. Zafra, and Z. Qui, “Modified Fictitious Play,” Naval Res. Logistics 43, 955–977 (1996).
A. Washburn, “A New Kind of Fictitious Play,” Naval Res. Logistics 48, 269–280 (2001).
V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].
V. I. Lebedev, Functional Analysis and Computational Mathematics (Fizmatlit, Moscow, 2005) [in Russian].
E. G. D’yakonov, “On Certain Classes of Saddle Gradient Methods,” in Computational Processes and Systems (Nauka, Moscow, 1987), No. 5, pp. 101–115 [in Russian].
R. Glowinski, J.-L. Lions, and R. Trémoliéres, Analyse numéerique des inéquations variationnelles (Dunod, Paris, 1976; Mir, Moscow, 1979).
B. N. Parlett, The Symmetric Eigenvalue Problem (Mir, Moscow, 1983; Prentice-Hall, Englewood Cliffs, N.J., 1980).
L. N. Vaserstein, Matrix Games, Linear Programming, and Linear Approximation, arXiv.org, math.cs/0609056v1[cs.GT], 2006.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to cherished memory of E.G. D’yakonov
Original Russian Text © E.V. Chizhonkov, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 8, pp. 1367–1380.
Rights and permissions
About this article
Cite this article
Chizhonkov, E.V. Iterative solution of matrix games using grid variational inequalities. Comput. Math. and Math. Phys. 50, 1299–1311 (2010). https://doi.org/10.1134/S0965542510080038
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542510080038