The Optimal Principle of Stable Solutions in Lexicographic Cooperative Games

Full Text (PDF, 1245KB), PP.11-18

Views: 0 Downloads: 0


Mindia E. Salukvadze 1,* Guram N. Beltadze 2

1. Georgian National Academy of Sciences, Georgian Technical University, Georgia, Tbilisi

2. Department of Control Systems, Georgian Technical University, Georgia, Tbilisi

* Corresponding author.


Received: 25 Dec. 2013 / Revised: 10 Jan. 2014 / Accepted: 10 Feb. 2014 / Published: 8 Mar. 2014

Index Terms

Game, Lexicography, Cooperative Game, Imputation, C-core, Stability, Stable Solutions


Neumann-Morgenstern’s solutions NM (v) as stable solution’s optimal principle is stated in a lexicographic v = (v1, v2 ,..., vm)T cooperative game. The conditions of NM (v) existence are proved for the cases, when: 1. v1 scalar cooperative game’s C-core C (v1) and NM (v1)solutions are equal; 2. Scalar cooperative v1 game’s C-core and NM (v1) solutions are different. In the first case the sufficient conditions are proved in order to say that a C-core C (v) of a lexicographic cooperative v game must not be empty and it should be coincided to NM (v) . In the second case the necessary condition of NM (v) existence is proved. In the case of the existence of NM (v) solutions their forms can be established. Some properties NM (v) of solutions are stated.

Cite This Paper

Mindia E. Salukvadze, Guram N. Beltadze, "The Optimal Principle of Stable Solutions in Lexicographic Cooperative Games", International Journal of Modern Education and Computer Science (IJMECS), vol.6, no.3, pp.11-18, 2014. DOI:10.5815/ijmecs.2014.03.02


[1]M. E. Salukvadze. Vectjr-Valued Optimization Problems in Control Theory. Academic Press, New York, 1979, 219 p.
[2]G.N.Beltadze. “Sets of equilibrium situations in lexicigraphic noncoalition games”. Bulletin of the the Academy of Sciences of the Georgian SSR, 98, № 1, 1980, 41-44 (in Russian).
[3]G.N.Beltadze. “A Mixed extension of finite noncoalicion lexicographic games”. Bulletin of the Academy of Sciences of the Georgian SSR, 98, № 2, 1980, 273-276 (in Russian).
[4]M. E. Salukvadze, G. Beltadze, F.Criado. “Diadic Theoretical Games Models of Decision - Making for the Lexicographic Vector Payoffs”. International Journal of Information Technology and Decision Making, Vol. 8, No.2,2009,193-216.
[5]G.N.Beltadze. “Lexicographic noncooperative game’s mixed extension with criteria”. International Journal of Systems and Software. Asian Research Publishing Network (ARPN) Publishers. Vol 1, # 8, November 2011,247-250.
[6]G.N. Beltadze. “Lexicographic Multistage Games With Perfect Information”. Informational and Computer Technologies –Theory and Practice: Proceedings of the International Scientific Conference ICTMC-2010 Devoted to the 80th Anniversary of I.V. Prangishvili. Editor Ivane Gorgidze. Nova Science Publishers, New York, 2012, 275-281.
[7]G.N. Beltadze. “Lexicographic Strategic Games’ Nonstandard Analisis”. International Journal of Intelligent Systems and Applications. Hong Kong, Volume 5, Number 7, 2013, 1-8.
[8]G.N. Beltadze. “Lexicographic Bimatrix Game’s Mixed Extension with Criteria”. Several Problems of Applied Mathematics end Mechanics. Mathematics Research Developments Dedicated to the 105 Birth Anniversary of Professor Alexi Gorgidze. Editors Ivane Gorgidze and Tamar Lominadze. Nowa Science Publishers, New York, 2013,137-143.
[9]Antonio Quesada. "Negative results in the theory of games with lexicographic utilities". Economics Bulletin, Universitat de València (Spain). Vol. 3, No. 20, 2003, 1−7.
[10]G.N.Beltadze. “Cooperative games with lexicographic paiofs”. Bulletin of the Academy Academy of Sciences of the Georgian SSR, 99, № 2, 1980, 309-312 (in Russian).
[11]G.N.Beltadze. “Foundations of Lexicographic Cooperative Game Theory”. International Journal Modern Education and Computer Science. Hong Kong, Volume 5, Number 3, 2013, 18-25.
[12]G. Owen. “Game Theory”. Third Edition. Academic Press, 1995, 459 p.
[13]N. N. Vorob’ev. “Game Theory. Lectures for Economists”. M.: Nauka, 1985, 271 p. (in Russian).
[14]Vinay Kumar. “Mathematics Is Science: A Topic Revisited in Context of FCS of India”. International Journal of Modern Education and Computer Science. Hong Kong, Volume 4, Number 6, 2012, 17-26.
[15]Jiang-Xia Nan, Deng-Feng Li and Mao-Jun Zhang. “A Lexicographic Method for Matrix Games with Payoffs of Triangular Intuitionistic Fuzzy Numbers”. International Journal of Computational Inteligence Systems. China, Volume 3, Number 3, 2010, 280-289.
[16]Barry O’Neill, Bezalel Peleg. “Lexicographic Composition of Simple Games”. Discusion Paper No. 415, The Hebrew University of Jerusalem, Center for the Study of Rationality, February February 2006, 25 p.