# A Difference-Index Based Ranking Bilinear Programming Approach to Solving Bimatrix Games with Payoffs of Trapezoidal Intuitionistic Fuzzy Numbers

Authors: Deng-Feng Li; Jie Yang
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

The aim of this paper is to develop a bilinear programming method for solving bimatrix games in which the payoffs are expressed with trapezoidal intuitionistic fuzzy numbers (TrIFNs), which are called TrIFN bimatrix games for short. In this method, we define the value index and ambiguity index for a TrIFN and propose a new order relation of TrIFNs based on the difference index of value index to ambiguity index, which is proven to be a total order relation. Hereby, we introduce the concepts of solutions of TrIFN bimatrix games and parametric bimatrix games. It is proven that any TrIFN bimatrix game has at least one satisfying Nash equilibrium solution, which is equivalent to the Nash equilibrium solution of corresponding parametric bimatrix game. The latter can be obtained through solving the auxiliary parametric bilinear programming model. The method proposed in this paper is demonstrated with a real example of the commerce retailers’ strategy choice problem.


---

## Body

## 1. Introduction

Bimatrix games are an important type of two-person nonzero-sum noncooperative games, which have been successfully applied to many different areas such as politics, economics, and management. The normal-form bimatrix games assume that the payoffs are represented with crisp values, which indicate that the payoffs are exactly known by players. However, players often are not able to evaluate exactly the payoffs due to imprecision or lack of available information in real game situations. In order to make bimatrix game theory more applicable to real competitive decision problems, the fuzzy set introduced by Zadeh [1] has been used to describe imprecise and uncertain information appearing in bimatrix problems. Using the ranking method of fuzzy numbers, Vidyottama et al. [2] studied the bimatrix game with fuzzy goals and fuzzy payoffs. Using the possibility measure of fuzzy numbers, Maeda [3] introduced two concepts of equilibrium for the bimatrix games with fuzzy payoffs. Bector and Chandra [4] studied bimatrix games with fuzzy payoffs and fuzzy goals based on some duality of fuzzy linear programming. Larbani [5] proposed an approach to solving fuzzy bimatrix games based on the idea of introducing “nature” as a player in fuzzy multiattribute decision-making problems. Atanassov [6] introduced the intuitionistic fuzzy (IF) set (IFS) by adding a nonmembership function, which seems to be suitable for expressing more abundant information. Trapezoidal intuitionistic fuzzy numbers (TrIFNs) are special cases of IFSs defined on the set of real numbers, which can deal with ill-known quantities effectively. And TrIFNs have been playing an important role in fuzzy optimization modeling and decision making [7–10].This paper will apply the TrIFNs to deal with imprecise quantities in bimatrix game problems, which are called TrIFN bimatrix games for short. Obviously, the TrIFN bimatrix game remarkably differs from fuzzy bimatrix game since the former uses both membership and nonmembership degrees to express the payoffs, while the latter only uses membership degrees to express the payoffs. However, the fuzzy bimatrix game models and methods cannot be directly used to solve TrIFN bimatrix games. Thus, we need to consider the order relation of TrIFNs. Ranking fuzzy numbers are difficult in nature, although there exist a large number of literature on ranking fuzzy numbers [11, 12], IF numbers (IFNs) or IFSs [13–18]. Based on the concept of value index and ambiguity index, we propose a difference-index based ranking method of TrIFNs in this paper. This ranking method has good properties such as the linearity. Hereby, a TrIFN bimatrix game is formulated and a solution method is developed on the difference-index based ranking bilinear programming.The rest of this paper is organized as follows. Section2 establishes a new order relation of TrIFNs based on the concepts of value index and ambiguity index. Section 3 formulates TrIFN bimatrix games and proposes corresponding solving methodology based on the constructed auxiliary parametric bilinear programming model, which is derived from the new order relation of TrIFNs and the bimatrix game model. In Section 4, the proposed models and method are illustrated with a real example of the commerce retailers’ strategy choice problem. Conclusion is drawn in Section 5.


## 2. Characteristics and the New Ranking Method of TrIFNs

### 2.1. The Definition and Operations of TrIFNs

A TrIFN is expressed asa~=〈(a_,a1,a2,a-);wa~,ua~〉, whose membership function and nonmembership function are defined as follows:
(1)μa~(x)={(x-a_)wa~(a1-a_)(a_≤x<a1)wa~(a1≤x≤a2)(a--x)wa~(a--a2)(a2<x≤a-)0(x<a_)or(x>a-),(2)υa~(x)={[a1-x+ua~(x-a_)](a1-a_)(a_≤x<a1)ua~(a1≤x≤a2)[x-a2+ua~(a--x)](a--a2)(a2<x≤a-)1(x<a_)or(x>a-),
respectively, where wA~ and uA~ represent the maximum membership degree and the minimum nonmembership degree such that they satisfy the following conditions: 0≤wA~≤1, 0≤uA~≤1, and 0≤wA~+uA~≤1.A TrIFNa~=〈(a_,a1,a2,a-);wa~,ua~〉 may express an approximate range of a closed interval [a1,a2], which is approximately equal to [a1,a2]. Namely, the ill-known quantity “approximate [a1,a2]” is expressed with any value between a_ and a- with different membership and nonmembership degrees. It is easily seen that μa~(x)+υa~(x)=1 for any x∈R if wa~=1 and ua~=0. Hence, the TrIFN a~=〈(a_,a1,a2,a-);wa~,ua~〉 degenerates to a~=〈(a_,a1,a2,a-);1,0〉, which is just about a trapezoid fuzzy number. On the other hand, if a1=a2, then a~=〈(a_,a1,a2,a-);wa~,ua~〉 degenerates to a~=〈(a_,a,a-);wa~,ua~〉, which is a TIFN, where a=a1=a2.In a similar way to the arithmetical operations of the trapezoid fuzzy numbers [19], the part arithmetical operations over TrIFNs are stipulated as follows:
(3)a~+b~=〈(a_+b_,a1+b1,a2+b2,a-+b-),wa~∧wb~,ua~∨ub~(a_+b_,a1+b1,a2+b2,a-+b-)〉,(4)γa~={〈(γa_,γa1,γa2,γa-);wa~,ua~〉(γ>0)〈(γa-,γa2,γa1,γa_);wa~,ua~〉(γ<0),
where λ≠0 is a real number and the symbols “∧” and “∨” are the minimum and maximum operators, respectively.


### 2.2. Value and Ambiguity of a TrIFN

Definition 1.
Aα-cut set of a TrIFN a~ is a crisp subset of R, which can be expressed as a~α={x∣μa~(x)≥α}, where 0≤α≤wa~. A β-cut set of a TrIFN a~ is a crisp subset of R, which can be expressed as a~β={x∣υa~(x)≤β}, where ua~≤β≤1. It directly follows that a~α and a~β are closed intervals, denoted by a~α=[La~(α),Ra~(α)] and a~β=[La~(β),Ra~(β)], respectively, which can be calculated as follows:
(5)a~α=[[(wa~-α)a_+αa1]wa~,[(wa~-α)a-+αa2]wa~],(6)a~β=[[(1-β)a1+(β-ua~)a_](1-ua~),[(1-β)a2+(β-ua~)a-](1-ua~)].Definition 2.
Leta~α and a~β be a α-cut number and β-cut number of a TrIFN a~, respectively. Then the values of the membership and nonmembership functions for the TrIFN a~ are defined as follows:
(7)Vμ(a~)=∫0wa~[La~(α)+Ra~(α)2]f(α)dα,(8)Vυ(a~)=∫ua~1[La~(β)+Ra~(β)2]g(β)dβ,
where f(α) is a nonnegative and nondecreasing function on the interval [0,wa~] with f(0)=0 and f(wa~)=1 and g(β) is a nonnegative and nonincreasing function on the interval [ua~,1] with g(ua~)=1 and g(1)=0. Obviously, f(α) and g(β) can be considered as weighting functions and have various specific forms in actual applications, which can be chosen according to the real-life situations. In the following, we choose f(α)=α/wa~ (α∈[0,wa~]) and g(β)=(1-β)/(1-ua~) (β∈[ua~,1]).The functionf(α) gives different weights to elements at different α-cuts, so that it can lessen the contribution of the lower α-cuts, since these cuts arising from values of μa~(x) have a considerable amount of uncertainty. Therefore, Vμ(a~) and Vυ(a~) synthetically reflect the information on membership and nonmembership degrees.According to (5) and (7), we can calculate the value of the membership function as follows:
(9)Vμ(a~)=∫0wa~[(wa~-α)a_+αa12wa~+(wa~-α)a-+αa22wa~]αwa~dα=(a_+2a1+2a2+a-)wa~12.In a similar way, according to (6) and (8), the value of the nonmembership function can be obtained as follows:
(10)Vυ(a~)=(a_+2a1+2a2+a-)(1-ua~)12.It is directly derived from the condition0≤wa~+ua~≤1 that 0≤Vμ(a~)≤Vυ(a~), which may be concisely expressed as an interval [Vμ(a~),Vυ(a~)]. It is easily proven that the following conclusion holds.Theorem 3.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Vμ(a~+b~)=Vμ(a~)+Vμ(b~).Proof.
According to (3) and (9), we can prove that
(11)Vμ(a~+b~)=(a_+2a1+2a2+a-)wa~12+(b_+2b1+2b2+b-)wb~12=Vμ(a~)+Vμ(b~).
In the same way, using (3) and (10), we can prove that Vυ(a~+b~)=Vυ(a~)+Vυ(b~).Definition 4.
Leta~α and a~β be a α-cut set and β-cut set of a TrIFN a~, respectively. The ambiguities of the membership and nonmembership functions for the TrIFN a~ are defined as follows:
(12)Aμ(a~)=∫0wa~(Ra~(α)-La~(α))f(α)dα,Aυ(a~)=∫ua~1(Ra~(β)-La~(β))g(β)dβ.
According to (5) and (6), (12) can be obtained as follows:
(13)Aμ(a~)=(a--a_+2a2-2a1)wa~6,(14)Aυ(a~)=(a--a_+2a2-2a1)(1-ua~)6.
Obviously,Aμ(a~)≤Aυ(a~), which can be expressed as an interval [Aμ(a~),Aυ(a~)].Theorem 5.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Aμ(a~+b~)=Aμ(a~)+Aμ(b~).Proof.
According to (3) and (13), we can prove that
(15)Aμ(a~+b~)=(a--a_+2a2-2a1)wa~6+(b--b_+2b2-2b1)wb~6=Aμ(a~)+Aμ(b~).
Similarly, using (3) and (14), we can prove that Aυ(a~+b~)=Aυ(a~)+Aυ(b~).


### 2.3. The Difference-Index Based Ranking Method

Based on the value and ambiguity of a TrIFNa~, its value index and an ambiguity index are defined as follows. Hereby, a new ranking method of TrIFNs is proposed.Definition 6.
Leta~ be a TrIFN. Its value index and an ambiguity index are defined as follows:
(16)Vλ(a~)=Vμ(a~)+λ(Vυ(a~)-Vμ(a~)),(17)Aλ(a~)=Aυ(a~)-λ(Aυ(a~)-Aμ(a~)),
where λ∈[0,1] is a weight which represents the decision maker’s preference information. λ∈[0,1/2) shows that decision maker prefers to uncertainty or negative feeling, who is a pessimist; λ∈(1/2,1] shows that the decision maker prefers to certainty or positive feeling, who is an optimist; λ=1/2 shows that the decision maker is a neutralist, between positive feeling and negative feeling. Therefore, the value-index and the ambiguity-index may reflect the decision maker’s subjectivity attitude to the TrIFN.A difference index of the value index to the ambiguity index for a TrIFNa~ is defined as follows:
(18)Dλ(a~)=Vλ(a~)-Aλ(a~).Theorem 7.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Dλ(a~+b~)=Dλ(a~)+Dλ(b~), where λ∈[0,1].Proof.
According to Theorems3 and 5, it is derived from (18) that
(19)Dλ(a~+b~)=Vλ(a~+b~)-Aλ(a~+b~)=(Vλ(a~)+Vλ(b~))-(Aλ(a~)+Aλ(b~))=(Vλ(a~)-Aλ(a~))+(Vλ(b~)-Aλ(b~))=Dλ(A~)+Dλ(b~).Theorem7 shows that the difference index Dλ(a~) is a linear function of any TrIFN. Furthermore, it can be easily seen that the larger the difference index, the bigger the TrIFN. Thus, we propose the difference index based ranking method of TrIFNs as follows.Definition 8.
Assume thatλ∈[0,1]. For any TrIFNs a~ and b~, we stipulate the following: (1)
Dλ(a~)>Dλ(b~) if and only if a~ is bigger than b~, denoted by a~>b~;(2)
Dλ(a~)=Dλ(b~) if and only if a~ is equal to b~, denoted by a~=b~;(3)
Dλ(a~)≥Dλ(b~) if and only if a~>b~ or a~=b~.The above ranking method has some useful properties, which satisfy five of the seven axioms proposed by Wang and Kerre [20] that serve as the reasonable properties for the ordering of fuzzy quantities. And the proposed  ranking method is two-index, which is used to aggregate both value index and the ambiguity index. Especially, this proposed ranking method has the linearity.


## 2.1. The Definition and Operations of TrIFNs

A TrIFN is expressed asa~=〈(a_,a1,a2,a-);wa~,ua~〉, whose membership function and nonmembership function are defined as follows:
(1)μa~(x)={(x-a_)wa~(a1-a_)(a_≤x<a1)wa~(a1≤x≤a2)(a--x)wa~(a--a2)(a2<x≤a-)0(x<a_)or(x>a-),(2)υa~(x)={[a1-x+ua~(x-a_)](a1-a_)(a_≤x<a1)ua~(a1≤x≤a2)[x-a2+ua~(a--x)](a--a2)(a2<x≤a-)1(x<a_)or(x>a-),
respectively, where wA~ and uA~ represent the maximum membership degree and the minimum nonmembership degree such that they satisfy the following conditions: 0≤wA~≤1, 0≤uA~≤1, and 0≤wA~+uA~≤1.A TrIFNa~=〈(a_,a1,a2,a-);wa~,ua~〉 may express an approximate range of a closed interval [a1,a2], which is approximately equal to [a1,a2]. Namely, the ill-known quantity “approximate [a1,a2]” is expressed with any value between a_ and a- with different membership and nonmembership degrees. It is easily seen that μa~(x)+υa~(x)=1 for any x∈R if wa~=1 and ua~=0. Hence, the TrIFN a~=〈(a_,a1,a2,a-);wa~,ua~〉 degenerates to a~=〈(a_,a1,a2,a-);1,0〉, which is just about a trapezoid fuzzy number. On the other hand, if a1=a2, then a~=〈(a_,a1,a2,a-);wa~,ua~〉 degenerates to a~=〈(a_,a,a-);wa~,ua~〉, which is a TIFN, where a=a1=a2.In a similar way to the arithmetical operations of the trapezoid fuzzy numbers [19], the part arithmetical operations over TrIFNs are stipulated as follows:
(3)a~+b~=〈(a_+b_,a1+b1,a2+b2,a-+b-),wa~∧wb~,ua~∨ub~(a_+b_,a1+b1,a2+b2,a-+b-)〉,(4)γa~={〈(γa_,γa1,γa2,γa-);wa~,ua~〉(γ>0)〈(γa-,γa2,γa1,γa_);wa~,ua~〉(γ<0),
where λ≠0 is a real number and the symbols “∧” and “∨” are the minimum and maximum operators, respectively.


## 2.2. Value and Ambiguity of a TrIFN

Definition 1.
Aα-cut set of a TrIFN a~ is a crisp subset of R, which can be expressed as a~α={x∣μa~(x)≥α}, where 0≤α≤wa~. A β-cut set of a TrIFN a~ is a crisp subset of R, which can be expressed as a~β={x∣υa~(x)≤β}, where ua~≤β≤1. It directly follows that a~α and a~β are closed intervals, denoted by a~α=[La~(α),Ra~(α)] and a~β=[La~(β),Ra~(β)], respectively, which can be calculated as follows:
(5)a~α=[[(wa~-α)a_+αa1]wa~,[(wa~-α)a-+αa2]wa~],(6)a~β=[[(1-β)a1+(β-ua~)a_](1-ua~),[(1-β)a2+(β-ua~)a-](1-ua~)].Definition 2.
Leta~α and a~β be a α-cut number and β-cut number of a TrIFN a~, respectively. Then the values of the membership and nonmembership functions for the TrIFN a~ are defined as follows:
(7)Vμ(a~)=∫0wa~[La~(α)+Ra~(α)2]f(α)dα,(8)Vυ(a~)=∫ua~1[La~(β)+Ra~(β)2]g(β)dβ,
where f(α) is a nonnegative and nondecreasing function on the interval [0,wa~] with f(0)=0 and f(wa~)=1 and g(β) is a nonnegative and nonincreasing function on the interval [ua~,1] with g(ua~)=1 and g(1)=0. Obviously, f(α) and g(β) can be considered as weighting functions and have various specific forms in actual applications, which can be chosen according to the real-life situations. In the following, we choose f(α)=α/wa~ (α∈[0,wa~]) and g(β)=(1-β)/(1-ua~) (β∈[ua~,1]).The functionf(α) gives different weights to elements at different α-cuts, so that it can lessen the contribution of the lower α-cuts, since these cuts arising from values of μa~(x) have a considerable amount of uncertainty. Therefore, Vμ(a~) and Vυ(a~) synthetically reflect the information on membership and nonmembership degrees.According to (5) and (7), we can calculate the value of the membership function as follows:
(9)Vμ(a~)=∫0wa~[(wa~-α)a_+αa12wa~+(wa~-α)a-+αa22wa~]αwa~dα=(a_+2a1+2a2+a-)wa~12.In a similar way, according to (6) and (8), the value of the nonmembership function can be obtained as follows:
(10)Vυ(a~)=(a_+2a1+2a2+a-)(1-ua~)12.It is directly derived from the condition0≤wa~+ua~≤1 that 0≤Vμ(a~)≤Vυ(a~), which may be concisely expressed as an interval [Vμ(a~),Vυ(a~)]. It is easily proven that the following conclusion holds.Theorem 3.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Vμ(a~+b~)=Vμ(a~)+Vμ(b~).Proof.
According to (3) and (9), we can prove that
(11)Vμ(a~+b~)=(a_+2a1+2a2+a-)wa~12+(b_+2b1+2b2+b-)wb~12=Vμ(a~)+Vμ(b~).
In the same way, using (3) and (10), we can prove that Vυ(a~+b~)=Vυ(a~)+Vυ(b~).Definition 4.
Leta~α and a~β be a α-cut set and β-cut set of a TrIFN a~, respectively. The ambiguities of the membership and nonmembership functions for the TrIFN a~ are defined as follows:
(12)Aμ(a~)=∫0wa~(Ra~(α)-La~(α))f(α)dα,Aυ(a~)=∫ua~1(Ra~(β)-La~(β))g(β)dβ.
According to (5) and (6), (12) can be obtained as follows:
(13)Aμ(a~)=(a--a_+2a2-2a1)wa~6,(14)Aυ(a~)=(a--a_+2a2-2a1)(1-ua~)6.
Obviously,Aμ(a~)≤Aυ(a~), which can be expressed as an interval [Aμ(a~),Aυ(a~)].Theorem 5.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Aμ(a~+b~)=Aμ(a~)+Aμ(b~).Proof.
According to (3) and (13), we can prove that
(15)Aμ(a~+b~)=(a--a_+2a2-2a1)wa~6+(b--b_+2b2-2b1)wb~6=Aμ(a~)+Aμ(b~).
Similarly, using (3) and (14), we can prove that Aυ(a~+b~)=Aυ(a~)+Aυ(b~).


## 2.3. The Difference-Index Based Ranking Method

Based on the value and ambiguity of a TrIFNa~, its value index and an ambiguity index are defined as follows. Hereby, a new ranking method of TrIFNs is proposed.Definition 6.
Leta~ be a TrIFN. Its value index and an ambiguity index are defined as follows:
(16)Vλ(a~)=Vμ(a~)+λ(Vυ(a~)-Vμ(a~)),(17)Aλ(a~)=Aυ(a~)-λ(Aυ(a~)-Aμ(a~)),
where λ∈[0,1] is a weight which represents the decision maker’s preference information. λ∈[0,1/2) shows that decision maker prefers to uncertainty or negative feeling, who is a pessimist; λ∈(1/2,1] shows that the decision maker prefers to certainty or positive feeling, who is an optimist; λ=1/2 shows that the decision maker is a neutralist, between positive feeling and negative feeling. Therefore, the value-index and the ambiguity-index may reflect the decision maker’s subjectivity attitude to the TrIFN.A difference index of the value index to the ambiguity index for a TrIFNa~ is defined as follows:
(18)Dλ(a~)=Vλ(a~)-Aλ(a~).Theorem 7.
Assume thata~ and b~ are two TrIFNs with wa~=wb~ and ua~=ub~. Then, Dλ(a~+b~)=Dλ(a~)+Dλ(b~), where λ∈[0,1].Proof.
According to Theorems3 and 5, it is derived from (18) that
(19)Dλ(a~+b~)=Vλ(a~+b~)-Aλ(a~+b~)=(Vλ(a~)+Vλ(b~))-(Aλ(a~)+Aλ(b~))=(Vλ(a~)-Aλ(a~))+(Vλ(b~)-Aλ(b~))=Dλ(A~)+Dλ(b~).Theorem7 shows that the difference index Dλ(a~) is a linear function of any TrIFN. Furthermore, it can be easily seen that the larger the difference index, the bigger the TrIFN. Thus, we propose the difference index based ranking method of TrIFNs as follows.Definition 8.
Assume thatλ∈[0,1]. For any TrIFNs a~ and b~, we stipulate the following: (1)
Dλ(a~)>Dλ(b~) if and only if a~ is bigger than b~, denoted by a~>b~;(2)
Dλ(a~)=Dλ(b~) if and only if a~ is equal to b~, denoted by a~=b~;(3)
Dλ(a~)≥Dλ(b~) if and only if a~>b~ or a~=b~.The above ranking method has some useful properties, which satisfy five of the seven axioms proposed by Wang and Kerre [20] that serve as the reasonable properties for the ordering of fuzzy quantities. And the proposed  ranking method is two-index, which is used to aggregate both value index and the ambiguity index. Especially, this proposed ranking method has the linearity.


## 3. Bilinear Programming Models for TrIFN Bimatrix Games

### 3.1. Bimatrix Games and Auxiliary Bilinear Programming Models

Assume thatS1={α1,α2,…,αm} and S2={β1,β2,…,βn} are sets of pure strategies for players I and II, respectively. The payoff matrices of players I and II are expressed with A=(aij)m×n and B=(bij)m×n, respectively. The vectors y=(y1,y2,…,ym)T and z=(z1,z2,…,zn)T are mixed strategies for players I and II, where yi  (i=1,2,…,m) and zj  (j=1,2,…,n) are probabilities in which players I and II choose their pure strategies αi∈S1  (i=1,2,…,m) and βj∈S2  (j=1,2,…,n), respectively; the symbol “T” is the transpose of a vector/matrix. Sets of all mixed strategies for I and II are denoted by Y and Z; that is, Y={y∣∑i=1myi=1,yi≥0(i=1,2,…,m)}, and Z={z∣∑j=1nzj=1,zj≥0(j=1,2,…,n)}, respectively. Thus, a two-person nonzero-sum finite game may be expressed with (Y,Z,A,B). In the sequent, such a game usually is simply called the bimatrix game (A,B) in which both players want to maximize his/her own payoffs. When I chooses any mixed strategy y∈Y and II chooses any mixed strategy z∈Z, the expected payoffs of I and  II can be computed as E1(y,z)=yTAz=∑i=1m∑j=1nyiaijzj and E2(y,z)=yTBz=∑i=1m∑j=1nyibijzj, respectively.Definition 9 (see [21]).
If there is a pair(y*,z*)∈Y×Z, such that yTAz*≤y*TAz* for any y∈Y and y*TBz≤y*TBz* for any z∈Z, then (y*,z*) is called a Nash equilibrium point of the bimatrix game (A,B), y* and z* are called Nash equilibrium strategies of players I and II, u*=y*TAz* and v*=y*TBz* are called Nash equilibrium values of players I and II, respectively, and (y*T,z*T,u*,v*) is called a Nash equilibrium solution of the bimatrix game (A,B).The following theorem guarantees the existence of Nash equilibrium solutions of any bimatrix game.Theorem 10 (see [22]).
Any bimatrix game(A,B) has at least one Nash equilibrium solution.A Nash equilibrium solution of any bimatrix game(A,B) can be obtained by solving the bilinear programming model stated as in the following Theorem 11.Theorem 11 (see [23]).
Let(A,B) be any bimatrix game. (y*T,z*T,u*,v*) is a Nash equilibrium solution of the bimatrix game (A,B) if and only if it is a solution of the bilinear programming model, which is shown as follows:
(20)max{yT(A+B)z-u-v}s.t.{Az≤uemBTy≤venyTem=1zTen=1y≥0,z≥0.
Furthermore, if (y*T,z*T,u*,v*) is a solution of the above bilinear programming model, then u*=y*TAz*, v*=y*TBz*, and y*T(A+B)z*-u*-v*=0.


### 3.2. Models and Method for TrIFN Bimatrix Games

Let us consider a TrIFN bimatrix game, where sets of pure strategiesS1 and S2 and sets of mixed strategies Y and Z for players I and II are defined as in the above sections. If player I chooses any pure strategy αi∈S1  (i=1,2,…,m) and player II chooses any pure strategy βj∈S2  (j=1,2,…,n), then at the situation (αi,βj) players I and II gain payoffs, which are expressed with TrIFNs A~ij(αi,βj)  =  {〈(αi,βj);(a_ij,a1ij,a2ij,a-ij),wA~ij,uA~ij〉} and B~ij(αi,βj)  =  {〈(αi,βj);(b_ij,b1ij,b2ij,b-ij),wA~ij,uA~ij〉} (i=1,2,…  ,  m;  j=1,2,…,n), respectively. Thus, the payoff matrices of players I and II are expressed as A~=(A~ij(αi,βj))m×n and B~=(B~ij(αi,βj))m×n, respectively.As stated earlier, the above payoffsA~ij(αi,βj) and B~ij(αi,βj) (i=1,2,…,m; j=1,2,…,n) of players I and II are TrIFNs in the finite universal set X′={(αi,βj)∣αi∈S1(i=1,2,…,m)  ,  βj∈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; that is, player I has pure strategies α1 and α2 and player II has pure strategies β1 and β2. Then, the universal set is X0′={(α1,β1),(α1,β2),(α2,β1),(α2,β2)}, which has four elements (i.e., situations). In the sequel, the above TrIFN bimatrix game is simply denoted by (A~,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 E~1(y,z)=yTA~z. According to the operations of TrIFNs, the expected payoff E~1(y,z) of player I is a TrIFN and can be calculated: E~1(y,z)  =  {〈(y-,z-),∑i=1n∑j=1nyi(a_ij,a1ij,a2ij,a-ij)zj;   min{wa~ij},max{ua~ij}〉}, where (y-,z-) represents a mixed situation, which corresponds to the mixed strategies y and z.Similarly, the expected payoff of player II isE~2(y,z)  =  yTB~z, which can be calculated: E~2(y,z)=∑i=1n∑j=1nyi(b_ij,b1ij,b2ij,b-ij)zj;min{wa~ij},max{ua~ij}〉}, where (y-′,z-′) represents a mixed situation, which corresponds to the mixed strategies y and z.Definition 12.
Assume that there is a pair(y*,z*)∈Y×Z. If any y∈Y and z∈Z satisfy yTA~z*≤TrIFNy*TA~z* and y*TB~z≤TrIFNy*TB~z*, then (y*,z*) is called a Nash equilibrium point of the TrIFN bimatrix game (A~,B~), y* and z* are called Nash equilibrium strategies of players I and II, u~*=y*TA~z* and v~*=y*TB~z* are called Nash equilibrium values of players I and II, respectively, and (y*,z*,u~*,v~*) is called a Nash equilibrium solution of the TrIFN bimatrix game (A~,B~).Stated as earlier, however, player I’s expected payoffyTA~z and player II’s expected payoff yTB~z are TrIFNs. Therefore, there are no commonly used concepts of solutions of TrIFN 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 TrIFN bimatrix games is very difficult. In the sequel, we use the ranking function Dλ to develop a new method for solving the TrIFN bimatrix game (A~,B~).Using the ranking function of TrIFNs given by (18), we can transform the TrIFN payoff matrices A~ and B~ of players I and II into the payoff matrices as follows:
(21)A~λ1=Dλ1((A~ij(αi,βj))m×n)=(Dλ1(A~ij(αi,βj)))m×n,B~λ2=Dλ2((B~ij(αi,βj))m×n)=(Dλ2(B~ij(αi,βj)))m×n,
where λ1∈[0,1], λ2∈[0,1], Dλ1(A~ij(αi,βj))=VA~ij(αi,βj)-AA~ij(αi,βj), and Dλ2(B~ij(αi,βj))  =  VB~ij(αi,βj)-AB~ij(αi,βj) (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(A~λ1,B~λ2), where the pure (or mixed) strategy sets of players I and II are S1 and S2 (or Y and Z) defined as mentioned above. Then, the TrIFN bimatrix game (A~,B~) is transformed into the parametric bimatrix game (A~λ1,B~λ2). Hereby, according to Definitions 8–12 and Theorem 7, we can give the definition of satisfying Nash equilibrium solutions of the TrIFN bimatrix game (A~λ1,B~λ2) as follows.Definition 13.
For given parametersλ1∈[0,1] and λ2∈[0,1], if there is a pair (y*,z*)∈Y×Z, such that any y∈Y and z∈Z satisfy the following conditions: yTA~λ1z*≤y*TA~λ1z* and y*TB~λ2z≤y*TB~λ2z*, then (y*,z*) is called a satisfying Nash equilibrium point of the TrIFN bimatrix game (A~λ1,B~λ2), y* and z* are called satisfying Nash equilibrium strategies of players I and II, u*(λ1)=y*TA~λ1z* and v*(λ2)=y*TB~λ2z* are called satisfying equilibrium values of players I and II, respectively, and (y*,z*,u*(λ1),v*(λ2)) is called a satisfying Nash equilibrium solution of the TrIFN bimatrix game (A~λ1,B~λ2).It can be easily seen from the ranking function given by (18) and Theorem 7 that Definitions 12 and 13 are equivalent in the sense of the order relation defined by Definition 8. Thus, for given parameters λ1∈[0,1] and λ2∈[0,1], according to Theorem 10, the parametric bimatrix game (A~λ1,B~λ2) has at least one Nash equilibrium solution. Namely, the TrIFN bimatrix game (A~λ1,B~λ2) has at least one satisfying Nash equilibrium solution, which can be obtained through solving the following parametric bilinear programming model according to Theorem 11:
(22)max{∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))hhhhhhhhhhh+Vλ2(B~ij(αi,βj))-Aλ2(B~ij(αi,βj))]zjh-u(λ1)-v(λ2)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vλ1(A~ij(αi,βj))h-Aλ1(A~ij(αi,βj))]zj≤u(λ1)(i=1,2,…,m)∑i=1m[Vλ2(B~ij(αi,βj))h-Aλ2(B~ij(αi,βj))]yi≤v(λ2)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1v(λ2)≥0,u(λ1)≥0yi≥0(i=1,2,…,m),zj≥0(j=1,2,…,n),
where yi (i=1,2,…,m), zj (j=1,2,…,n), u(λ1), and v(λ2) are decision variables.According to Theorem11, if (y*,z*,u*(λ1),v*(λ2)) is a solution of the above parametric bilinear programming model (i.e., (22)), then u*(λ1)=y*TA~λ1z*  =  ∑j=1n∑i=1m[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))]yi*zj*, v*(λ2)   =  y*TB~λ2z*  =  ∑j=1n∑i=1m[Vλ2(B~ij(αi,βj))-Aλ2(B~ij(αi,βj))]yi*zj*, and y*T(Dλ1(A~)+Dλ2(B~))z*-u*(λ1)-v*(λ2)=0.Notice thatyi*≥0, zj*≥0, and Vλ(a~) and Aλ(a~) are, respectively, continuous nondecreasing and nonincreasing functions of the parameter λ∈[0,1] if a~ is a nonnegative TrIFN.  Then, u*(λ1) and v*(λ2) are monotonic and nondecreasing 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 TrIFNs {〈(y-*,z-*),u*(0),1-u*(1)〉} and {〈(y-*,z-*),v*(0),1-v*(1)〉}, where (y-*,z-*) represents a mixed situation. Thus, the u~*(y-*,z-*) and v~*(y-*,z-*) are Nash equilibrium values of players I and II, respectively.In particular, for the parametersλ1=0 and λ2=0, (22) becomes the bilinear programming model as follows:
(23)max{∑j=1n∑i=1myi[Vμ(A~ij(αi,βj))-Aυ(A~ij(αi,βj))hhhhhhhhhhh+Vμ(B~ij(αi,βj))-Aυ(B~ij(αi,βj))]zjh-u(0)-v(0)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vμ(A~ij(αi,βj))h-Aυ(A~ij(αi,βj))]zj≤u(0)(i=1,2,…,m)∑i=1m[Vμ(B~ij(αi,βj))h-Aυ(B~ij(αi,βj))]yi≤v(0)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1zj≥0,yi≥0(i,j=1,2,…,n)u(0)≥0,v(0)≥0,
where u(0)=Vμ(u~(y-,z-))-Aυ(u~(y-,z-)), v(0)=Vμ(v~(y-,z-))-Aυ(v~(y-,z-)), yi (i=1,2,…,m), zj (j=1,2,…,n), u(0), and v(0) are decision variables. The solution of (23) can be obtained by Lemke-Howson’s algorithm (Lemke and Howson [24]) and denoted by (y*T  ,  z*T  ,  Vμ(u~*(y-*,z-*))-Aυ(u~*(y-*,z-*))  ,  Vμ(v~*(y-*,z-*))-Aυ(v~*(y-*,z-*))).Similarly, for the parametersλ1=1 and λ2=1, (22) becomes the bilinear programming model as follows:
(24)max{∑j=1n∑i=1myi[Vυ(A~ij(αi,βj))-Aμ(A~ij(αi,βj))hhhhhhhhhhhh+Vυ(B~ij(αi,βj))-Aμ(B~ij(αi,βj))]zjh-u(1)-v(1)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vμ(A~ij(αi,βj))h-Aυ(A~ij(αi,βj))]zj≤u(1)(i=1,2,…,m)∑i=1m[Vμ(B~ij(αi,βj))h-Aυ(B~ij(αi,βj))]yi≤v(0)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1zj≥0,yi≥0(i,j=1,2,…,n)u(1)≥0,v(1)≥0,
where u(1)=Vυ(u~*(y-*,z-*))-Aμ(u~*(y-*,z-*)), v(1)=Vυ(v~*(y-*,z-*))-Aμ(v~*(y-*,z-*)), yi (i=1,2,…,m), zj (j=1,2,…,n), u(1), and v(1) are decision variables. Likewise, the solution of (24) can be obtained by Lemke-Howson’s algorithm (Lemke and Howson [24]) and denoted by (y′*T,z′*T,Vυ(u~*(y-*,z-*))-Aμ(u~*(y-*,z-*)),Vυ(v~*(y-*,z-*))-Aμ(v~*(y-*,z-*))).Thus, we can explicitly obtain the satisfying Nash equilibrium values and corresponding satisfying Nash equilibrium strategies of players I and II through solving the derived two bilinear programming models (i.e., (23) and (24)). Furthermore, according to (22), 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].


## 3.1. Bimatrix Games and Auxiliary Bilinear Programming Models

Assume thatS1={α1,α2,…,αm} and S2={β1,β2,…,βn} are sets of pure strategies for players I and II, respectively. The payoff matrices of players I and II are expressed with A=(aij)m×n and B=(bij)m×n, respectively. The vectors y=(y1,y2,…,ym)T and z=(z1,z2,…,zn)T are mixed strategies for players I and II, where yi  (i=1,2,…,m) and zj  (j=1,2,…,n) are probabilities in which players I and II choose their pure strategies αi∈S1  (i=1,2,…,m) and βj∈S2  (j=1,2,…,n), respectively; the symbol “T” is the transpose of a vector/matrix. Sets of all mixed strategies for I and II are denoted by Y and Z; that is, Y={y∣∑i=1myi=1,yi≥0(i=1,2,…,m)}, and Z={z∣∑j=1nzj=1,zj≥0(j=1,2,…,n)}, respectively. Thus, a two-person nonzero-sum finite game may be expressed with (Y,Z,A,B). In the sequent, such a game usually is simply called the bimatrix game (A,B) in which both players want to maximize his/her own payoffs. When I chooses any mixed strategy y∈Y and II chooses any mixed strategy z∈Z, the expected payoffs of I and  II can be computed as E1(y,z)=yTAz=∑i=1m∑j=1nyiaijzj and E2(y,z)=yTBz=∑i=1m∑j=1nyibijzj, respectively.Definition 9 (see [21]).
If there is a pair(y*,z*)∈Y×Z, such that yTAz*≤y*TAz* for any y∈Y and y*TBz≤y*TBz* for any z∈Z, then (y*,z*) is called a Nash equilibrium point of the bimatrix game (A,B), y* and z* are called Nash equilibrium strategies of players I and II, u*=y*TAz* and v*=y*TBz* are called Nash equilibrium values of players I and II, respectively, and (y*T,z*T,u*,v*) is called a Nash equilibrium solution of the bimatrix game (A,B).The following theorem guarantees the existence of Nash equilibrium solutions of any bimatrix game.Theorem 10 (see [22]).
Any bimatrix game(A,B) has at least one Nash equilibrium solution.A Nash equilibrium solution of any bimatrix game(A,B) can be obtained by solving the bilinear programming model stated as in the following Theorem 11.Theorem 11 (see [23]).
Let(A,B) be any bimatrix game. (y*T,z*T,u*,v*) is a Nash equilibrium solution of the bimatrix game (A,B) if and only if it is a solution of the bilinear programming model, which is shown as follows:
(20)max{yT(A+B)z-u-v}s.t.{Az≤uemBTy≤venyTem=1zTen=1y≥0,z≥0.
Furthermore, if (y*T,z*T,u*,v*) is a solution of the above bilinear programming model, then u*=y*TAz*, v*=y*TBz*, and y*T(A+B)z*-u*-v*=0.


## 3.2. Models and Method for TrIFN Bimatrix Games

Let us consider a TrIFN bimatrix game, where sets of pure strategiesS1 and S2 and sets of mixed strategies Y and Z for players I and II are defined as in the above sections. If player I chooses any pure strategy αi∈S1  (i=1,2,…,m) and player II chooses any pure strategy βj∈S2  (j=1,2,…,n), then at the situation (αi,βj) players I and II gain payoffs, which are expressed with TrIFNs A~ij(αi,βj)  =  {〈(αi,βj);(a_ij,a1ij,a2ij,a-ij),wA~ij,uA~ij〉} and B~ij(αi,βj)  =  {〈(αi,βj);(b_ij,b1ij,b2ij,b-ij),wA~ij,uA~ij〉} (i=1,2,…  ,  m;  j=1,2,…,n), respectively. Thus, the payoff matrices of players I and II are expressed as A~=(A~ij(αi,βj))m×n and B~=(B~ij(αi,βj))m×n, respectively.As stated earlier, the above payoffsA~ij(αi,βj) and B~ij(αi,βj) (i=1,2,…,m; j=1,2,…,n) of players I and II are TrIFNs in the finite universal set X′={(αi,βj)∣αi∈S1(i=1,2,…,m)  ,  βj∈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; that is, player I has pure strategies α1 and α2 and player II has pure strategies β1 and β2. Then, the universal set is X0′={(α1,β1),(α1,β2),(α2,β1),(α2,β2)}, which has four elements (i.e., situations). In the sequel, the above TrIFN bimatrix game is simply denoted by (A~,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 E~1(y,z)=yTA~z. According to the operations of TrIFNs, the expected payoff E~1(y,z) of player I is a TrIFN and can be calculated: E~1(y,z)  =  {〈(y-,z-),∑i=1n∑j=1nyi(a_ij,a1ij,a2ij,a-ij)zj;   min{wa~ij},max{ua~ij}〉}, where (y-,z-) represents a mixed situation, which corresponds to the mixed strategies y and z.Similarly, the expected payoff of player II isE~2(y,z)  =  yTB~z, which can be calculated: E~2(y,z)=∑i=1n∑j=1nyi(b_ij,b1ij,b2ij,b-ij)zj;min{wa~ij},max{ua~ij}〉}, where (y-′,z-′) represents a mixed situation, which corresponds to the mixed strategies y and z.Definition 12.
Assume that there is a pair(y*,z*)∈Y×Z. If any y∈Y and z∈Z satisfy yTA~z*≤TrIFNy*TA~z* and y*TB~z≤TrIFNy*TB~z*, then (y*,z*) is called a Nash equilibrium point of the TrIFN bimatrix game (A~,B~), y* and z* are called Nash equilibrium strategies of players I and II, u~*=y*TA~z* and v~*=y*TB~z* are called Nash equilibrium values of players I and II, respectively, and (y*,z*,u~*,v~*) is called a Nash equilibrium solution of the TrIFN bimatrix game (A~,B~).Stated as earlier, however, player I’s expected payoffyTA~z and player II’s expected payoff yTB~z are TrIFNs. Therefore, there are no commonly used concepts of solutions of TrIFN 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 TrIFN bimatrix games is very difficult. In the sequel, we use the ranking function Dλ to develop a new method for solving the TrIFN bimatrix game (A~,B~).Using the ranking function of TrIFNs given by (18), we can transform the TrIFN payoff matrices A~ and B~ of players I and II into the payoff matrices as follows:
(21)A~λ1=Dλ1((A~ij(αi,βj))m×n)=(Dλ1(A~ij(αi,βj)))m×n,B~λ2=Dλ2((B~ij(αi,βj))m×n)=(Dλ2(B~ij(αi,βj)))m×n,
where λ1∈[0,1], λ2∈[0,1], Dλ1(A~ij(αi,βj))=VA~ij(αi,βj)-AA~ij(αi,βj), and Dλ2(B~ij(αi,βj))  =  VB~ij(αi,βj)-AB~ij(αi,βj) (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(A~λ1,B~λ2), where the pure (or mixed) strategy sets of players I and II are S1 and S2 (or Y and Z) defined as mentioned above. Then, the TrIFN bimatrix game (A~,B~) is transformed into the parametric bimatrix game (A~λ1,B~λ2). Hereby, according to Definitions 8–12 and Theorem 7, we can give the definition of satisfying Nash equilibrium solutions of the TrIFN bimatrix game (A~λ1,B~λ2) as follows.Definition 13.
For given parametersλ1∈[0,1] and λ2∈[0,1], if there is a pair (y*,z*)∈Y×Z, such that any y∈Y and z∈Z satisfy the following conditions: yTA~λ1z*≤y*TA~λ1z* and y*TB~λ2z≤y*TB~λ2z*, then (y*,z*) is called a satisfying Nash equilibrium point of the TrIFN bimatrix game (A~λ1,B~λ2), y* and z* are called satisfying Nash equilibrium strategies of players I and II, u*(λ1)=y*TA~λ1z* and v*(λ2)=y*TB~λ2z* are called satisfying equilibrium values of players I and II, respectively, and (y*,z*,u*(λ1),v*(λ2)) is called a satisfying Nash equilibrium solution of the TrIFN bimatrix game (A~λ1,B~λ2).It can be easily seen from the ranking function given by (18) and Theorem 7 that Definitions 12 and 13 are equivalent in the sense of the order relation defined by Definition 8. Thus, for given parameters λ1∈[0,1] and λ2∈[0,1], according to Theorem 10, the parametric bimatrix game (A~λ1,B~λ2) has at least one Nash equilibrium solution. Namely, the TrIFN bimatrix game (A~λ1,B~λ2) has at least one satisfying Nash equilibrium solution, which can be obtained through solving the following parametric bilinear programming model according to Theorem 11:
(22)max{∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))hhhhhhhhhhh+Vλ2(B~ij(αi,βj))-Aλ2(B~ij(αi,βj))]zjh-u(λ1)-v(λ2)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vλ1(A~ij(αi,βj))h-Aλ1(A~ij(αi,βj))]zj≤u(λ1)(i=1,2,…,m)∑i=1m[Vλ2(B~ij(αi,βj))h-Aλ2(B~ij(αi,βj))]yi≤v(λ2)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1v(λ2)≥0,u(λ1)≥0yi≥0(i=1,2,…,m),zj≥0(j=1,2,…,n),
where yi (i=1,2,…,m), zj (j=1,2,…,n), u(λ1), and v(λ2) are decision variables.According to Theorem11, if (y*,z*,u*(λ1),v*(λ2)) is a solution of the above parametric bilinear programming model (i.e., (22)), then u*(λ1)=y*TA~λ1z*  =  ∑j=1n∑i=1m[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))]yi*zj*, v*(λ2)   =  y*TB~λ2z*  =  ∑j=1n∑i=1m[Vλ2(B~ij(αi,βj))-Aλ2(B~ij(αi,βj))]yi*zj*, and y*T(Dλ1(A~)+Dλ2(B~))z*-u*(λ1)-v*(λ2)=0.Notice thatyi*≥0, zj*≥0, and Vλ(a~) and Aλ(a~) are, respectively, continuous nondecreasing and nonincreasing functions of the parameter λ∈[0,1] if a~ is a nonnegative TrIFN.  Then, u*(λ1) and v*(λ2) are monotonic and nondecreasing 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 TrIFNs {〈(y-*,z-*),u*(0),1-u*(1)〉} and {〈(y-*,z-*),v*(0),1-v*(1)〉}, where (y-*,z-*) represents a mixed situation. Thus, the u~*(y-*,z-*) and v~*(y-*,z-*) are Nash equilibrium values of players I and II, respectively.In particular, for the parametersλ1=0 and λ2=0, (22) becomes the bilinear programming model as follows:
(23)max{∑j=1n∑i=1myi[Vμ(A~ij(αi,βj))-Aυ(A~ij(αi,βj))hhhhhhhhhhh+Vμ(B~ij(αi,βj))-Aυ(B~ij(αi,βj))]zjh-u(0)-v(0)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vμ(A~ij(αi,βj))h-Aυ(A~ij(αi,βj))]zj≤u(0)(i=1,2,…,m)∑i=1m[Vμ(B~ij(αi,βj))h-Aυ(B~ij(αi,βj))]yi≤v(0)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1zj≥0,yi≥0(i,j=1,2,…,n)u(0)≥0,v(0)≥0,
where u(0)=Vμ(u~(y-,z-))-Aυ(u~(y-,z-)), v(0)=Vμ(v~(y-,z-))-Aυ(v~(y-,z-)), yi (i=1,2,…,m), zj (j=1,2,…,n), u(0), and v(0) are decision variables. The solution of (23) can be obtained by Lemke-Howson’s algorithm (Lemke and Howson [24]) and denoted by (y*T  ,  z*T  ,  Vμ(u~*(y-*,z-*))-Aυ(u~*(y-*,z-*))  ,  Vμ(v~*(y-*,z-*))-Aυ(v~*(y-*,z-*))).Similarly, for the parametersλ1=1 and λ2=1, (22) becomes the bilinear programming model as follows:
(24)max{∑j=1n∑i=1myi[Vυ(A~ij(αi,βj))-Aμ(A~ij(αi,βj))hhhhhhhhhhhh+Vυ(B~ij(αi,βj))-Aμ(B~ij(αi,βj))]zjh-u(1)-v(1)∑j=1n∑i=1myi[Vλ1(A~ij(αi,βj))-Aλ1(A~ij(αi,βj))}s.t.{∑j=1n[Vμ(A~ij(αi,βj))h-Aυ(A~ij(αi,βj))]zj≤u(1)(i=1,2,…,m)∑i=1m[Vμ(B~ij(αi,βj))h-Aυ(B~ij(αi,βj))]yi≤v(0)(j=1,2,…,n)y1+y2+⋯+ym=1z1+z2+⋯+zn=1zj≥0,yi≥0(i,j=1,2,…,n)u(1)≥0,v(1)≥0,
where u(1)=Vυ(u~*(y-*,z-*))-Aμ(u~*(y-*,z-*)), v(1)=Vυ(v~*(y-*,z-*))-Aμ(v~*(y-*,z-*)), yi (i=1,2,…,m), zj (j=1,2,…,n), u(1), and v(1) are decision variables. Likewise, the solution of (24) can be obtained by Lemke-Howson’s algorithm (Lemke and Howson [24]) and denoted by (y′*T,z′*T,Vυ(u~*(y-*,z-*))-Aμ(u~*(y-*,z-*)),Vυ(v~*(y-*,z-*))-Aμ(v~*(y-*,z-*))).Thus, we can explicitly obtain the satisfying Nash equilibrium values and corresponding satisfying Nash equilibrium strategies of players I and II through solving the derived two bilinear programming models (i.e., (23) and (24)). Furthermore, according to (22), 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].


## 4. An Example of a Commerce Retailers’ Strategy Choice Problem

Let us consider the case of two commerce retailersR1 and R2 (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 intuitions. And assume that commerce retailers R1 and R2 are rational; that is, they will choose optimal strategies to maximize their own profits without cooperation. Suppose that retailer R1 has two pure strategies: establishing a scientific and rational service system α1 and providing customers with satisfaction products α2. Retailer R2 possesses the same pure strategies as retailer R1; that is, the options of retailer R2 are establishing a scientific and rational service system β1 and providing customers with satisfaction products β2.Let us consider the following specific TrIFN bimatrix game for this scenario, where the payoff matrices of commerce retailersR1 and R2 are expressed with TrIFNs as follows:(25)A~=({〈(α1,β1),(50,60,70,80),0.8,0.1〉}{〈(α1,β2),(30,40,50,60),0.4,0.3〉}{〈(α2,β1),(20,30,40,50),0.5,0.4〉}{〈(α2,β2),(40,50,60,70),0.6,0.2〉}),B~=({〈(α1,β1),(40,50,60,70),0.7,0.1〉}{〈(α1,β2),(30,40,50,60),0.7,0.2〉}{〈(α2,β1),(20,30,40,50),0.5,0.3〉}{〈(α2,β2),(50,60,70,80),0.8,0.1〉}),where the TrIFN {〈(α1,β1),(50,60,70,80),0.8,0.1〉} in the payoff matrix A~ means that the payoff is between 50 and 80, and satisfaction (or membership) degree of customers is 0.8, and the dissatisfaction (or nonmembership) degree of customers is 0.1 for commerce retailer R1 if he/she adopts the pure strategy α1 and commerce retailer R2 also adopts the pure strategy β1. Other entries in the TrIFN payoff matrices A~ and B~ can be similarly explained.According to (16) and (17), respectively, the value indexes and the ambiguity-indexes of A~(αi,βj)), i=1,2, j=1,2, can be obtained as follows:
(26)Vλ1(A~(α1,β1))=26+3.25λ1,Aλ1(A~(α1,β1))=7.5-0.83λ1,Vλ1(A~(α1,β2))=9+6.75λ1,Aλ1(A~(α1,β2))=5.83-2.5λ1,Vλ1(A~(α2,β1))=8.75+1.75λ1,Aλ1(A~(α2,β1))=5-0.83λ1,Vλ1(A~(α2,β2))=16+5.5λ1,Aλ1(A~(α2,β2))=6.67-1.67λ1.According to (18), the differences of the value indexes to the ambiguity indexes for the TrIFNs A~(αi,βj)), i=1,2, j=1,2, can be obtained as follows:
(27)Dλ1(A~(α1,β1))=Vλ1(A~(α1,β1))-Aλ1(A~(α1,β1))=18.5+4.08λ1,Dλ1(A~(α1,β2))=Vλ1(A~(α1,β2))-Aλ1(A~(α1,β2))=3.17+9.25λ1,Dλ1(A~(α2,β1))=Vλ1(A~(α2,β1))-Aλ1(A~(α2,β1))=3.75+2.58λ1,Dλ1(A~(α2,β2))=Vλ1(A~(α2,β2))-Aλ1(A~(α2,β2))=9.83+7.17λ1.In the same way, we can obtain that the differences of the value indexes to the ambiguity indexes for the TrIFNsB~(αi,βj))  (i=1,2,j=1,2) can be obtained as follows:
(28)Dλ2(B~(α1,β1))=Vλ2(B~(α1,β1))-Aλ2(B~(α1,β1))=11.75+7.17λ2,Dλ2(B~(α1,β2))=Vλ2(B~(α1,β2))-Aλ2(B~(α1,β2))=9.08+3.08λ2,Dλ2(B~(α2,β1))=Vλ2(B~(α2,β1))-Aλ2(B~(α2,β1))=2.92+5.17λ2,Dλ2(B~(α2,β2))=Vλ2(B~(α2,β2))-Aλ2(B~(α2,β2))=18.5+4.08λ2.Using (22), the parametric bilinear programming model is constructed as follows:
(29)max{(30.25+4.08λ1+7.17λ2)y1z1hhhh+(12.25+9.25λ1+3.08λ2)y1z2hhhh+(6.67+2.58λ1+5.17λ2)y2z1hhhh+(28.33+7.17λ1+4.08λ2)y2z2-u(λ1)-v(λ2)}s.t.{(18.5+4.08λ1)z1+(3.17+9.25λ1)z2≤u(λ1)(3.75+2.58λ1)z1+(9.83+7.17λ1)z2≤u(λ1)(11.75+7.17λ2)y1+(9.08+3.08λ2)y2≤v(λ2)(2.92+5.17λ2)y1+(18.5+4.08λ2)y2≤v(λ2)y1+y2=1z1+z2=1u(λ1)≥0,v(λ2)≥0yi≥0(i=1,2),zj≥0(j=1,2).It is easily seen that, for the parametersλ1∈[0,1] and λ2∈[0,1], solving (29), we can obtain the satisfying Nash equilibrium values and corresponding satisfying Nash equilibrium strategies of commerce retailers R1 and R2, respectively, depicted as in Tables 1, 2, 3, 4, 5, and 6.Table 1
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
0
0
(
0.484,0.516
)
7.94
(
0.311,0.689
)
10.46
0
0.3
(
0.492,0.508
)
7.94
(
0.311,0.689
)
11.98
0
0.5
(
0.498,0.502
)
7.94
(
0.311,0
.
689
)
12.98
0
0.8
(
0.495,0.505
)
7.94
(
0.311,0
.
689
)
14.48
0
1
(
0.490,0.510
)
7.94
(
0.311,0
.
689
)
15.47Table 2
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
0.5
0
(
0.484,0.516
)
11.19
(
0.266,0.734
)
10.46
0.5
0.3
(
0.492,0.508
)
11.19
(
0.266,0.734
)
11.98
0.5
0.5
(
0.498,0.502
)
11.19
(
0.266,0.734
)
12.98
0.5
0.8
(
0.495,0.505
)
11.19
(
0.266,0.734
)
14.48
0.5
1
(
0.490,0.510
)
11.19
(
0.266,0.734
)
15.47Table 3
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
1
0
(
0.484,0.516
)
14.65
(
0.220,0.780
)
10.46
1
0.3
(
0.492,0.508
)
14.65
(
0.220,0.780
)
11.98
1
0.5
(
0.498,0.502
)
14.65
(
0.220,0.780
)
12.98
1
0.8
(
0.495,0.505
)
14.65
(
0.220,0.780
)
14.48
1
1
(
0.490,0.510
)
14.65
(
0.220,0.780
)
15.47Table 4
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
0
0
(
0.516,0.484
)
7.94
(
0.311,0.689
)
10.46
0.3
0
(
0.516,0.484
)
9.86
(
0.284,0.716
)
10.46
0.5
0
(
0.516,0.484
)
11.19
(
0.266,0.734
)
10.46
0.8
0
(
0.516,0.484
)
13.24
(
0.239,0.761
)
10.46
1
0
(
0.516,0.484
)
14.65
(
0.220,0.780
)
10.46Table 5
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
0
0.5
(
0.502,0.498
)
7.94
(
0.311,0.689
)
12.99
0.3
0.5
(
0.502,0.498
)
9.86
(
0.284,0.716
)
12.99
0.5
0.5
(
0.502,0.498
)
11.19
(
0.266,0.734
)
12.99
0.8
0.5
(
0.502,0.498
)
13.24
(
0.239,0.761
)
12.99
1
0.5
(
0.502,0.498
)
14.65
(
0.220,0.780
)
12.99Table 6
Satisfying Nash equilibrium values and corresponding strategies of commerce retailers.
Parameters
R
1
R
2
λ
1
λ
2
y
*
T
u
*
(
λ
1
)
z
*
T
v
*
(
λ
2
)
0
1
(
0.490,0.510
)
7.94
(
0.311,0.689
)
15.47
0.3
1
(
0.490,0.510
)
9.86
(
0.284,0.716
)
15.47
0.5
1
(
0.490,0.510
)
11.19
(
0.266,0.734
)
15.47
0.8
1
(
0.490,0.510
)
13.24
(
0.239,0.761
)
15.47
1
1
(
0.490,0.510
)
14.65
(
0.220,0.780
)
15.47It can be easily seen from Tables1–3 (or Tables 4–6) that the satisfying Nash equilibrium value of a player (i.e., I/commerce retailer R1 or II/R2) only depends on his/her own preference/parameter regardless of other player’s preferences/parameters. However, strategy choice of a player is only affected by other player’s preferences/parameters.


## 5. Conclusion

In some situations, 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 the above, we model TrIFN bimatrix games and develop the parametric bilinear programming models and method by using the new order relation of TrIFNs given in this paper. The developed models and method can simplify the calculation of Nash equilibrium solutions of TrIFN bimatrix games.Furthermore, it is easy to see that the models and method proposed in this paper may be extended to TrIFN multiobjective bimatrix games. And more effective methods of TrIFN bimatrix 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.



---

*Source: 697261-2013-12-22.xml*