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International Journal of Intelligent Systems and Applications(IJISA)

ISSN: 2074-904X (Print), ISSN: 2074-9058 (Online)

Published By: MECS Press

IJISA Vol.5, No.7, Jun. 2013

Lexicographic Strategic Games' Nonstandard Analysis

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Author(s)

Guram N. Beltadze

Index Terms

Game, Noncooperative, Lexicographic, Strategy, Situation, Standard, Nonstandard, Equilibrium, A Saddle Point

Abstract

A new concept of a mixed strategy is given for m-dimensional lexicographic noncooperative Γ(Γ^0,Γ^1,...,Γ^(m-1)) game when on a set of pure strategies m-dimensional probability distributions are given. In this case each Γ^k(k=0,1,...,m-1) criteria of Γ game corresponds to its probability distributions on sets of pure strategies. Besides, a lexicographic m-dimensional order relation is given on set of -dimensional probability distribution. The given construction is made by the methodology of nonstandard analysis Therefore, the given mixed strategy is called a nonstandard mixed strategy, and a lexicographic game in such strategies is called a nonstandard mixed extension. An equilibrium situation in mixed strategies is defined in Γ game. A nonstandard mixed extension of lexicographic matrix games is studied thoroughly. In such games, necessary and sufficient conditions of the existence of a saddle point are proved. The analyzed examples show that if in a lexicographic matrix game doesn’t exist a saddle point in standard mixed strategies then a saddle point maybe doesn’t exist in nonstandard mixed strategies. If in a lexicographic matrix game doesn’t exist a saddle point in standard mixed strategies then there can be existed a saddle point in nonstandard mixed strategies. Thus, lexicographic games’ nonstandard mixed distribution is a generalization of a standard mixed extension.

Cite This Paper

Guram N. Beltadze,"Lexicographic Strategic Games' Nonstandard Analysis", International Journal of Intelligent Systems and Applications(IJISA), vol.5, no.7, pp.1-8, 2013.DOI: 10.5815/ijisa.2013.07.01

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