A mean-area ranking based non-linear programming approach to solve intuitionistic fuzzy bi-matrix games

Abstract

The aim of this paper is to develop a new methodology for solving bi-matrix games with payoffs of Atanassov’s intuitionistic fuzzy (IF) numbers (IFNs), which are called IF bi-matrix games for short. In this methodology, we propose a weighted mean-area ranking method of IFNs, which is proven to satisfy the linearity. Hereby, the concept of Pareto optimal solution of IF bi-matrix games is introduced and the Pareto optimal solution can be obtained through solving the parameterized non-linear programming model, which is derived from an IF mathematical programming model based on the proposed weighted mean-area ranking method of IFNs. Validity and applicability of the model and method proposed in this paper are illustrated with a practical example of two commerce retailers’ strategy choice problem.

Keywords

Intuitionistic fuzzy number (IFN)mean-area ranking method intuitionistic fuzzy bi-matrix game intuitionistic fuzzy mathematical programming

1 Introduction

The bi-matrix game is a kind of two person non-zero-sum games. In a bi-matrix game, there are two players who effectively make their moves simultaneously without knowing the other player’s action. The normal form of such a game can be described by two matrices with matrix A describing the payoffs of player I and matrix B describing the payoffs of player II. And the zero-sum game is a special case of the bi-matrix game if A + B = 0. The bi-matrix game is an important type of two person non-zero-sum games, which have been successfully applied to different areas, such as politics, economics, and management. In real game situations, due to lack of information or imprecision of the available information, players could only estimate the payoff value approximately with some imprecise degree. In order to make bi-matrix game theory more applicable to real competitive decision problem, the fuzzy set [1] has been used to describe imprecise and uncertain information appearing in bi-matrix problems. Using the possibility measure of fuzzy numbers, Maeda [2] introduced two concepts of equilibrium for the bi-matrix games with fuzzy payoffs. According to the ranking method of fuzzy numbers, Vijay et al. [3] studied the bi-matrix games with fuzzy goals and fuzzy payoffs. Bector and Chandra [4] studied the bi-matrix games with fuzzy payoffs and fuzzy goals based on some duality of fuzzy linear programming. Larbani [5] proposed an approach to solve fuzzy bi-matrix games based on the idea of introducing “nature” as a third player in fuzzy multi-attribute decision making problem.

However, the fuzzy set uses only a membership function to indicate a degree of membership to the fuzzy set under consideration. A degree of non-membership is just automatically the complement to 1. Atanassov [6] introduced the concept of an intuitionistic fuzzy (IF) set (IFS), which is characterized by two functions expressing the degree of membership and the degree of non-membership, respectively. The IFS has been applied to some areas, such as decision analysis [7, 8], logic programming [9], topology [10], medical diagnosis [11], and pattern recognition [12], although there is a debate on the nomenclature of the IFS [13 –15]. However, in the sequent, the term “IF” is used to represent “intuitionistic fuzzy” in the sense of Atanassov [6] to avoid being involved in this discussion. The idea of the IFSs made the description more close to the actual situation in fuzzy bi-matrix games. As far as we know, however, there exists less investigation on bi-matrix games using the IFS. Li [16] proposed a new order relation of trapezoidal IF numbers (TrIFNs) based on the difference-index of value-index to ambiguity-index and hereby developed a bi-linear programming method for solving bi-matrix games in which the payoffs are expressed with TrIFNs. Fan [17] proved the existence of the Nash equilibrium of bi-matrix games with payoffs of IFSs by using the fixed point theory and discussed two kinds of nonlinear programming algorithms to compute this Nash equilibrium of bi-matrix games with payoffs of IFSs. Yang [18] proposed a non-linear programming model to compute the mixed Nash equilibrium solution of multi-objective bi-matrix games in the IF environment based on score function of IFSs. The aim of this paper is to study bi-matrix games with payoffs of IF numbers (IFNs), which are called IF bi-matrix games for short. The ranking method of IFNs is important to obtain the solution of IF bi-matrix games. Different from some ranking methods on IFNs [19 –26], this paper proposes a weighted mean-area ranking method of IFNs and proves its some useful and important properties. Besides, the ranking method satisfies six of seven axioms proposed by Wang and Kerre [27], which serve as the reasonable for the ordering of fuzzy quantities. Hereby an IF bi-matrix game is formulated and the parameterized non-linear programming method is developed to solve such IF bi-matrix games.

The rest of this paper is organized as follows. Section 2 presents the preliminaries of IFNs. Section 3 proposes a weighted mean-area ranking method of IFNs and discusses its some properties. Section 4 defines Pareto optimal solutions of IF bi-matrix games, which can be obtained by solving the parameterized non-linear programming models derived from IF mathematical programming models based on the proposed weighted mean-area ranking method. In Section 5, the models and method proposed in this paper are illustrated with a real example of the commerce retailers’ strategy choice problem. Conclusion is given in Section 6.

2 Some definitions and notations

How to express an ill-known quantity using the IFS is very important in game modeling. In this section, we review IFSs [28] and IFNs [29, 30].

2.1 The definition of IFNs

Definition 1. [31] An IFN $\tilde{A}$ is defined as a special IFS on the real number set R, whose membership function $μ_{\tilde{A}} : R \to [0, 1]$ and non-membership function $υ_{\tilde{A}} : R \to [0, 1]$ should satisfy the four conditions (1–4) as follows [26, 31]:

There exist at least two real numbers $x_{0}^{'} \in R$ and $x_{0}^{″} \in R$ such that $μ_{\tilde{A}} (x_{0}^{'}) = 1$ and $υ_{\tilde{A}} (x_{0}^{″}) = 0$ ;

$μ_{\tilde{A}}$ is quasi concave and upper semi-continuous on R;

$υ_{\tilde{A}}$ is quasi convex and lower semi-continuous on R;

The support sets ${x | μ_{\tilde{A}} (x) > 0, x \in R}$ and ${x | υ_{\tilde{A}} (x) < 1, x \in R}$ are bounded.

According to Definition 1, we can easily define the follow general IFN.

Definition 2. [32] A general IFN $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}), f_{l}, f_{r}; ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}), g_{l}, g_{r} 〉$ is a special IFS on the real number set R, whose membership and non-membership function are defined as follows: $μ_{\tilde{A}} (x) = {\begin{matrix} 0 & (x < {\underline{a}}_{1}) \\ f_{l} (x) & ({\underline{a}}_{1} \leq x < a_{1 l}) \\ 1 & (a_{1 l} \leq x \leq a_{1 r}) \\ f_{r} (x) & (a_{1 r} < x \leq {\bar{a}}_{1}) \\ 0 & (x > {\bar{a}}_{1}) \end{matrix}$ (1) and $υ_{\tilde{A}} (x) = {\begin{matrix} 1 & (x < {\underline{a}}_{2}) \\ g_{l} (x) & ({\underline{a}}_{2} \leq x < a_{2 l}) \\ 0 & (a_{2 l} \leq x \leq a_{2 r}) \\ g_{r} (x) & (a_{2 r} < x \leq {\bar{a}}_{2}) \\ 1 & (x > {\bar{a}}_{2}), \end{matrix}$ (2) respectively, depicted as in Fig. 1, where ${\underline{a}}_{2} \leq {\underline{a}}_{1} \leq a_{2 l} \leq a_{1 l} \leq a_{1 r} \leq a_{2 r} \leq {\bar{a}}_{1} \leq {\bar{a}}_{2}$ . f_l and g_r are non-decreasing and piecewise upper semi-continuous functions; f_r and g_l are non-increasing and piecewise lower semi-continuous functions.

Fig.1

An IFN.

Let $π_{\tilde{A}} (x) = 1 - μ_{\tilde{A}} (x) - υ_{\tilde{A}} (x)$ , which is called the IF index of an element x in the IFN $\tilde{A}$ . It is the degree of indeterminacy membership of the element x to $\tilde{A}$ .

Due to the IFN is the subjective reflection of objective things, the membership function and non-membership function expression forms are not unique. According to the specific problem, we can adopt the corresponding method to construct the membership function and non-membership function. There are some commonly used methods of constructing membership function and non-membership such as intuitive method, fuzzy statistics, fuzzy distribution, and so on.

Particularly, when f_l (x), f_r (x), g_l (x) and g_r (x) are linear functions, we have the notation of TrIFN as follows:

Definition 3. Let $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}); ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}) 〉$ is a TrIFN on the real number set R, whose membership and non-membership function are defined as follows: $μ_{\tilde{A}} (x) = {\begin{matrix} 0 & (x < {\underline{a}}_{1}) \\ (x - {\underline{a}}_{1}) / (a_{1 l} - {\underline{a}}_{1}) & ({\underline{a}}_{1} \leq x < a_{1 l}) \\ 1 & (a_{1 l} \leq x \leq a_{1 r}) \\ ({\bar{a}}_{1} - x) / ({\bar{a}}_{1} - a_{1 r}) & (a_{1 r} < x \leq {\bar{a}}_{1}) \\ 0 & (x > {\bar{a}}_{1}) \end{matrix}$ (3) and $υ_{\tilde{A}} (x) = {\begin{matrix} 1 & (x < {\underline{a}}_{2}) \\ (a_{2 l} - x) / (a_{2 l} - {\underline{a}}_{2}) & ({\underline{a}}_{2} \leq x < a_{2 l}) \\ 0 & (a_{2 l} \leq x \leq a_{2 r}) \\ (x - a_{2 r}) / ({\bar{a}}_{2} - a_{2 r}) & (a_{2 r} < x \leq {\bar{a}}_{2}) \\ 1 & (x > {\bar{a}}_{2}), \end{matrix}$ (4) respectively, depicted as in Fig. 2, where ${\underline{a}}_{2} \leq {\underline{a}}_{1} \leq a_{2 l} \leq a_{1 l} \leq a_{1 r} \leq a_{2 r} \leq {\bar{a}}_{1} \leq {\bar{a}}_{2}$ .

Fig.2

A TrIFN.

Further, if a_2l = a_2r (hereby a_1l = a_1r), i.e., ${\underline{a}}_{2} \leq {\underline{a}}_{1} \leq a \leq {\bar{a}}_{1} \leq {\bar{a}}_{2}$ , where a = a_2l = a_2r = a_1l = a_1r, then the TrIFN $\tilde{A}$ is degenerated to the triangular IFN (TIFN) [22], denoted by $\tilde{A} = 〈 ({\underline{a}}_{1}, a, {\bar{a}}_{1}); ({\underline{a}}_{2}, a, {\bar{a}}_{2}) 〉$ .

Obviously, if ${\underline{a}}_{2} = {\underline{a}}_{1}$ , a_2l = a_1l, a_2r = a_1r and ${\bar{a}}_{2} = {\bar{a}}_{1}$ , then $μ_{\tilde{A}} (x) + υ_{\tilde{A}} (x) = 1$ for all x ∈ R. In this case, the TrIFN $\tilde{A}$ is degenerated to $\tilde{A} = ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1})$ or $\tilde{A} = ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2})$ , which is just the trapezoidal fuzzy number (TrFN) [30, 33]. Therefore, the TrIFN are a generalization of the TrFN.

2.2 Cut sets of IFNs and the addition and scalar multiplication

Definition 4. A α-cut set of the general IFN $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}), f_{l}, f_{r}; ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}), g_{l}, g_{r} 〉$ is defined for ${\tilde{A}}^{α} = {x | μ_{\tilde{A}} (x) \geq α}$ , where α ∈ [0, 1].

According to Definition 2, it can be easily seen that ${\tilde{A}}^{α}$ is a closed interval, denoted by ${\tilde{A}}^{α} = [L^{α} (\tilde{A}), R^{α} (\tilde{A})]$ . It directly follows from Equation (1) that ${\tilde{A}}^{α} = [L^{α} (\tilde{A}), R^{α} (\tilde{A})] = [f_{l}^{- 1} (α), f_{r}^{- 1} (α)],$ (5) where $f_{l}^{- 1}$ and $f_{r}^{- 1}$ are the inverse functions of f_l and f_r, respectively.

Definition 5. A β-cut set of the general IFN $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}), f_{l}, f_{r}; ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}), g_{l}, g_{r} 〉$ is defined for ${\tilde{A}}_{β} = {x | υ_{\tilde{A}} (x) \leq β}$ , where β ∈ [0, 1].

Obviously, ${\tilde{A}}_{β}$ is a closed interval, denoted by ${\tilde{A}}_{β} = [L_{β} (\tilde{A}), R_{β} (\tilde{A})]$ . It directly follows from Equation (2) that ${\tilde{A}}_{β} = [L_{β} (\tilde{A}), R_{β} (\tilde{A})] = [g_{l}^{- 1} (β), g_{r}^{- 1} (β)],$ (6) where $g_{l}^{- 1}$ and $g_{r}^{- 1}$ are the inverse functions of g_l and g_r, respectively.

According to Definitions 3–5, any α-cut set and β-cut set of a TrIFN $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}); ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}) 〉$ are easily calculated as follows:

$\begin{matrix} [L^{α} (\tilde{A}), R^{α} (\tilde{A})] \\ = [{\underline{a}}_{1} + (a_{1 l} - {\underline{a}}_{1}) α, {\bar{a}}_{1} - ({\bar{a}}_{1} - a_{1 r}) α] \end{matrix}$ (7) and

$\begin{matrix} [L_{β} (\tilde{A}), R_{β} (\tilde{A})] \\ = [a_{2 l} - (a_{2 l} - {\underline{a}}_{2}) β, a_{2 r} + ({\bar{a}}_{2} - a_{2 r}) β] \end{matrix}$ (8) respectively.

According to the arithmetic operations of intervals [30] and the above concept of cut-sets of general IFN, the addition and scalar multiplication of cut-sets of general IFN are defined as follows.

Definition 6. Let $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}), f_{l}, f_{r}; ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}), g_{l}, g_{r} 〉$ and $\tilde{B} = 〈 ({\underline{b}}_{1}, b_{1 l}, b_{1 r}, {\bar{b}}_{1}), f_{l}^{'}, f_{r}^{'}; ({\underline{b}}_{2}, b_{2 l}, b_{2 r}, {\bar{b}}_{2}), g_{l}^{'}, g_{r}^{'} 〉$ be two general IFN. The sum of $\tilde{A}$ and $\tilde{B}$ is defined as a general IFN $\tilde{A} + \tilde{B}$ , whose α-cut set and β-cut set are given as follows:

$\begin{matrix} (\tilde{A} + \tilde{B})^{α} & = & {\tilde{A}}^{α} + {\tilde{B}}^{α} \\ = & [L^{α} (\tilde{A}) + L^{α} (\tilde{B}), R^{α} (\tilde{A}) + R^{α} (\tilde{B})] \\ = & [f_{l}^{- 1} (α) + f_{l}^{' - 1} (α), f_{r}^{- 1} (α) + f_{r}^{' - 1} (α)] \end{matrix}$ (9) and

$\begin{matrix} (\tilde{A} + \tilde{B})_{β} & = & {\tilde{A}}_{β} + {\tilde{B}}_{β} \\ = & [L_{β} (\tilde{A}) + L_{β} (\tilde{B}), R_{β} (\tilde{A}) + R_{β} (\tilde{B})] \\ = & [g_{l}^{- 1} (β) + g_{l}^{' - 1} (β), g_{r}^{- 1} (β) + g_{r}^{' - 1} (β)] \end{matrix}$ (10) respectively, where α ∈ [0, 1] and β ∈ [0, 1].

Definition 7. Let $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}), f_{l}, f_{r}; ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}), g_{l}, g_{r} 〉$ be a general IFN. The scalar multiplication of $\tilde{A}$ and any real number ρ is defined as a general IFN $ρ \tilde{A}$ , whose α-cut set and β-cut set are given as follows: $(ρ \tilde{A})^{α} = ρ {\tilde{A}}^{α} = {\begin{matrix} [ρ f_{l}^{- 1} (α), ρ f_{r}^{- 1} (α)] (ρ \geq 0) \\ [ρ f_{r}^{- 1} (α), ρ f_{l}^{- 1} (α)] (ρ < 0) \end{matrix}$ (11) and $(ρ \tilde{A})_{β} = ρ {\tilde{A}}_{β} = {\begin{matrix} [ρ g_{l}^{- 1} (β), ρ g_{r}^{- 1} (β)] (ρ \geq 0) \\ [ρ g_{r}^{- 1} (β), ρ g_{l}^{- 1} (β)] (ρ < 0) \end{matrix}$ (12) respectively, where α ∈ [0, 1] and β ∈ [0, 1].

3 The mean-area ranking method of general IFNs and properties

In this section, the ranking method of general IFNs based on weighted mean-areas is defined and its properties are discussed.

Let $m ({\tilde{A}}^{α})$ and $m ({\tilde{A}}_{β})$ be mean values of the intervals ${\tilde{A}}^{α}$ and ${\tilde{A}}_{β}$ , i.e.,

$\begin{matrix} m ({\tilde{A}}^{α}) & = & (L^{α} (\tilde{A}) + R^{α} (\tilde{A})) / 2 \\ = & (f_{l}^{- 1} (α) + f_{r}^{- 1} (α)) / 2 \end{matrix}$ (13) and

$\begin{matrix} m ({\tilde{A}}_{β}) & = & (L_{β} (\tilde{A}) + R_{β} (\tilde{A})) / 2 \\ = & (g_{l}^{- 1} (α) + g_{r}^{- 1} (α)) / 2 . \end{matrix}$ (14)

Definition 8. The average area of the membership function $μ_{\tilde{A}} (x)$ and the average area of the non-membership function $υ_{\tilde{A}} (x)$ for the general IFN $\tilde{A}$ are defined as follows: $S_{μ} (\tilde{A}) = \int_{0}^{1} m ({\tilde{A}}^{α}) d α$ (15) and $S_{υ} (\tilde{A}) = \int_{0}^{1} m ({\tilde{A}}_{β}) d β$ (16) respectively.

Let $S_{λ} (\tilde{A}) = λ S_{μ} (\tilde{A}) + (1 - λ) S_{υ} (\tilde{A}),$ (17) where λ ∈ [0, 1] is weight, which represents the decision maker’s preference information about the average area of the membership function $μ_{\tilde{A}} (x)$ and the average area of the non-membership function $υ_{\tilde{A}} (x)$ for the general IFN $\tilde{A}$ . λ ∈ (0.5, 1) means players pay more attention to membership degree. λ ∈ (0, 0.5) means players pay more attention to non-membership degree. λ = 1 means players only consider membership degree. λ = 0 means players only consider non-membership degree. λ = 0.5 means players hold a neutral attitude. $S_{λ} (\tilde{A})$ is called weighted mean-area of the general IFN $\tilde{A}$ .

Specially, for any TrIFN $\tilde{A} = 〈 ({\underline{a}}_{1}, a_{1 l}, a_{1 r}, {\bar{a}}_{1}); ({\underline{a}}_{2}, a_{2 l}, a_{2 r}, {\bar{a}}_{2}) 〉$ , it is easily derived from Equations (7–8) and Definition 8 that $S_{μ} (\tilde{A}) = ({\underline{a}}_{1} + a_{1 l} + a_{1 r} + {\bar{a}}_{1}) / 4$ and $S_{υ} (\tilde{A}) = ({\underline{a}}_{2} + a_{2 l} + a_{2 r} + {\bar{a}}_{2}) / 4 .$

Then, the weighted mean-area of the TrIFN $\tilde{A}$ is calculated as follows:

$\begin{matrix} S_{λ} (\tilde{A}) & = & λ ({\underline{a}}_{1} + a_{1 l} + a_{1 r} + {\bar{a}}_{1}) / 4 \\ + (1 - λ) ({\underline{a}}_{2} + a_{2 l} + a_{2 r} + {\bar{a}}_{2}) / 4 . \end{matrix}$ (18)

Theorem 1.Assume that $\tilde{A}$ and $\tilde{B}$ are two general IFNs. Then, for any real numberρ ∈ R, the following equality is always valid: $S_{λ} (ρ \tilde{A} + \tilde{B}) = ρ S_{λ} (\tilde{A}) + S_{λ} (\tilde{B}) .$

Proof. According to Equations (9) and (11), for any α ∈ [0, 1], we have $(ρ \tilde{A} + \tilde{B})^{α} = ρ {\tilde{A}}^{α} + {\tilde{B}}^{α}$ . Hence, we have $\begin{matrix} L^{α} (ρ \tilde{A} + \tilde{B}) + R^{α} (ρ \tilde{A} + \tilde{B}) \\ = L^{α} (ρ \tilde{A}) + L^{α} (\tilde{B}) + R^{α} (ρ \tilde{A}) + R^{α} (\tilde{B}) \\ = ρ (L^{α} (\tilde{A}) + R^{α} (\tilde{A})) + L^{α} (\tilde{B}) + R^{α} (\tilde{B}) . \end{matrix}$

Combining with Equations (13 and 15), we have

$\begin{matrix} S_{μ} (ρ \tilde{A} + \tilde{B}) \\ = \int_{0}^{1} [(L^{α} (ρ \tilde{A} + \tilde{B}) + L^{α} (ρ \tilde{A} + \tilde{B})) / 2] d α \\ = ρ \int_{0}^{1} [L^{α} (\tilde{A}) + R^{α} (\tilde{A})] d α \\ + \int_{0}^{1} [L^{α} (\tilde{B}) + R^{α} (\tilde{B})] d α \\ = ρ S_{μ} (\tilde{A}) + S_{μ} (\tilde{B}) . \end{matrix}$ (19)

For any β ∈ [0, 1], according to Equations (10 and 12), it easily follows that $(ρ \tilde{A} + \tilde{B})_{β} = ρ {\tilde{A}}_{β} + {\tilde{B}}_{β}$ .

Similarly, according to Equations (14 and 16), we can prove that $S_{υ} (ρ \tilde{A} + \tilde{B}) = ρ S_{υ} (\tilde{A}) + S_{υ} (\tilde{B}) .$ (20)

According to Equations (19 and 20), it is derived from Equation (17) that $\begin{matrix} S_{λ} (ρ \tilde{A} + \tilde{B}) \\ = λ S_{μ} (ρ \tilde{A} + \tilde{B}) + (1 - λ) S_{υ} (ρ \tilde{A} + \tilde{B}) \\ = λ [ρ S_{μ} (\tilde{A}) + S_{μ} (\tilde{B})] \\ + (1 - λ) [ρ S_{υ} (\tilde{A}) + S_{υ} (\tilde{B})] \\ = ρ [λ S_{μ} (\tilde{A}) + (1 - λ) S_{υ} (\tilde{A})] \\ + λ S_{μ} (\tilde{B}) + (1 - λ) S_{υ} (\tilde{B}) \\ = ρ S_{λ} (\tilde{A}) + S_{λ} (\tilde{B}) . □ \end{matrix}$

Theorem 1 shows that the weighted mean-area of general IFNs is linear. Further, $S_{λ} (\tilde{A})$ synthetically reflects the information on the membership degree and the non-membership degree. The bigger the value $S_{λ} (\tilde{A})$ the larger the general IFN $\tilde{A}$ . Then, a ranking method of IFNs based on weighted mean-area is given as follows.

Definition 9. Let $S_{λ} (\tilde{A})$ and $S_{λ} (\tilde{B})$ be weighted mean-areas of general IFNs $\tilde{A}$ and $\tilde{B}$ respectively, λ ∈ [0, 1], then

(1) $S_{λ} (\tilde{A}) > S_{λ} (\tilde{B})$ if and only if $\tilde{A}$ is larger than $\tilde{B}$ , denoted by $\tilde{A} >_{IF} \tilde{B}$ ;

(2) $S_{λ} (\tilde{A}) = S_{λ} (\tilde{B})$ if and only if $\tilde{A}$ is equal to $\tilde{B}$ , denoted by $\tilde{A} =_{IF} \tilde{B}$ ;

(3) $\tilde{A} \geq_{IF} \tilde{B}$ if and only if $\tilde{A} >_{IF} \tilde{B}$ or $\tilde{A} =_{IF} \tilde{B}$ .

The symbol “>_IF” is an IF version of the order relation “>” in the real number set and has the linguistic interpretation “essentially larger than”. The symbols “<_IF” and “= _IF” are explained similarly.

4 Intuitionistic fuzzy mathematical programming model for intuitionistic fuzzy bi-matrix games

Let us consider any bi-matrix games with payoffs of general IFNs as follows. Assume that S₁ = {δ₁, δ₂, ⋯ , δ_m} and S₂ = {σ₁, σ₂, ⋯ , σ_n} are sets of pure strategies for players I and II, respectively. The vectors x = (x₁, x₂, ⋯ , x_m) ^T and y = (y₁, y₂, ⋯ , y_n) ^T are mixed strategies for players I and II, respectively, where x_i (i = 1, 2, ⋯ , m) and y_j (j = 1, 2, ⋯ , n) are probabilities in which players I and II choose their pure strategies δ_i ∈ S₁ (i = 1, ⋯ , m) and σ_j ∈ S₂ (j = 1, ⋯ , n), respectively. Sets of mixed strategies for players I and II are denoted by X and Y, where $X = {x | \sum_{i = 1}^{m} x_{i} = 1, x_{i} \geq 0 (i = 1, 2, \dots, m)}$ and $Y = {y | \sum_{j = 1}^{n} y_{j} = 1, y_{j} \geq 0 (j = 1, 2, \dots, n)}$ , respectively. If player I choose any pure strategy δ_i ∈ S₁ and player II choose any pure strategy σ_j ∈ S₂, then at the situation (δ_i, σ_j) players I and II gain payoffs, which are expressed with general IFNs as follows: $\begin{matrix} {\tilde{A}}_{ij} & = & < ({\underline{a}}_{1 ij}, a_{1 ijl}, a_{1 ijr}, {\bar{a}}_{1 ij}), f_{ijl}, f_{ijr}; \\ ({\underline{a}}_{2 ij}, a_{2 ijl}, a_{2 ijr}, {\bar{a}}_{2 ij}), g_{ijl}, g_{ijr} > \end{matrix}$ and $\begin{matrix} {\tilde{B}}_{ij} & = & < ({\underline{b}}_{1 ij}, b_{1 ijl}, b_{1 ijr}, {\bar{b}}_{1 ij}), f_{ijl}^{'}, f_{ijr}^{'}; \\ ({\underline{b}}_{2 ij}, b_{2 ijl}, b_{2 ijr}, {\bar{b}}_{2 ij}), g_{ijl}^{'}, g_{ijr}^{'} > \end{matrix}$ (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n), respectively. Thus, the payoff matrices of players I and II are expressed as $\tilde{A} = ({\tilde{A}}_{ij})_{m \times n}$ and $\tilde{B} = ({\tilde{B}}_{ij})_{m \times n}$ , respectively. In the sequel, the above IF bi-matrix game is simply denoted by $(\tilde{A}, \tilde{B})$ for short.

If player I chooses any mixed strategy x ∈ X and player II chooses any mixed strategy y ∈ Y, then the expected payoffs of players I and II are defined as general IFN ${\tilde{E}}_{1} (x, y) = x^{T} \tilde{Ay}$ and ${\tilde{E}}_{2} (x, y) = x^{T} \tilde{By}$ , respectively.

Definition 10. (Pareto optimal solution) Assume that there is a pair (x^*, y^*) ∈ X × Y. A strategy x^* ∈ X is the Pareto optimal strategy for player I if there is no x ∈ X such that $x^{T} {\tilde{Ay}}^{*} \geq_{IF} x^{* T} {\tilde{Ay}}^{*} .$

A strategy y^* ∈ Y is the Pareto optimal strategy for player II if there is no y ∈ Y such that $x^{* T} \tilde{By} \geq_{IF} x^{* T} {\tilde{By}}^{*} .$ ${\tilde{u}}^{*} = x^{* T} {\tilde{Ay}}^{*}$ and ${\tilde{ν}}^{*} = x^{* T} {\tilde{By}}^{*}$ are called optimal values of players I and II, respectively. $(x^{*}, y^{*}, {\tilde{u}}^{*}, {\tilde{ν}}^{*})$ is called a Pareto optimal solution of the IF bi-matrix game $(\tilde{A}, \tilde{B})$ .

Theorem 2.Let $(\tilde{A}, \tilde{B})$ be any IF bi-matrix game. $(x^{*}, y^{*}, {\tilde{u}}^{*}, {\tilde{ν}}^{*})$ is a Pareto optimal solution of the IF bi-matrix game $(\tilde{A}, \tilde{B})$ if and only if it is a solution of the IF programming model as follow:

$\begin{matrix} max {x^{T} (\tilde{A} + \tilde{B}) y - \tilde{u} - \tilde{ν}} \\ s . t . {\begin{matrix} \tilde{Ay} \leq_{IF} {\tilde{ue}}_{m} \\ {\tilde{B}}^{T} x \leq_{IF} \tilde{ν} e_{n} \\ x^{T} e_{m} = 1 \\ y^{T} e_{n} = 1 \\ x \geq 0, y \geq 0 \end{matrix} \end{matrix}$ (21) where e_m is an m-dimensional vector of ones, e_n having a dimension n. Furthermore, if $(x^{*}, y^{*}, {\tilde{u}}^{*}, {\tilde{ν}}^{*})$ is a solution of the above programming model, then $x^{* T} (\tilde{A} + \tilde{B}) y^{*} - {\tilde{u}}^{*} - {\tilde{ν}}^{*} =_{IF} 0$ .

Proof. According to the constrains of the programming model (21), $x^{T} \tilde{Ay} \leq_{IF} - x^{T} {\tilde{ue}}_{m}$ , $y^{T} \tilde{Bx} \leq_{IF} - y^{T} \tilde{ν} e_{n}$ , x^Te_m = 1, and y^Te_n = 1, we have $x^{T} \tilde{Ay} \leq_{IF} - \tilde{u}$ and $y^{T} \tilde{Bx} \leq_{IF} - \tilde{ν}$ . So we have $x^{* T} (\tilde{A} + \tilde{B}) y^{*} - {\tilde{u}}^{*} - {\tilde{ν}}^{*} \leq_{IF} 0$ .

According to Definition 10, the Pareto optimal solution of the IF bi-matrix game can be obtained by solving the mathematical model as follows:

$\begin{matrix} max {x^{T} {\tilde{Ay}}^{*} + x^{* T} \tilde{By}} \\ s . t . {\begin{matrix} x^{T} e_{m} = 1 \\ y^{T} e_{n} = 1 \\ x \geq 0, y \geq 0 \end{matrix} \end{matrix}$ (22)

Let $\tilde{u} =_{IF} max_{x \in X} x^{T} {\tilde{Ay}}^{*}$ and $\tilde{ν} =_{IF} max_{y \in Y} x^{* T} \tilde{By}$ . The constraint $\tilde{u} \geq_{IF} x^{T} {\tilde{Ay}}^{*} \geq_{IF} x^{T} \tilde{Ay}$ is also true for all x ≥ 0. So we have ${\tilde{ue}}_{m} \geq_{IF} \tilde{Ay}$ . Also we have the constraints $\tilde{ν} \geq_{IF} x^{* T} \tilde{By} \geq_{IF} x^{T} \tilde{By}$ for all y ≥ 0. Then, we have $\tilde{ν} e_{n} \geq_{IF} {\tilde{B}}^{T} x$ .

Then, Equation (22) can be transformed into the following IF programming model:

$\begin{matrix} min_{IF} {(\tilde{u} - x^{T} \tilde{Ay}) + (\tilde{ν} - x^{T} \tilde{By})} \\ s . t . {\begin{matrix} \tilde{Ay} \leq_{IF} {\tilde{ue}}_{m} \\ {\tilde{B}}^{T} x \leq_{IF} \tilde{ν} e_{n} \\ x^{T} e_{m} = 1 \\ y^{T} e_{n} = 1 \\ x \geq 0, y \geq 0 \end{matrix} \end{matrix}$ (23)

Thus, the IF programming problem of Theorem 2 is deduced.

Using the ranking method of general IFNs in Definition 9 and Theorem 2, the IF mathematical programming model (21) can be converted into the parameterized non-linear programming model as follow: $\begin{matrix} max {\sum_{j = 1}^{n} \sum_{i = 1}^{m} x_{i} (S_{λ} ({\tilde{A}}_{ij}) + S_{λ} ({\tilde{B}}_{ij})) y_{j} - S_{λ} (\tilde{u}) - S_{λ} (\tilde{ν})} \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} S_{λ} ({\tilde{A}}_{ij}) y_{j} \leq S_{λ} (\tilde{u}) (i = 1, 2, \dots, m) \\ \sum_{i = 1}^{m} S_{λ} ({\tilde{B}}_{ij})^{T} x_{i} \leq S_{λ} (\tilde{ν}) (j = 1, 2, \dots, n) \\ x_{1} + x_{2} + \dots + x_{m} = 1 \\ y_{1} + y_{2} + \dots + y_{n} = 1 \\ x_{i} \geq 0 (i = 1, 2, \dots, m), y_{j} \geq 0 (j = 1, 2, \dots, n) \end{matrix} \end{matrix}$ (24) ∥

If $(x^{*}, y^{*}, S_{λ} ({\tilde{u}}^{*}), S_{λ} ({\tilde{ν}}^{*}))$ is an optimal solution of the parameterized non-linear programming model (24), then (x^*, y^*) is a Pareto optimal strategy of the IF bi-matrix game, and $S_{λ} ({\tilde{u}}^{*})$ and $S_{λ} ({\tilde{ν}}^{*})$ are the weighted mean areas of Pareto optimal values of player I and II respectively.

5 An example of two commerce retailers’ strategy choice problem

Let us consider the case of two commerce retailers P₁ and P₂ (i.e., players I and II) making a decision aiming to enhance the satisfaction degrees of customers. As players’ judgments for the satisfaction degrees of customers including preference and experience are often vague and players estimate them with their intuition. And assume that commerce retailers P₁ and P₂ are rational, i.e., they will choose optimal strategies to maximize their own profits without cooperation. Suppose that retailer P₁ has two pure strategies: establishing a scientific and rational service system δ₁ and providing customers with satisfaction products δ₂. Retailer P₂ possesses the same pure strategies as P₁, i.e., the options of retailer P₂ are: establishing a scientific and rational service system σ₁ and providing customers with satisfaction products σ₂.

Let us consider the following IF bi-matrix game for this scenario, where the payoff matrices of commerce retailers P₁ and P₂ are expressed with IFNs as follows: $\tilde{A} = (\begin{matrix} {\tilde{A}}_{11} & {\tilde{A}}_{12} \\ {\tilde{A}}_{21} & {\tilde{A}}_{22} \end{matrix}), \tilde{B} = (\begin{matrix} {\tilde{B}}_{11} & {\tilde{B}}_{12} \\ {\tilde{B}}_{21} & {\tilde{B}}_{22} \end{matrix}),$ where ${\tilde{A}}_{11} = < (184, 192, 196); (180, 192, 204) >$ is a TIFN, whose membership and non-member functions are given as follows: $μ_{{\tilde{A}}_{11}} (x) = {\begin{matrix} 0 & (x < 184) \\ (x - 184) / 8 & (184 \leq x < 192) \\ 1 & (x = 192) \\ (196 - x) / 4 & (192 < x \leq 196) \\ 0 & (x > 196) \end{matrix}$ and $υ_{{\tilde{A}}_{11}} (x) = {\begin{matrix} 1 & (x < 180) \\ (192 - x) / 12 & (180 \leq x < 192) \\ 0 & (x = 192) \\ (x - 192) / 12 & (192 < x \leq 204) \\ 1 & (x > 204) \end{matrix}$ respectively;

${\tilde{A}}_{12} = < (160, 170, 180, 190), f_{12 l}, f_{12 r}; (155, 165, 180, 200),$ g_12l, g_12r> is an IFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{A}}_{12}} (x) = {\begin{matrix} 0 & (x < 160) \\ (x - 160)^{2} / 100 & (160 \leq x < 170) \\ 1 & (170 \leq x \leq 180) \\ (190 - x) / 10 & (180 < x \leq 190) \\ 0 & (x > 190) \end{matrix}$ and $υ_{{\tilde{A}}_{12}} (x) = {\begin{matrix} 1 & (x < 155) \\ (165 - x)^{2} / 100 & (155 \leq x < 165) \\ 0 & (165 \leq x \leq 180) \\ (x - 180) / 20 & (180 < x \leq 200) \\ 1 & (x > 200) \end{matrix},$ respectively;

${\tilde{A}}_{21} = < (145, 155, 190); (140, 155, 190) >$ is a TIFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{A}}_{21}} (x) = {\begin{matrix} 0 & (x < 145) \\ (x - 145) / 10 & (145 \leq x < 155) \\ 1 & (x = 155) \\ (190 - x) / 35 & (155 < x \leq 190) \\ 0 & (x > 190) \end{matrix}$ and $υ_{{\tilde{A}}_{21}} (x) = {\begin{matrix} 1 & (x < 140) \\ (155 - x) / 15 & (140 \leq x < 155) \\ 0 & (x = 155) \\ (x - 155) / 35 & (155 < x \leq 190) \\ 1 & (x > 190) \end{matrix},$ respectively;

${\tilde{A}}_{22} = < (140, 150, 170, 190); (130, 145, 180, 190) >$ is a TrIFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{A}}_{22}} (x) = {\begin{matrix} 0 & (x < 140) \\ (x - 140) / 10 & (140 \leq x < 150) \\ 1 & (150 \leq x \leq 170) \\ (190 - x) / 20 & (170 < x \leq 190) \\ 0 & (x > 190) \end{matrix},$ and $υ_{{\tilde{A}}_{22}} (x) = {\begin{matrix} 1 & (x < 130) \\ (145 - x) / 15 & (130 \leq x < 145) \\ 0 & (145 \leq x \leq 180) \\ (x - 180) / 10 & (180 < x \leq 190) \\ 1 & (x > 190) \end{matrix},$ respectively;

${\tilde{B}}_{11} = < (150, 170, 180), f_{11 l}^{'}, f_{11 r}^{'}; (140, 165, 175, 185),$ $g_{11 l}^{'}, g_{11 r}^{'} >$ is an IFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{B}}_{11}} (x) = {\begin{matrix} 0 & (x < 150) \\ (x - 150) / 20 & (150 \leq x < 170) \\ 1 & (x = 170) \\ (180 - x)^{2} / 100 & (170 < x \leq 180) \\ 0 & (x > 180) \end{matrix}$ and $υ_{{\tilde{B}}_{11}} (x) = {\begin{matrix} 1 & (x < 140) \\ (165 - x) / 25 & (140 \leq x < 165) \\ 0 & (165 \leq x \leq 175) \\ (x - 175)^{2} / 100 & (175 < x \leq 185) \\ 1 & (x > 185) \end{matrix},$ respectively;

${\tilde{B}}_{12} = < (140, 150, 160, 170); (135, 150, 160, 175) >$ is a TrIFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{B}}_{12}} (x) = {\begin{matrix} 0 & (x < 140) \\ (x - 140) / 10 & (140 \leq x < 150) \\ 1 & (150 \leq x \leq 160) \\ (170 - x) / 10 & (160 < x \leq 170) \\ 0 & (x > 170) \end{matrix}$ and $υ_{{\tilde{B}}_{12}} (x) = {\begin{matrix} 1 & (x < 135) \\ (150 - x) / 15 & (135 \leq x < 150) \\ 0 & (150 \leq x \leq 160) \\ (x - 160) / 15 & (160 < x \leq 175) \\ 1 & (x > 175) \end{matrix},$ respectively;

${\tilde{B}}_{21} = < (130, 150, 160); (125, 150, 170) >$ is a TIFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{B}}_{21}} (x) = {\begin{matrix} 0 & (x < 1 30) \\ (x - 130) / 20 & (130 \leq x < 150) \\ 1 & (x = 150) \\ (160 - x) / 10 & (150 < x \leq 160) \\ 0 & (x > 160) \end{matrix}$ and $υ_{{\tilde{B}}_{21}} (x) = {\begin{matrix} 1 & (x < 125) \\ (150 - x) / 25 & (125 \leq x < 150) \\ 0 & (x = 150) \\ (x - 150) / 20 & (150 < x \leq 170) \\ 1 & (x > 170) \end{matrix},$ respectively;

${\tilde{B}}_{22} = < (144, 164, 170); (138, 164, 180) >$ is a TIFN, whose membership and non-membership functions are given as follows: $μ_{{\tilde{B}}_{22}} (x) = {\begin{matrix} 0 & (x < 144) \\ (x - 144) / 20 & (144 \leq x < 164) \\ 1 & (x = 164) \\ (170 - x) / 6 & (164 < x \leq 170) \\ 0 & (x > 170) \end{matrix}$ and $υ_{{\tilde{B}}_{22}} (x) = {\begin{matrix} 1 & (x < 138) \\ (164 - x) / 26 & (138 \leq x < 164) \\ 0 & (x = 164) \\ (x - 164) / 16 & (164 < x \leq 180) \\ 1 & (x > 180) \end{matrix},$ respectively.

The TIFN ${\tilde{A}}_{11} = < (184, 192, 196); (180, 192, 204) >$ in payoff matrix $\tilde{A}$ means that the payoff is about 192 if commerce retailer P₁ adopts the pure strategy δ₁ and commerce retailer P₂ adopts the pure strategy σ₁. Other entries in the IF payoff matrices $\tilde{A}$ and $\tilde{B}$ can be explained similarly.

According to Definition 8, the weighted mean-areas of IF matrices $\tilde{A}$ and $\tilde{B}$ can be obtained as follows: $\begin{matrix} S_{λ} (\tilde{A}) \\ = (\begin{matrix} S_{λ} ({\tilde{A}}_{11}) & S_{λ} ({\tilde{A}}_{12}) \\ S_{λ} ({\tilde{A}}_{21}) & S_{λ} ({\tilde{A}}_{22}) \end{matrix}) \\ = (\begin{matrix} 192 - λ & (20 / 3 - 27.5) λ + 17.5 - 10 / 3 \\ 160 + 1 . 25 λ & 161 . 25 + 1 . 25 λ \end{matrix}) \end{matrix}$ and $\begin{matrix} S_{λ} (\tilde{B}) \\ = (\begin{matrix} S_{λ} ({\tilde{B}}_{11}) & S_{λ} ({\tilde{B}}_{12}) \\ S_{λ} ({\tilde{B}}_{21}) & S_{λ} ({\tilde{B}}_{22}) \end{matrix}) \\ = (\begin{matrix} (6 . 25 - 20 / 3) λ + 163 . 75 + 10 / 3 & 155 \\ 148 . 75 - 1 . 25 λ & 161 . 5 - λ \end{matrix}) . \end{matrix}$

Then, according to Theorem 2 and Equation (24), the parameterized non-linear programming model is constructed as follow:

$\begin{matrix} max {[(5.25 - 20 / 3) λ + 355.75 + 10 / 3] x_{1} y_{1} \\ + [(20 / 3 - 27.5) λ + 332.5 - 10 / 3] x_{1} y_{2} \\ + 308.75 x_{2} y_{1} + (0.25 λ + 322.75) x_{2} y_{2} - S_{λ} (\tilde{u}) \\ - S_{λ} (\tilde{ν})} \\ s . t . {\begin{matrix} (192 - λ) y_{1} + [(20 / 3 - 27.5) λ \\ + 177.5 - 10 / 3] y_{2} \leq S_{λ} (\tilde{u}), \\ (160 + 1.25 λ) y_{1} + (161.25 + 1.25 λ) y_{2} \\ \leq S_{λ} (\tilde{u}), \\ [(6.25 - 20 / 3) λ + 163.75 + 10 / 3] x_{1} \\ + (148.75 - 1.25 λ) x_{2} \leq S_{λ} (\tilde{ν}), \\ 155 x_{1} + (161.5 - λ) x_{2} \leq S_{λ} (\tilde{ν}), \\ x_{1} + x_{2} = 1, \\ y_{1} + y_{2} = 1, \\ x_{1}, x_{2}, y_{1}, y_{2} \geq 0 . \end{matrix} \end{matrix}$ (25)

For the given value λ = 0.5, the optimal solution of Equation (25) can be obtained $(x^{*}, y^{*}, S_{λ} ({\tilde{u}}^{*}), S_{λ} ({\tilde{ν}}^{*}))$ , where x^* = (x₁, x₂) ^T = (0.52, 0.48) ^T, y^* = (y₁, y₂) ^T = (0, 1) ^T, $S_{λ} ({\tilde{u}}^{*}) = 163.75$ , and $S_{λ} ({\tilde{ν}}^{*}) = 157.88$ . Thus, the Pareto optimal strategy x^* and the mean-area of the Pareto optimal value ${\tilde{u}}^{*}$ of player I are obtained as follows: $x^{*} = (x_{1}, x_{2})^{T} = (0.52, 0.48)^{T}$ and $S_{λ} ({\tilde{u}}^{*}) = 163.75 .$

Similarly, the Pareto optimal strategy y^* and the mean-area of the Pareto optimal value ${\tilde{ν}}^{*}$ of player II are obtained as follows: $y^{*} = (y_{1}, y_{2})^{T} = (0, 1)^{T} and S_{λ} ({\tilde{ν}}^{*}) = 157.88 .$

Analogously, for other given value λ ∈ [0, 1], we can obtain the Pareto optimal strategies for the players P₁ and P₂ and corresponding weighted mean-areas of the optimal values through solving Equation (25), depicted as in Table 1.

Table 1

Pareto optimal strategies of P₁ and P₂ and weighted mean-areas of optimal values

	Player P₁		Player P₂
λ	x ^*	$S_{λ} ({\tilde{u}}^{*})$	y ^*	$S_{λ} ({\tilde{ν}}^{*})$
0.0	(0 . 51, 0.49)	174.17	(0, 1)	158.16
0.1	(0 . 51, 0.49)	172.08	(0, 1)	158.11
0.2	(0.52, 0.48)	170.00	(0, 1)	158.05
0.3	(0.52, 0.48)	167.92	(0, 1)	157.99
0.4	(0 . 52, 0 . 48)	165.83	(0, 1)	157.94
0.5	(0.52, 0.48)	163.75	(0, 1)	157.88
0.6	(0.52, 0.48)	161.99	(0.01, 0.99)	157.82
0.7	(0.52, 0.48)	162.03	(0.08, 0.92)	157.77
0.8	(0.52, 0.48)	162.08	(0.14, 0.86)	157.71
0.9	(0.52, 0.48)	162.14	(0.19, 0.81)	157.66
1.0	(0 . 53, 0 . 47)	162.21	(0.24, 0.76)	157.60

It is easy to see from Table 1 that the Pareto optimal strategies for the players P₁ and P₂ and corresponding weighted mean-areas of the optimal values change with the change of the parameter λ. That is to say, the Pareto optimal solutions of IF bi-matrix game vary with the players’ preference information about the membership function and the non-membership function. And strategy choice of a player is affected by other player’ preference.

6 Conclusion

In some situations, determining payoffs of bi-matrix games precisely depends on players’ judgment and intuition, which are often vague and not easy to be represented with crisp values and fuzzy numbers. In the above, we model IF bi-matrix games and develop the parametrized non-linear programming model, which is derived from an IF mathematical programming model by using the ranking order relation of IFNs given in this paper. In particular, the ranking order relation satisfies the six properties proposed by Wang and Kerre [27] and has a natural appealing feature, i.e., the linearity, which can be easily applied to real game problems.

In addition, real numbers and fuzzy numbers (e.g., TrFNs, TFNs, interval-valued fuzzy numbers/intervals) are only degenerate cases of general IFNs. Therefore, contrasted with existing methods, the models and method proposed in this paper can solve not only bi-matrix game problems with various forms of general IFNs but also bi-matrix game problems with special IFN such as TrIFNs, TIFNs, TrFNs, FNs, interval-valued fuzzy numbers, and real numbers. Thus, the developed models and method not only are of universality and applicability but also are feasible and effective. Furthermore, it is easy to see that the models and method proposed in this paper may be extended to IF multi-objective bi-matrix games. And more effective methods of IF bi-matrix games will be investigated in the near future. Also the proposed models and method may be applied to solving many competitive decision problems in similar fields such as management, supply chain, and advertising although they are illustrated with the example of the commerce retailers’ strategy choice problem in this paper.

Footnotes

Acknowledgments

This research was sponsored by the National Natural Science Foundation of China (No. 71231003), “Science and Technology Innovation Team Cultivation Plan of Colleges and Universities in Fujian Province”, the National Natural Science Foundation of China (Nos. 71561008, 71461005) and the Science Foundation of Guangxi Province in China (No. 2014GXNSFAA118010).

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