Connection between matrix games and linear programming

The aim of this paper is to develop an effective method for solving matrix game with payoffs of trapezoidal fuzzy numbers (TrFNs). The method always assures that players’ gain-floor and loss-ceiling have a common TrFN-type fuzzy value and hereby any matrix game with payoffs of TrFNs has a TrFN-type fuzzy value. The matrix game is first converted to a fuzzy linear programming problem, which is converted to three different optimization problems, which are then solved to get the optimum value of the game. The proposed method has an edge over other method as this focuses only on matrix games with payoff element as symmetric trapezoidal fuzzy number, which might not always be the case. A numerical example is given to illustrate the method.

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Linear programming and matrix games

Introduction to Game Theory - Texts and Readings in Mathematics ◽

10.1007/978-93-86279-17-0_6 ◽

2003 ◽

pp. 45-48

Author(s):

Stef Tijs

Keyword(s):

Linear Programming ◽

Matrix Games

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Modified fictitious play for solving matrix games and linear-programming problems

Computers & Operations Research ◽

10.1016/0305-0548(94)00075-j ◽

1995 ◽

Vol 22 (9) ◽

pp. 893-903 ◽

Cited By ~ 9

Author(s):

Saul I. Gass ◽

Pablo M.R. Zafra

Keyword(s):

Linear Programming ◽

Fictitious Play ◽

Matrix Games ◽

Linear Programming Problems

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Matrix Games with Interval-Valued 2-Tuple Linguistic Information

Games ◽

10.3390/g9030062 ◽

2018 ◽

Vol 9 (3) ◽

pp. 62 ◽

Cited By ~ 2

Author(s):

Anjali Singh ◽

Anjana Gupta

Keyword(s):

Linear Programming ◽

Lower Bound ◽

Real World ◽

Upper Bound ◽

Duality Theorem ◽

Matrix Game ◽

Matrix Games ◽

Linguistic Information ◽

The Real ◽

Interval Valued

In this paper, a two-player constant-sum interval-valued 2-tuple linguistic matrix game is construed. The value of a linguistic matrix game is proven as a non-decreasing function of the linguistic values in the payoffs, and, hence, a pair of auxiliary linguistic linear programming (LLP) problems is formulated to obtain the linguistic lower bound and the linguistic upper bound of the interval-valued linguistic value of such class of games. The duality theorem of LLP is also adopted to establish the equality of values of the interval linguistic matrix game for players I and II. A flowchart to summarize the proposed algorithm is also given. The methodology is then illustrated via a hypothetical example to demonstrate the applicability of the proposed theory in the real world. The designed algorithm demonstrates acceptable results in the two-player constant-sum game problems with interval-valued 2-tuple linguistic payoffs.

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LEXICOGRAPHIC METHOD FOR MATRIX GAMES WITH PAYOFFS OF TRIANGULAR FUZZY NUMBERS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488508005327 ◽

2008 ◽

Vol 16 (03) ◽

pp. 371-389 ◽

Cited By ~ 19

Author(s):

DENG-FENG LI

Keyword(s):

Linear Programming ◽

Fuzzy Numbers ◽

Optimization Problems ◽

Fuzzy Optimization ◽

Order Relation ◽

Programming Models ◽

Matrix Game ◽

Triangular Fuzzy Numbers ◽

Matrix Games ◽

Lexicographic Method

The purpose of the paper is to study how to solve a type of matrix games with payoffs of triangular fuzzy numbers. In this paper, the value of a matrix game with payoffs of triangular fuzzy numbers has been considered as a variable of the triangular fuzzy number. First, based on two auxiliary linear programming models of a classical matrix game and the operations of triangular fuzzy numbers, fuzzy optimization problems are established for two players. Then, based on the order relation of triangular fuzzy numbers the fuzzy optimization problems for players are decomposed into three-objective linear programming models. Finally, using the lexicographic method maximin and minimax strategies for players and the fuzzy value of the matrix game with payoffs of triangular fuzzy numbers can be obtained through solving two corresponding auxiliary linear programming problems, which are easily computed using the existing Simplex method for the linear programming problem. It has been shown that the models proposed in this paper extend the classical matrix game models. A numerical example is provided to illustrate the methodology.

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