Modified difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers

Abstract

Li and Yang [D.F. Li and J. Yang, A difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers, Journal of Applied Mathematics 2013 (2013), 1–10] pointed out that there is no method in the literature for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers and proposed a method for the same. In this paper, it is pointed out that Li and Yang have considered some mathematical incorrect assumptions in their proposed method. For resolving the shortcomings of Li and Yang’s method, a new method (named as Mehar method) is proposed. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method.

Keywords

Bimatrix game trapezoidal intuitionistic fuzzy numbers difference-index based ranking bilinear programming

1 Introduction

Game theory is a mathematical tool to describe strategic interactions among multiple decision makers who behave rationale. The concept of game theory was introduced by Von Neumann and Morgenstern [1] in 1944. Since, then many diverse kind of mathematical games and their solution concept have been proposed [2]. Games are broadly classified into two major categories: cooperative games and non-cooperative games. Cooperation may exist in games, but in most cases, non-cooperation is more attractive because it is more realistic, especially in the presence of competition between players. In non-cooperative games, one important class of games is bimatrix games (or two-person nonzero-sum games) [3].

In the classical (or crisp) game theory, usually the payoffs of players are represented by real numbers. But in real life games, there is need to represent the player’s payoffs by their subjective judgments (or opinions) about competitive situations (or outcomes) instead of real numbers. In this case, the fuzzy set [4] may be used to express the judgments of players. Fuzzy sets are designed to handle aforementioned uncertainties by attributing a degree, called the membership degree, to which an object belongs to a set. The degree to which it does not belong to the same set is taken as one minus the membership degree, and is termed as the non-membership degree.

In real decision making problems there are instances where not only the degree to which an object belongs to a set is known but in addition a degree to which the same object does not belong to the set is also known. Atanassov [5] proposed an interesting generalization of fuzzy sets called intuitionistic fuzzy sets to capture this aspect of human behavior. Intuitionistic fuzzy sets are characterized by two membership functions, one for the degree of belongingness and the other for the degree of non-belongingness. These membership functions are defined such that for each element of the universe the sum of their degrees is less than or equal to one rather than being one as in classical fuzzy set theory. In the last decades, game theory in fuzzy/intuitionistic fuzzy environment has been extensively studied by many researchers[6 –22].

Li and Yang [15] pointed out that, there is no method in the literature for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers. To fill this gap, Li and Yang [15] proposed a method for solving such bimatrix games in which payoffs are represented by trapezoidal intuitionistic fuzzy numbers. After a deep study of Li and Yang’s method [15], it is noticed that Li and Yang [15] have considered some mathematical incorrect assumptions in their proposed method and hence, it is not genuine to use Li and Yang’s method [15] for finding the solution of bimatrix games with intuitionistic fuzzy payoffs. In this paper, the mathematical incorrect assumptions, considered by Li and Yang [15], are pointed out and a new method (named as Mehar method) is proposed for solving bimatrix games with intuitionistic fuzzy payoffs. Also, the exact optimal solution of the numerical problem, solved by Li and Yang by their proposed method, is obtained by the proposed Mehar method.

This paper is organized as follows. In Section 2, some basic concepts related to trapezoidal intuitionistic fuzzy numbers are represented. In Section 3, the method, used by Li and Yang [15], for obtaining mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs is presented as well as mathematical incorrect assumptions, used by Li and Yang [15], are pointed out. The correct mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs and a new method (named as Mehar method) for solving bimatrix games with intuitionistic fuzzy payoffs are proposed in Section 4. The convergence criterion of bimatrix games with intuitionistic fuzzy games is also discussed in Section 4. In Section 5, the exact optimal solution of an existing problem is obtained by proposed Mehar method. Section 6 concludes the paper.

2 Preliminaries

In this section, arithmetic operations of trapezoidal intuitionistic fuzzy numbers, difference-index based ranking method for comparing trapezoidal intuitionistic fuzzy numbers and the method for finding the maximum of trapezoidal intuitionistic fuzzy numbers, required to point out the mathematical incorrect assumptions considered by Li and Yang [15] and to propose a new method (named as Mehar method) for solving bimatrix games with intuitionistic fuzzy payoffs, are presented [15].

2.1 Arithmetic operations of trapezoidal intuitionistic fuzzy numbers

In this section, arithmetic operations of trapezoidal intuitionistic fuzzy numbers are presented [15, Section 2.1, pp. 2].

If ${\tilde{a}}_{i} = 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉, i = 1, 2, \dots, n$ are n-trapezoidal intuitionistic fuzzy numbers.

Then, $\begin{matrix} (i) \sum_{i = 1}^{n} {\tilde{a}}_{i} = \sum_{i = 1}^{n} 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉 \\ = 〈 (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, max_{1 \leq i \leq n} {u_{{\tilde{a}}_{i}}} 〉 \\ = 〈 (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} 〉 . \\ (ii) {\tilde{a}}_{i} - {\tilde{a}}_{j} = 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉 - 〈 ({\underline{a}}_{j}, a_{j}^{L}, a_{j}^{U}, {\bar{a}}_{j}); w_{{\tilde{a}}_{j}}, u_{{\tilde{a}}_{j}} 〉 \\ = 〈 ({\underline{a}}_{i} - {\bar{a}}_{j}, a_{i}^{L} - a_{j}^{U}, a_{i}^{U} - a_{j}^{L}, {\bar{a}}_{i} - {\underline{a}}_{j}); \\ min {w_{{\tilde{a}}_{i}}, w_{{\tilde{a}}_{j}}}; min {(1 - u_{{\tilde{a}}_{i}}), (1 - u_{{\tilde{a}}_{j}})} 〉 . \\ (iii) λ {\tilde{a}}_{i} = λ 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉 \\ = {〈 (λ {\underline{a}}_{i}, λ a_{i}^{L}, λ a_{i}^{U}, λ {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉, λ \geq 0, \\ 〈 (λ {\bar{a}}_{i}, λ a_{i}^{U}, λ a_{i}^{L}, λ {\underline{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉, λ < 0 . \end{matrix}$

2.2 The difference-index based ranking method

In this section, existing difference-index based ranking method [23], used by Li and Yang [15] for comparing trapezoidal intuitionistic fuzzy numbers, is presented [15, Section 2.3, Definition 8, pp. 4].

If ${\tilde{a}}_{1} = 〈 ({\underline{a}}_{1}, a_{1}^{L}, a_{1}^{U}, {\bar{a}}_{1}); w_{{\tilde{a}}_{1}}, u_{{\tilde{a}}_{1}} 〉$ and ${\tilde{a}}_{2} = 〈 ({\underline{a}}_{2},$ $a_{2}^{L}, a_{2}^{U}, {\bar{a}}_{2}); w_{{\tilde{a}}_{2}}, u_{{\tilde{a}}_{2}} 〉$ are two trapezoidal intuitionistic fuzzy numbers and λ ∈ [0, 1]. Then,

${\tilde{a}}_{1} > {\tilde{a}}_{2}$ if and only if $D_{λ} ({\tilde{a}}_{1}) > D_{λ} ({\tilde{a}}_{2})$ .

${\tilde{a}}_{1} \geq {\tilde{a}}_{2}$ if and only if $D_{λ} ({\tilde{a}}_{1}) \geq D_{λ} ({\tilde{a}}_{2})$ .

${\tilde{a}}_{1} = {\tilde{a}}_{2}$ if and only if $D_{λ} ({\tilde{a}}_{1}) = D_{λ} ({\tilde{a}}_{2})$ .

where,

\begin{matrix} D_{λ} ({\tilde{a}}_{i}) = V_{λ} ({\tilde{a}}_{i}) - A_{λ} ({\tilde{a}}_{i}) \\ = (\begin{matrix} V_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) - \\ A_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) \end{matrix}) \\ = (\begin{matrix} (λ (1 - u_{{\tilde{a}}_{i}}) + (1 - λ) (w_{{\tilde{a}}_{i}})) (\frac{{\underline{a}}_{i} + 2 a_{i}^{L} + 2 a_{i}^{U} + {\bar{a}}_{i}}{12}) - \\ (λ (w_{{\tilde{a}}_{i}}) + (1 - λ) (1 - u_{{\tilde{a}}_{i}})) (\frac{{\bar{a}}_{i} - {\underline{a}}_{i} + 2 a_{i}^{U} - 2 a_{i}^{L}}{12}) \end{matrix}) . \end{matrix}

2.3 Maximum of trapezoidal intuitionistic fuzzy numbers

In this section, the method, used by Li and Yang [15], for finding the maximum of trapezoidal intuitionistic fuzzy numbers, is presented.

It is obvious from Section 2.2 that if ${\tilde{a}}_{i} = 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉, i = 1, 2, \dots, n$ aren-trapezoidal intuitionistic fuzzy numbers and λ ∈ [0, 1]. Then, maximum of these numbers can be obtained as follows:

Step 1: Find $\begin{matrix} D_{λ} ({\tilde{a}}_{i}) = V_{λ} ({\tilde{a}}_{i}) - A_{λ} ({\tilde{a}}_{i}) \\ = (\begin{matrix} V_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) - \\ A_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) \end{matrix}) \end{matrix}$ $= (\begin{matrix} (λ (1 - u_{{\tilde{a}}_{i}}) + (1 - λ) (w_{{\tilde{a}}_{i}})) (\frac{{\underline{a}}_{i} + 2 a_{i}^{L} + 2 a_{i}^{U} + {\bar{a}}_{i}}{12}) - \\ (λ (w_{{\tilde{a}}_{i}}) + (1 - λ) (1 - u_{{\tilde{a}}_{i}})) (\frac{{\bar{a}}_{i} - {\underline{a}}_{i} + 2 a_{i}^{U} - 2 a_{i}^{L}}{12}) \end{matrix}) .$

Step 2: Find $max_{1 \leq i \leq n} {D_{λ} ({\tilde{a}}_{i})}$ .

Step 3: If $max_{1 \leq i \leq n} {D_{λ} ({\tilde{a}}_{i})} = D_{λ} ({\tilde{a}}_{p})$ (say). Then, $max_{1 \leq i \leq n} {{\tilde{a}}_{i}} = {\tilde{a}}_{p}$ .

3 Flaws in the existing mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs

The aim of this section is to point out that Li and Yang [15] have used some mathematical incorrect assumption to obtain the mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs. To point out the same, it is necessary to explain the procedure followed by Li and Yang [15] to obtain this mathematical formulation. So, firstly the procedure followed by Li and Yang [15], to obtain the mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs, is presented and then the mathematical incorrect assumptions, considered by Li and Yang [15], are pointed out.

3.1 Mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs

Mangasarian and Stone [3] proved that a Nash equilibrium point (a pair of strategies where the objectives of both the players are fulfilled simultaneously) for bimatrix games (two person nonzero-sum games) can be obtained by solving a quadratic programming problem. On the same direction, Li and Yang [15] obtained a quadratic problem P1 to obtain a Nash equilibrium point for bimatrix games with intuitionistic fuzzy payoffs.

Problem P1 [15 equation 22 Section 3.2, pp.5] $\begin{matrix} Maximize (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} (D_{λ_{1}} ({\tilde{a}}_{ij}) + D_{λ_{2}} ({\tilde{b}}_{ij})) z_{j} - u - v) \\ Subject to \\ \sum_{j = 1}^{n} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j} - u \leq 0, i = 1, 2, \dots, m; \end{matrix}$ $\begin{matrix} \sum_{i = 1}^{m} D_{λ_{2}} ({\tilde{b}}_{ij}) y_{i} - v \leq 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} y_{i} = 1; \\ \sum_{j = 1}^{n} z_{j} = 1; \\ y_{i} \geq 0, i = 1, 2, \dots, m; \\ z_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$ To point out the mathematical incorrect assumption considered by Li and Yang [15], it is necessary to explain the method, followed by Li and Yang [15], to obtain the mathematical formulation i.e. problem P1 of bimatrix games with intuitionistic fuzzy payoffs. Therefore, in this section, the same is presented.

Let player 1 and player 2 have mixed strategies as y _i, i = 1, 2, …, m and z _j, j = 1, 2, …, n respectively. Let ${\tilde{A}}_{m \times n} = ({\tilde{a}}_{ij})_{m \times n}$ and ${\tilde{B}}_{m \times n} = ({\tilde{b}}_{ij})_{m \times n}$ be intuitionistic fuzzy payoff matrices of player 1 and player 2 respectively. Player 1 maximizes profit over rows of fuzzy matrix ${\tilde{A}}_{m \times n} = ({\tilde{a}}_{ij})_{m \times n}$ and player 2 maximizes profit over columns of fuzzy matrix ${\tilde{B}}_{m \times n} = ({\tilde{b}}_{ij})_{m \times n}$ .

Therefore, the objective of player 1 is to $\begin{matrix} Maximize (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{a}}_{ij} z_{j}) \\ Subject to \\ \sum_{i = 1}^{m} y_{i} = 1, \\ y_{i} \geq 0, i = 1, 2, \dots, m . \end{matrix}$

and

the objective of player 2 is to $\begin{matrix} Maximize (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} {\tilde{b}}_{ij} z_{j}) \\ Subject to \\ \sum_{j = 1}^{n} z_{j} = 1, \\ z_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

Using Section 2.3, the objective of player 1 is to $\begin{matrix} Maximize (D_{λ_{1}} (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{a}}_{ij} z_{j})) \\ Subject to \\ \sum_{i = 1}^{m} y_{i} = 1, \\ y_{i} \geq 0, i = 1, 2, \dots, m . \end{matrix}$

and the objective of player 2 is to $\begin{matrix} Maximize (D_{λ_{2}} (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} {\tilde{b}}_{ij} z_{j})) \\ Subject to \\ \sum_{j = 1}^{n} z_{j} = 1, \\ z_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

Li and Yang [15] assumed that to Maximize $(D_{λ_{1}} (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{a}}_{ij} z_{j}))$ is equivalent to Maximize $(\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j})$ and to Maximize $(D_{λ_{2}} (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} {\tilde{b}}_{ij} z_{j}))$ is equivalent to Maximize $(\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} D_{λ_{2}} ({\tilde{b}}_{ij}) z_{j})$ .

Therefore, the objective of player 1 is to $\begin{matrix} Maximize (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j}) \\ Subject to \\ \sum_{i = 1}^{m} y_{i} = 1, \\ y_{i} \geq 0, i = 1, 2, \dots, m . \end{matrix}$

and the objective of player 2 is to $\begin{matrix} Maximize (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} D_{λ_{2}} ({\tilde{b}}_{ij}) z_{j}) \\ Subject to \\ \sum_{j = 1}^{n} z_{j} = 1, \\ z_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

According to Mangasarin and Stone [3], the point $(y_{i}^{0}, z_{j}^{0})$ will be a Nash equilibrium point (a pair of strategies $y_{i}^{0}, z_{j}^{0}$ where the objectives of both the players are fulfilled simultaneously) if there exist real numbers u ⁰, v ⁰ such that $y_{i}^{0}, z_{j}^{0}, u^{0}, v^{0}$ satisfy the following conditions: $\begin{matrix} \sum_{i = 1}^{m} \sum_{j = 1}^{n} (y_{i}^{0} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j}^{0}) - u^{0} = 0; \\ \sum_{j = 1}^{n} \sum_{i = 1}^{m} (y_{i}^{0} D_{λ_{2}} ({\tilde{b}}_{ij}) z_{j}^{0}) - v^{0} = 0; \\ \sum_{j = 1}^{n} (D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j}^{0}) - u^{0} \leq 0, i = 1, 2, \dots, m; \\ \sum_{i = 1}^{m} (D_{λ_{2}} ({\tilde{b}}_{ij}) y_{i}^{0}) - v^{0} \leq 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} y_{i}^{0} = 1; \\ \sum_{j = 1}^{n} z_{j}^{0} = 1; \\ y_{i}^{0} \geq 0, i = 1, 2, \dots, m; \\ z_{j}^{0} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

Also, according to Mangasarin and Stone [3], the values of $y_{i}^{0}, z_{j}^{0}, u^{0}, v^{0}$ which will satisfy the above conditions will be optimal solution of the quadratic programming problem P1.

3.2 Mathematical incorrect assumption considered by Li and Yang

It is obvious from Section 3.1 that to obtain the mathematical formulation i.e. problem P1, Li and Yang [15] have assumed that to Maximize $(D_{λ_{1}} (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{a}}_{ij} z_{j}))$ is equivalent to Maximize $(\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j})$ and to Maximize $(D_{λ_{2}} (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} {\tilde{b}}_{ij} z_{j}))$ is equivalent to Maximize $(\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} D_{λ_{2}} ({\tilde{b}}_{ij}) z_{j})$ .

However,

$\begin{matrix} D_{λ} (\sum_{i = 1}^{n} {\tilde{a}}_{i}) \\ = D_{λ} (\sum_{i = 1}^{n} 〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) \\ = D_{λ} (〈 \begin{matrix} (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, \\ max_{1 \leq i \leq n} {u_{{\tilde{a}}_{i}}} \end{matrix} 〉) \\ = D_{λ} (〈 \begin{matrix} (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, \\ min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} \end{matrix} 〉) \\ = V_{λ} (〈 \begin{matrix} (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, \\ min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} \end{matrix} 〉) \\ - A_{λ} (〈 \begin{matrix} (\sum_{i = 1}^{n} {\underline{a}}_{i}, \sum_{i = 1}^{n} a_{i}^{L}, \sum_{i = 1}^{n} a_{i}^{U}, \sum_{i = 1}^{n} {\bar{a}}_{i}); min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}}, \\ min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} \end{matrix} 〉) \end{matrix} par$ $\begin{matrix} \begin{matrix} = (\begin{matrix} λ min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} + \\ (1 - λ) min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}} \end{matrix}) \\ (\frac{\sum_{i = 1}^{n} {\underline{a}}_{i} + 2 \sum_{i = 1}^{n} a_{i}^{L} + 2 \sum_{i = 1}^{n} a_{i}^{U} + \sum_{i = 1}^{n} {\bar{a}}_{i}}{12}) \end{matrix} \end{matrix}$

$\begin{matrix} - (\begin{matrix} λ min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}} + \\ (1 - λ) min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})} \end{matrix}) \\ (\frac{\sum_{i = 1}^{n} {\bar{a}}_{i} - \sum_{i = 1}^{n} {\underline{a}}_{i} + 2 \sum_{i = 1}^{n} a_{i}^{U} - 2 \sum_{i = 1}^{n} a_{i}^{L}}{12}) \end{matrix}$ (1) and $\begin{matrix} \sum_{i = 1}^{n} D_{λ} ({\tilde{a}}_{i}) \\ = \sum_{i = 1}^{n} D_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) \\ = \sum_{i = 1}^{n} (\begin{matrix} V_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) - \\ A_{λ} (〈 ({\underline{a}}_{i}, a_{i}^{L}, a_{i}^{U}, {\bar{a}}_{i}); w_{{\tilde{a}}_{i}}, u_{{\tilde{a}}_{i}} 〉) \end{matrix}) \\ = \sum_{i = 1}^{n} (\begin{matrix} ((λ (1 - u_{{\tilde{a}}_{i}}) + (1 - λ) (w_{{\tilde{a}}_{i}})) (\frac{{\underline{a}}_{i} + 2 a_{i}^{L} + 2 a_{i}^{U} + {\bar{a}}_{i}}{12})) - \\ ((λ (w_{{\tilde{a}}_{i}}) + (1 - λ) (1 - u_{{\tilde{a}}_{i}})) (\frac{{\bar{a}}_{i} - {\underline{a}}_{i} + 2 a_{i}^{U} - 2 a_{i}^{L}}{12})) \end{matrix}) \end{matrix}$ (2)

It is obvious from (1) and (2) that $D_{λ} (\sum_{i = 1}^{n} {\tilde{a}}_{i}) \neq \sum_{i = 1}^{n} D_{λ} ({\tilde{a}}_{i})$ . Li and Yang [15, Theorem 7, pp. 3] have also pointed out that $D_{λ} (\sum_{i = 1}^{n} {\tilde{a}}_{i}) = \sum_{i = 1}^{n} D_{λ} ({\tilde{a}}_{i})$ if and only if for $w_{{\tilde{a}}_{i}}, i = 1, 2,, \dots n$ and $u_{{\tilde{a}}_{i}}, i = 1, 2, \dots, n$ , the conditions $w_{{\tilde{a}}_{1}} = w_{{\tilde{a}}_{2}} = \dots = w_{{\tilde{a}}_{n}}$ and $u_{{\tilde{a}}_{1}} = u_{{\tilde{a}}_{2}} = \dots = u_{{\tilde{a}}_{n}}$ will be satisfied.

This clearly indicates that to Maximize $(D_{λ_{1}} (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{a}}_{ij} z_{j}))$ is not equivalent to Maximize $(\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} D_{λ_{1}} ({\tilde{a}}_{ij}) z_{j})$ and to Maximize $(D_{λ_{2}} (\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} {\tilde{b}}_{ij} z_{j}))$ is not equivalent to Maximize $(\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} D_{λ_{2}} ({\tilde{b}}_{ij}) z_{j})$ .

Since, Li and Yang [15] have considered the above mentioned mathematical incorrect assumption for obtaining the mathematical formulation i.e. problem P1, therefore, the mathematical formulation i.e. problem P1 and hence the method, proposed by Li and Yang [15] for solving bimatrix games with intuitionistic fuzzy payoffs based on the mathematical formulation i.e. problem P1, is notvalid.

4 Exact solution of bimatrix games with intuitionistic fuzzy payoffs

Li and Yang [15] claimed that optimal solution {y _i, z _j, u, v} of any bimatrix games with intuitionistic

fuzzy payoffs can be obtained by solving mathematical programming problem P1. However, as discussed in Section 3.2 that the problem P1 is not the exact mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs. Therefore, it is not genuine to use Li and Yang’s method [15] for solving bimatrix games with intuitionistic payoffs. In this section, the exact mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs is proposed. Also, a new method (named as Mehar method) to find the exact solution of bimatrix games with intuitionistic fuzzy payoffs is proposed. The necessary and sufficient condition for the convergence of bimatrix games with intuitionistic fuzzy payoffs is also discussed.

4.1 Correct mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs

It can be easily verified from Section 2.1 and Section 2.2 that $D_{λ} (\sum_{i = 1}^{n} {\tilde{a}}_{i}) = \sum_{i = 1}^{n} M_{λ} ({\tilde{a}}_{i})$ ,

where,

$\begin{matrix} M_{λ} ({\tilde{a}}_{i}) = (λ ξ + (1 - λ) κ) (\frac{{\underline{a}}_{i} + 2 a_{i}^{L} + 2 a_{i}^{U} + {\bar{a}}_{i}}{12}) \\ - (λ κ + (1 - λ) ξ) (\frac{{\bar{a}}_{i} - {\underline{a}}_{i} + 2 a_{i}^{U} - 2 a_{i}^{L}}{12}), \\ ξ = min_{1 \leq i \leq n} {(1 - u_{{\tilde{a}}_{i}})}, κ = min_{1 \leq i \leq n} {w_{{\tilde{a}}_{i}}} . \end{matrix} par$ Therefore, using the relation $D_{λ} (\sum_{i = 1}^{n} {\tilde{a}}_{i}) = \sum_{i = 1}^{n} M_{λ} ({\tilde{a}}_{i})$ in the procedure, for obtaining mathematical formulation of bimatrix games with intuitionistic fuzzy payoffs described in Section 3.1, the correct mathematical formulation i.e. problem P2 of bimatrix games with intuitionistic fuzzy payoffs is obtained.

Problem P2

$\begin{matrix} Maximize (\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} (M_{λ_{1}} ({\tilde{a}}_{ij}) + M_{λ_{2}} ({\tilde{b}}_{ij})) z_{j} - u - v) \\ Subject to \\ \sum_{j = 1}^{n} M_{λ_{1}} ({\tilde{a}}_{ij}) z_{j} - u \leq 0, i = 1, 2, \dots, m; \\ \sum_{i = 1}^{m} M_{λ_{2}} ({\tilde{b}}_{ij}) y_{i} - v \leq 0, j = 1, 2, \dots, n; \\ \sum_{i = 1}^{m} y_{i} = 1; \\ \sum_{j = 1}^{n} z_{j} = 1; \\ y_{i} \geq 0, i = 1, 2, \dots, m; \\ z_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$ where $\begin{matrix} M_{λ_{1}} ({\tilde{a}}_{ij}) = (λ_{1} α + (1 - λ_{1}) β) (\frac{{\underline{a}}_{ij} + 2 a_{ij}^{L} + 2 a_{ij}^{U} + {\bar{a}}_{ij}}{12}) \\ - (λ_{1} β + (1 - λ_{1}) α) (\frac{{\bar{a}}_{ij} - {\underline{a}}_{ij} + 2 a_{ij}^{U} - 2 a_{ij}^{L}}{12}), \end{matrix}$ $\begin{matrix} M_{λ_{2}} ({\tilde{b}}_{ij}) = (λ_{2} ψ + (1 - λ_{2}) τ) (\frac{{\underline{b}}_{ij} + 2 b_{ij}^{L} + 2 b_{ij}^{U} + {\bar{b}}_{ij}}{12}) \\ - (λ_{2} τ + (1 - λ_{2}) ψ) (\frac{{\bar{b}}_{ij} - {\underline{b}}_{ij} + 2 b_{ij}^{U} - 2 b_{ij}^{L}}{12}), \\ α = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {(1 - u_{{\tilde{a}}_{ij}})}, β = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {w_{{\tilde{a}}_{ij}}}, \\ ψ = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {(1 - u_{{\tilde{b}}_{ij}})}, τ = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {w_{{\tilde{b}}_{ij}}} . \end{matrix}$

4.2 Proposed Mehar method

In this section, a new method (named as Mehar method) is proposed for solving bimatrix games with intuitionistic fuzzy payoffs.

The steps of the proposed Mehar method are as follows:

Step 1: Find the values of $\begin{matrix} α = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {(1 - u_{{\tilde{a}}_{ij}})}, β = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {w_{{\tilde{a}}_{ij}}}, \\ ψ = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {(1 - u_{{\tilde{b}}_{ij}})}, τ = min_{\binom{1 \leq i \leq m}{1 \leq j \leq n}} {w_{{\tilde{b}}_{ij}}} . \end{matrix}$

Step 2: Find the values of $\begin{matrix} M_{λ_{1}} ({\tilde{a}}_{ij}) = (λ_{1} α + (1 - λ_{1}) β) (\frac{{\underline{a}}_{ij} + 2 a_{ij}^{L} + 2 a_{ij}^{U} + {\bar{a}}_{ij}}{12}) \\ - (λ_{1} β + (1 - λ_{1}) α) (\frac{{\bar{a}}_{ij} - {\underline{a}}_{ij} + 2 a_{ij}^{U} - 2 a_{ij}^{L}}{12}), \\ M_{λ_{2}} ({\tilde{b}}_{ij}) = (λ_{2} ψ + (1 - λ_{2}) τ) (\frac{{\underline{b}}_{ij} + 2 b_{ij}^{L} + 2 b_{ij}^{U} + {\bar{b}}_{ij}}{12}) \\ - (λ_{2} τ + (1 - λ_{2}) ψ) (\frac{{\bar{b}}_{ij} - {\underline{b}}_{ij} + 2 b_{ij}^{U} - 2 b_{ij}^{L}}{12}) . \end{matrix}$

Step 3: Put the values of $M_{λ_{1}} ({\tilde{a}}_{ij})$ and $M_{λ_{2}} ({\tilde{b}}_{ij})$ , obtained from Step 2, in problem P2.

Step 4: Find the optimal solution {y _i, z _j, u, v} of the problem, obtained in Step 3, by choosing λ ₁ ∈ [0, 1] and λ ₂ ∈ [0, 1] .

4.3 Convergence of the new proposed Mehar method

Mangasarin and Stone [3, Section II, pp. 349-350] proved that a bimatrix game with crisp payoffs can be formulated into a crisp quadratic programming problem. Mangasarin and Stone [3, Section II, pp. 350-351] also proved that the necessary and sufficient condition for the convergence of a bimatrix game with crisp payoffs is that the global optimal value of the corresponding crisp quadratic programming problem is zero.

Following the procedure of Mangasarin and Stone [3, Section II, pp. 349-350] and replacing crisp payoffs a _ij and b _ij by $D_{λ_{1}} ({\tilde{a}}_{ij})$ and $D_{λ_{2}} ({\tilde{b}}_{ij})$ respectively, Li and Yang [15] proved that a bimatrix game with intuitionistic fuzzy payoffs can be formulated into a quadratic programming problem P1. However, in Section 3.2, it is pointed out that the problem P1 is not the exact mathematical formulation of a bimatrix game with intuitionistic fuzzy payoffs. Also, in Section 4.1, it is pointed out that the exact mathematical formulation of a bimatrix game with intuitionistic fuzzy payoffs can be obtained by following the procedure of Mangasarin and Stone [3, Section II, pp. 349-350] and replacing crisp payoffs a _ij and b _ij by $M_{λ_{1}} ({\tilde{a}}_{ij})$ and $M_{λ_{2}} ({\tilde{b}}_{ij})$ respectively instead of replacing a _ij and b _ij by $D_{λ_{1}} ({\tilde{a}}_{ij})$ and $D_{λ_{2}} ({\tilde{b}}_{ij})$ respectively.

Similarly, replacing crisp payoffs a _ij and b _ij by $M_{λ_{1}} ({\tilde{a}}_{ij})$ and $M_{λ_{1}} ({\tilde{b}}_{ij})$ respectively in the existing proof [3, Section II, pp. 350-351], it can be easily proved that the necessary and sufficient condition for the convergence of a bimatrix games with intuitionistic fuzzy payoffs is that the global optimal value of the corresponding mathematical quadratic programming problem P2 is zero.

5 Exact optimal solution of existing commerce retailer’s strategy choice problem

Li and Yang [15, Section 4, pp. 6-8] solved a commerce retailer’s strategy choice problem to illustrate their proposed method by considering $\tilde{A} = (\begin{matrix} 〈 \begin{matrix} (50, 60, 70, 80), \\ 0.8, 0.1 \end{matrix} 〉 & 〈 \begin{matrix} (30, 40, 70, 80), \\ 0.4, 0.3 \end{matrix} 〉 \\ 〈 \begin{matrix} (20, 30, 40, 50), \\ 0.5, 0.4 \end{matrix} 〉 & 〈 \begin{matrix} (40, 50, 60, 70), \\ 0.6, 0.2 \end{matrix} 〉 \end{matrix}) and$ $\tilde{B} = (\begin{matrix} 〈 \begin{matrix} (40, 50, 60, 70); \\ 0.7, 0.1 \end{matrix} 〉 & 〈 \begin{matrix} (30, 40, 50, 60); \\ 0.7, 0.2 \end{matrix} 〉 \\ 〈 \begin{matrix} (20, 30, 40, 50); \\ 0.5, 0.3 \end{matrix} 〉 & 〈 \begin{matrix} (50, 60, 70, 80); \\ 0.8, 0.1 \end{matrix} 〉 \end{matrix}) as$ intuitionistic fuzzy payoffs matrices of first and second commercial retailer respectively. However, as discussed in Section 3.2, that the existing method [15] is not valid. So, the intuitionistic fuzzy optimal solution of commerce retailer’s strategy choice problem, obtained by Li and Yang [15], is not exact. Also it can be easily verified that the value of objective function of problem P1 corresponding to the existing optimal solution, obtained by Li and Yang [15] shown in Table 1 to Table 12, is not zero. While, according to Mangasarin and Stone [3], an optimal solution of problem P1 will be Nash equilibrium if and only if the value of objective function of problem P1 corresponding to that optimal solution is zero. This clearly indicates that the optimal solution of commerce retailer’s strategy choice problem, obtained by Li and Yang [15] shown in Table 1 to Table 12, are actually not Nash equilibrium points.

In this section, the exact optimal solution (Nash equilibrium points) of this existing problem is obtained by the proposed Mehar method.

Using the proposed Mehar method, the exact optimal solution of this problem can be obtained as follows:

Step 1: Using Step 1 of proposed Mehar method $\begin{matrix} α = min {(1 - 0.1), (1 - 0.3), (1 - 0.4), (1 - 0.2)} \\ = min {0.9, 0.7, 0.6, 0.8} = 0.6, \\ β = min {0.8, 0.4, 0.5, 0.6} = 0.4, \\ ψ = min {(1 - 0.1), (1 - 0.2), (1 - 0.3), (1 - 0.1)} \\ = min {0.9, 0.8, 0.7, 0.9} = 0.7, \\ τ = min {0.7, 0.7, 0.5, 0.8} = 0.5 . \end{matrix}$

Step 2: Using Step 2 of the proposed Mehar method, $\begin{matrix} M_{λ 1} ({\tilde{a}}_{11}) & = & M_{λ 1} (〈 (50, 60, 70, 80), 0.8, 0.1 〉) \\ = & (0.2 λ_{1} + 0.4) (\frac{390}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ 1} ({\tilde{a}}_{12}) & = & M_{λ 1} (〈 (30, 40, 70, 80), 0.4, 0.3 〉) \\ = & (0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{110}{12}) \\ M_{λ 1} ({\tilde{a}}_{21}) & = & M_{λ 1} (〈 (20, 30, 40, 50), 0.5, 0.4 〉) \\ = & (0.2 λ_{1} + 0.4) (\frac{210}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ 1} ({\tilde{a}}_{22}) & = & M_{λ 1} (〈 (40, 50, 60, 70), 0.6, 0.2 〉) \\ = & (0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ_{2}} ({\tilde{b}}_{11}) & = & M_{λ_{2}} (〈 (40, 50, 60, 70); 0.7, 0.1 〉) \\ = & (0.2 λ_{1} + 0.5) (\frac{330}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ_{2}} ({\tilde{b}}_{12}) & = & M_{λ_{2}} (〈 (30, 40, 50, 60); 0.7, 0.2 〉) \\ = & (0.2 λ_{1} + 0.5) (\frac{270}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ_{2}} ({\tilde{b}}_{21}) & = & M_{λ_{2}} (〈 (20, 30, 40, 50); 0.5, 0.3 〉) \\ = & (0.2 λ_{1} + 0.5) (\frac{210}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12}) \\ M_{λ_{2}} ({\tilde{b}}_{22}) & = & M_{λ_{2}} (〈 (50, 60, 70, 80); 0.8, 0.1 〉) \\ = & (0.2 λ_{1} + 0.5) (\frac{390}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12}) . \end{matrix}$

Step 3: Putting the values, obtained in Step 2, in problem P2, then problem P2 can be transformed into problem P3.

Problem P3 $Maximize (\begin{matrix} ((0.2 λ_{1} + 0.4) (\frac{390}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) y_{1} z_{1} \\ + ((0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{110}{12})) y_{1} z_{2} \\ + ((0.2 λ_{1} + 0.4) (\frac{210}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} z_{1} \\ + ((0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} z_{2} \\ + ((0.2 λ_{1} + 0.5) (\frac{330}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{1} z_{1} \\ + ((0.2 λ_{1} + 0.5) (\frac{270}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{1} z_{2} \\ + ((0.2 λ_{1} + 0.5) (\frac{210}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} z_{1} \\ + ((0.2 λ_{1} + 0.5) (\frac{390}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} z_{2} \\ - u - v \end{matrix})$ $\begin{matrix} Subject to \\ (\begin{matrix} ((0.2 λ_{1} + 0.4) (\frac{390}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) z_{1} + \\ ((0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{110}{12})) z_{2} \end{matrix}) \leq u; \\ (\begin{matrix} ((0.2 λ_{1} + 0.4) (\frac{210}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) z_{1} + \\ ((0.2 λ_{1} + 0.4) (\frac{330}{12}) - (0.6 - 0.2 λ_{1}) (\frac{50}{12})) z_{2} \end{matrix}) \leq u; \\ (\begin{matrix} ((0.2 λ_{1} + 0.5) (\frac{330}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{1} + \\ ((0.2 λ_{1} + 0.5) (\frac{210}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} \end{matrix}) \leq v; \\ (\begin{matrix} ((0.2 λ_{1} + 0.5) (\frac{270}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{1} + \\ ((0.2 λ_{1} + 0.5) (\frac{390}{12}) - (0.7 - 0.2 λ_{1}) (\frac{50}{12})) y_{2} \end{matrix}) \leq v; \end{matrix}$ $\begin{matrix} y_{1} + y_{2} = 1; \\ z_{1} + z_{2} = 1; \\ y_{1} \geq 0, y_{2} \geq 0, z_{1} \geq 0, z_{2} \geq 0 . \end{matrix}$

Step 4: The exact optimal solution of commerce retailer’s strategy choice problem with intuitionistic fuzzy payoffs, obtained by solving problem P3, as well as the incorrect optimal solution of the same problem, obtained by Li and Yang [15], are shown in Table 1 to Table 12.

6 Conclusion

On the basis of the present study, it can be concluded that some mathematical incorrect assumptions are used in the existing method [15] and hence, it is not genuine to use the existing method [15] for solving such bimatrix games in which payoffs are represented by intuitionistic fuzzy numbers. Also, it can be concluded that in the Mehar method, proposed in this paper, no mathematical incorrect assumption is considered and hence, Mehar method should be used for solving these type of problems.

Footnotes

Acknowledgments

The authors would like to thank Editor-in-Chief, Associate Editor and valuable reviewers. They also appreciate the constructive suggestions from the anonymous referees which have led to an improvement in both the quality and clarity of the paper. The first and second author would like to acknowledge the adolescent inner blessings of Mehar (lovely daughter of cousin sister of Dr. Amit Kumar). They believe that Mata Vaishno Devi has appeared on the earth in the form of Mehar and without her blessings it was not possible to think the ideas presented in this paper. The first author also acknowledge the financial support given to her by Department of Science and Technology under INSPIRE Programme for research students [IF130759] to complete Doctoral studies and would like to thank God for not letting her down at the time of crisis and showing her the silver lining in the dark clouds.

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