Characterization of the Equilibrium Strategy of Fuzzy Bimatrix Games Based on L-R Fuzzy Variables

Abstract

This paper deals with bimatrix games in uncertainty environment based on several types of ordering, which Maeda proposed. But Maeda’s models was just made based on symmetrical triangle fuzzy variable. In this paper, we generalized Maeda’s model to the non-symmetrical environment. In other words, we investigated the fuzzy bimatrix games based on nonsymmetrical 𝐿-𝑅 fuzzy variables. Then the pseudoinverse of a nonconstant monotone function was given and the concept of crisp parametric bimatrix games was introduced. At last, the existence condition of Nash equilibrium strategies of the fuzzy bimatrix games is proposed and (weak) Pareto equilibrium of the fuzzy bimatrix games was obtained through the Nash equilibrium of the crisp parametric bimatrix.

1. Introduction

Nash presented noncooperative game theory [1, 2] based on the assumption that each player has a well-defined quantitative utility function over a set of the player’s strategy, each player attempts to optimize his own expected payoffs, and each is assumed to know the extensive game completely. In this paper, above assumption had been violated involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate payoff function for each player in these types of problems. In other words, while modeling the noncooperative games, each player can not give the exact payoffs in practice because of the complexity and uncertainty of the problems. So the gain or payoff function is not always evaluated by a crisp number. It should be formulated semantically, in such terms as excellent, good, sufficiently reliable, durable, or resistant. Fortunately, in this paper, we have another utility to measure the uncertainty: the fuzzy set theory. Indeed, a fuzzy variable is used to present the payoffs of the players. For example, one player’s payoff is about 100 thousand dollars. Since the expected payoffs of the player are fuzzy variable, we should define new concepts of equilibrium strategy to investigate their properties.

Butnariu [3] started with the work of noncooperative fuzzy games, he proposed a game to be fuzzy when the players have fuzzy preferences. Campos [4] modeled the two-person zero-sum games with fuzzy payoffs by fuzzy programming and transformed the fuzzy programming into a linear programming problems through Yager’s fuzzy variable ordering method [5]. Nishizaki and Sakawa [6, 7] investigated single- and multiobjective games and presented max-min algorithm with respect to a degree of attainment of the aggregated fuzzy goal. Through maximizing the minimum of the fuzzy expected payoff and fuzzy goal, they transform their model into a fractional programming problem and compute the fractional programming by a relaxed method., Bector et al. [8, 9] and Vijay e al. [10, 11] modeled the noncooperative games in uncertainty by fuzzy programming problem and computed their model by the fuzzy dual programming. Takashi [12, 13] presented three kinds of equilibrium strategies of fuzzy matrix games based on especial symmetric triangular fuzzy variable and investigated the existence condition of these equilibrium strategies. But it is partial acceptance to modeling payoffs by symmetric triangular fuzzy variable.

This paper was going to generalize Maeda’s model and investigate all types of equilibrium strategies based on more general asymmetric 𝐿-𝑅 fuzzy variables. Then we introduce the crisp parametric matrix games and its equilibrium strategies in order to find the Pareto equilibrium strategies of fuzzy bimatrix games. For that purpose, this paper is organized as follows. In Section 2, we introduce some basic definitions and notations about fuzzy set theory. Then, several fuzzy ordering that Ramík [14] and Maeda presented were introduced. And the pseudoinverse of a nonconstant monotone function was given. In Section 3, we focus on the condition of different equilibrium strategies. Especially, the relation between the Nash equilibrium strategies of crisp bimatrix parametric games and Pareto equilibrium strategies of fuzzy bimatrix games is established.

2. Preliminaries

Definition 2.1. A 𝐿-𝑅 fuzzy variable 𝑎̃ is a fuzzy set defined on the space of real number, whose membership functions 𝜇_̃𝑎∶ℛ→[0,1] as following 𝜇_̃𝑎(𝑥)=⎧⎪⎨⎪⎩𝐿𝑎−𝑥𝑅𝑥−𝑙𝑎,𝑥≤𝑎,𝑙≥0,𝑟,𝑥≥𝑎,𝑟≥0,(2.1) where, 𝐿,𝑅∶ℛ→[0,1] are not constant and left continuous function, they satisfy the followings: (1) 𝐿(𝑥)=𝐿(−𝑥), 𝑅(𝑥)=𝑅(−𝑥); (2) 𝐿(0)=𝑅(0)=1, 𝐿(1)=𝑅(1)=0; (3) they are nonincreasing on [0,+∞). The 𝐿-𝑅 fuzzy variable is denoted by 𝑎̃=(𝑎,𝑙,𝑟)_𝐿-𝑅, 𝑎, 𝑙 and 𝑟 are called the center, left extension, and right extension, respectively.

Remark 2.2. Notice that the domains of 𝐿(𝑡) and 𝑅(𝑡), which are defined in Definition 2.1, are [0,+∞), respectively. Moreover, for 𝑡∈[1,+∞), 𝐿(𝑡)=𝑅(𝑡)=0 holds.
We denote the 𝐿-𝑅 fuzzy variable set as ℱ, for 𝑎̃∈ℱ, 𝛼∈[0,1], 𝑎_𝛼̃≜{𝑥∣𝜇_̃𝑎(𝑥)≥𝛼,𝑥∈ℛ} is called as 𝛼-level set of 𝑎̃. 𝑎₀̃≜{𝑥∣𝜇_̃𝑎(𝑥)>0,𝑥∈ℛ} is called support of 𝑎̃. In the following of the paper, we denote 𝑎^𝑅𝛼≜sup𝑎_𝛼̃, 𝑎^𝐿𝛼≜inf𝑎_𝛼̃, and 𝑎_𝛼̃=[𝑎^𝐿𝛼,𝑎^𝑅𝛼].

Definition 2.3. Let 𝑓∶[𝑎,𝑏]→[𝑐,𝑑] be a monotone function, where [𝑎,𝑏] and [𝑐,𝑑] are closed subintervals of the extended real line [−∞,+∞]. The pseudoinverse 𝑓⁽⁻¹⁾(𝑦)∶[𝑐,𝑑]→[𝑎,𝑏] of 𝑓 is defined by 𝑓⁽⁻¹⁾(𝑦)=⎧⎪⎨⎪⎩sup{𝑥∈[𝑎,𝑏]∣𝑓(𝑥)<𝑦},sup{𝑥∈[𝑎,𝑏]∣𝑓(𝑥)≥𝑦},if𝑓(𝑎)<𝑓(𝑏),𝑎,if𝑓(𝑎)>𝑓(𝑏),if𝑓(𝑎)=𝑓(𝑏).(2.2)

Example 2.4. Let 𝑅(𝑡) be a function defined as Definition 2.1, the graph of 𝑅(𝑡) is given in Figure 1. The graph of the pseudoinverse 𝑅⁽⁻¹⁾(𝑡) is given in Figure 2. These pictures also indicate how to construct the pseudoinverse of a non-constant monotone function.

Figure 1 

Monotone function 𝑅(𝑡).

Figure 2 

𝑅(𝑡) and its pseudoinverse.

Remark 2.5. According to Remark 2.2 and Definition 2.3, the domains of 𝐿⁽⁻¹⁾(𝑡) and 𝑅⁽⁻¹⁾(𝑡) are [0,1].
Next, we introduce the extension principle that Zadeh proposed in [15], which has become an important tool in fuzzy theory and its applications.

Definition 2.6. (i) Let 𝑋, 𝑌 be two crisp set, ℱ(𝑋) and ℱ(𝑌) are two fuzzy variable set defined on 𝑋 and 𝑌. The function 𝑓∶𝑋→𝑌 induces another function 𝑓∶ℱ(𝑋)→ℱ(𝑌) defined on each fuzzy set 𝑢̃ on 𝑋 by 𝑓(𝑢)̃(𝑦)=sup_{𝑥∈𝑋,𝑓(𝑥)=𝑦}𝑢(𝑥).(2.3) Let 𝑋_𝑖, 𝑖=1,2,…,𝑛, 𝑌 be crisp set, ℱ(∏^𝑛𝑖=1𝑋_𝑖) and ℱ(𝑌) are two fuzzy variable set defined on ∏^𝑛𝑖=1𝑋_𝑖 and 𝑌. The function 𝑓∶𝑋→𝑌 induces another function 𝑓∶ℱ(∏^𝑛𝑖=1𝑋_𝑖)→ℱ(𝑌) defined on each fuzzy set on ∏^𝑛𝑖=1𝑋_𝑖 by 𝑓𝑢₁̃,𝑢₂̃,…,𝑢_𝑛̃(𝑦)=sup_{𝑓𝑥1,𝑥2,…,𝑥𝑛=𝑦}min_𝑖𝑢_𝑖𝑥_𝑖.(2.4)

Based on the extension principle, we introduce the following definitions.

Definition 2.7. Let 𝑎,̃𝑏̃∈ℱ be 𝐿-𝑅 fuzzy variables, 𝑐∈ℛ, then, the membership function of the sum of two fuzzy variables 𝑎̃ and 𝑏̃, the scalar product of 𝑎̃ and 𝑐 are defined as following:(i)𝜇_̃𝑎+𝑏̃(𝑥)=sup^{𝑥=𝜇+𝜐}min{𝜇_̃𝑎,𝜇_̃𝑏},(ii)𝜇_𝑐𝑎(̃𝑥)=max{sup^{𝑥=𝑐𝜇}𝜇_̃𝑎,0}, with sup{𝜙}=−∝.

Lemma 2.8. Let ã‌=(𝑎,𝑙,𝑟)_𝐿-𝑅, 𝑏̃=(𝑏,𝑚,𝑛)_𝐿-𝑅∈ℱ be 𝐿-𝑅 fuzzy variables, 𝑐∈ℛ⁺, it holds that(i)𝑐𝑎̃=(𝑐𝑎,𝑐𝑙,𝑐𝑟)_𝐿-𝑅,(ii)𝑎̃+𝑏̃=(𝑎+𝑏,𝑙+𝑚,𝑟+𝑛)_𝐿-𝑅.

Definition 2.9. Let ℛ^𝑛 be 𝑛-dimensional Euclidean space, 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℛ^𝑛, 𝑥_𝑖∈ℛ (𝑖=1,2,…,𝑛).(i)𝑥≦𝑦 if and only if 𝑥_𝑖⩽𝑦_𝑖 holds,(ii)𝑥≤𝑦 if and only if 𝑥≦𝑦 and 𝑥≠𝑦 holds,(iii)𝑥<𝑦 if and only if 𝑥_𝑖<𝑦_𝑖 holds.

Definition 2.10 (see [14]). Let 𝑎,̃𝑏̃∈ℱ be 𝐿-𝑅 fuzzy variables, then,(i)𝑏̃ dominates 𝑎̃ (denote by 𝑎̃⪷𝑏̃) if and only if (𝑎^𝐿𝛼,𝑎^𝑅𝛼)≦(𝑏^𝐿𝛼,𝑏^𝑅𝛼) holds with 𝛼∈[0,1],(ii)𝑏̃ strictly dominates 𝑎̃ (denote by 𝑎̃≼𝑏̃) if and only if 𝑎̃⪷𝑏̃ and (𝑎^𝐿𝛼,𝑎^𝑅𝛼)≤(𝑏^𝐿𝛼,𝑏^𝑅𝛼) holds with 𝛼∈[0,1),(iii)𝑏̃ strongly dominates 𝑎̃ (denote by 𝑎̃≺𝑏̃) if and only if (𝑎^𝐿𝛼,𝑎^𝑅𝛼)<(𝑏^𝐿𝛼,𝑏^𝑅𝛼) holds with 𝛼∈[0,1],(iv)𝑏̃ is equal to 𝑎̃ (denote by 𝑎̃=𝑏̃) if and only if (𝑎^𝐿𝛼,𝑎^𝑅𝛼)=(𝑏^𝐿𝛼,𝑏^𝑅𝛼) holds with 𝛼∈[0,1].

Theorem 2.11. Let 𝑎̃=(𝑎,𝑙,𝑟)_𝐿-𝑅,𝑏̃=(𝑏,𝑚,𝑛)_𝐿-𝑅∈ℱ be 𝐿-𝑅 fuzzy variables,(i)𝑎̃⪷𝑏̃ if and only if max{𝑚−𝑙,0}⩽𝑏−𝑎 and max{𝑟−𝑛,0}⩽𝑏−𝑎 hold,(ii)𝑎̃≺𝑏̃ if and only if max{𝑚−𝑙,0}<𝑏−𝑎 and max{𝑟−𝑛,0}<𝑏−𝑎 hold.

Proof. Just prove (i), the proof of (ii) is similar to (i). Let 𝜇_̃𝑎(𝑥)=⎧⎪⎨⎪⎩𝐿𝑎−𝑙𝑥𝑅𝑥−𝑎,𝑥≤𝑎,𝑙≥0,𝑟,𝑥≥𝑎,𝑟≥0,𝜇^̃𝑏(𝑥)=⎧⎪⎨⎪⎩𝐿𝑅𝑏𝑥−𝑚−𝑥𝑏,𝑥≤𝑏,𝑚≥0,𝑛,𝑥≥𝑏,𝑛≥0,(2.5) be the membership function of fuzzy variables 𝑎̃ and 𝑏̃ (see Figure 3), respectively.
From Definition 2.1, for 𝛼∈[0,1], it holds that 𝛼=𝐿𝑎−𝑥𝑙=𝐿𝑏−𝑚𝑦=𝑅𝑥−𝑟𝑎=𝑅𝑥−𝑛𝑏.(2.6) By 𝐿(0)=𝑅(0)=1, setting 𝛼=1 it holds that 𝑎^𝐿1=𝑎, 𝑏^𝑅1=𝑏. Combining these with Definition 2.10, it implies that 𝑎≤𝑏.(2.7) By 𝐿(1)=𝑅(1)=0, setting 𝛼=0, it holds that 𝑎^𝐿0=𝑎−𝑙, 𝑎^𝑅0=𝑎+𝑟; 𝑏^𝐿0=𝑏−𝑚, 𝑏^𝑅0=𝑏+𝑛. From Definition 2.10, it implies 𝑎^𝐿0=𝑎−𝑙≤𝑏^𝐿0=𝑏−𝑚,𝑎^𝑅0=𝑎+𝑟≤𝑏^𝑅0=𝑏+𝑛.(2.8) Combining (2.7) and (2.8), we have mmaaxx{{𝑚𝑟−−𝑛𝑙,,00}}⩽⩽𝑏𝑏−−𝑎𝑎.,(2.9)
Conversely, for any 𝛼∈[0,1], it holds (2.6). Using the pseudoinverse of 𝐿 and 𝑅, we obtain 𝑏𝑎̃_𝛼̃=𝑎^𝐿𝛼,𝑎^𝑅𝛼=𝑎−𝑙𝐿⁽⁻¹⁾(𝛼),𝑎+𝑟𝑅⁽⁻¹⁾(𝛼)_𝛼,=𝑏^𝐿𝛼,𝑏^𝑅𝛼=𝑏−𝑚𝐿⁽⁻¹⁾(𝛼),𝑏+𝑛𝑅⁽⁻¹⁾(𝛼).(2.10) Then 𝑏𝑏^𝐿𝛼−𝑎^𝐿𝛼=𝑏−𝑚𝐿⁽⁻¹⁾(𝛼)−𝑎+𝑙𝐿⁽⁻¹⁾(𝛼)=(𝑏−𝑎)−(𝑚−𝑙)𝐿⁽⁻¹⁾(𝛼),_𝑅𝛼−𝑎^𝑅𝛼=𝑏+𝑛𝑅⁽⁻¹⁾(𝛼)−𝑎−𝑟𝑅⁽⁻¹⁾(𝛼)=(𝑏−𝑎)−(𝑟−𝑛)𝑅⁽⁻¹⁾(𝛼).(2.11) From (i) of this Theorem and Remark 2.5, it obviously that 𝑎^𝐿𝛼≤𝑏^𝐿𝛼 and 𝑎^𝑅𝛼≤𝑏^𝑅𝛼, in other words, 𝑎̃⪷𝑏̃.

Figure 3 

the membership function of fuzzy variables 𝑎̃ and 𝑏̃.

Lemma 2.12 (see [16]). Let 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛) be any usual continuous function with respect to (𝑥₁,𝑥₂,…,𝑥_𝑛), 𝑥̃_𝑖, 𝑖=1,2,…,𝑛, be fuzzy variables, and 𝑥̃_𝑖𝛼=[𝑥^𝐿𝛼𝑖,𝑥^𝐿𝛼𝑖], 𝑖=1,2,…,𝑛, be corresponding 𝛼-level sets, respectively. Then, for the 𝛼-level set of the function 𝑓(𝑥̃₁,𝑥̃₂,…,𝑥̃_𝑛) with fuzzy variables, the following relation holds: 𝑓𝑥̃¹,𝑥̃₂,…,𝑥̃_𝑛_𝛼=𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∈ℛ∣𝑥_𝑖∈𝑥^𝐿𝑖𝛼,𝑥^𝑅𝑖𝛼,𝑖=1,2,…,𝑛.(2.12) In short, one denotes {𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℛ∣𝑥_𝑖∈[𝑥^𝐿𝑖𝛼,𝑥^𝑅𝑖𝛼],𝑖=1,2,…,𝑛} by 𝑓([𝑥^𝐿𝑖𝛼,𝑥^𝑅𝑖𝛼]), 𝑖=1,2,…,𝑛.

3. Equilibrium Strategies of Bimatrix Games with Fuzzy Payoffs

When we apply the game theory to model some practical problems which we encounter in real world, we have to find the values of payoffs exactly. However, it is difficult to know the exact values of payoffs and we just know the values of payoffs approximately. In such situations, we should model the payoffs of the player as a fuzzy variable, so the expected payoffs of the game should be fuzzy valued. Since there are no concepts of equilibrium strategies to be accepted widely, it is an important task to define the concepts of equilibrium strategies and investigate their properties in fuzzy viroment. In this paper, the payoffs of strategy pair (𝑖,𝑗)∈𝑆_𝐼×𝑆_𝐽 will be modeled as 𝐿-𝑅 fuzzy variable, such as 𝑎̃=(𝑎,𝑙,𝑟)_𝐿-𝑅, 𝑏̃=(𝑏,ℎ,𝑧)_𝐿-𝑅∈ℱ.

Definition 3.1. Let 𝑀={1,2,…,𝑚} and 𝑁={1,2,…,𝑛} be sets of all pure strategies of Player 𝐼 and Player 𝐽, respectively. Their mixed strategies are probability distribution on their pure strategies. The set of mixed strategies for Player I is represented by 𝑆_𝐼=𝑥₁,𝑥₂,…,𝑥_𝑚∈ℛ^𝑚∣𝑥_𝑖⩾0,𝑖=1,2,…,𝑚,^𝑚_𝑖=1𝑥_𝑖=1,(3.1) where ℛ^𝑚 is a set of 𝑚-dimensional real numbers space. Similarly, the set of mixed strategies for Player 𝐽 is represented by 𝑆_𝐽=𝑦₁,𝑦₂,…,𝑦_𝑛∈ℛ^𝑛∣𝑦_𝑖⩾0,𝑖=1,2,…,𝑛,^𝑛_𝑖=1𝑦_𝑖=1.(3.2)

Definition 3.2. Let Player 𝐼 choose a mixed strategy 𝑥∈𝑆_𝐼 and Player 𝐽 choose a mixed strategy 𝑦∈𝑆_𝐽. A game is said to be fuzzy bimatrix game if 𝑎̃_𝑖𝑗 represents the income of player 𝐼 and 𝑏̃_𝑖𝑗 represent the incomes of player 𝐽, denoted by Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵), where, the membership function of the payoffs will be given based on the experts’ assessment. The expected value 𝐸(𝑥,𝑦)=𝑥^𝑇𝐴𝑦=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑥_𝑖𝑎̃_𝑖𝑗𝑦_𝑗 and 𝐸(𝑥,𝑦)=𝑥^𝑇𝐵𝑦=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑥_𝑖𝑏̃_𝑖𝑗𝑦_𝑗 are called the expected of players. The matrix: 𝐴=⎛⎜⎜⎜⎜⎝𝑎̃¹¹⋯𝑎̃_1𝑛𝑎⋮̃_𝑚1⋱⋮⋯𝑎̃_𝑚𝑛⎞⎟⎟⎟⎟⎠,𝐵=⎛⎜⎜⎜⎜⎝𝑏̃₁₁⋯𝑏̃_1𝑛𝑏⋮̃⋱⋮_𝑚1⋯𝑏̃_𝑚𝑛⎞⎟⎟⎟⎟⎠(3.3) is payoff matrix of Player 𝐼 and Player 𝐽, respectively.

In the following of this paper, we denote 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑛, 𝐿=(𝑙_𝑖𝑗)_𝑚×𝑛, 𝑅=(𝑟_𝑖𝑗)_𝑚×𝑛, 𝐴^𝐿0=𝐴−𝐿, 𝐴^𝑅0=𝐴+𝑅, 𝐵=(𝑏_𝑖𝑗)_𝑚×𝑛, 𝐻=(ℎ_𝑖𝑗)_𝑚×𝑛, 𝑍=(𝑧_𝑖𝑗)_𝑚×𝑛, 𝐵^𝐿0=𝐵−𝐻 and 𝐵^𝑅0=𝐵+𝑍.

Definition 3.3 (see [12]). A pair(𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is called Nash equilibrium strategy of fuzzy bimatrix game Γ, if it holds that(i)𝑥^𝑇𝐴𝑦^∗⪷𝑥^∗𝑇𝐴𝑦^∗,𝑥∈𝑆_𝐼, (ii)𝑥^∗𝑇𝐵𝑦⪷x^∗𝑇𝐵𝑦^∗,𝑦∈𝑆_𝐽.

Definition 3.4 (see [12]). A pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is called Pareto equilibrium strategy of fuzzy bimatrix game Γ, if it holds that(i)there exists no 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴𝑦^∗≼𝑥^𝑇𝐴𝑦^∗,(ii)there exists no 𝑦∈𝑆_𝐽 such that 𝑥^∗𝑇𝐵𝑦^∗≼𝑥^∗𝑇𝐵𝑦.

Definition 3.5 (see [12]). A pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is called weak Pareto equilibrium strategy of fuzzy bimatrix gameΓ, if it holds that(i)there exists no 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴𝑦^∗≺𝑥^𝑇𝐴𝑦^∗,(ii)there exists no 𝑦∈𝑆_𝐽 such that 𝑥^∗𝑇𝐵𝑦^∗≺𝑥^∗𝑇𝐵𝑦.

It is obvious that, if the payoffs of fuzzy bimatrix games are all crisp number, then the above three definitions coincide with the equilibrium of a crisp bimatrix games. Therefore, these definitions are natural generalization of the crisp bimatrix games.

Lemma 3.6. Let 𝐴=(𝑎̃_𝑖𝑗)_𝑚×𝑛 be one of fuzzy payoff matrix of fuzzy bimatrix game Γ, for 𝑥∈𝑆_𝐼, 𝑦∈𝑆_𝐽, 𝛼∈[0,1], it holds that(i)𝑥^𝑇𝐴𝑦∈ℱ, (ii)(𝑥^𝑇𝐴𝑦)_𝛼=[𝑥^𝑇𝐴^𝐿𝛼𝑦,𝑥^𝑇𝐴^𝑅𝛼𝑦].

Proof. It is easy to obtain (i) from Lemma 2.8. Hence we just prove (ii). Since 𝑥^𝑇𝐴𝑦 is a liner function with respect to 𝑎_𝑖𝑗, hence it is continuous and increased with respect to 𝑎_𝑖𝑗, from Lemma 2.12, it is obvious that 𝑥^𝑇𝐴𝑦_𝛼=^𝑚_𝑛𝑖=1_𝑗=1𝑥_𝑖𝑎̃_𝑖𝑗𝑦_𝑗_𝛼=^𝑚_𝑛𝑖=1_𝑗=1=𝑡_𝑖𝑗∣𝑡_𝑖𝑗∈𝑥_𝑖𝑎̃_𝑖𝑗𝑦_𝑗_𝛼,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛_𝑚_𝑛𝑖=1_𝑗=1=𝑥_𝑖𝑝_𝑖𝑗𝑦_𝑗∣𝑝_𝑖𝑗∈𝑎̃_𝑖𝑗_𝛼,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛_𝑚_𝑛𝑖=1_𝑗=1=𝑥_𝑖𝑝_𝑖𝑗𝑦_𝑗∣𝑝_𝑖𝑗∈𝑎_𝑖𝑗−𝐿⁽⁻¹⁾(𝛼)𝑙_𝑖𝑗,𝑎_𝑖𝑗+𝑅⁽⁻¹⁾(𝛼)𝑟_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛_𝑚_𝑛𝑖=1_𝑗=1𝑥_𝑖𝑎_𝑖𝑗−𝐿⁽⁻¹⁾(𝛼)𝑙_𝑖𝑗𝑦_𝑗,^𝑚_𝑛𝑖=1_𝑗=1𝑥_𝑖𝑎_𝑖𝑗+𝑅⁽⁻¹⁾(𝛼)𝑟_𝑖𝑗𝑦_𝑗=𝑥_𝑇𝐴^𝐿𝛼𝑦,𝑥^𝑇𝐴^𝑅𝛼𝑦.(3.4) The theorem is proved.

Theorem 3.7. Let Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) be a bimatrix game with fuzzy payoffs, a pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the Nash equilibrium strategy of game Γ if and only if the following inequalities hold with every 𝑥∈𝑆_𝐼, 𝑦∈𝑆_𝐽𝑥^𝑇𝐴𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗,𝑥^∗𝑇𝐵𝑦⩽𝑥^∗𝑇𝐵𝑦^∗,(3.5)𝑥^𝑇𝐴𝑦^∗−𝑥^𝑇𝐿𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^∗𝑇𝐿𝑦^∗,𝑥^∗𝑇𝐵𝑦−𝑥^∗𝑇𝐻𝑦⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐻𝑦^∗,(3.6)𝑥^𝑇𝐴𝑦^∗+𝑥^𝑇𝑅𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗+𝑥^∗𝑇𝑅𝑦^∗,𝑥^∗𝑇𝐵𝑦+𝑥^∗𝑇𝑍𝑦⩽𝑥^∗𝑇𝐵𝑦^∗+𝑥^∗𝑇𝑍𝑦^∗.(3.7)

Proof. Let the pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 be Nash equilibrium of game Γ. According to Theorem 2.11 and Definition 3.3, we have max𝑥^∗𝑇𝐿𝑦^∗−𝑥^𝑇𝐿𝑦^∗,0⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,(3.8)max𝑥^𝑇𝑅𝑦^∗−𝑥^∗𝑇𝑅𝑦^∗,0⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,(3.9)max𝑥^∗𝑇𝐻𝑦^∗−𝑥^∗𝑇𝐻𝑦,0⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐵𝑦,(3.10)max𝑥^∗𝑇𝑍𝑦−𝑥^∗𝑇𝑍𝑦^∗,0⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐵𝑦.(3.11) From (3.8), it indicates that 𝑥^∗𝑇𝐿𝑦^∗−𝑥^𝑇𝐿𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,0⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗.(3.12) By rearranging, it holds 𝑥^𝑇𝐴𝑦^∗−𝑥^𝑇𝐿𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^∗𝑇𝐿𝑦^∗,𝑥^𝑇𝐴𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗.(3.13) In the same way, form (3.9), it holds that 𝑥^𝑇𝑅𝑦^∗−𝑥^∗𝑇𝑅𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗.(3.14) By rearranging, it holds 𝑥^𝑇𝑅𝑦^∗+𝑥^𝑇𝐴𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗+𝑥^∗𝑇𝑅𝑦^∗.(3.15)
By a similar way, from form (3.10) and (3.11), we have 𝑥^∗𝑇𝐵𝑦−𝑥𝑥^∗𝑇_∗𝑇𝐻𝑦⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐻𝑦^∗,𝑥^∗𝑇𝐵𝑦⩽𝑥^∗𝑇𝐵𝑦^∗,𝑍𝑦+𝑥^∗𝑇𝐵𝑦⩽𝑥^∗𝑇𝐵𝑦^∗+𝑥^∗𝑇𝑍𝑦^∗.(3.16) By rearranging (3.13), (3.15), and (3.16) we get (3.5), (3.6), and (3.7).
Otherwise, according to (3.5), it holds, 0⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,(3.17)0⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^∗𝑇𝐵𝑦.(3.18) From (3.6), it implies that 𝑥^∗𝑇𝐿𝑦^∗−𝑥^𝑇𝐿𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,(3.19)𝑥^∗𝑇𝐻𝑦^∗−𝑥^𝑇𝐻𝑦^∗⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^𝑇𝐵𝑦^∗.(3.20) From (3.7), it follows that 𝑥^𝑇𝑅𝑦^∗−𝑥^∗𝑇𝑅𝑦^∗⩽𝑥^∗𝑇𝐴𝑦^∗−𝑥^𝑇𝐴𝑦^∗,(3.21)𝑥^𝑇𝑍𝑦^∗−𝑥^∗𝑇𝑍𝑦^∗⩽𝑥^∗𝑇𝐵𝑦^∗−𝑥^𝑇𝐵𝑦^∗.(3.22) Combining (3.17), (3.19), and (3.21), it holds (3.8) and (3.9). Combining (3.18), (3.20), and (3.22), it holds (3.10) and (3.11). Therefore, we have Definition 3.3 by Theorem 2.11.

By the above result, from 𝑥^𝑇𝐴𝑦−𝑥^𝑇𝐿𝑦=𝑥^𝑇𝐴^𝐿0𝑦, 𝑥^𝑇𝐴𝑦+𝑥^𝑇𝑅𝑦=𝑥^𝑇𝐴^𝑅0𝑦, 𝑥^𝑇𝐵𝑦−𝑥^𝑇𝐻𝑦=𝑥^𝑇𝐵^𝐿0𝑦, 𝑥^𝑇𝐵𝑦+𝑥^𝑇𝑍𝑦=𝑥^𝑇𝐵^𝑅0𝑦 with Theorem 3.7, we conclude that a fuzzy bimatrix game is equivalent to the following three crisp matrix games Γ_𝑙≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴^𝐿0,𝐵^𝐿0), Γ_𝑐≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵), Γ_𝑟≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴^𝑅0,𝐵^𝑅0). Then the following holds.

Corollary 3.8. A pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is equilibrium strategy of bimatrix game Γ if and only if the pair is still to be the equilibrium strategy of these games Γ_𝑙, Γ_c, and Γ_𝑟.

Remark 3.9. From above existence conditions of equilibrium strategy, it is obvious that the equilibrium strategy of fuzzy bimatrix games still is equilibrium strategy of three crisp bimatrix games. It is difficult to satisfy these conditions at the same time, but in following cases, there exists a equilibrium strategy of the bimatrix games with fuzzy payoffs.

Case 1. The payoff matrix of the bimatrix games are as follows: 𝑎_𝑖𝑗=𝑘₁𝑙_𝑖𝑗,𝑎_𝑖𝑗=𝑘₂𝑟_𝑖𝑗,𝑏_𝑖𝑗=𝑘₃ℎ_𝑖𝑗,𝑏_𝑖𝑗=𝑘₄𝑧_𝑖𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,(3.23) where 𝑘₁,𝑘₂,𝑘₃,𝑘₄∈(0,1]. In other words, 𝐴=𝑘₁𝐿, 𝐴=𝑘₂𝑅, 𝐵=𝑘₃𝐻, 𝐵=𝑘₄𝑍.

Case 2. If 𝑙_𝑖𝑗=𝑙,𝑟_𝑖𝑗=𝑟,ℎ_𝑖𝑗=ℎ,𝑧_𝑖𝑗=𝑧,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, where 𝑙,𝑟,ℎ,𝑧∈ℛ.

Remark 3.10. In these cases, the fuzzy bimatrix game Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) is equivalent to the crisp bimatrix game Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵).

Theorem 3.11. Let Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) be a fuzzy bimatrix game, 𝑇_𝐼(Γ), and 𝑇_𝐽(Γ) are the sets of the optimal strategy of Player 𝐼 and Player 𝐽, respectively. Then, 𝑇_𝐼(Γ), and 𝑇_𝐽(Γ) are both nonempty closed convex set if they are nonempty.

Proof. We just present the proof of the closed convex set of 𝑇_𝐼(Γ), the proof of the closed convex set of 𝑇_𝐽(Γ) is similar to the proof of 𝑇_𝐼(Γ).
Let us assume 𝑥^∗∈𝑇_𝐼(Γ) and 𝑦^∗∈𝑇_𝐽(Γ), then the pair (𝑥^∗,𝑦^∗) is a equilibrium strategy of the fuzzy bimatrix games. From Definition 3.3, it is obvious that 𝑥𝑥^∗𝑇_𝑇𝐴𝑦⪷𝑥^∗𝑇𝐴𝑦^∗,𝑦∈𝑆_𝐽,𝐵𝑦^∗⪷𝑥^∗𝑇𝐵𝑦^∗,𝑥∈𝑆_𝐼.(3.24) By Theorem 3.7 and Remark 3.9, the pair (𝑥^∗,𝑦^∗) is also the equilibrium strategy of three crisp bimatrix games Γ_𝑙≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴^𝐿0,𝐵^𝐿0), Γ_𝑐≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵), Γ_𝑟≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴^𝑅0,𝐵^𝑅0). In other words, 𝑥^∗∈𝑇_𝐼(Γ_𝑙), 𝑥^∗∈𝑇_𝐼(Γ_𝑐), 𝑥^∗∈𝑇_𝐼(Γ_𝑟), then, 𝑥^∗∈𝑇_𝐼(Γ_𝑙)⋂𝑇_𝐼(Γ_𝑐)⋂𝑇_𝐼(Γ_𝑟).
On the other hand, 𝑇_𝐼(Γ_𝑙), 𝑇_𝐼(Γ_𝑐), and 𝑇_𝐼(Γ_𝑙) are all closed convex set. Therefore, 𝑇_𝐼(Γ) is a closed convex set.

Theorem 3.12. Let Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) be a fuzzy bimatrix game, a pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the Pareto equilibrium strategy of game Γ if and only if(i)there exists no 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴^𝐿0,𝐴^𝑅0𝑦^∗≤𝑥^𝑇𝐴^𝐿0,𝐴^𝑅0𝑦^∗,𝑥^∗𝑇𝐴𝑦^∗≤𝑥^𝑇𝐴𝑦^∗,(3.25)(ii)there exists no 𝑦∈𝑆_𝐽 such that 𝑥^∗𝑇𝐵^𝐿0,𝐵^𝑅0𝑦^∗≤𝑥^∗𝑇𝐵^𝐿0,𝐵^𝑅0𝑦,𝑥^∗𝑇𝐵𝑦^∗≤𝑥^∗𝑇𝐵𝑦.(3.26)

Proof. Let 𝜇𝑎̃_𝑖𝑗(𝑥)=⎧⎪⎨⎪⎩𝐿𝑎^𝑖𝑗−𝑥𝑙_𝑖𝑗,𝑥≤𝑎^𝑖𝑗𝑅𝑥−𝑎_𝑖𝑗,𝑙_𝑖𝑗≥0,𝑟_𝑖𝑗,𝑥≥𝑎^𝑖𝑗,𝑟_𝑖𝑗≥0,(3.27) be the membership function of 𝑎̃_𝑖𝑗, which is the payoff of Player 𝐼 based on pure strategy pair (𝑖,𝑗). If the pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the Pareto equilibrium of the game Γ. we assume there exists a strategy 𝑥∈𝑆_𝐼 such that (3.25) holds, that is to say, 𝑥^∗𝑇(𝐴−𝐿,𝐴+𝑅)𝑦^∗≤𝑥^𝑇(𝐴−𝐿,𝐴+𝑅)𝑦^∗,𝑥^∗𝑇𝐴𝑦^∗≤𝑥^𝑇𝐴𝑦^∗,(3.28) It implies the following from above: 𝑥^∗𝑇(𝐴−𝐿)𝑦^∗≤𝑥^𝑇(𝐴−𝐿)𝑦^∗,𝑥^∗𝑇(𝐴+𝑅)𝑦^∗≤𝑥^𝑇(𝐴+𝑅)𝑦^∗.(3.29) Furthermore, the above two inequalities do not occur simultaneously. Let 𝛼∈[0,1], then 𝐿⁽⁻¹⁾(𝛼)∈[0,1] and 𝑅⁽⁻¹⁾(𝛼)∈[0,1] based on Remark 2.5. From above inequalities, it indicates that 𝑥≤^∗𝑇1−𝐿⁽⁻¹⁾(𝛼)𝐴+𝐿⁽⁻¹⁾(𝛼)(𝐴−𝐿)𝑦^∗,𝑥^∗𝑇1−𝑅⁽⁻¹⁾(𝛼)𝐴+𝑅⁽⁻¹⁾(𝛼)(𝐴+𝑅)𝑦^∗𝑥^𝑇1−𝐿⁽⁻¹⁾(𝛼)𝐴+𝐿⁽⁻¹⁾(𝛼)(𝐴−𝐿)𝑦^∗,𝑥^𝑇1−𝑅⁽⁻¹⁾(𝛼)𝐴+𝑅⁽⁻¹⁾(𝛼)(𝐴+𝑅)𝑦^∗.(3.30) By rearranging, it implies 𝑥^∗𝑇≤𝐴−𝐿⁽⁻¹⁾(𝛼)𝐿𝑦^∗,𝑥^∗𝑇𝐴+𝑅⁽⁻¹⁾(𝛼)𝑅𝑦^∗𝑥^𝑇𝐴−𝐿⁽⁻¹⁾(𝛼)𝐿𝑦^∗,𝑥^𝑇𝐴+𝑅⁽⁻¹⁾(𝛼)𝑅𝑦^∗,(3.31) that is, 𝑥^∗𝑇𝐴𝑦^∗≼𝑥^𝑇𝐴𝑦^∗, by the Definition 2.10. This is a contradiction.
By the manner similar to above, there exists no 𝑦 such that (3.26) holds.
Otherwise, let the pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 satisfies (3.25), (3.26). We assume that there exists a strategy 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴𝑦^∗≼𝑥^𝑇𝐴𝑦^∗ holds. From Definition 2.10, for 𝛼∈[0,1), it is obvious that 𝑥^∗𝑇𝐴^𝐿𝛼,𝐴^𝑅𝛼𝑦^∗≤𝑥^𝑇𝐴^𝐿𝛼,𝐴^𝑅𝛼𝑦^∗.(3.32) Let 𝛼 incline to 1 and set 𝛼=0, respectively, it holds 𝑥^∗𝑇𝐴^𝐿0,𝐴^𝑅0𝑦^∗≤𝑥^𝑇𝐴^𝐿0,𝐴^𝑅0𝑦^∗,𝑥^∗𝑇𝐴𝑦^∗≤𝑥^𝑇𝐴𝑦^∗.(3.33) This is a contradiction.
By the similar way, we have there exists no 𝑦∈𝑆_𝐽 such that 𝑥^∗𝑇𝐵𝑦≼𝑥^∗𝑇𝐵𝑦^∗.

Theorem 3.13. Let Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) be a fuzzy bimatrix game, a pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the weak Pareto equilibrium strategy of game Γ if and only if(i) there exists no 𝑥∈𝑆_𝐼 such that𝑥^∗𝑇𝐴^𝐿0,𝐴,𝐴^𝑅0𝑦^∗<𝑥^𝑇𝐴^𝐿0,𝐴,𝐴^𝑅0𝑦^∗,(3.34)(ii) there exists no 𝑦∈𝑆_𝐽 such that𝑥^∗𝑇𝐵^𝐿0,𝐵,𝐵^𝑅0𝑦^∗<𝑥^∗𝑇𝐵^𝐿0,𝐵,𝐵^𝑅0𝑦.(3.35)

Proof. The proof of this theorem is similar to the Theorem 3.12.

In the following of this paper, we will characterize the crisp parametric matrix games. Then other types of equilibrium strategy of fuzzy bimatrix games will be investigated through the crisp parametric matrix games.

Let Γ≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴,𝐵) be a fuzzy bimatrix game, while Player 𝐼 chooses pure strategy 𝑖 and Player 𝐽 chooses pure strategy 𝑗, we set (1−𝜆)(𝑎_𝑖𝑗−𝑙_𝑖𝑗)+𝜆(𝑎_𝑖𝑗+𝑟_𝑖𝑗) to be the payoffs of player 𝐼 and −[(1−𝜇)(𝑏_𝑖𝑗−ℎ_𝑖𝑗)+𝜇(𝑏_𝑖𝑗+𝑧_𝑖𝑗)] to be the payoffs of player 𝐽, where 𝜆,𝜇∈[0,1]. The payoff matrixes of player 𝐼 and 𝐽 are 𝐴(𝜆)=(1−𝜆)(𝐴−𝐿)+𝜆(𝐴+𝑅),𝐵(𝜇)=(1−𝜇)(𝐵−𝐻)+𝜇(𝐵+𝑍).(3.36) Then, we have the crisp parametric matrix game Γ(𝜆,𝜇)≡({𝐼,𝐽},𝑆_𝐼,𝑆_𝐽,𝐴(𝜆),𝐵(𝜇)), where 𝜆 and 𝜇 are parameters.

Definition 3.14. For 𝜆,𝜇∈[0,1], the pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is called to be the Nash equilibrium of the crisp parametric bimatrix game Γ(𝜆,𝜇), if it holds that(i)𝑥^𝑇𝐴(𝜆)𝑦^∗⩽𝑥^∗𝑇𝐴(𝜆)𝑦^∗, 𝑥∈𝑆_𝐼,(ii)𝑥^∗𝑇𝐵(𝜇)𝑦⩽𝑥^∗𝑇𝐵(𝜇)𝑦^∗, 𝑦∈𝑆_𝐽.

Lemma 3.15. There exists at least one Nash equilibrium strategy to all crisp parametric bimatrix game Γ(𝜆,𝜇) with 𝜆,𝜇∈[0,1].

Theorem 3.16. A pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the Pareto equilibrium strategy of fuzzy bi-matrix game Γ, it is necessary and sufficient that there exist 𝜆,𝜇∈(0,1) such that (𝑥^∗,𝑦^∗) is the Nash equilibrium strategy of crisp bimatrix game Γ(𝜆,𝜇).

Proof. Let (𝑥^∗,𝑦^∗) be the Nash equilibrium strategy of bimatrix game Γ(𝜆₀,𝜇₀), where 𝜆₀,𝜇₀∈(0,1). For 𝑥∈𝑆_𝐼, from Definition 3.14(i), it holds 1−𝜆₀𝑥^𝑇𝐴^𝐿0𝑦^∗+𝜆₀𝑥^𝑇𝐴^𝑅0𝑦^∗⩽1−𝜆₀𝑥^∗𝑇𝐴^𝐿0𝑦^∗+𝜆₀𝑥^∗𝑇𝐴^𝑅0𝑦^∗.(3.37) For 𝑦∈𝑆_𝐽, from Definition 3.14(ii), it holds 1−𝜇₀𝑥^∗𝑇𝐵^𝐿0𝑦+𝜇₀𝑥^∗𝑇𝐵^𝑅0𝑦⩽1−𝜇₀𝑥^∗𝑇𝐵^𝐿0𝑦^∗+𝜇₀𝑥^∗𝑇𝐵^𝑅0𝑦^∗.(3.38)
First, we assume there exists 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴𝑦^∗≼𝑥^𝑇𝐴𝑦^∗ holds. From Definition 2.10, it holds 𝑥^∗𝑇𝐴^𝐿0𝑦^∗,𝑥^∗𝑇𝐴^𝑅0𝑦^∗≤𝑥^𝑇𝐴^𝐿0𝑦^∗,𝑥^𝑇𝐴^𝑅0𝑦^∗.(3.39) Because 𝑥^∗𝑇𝐴^𝐿0𝑦^∗=𝑥^𝑇𝐴^𝐿0𝑦^∗ and 𝑥^∗𝑇𝐴^𝑅0𝑦^∗=𝑥^𝑇𝐴^𝑅0𝑦^∗ do not occur simultaneously, we have 1−𝜆₀𝑥^∗𝑇𝐴^𝐿0𝑦^∗+𝜆₀𝑥^∗𝑇𝐴^𝑅0𝑦^∗<1−𝜆₀𝑥^𝑇𝐴^𝐿0𝑦^∗+𝜆₀𝑥^𝑇𝐴^𝑅0𝑦^∗,𝜆₀∈(0,1).(3.40) This contradicts (3.37).
By the similar way, we can conclude a contradiction to (3.38).
On the other hand, let (𝑥^∗,𝑦^∗) be the Pareto equilibrium strategy of fuzzy bimatrix game Γ such that conditions (i) and (ii) of Definition 3.4 hold. It show that there is no 𝑥∈𝑆_𝐼 such that 𝑥^∗𝑇𝐴^𝐿0𝑦^∗,𝑥^∗𝑇𝐴^𝑅0𝑦^∗≤𝑥^𝑇𝐴^𝐿0𝑦^∗,𝑥^𝑇𝐴^𝑅0𝑦^∗(3.41) holds. Then, one of the following cases happens,
Case 1. 𝑥^∗𝑇(𝐴−𝐿)𝑦^∗>𝑥^𝑇(𝐴−𝐿)𝑦^∗or𝑥^∗𝑇(𝐴+𝑅)𝑦^∗>𝑥^𝑇(𝐴+𝑅)𝑦^∗;(3.42)Case 2. 𝑥^∗𝑇(𝐴−𝐿)𝑦^∗=𝑥^𝑇(𝐴−𝐿)𝑦^∗,𝑥^∗𝑇(𝐴+𝑅)𝑦^∗=𝑥^𝑇(𝐴+𝑅)𝑦^∗.(3.43)
If Case 1 happens, without loss of generality, we set 𝑥^∗𝑇(𝐴−𝐿)𝑦^∗>𝑥^𝑇(𝐴−𝐿)𝑦^∗. It is obviously that there exist 𝜆₀∈(0,1) tended to zero such that 1−𝜆₀𝑥^∗𝑇(𝐴−𝐿)𝑦^∗+𝜆₀𝑥^∗𝑇(𝐴+𝑅)𝑦^∗>1−𝜆₀𝑥^𝑇(𝐴−𝐿)𝑦^∗+𝜆₀𝑥^𝑇(𝐴+𝑅)𝑦^∗,(3.44) If Case 2 happens, for any 𝜆∈(0,1)(1−𝜆)𝑥^∗𝑇(𝐴−𝐿)𝑦^∗+𝜆𝑥^∗𝑇(𝐴+𝑅)𝑦^∗≥(1−𝜆)𝑥^𝑇(𝐴−𝐿)𝑦^∗+𝜆𝑥^𝑇(𝐴+𝑅)𝑦^∗.(3.45) Both Cases 1 and 2 show that there exists 𝜆₀∈(0,1) such that Definition 3.14 (i) holds.
By the similar way, we have Definition 3.14 (ii).

Theorem 3.17. A pair (𝑥^∗,𝑦^∗)∈𝑆_𝐼×𝑆_𝐽 is the Weak Pareto equilibrium strategy of fuzzy bimatrix game Γ, it is necessary and sufficient that there exist 𝜆,𝜇∈[0,1] such that (𝑥^∗,𝑦^∗) is the Nash equilibrium strategy of crisp bimatrix game Γ(𝜆,𝜇).

Proof. The proof of this theorem is similar to the Theorem 3.16.

From Theorems 3.16, 3.17 and Lemma 3.15, it is easy to get the followings.

Theorem 3.18. Let Γ be fuzzy bimatrix games.The followings hold (i)there exists at least one Pareto equilibrium strategy to game Γ;(ii)there exists at least one weak Pareto equilibrium strategy to game Γ.

4. Conclusion

This paper developed a new theoretical approach to deal with the bimatrix games with fuzzy payoffs. Especially, Maeda’s models were generalized from symmetric triangular variable field to asymmetric L-R fuzzy variable field. By investigating crisp parametric bimatrix games, we presented a method to figure out the Nashe quilibrium strategy and (weak) Pareto equilibrium strategy of fuzzy bimatrix games. It is also easy to find that all kinds of equilibrium strategy that we defined are nature extensions of the Nash equilibrium strategy of crisp bimatrix games.

Recommendations for future work include the development of a procedure to extend the concept of two-person fuzzy games to n-person fuzzy games. Furthermore, incorporating the concept of multiobjective decision making with the definition of fuzzy noncooperative games would allow us to establish the fuzzy multiobjective games, which is desirable for situations involving complex multiobjective decision-making problems.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 71161001) and the Science Foundation of Beifang University of Nationalities (no. 72010Y041).