Comparison of Nash and Berge Equilibrium’s in Bimatrix Game
DOI:
https://doi.org/10.5564/jimdt.v5i1.3204Keywords:
Nash equilibrium, Berge equilibrium, global solution, global optimal conditionAbstract
Game theory has numerous applications in applied mathematics, economics, and decision theory. There are several books and articles that deal with Nash and Berge equilibriums. To our knowledge, there are no comparisons or conclusive results related to the optimal decision-making between Nash and Berge equilibriums. We provide numerical experiments for both equilibria.
Биматрицан Тоглоом Дахь Нэш, Бержийн Тэнцвэрийн Харьцуулалт
Хураангуй: Тоглоомын онол нь эдийн засаг, шийдвэр гаргалтын онол, бизнес, улс төр, хэрэглээний математик зэрэг салбарт хэрэглээ ихтэй. Бержийн тэнцвэрийн талаар хэд хэдэн судалгаа, зохиолууд байдаг боловч бидний одоогийн судалснаар түүний Бержийн тэнцвэрийн оновчтой шийдийн хувьд Нэшийн тэнцвэртэй харьцуулсан судалгааны ажил байхгүй байна. Бидний ажил нь энэ харьцуулалтыг хийж Берж ба Нэшийн тэнцвэрүүдийн хувьд тоглогчдын хожлын утгын хувьд харьцуулсан дүгнэлт гаргах зорилготой. Тоон туршилт хийж үр дүнг гаргасан.
Түлхүүр үгс: Бержийн ба Нэшийн тэнцвэр, локал ба глобал оновчтой шийд, глобал оновчтой
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References
K.Y. Abalo, M.M. Kostreva, "Some existence theorems of nash and berge equilibria," Appl. Math. Lett., Vol. 17, pp. 569–573, 2004, https://doi.org/10.1016/S0893-9659(04)90127-9.
K.Y. Abalo, M.M. Kostreva, "Berge equilibrium: some recent results from fixed-point theorems," Appl. Math. Comput., Vol. 169, pp. 624–638, 2005, https://doi.org/10.1016/j.amc.2004.09.080.
A.M. Colman, T.W. K¨orner, O. Musy, T. Tazda¨ıt," Mutual Support in Games: Some Properties of Berge Equilibria,"Journal of Mathematical Psychology, Vol. 55, no. 2, pp. 166–175, 2011, https://doi.org/10.1016/j.jmp.2011.02.001.
K.Y. Abalo, M. M. Kostreva, "Fixed Points, Nash Games and Their Organizations," Topol. Methods Nonlinear Anal., tmna., Vol. 8, no. 1, pp. 205–215, 1996, https://doi.org/10.12775/TMNA.1996.029.
A.S. Antipin, "Equilibrium programming: models and solution methods," The Bulletin of Irkutsk State University. Series Mathematics, Vol. 2, pp. 8–36, 2009.
J.P. Aubin, "Optima and Equilibria," Springer-Verlag, Berlin, 1998.
C. Berge, "Theorie Generale des Jeux N-personnes," Gauthier Villars, Paris, 1957.
A.M. Colman, T.W. Korner, O. Musy, T. Tazdait, "Mutual support in games: some properties of Berge equilibria," J. Math. Psychol., Vol. 55, no. 2, pp. 166–175, 2011, https://doi.org/10.1016/j.jmp.2011.02.001.
B. Crettez, "On sugdens mutually benecial practice and Berge equilibrium," Int. Rev. Econ., Vol. 64, pp. 357–366, 2017, https://doi.org/10.1007/s12232-017-0278-3.
J. Dickhaut, T. Kaplan, "A program for finding Nash equilibria," Mathematica J., Vol. 1, pp. 87–93, 1991.
R. Enkhbat, N. Tungalag, A. Gornov, A. Anikin, "The curvilinear search algorithm for solving three-person game," Proc. DOOR 2016. CEUR-WS, Vol. 1623, pp. 574–583, 2016.
R. Enkhbat, S. Batbileg, A. Anikin, N. Tungalag, A. Gornov, "A note on solving 5-person game," Advanced Modeling and Optimization, Vol. 19, pp. 227–232, 2017.
R. Enkhbat, S. Batbileg, N. Tungalag, A. Anikin, A. Gornov, "Computational method for solving n-person game," The Bulletin of Irkutsk State University, Series Mathematics, Vol. 20, pp. 109–121, 2017, https://doi.org/10.26516/1997-7670.2017.20.109.
R. Enkhbat, "A note on anti-Berge equilibrium for bimatrix game," The Bulletin of Irkutsk State University, Series Mathematics, Vol. 36, pp. 3–13, 2021, https://doi.org/10.26516/1997- 7670.2021.36.3.
R. Enkhbat, S. Batbileg, "Optimization approach to Berge equilibrium for bimatrix game," Optimization Letters, Vol. 15, pp. 711–718 2021, https://doi.org/10.1007/s11590-020- 01688-8.
H. Mills, "Equilibrium points in finite games," J. Soc. Indust. Appl. Mathemat., Vol. 8, no. 2, pp. 397–402, 1960, https://doi.org/10.1137/0108026.
J.T. Howson, "Equilibria of polymatrix games," Management Sci., Vol. 18, no. 5, pp. 312–318, 1972, https://doi.org/10.1287/mnsc.18.5.312.
C.E. Lemke, T.T. Howson, "Equilibrium points of bimatrix games," SIAM J. Appl. Math., Vol. 12, pp. 413–423, 1961, https://doi.org/10.1137/0112033.
O.L. Mangasarian, "Equilibrium points of bimatrix games," J. Soc. Indust. Appl. Mathemat., Vol. 12, pp. 778–780, 1964, https://doi.org/10.1137/0112064.
O.L. Mangasarian, H. Stone, "Two-person nonzero games and quadratic programming," J. Mathemat. Anal. Appl., Vol. 9, pp. 348–355, 1964, https://doi.org/10.1016/0022- 247X(64)90021-6.
R.D. McKelvey, A. McLennan, "Computation of equilibria in finite games," Handbook of Computational Economics, Vol. 1, pp. 87–142, 1996, https://doi.org/10.1016/S1574- 0021(96)01004-0.
Minarchenko. I., "Search of Nash equilibrium in quadratic nonconvex game with weighted potential," S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
Minarchenko. I., "Search of Nash Equilibrium in Quadratic n-person Game," International Conference on Discrete Optimization and Operations Research.
G. Owen, "Game Theory," Saunders, Philadelphia, 1971.
J.F. Nash, "Equilibrium points in n-person games," Proc. of the Nat. Acad. of Sci. USA, Vol. 36, pp. 48–49, 1950, https://doi.org/10.1073/pnas.36.1.48.
J.F. Nash, "Non-cooperative games," Annals of Mathematics, Vol. 54, pp. 286–295, 1951, https://doi.org/10.2307/1969529.
J.V. Neumann, O. Morgenstern, "Theory of Games and Economic Behavior," Princeton University Press, Princeton, 1944.
A.V. Orlov, A.S. Strekalovsky, S. Batbileg, "On computational search for Nash equilibrium in hexamatrix games," Optimization Letters, Vol. 10, pp. 369–381, 2014, https://doi.org/10.1007/s11590-014-0833-8.
M.E. Salukvadze, I. Zhukovskiy, "The Berge Equilibrium: a Game-theoretic Framework for the Golden Rule of Ethics", Birkhauser, Basel, 2020, https://doi.org/10.1007/978-3-030-25546-6.
A.S. Strekalovsky, "Global optimality conditions for optimal control problems with functions of A.D. Alexandrov," J. Optim. Theory Appl. Vol. 159, pp. 297–321, 2013, https://doi.org/10.1007/s10957-013-0355-z.
A.S. Strekalovsky, R. Enkhbat, "Polymatrix games and optimization problems," Automation and Remote Control, Vol. 75, pp. 632–645, 2014, https://doi.org/10.1134/S0005117914040043.
A.S. Strekalovsky, A.V. Orlov, "Bimatrix Game and Bilinear Programming,"Nauka, Moscow, 2007.
A.S. Strekalovsky, I.M. Minarchenko, "A local search method for optimization problem with D.C. inequality constraints," Appl. Math. Model., Vol. 58, pp. 229–244, 2018, https://doi.org/10.1016/j.apm.2017.07.031.
A. Tucker, "A two-person dilemma," Stanford University /Edit.E.Rassmussen/, Readings in Games and Information, pp. 7–8, 1950.
K.S. Vaisman, "Berge Equilibrium," Ph. D. thesis. St. Petersburg State Univ., 1995.
N.N. Vorobyev, "Noncooperative Games," Nauka, Moscow, 1984.
E.B. Yanovskaya, "Equilibrium points in polymatrix games," Latvian Math. Col lect., Vol. 8, pp. 381–384, 1968, https://doi.org/10.15388/LMJ.1968.20224.
V.I. Zhukovskiy, "Some problems of non-antagonistic differential games," Mathematical methods in operation research, Bulgarian Academy of Sciences Publ., pp. 103–195, 1985.
V.I. Zhukovskii, A.A. Chikrii, "Linear-quadratic Differential Games," Naukova Dumka Publ., Kiev, 1994.
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