The Solution of Scalar Bimatrix Games in Preferred Pure Strategies

Full Text (PDF, 444KB), PP.1-7

Views: 0 Downloads: 0


Guram N. Beltadze 1,*

1. Departament Artifical Intelligence Georgian Technical University, Georgia, Tbilisi, 0175, str. Kostava 77

* Corresponding author.


Received: 16 Nov. 2019 / Revised: 1 Dec. 2019 / Accepted: 25 Dec. 2019 / Published: 8 Jun. 2020

Index Terms

Bimatrix game, Equilibrium, Preferred, Pure strategy, Domination, Orientation, Guaranted level


In the present work the task of finding Nash equi-librium situation or finding the most preffered pure strategies in finite scalar m x n  bimatrix Γ(A, B)  game is studied. The problem of finding an equilibrium in the  Γ(A, B)  game has a long history, but due to the complexity of the known algorithms and methods, that cause various problems, its study is being continued today. Our  approach is different from these methods. In order to find an equilibrium situation in the pure strategies, we mean making a prediction by each player about the second player's behavior for choosing preferred pure strategy. Therefore we use a sequential strictly and weakly procedure of dominance for comparing of two pure strategies. Only in the case of strictly dominance the line of dominance has no meaning in order to maintain an equilibrium situation in pure strategies. We don’t have such kind of situation in every game. In this kind of game each player can act by different principles to define the most preferred pure strategy, that is based on making a prediction about his partner’s behavior by the player and that means the orientation on guaranteed levels in concrete situations. We mean the orientation in A and B  matrix games average V(A) and V( BT)  playoffs obtained by the players using maximum optimal mixed strategies. Each player’s decisions are discussed about the usage the preferred pure strategies related to his partner’s actions. By using them he will gain much more, then in an equilibrium situation. All kinds of other actions are also discussed. Acceptable results are used for solution of high ranged  m x n  (m>2, n>2) Bimatrix  Γ(A, B)  games.

Cite This Paper

Guram N. Beltadze, " The Solution of Scalar Bimatrix Games in Preferred Pure Strategies", International Journal of Modern Education and Computer Science(IJMECS), Vol.12, No.3, pp. 1-7, 2020.DOI: 10.5815/ijmecs.2020.03.01


[1]J. von Neuman, O. Morgenstern. “Theory of Games and Economic Behavior”.  Prinston  University Press, 1944, 625 p. 

[2]G. Owen.  “Game Theory”. Academic Press, Third Edition, 1995, 459 p. 

[3]N. N.Vorob’ev. “Foundations of Game Theory. Nonco-operative Games”. Birkhauser Verlang, Basel – Boston  – Berlin, 1994, 496 p. 

[4]G. Beltadze. “Game  theory: A mathematikal theory of correlations and equilibrium”. Georgian Technical Univer-sity, Tbilisi, 2016, 505 p. (in Georgian). 

[5]Herbert Gixtis. “Game theory evolving”, Second edition. Princeton University Press, 2009, 409 p. 

[6]Erich  Prisner. “Game Theory Through Examples”.  Franklin University Switzerland. MAA, Published and Distributed by The Mathematical Association of America, 2014, 308 p. 

[7]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, pp. 137-143. 

[8]N. N. Vorob'ev. "Equilibrium situations in bimatrix games'.  Teor. Veroztnost i Primenen. 3, no. 3 1958, pp. 318-331 (in Russian).

[9]H.W.Kuhn. "An algoritm for equilibrium points in bimat-rix games". Proc. Nat. Acad. Sci. USA 47, 1961, pp. 1657- -1662. 

[10]O.L.Mangasarian. "Equilibrium points in bimatrix games". Journal Soc. Industr. Appl. Math., Vol 12, 1964, pp.  778- 780. 

[11]C. E. Lemke and J.T. Howson. "Equilibrium points of bimatrix games". Journal of the Society for Industrial and Applied Mathematics, Vol 12, no. 2, 1964, pp. 413-423. 

[12]J. Rosenmuller. "One generalization of the Lemke- Howson algorithm to noncooperative N-person games. SIAM Journal on Applied Mathematics, Vol 21, no. 1 (1971), 73-79.

[13]G.N.Beltadze. “Game Theory - basis of Higher Education and Teaching Organization“.  International  Journal  of Modern Education and Computer  Science (IJMECS). Hong Kong, Volume 8, Number 6, 2015, pp. 41-49. 

[14]M. E. Salukvadze, G. N.Beltadze. "Strategies of Nonsoli-dary Behavior in Teaching Organization". International Journal of  Modern  Education and Computer Science (IJMECS). Hong Kong, Volume 9, Number 4, 2017, pp. 12-18. 

[15]C.E. Lemke. "Bimatrix equilibrium points and mathemati- cal programming". Managment  Schience. Vol 11, no. 7, New York, 1965, pp. 681-689. 

[16]R. Savani. "Finding Nash equilibria of bimatrix games". A thesis of PhD. London School of Economics and Political Science, 2009, 116 p. 

[17]Anne Verena Balthasar. "Geometry and equilibria in bimatrix games". A thesis submitted for the degree of Doctor  of  Philosophy. Department of Mathematics London School of Economics and Political Science, 2009, 107 p.