A new approach for the solution of fuzzy games

In this paper, a new approach is proposed to solve the games with imprecise entries in its payoff matrix. All these imprecise entries are assumed to be trapezoidal fuzzy numbers. Also the proposed approach provides fuzzy optimal solution of the fuzzy valued game without converting to classical version. A numerical example is provided.


Introduction
Game theory is a method for the study of decision-making in situations of conflictsand sometimes cooperation."Game theory provides a mathematical process for selecting an optimal strategy."It was developed to quantify, model and explain human behavior under conflicts between individuals and public interests. A player in a game is an autonomous decision-making unit. A strategy is a decision rule that specifies how the player will act in every possible circumstance.The mathematical treatment of the Game Theory was made available in 1944 by John Von Newmann et.al [10]through their book "Theory of Games and Economic Behavior".The Von Newmann's approach to solve the Game Theory problems was based on the principle of best out of the worst i.e., he utilized the idea of minimization of the maximum losses. Most of the competitive game theory problems can be handled by this principle."However, in real life situations, the information available is of imprecise nature and there is an inherent degree of vagueness or uncertainty present in the system under consideration."Hence the classical mathematical techniques may not be useful to formulate and solve the real world problems. In such situations, the fuzzy sets introduced by Zadeh [11]in 1965 provide effective and efficient tools and techniques to handle these problems. Many authors such as Campos [2], Sakawa et.al [6], Selvakumari et.al [8], Thirucheran et.al [9] etc., have studied fuzzygames."Sakawa et.al [6] introduced max-min solution procedure for multi-objective fuzzy games.Charilas et.al [3] collected applications of game theory in wireless networking and presents them in a layered perspective, emphasizing on which fields game theory could be effectively applied. Bompard et.al [1] presented a medium run electricity market simulator based on game theory."Fiestras-Janeiro et.al [4]provides a review of the applications of cooperative game theory in the management of centralized inventory systems.Madani [5] reviewed applicability of game theory to water resources management and conflict resolution through a series of non-cooperative water resource games."His paper illustrates the dynamic structure of water resource problems and the importance of considering the game's evolution path while studying such problems.In this paper, we have proposed a new approach based on the principle of dominance for the fuzzy optimal solution of the fuzzy valued game without converting to its equivalent crisp form." The rest of this paper is organized as follows. In section 2,"we recall the basic concepts and the results of trapezoidal fuzzy numbers and their arithmetic operations. In section 3, we have proposed a matrix method for the solution of fuzzy games. In section 4, Numerical example is provided to illustrate the efficiency of the proposed method. Section 5 gives the conclusion of this Paper.  has the following characteristics: , we have the following comparison:

Definition 2.4.Arithmetic Operations on Trapezoidal Fuzzy Numbers
For arbitrary trapezoidal fuzzy numbers

Matrixgames
A finite two-person zero-sum game which is represented in matrix form is called a matrix game. This is a direct consequence of the fact that two opponents with exactly opposite interests play a game under a finite number of strategies, independently of his or her opponent's action. Once both players each make an action, their decisions are disclosed. A payment is made from one player to the other based on the outcome, such that the gain of one player equals the loss of the other, resulting in a net payoff summing to zero.

The Fuzzy Payoff Matrix
"The problem that we are aiming to solve is a two player zero sumfuzzy game in which the entries in the payoff matrix A are trapezoidal fuzzy number. The fuzzy payoff matrix is

The Principle of Dominance (Dominant Strategy or Dominance Method)
Step 1. Represent the trapezoidal fuzzy numbers in the fuzzy game problem in its parametric form.
Step 2. Identify any two rows. If the elements in one row is found to be less than or equal to the corresponding elements in the other row that row dominates the other.
Step 3. Consider the elements in any column which are greater than or equal to the corresponding elements in any other column then that column is dominated.

Conclusion
In this paper we have obtained the optimum solution of fuzzy game without converting to classical form by applying a new ranking method and a new fuzzy arithmetic on the parametric form of trapezoidal fuzzy numbers. It is evident from the above example that the proposed method is capable of giving fuzzy solution for the fuzzy matrix game.