Parameterized bilinear programming methodology for solving triangular intuitionistic fuzzy number bimatrix games

Abstract

The aim of this paper is to develop a new methodology for solving bimatrix games with payoffs of triangular intuitionistic fuzzy numbers (TIFNs), which are called TIFN bimatrix games for short. In this methodology, we define the concepts of the value-index and ambiguity-index and hereby develop a difference-index based ranking method, which is proven to be a total order. The parameterized bilinear programming models are derived from a pair of auxiliary TIFN mathematical programming models, which are used to determine solutions of TIFN bimatrix games. Validity and applicability of the models and method proposed in this paper are illustrated with a practical example.

Keywords

Bimatrix games triangular intuitionistic fuzzy numbers ranking bilinear programming fuzzy set

1 Introduction

In many real-life game problems, players are not able to evaluate exactly the outcomes of games due to the lack of information or imprecise information. In order to make bimatrix game theory more applicable to real competitive decision problems, the fuzzy set [13] was introduced to describe imprecise and uncertain information appearing in bimatrix problems. Using the ranking method of fuzzy numbers, Vidyottama et al. [19] studied the bimatrix games with fuzzy goals and fuzzy payoffs. Using the possibility measure of fuzzy numbers, Meada [17] introduced two concepts of equilibrium for the bimatrix games with fuzzy payoffs. Bector and Chandra [1] studied bimatrix games with fuzzy payoffs and fuzzy goals based on some duality of fuzzy linear programming. Larbani [14] proposed an approach to solve fuzzy bimatrix games based on the idea of introducing “nature” player in fuzzy multi-attribute decision making problems. However, the intuitionistic fuzzy set (IFS) by adding a non-membership function seems to be suitable for expressing more abundant information [12]. Triangular intuitionistic fuzzy numbers (TIFNs) are special cases of IFSs defined on the set of real numbers, which may easily deal with ill-known information or quantities. TIFNs play an important role in fuzzy decision making [7 , 21].

This paper will apply the TIFNs to deal with imprecise information in bimatrix game problems, which are called TIFN bimatrix games for short. Obviously, TIFN bimatrix games remarkably differ from ordinary fuzzy bimatrix games since the former uses both membership and non-membership degrees to express the payoffs while the latter only uses membership degrees to express the payoffs. So the ordinary models and methods of fuzzy bimatrix games cannot be directly used to solve TIFN bimatrix games. In order to solve TIFN matrix games, the order relation of TIFNs is necessarily, and ranking TIFNs is difficult in nature [2 , 18]. Thus, we propose a difference-index based ranking method of TIFNs based on the concept of value-index and ambiguity-index in this paper. The ranking method with two-index can aggregate both value and ambiguity. And it can take into consideration the subjective attitude of the players adequately. Besides, the ranking method satisfies the most properties proposed by Wang and Kerre [20], which serve as the reasonable for the ordering of fuzzy quantities. Especially, this method has a natural appealing feature, i.e., the linearity, which can be easily applied to real game problems. Further, we derive a pair of parameterized bilinear programming models from the auxiliary mathematical programming models of TIFN bimatrix games.

The rest of this paper is organized as follows. Section 2 establishes a new order relation of TIFNs based on the concepts of value-index and ambiguity-index. Section 3 formulates TIFN bimatrix games and proposes corresponding methodology based on the constructed auxiliary parametric bilinear programming model. In Section 4, the proposed model and method are illustrated with a real example of the commerce retailers’ strategy choice problem. Conclusion is given in Section 5.

2 TIFNs and cut sets

2.1 The definition of TIFNs

How to express an ill-known quantity using the IFS is very important in game modeling. Inspired by the concept of the fuzzy number [8], an intuitionistic fuzzy number $\tilde{A}$ 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 [2, 10]:

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

$μ_{\tilde{A}}$ is quasi convex 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.

From the above definition of an intuitionistic fuzzy number, we can easily construct a TIFN $\tilde{A} = < ({\underline{a}}_{1}, a, {\bar{a}}_{1}); ({\underline{a}}_{2}, a, {\bar{a}}_{2}) >$ , depicted as in Fig. 1, whose membership and non-membership functions are given as follows:

$μ_{\tilde{A}} (x) = {\begin{matrix} 0 & (x < {\underline{a}}_{1}) \\ (x - {\underline{a}}_{1}) / (a - {\underline{a}}_{1}) & ({\underline{a}}_{1} \leq x < a) \\ 1 & (x = a) \\ ({\bar{a}}_{1} - x) / ({\bar{a}}_{1} - a) & (a < x \leq {\bar{a}}_{1}) \\ 0 & (x > {\bar{a}}_{1}) \end{matrix}$ (1) $υ_{\tilde{A}} (x) = {\begin{matrix} 1 & (x < {\underline{a}}_{2}) \\ (a - x) / (a - {\underline{a}}_{2}) & ({\underline{a}}_{2} \leq x < a) \\ 0 & (x = a) \\ (x - a) / ({\bar{a}}_{2} - a) & (a < x \leq {\bar{a}}_{2}) \\ 1 & (x > {\bar{a}}_{2}) . \end{matrix}$ (2) respectively, where $[{\underline{a}}_{1}, {\bar{a}}_{1}]$ and $[{\underline{a}}_{2}, {\bar{a}}_{2}]$ are called the mean intervals of the TIFN $\tilde{A}$ for the membership and non-membership functions, respectively.

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

If ${\underline{a}}_{2} \geq 0$ , then $\tilde{A} = < ({\underline{a}}_{1}, a, {\bar{a}}_{1}); ({\underline{a}}_{2}, a, {\bar{a}}_{2}) >$ is called the non-negative TIFN, denoted by $\tilde{A} \geq 0$ . Conversely, if ${\bar{a}}_{2} \leq 0$ , then $\tilde{A}$ is called the non-positive TIFN, denoted by $\tilde{A} \leq 0$ . Further, $\tilde{A}$ is called the positive TIFN if ${\underline{a}}_{2} \geq 0$ and ${\bar{a}}_{2} > 0$ , denoted by $\tilde{A} > 0$ . Likewise, $\tilde{A}$ is called the negative TIFN if ${\underline{a}}_{2} < 0$ and ${\bar{a}}_{2} \leq 0$ , denoted by $\tilde{A} < 0$ .

Generally, arithmetic operations of TIFNs can be derived from the extension principle of IFSs [4 , 7]. In the following, we discuss the addition and scalar multiplication of TIFNs based on the concept of cut sets [13].

2.2 Cut sets of TIFN and operations

For any α ∈ [0, 1], a α-cut set of a TIFN $\tilde{A}$ can be expressed as a crisp subset of R, denoted by ${\tilde{A}}_{α} = {x | μ_{\tilde{A}} (x) \geq α, x \in R}$ . According to the definition of the TIFN, 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 $[L_{α} (\tilde{A}), R_{α} (\tilde{A})] = [{\underline{a}}_{1} + (a - {\underline{a}}_{1}) α, {\bar{a}}_{1} - ({\bar{a}}_{1} - a) α]$ (3)

Likewise, for any β ∈ [0, 1], a β-cut set of a TIFN $\tilde{A}$ can be expressed as a crisp subset of R, denoted by ${\tilde{A}}_{β} = {x | υ_{\tilde{A}} (x) \leq β, x \in R}$ . Obviously, ${\tilde{A}}_{β}$ is a closed interval, denoted by ${\tilde{A}}_{β} = [L_{β} (\tilde{A}), R_{β} (\tilde{A})]$ . It is directly derived from Equation (2) that $[L_{β} (\tilde{A}), R_{β} (\tilde{A})] = [a - (a - {\underline{a}}_{2}) β, a + ({\bar{a}}_{2} - a) β]$ (4)

According to the arithmetic operations of intervals and the above concept of cut sets of TIFNs, we can define the addition and scalar multiplication of TIFNs.

Specifically, for any TIF $\tilde{A} = < ({\underline{a}}_{1}, a, {\bar{a}}_{1});$ $({\underline{a}}_{2}, a, {\bar{a}}_{2}) >$ and ${\tilde{A}}^{'} = < ({\underline{a}}_{1}^{'}, a^{'}, {\bar{a}}_{1}^{'}); ({\underline{a}}_{2}^{'}, a^{'}, {\bar{a}}_{2}^{'}) >$ , the sum of $\tilde{A}$ and ${\tilde{A}}^{'}$ is defined as a TIFN $\tilde{A} + {\tilde{A}}^{'}$ , whose α-cut set and β-cut set are given as follows: $\begin{matrix} (\tilde{A} + {\tilde{A}}^{'})_{α} = {\tilde{A}}_{α} + {\tilde{A}}_{α}^{'} \\ = [L_{α} (\tilde{A}) + L_{α} ({\tilde{A}}^{'}), R_{α} (\tilde{A}) + R_{α} ({\tilde{A}}^{'})] \end{matrix}$ (5) $\begin{matrix} (\tilde{A} + {\tilde{A}}^{'})_{β} = {\tilde{A}}_{β} + {\tilde{A}}_{β}^{'} \\ = [L_{β} (\tilde{A}) + L_{β} ({\tilde{A}}^{'}), R_{β} (\tilde{A}) + R_{β} ({\tilde{A}}^{'})], \end{matrix}$ (6) respectively, where α ∈ [0, 1] and β ∈ [0, 1].

The scalar multiplication of $\tilde{A}$ and any real number ρ is defined as a TIFN $ρ \tilde{A}$ , whose α-cut set and β-cut set are given as follows: $(ρ \tilde{A})_{α} = ρ {\tilde{A}}_{α} = {\begin{matrix} [ρ L_{α} (\tilde{A}), ρ R_{α} (\tilde{A})] (ρ \geq 0) \\ [ρ R_{α} (\tilde{A}), L_{α} (\tilde{A})] (ρ < 0) \end{matrix}$ (7) $(ρ \tilde{A})_{β} = ρ {\tilde{A}}_{β} = {\begin{matrix} [ρ L_{β} (\tilde{A}), ρ R_{β} (\tilde{A})] (ρ \geq 0) \\ [ρ R_{β} (\tilde{A}), L_{β} (\tilde{A})] (ρ < 0) \end{matrix}$ (8) respectively.

3 The new ranking method of TIFNs and properties

3.1 The value-index and ambiguity-indexof a TIFN

For any TIFN $\tilde{A} = < ({\underline{a}}_{1}, a, {\bar{a}}_{1}); ({\underline{a}}_{2}, a, {\bar{a}}_{2}) >$ , its values of the membership and non-membership functions are defined as follows: $V_{μ} (\tilde{A}) = \int_{0}^{1} [(L_{α} (\tilde{A}) + R_{α} (\tilde{A})) / 2] f (α) d α$ (9)

and $V_{υ} (\tilde{A}) = \int_{0}^{1} [(L_{β} (\tilde{A}) + R_{β} (\tilde{A})) / 2] g (β) d β$ (10) respectively, where f (α) is a non-negative and non-decreasing function on the interval [0, 1], which should satisfy the conditions: f (0) =0 and f (1) =1; g (β) is a non-negative and non-increasing function on the interval [0, 1], which should meet the conditions: g (0) =1 and g (1) =0.

f (α) and g (β) may reflect the attitude of players (or decision makers) towards uncertainty. f (α) gives different weights to elements at the α-cut sets of the TIFN $\tilde{A}$ so that the contribution of the lower α-cut sets can be lessened due to the fact that these cut sets arising from $μ_{\tilde{A}} (x)$ have a considerable amount of uncertainty. Therefore, $V_{μ} (\tilde{A})$ synthetically reflects the membership degrees of $\tilde{A}$ . Likewise, g (β) can lessen the contribution of the higher β-cut sets of $\tilde{A}$ since these cut sets arising from $υ_{\tilde{A}} (x)$ have a considerable amount of uncertainty. $V_{υ} (\tilde{A})$ synthetically reflects the non-membership degrees of $\tilde{A}$ . f (α) and g (β) may be considered as weighting functions, which are specifically chosen according to need in real situations. Jafarian and Rezvani [9] gave more explanations and specific forms of the functions f (α) and g (β), respectively. For example, f (α) = α and g (β) =1 - β are simpler forms of such functions.

It is easy to see from Equations (9) and (10) that $V_{μ} (\tilde{A}) \geq 0$ and $V_{υ} (\tilde{A}) \geq 0$ for any TIFN $\tilde{A} \geq 0$ .

Likewise, the ambiguities of the membership and non-membership functions for any TIFN $\tilde{A}$ are defined as follows: $W_{μ} (\tilde{A}) = \int_{0}^{1} (R_{α} (\tilde{A}) - L_{α} (\tilde{A})) f (α) d α$ (11) $W_{υ} (\tilde{A}) = \int_{0}^{1} (R_{β} (\tilde{A}) - L_{β} (\tilde{A})) g (β) d β$ (12)

Clearly, $R_{α} (\tilde{A}) - L_{α} (\tilde{A})$ and $R_{β} (\tilde{A}) - L_{β} (\tilde{A})$ are just the lengths of the intervals $A_{α}$ and ${\tilde{A}}_{β}$ . Thus, $W_{μ} (\tilde{A})$ and $W_{υ} (\tilde{A})$ basically measure how much there is uncertainty in $\tilde{A}$ .

If we take f (α) = α (α ∈ [0, 1]) and g (β) =1 - β (β ∈ [0, 1]). In this case, for any TIFN $\tilde{A} = < ({\underline{a}}_{1}, a, {\bar{a}}_{1}); ({\underline{a}}_{2}, a, {\bar{a}}_{2}) >$ , it is easily derived from Equations (9–12) that $\begin{matrix} V_{μ} (\tilde{A}) & = & ({\underline{a}}_{1} + 4 a + {\bar{a}}_{1}) / 12, \\ V_{υ} (\tilde{A}) & = & ({\underline{a}}_{2} + 4 a + {\bar{a}}_{2}) / 12, \\ W_{μ} (\tilde{A}) & = & ({\bar{a}}_{1} - {\underline{a}}_{1}) / 6, \\ W_{υ} (\tilde{A}) & = & ({\bar{a}}_{2} - {\underline{a}}_{2}) / 6 . \end{matrix}$

It can be easily seen from Equations (11) and (12) that $W_{μ} (\tilde{A}) \geq 0$ and $W_{υ} (\tilde{A}) \geq 0$ for any TIFN $\tilde{A}$ . Further, we can draw the following conclusion, which is summarized as in Theorem 1.

Theorem 1:Assume that $\tilde{A}$ and ${\tilde{A}}^{'}$ are any two TIFN. Then, for any real numberρ ∈ R, the following equalities are always valid: $\begin{matrix} V_{μ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ V_{μ} (\tilde{A}) + V_{μ} ({\tilde{A}}^{'}), \\ V_{υ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ V_{υ} (\tilde{A}) + V_{υ} ({\tilde{A}}^{'}), \\ W_{μ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ W_{μ} (\tilde{A}) + W_{μ} ({\tilde{A}}^{'}) \\ W_{υ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ W_{υ} (\tilde{A}) + W_{υ} ({\tilde{A}}^{'}) . \end{matrix}$

Proof: See Appendix A.

Theorem 1 shows that the values and ambiguities of any TIFN are linear.

The value-index and ambiguity-index of any TIFN $\tilde{A}$ are defined as follows: $V_{λ} (\tilde{A}) = λ V_{υ} (\tilde{A}) + (1 - λ) V_{μ} (\tilde{A})$ (13) $W_{λ} (\tilde{A}) = λ W_{μ} (\tilde{A}) + (1 - λ) W_{υ} (\tilde{A})$ (14) respectively, where λ ∈ [0, 1] is the weight which represents the attitude or preference information of players. λ ∈ [0, 1/2) shows that the player prefers to uncertainty or negative feeling; λ ∈ (1/2, 1] shows that the player prefers to certainty or positive feeling; λ = 1/2 shows that the player is indifferent between positive feeling and negative feeling. Therefore, the value-index and ambiguity-index may reflect players’ attitude or preference to the TIFN.

Theorem 2:Assume that $\tilde{A}$ and ${\tilde{A}}^{'}$ are any TIFN. Then, for any real numberρ ∈ R, the following equalities are always valid: $\begin{matrix} V_{λ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ V_{λ} (\tilde{A}) + V_{λ} ({\tilde{A}}^{'}) \\ W_{λ} (ρ \tilde{A} + {\tilde{A}}^{'}) & = & ρ W_{λ} (\tilde{A}) + W_{λ} ({\tilde{A}}^{'}) . \end{matrix}$

Proof: See Appendix B.

Theorem 2 shows that the value-index and ambiguity-index of any TIFN are linear.

3.2 The difference-index ranking methodof TIFNs and properties

It can be seen from Equations (13) and (14) that the larger the value-index and the smaller the ambiguity-index the bigger the TIFN. Therefore, we define a new ranking-index of any TIFN based on difference of the value-index and ambiguity-index as follows: $D_{λ} (\tilde{A}) = V_{λ} (\tilde{A}) - W_{λ} (\tilde{A})$ (15) which is usually called the difference-index of the TIFN $\tilde{A}$ for short.

Theorem 3:Assume that $\tilde{A}$ and ${\tilde{A}}^{'}$ are any TIFN. Then, for any real numberρ ∈ R, the following equality is always valid: $D_{λ} (ρ \tilde{A} + {\tilde{A}}^{'}) = ρ D_{λ} (\tilde{A}) + D_{λ} ({\tilde{A}}^{'}) .$

Proof: See Appendix C.

Theorem 3 shows that the difference-index of any TIFN is linear. Further, it can be easily seen from Equation (15) that the larger the difference-index the bigger the TIFN. Thus, the difference-index based ranking method is proposed as follows.

Definition 1: Assume that λ ∈ [0, 1] is any real number. For any TIFNs $\tilde{A}$ and ${\tilde{A}}^{'}$ , we stipulate:

$D_{λ} (\tilde{A}) > D_{λ} ({\tilde{A}}^{'})$ if and only if $\tilde{A}$ is larger than ${\tilde{A}}^{'}$ , denoted by $\tilde{A} > {\tilde{A}}^{'}$ ;

$D_{λ} (\tilde{A}) = D_{λ} ({\tilde{A}}^{'})$ if and only if $\tilde{A}$ is equal to ${\tilde{A}}^{'}$ , denoted by $\tilde{A} = {\tilde{A}}^{'}$ ;

$\tilde{A} \geq {\tilde{A}}^{'}$ if and only if $\tilde{A} > {\tilde{A}}^{'}$ or $\tilde{A} = {\tilde{A}}^{'}$ .

The above ranking method has some useful properties, which are summarized as in Theorem 4.

Theorem 4: The difference-index based ranking method of TIFNs has the six properties as follows:

(P1) For any TIFN $\tilde{A}$ , $\tilde{A} \geq \tilde{A}$ is always valid;

(P2) For any TIFNs $\tilde{A}$ and ${\tilde{A}}^{'}$ , if $\tilde{A} \geq {\tilde{A}}^{'}$ and ${\tilde{A}}^{'} \geq \tilde{A}$ , then $\tilde{A} = {\tilde{A}}^{'}$ ;

(P3) For any TIFNs $\tilde{A}$ , ${\tilde{A}}^{'}$ and ${\tilde{A}}^{''}$ , if $\tilde{A} \geq {\tilde{A}}^{'}$ and ${\tilde{A}}^{'} \geq {\tilde{A}}^{''}$ , then $\tilde{A} \geq {\tilde{A}}^{''}$ ;

(P4) Assume that F₁ and F₂ are arbitrary finite subsets of TIFNs. For any TIFN $\tilde{A} \in F_{1} \cap F_{2}$ and ${\tilde{A}}^{'} \in F_{1} \cap F_{2}$ , then $\tilde{A} > {\tilde{A}}^{'}$ on F₁ if and only if $\tilde{A} > {\tilde{A}}^{'}$ on F₂;

(P5) For any TIFNs $\tilde{A}$ , ${\tilde{A}}^{'}$ , if $inf {supp (\tilde{A})} \geq sup {supp ({\tilde{A}}^{'})}$ , then $\tilde{A} \geq {\tilde{A}}^{'}$ .

(P5’) For any TIFNs $\tilde{A}$ , ${\tilde{A}}^{'}$ , if $inf {supp (\tilde{A})} > sup {supp ({\tilde{A}}^{'})}$ , then $\tilde{A} > {\tilde{A}}^{'}$ .

(P6) For any TIFNs $\tilde{A}$ and ${\tilde{A}}^{'}$ , if $\tilde{A} \geq {\tilde{A}}^{'}$ , then $\tilde{A} + {\tilde{A}}^{''} \geq {\tilde{A}}^{'} + {\tilde{A}}^{''}$ for any TIFN ${\tilde{A}}^{''}$ ;

(P6’) For any TIFNs $\tilde{A}$ and ${\tilde{A}}^{'}$ , if $\tilde{A} > {\tilde{A}}^{'}$ , then $\tilde{A} + {\tilde{A}}^{''} > {\tilde{A}}^{'} + {\tilde{A}}^{''}$ for any TIFN ${\tilde{A}}^{''}$ .

Proof: See Appendix D.

Remark 1: Wang and Kerre [20] proposed seven axioms, which serve as the reasonable properties to figure out the rationality of a ranking method for the ordering of fuzzy quantities. The above properties correspond to the six axioms, respectively. It can be easily seen from the properties of Theorem 4 that the difference-index based ranking method of TIFNs is a total order.

4 Parameterized bilinear programming models for TIFN bimatrix games

Let us consider a TIFN bimatrix game, where sets of pure strategies are S₁ and S₂ and sets of mixed strategies are Y and Z for players I and II. If player I chooses any pure strategy α_i ∈ S₁ (i = 1, 2, ⋯ , m) and player II chooses any pure strategy β_j ∈ S₂ (j = 1, 2, ⋯ , n), then at the situation (α_i, β_j) players I and II gain payoffs, which are expressed with TIFNs ${\tilde{A}}_{ij} (α_{i}, β_{j}) = {< (α_{i}, β_{j}); ({\underline{a}}_{1 ij}, a_{ij}, {\bar{a}}_{1 ij}); ({\underline{a}}_{2 ij}, a_{ij}, {\bar{a}}_{2 ij}) >}$ (i =1, 2, ⋯ , m; j = 1, 2, ⋯ , n), where ${\underline{a}}_{2 ij} \leq {\underline{a}}_{1 ij} \leq a_{ij} \leq {\bar{a}}_{1 ij} \leq {\bar{a}}_{2 ij}$ and ${\tilde{B}}_{ij} (α_{i}, β_{j}) = {< (α_{i},$ $β_{j}); ({\underline{b}}_{1 ij}, b_{ij}, {\bar{b}}_{1 ij}); ({\underline{b}}_{2 ij}, b_{ij}, {\bar{b}}_{2 ij}) >}$ (i = 1, 2, ⋯, m; j = 1, 2, ⋯ , n), where ${\underline{b}}_{2 ij} \leq {\underline{b}}_{1 ij} \leq b_{ij} \leq {\bar{b}}_{1 ij} \leq {\bar{b}}_{2 ij}$ . Thus, the payoff matrices of players I and II are expressed as $\tilde{A} = ({\tilde{A}}_{ij} (α_{i}, β_{j}))_{m \times n}$ and $\tilde{B} = ({\tilde{B}}_{ij} (α_{i}, β_{j}))_{m \times n}$ , respectively.

The above payoffs ${\tilde{A}}_{ij} (α_{i}, β_{j})$ and ${\tilde{B}}_{ij} (α_{i}, β_{j})$ (i = 1, 2, ⋯ , m; j = 1, 2, ⋯ , n) of players I and II are TIFNs in the finite universal set X′ = {(α_i, β_j) |α_i ∈ S₁ (i = 1, 2, ⋯ , m) , β_j ∈ S₂ (j = 1, 2, ⋯ , n)}. For instance, let us consider a simple example in which there are two pure strategies for both players I and II, i.e., player I has pure strategies α₁ and α₂ and player II has pure strategies β₁ and β₂. Then, the universal set is $X_{0}^{'} = {(α_{1}, β_{1}), (α_{1}, β_{2}), (α_{2}, β_{1}), (α_{2}, β_{2})}$ , which has four elements (i.e., situations). In the sequel, the above TIFN bimatrix game is simply denoted by $(\tilde{A}, \tilde{B})$ for short.

If player I chooses any mixed strategy y ∈ Y and player II chooses any mixed strategy z ∈ Z, then the expected payoff of player I is ${\tilde{E}}_{1} (y, z) = y^{T} \tilde{A} z = \sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} {\tilde{A}}_{ij} z_{j}$ , whose α-cut set and β-cut set can be computed as follows: $\begin{matrix} (\tilde{E} (y, z))_{α} \\ = [\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} L_{α} ({\tilde{A}}_{ij}) z_{j}, \sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} R_{α} ({\tilde{A}}_{ij}) z_{j}], \end{matrix}$ $\begin{matrix} (\tilde{E} (y, z))_{β} \\ = [\sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} L_{β} ({\tilde{A}}_{ij}) z_{j}, \sum_{i = 1}^{m} \sum_{j = 1}^{n} y_{i} R_{β} ({\tilde{A}}_{ij}) z_{j}], \end{matrix}$ respectively, where α ∈ [0, 1] and β ∈ [0, 1].

According to the operations of TIFNs, the expected payoff ${\tilde{E}}_{1} (y, z)$ of player I is a TIFN and can be calculated: $\begin{matrix} {\tilde{E}}_{1} (y, z) = {< (\bar{y}, \bar{z}) : \sum_{i = 1}^{n} \sum_{j = 1}^{n} y_{i} ({\underline{a}}_{1 ij}, a_{ij}, {\bar{a}}_{1 ij}) z_{j}; \\ \sum_{i = 1}^{n} \sum_{j = 1}^{n} y_{i} ({\underline{a}}_{2 ij}, a_{ij}, {\bar{a}}_{2 ij}) z_{j} >} \end{matrix}$ where $(\bar{y}, \bar{z})$ represents a mixed situation, which corresponds to the mixed strategies y and z .

Similarly, the expected payoff of player II is ${\tilde{E}}_{2} (y, z) = y^{T} \tilde{Bz}$ , which can be calculated as follows: $\begin{matrix} {\tilde{E}}_{2} (y, z) = {< (\bar{y}, \bar{z}) : \sum_{i = 1}^{n} \sum_{j = 1}^{n} y_{i} ({\underline{b}}_{1 ij}, b_{ij}, {\bar{b}}_{1 ij}) z_{j}; \\ \sum_{i = 1}^{n} \sum_{j = 1}^{n} y_{i} ({\underline{b}}_{2 ij}, b_{ij}, {\bar{b}}_{2 ij}) z_{j} >} \end{matrix}$

Definition 2. Assume that there is a pair ( y ^*, z ^*) ∈ Y × Z. If any y ∈ Y and z ∈ Z satisfy $y^{T} {\tilde{Az}}^{*} \leq y^{* T} {\tilde{Az}}^{*}$ and $y^{* T} \tilde{Bz} \leq y^{* T} {\tilde{Bz}}^{*}$ , then ( y ^*, z ^*) is called a Nash equilibrium point of the TIFN bimatrix game $(\tilde{A}, \tilde{B})$ , y^* and z ^* are called Nash equilibrium strategies of players I and II, ${\tilde{u}}^{*} = y^{* T} {\tilde{Az}}^{*}$ and ${\tilde{v}}^{*} = y^{* T} {\tilde{Bz}}^{*}$ are called Nash equilibrium values of players I and II, respectively. $(y^{*}, z^{*}, {\tilde{u}}^{*}, {\tilde{v}}^{*})$ is called a Nash equilibrium solution of the TIFN bimatrix game $(\tilde{A}, \tilde{B})$ .

Stated as earlier, however, player I’s expected payoff $y^{T} \tilde{Az}$ and player II’s expected payoff $y^{T} \tilde{Bz}$ are TIFNs. Therefore, there are no commonly-used concepts of solutions of the TIFN bimatrix games. Furthermore, it is not easy to compute the membership degrees and the nonmembership degrees of players’ expected payoffs. As a result, solving Nash equilibrium solutions of TIFN bimatrix games are very difficult. In the sequel, we use the ranking function D_λ to develop a new method for solving the TIFN bimatrix game $(\tilde{A}, \tilde{B})$ .

Using the ranking function of TIFNs given by Equation (15), we can transform the TIFN payoff matrices $\tilde{A}$ and $\tilde{B}$ of players I and II into the payoff matrices as follows: $\begin{matrix} {\tilde{A}}_{λ_{1}} = D_{λ_{1}} (({\tilde{A}}_{ij} (α_{i}, β_{j}))_{m \times n}) \\ = (D_{λ_{1}} ({\tilde{A}}_{ij} (α_{i}, β_{j})))_{m \times n} \end{matrix}$ (16) $\begin{matrix} {\tilde{B}}_{λ_{2}} = D_{λ_{2}} (({\tilde{B}}_{ij} (α_{i}, β_{j}))_{m \times n}) \\ = (D_{λ_{2}} ({\tilde{B}}_{ij} (α_{i}, β_{j})))_{m \times n} \end{matrix}$ (17) where λ₁ ∈ [0, 1], λ₂ ∈ [0, 1], $D_{λ_{1}} ({\tilde{A}}_{ij} (α_{i}, β_{j})) = V_{{\tilde{A}}_{ij} (α_{i}, β_{j})} - A_{{\tilde{A}}_{ij} (α_{i}, β_{j})}$ and $D_{λ_{2}} ({\tilde{B}}_{ij} (α_{i}, β_{j})) = V_{{\tilde{B}}_{ij}}$ $(α_{i}, β_{j}) - A_{{\tilde{B}}_{ij} (α_{i}, β_{j})}$ (i = 1, 2, ⋯ , m; j = 1, 2, ⋯, n).

According to the above usage and notations, the above parametric bimatrix game can be simply denoted by $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ , where the pure (or mixed) strategy sets of players I and II are S₁ and S₂ (or Y and Z) defined as the above. Then, the TIFN bimatrix game $(\tilde{A}, \tilde{B})$ is transformed into the parametric bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ . Hereby, according to Definitions 1–3 and Theorem 3, we can give the definition of satisfying Nash equilibrium solutions of the TIFN bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ as follows.

Definition 3. For given parameters λ₁ ∈ [0, 1] and λ₂ ∈ [0, 1], if there is a pair ( y ^*, z ^*) ∈ Y × Z so that any y ∈ Y and z ∈ Z satisfy the following conditions: $y^{T} {\tilde{A}}_{λ_{1}} z^{*} \leq y^{* T} {\tilde{A}}_{λ_{1}} z^{*}$ and $y^{* T} {\tilde{B}}_{λ_{2}} z \leq y^{* T} {\tilde{B}}_{λ_{2}} z^{*}$ , then ( y ^*, z ^*) is called a satisfying Nash equilibrium point of the TIFN bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ , y ^* and z ^* are called satisfying Nash equilibrium strategies of players I and II, $u^{*} (λ_{1}) = y^{* T} {\tilde{A}}_{λ_{1}} z^{*}$ and $v^{*} (λ_{2}) = y^{* T} {\tilde{B}}_{λ_{2}} z^{*}$ are called satisfying equilibrium values of players I and II, respectively. ( y ^*, z ^*, u^* (λ₁) , v^* (λ₂)) is called a satisfying Nash equilibrium solution of the TIFN bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ .

It can be easily seen from the ranking function given by Equation (15) and Theorem 4 that Definitions 3 and 4 are equivalent in the sense of the order relation defined by Definition 1. Thus, for given parameters λ₁ ∈ [0, 1] and λ₂ ∈ [0, 1], according to Theorem 4, the parametric bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ has at least one Nash equilibrium solution. Namely, the TIFN bimatrix game $({\tilde{A}}_{λ_{1}}, {\tilde{B}}_{λ_{2}})$ has at least one satisfying Nash equilibrium solution, which can be obtained through solving the following parametric nonlinear programming model: $\begin{matrix} max {\sum_{j = 1}^{n} \sum_{i = 1}^{m} y_{i} [D_{λ 1} ({\tilde{A}}_{ij} (α_{i}, β_{j})) + D_{λ 2} ({\tilde{B}}_{ij} (α_{i}, β_{j}))] z_{j} \\ - u (λ_{1}) - v (λ_{2})} \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} [D_{λ 1} ({\tilde{A}}_{ij} (α_{i}, β_{j}))] z_{j} \leq u (λ_{1}) (i = 1, 2, \dots, m) \\ \sum_{i = 1}^{m} [D_{λ 2} ({\tilde{B}}_{ij} (α_{i}, β_{j}))] y_{i} \leq v (λ_{2}) (j = 1, 2, \dots, n) \\ y_{1} + y_{2} + \dots + y_{m} = 1 \\ z_{1} + z_{2} + \dots + z_{n} = 1 \\ v (λ_{2}) \geq 0, u (λ_{1}) \geq 0 \\ y_{i} \geq 0 (i = 1, 2, \dots, m) \\ z_{j} \geq 0 (j = 1, 2, \dots, n), \end{matrix} \end{matrix}$ (18) where y_i (i = 1, 2, ⋯ , m), z_j (j = 1, 2, ⋯ , n), u (λ₁) and v (λ₂) are decision variables.

According to Nash equilibrium solution, if( y ^*, z ^*, u^* (λ₁) , v^* (λ₂)) is a solution of the aboveparametric nonlinear programming model (i.e.,Equation (18)), then $\begin{matrix} u^{*} (λ_{1}) = y^{* T} {\tilde{A}}_{λ_{1}} z^{*} = \sum_{j = 1}^{n} \sum_{i = 1}^{m} [V_{λ 1} ({\tilde{A}}_{ij} (α_{i}, β_{j})) \\ - W_{λ 1} ({\tilde{A}}_{ij} (α_{i}, β_{j}))] y_{i}^{*} z_{j}^{*}, \\ v^{*} (λ_{2}) = y^{* T} {\tilde{B}}_{λ_{2}} z^{*} = \sum_{j = 1}^{n} \sum_{i = 1}^{m} [V_{λ 2} ({\tilde{B}}_{ij} (α_{i}, β_{j})) \\ - W_{λ 2} ({\tilde{B}}_{ij} (α_{i}, β_{j}))] y_{i}^{*} z_{j}^{*} \end{matrix}$ and $y^{* T} (D_{λ 1} (\tilde{A}) + D_{λ 2} (\tilde{B})) z^{*} - u^{*} (λ_{1}) - v^{*} (λ_{2}) = 0$ .

Noticing that $y_{i}^{*} \geq 0$ , $z_{j}^{*} \geq 0$ , and $V_{λ} (\tilde{a})$ and $A_{λ} (\tilde{a})$ are respectively continuous non-decreasing and non-increasing functions of the parameter λ ∈ [0, 1] if $\tilde{a}$ is a non-negative TIFN. Then, u^* (λ₁) and v^* (λ₂) are monotonic and non-decreasing functions of the parameters λ₁ ∈ [0, 1] and λ₂ ∈ [0, 1], respectively. Thus, the satisfying Nash equilibrium values of players I and II are obtained as [u^* (0) , u^* (1)] and [v^* (0) , v^* (1)], respectively, and can be written as the TIFNs ${< ({\bar{y}}^{*}, {\bar{z}}^{*}), u^{*} (0), 1 - u^{*} (1) >}$ and ${< ({\bar{y}}^{*}, {\bar{z}}^{*}), v^{*} (0), 1 - v^{*} (1) >}$ , where $({\bar{y}}^{*}, {\bar{z}}^{*})$ represents a mixed situation. Thus, ${\tilde{u}}^{*} ({\bar{y}}^{*}, {\bar{z}}^{*})$ and ${\tilde{v}}^{*} ({\bar{y}}^{*}, {\bar{z}}^{*})$ are Nash equilibrium values of players I and II, respectively.

Thus, according to Equation (18), any satisfying Nash equilibrium values and corresponding satisfying Nash equilibrium strategies of players I and II can be obtained through choosing different parameters λ₁ ∈ [0, 1] and λ₂ ∈ [0, 1].

5 An application to the strategy choice problem

Let us consider the case of two manufacturers P₁ and P₂ making a decision aiming to enhance the satisfaction degrees of customers. And assume that manufacturers P₁ and P₂ are rational, i.e., they will choose optimal strategies to maximize their own profits without cooperation. Suppose that manufacturer P₁ has two pure strategies: establishing a scientific and rational service system α₁ and providing customers with satisfaction product α₂. Manufacturer P₂ possesses the same pure strategies as manufacturer P₁, i.e., the options of manufacturer P₂ are: establishing a scientific and rational service system β₁ and providing customers with satisfaction product β₂. Due to lack of information or imprecision of the available information, the players’ judgments for the satisfaction degrees of customers including preference and experience are often vague and players estimate them with their intuitions, so the players usually are not able to forecast the sales amount of their companies exactly, which may be some hesitation about the estimation of the sales amount. In order to deal with the uncertainty, TIFNs are used to express the sales amount of the product. The payoff matrices of manufacturers P₁ and P₂ are expressed as follows:

By using Equation (18), the parametric bilinear programming model is constructed as follows: $\begin{matrix} \begin{matrix} \begin{matrix} α_{1} & α_{2} \end{matrix} \\ \tilde{A} = \begin{matrix} β_{1} \\ β_{2} \end{matrix} (\begin{matrix} < (160, 170, 180); (140, 170, 190) > & < (130, 150, 165); (120, 150, 170) > \\ < (140, 150, 165); (130, 150, 170) > & < (160, 170, 185); (140, 170, 185) > \end{matrix}) \\ \tilde{B} = \begin{matrix} β_{1} \\ β_{2} \end{matrix} (\begin{matrix} < (160, 170, 190); (145, 170, 200) > & < (140, 150, 160); (120, 150, 170) > \\ < (135, 140, 160); (120, 140, 165) > & < (150, 170, 180); (140, 170, 190) > \end{matrix}) \end{matrix} \end{matrix}$ where ${\tilde{A}}_{11} = < (160, 170, 180); (140, 170, 190) >$ is a TIFN, whose satisfaction (or membership) degree and the dissatisfaction (or non-membership) degree functions are given as follows: $μ_{{\tilde{A}}_{11}} (x) = {\begin{matrix} 0 & (x < 160) \\ (x - 160) / 10 & (160 \leq x < 170) \\ 1 & (x = 170) \\ (180 - x) / 10 & (170 < x \leq 180) \\ 0 & (x > 180) \end{matrix}$ $υ_{{\tilde{A}}_{11}} (x) = {\begin{matrix} 1 & (x < 140) \\ (170 - x) / 30 & (140 \leq x < 170) \\ 0 & (x = 170) \\ (x - 170) / 20 & (170 < x \leq 190) \\ 1 & (x > 190) \end{matrix}$ respectively. Other items in payoff matrices $\tilde{A}$ and $\tilde{B}$ can be similarly explained.

Taking f (α) = α (α ∈ [0, 1]) and g (β) =1 - β (β ∈ [0, 1]), and by using Equations (9–15), the differences of the value-indexes to the ambiguity-indexes for the TIFNs $\tilde{A}$ and $\tilde{B}$ can be obtained as follows: $\begin{matrix} D_{λ 1} (\tilde{A} (α_{1}, β_{1})) = 76.67 + 4.17 λ_{1} \\ D_{λ 1} (\tilde{A} (α_{1}, β_{2})) = 66.25 + 2.08 λ_{1} \\ D_{λ 1} (\tilde{A} (α_{2}, β_{1})) = 68.75 + 2.08 λ_{1} \\ D_{λ 1} (\tilde{A} (α_{2}, β_{2})) = 77.91 + 1.67 λ_{1} \\ D_{λ 2} (\tilde{B} (α_{1}, β_{1})) = 76.67 + 3.75 λ_{2} \\ D_{λ 2} (\tilde{B} (α_{1}, β_{2})) = 66.67 + 4.17 λ_{2} \\ D_{λ 2} (\tilde{B} (α_{2}, β_{1})) = 63.75 + 2.92 λ_{2} \\ D_{λ 2} (\tilde{B} (α_{2}, β_{2})) = 75.83 + 3.33 λ_{2} \end{matrix}$

$\begin{matrix} max {(153.3 + 4.17 λ_{1} + 3.75 λ_{2}) y_{1} z_{1} + (132.92 \\ + 2.08 λ_{1} + 4.17 λ_{2}) y_{1} z_{2} + (132.5 + 2.08 λ_{1} 1 pt 1 pt 1 pt + 2.92 λ_{2}) \\ y_{2} z_{1} + (153.75 + 1.67 λ_{1} + 3.33 λ_{2}) y_{2} z_{2} - u (λ_{1}) - v (λ_{2})} \\ s . t . {\begin{matrix} (76.67 + 4.17 λ_{1}) z_{1} + (66.25 + 2.08 λ_{1}) z_{2} \leq u (λ_{1}) \\ (68.75 + 2.08 λ_{1}) z_{1} + (77.91 + 1.67 λ_{1}) z_{2} \leq u (λ_{1}) \\ (76.67 + 3.75 λ_{2}) y_{1} + (66.67 + 4.17 λ_{2}) y_{2} \leq v (λ_{2}) \\ (63.75 + 2.92 λ_{2}) y_{1} + (75.83 + 3.33 λ_{2}) y_{2} \leq v (λ_{2}) \\ y_{1} + y_{2} = 1 \\ z_{1} + z_{2} = 1 \\ u (λ_{1}) \geq 0, v (λ_{2}) \geq 0 \\ y_{i} \geq 0 (i = 1, 2) \\ z_{j} \geq 0 (j = 1, 2) . \end{matrix} \end{matrix}$ (19)

It is easily seen that, for the parameters λ₁ ∈ [0, 1] and λ₂ ∈ [0, 1], solving Equation (19), we can obtain the satisfying Nash equilibrium values and corresponding satisfying Nash equilibrium strategies of manufacturers P₁ and P₂, respectively, depicted as in Tables 1–3.

It can be easily seen from Tables 1–3 that the satisfying Nash equilibrium value of a player (i.e., manufacturer P₁ or P₂) depends on both their preference/parameter: (1) Strategy choice of a player is only affected by other player’ preference/parameter, and it shown a negative correlation. That is to say, the bigger other player’s risk preference/parameter the smaller the player’s optimal strategy value. (2) From the optimal strategy, a player’s optimal expected payoff value is only affected by his preference/parameter, and showed a positive correlation. That is to say, the bigger a player’s risk preference/parameter the bigger the player’s optimal expected pay off value.

6 Conclusion

Determining payoffs of bimatrix games absolutely depends on players’ judgments and intuition, which are often vague and not easy to be represented with crisp values and fuzzy numbers. In this paper, we formulate bimatrix games with payoffs of TIFNs and propose corresponding parameterized bilinear programming method. The highlights include: (1) We propose the concepts of TIFNs and the difference-index of the value-index to the ambiguity-index, and the new ranking method based on the difference-index is a total order, which satisfies the 6 properties proposed by Wang and Kerre [20] and possesses some useful properties such as the linearity. (2) A pair of parameterized bilinear programming models is derived from the auxiliary TIFN mathematical programming models of any TIFN bimatrix games.

Obviously, for any payoffs with various forms of TIFNs, using the difference-index defined in this paper, we easily construct a pair of parameterized bilinear programming models. Hereby, for any given parameter λ ∈ [0, 1], the parameterized bilinear programming models become a pair of primal-dual linear programming models, which may be easily solved by using the bilinear programming method. Furthermore, it is easy to see that the derived parameterized bilinear programming models for TIFN bimatrix games are an extension of the linear programming models for fuzzy matrix games. Therefore, effective and efficient methods for explicitly determining values of TIFN matrix games will be investigated in the near future.

Footnotes

Acknowledgments

This research was supported by the Key Program of National Natural Science Foundation of China (No. 71231003), and the National Natural Science Foundation of China (Nos. 71561008, 71171055), and the Natural Science Foundation of Fujian Province (No. 2016J05169).

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