Mindia E. Salukvadze; Guram N. Beltadze

Stochastic Game with Lexicographic Payoffs

Full Text (PDF, 701KB), PP.10-17

Views: 0 Downloads: 0

Author(s)

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

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

2. Departaments Control Systems and Artifical Intelligence Georgian Technical University, Georgia, Tbilisi, 0175, str. Kostava 77

* Corresponding author.

DOI: https://doi.org/10.5815/ijmecs.2018.04.02

Received: 22 Dec. 2017 / Revised: 5 Jan. 2018 / Accepted: 30 Jan. 2018 / Published: 8 Apr. 2018

Index Terms

Lexicographic, Stochastic game, Equilibrium situation, Affine game

Abstract

Stochastic games are discussed as a priva-te class of a general dynamic games. A certain class of lexicographic noncooperative games is studied - lexi-cographic stochastic matrix games . The problem of the existence of Nash equilibrium is studied with two analyses - standard and nonstandard way. Standard means using the same kind of mixed strategies in case of scalar games. In this case in lexi-cographic stochastic matrix game Nash equilibrium may not be existed. Its existence takes place in relevant stochastic affine matrix game to the existence of Nash equilibrium. In game a set of Nash equi-librium is given by means of relevant stochastic affine matrix game's set of equilibrium. The sufficient condi-tions of the existance such affine game is proved. In nonstandard way of analyses we use such mixed stra-tegies, they use components with lexicog-raphic probabilites. In this case the kinds of subsets of a set of equilibrium in game are described.

Cite This Paper

Mindia E. Salukvadze, Guram N. Beltadze, " Stochastic Game with Lexicographic Payoffs", International Journal of Modern Education and Computer Science(IJMECS), Vol.10, No.4, pp. 10-17, 2018. DOI:10.5815/ijmecs.2018.04.02

Reference

[1]L. S. Shapley. "Stochastic games". Proc. Nat. Acad. Science. - 1953. Vol. 39. pp. 1095-1100.
[2]"Stochastic Games and Applications". A. Neyman, S. Sorin, eds. Kluwer Academic Press, 2003.
[3]A. Condon. "The complexity of stochastic games" Information and Computation, 96: 1992, pp. 203–224.
[4]N. Vieille. "Stochastic games: Recent results". In: Handbook of Game Theory. - Elsevier Science, 2002, pp. 1833-1850.
[5]A. Rabah. "Stochastic Games in Economics: The Lattice-Theoretic Approach". Stochastic Games and Applications, A. Neyman, S. Sorin, eds. Kluwer Academic Press, 2003, pp. 443-453.
[6]P.A. Abdullar, N.B. Henda, L.d. Alfaro, R. Mayr, S. Sandberg. "Stochastic Games with lossy Channels". Foundations of Software Science and Computational Structures: 11 th International Conference, ETAPS 2008, Budapest, Hungary, March 29-April 6, 2008, Proceedings, pp. 35-49.
[7]M. Baykal-Gursoy. "Tqo-person zero-sum stochastic games". Annals of operations Research, December 1991, Volume 28, Issue 1, pp. 135-152.
[8]J. Flesch, F. Thuijsman, J. Vrieze. "Stationary Strategies in zero-sum stochastic games". International Game Theory Review, December 2001, Vol. 03, No. 04: pp. 283-290.
[9]G. Owen. “Game Theory”. Third Edition. Academic Press, 1995, 459 p.
[10]G. N. Beltadze. "Sets of equilibrium situations in lexi- cographic noncoalition games". Bulletin of the Acade- my of sciences of the Georgian SSR, 98, № 1,1980, pp.41-44 (in Russian).
[11]G. N. Beltadze. "A mixed extension of finite noncoalition lexicographic games". Bulletin of the Academy of sciences of the Georgian SSR, 98, № 2, 1980, pp. 273-276 (in Russian).
[12]G.N.Beltadze. "Analysis of the infinite dimensional lexicographic games". Bulletin of the Academy of sciences of Georgian, 141, № 2, 1991, pp. 241-244 (in Russian).
[13]G. N. Beltadze, A.L.Topchishvili. "Multicriteria nonco- operative games with strictly ordered criteria". A.Gop-fert, J. Seelender, Chr. Tammer (Eds). Methods of Mul- ticriteria Decision Theory, Proceedings of the 6 th Work- shop of the DGOR -Working Group Multicriteria Op- timization and Decision, Frankfurt, 1997, pp. 69-86.
[14]M. E. Salukvadze, G.N. Beltadze, F. Criado. "Dyadic theoretical games models of decision – making for the lexicographic vector payoffs". International Journal of information Technology and Decision Making, Vol. 8, Issue 2, 2009, pp. 193-216.
[15]G. N. Beltadze."Lexicographic non-cooperative game's mixed extension with criteria". International Journal of Systems and Sofware ARPN Publishers, Vol 1, № 8, November 2011, pp. 247- 250.
[16]G.N. Beltadze. "Lexicographic Multistage Games with Perfect Information". Informational and Communica- tion technologies - Theory and Practice: Proceedings of the International Scientific Conference ICTMC- 2010 Devoted to the 80th Anniversary of I.V. Prangishvili. Nova Publishers, 664 pp. USA, 2012, pp. 275- 281.
[17]G. N. Beltadze. “Lexicographic Strategic Games’ Non- standard Analisis”. International Journal of Intelligent Systems and Applications. Hong Kong, Volume 5, Number 7, 2013, pp. 1-8.
[18]G. N. Beltadze, J. A. Giorgobiani. "Shapley’s Axioma- tics for Lexicographic Cooperative Games'. Internati-onal Journal of Modern Education and Computer Science (IJMECS). Hong Kong, Volume 7, Number 8, August 2015, pp. 1-8.
[19]G. N. Beltadze, J. A. Giorgobiani. "The Stability of Equilibrium Situation in Lexicographic Strategic Games". International Journal of Modern Education and Computer Science (IJMECS). Hong Kong,Volume 8, Number 8, December 2016, pp. 38-45.
[20]M. Davis. "Applied Nonstandard Analisis". Courant Institute of Mathematical Sciences, New York University, 1977.

International Journal of Modern Education and Computer Science (IJMECS)