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## Archive of Issues

Russia Moscow
Year
2016
Volume
26
Issue
1
Pages
27-45
 Section Mathematics Title Altruistic (Berge) equilibrium in the model of Bertrand duopoly Author(-s) Bel'skikh Yu.A.a, Zhukovskii V.I.b, Samsonov S.P.b Affiliations Moscow State University of Technologies and Managementa, Lomonosov Moscow State Universityb Abstract In 1883 the French mathematician J. Bertrand (1822-1900) constructed the model of price competition on oligopoly market in which firms compete between themselves changing the price of goods. The mathematical model of Bertrand duopoly is represented by a non-cooperative game of two persons in normal form. Two equilibriums are formalized for it: Berge equilibrium (BE) and Nash equilibrium (NE). It is assumed that $a)$ maximal price and cost price of both players coincide (it's naturally for the market of one product); $b)$ the coalition of two players is prohibited (this is non-cooperative character of the game); $c)$ the price is higher than the cost price for otherwise the sellers (players) would hardly appear on the market. In the present article for almost all values of parameters of the model (except the measure-null) the constructive method of the choice of concrete equilibrium (BE or NE) depending on the maximal price of the product established in the market is suggested. Keywords non-cooperative game, Nash equilibrium, Berge equilibrium, model of Bertrand duopoly UDC 519.833 MSC 91A10, 91B26 DOI 10.20537/vm160103 Received 25 November 2015 Language Russian Citation Bel'skikh Yu.A., Zhukovskii V.I., Samsonov S.P. Altruistic (Berge) equilibrium in the model of Bertrand duopoly, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 27-45. References Bertrand J. Book review of theorie mathematique de la richesse sociale and of recherches sur les principes mathematiques de la theorie des richesses, Journal des Savants, 1883, vol. 67, pp. 499-508. Cournot A.A. Recherches sur les principes mathematiques de la theorie des richesses, Paris: Hachette, 1838. Nash J.F. Equilibrium points in $N$-person games, Proc. Natl. Acad. Sci. USA, 1950, vol. 36, no. 8, pp. 48-49. Vaisman K.S. The Berge equilibrium, Abstract of Cand. Sci. (Phys.-Math.) Dissertation, St. Petersburg, 1995, 16 p (in Russian). Vaisman K.S. The Berge equilibrium, in book: Zhukovskii V.I., Chikrii A.A. Lineino-kvadratichnye differensial'nye igry (Linear-quadratic differential games), Kiev: Naukova Dumka, 1994, Section 3.2, pp. 119-142 (in Russian). Zhukovskii V.I., Salukvadze M.E., Vaisman K.S. The Berge equilibrium: Preprint, Tbilisi: Institute of Control Systems, 1994, 28 p. Berge C. Theorie generale des jeux $n$ personnes, Paris: Gauthier-Villars, 1957, 114 p. Translated under the title Obshchaya teoriya igr neskol'kikh lits, Moscow: Fizmatgiz, 1961, 126 p. Shubik M. Review of C. Berge “General theory of $n$-person games”, Econometrica, 1961, vol. 29, no. 4, p. 821. Zhukovskii V.I., Kudryavtsev K.N., Gorbatov A.S. The Berge equilibrium in Cournot's model of oligopoly, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, no. 2, pp. 147-156 (in Russian). Colman A.M., Körner T.W., Musy O., Tazdait T. Mutual support in games: some properties of Berge equilibria, Journal of Mathematical Psychology, 2011, vol. 55, issue 2, p. 166-175. Mashchenko S.O. The concept of Nash equilibrium and its development, Zh. Obchysl. Prykl. Mat., 2012, no. 1, pp. 40-61 (in Russian). Zhukovskiy V.I., Topchishvili A., Sachkov S.N. Application of probability measures to the existence problem of Berge-Vaisman guaranteed equilibrium, Model Assisted Statistics and Applications, 2014, vol. 9, no. 3, pp. 223-239. Zhukovskii V.I., Sachkov S.N. Bilanciamento confilitti friendly, Italian Science Review, 2014, vol. 18, no. 9, pp. 169-179. Zhukovskii V.I., Sachkov S.N. About an unusual but friendly method of balancing of conflicts, International Independent Institute of Mathematics and Systems. Monthly Scientific Journal, 2014, no. 10, pp. 61-64 (in Russian). Zhukovskii V.I., Sachkov S.N., Gorbatov A.S. Mathematical model of the “Golden rule”, Science, Technology and Life-2014. Proceedings of the International Scientific Conference, Czech Republic, Karlovy Vary, 27-28 December 2014, pp. 16-23. Zhukovskii V.I., Chikrii A.A., Soldatova N.G. The Berge equilibrium in the conflicts under uncertainty, XII Vserossiiskoe soveshchanie po problemam upravleniya (VSPU–2014): Trudy (Proc. XII All-Russia Conf. on Control Problems (RCCP–2014)), Moscow: Inst. of Control Problems, 2014, pp. 8290-8302 (in Russian). Full text