Abstract
Consideration was given to the properties of the polymatrix game, a finite noncooperative game of N players (N ⩾ 3). A theorem of reduction of the search for Nash equilibria to an optimization problem was proved. This clears the way to the numerical search of equilibria. Additionally, a simple proof of the Nash theorem of existence of equilibrium in the polymatrix game was given using the theorem of existence of solution in the optimization problem.
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References
Neumann, J. and Morgenshtern, O., Theory of Games and Economic Behavior, Princeton: Princeton Univ. Press, 1953. Translated under the title Teoriya igr i ekonomicheskoe povedenie, Moscow: Nauka, 1970.
Nash, J.F., Equilibrium Points in n-Person Games, Proc. Nat. Acad. Sci. USA, 1950, vol. 36, pp. 48–49.
Vasin, A.A. and Morozov, V.V., Teoriya igr i modeli mathematicheskoi ekonomiki. Uch. pos. (Textbook of the Game Theory and Models of Mathematical Economics), Moscow: MAKS Press, 2005.
Vorob’ev, N.N., Teoriya igr dlya ekonomistov i kibernetikov (Game Theory for Economists and Cyberneticians), Moscow: Nauka, 1985.
Germei’er, Yu.B., Igry s neprotivopolozhnymi interesami (Games with Noncontradictory Interests), Moscow: Nauka, 1976.
Kuhn, H.W., An Algorithm for Equilibrium Points in Bimatrix Games, Proc. Nat. Acad. Sci., 1961, vol. 47, pp. 1657–1662.
Lemke, C.E. and Howson, T.T., Equilibrium Points of Bimatrix Games, SIAM J. Appl. Math., 1961, vol. 12, pp. 413–423.
Dickhaut, J. and Kaplan, T., A Program for Finding Nash Equilibria, Math. J., 1991, no. 1, pp. 87–93.
McKelvey, R.D. and McLennan, A., Computation of Equilibria in Finite Games, in Handbook of Computational Economics, Amman, H.M., Kendrick, D.A., and Rust, J., Eds., Amsterdam: Elsevier, 1996, vol. 1, pp. 87–142.
Howson, J.T., Equilibria of Polymatrix Games, Manag. Sci., 1972, vol. 18, pp. 312–318.
Audet, C., Belhaiza, S., and Hansen, P., Enumeration of All the Extreme Equilibria in Game Theory: Bimatrix and Polymatrix Games, J. Optim. Theory Appl., 2006, vol. 129, no. 3, pp. 349–372.
Strekalovskii, A.S. and Orlov, A.V., Bimatrichnye igry i bilineinoe programmirovanie (Bymatrix Games and Bilinear Programming), Moscow: Fizmatlit, 2007.
Vasil’ev, I.L., Klimentova, K.B., and Orlov, A.V., Parallel Global Search of Equilibrium Situations in the Bimatrix Games, Vychisl. Metod. Progr., 2007, vol. 8, pp. 233–243.
Yanovskaya, E.B., Equilibrium Points in Polymatrix Games, Latvian Math. Collect., 1968, vol. 8, pp. 381–384.
Mills, H., Equilibrium Points in Finite Games, J. Soc. Ind. Appl. Math., 1960, vol. 8, no. 2, pp. 397–402.
Mangasarian, O.L., Equilibrium Points in Bimatrix Games, J. Soc. Ind. Appl. Math., 1964, vol. 12, pp. 778–780.
Mangasarian, O.L. and Stone, H., Two-Person Nonzero Games and Quadratic Programming, J. Math. Anal. Appl., 1964, no. 9, pp. 348–355.
Strekalovskii, A.S., Elementy nevypukloi optimizatsii (Elements of Nonconvex Optimization), Novosibirsk: Nauka, 2003.
Pang, J.-S., Three Modelling Paradigms in Mathematical Programming, Math. Program., Ser. B, 2010, vol. 125, pp. 297–323.
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Original Russian Text © A.S. Strekalovskii, R. Enkhbat, 2014, published in Avtomatika i Telemekhanika, 2014, No. 4, pp. 51–66.
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Strekalovskii, A.S., Enkhbat, R. Polymatrix games and optimization problems. Autom Remote Control 75, 632–645 (2014). https://doi.org/10.1134/S0005117914040043
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DOI: https://doi.org/10.1134/S0005117914040043