A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts

Abstract

In the word of uncertain competitive situations everything is in the state of flux. Under such situations knowing the exact outcomes of mixed strategies adopted by a player is nearly impossible. It is highly rational to assume that no two experts will project the similar fuzzy payoffs for mix of strategies used. Aggregation of expert’s judgement becomes utmost important before solving such competitive situations. Considering this the present paper proposes a method to solve intuitionistic fuzzy game problems by using aggregation operators on payoff judgments of more than one expert. The proposed method significantly adds to the existing literature by overcoming the limitation of Li’s existing method that considers only one expert’s opinion for solving intuitionistic fuzzy game problems. Illustrative example has been given for showing the superiority of the proposed method.

Keywords

Aggregation operators intuitionistic fuzzy sets fuzzy matrix game

1 Introduction

Game theory deals with the study of decision making in competitive situations in the various fields viz. economics, management, defence, political science, etc. Game theory emphasizes on minimizing the losses and maximizing the gain of two or more opponents with conflicting interests and uses their best strategy out of finite alternatives. Von Neumann, a Mathematician and the Economist, Morgenstren [22] originated mathematical theory of games of strategies and their approach is based on the theory of minimization of the maximum losses. Nash [21] proved that the finite game problem always has a point of equilibrium at which all players choose their best alternatives, when the opponent’s strategies are given. However due to ambiguity and lack of certainty in environment the accessibility of exact payoffs is a big problem. So fuzzy set theory can be modelled to deal with such type of situations. Zadeh [26] was the first to initiate fuzzy set theory. Fuzzy set theory deals only with membership functions which consider degree of satisfaction and degree of unsatisfaction. But in real life situations both membership functions and non membership functions are important to deal with vague problems. Thus to deal with three states (i.e. support, opposition and neutrality) and impreciseness in the payoffs, fuzzy set theory is extended to intuitionistic fuzzy set (IFSs) theory by Atanassov [1 –3].

There are various methods proposed to solve mathematical programming problems of intuitionistic fuzzy matrix games. Bellman and Zadeh [4] define fuzzy goals and fuzzy constraints in the form of fuzzy sets and their intersection represents the fuzzy decision. Li [14] proposed an effective method for each player and established multi objective models to solve fuzzy matrix games. Cevikel and Ahlatcioglu [7] developed two models for solving matrix games with fuzzy goals and fuzzy payoffs. Chakeri and Sheikholeslam [8] developed fuzzy Nash equilibrium to establish a graded depiction of Nash equilibriums in crisp and fuzzy games.

Li and Cheng [17] introduced a computational procedure of fuzzy constrained matrix games and also proposed auxiliary fuzzy linear programming for each player. Li and Nan [18] proved that there are solutions of matrix games with payoffs of IFSs and obtained by solving a pair of nonlinear programming models derived from two auxiliary nonlinear bi-objective programming models. Nan and Li [20] described the matrix games in which payoffs are represented in real numbers but goals of players are expressed with intuitionistic fuzzy sets. In their paper auxiliary linear programming models are used to solve matrix game problems with Intuitionistic Fuzzy (IF) goals set.

Vijay et al. [23] developed fuzzy relation approach for a matrix game with fuzzy goals and fuzzy payoffs. Vijay et al. [24] proved that the matrix game with fuzzy payoffs and fuzzy goals are equivalent to a primal dual pair of fuzzy linear programming problems. Brikaa et al. [5] considered fuzzy rough constrained matrix games and developed an effective fuzzy multi-objective programming algorithm to solve such constrained matrix games. Brikaa et al. [6] introduced an approach based on indeterminacy resolving functions to solve multi-criteria matrix game with intuitionistic fuzzy goals. Li [16] developed the concept of alpha-matrix games and also solve four linear programming problems with upper and lower bounds of alpha cut to get fuzzy value and optimal strategies. Li [15] proposed various methods and the solutions of linear and non linear programming problems of matrix games with payoffs of intuitionistic fuzzy sets. Li et al. [19] described the concept of solution of matrix game with interval payoffs and define procedure of solution of bi objective linear programming models derived from interval programming problems.

Aggregation is an essential tool which is used to combine the information and convert several input values into a single output. Aggregation operators play an important role in optimization problems. Xu [25] developed a method to compare two intuitionistic fuzzy values and introduce aggregation operators. A lot of work has been done in the area of decision making problems by using aggregation operators [9 –13]. After reviewing the literature it can be found that various methods have proposed to solve fuzzy matrix games in which either evaluation of the game is done by a single expert or aggregation data has been given. But in real world fuzzy game problem can be evaluated by group of experts rather than a single expert. Till now no work has been done for solving fuzzy matrix game problems having more than one expert who evaluate the company’s (players) payoffs with diverse strategies. Aggregation of experts’ judgment becomes utmost important for solving such competitive situations. In decision making problems, decision makers give their decisions and judgment only to some extent but not always exact. So there exists some hesitancy degree in order to solve decision making problems and such a hesitancy degree is expressed by using intuitionistic fuzzy sets. Due to competitive situations a company (players) never depends upon only one judgment but always takes opinion of more than one expert for the use of different strategies. So keeping in mind, in this paper, limitation of the existing method of Li [15], has been shown. Also to overcome the limitation of existing method of Li [15] a new method is proposed to solve intuitionistic fuzzy game problems with aggregation operators [12] by adopting the idea of more than one expert’s opinion. So in this paper aggregation operators are used to combine the payoffs which are given by different professionals and renovate it into intuitionistic fuzzy matrix game.

The rest of the paper is organized as follows: Section 2 reviews some basic definitions of intuitionistic fuzzy sets and aggregation operators. Section 3 describes fuzzy matrix game problem and its expected payoffs. In Section 4, existing method of Li [15] is defined. Section 5 represents limitation of existing method [15] with the help of an example. In Section 6, a new method is proposed to solve intuitionistic fuzzy game problems by aggregating the opinions of different experts in the form of intuitionistic fuzzy sets and convert into aggregated intuitionistic fuzzy matrix game. The proposed method has been described in Section 7 with the help of an example. Finally, a concrete conclusion has been given in Section 8.

2 Preliminaries

Definition 2.1 [1]: Let X be a non empty set of the universe. Then an intuitionistic fuzzy set $\tilde{A}$ in X is a set $\tilde{A} = {(x, μ_{\tilde{A}} (x)$ , $v_{\tilde{A}} (x)$ ); x ∈ X} where $μ_{\tilde{A}}$ (x) is termed as the grade of membership of x in A and where $ν_{\tilde{A}}$ (x) is termed as the grade of non membership of x in A. Here $μ_{\tilde{A}}, ν_{\tilde{A}} (x)$ : X ⟶ M is a function from X to a space M which is called membership space, where M contains only two points 0 and 1 with $0 ⩽ μ_{\tilde{A}}$ (x) $+ v_{\tilde{A}} (x) ⩽ 1 \forall x \in X$ . The degree of hesitation is given by $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}}$ (x) $- v_{\tilde{A}} (x)$ .

Definition 2.2 [1]: An intuitionistic fuzzy numbers (IFN) $\tilde{A}$ has the following properties:

$\tilde{A}$ is convex for membership function $μ_{\tilde{A}}$ (x), i.e. $μ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) ⩾ \min [μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2})]$ for x₁, x₂ ∈ R ; λ ∈ [0, 1]

$\tilde{A}$ is concave for non-membership function $ν_{\tilde{A}}$ (x), i.e. $υ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) ⩽ \max [υ_{\tilde{A}} (x_{1}), υ_{\tilde{A}} (x_{2})]$ for x₁, x₂ ∈ R ; λ ∈ [0, 1]

$\tilde{A}$ is normal.

Definition 2.3 [12]: Intuitionistic Fuzzy Einstein Interactive Weighted Averaging Operator (IFEIWA): In order to aggregate the different intuitionistic fuzzy numbers ${\tilde{x}}_{i} = 〈 μ_{i}, ν_{i} 〉$ , (i = 1, 2, 3, …, n) corresponding to weight w = (w₁, w₂, w₃, …, w_n) such that w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , then the weighted averaging operator is defined as $\begin{matrix} IFEIWA ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n}) = \\ 〈 \frac{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} - \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}}{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} - \prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}}}, \\ \frac{2 (\prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}} - \prod_{i = 1}^{n} (1 - μ_{i} - ν_{i})^{w_{i}})}{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} + \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}} 〉 \end{matrix}$

3 Fuzzy matrix game with intuitionistic payoffs

In real competitive decision problems, it is difficult to get exact payoffs of the players. But linguistic variable is a technique to describe the outcomes of game in simple manner. To deal with competitive problems under complex environment, Li and Nan [18] developed a relation between linguistic variable and intuitionistic fuzzy sets. Let M={γ₁, γ₂, …, γ_m} , N={δ₁, δ₂, …, δ_n} be the set of pure strategies for player A₁ and A₂ respectively. If player A₁ chooses any pure strategy γ_i ∈ M ; (i = 1, 2, 3, …, m) and player A₂ chooses any pure strategy δ_j ∈ N ; (j = 1, 2, 3, …, n), then the payoffs of player A₁ at (γ_i, δ_j) can be expressed with intuitionistic fuzzy set (μ_ij, ν_ij) and the payoffs of player A₂ at (γ_i, δ_j) is the negative of that of player A₁. In the concept of mixed strategies, let p_i; (i = 1, 2, 3, …, m) and q_j; (j = 1, 2, 3, …, n) be the probabilities of players A₁ and A₂ who chooses the pure strategy γ_i ∈ M and δ_j ∈ N to employ their strategies.

Denote $R = {p | \sum_{i = 1}^{m} p_{i} = 1; p_{i} ⩾ 0}$ and $S = {q | \sum_{j = 1}^{n} q_{j} = 1; q_{j} ⩾ 0}$ are mixed strategies of player A₁ and A₂ respectively. In any fuzzy matrix game, the expected payoffs of player A₁ with mixed strategies can be calculated as follows: $\begin{matrix} E (p, q) = (\begin{matrix} p_{1} & p_{2} & \dots & p_{m} \end{matrix}) \\ (\begin{matrix} 〈 μ_{11}, ν_{11} 〉 & \dots & 〈 μ_{1 n}, ν_{1 n} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}, ν_{m 1} 〉 & \dots & 〈 μ_{mn}, ν_{mn} 〉 \end{matrix}) (\begin{matrix} q_{1} \\ q_{2} \\ ⋮ \\ q_{n} \end{matrix}) \\ = (1 - \prod_{j = 1}^{n} \prod_{i = 1}^{m} {(1 - μ_{ij})}^{p_{i} q_{j}}, \prod_{j = 1}^{n} \prod_{i = 1}^{m} {(ν_{ij})}^{p_{i} q_{j}}) \end{matrix}$

4 Li’s method of matrix games with intuitionistic fuzzy sets

In this section existing method of Li [15] is presented. This method is applicable on the aggregated data of the fuzzy matrices or on the matrix whose payoffs are given by only one expert. This section emphasis on the steps to find solutions of linear and non linear programming models of matrix games in which payoffs are intuitionistic fuzzy sets. The steps of bi-objective mathematical programming models of both the players are as follows:

Step 1: The bi-objective mathematical programming model of player A₁ is as follow: $\begin{matrix} Z = max {σ}, min {τ} \\ s . t {\begin{matrix} 1 - \prod_{i = 1}^{m} (1 - μ_{ij})^{p_{i}} ⩾ σ (j = 1, 2, 3, \dots, n), \\ \prod_{i = 1}^{m} ν_{ij}^{p_{i}} ⩽ τ (j = 1, 2, 3, \dots, n), \\ 0 ⩽ σ + τ ⩽ 1, σ ⩾ 0, τ ⩾ 0, \\ \begin{matrix} \sum_{i = 1}^{m} p_{i} = 1, \\ p_{i} ⩾ 0, (i = 1, 2, 3, \dots, m) \end{matrix} \end{matrix} \end{matrix}$ where $max {σ} = max {min_{1 ⩽ j ⩽ n} (1 - \prod_{i = 1}^{m} (1 - μ_{ij})^{p_{i}})}$

and $\min {τ} = min {max_{1 ⩽ j ⩽ n} (\prod_{i = 1}^{m} ν_{ij}^{p_{i}})}$

Step 2: The aggregation of bi-objective mathematical programming model of player A₁ obtained in Step 1 can be converted into linear programming model is as follow: $min {w} = min {λ In (1 - σ) + (1 - λ) In (τ)}$ $s . t {\begin{matrix} \sum_{i = 1}^{m} [λ In (1 - μ_{ij}) + (1 - λ) In ν_{ij}] p_{i} ⩽ w \\ (j = 1, 2, 3, . . ., n) \\ \begin{matrix} \begin{matrix} \sum_{i = 1}^{m} p_{i} = 1, \\ p_{i} ⩾ 0 (i = 1, 2, 3, \dots, m), \end{matrix} \\ w is an unrestricted \end{matrix} \\ 0 ⩽ 1 - σ ⩽ 1, 0 ⩽ τ ⩽ 1, λ \in [0, 1] \end{matrix}$

Step 3: The bi-objective Mathematical programming model of player A₂ is constructed as follows: $Z^{*} = min {ξ}, max {ζ}$ $s . t {\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - μ_{ij})}^{q_{j}} ⩽ ξ (i = 1, 2, 3, \dots, m), \\ \prod_{j = 1}^{n} ν_{ij}^{q_{j}} ⩾ ζ (i = 1, 2, 3, \dots, m), \\ 0 ⩽ ξ + ζ ⩽ 1, ξ ⩾ 0, ζ ⩾ 0, \\ \begin{matrix} \sum_{j = 1}^{n} q_{j} = 1, \\ q_{j} ⩾ 0, (j = 1, 2, 3, \dots, n) \end{matrix} \end{matrix}$ where $min {ξ} = min {max_{1 ⩽ i ⩽ m} (1 - \prod_{j = 1}^{n} (1 - μ_{ij})^{q_{j}})}$

and $max {ζ} = max {min_{1 ⩽ i ⩽ m} \prod_{j = 1}^{n} {(ν_{ij})}^{q_{j}}}$

Step 4: The aggregation of bi-objective mathematical programming model of player A₂ obtained in Step 3 is equivalent to the linear programming model is as follows: $max {u} = max {λ In (1 - ξ) + (1 - λ) In (ζ)}$ $s . t {\begin{matrix} \begin{matrix} \sum_{j = 1}^{n} (λ In (1 - μ_{ij}) + (1 - λ) In ν_{ij}) q_{j} ⩾ u \\ (i = 1, 2, 3, . . ., m) \\ \begin{matrix} \sum_{j = 1}^{n} q_{j} = 1 \\ q_{j} ⩾ 0 (j = 1, 2, 3, . . ., n) \\ u is an unrestricted \end{matrix} \\ 0 ⩽ 1 - ξ ⩽ 1, 0 ⩽ ζ ⩽ 1, λ \in [0, 1] \end{matrix} \end{matrix}$

Step 5: From the above steps, it is clear that Step 2 and Step 4 are primal dual pair of linear programming problems. Solve the dual pair of Linear Programming Problems of player A₁ and A₂, which are obtained in Step 2 and Step 4. Then the value of the game and the optimal strategies of both the players are obtained.

5 Limitation of Li’s method

In this section the limitation of existing method of Li [15] is described. In the competitive business world the companies are always uncertain about the customer’s responses to their strategies, they always estimate them on the basis of their intuition. The varying customer’s responses to the strategies adopted by players can be appropriately expressed using intuitionistic fuzzy sets, i.e positive (satisfaction), negative (dissatisfaction) and indifferent. Further in order to save themselves from the erroneous estimation of single individual the companies generally solicit the opinion of more than one industry expert or independent survey agencies. Then the aggregation of solicited opinions is done by assigning weights to individual opinions respective to their stature. Finally the outcome of competitive situation is obtained by applying the game theory principles to the aggregated pay off matrix. Such type of some problems in which opinions of more than one expert are sought, cannot be solved by Li [15] as shown below in an example.

Example 5.1: Let Company A₁ and Company A₂ be the two leading car manufacturing companies in India, badly hit by the Covid-19 led economic recession. Each of these companies has three strategies to protect and gain the market share in this difficult time. The Strategy I is to tie up with major vehicle financiers for providing very liberal financing to the new car buyers. Strategy II is to provide an exclusive five year warranty and free service to the vehicle sold. Strategy III is to provide handsome exchange bonus on the sale of old car by the customer to get the new one. Further, since both the companies view this environment as highly uncertain, so they sought the opinion of two independent experts, namely E1 and E2 on the possible payoffs for company A₁ for three combination of strategies adopted by two car manufacturing companies. The two experts have depicted the customer responses to the three combination of strategies in terms of intuitionistic fuzzy sets, i.e positive (satisfaction), negative (dissatisfaction) and indifferent. According to the relative importance of the opinion of two experts, their opinions are assigned a weightage of 0.7 and 0.3 respectively. The payoff matrices as obtained for company A₁ according to the opinion of two experts E1 and E2 are as below: $\begin{matrix} \begin{matrix} I & II & III \end{matrix} \\ P^{E_{1}} = \begin{matrix} I \\ II \\ III \end{matrix} (\begin{matrix} 〈 0.7, 0.2 〉 & 〈 0.6, 0.3 〉 & 〈 0.9, 0.05 〉 \\ 〈 0.6, 0.15 〉 & 〈 0.5, 0.4 〉 & 〈 0.4, 0.4 〉 \\ 〈 0.85, 0.05 〉 & 〈 0.8, 0.1 〉 & 〈 0.7, 0.3 〉 \end{matrix}) \end{matrix}$ $\begin{matrix} \begin{matrix} I & II & III \end{matrix} \\ P^{E_{2}} = \begin{matrix} I \\ II \\ III \end{matrix} (\begin{matrix} 〈 0.85, 0.1 〉 & 〈 0.7, 0.2 〉 & 〈 0.8, 0.1 〉 \\ 〈 0.5, 0.1 〉 & 〈 0.6, 0.2 〉 & 〈 0.5, 0.2 〉 \\ 〈 0.7, 0.1 〉 & 〈 0.9, 0.05 〉 & 〈 0.6, 0.2 〉 \end{matrix}) \end{matrix}$

Thus the existing method of Li [15] cannot be applied to the above example in which the company A₁ has obtained the payoff matrices based on the opinion of two different experts.

6 Proposed method to solve fuzzy matrix game based on the evaluation of different experts

Due to competitive situations a player (company) never depends upon only one judgment or opinion but always considers opinion of more than one expert. In this situation, aggregation is an important tool used to aggregate the different opinions of multiple decision makers. Li [15] method is applicable only on aggregated data or data given by one expert. But it is not applicable on the problems where opinions are given by more than one expert. To keep it in mind, we have proposed a new method to solve fuzzy matrix games in which there are more than one expert judging the game problems. The steps of proposed method are as under:

Step 1: Let A₁ and A₂ be two players. Let M ={ γ₁, γ₂, …, γ_m } and N = {δ₁, δ₂, …, δ_n} be the set of pure strategies of player A₁ and A₂ respectively. Let {E₁, E₂, … , E_n } be the set of experts who gave their opinions about the strategies of the players. Thus the payoffs of players A₁ given by different experts with mxn strategies can be expressed in the intutionistic fuzzy game matrices as follows: $P^{E_{1}} = (\begin{matrix} 〈 μ_{11}^{E_{1}}, ν_{11}^{E_{1}} 〉 & \dots & 〈 μ_{1 n}^{E_{1}}, ν_{1 n}^{E_{1}} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}^{E_{1}}, ν_{m 1}^{E_{1}} 〉 & \dots & 〈 μ_{mn}^{E_{1}}, ν_{mn}^{E_{1}} 〉 \end{matrix})$ $P^{E_{2}} = (\begin{matrix} 〈 μ_{11}^{E_{2}}, ν_{11}^{E_{2}} 〉 & \dots & 〈 μ_{1 n}^{E_{2}}, ν_{1 n}^{E_{2}} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}^{E_{2}}, ν_{m 1}^{E_{2}} 〉 & \dots & 〈 μ_{mn}^{E_{2}}, ν_{mn}^{E_{2}} 〉 \end{matrix})$ $⋮$ $⋮$ $P^{E_{n}} = (\begin{matrix} 〈 μ_{11}^{E_{n}}, ν_{11}^{E_{n}} 〉 & \dots & 〈 μ_{1 n}^{E_{n}}, ν_{1 n}^{E_{n}} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}^{E_{n}}, ν_{m 1}^{E_{n}} 〉 & \dots & 〈 μ_{mn}^{E_{n}}, ν_{mn}^{E_{n}} 〉 \end{matrix})$ Step 2: Now aggregate the above matrices in which payoffs are given by different experts with the help of Intuitionistic Fuzzy Einstein Interactive Weighted Averaging Operator (IFEIWA) [12]. In IFEIWA if ${\tilde{x}}_{i} = 〈 μ_{i}, ν_{i} 〉$ ; i = 1, 2, 3, …, n be intuitionistic fuzzy sets and w_i ∈ [0, 1] be the weights of the experts then $\begin{matrix} IFEIWA ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n}) = \\ 〈 \frac{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} - \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}}{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} + \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}}, \\ \frac{2 (\prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}} - \prod_{i = 1}^{n} (1 - μ_{i} - ν_{i})^{w_{i}})}{\prod_{i = 1}^{n} (1 + μ_{i})^{w_{i}} + \prod_{i = 1}^{n} (1 - μ_{i})^{w_{i}}} 〉 \end{matrix}$ After aggregating the intuitionistic fuzzy payoffs of the fuzzy game matrices, the resultant aggregated fuzzy matrix P^{E
^A} is $P^{E^{A}} = (\begin{matrix} 〈 μ_{11}^{E^{A}}, ν_{11}^{E^{A}} 〉 & \dots & 〈 μ_{1 n}^{E^{A}}, ν_{1 n}^{E^{A}} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}^{E^{A}}, ν_{m 1}^{E^{A}} 〉 & \dots & 〈 μ_{mn}^{E^{A}}, ν_{mn}^{E^{A}} 〉 \end{matrix})$

Step 3: Now apply Step 2 and Step 4 of existing method of Li [15] (as described in Section 4) to convert the above aggregated fuzzy game matrix into two linear programming problems of both the players A₁ and A₂.

Step 4: Now apply MATLAB 7.0 to solve the dual pair of linear programming problems of the players A₁ and A₂. Then the value of the game and the optimal strategies of both the players are obtained.

The steps of proposed methods are also shown below in the flow chart

Validity of the proposed method

Every person has biases. So the opinion sought from an expert might be influenced by one’s biases which may affect the quality of decision. Seeking opinion of more than one expert is better as it allows hearing the perspectives of people with different background, experiences and thinking style. It also neutralizes the effect of the personal biases of different experts and hence improve upon the quality of decision taken. So the proposed method allows for incorporating the opinion of n experts in decision making and hence is significant improvement over the existing method.

7 Numerical example

The Example 5.1 discussed in the Section 5 is solved here with the help of proposed method.

Step 1: The intuitionistic fuzzy payoffs matrices (as in Example 5.1) are as $\begin{matrix} \begin{matrix} I & II & III \end{matrix} \\ P^{E_{1}} = \begin{matrix} I \\ II \\ III \end{matrix} (\begin{matrix} 〈 0.7, 0.2 〉 & 〈 0.6, 0.3 〉 & 〈 0.9, 0.05 〉 \\ 〈 0.6, 0.15 〉 & 〈 0.5, 0.4 〉 & 〈 0.4, 0.4 〉 \\ 〈 0.85, 0.05 〉 & 〈 0.8, 0.1 〉 & 〈 0.7, 0.3 〉 \end{matrix}) \end{matrix}$ $\begin{matrix} \begin{matrix} I & II & III \end{matrix} \\ P^{E_{2}} = \begin{matrix} I \\ II \\ III \end{matrix} (\begin{matrix} 〈 0.85, 0.1 〉 & 〈 0.7, 0.2 〉 & 〈 0.8, 0.1 〉 \\ 〈 0.5, 0.1 〉 & 〈 0.6, 0.2 〉 & 〈 0.5, 0.2 〉 \\ 〈 0.7, 0.1 〉 & 〈 0.9, 0.05 〉 & 〈 0.6, 0.2 〉 \end{matrix}) \end{matrix}$

Step 2: Use the IFEIWA operator to aggregate all the payoffs given by experts E₁ and E₂ with weights 0.7 and 0.3 respectively. After applying aggregation operator on the above payoffs given by two experts, we get the intuitionistic fuzzy values ${\tilde{a}}_{ij}^{'} s$ which represents the aggregated values of the payoff matrix at the pure strategies γ_i ∈ M and δ_j ∈ N are as follows: ${\tilde{a}}_{11} = (0.75, 0.16)$ Where $\begin{matrix} {\tilde{a}}_{11} = 〈 \frac{(1 + 0.7)^{0.7} (1 + 0.85)^{0.3} - (1 - 0.7)^{0.7} (1 - 0.85)^{0.3}}{(1 + 0.7)^{0.7} (1 + 0.85)^{0.3} + (1 - 0.7)^{0.7} (1 - 0.85)^{0.3}}, \\ \frac{2 [(1 - 0.7)^{0.7} (1 - 0.85)^{0.3} - (1 - 0.7 - 0.2)^{0.7} (1 - 0.85 - 0.1)^{0.3}]}{(1 + 0.7)^{0.7} (1 + 0.85)^{0.3} + (1 - 0.7)^{0.7} (1 - 0.85)^{0.3}} 〉 \end{matrix}$

Similarly, we can find all the aggregated payoffs which are as under:

${\tilde{a}}_{12} = (0.63, 0.27),$ ${\tilde{a}}_{13} = (0.88, 0.06),$ ${\tilde{a}}_{21} = (0.57, 0.14),$ ${\tilde{a}}_{22} = (0.53, 0.35), {\tilde{a}}_{23} = (0.43, 0.34),$ ${\tilde{a}}_{31} = (0.81, 0.062),$ ${\tilde{a}}_{32} = (0.84, 0.082)$ , ${\tilde{a}}_{33} = (0.67, 0.33)$

Thus the aggregated payoff matrix with intuitionistic payoffs is as under: $\begin{matrix} \begin{matrix} I & II & III \end{matrix} \\ P^{E^{A}} = \begin{matrix} I \\ II \\ III \end{matrix} (\begin{matrix} 〈 0.75, 0.16 〉 & 〈 0.63, 0.27 〉 & 〈 0.88, 0.06 〉 \\ 〈 0.57, 0.14 〉 & 〈 0.53, 0.35 〉 & 〈 0.43, 0.34 〉 \\ 〈 0.81, 0.062 〉 & 〈 0.84, 0.082 〉 & 〈 0.67, 0.33 〉 \end{matrix}) \end{matrix}$ (1)

Step 3: Convert the above aggregated matrix into two linear programming problems of player A₁ and A₂ which are dual to each other. Thus the linear programming models of player A₁ and A₂ are constructed as under:

For Player A₁

The linear programming problem of equation (1) of player A₁ is as under: $Z = min {w}$ subject to: ${\begin{matrix} {[λ In (1 - 0.75) + (1 - λ) In (0.16)] p_{1} + [λ In (1 - 0.57) + (1 - λ) In (0.14)] p_{2} + [λ In (1 - 0.81) + (1 - λ) In (0.062)] p_{3}} ⩽ w, \\ {[λ In (1 - 0.63) + (1 - λ) In (0.27)] p_{1} + [λ In (1 - 0.53) + (1 - λ) In (0.35)] p_{2} + [λ In (1 - 0.84) + (1 - λ) In (0.082)] p_{3}} ⩽ w, \\ {[λ In (1 - 0.88) + (1 - λ) In (0.06)] p_{1} + [λ In (1 - 0.43) + (1 - λ) In (0.34)] p_{2} + [λ In (1 - 0.67) + (1 - λ) In (0.33)] p_{3}} ⩽ w, \\ \begin{matrix} p_{1} + p_{2} + p_{3} = 1, \\ p_{i} ⩾ 0; i = 1, 2, 3 and w is an unresricted \end{matrix} \end{matrix}$ Thus the above problem can be rewritten as the following system of inequalities: $Z = min {w}$ subject to: ${\begin{matrix} {[λ In (0 . 25) + (1 - λ) In (0 . 16)] p_{1} + [λ In (0 . 43) + (1 - λ) In (0 . 14)] p_{2} + [λ In (0 . 19) + (1 - λ) In (0 . 062)] p_{3}} - w ⩽ 0, (i) \\ {[λ In (0 . 37) + (1 - λ) In (0 . 27)] p_{1} + [λ In (0 . 47) + (1 - λ) In (0 . 35)] p_{2} + [λ In (0 . 16) + (1 - λ) In (0 . 082)] p_{3}} - w ⩽ 0, (ii) \\ {[λ In (0 . 12) + (1 - λ) In (0 . 06)] p_{1} + [λ In (0 . 57) + (1 - λ) In (0 . 34)] p_{2} + [λ In (0 . 33) + (1 - λ) In (0 . 33)] p_{3}} - w ⩽ 0, (iii) \\ \begin{matrix} p_{1} + p_{2} + p_{3} = 1, (iv) \\ p_{i} ⩾ 0; i = 1, 2, 3 and w is an unrestricted \end{matrix} \end{matrix}$ (2)

For Player A₂

The linear programming problem of player A₂ which is dual of equation (2) is as under:

Let $q_{1}^{'}, q_{2}^{'}, q_{3}^{'}, u$ be the dual variables corresponding to the equations (i), (ii), (iii), (iv) respectively $Z^{*} = max {0 q_{1}^{'} + 0 q_{2}^{'} + 0 q_{3}^{'} + 1 u}$ subject to: ${\begin{matrix} {[λ In (0.25) + (1 - λ) In (0.16)] q_{1}^{'} + [λ In (0.37) + (1 - λ) In (0 . 27)] q_{2}^{'} + [λ In (0.12) + (1 - λ) In (0 . 06)] q_{3}^{'}} + u ⩽ 0, \\ {[λ In (0.43) + (1 - λ) In (0 . 14)] q_{1}^{'} + [λ In (0.47) + (1 - λ) In (0 . 35)] q_{2}^{'} + [λ In (0.57) + (1 - λ) In (0 . 34)] q_{3}^{'}} + u ⩽ 0, \\ {[λ In (0.19) + (1 - λ) In (0 . 062)] q_{1}^{'} + [λ In (0.16) + (1 - λ) In (0 . 082)] q_{2}^{'} + [λ In (0.33) + (1 - λ) In (0 . 33)] q_{3}^{'}} + u ⩽ 0, \\ \begin{matrix} - q_{1}^{'} - q_{2}^{'} - q_{3}^{'} = 1, \\ q_{j}^{'} ⩽ 0; j = 1, 2, 3 and u is an unrestricted \end{matrix} \end{matrix}$ Thus the above problem can be rewritten as follows: $Z^{*} = max {u}$ subject to: ${\begin{matrix} {[λ In (0 . 25) + (1 - λ) In (0 . 16)] q_{1} + [λ In (0 . 37) + (1 - λ) In (0 . 27)] q_{2} + [λ In (0 . 12) + (1 - λ) In (0 . 06)] q_{3}} ⩾ u, \\ {[λ In (0 . 43) + (1 - λ) In (0 . 14)] q_{1} + [λ In (0 . 47) + (1 - λ) In (0 . 35)] q_{2} + [λ In (0 . 57) + (1 - λ) In (0 . 034)] q_{3}} ⩾ u, \\ {[λ In (0 . 19) + (1 - λ) In (0 . 062)] q_{1} + [λ In (0 . 16) + (1 - λ) In (0 . 082)] q_{2} + [λ In (0 . 33) + (1 - λ) In (0 . 33)] q_{3}} ⩾ u, \\ \begin{matrix} q_{1} + q_{2} + q_{3} = 1, \\ q_{j} ⩾ 0; j = 1, 2, 3 and u is an unrestricted where q_{j} = - q_{j}^{'} \end{matrix} \end{matrix}$ (3)

Step 4: For different values ofλ ∈ [0, 1], apply MATLAB 7.0 to solve linear programming problem of (2) and (3) for both the players. The optimal solution is shown below in Table 7.1

Table 7.1

Optimal solution and expected payoffs of aggregated fuzzy matrix game of problem (2) and (3)

λ	p ^*T	w ^*	q ^*T	u ^*	E (p^, q^) = p^P^{E ^A}q^T
0	(0.4807, 0, 0.5193)	(-1.9282)	(0, 0.5886, 0.4114)	(-1.9282)	(0.776338, 0.145415)
0.2	(0.4684, 0, 0.5316)	(-1.8422)	(0, 0.5828, 0.4172)	(- 1.8422)	(0.776735, 0.145445)
0.3	(0.4615, 0, 0.5385)	(- 1.7994)	(0, 0.5796, 0.4204)	(- 1.7994)	(0.77693, 0.145487)
0.5	(0.4459, 0, 0.5541)	(- 1.7142)	(0, 0.5723, 0.4274)	(- 1.7142)	(0.777201, 0.145735)
0.7	(0.4272, 0, 0.5728)	(- 1.6297)	(0, 0.5636, 0.4364)	(- 1.6297)	(0.77763, 0.145979)
0.8	(0.4165, 0, 0.5835)	(- 1.5877)	(0, 0.5586, 0.4414)	(- 1.5877)	(0.777755, 0.146228)
0.9	(0.4046, 0, 0.5954)	(- 1.5460)	(0, 0.5530, 0.4470)	(- 1.5460)	(0.777843, 0.14656)
1	(0.3913, 0, 0.6087)	(- 1.5045)	(0, 0.5468, 0.4532)	(- 1.5045)	(0.777877, 0.146996)

Thus we obtain the maximin strategies p^* of player A₁ and minimax strategies q^* of player A₂ as well as expected payoffs of the fuzzy matrix game P^{E
^A}. For example, if λ = 0.5, the maximin strategies p^* of player A₁ is (0.4459, 0, 0.5541) and minimax strategies q^* of player A₂ is (0, 0.5723, 0.4274) and the corresponding expected payoffs are (0.777201, 0.145735). So the result obtained at λ = 0.5 reveals that Company A₁ will use his Ist strategy of providing very liberal financing to the new car buyers with probability of 0.4459 and also uses his IIIrd strategy to provide exchange bonus on the sale of old car with probability of 0.5541. On the other hand Company A₂ will use his IInd strategy to provide an exclusive five year warranty and free service with probability of 0.5723 and also uses IIIrd strategy to give a handsome exchange bonus on the sale of old car with probability of 0.4274.

8 Conclusion

In this paper limitation of existing method of Li [15] has been pointed out. To overcome this limitation we have proposed a method to solve fuzzy matrix game problem with payoffs as intuitionistic fuzzy sets. The major advantage of this method is that it is used to solve fuzzy matrix games in which more than one expert gives their opinion about the strategies. A practical numerical example has also be given to show the validity of the proposed method. Furthermore in future this method will be more applicable to solve practical problems of game theory in day today life.

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A new method to solve intuitionistic fuzzy matrix game based on the evaluation of different experts

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Fuzzy matrix game with intuitionistic payoffs

4 Li’s method of matrix games with intuitionistic fuzzy sets

5 Limitation of Li’s method

6 Proposed method to solve fuzzy matrix game based on the evaluation of different experts

Validity of the proposed method

7 Numerical example

For Player A1

For Player A2

References

For Player A₁

For Player A₂