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Section
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Mathematics
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Title
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The Berge equilibrium in Cournot's model of oligopoly
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Author(-s)
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Zhukovskii V.I.a,
Kudryavtsev K.N.b,
Gorbatov A.S.a
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Affiliations
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Lomonosov Moscow State Universitya,
South Ural State Universityb
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Abstract
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In many large areas of the economy (such as metallurgy, oil production and refining, electronics), the main competition takes place among several companies that dominate the market. The first models of such markets - oligopolies were described more than a hundred years ago in articles by Cournot, Bertrand, Hotelling. Modeling of oligopolies continues in many modern works. Moreover, in 2014 Nobel Prize in Economics “for his analysis of market power and regulation in sectors with few large companies” was received by Jean Tirole - the author of one of the best modern textbooks on the theory of imperfect competition “The Theory of Industrial Organization”.
The main idea of all these publications, studying the behavior of oligopolies, is that every company is primarily concerned with its profits. This approach meets the concept of Nash equilibrium and is actively used in modeling the behavior of players in a competitive market. The exact opposite of such “selfish” equilibrium is “altruistic” concept of Berge equilibrium. In this approach, each player, without having to worry about himself, choose his actions (strategies) trying to maximize the profits of all other market participants. This concept called Berge equilibrium appeared in Russia in 1994 in reference to the France Claude Berge monograph published in 1957. The first works on the concept of Berge equilibrium belong to K.S. Vaisman and V.I. Zhukovskii. Once outside Russia, the concept of “Berge equilibrium” is slowly gaining popularity. To day, the number of publications related to this balance is already measured in tens. However, all of these items are limited to purely theoretical issues, or, in general, to psychology applications. Works devoted to the study of Berge equilibrium in economic problems, were not seen until now. It's probably a consequence of Martin Shubik's review (“… no attention is paid to the application to the economy. … the book is of little interest for economists”) of the Berge's book, it “scared” economists for a long time. However, it is not so simple. In this article, Berge equilibrium is considered in Cournot oligopoly, its relation to Nash equilibrium is studied. Cases are revealed in which players gain more profit by following the concept of Berge equilibrium, than by using strategies dictated by Nash equilibrium.
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Keywords
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Cournot's oligopoly, Berge equilibrium, Nash equilibrium, non-cooperative game
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UDC
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519.833
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MSC
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91A10, 91B26
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DOI
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10.20537/vm150201
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Received
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18 May 2015
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Language
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Russian
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Citation
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Zhukovskii V.I., Kudryavtsev K.N., Gorbatov A.S. The Berge equilibrium in Cournot's model of oligopoly, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 147-156.
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References
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- Cournot A. Recherches sur les principes mathematiques de la theorie des richesses, Paris: Hachette, 1838, 198 p.
- 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.
- Hotelling H. Stability in competition, Economic Journal, 1929, vol. 39, pp. 41-57.
- Tirole J. The theory of industrial organization, Cambridge: MIT press, 1988, 479 p.
- Nash J.F. Equilibrium points in $N$-person games, Proc. Nat. Academ. Sci. USA, 1950, vol. 36, pp. 48-49.
- Berge C. Theorie generale des jeux a $n$ personnes games, Paris: Gauthier-Villars, 1957, 114 p.
- Vaisman K.S. The Berge equilibrium, Abstract of Cand. Sci. (Phys.-Math.) Dissertation, St. Petersburg, 1995, 15 p (in Russian).
- Vaisman K.S. The Berge equilibrium for linear-quadratic differential game, Multiple criteria problems under uncertainty: Abstracts of the Third International Workshop, Orekhovo-Zuevo, Russia, 1994, p. 96.
- Zhukovskii V.I., Chikrii A.A. Lineino-kvadratichnye differentsial'nye igry (Linear-quadratic differential games, Kiev: Naukova Dumka, 1994, 319 p.
- Shubik M. Review of C. Berge “General theory of $n$-person games”, Econometrica, 1961, vol. 29, no. 4, p. 821.
- 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, no. 2, pp. 166-175.
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