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Journal of Applied Mathematics
Volume 2012, Article ID 824790, 15 pages
http://dx.doi.org/10.1155/2012/824790
Research Article

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

School of Management, Beifang University of Nationalities, Yinchuan 750021, China

Received 25 February 2012; Accepted 30 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Cun-lin Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ̃𝑎 is a fuzzy set defined on the space of real number, whose membership functions 𝜇̃𝑎[0,1] as following 𝜇̃𝑎𝐿(𝑥)=𝑎𝑥𝑙𝑅,𝑥𝑎,𝑙0,𝑥𝑎𝑟,𝑥𝑎,𝑟0,(2.1) where, 𝐿,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 ̃𝑎=(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 , for ̃𝑎, 𝛼[0,1], ̃𝑎𝛼{𝑥𝜇̃𝑎(𝑥)𝛼,𝑥} is called as 𝛼-level set of ̃𝑎. ̃𝑎0{𝑥𝜇̃𝑎(𝑥)>0,𝑥} is called support of ̃𝑎. In the following of the paper, we denote 𝑎𝑅𝛼sup̃𝑎𝛼, 𝑎𝐿𝛼inf̃𝑎𝛼, and ̃𝑎𝛼=[𝑎𝐿𝛼,𝑎𝑅𝛼].

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{𝑥𝑎,𝑏𝑓(𝑥)<𝑦},if[]𝑓(𝑎)<𝑓(𝑏),sup{𝑥𝑎,𝑏𝑓(𝑥)𝑦},if𝑓(𝑎)>𝑓(𝑏),𝑎,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.

824790.fig.001
Figure 1: Monotone function 𝑅(𝑡).
824790.fig.002
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, (𝑋) and (𝑌) are two fuzzy variable set defined on 𝑋 and 𝑌. The function 𝑓𝑋𝑌 induces another function 𝑓(𝑋)(𝑌) defined on each fuzzy set ̃𝑢 on 𝑋 by 𝑓(̃𝑢)(𝑦)=sup𝑥𝑋,𝑓(𝑥)=𝑦𝑢(𝑥).(2.3) Let 𝑋𝑖, 𝑖=1,2,,𝑛, 𝑌 be crisp set, (𝑛𝑖=1𝑋𝑖) and (𝑌) are two fuzzy variable set defined on 𝑛𝑖=1𝑋𝑖 and 𝑌. The function 𝑓𝑋𝑌 induces another function 𝑓(𝑛𝑖=1𝑋𝑖)(𝑌) defined on each fuzzy set on 𝑛𝑖=1𝑋𝑖 by 𝑓̃𝑢1,̃𝑢2,,̃𝑢𝑛(𝑦)=sup𝑓𝑥1,𝑥2,,𝑥𝑛=𝑦min𝑖𝑢𝑖𝑥𝑖.(2.4)

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

Definition 2.7. Let ̃̃𝑎,𝑏 be 𝐿-𝑅 fuzzy variables, 𝑐, then, the membership function of the sum of two fuzzy variables ̃𝑎 and ̃𝑏, the scalar product of ̃𝑎 and 𝑐 are defined as following:(i)𝜇̃𝑎+̃𝑏(𝑥)=sup𝑥=𝜇+𝜐min{𝜇̃𝑎,𝜇̃𝑏},(ii)𝜇𝑐̃𝑎(𝑥)=max{sup𝑥=𝑐𝜇𝜇̃𝑎,0}, with sup{𝜙}=.

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

Definition 2.9. Let 𝑛 be 𝑛-dimensional Euclidean space, 𝑥=(𝑥1,𝑥2,,𝑥𝑛)𝑛, 𝑥𝑖 (𝑖=1,2,,𝑛).(i)𝑥𝑦 if and only if 𝑥𝑖𝑦𝑖 holds,(ii)𝑥𝑦 if and only if 𝑥𝑦 and 𝑥𝑦 holds,(iii)𝑥<𝑦 if and only if 𝑥𝑖<𝑦𝑖 holds.

Definition 2.10 (see [14]). Let ̃̃𝑎,𝑏 be 𝐿-𝑅 fuzzy variables, then,(i)̃𝑏 dominates ̃𝑎 (denote by ̃𝑏̃𝑎) if and only if (𝑎𝐿𝛼,𝑎𝑅𝛼)(𝑏𝐿𝛼,𝑏𝑅𝛼) holds with 𝛼[0,1],(ii)̃𝑏 strictly dominates ̃𝑎 (denote by ̃𝑏̃𝑎) if and only if ̃𝑏̃𝑎 and (𝑎𝐿𝛼,𝑎𝑅𝛼)(𝑏𝐿𝛼,𝑏𝑅𝛼) holds with 𝛼[0,1),(iii)̃𝑏 strongly dominates ̃𝑎 (denote by ̃𝑏̃𝑎) if and only if (𝑎𝐿𝛼,𝑎𝑅𝛼)<(𝑏𝐿𝛼,𝑏𝑅𝛼) holds with 𝛼[0,1],(iv)̃𝑏 is equal to ̃𝑎 (denote by ̃𝑏̃𝑎=) if and only if (𝑎𝐿𝛼,𝑎𝑅𝛼)=(𝑏𝐿𝛼,𝑏𝑅𝛼) holds with 𝛼[0,1].

Theorem 2.11. Let ̃𝑎=(𝑎,𝑙,𝑟)𝐿-𝑅,̃𝑏=(𝑏,𝑚,𝑛)𝐿-𝑅 be 𝐿-𝑅 fuzzy variables,(i)̃𝑏̃𝑎 if and only if max{𝑚𝑙,0}𝑏𝑎 and max{𝑟𝑛,0}𝑏𝑎 hold,(ii)̃𝑏̃𝑎 if and only if max{𝑚𝑙,0}<𝑏𝑎 and max{𝑟𝑛,0}<𝑏𝑎 hold.

Proof. Just prove (i), the proof of (ii) is similar to (i). Let 𝜇̃𝑎𝐿(𝑥)=𝑎𝑥𝑙,𝑅𝑥𝑎,𝑙0,𝑥𝑎𝑟,𝜇𝑥𝑎,𝑟0,̃𝑏𝐿(𝑥)=𝑏𝑥𝑚𝑅,𝑥𝑏,𝑚0,𝑥𝑏𝑛,𝑥𝑏,𝑛0,(2.5) be the membership function of fuzzy variables ̃𝑎 and ̃𝑏 (see Figure 3), respectively.
From Definition 2.1, for 𝛼[0,1], it holds that 𝛼=𝐿𝑎𝑥𝑙=𝐿𝑏𝑦𝑚=𝑅𝑥𝑎𝑟=𝑅𝑥𝑏𝑛.(2.6) By 𝐿(0)=𝑅(0)=1, setting 𝛼=1 it holds that 𝑎𝐿1=𝑎, 𝑏𝑅1=𝑏. Combining these with Definition 2.10, it implies that 𝑎𝑏.(2.7) By 𝐿(1)=𝑅(1)=0, setting 𝛼=0, it holds that 𝑎𝐿0=𝑎𝑙, 𝑎𝑅0=𝑎+𝑟; 𝑏𝐿0=𝑏𝑚, 𝑏𝑅0=𝑏+𝑛. From Definition 2.10, it implies 𝑎𝐿0=𝑎𝑙𝑏𝐿0=𝑏𝑚,𝑎𝑅0=𝑎+𝑟𝑏𝑅0=𝑏+𝑛.(2.8) Combining (2.7) and (2.8), we have max{𝑚𝑙,0}𝑏𝑎,max{𝑟𝑛,0}𝑏𝑎.(2.9)
Conversely, for any 𝛼[0,1], it holds (2.6). Using the pseudoinverse of 𝐿 and 𝑅, we obtain ̃𝑎𝛼=𝑎𝐿𝛼,𝑎𝑅𝛼=𝑎𝑙𝐿(1)(𝛼),𝑎+𝑟𝑅(1),̃𝑏(𝛼)𝛼=𝑏𝐿𝛼,𝑏𝑅𝛼=𝑏𝑚𝐿(1)(𝛼),𝑏+𝑛𝑅(1).(𝛼)(2.10) Then 𝑏𝐿𝛼𝑎𝐿𝛼=𝑏𝑚𝐿(1)(𝛼)𝑎+𝑙𝐿(1)(𝛼)=(𝑏𝑎)(𝑚𝑙)𝐿(1)𝑏(𝛼),𝑅𝛼𝑎𝑅𝛼=𝑏+𝑛𝑅(1)(𝛼)𝑎𝑟𝑅(1)(𝛼)=(𝑏𝑎)(𝑟𝑛)𝑅(1)(𝛼).(2.11) From (i) of this Theorem and Remark 2.5, it obviously that 𝑎𝐿𝛼𝑏𝐿𝛼 and 𝑎𝑅𝛼𝑏𝑅𝛼, in other words, ̃𝑏̃𝑎.

824790.fig.003
Figure 3: the membership function of fuzzy variables ̃𝑎 and ̃𝑏.

Lemma 2.12 (see [16]). Let 𝑓(𝑥1,𝑥2,,𝑥𝑛) be any usual continuous function with respect to (𝑥1,𝑥2,,𝑥𝑛), ̃𝑥𝑖, 𝑖=1,2,,𝑛, be fuzzy variables, and ̃𝑥𝑖𝛼=[𝑥𝐿𝛼𝑖,𝑥𝐿𝛼𝑖], 𝑖=1,2,,𝑛, be corresponding 𝛼-level sets, respectively. Then, for the 𝛼-level set of the function 𝑓(̃𝑥1,̃𝑥2,,̃𝑥𝑛) with fuzzy variables, the following relation holds: 𝑓̃𝑥1,̃𝑥2,,̃𝑥𝑛𝛼=𝑓𝑥1,𝑥2,,𝑥𝑛𝑥𝑖𝑥𝐿𝑖𝛼,𝑥𝑅𝑖𝛼.,𝑖=1,2,,𝑛(2.12) In short, one denotes {𝑓(𝑥1,𝑥2,,𝑥𝑛)𝑥𝑖[𝑥𝐿𝑖𝛼,𝑥𝑅𝑖𝛼],𝑖=1,2,,𝑛} by 𝑓([𝑥𝐿𝑖𝛼,𝑥𝑅𝑖𝛼]), 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)𝑆𝐼×𝑆𝐽 will be modeled as 𝐿-𝑅 fuzzy variable, such as ̃𝑎=(𝑎,𝑙,𝑟)𝐿-𝑅, ̃𝑏=(𝑏,,𝑧)𝐿-𝑅.

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 𝑆𝐼=𝑥1,𝑥2,,𝑥𝑚𝑚𝑥𝑖0,𝑖=1,2,,𝑚,𝑚𝑖=1𝑥𝑖,=1(3.1) where 𝑚 is a set of 𝑚-dimensional real numbers space. Similarly, the set of mixed strategies for Player 𝐽 is represented by 𝑆𝐽=𝑦1,𝑦2,,𝑦𝑛𝑛𝑦𝑖0,𝑖=1,2,,𝑛,𝑛𝑖=1𝑦𝑖.=1(3.2)

Definition 3.2. Let Player 𝐼 choose a mixed strategy 𝑥𝑆𝐼 and Player 𝐽 choose a mixed strategy 𝑦𝑆𝐽. A game is said to be fuzzy bimatrix game if ̃𝑎𝑖𝑗 represents the income of player 𝐼 and ̃𝑏𝑖𝑗 represent the incomes of player 𝐽, denoted by Γ({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵), where, the membership function of the payoffs will be given based on the experts’ assessment. The expected value 𝐸(𝑥,𝑦)=𝑥𝑇𝐴𝑦=𝑚𝑖=1𝑛𝑗=1𝑥𝑖̃𝑎𝑖𝑗𝑦𝑗 and 𝐸(𝑥,𝑦)=𝑥𝑇𝐵𝑦=𝑚𝑖=1𝑛𝑗=1𝑥𝑖̃𝑏𝑖𝑗𝑦𝑗 are called the expected of players. The matrix: 𝐴=̃𝑎11̃𝑎1𝑛̃𝑎𝑚1̃𝑎𝑚𝑛,̃𝑏𝐵=11̃𝑏1𝑛̃𝑏𝑚1̃𝑏𝑚𝑛(3.3) is payoff matrix of Player 𝐼 and Player 𝐽, respectively.

In the following of this paper, we denote 𝐴=(𝑎𝑖𝑗)𝑚×𝑛, 𝐿=(𝑙𝑖𝑗)𝑚×𝑛, 𝑅=(𝑟𝑖𝑗)𝑚×𝑛, 𝐴𝐿0=𝐴𝐿, 𝐴𝑅0=𝐴+𝑅, 𝐵=(𝑏𝑖𝑗)𝑚×𝑛, 𝐻=(𝑖𝑗)𝑚×𝑛, 𝑍=(𝑧𝑖𝑗)𝑚×𝑛, 𝐵𝐿0=𝐵𝐻 and 𝐵𝑅0=𝐵+𝑍.

Definition 3.3 (see [12]). A pair(𝑥,𝑦)𝑆𝐼×𝑆𝐽 is called Nash equilibrium strategy of fuzzy bimatrix game Γ, if it holds that(i)𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,𝑥𝑆𝐼, (ii)𝑥𝑇𝐵𝑦x𝑇𝐵𝑦,𝑦𝑆𝐽.

Definition 3.4 (see [12]). A pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is called Pareto equilibrium strategy of fuzzy bimatrix game Γ, if it holds that(i)there exists no 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(ii)there exists no 𝑦𝑆𝐽 such that 𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.

Definition 3.5 (see [12]). A pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is called weak Pareto equilibrium strategy of fuzzy bimatrix gameΓ, if it holds that(i)there exists no 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(ii)there exists no 𝑦𝑆𝐽 such that 𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.

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=(̃𝑎𝑖𝑗)𝑚×𝑛 be one of fuzzy payoff matrix of fuzzy bimatrix game Γ, for 𝑥𝑆𝐼, 𝑦𝑆𝐽, 𝛼[0,1], it holds that(i)𝑥𝑇𝐴𝑦, (ii)(𝑥𝑇𝐴𝑦)𝛼=[𝑥𝑇𝐴𝐿𝛼𝑦,𝑥𝑇𝐴𝑅𝛼𝑦].

Proof. It is easy to obtain (i) from Lemma 2.8. Hence we just prove (ii). Since 𝑥𝑇𝐴𝑦 is a liner function with respect to 𝑎𝑖𝑗, hence it is continuous and increased with respect to 𝑎𝑖𝑗, from Lemma 2.12, it is obvious that 𝑥𝑇𝐴𝑦𝛼=𝑚𝑛𝑖=1𝑗=1𝑥𝑖̃𝑎𝑖𝑗𝑦𝑗𝛼=𝑚𝑛𝑖=1𝑗=1𝑡𝑖𝑗𝑡𝑖𝑗𝑥𝑖̃𝑎𝑖𝑗𝑦𝑗𝛼=,𝑖=1,2,,𝑚,𝑗=1,2,,𝑛𝑚𝑛𝑖=1𝑗=1𝑥𝑖𝑝𝑖𝑗𝑦𝑗𝑝𝑖𝑗̃𝑎𝑖𝑗𝛼=,𝑖=1,2,,𝑚,𝑗=1,2,,𝑛𝑚𝑛𝑖=1𝑗=1𝑥𝑖𝑝𝑖𝑗𝑦𝑗𝑝𝑖𝑗𝑎𝑖𝑗𝐿(1)(𝛼)𝑙𝑖𝑗,𝑎𝑖𝑗+𝑅(1)(𝛼)𝑟𝑖𝑗=,𝑖=1,2,,𝑚,𝑗=1,2,,𝑛𝑚𝑛𝑖=1𝑗=1𝑥𝑖𝑎𝑖𝑗𝐿(1)(𝛼)𝑙𝑖𝑗𝑦𝑗,𝑚𝑛𝑖=1𝑗=1𝑥𝑖𝑎𝑖𝑗+𝑅(1)(𝛼)𝑟𝑖𝑗𝑦𝑗=𝑥𝑇𝐴𝐿𝛼𝑦,𝑥𝑇𝐴𝑅𝛼𝑦.(3.4) The theorem is proved.

Theorem 3.7. Let Γ({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵) be a bimatrix game with fuzzy payoffs, a pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is the Nash equilibrium strategy of game Γ if and only if the following inequalities hold with every 𝑥𝑆𝐼, 𝑦𝑆𝐽𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦,(3.5)𝑥𝑇𝐴𝑦𝑥𝑇𝐿𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐿𝑦,𝑥𝑇𝐵𝑦𝑥𝑇𝐻𝑦𝑥𝑇𝐵𝑦𝑥𝑇𝐻𝑦,(3.6)𝑥𝑇𝐴𝑦+𝑥𝑇𝑅𝑦𝑥𝑇𝐴𝑦+𝑥𝑇𝑅𝑦,𝑥𝑇𝐵𝑦+𝑥𝑇𝑍𝑦𝑥𝑇𝐵𝑦+𝑥𝑇𝑍𝑦.(3.7)

Proof. Let the pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 be Nash equilibrium of game Γ. According to Theorem 2.11 and Definition 3.3, we have 𝑥max𝑇𝐿𝑦𝑥𝑇𝐿𝑦,0𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.8)𝑥max𝑇𝑅𝑦𝑥𝑇𝑅𝑦,0𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.9)𝑥max𝑇𝐻𝑦𝑥𝑇𝐻𝑦,0𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦,(3.10)𝑥max𝑇𝑍𝑦𝑥𝑇𝑍𝑦,0𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.(3.11) From (3.8), it indicates that 𝑥𝑇𝐿𝑦𝑥𝑇𝐿𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,0𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦.(3.12) By rearranging, it holds 𝑥𝑇𝐴𝑦𝑥𝑇𝐿𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐿𝑦,𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦.(3.13) In the same way, form (3.9), it holds that 𝑥𝑇𝑅𝑦𝑥𝑇𝑅𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦.(3.14) By rearranging, it holds 𝑥𝑇𝑅𝑦+𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦+𝑥𝑇𝑅𝑦.(3.15)
By a similar way, from form (3.10) and (3.11), we have 𝑥𝑇𝐵𝑦𝑥𝑇𝐻𝑦𝑥𝑇𝐵𝑦𝑥𝑇𝐻𝑦,𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦,𝑥𝑇𝑍𝑦+𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦+𝑥𝑇𝑍𝑦.(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, 0𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.17)0𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.(3.18) From (3.6), it implies that 𝑥𝑇𝐿𝑦𝑥𝑇𝐿𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.19)𝑥𝑇𝐻𝑦𝑥𝑇𝐻𝑦𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.(3.20) From (3.7), it follows that 𝑥𝑇𝑅𝑦𝑥𝑇𝑅𝑦𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.21)𝑥𝑇𝑍𝑦𝑥𝑇𝑍𝑦𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.(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 𝑥𝑇𝐴𝑦𝑥𝑇𝐿𝑦=𝑥𝑇𝐴𝐿0𝑦, 𝑥𝑇𝐴𝑦+𝑥𝑇𝑅𝑦=𝑥𝑇𝐴𝑅0𝑦, 𝑥𝑇𝐵𝑦𝑥𝑇𝐻𝑦=𝑥𝑇𝐵𝐿0𝑦, 𝑥𝑇𝐵𝑦+𝑥𝑇𝑍𝑦=𝑥𝑇𝐵𝑅0𝑦 with Theorem 3.7, we conclude that a fuzzy bimatrix game is equivalent to the following three crisp matrix games Γ𝑙({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴𝐿0,𝐵𝐿0), Γ𝑐({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵), Γ𝑟({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴𝑅0,𝐵𝑅0). Then the following holds.

Corollary 3.8. A pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is equilibrium strategy of bimatrix game Γ if and only if the pair is still to be the equilibrium strategy of these games Γ𝑙, Γc, and Γ𝑟.

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: 𝑎𝑖𝑗=𝑘1𝑙𝑖𝑗,𝑎𝑖𝑗=𝑘2𝑟𝑖𝑗,𝑏𝑖𝑗=𝑘3𝑖𝑗,𝑏𝑖𝑗=𝑘4𝑧𝑖𝑗,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 𝑙𝑖𝑗=𝑙,𝑟𝑖𝑗=𝑟,𝑖𝑗=,𝑧𝑖𝑗=𝑧,𝑖=1,2,,𝑚,𝑗=1,2,,𝑛, where 𝑙,𝑟,,𝑧.

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

Theorem 3.11. Let Γ({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵) be a fuzzy bimatrix game, 𝑇𝐼(Γ), and 𝑇𝐽(Γ) are the sets of the optimal strategy of Player 𝐼 and Player 𝐽, respectively. Then, 𝑇𝐼(Γ), and 𝑇𝐽(Γ) are both nonempty closed convex set if they are nonempty.

Proof. We just present the proof of the closed convex set of 𝑇𝐼(Γ), the proof of the closed convex set of 𝑇𝐽(Γ) is similar to the proof of 𝑇𝐼(Γ).
Let us assume 𝑥𝑇𝐼(Γ) and 𝑦𝑇𝐽(Γ), then the pair (𝑥,𝑦) is a equilibrium strategy of the fuzzy bimatrix games. From Definition 3.3, it is obvious that 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,𝑦𝑆𝐽,𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦,𝑥𝑆𝐼.(3.24) By Theorem 3.7 and Remark 3.9, the pair (𝑥,𝑦) is also the equilibrium strategy of three crisp bimatrix games Γ𝑙({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴𝐿0,𝐵𝐿0), Γ𝑐({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵), Γ𝑟({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴𝑅0,𝐵𝑅0). In other words, 𝑥𝑇𝐼(Γ𝑙), 𝑥𝑇𝐼(Γ𝑐), 𝑥𝑇𝐼(Γ𝑟), then, 𝑥𝑇𝐼(Γ𝑙)𝑇𝐼(Γ𝑐)𝑇𝐼(Γ𝑟).
On the other hand, 𝑇𝐼(Γ𝑙), 𝑇𝐼(Γ𝑐), and 𝑇𝐼(Γ𝑙) are all closed convex set. Therefore, 𝑇𝐼(Γ) is a closed convex set.

Theorem 3.12. Let Γ({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵) be a fuzzy bimatrix game, a pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is the Pareto equilibrium strategy of game Γ if and only if(i)there exists no 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝐿0,𝐴𝑅0𝑦𝑥𝑇𝐴𝐿0,𝐴𝑅0𝑦,𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.25)(ii)there exists no 𝑦𝑆𝐽 such that 𝑥𝑇𝐵𝐿0,𝐵𝑅0𝑦𝑥𝑇𝐵𝐿0,𝐵𝑅0𝑦,𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.(3.26)

Proof. Let 𝜇̃𝑎𝑖𝑗𝐿𝑎(𝑥)=𝑖𝑗𝑥𝑙𝑖𝑗,𝑥𝑎𝑖𝑗,𝑙𝑖𝑗𝑅0,𝑥𝑎𝑖𝑗𝑟𝑖𝑗,𝑥𝑎𝑖𝑗,𝑟𝑖𝑗0,(3.27) be the membership function of ̃𝑎𝑖𝑗, which is the payoff of Player 𝐼 based on pure strategy pair (𝑖,𝑗). If the pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is the Pareto equilibrium of the game Γ. we assume there exists a strategy 𝑥𝑆𝐼 such that (3.25) holds, that is to say, 𝑥𝑇(𝐴𝐿,𝐴+𝑅)𝑦𝑥𝑇(𝐴𝐿,𝐴+𝑅)𝑦,𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦,(3.28) It implies the following from above: 𝑥𝑇(𝐴𝐿)𝑦𝑥𝑇(𝐴𝐿)𝑦,𝑥𝑇(𝐴+𝑅)𝑦𝑥𝑇(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 𝑥𝑇1𝐿(1)(𝛼)𝐴+𝐿(1)𝑦(𝛼)(𝐴𝐿),𝑥𝑇1𝑅(1)(𝛼)𝐴+𝑅(1)𝑦(𝛼)(𝐴+𝑅)𝑥𝑇1𝐿(1)(𝛼)𝐴+𝐿(1)𝑦(𝛼)(𝐴𝐿),𝑥𝑇1𝑅(1)(𝛼)𝐴+𝑅(1)𝑦(𝛼)(𝐴+𝑅).(3.30) By rearranging, it implies 𝑥𝑇𝐴𝐿(1)𝑦(𝛼)𝐿,𝑥𝑇𝐴+𝑅(1)𝑦(𝛼)𝑅𝑥𝑇𝐴𝐿(1)𝑦(𝛼)𝐿,𝑥𝑇𝐴+𝑅(1)𝑦(𝛼)𝑅,(3.31) that is, 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦, 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)𝑆𝐼×𝑆𝐽 satisfies (3.25), (3.26). We assume that there exists a strategy 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦 holds. From Definition 2.10, for 𝛼[0,1), it is obvious that 𝑥𝑇𝐴𝐿𝛼,𝐴𝑅𝛼𝑦𝑥𝑇𝐴𝐿𝛼,𝐴𝑅𝛼𝑦.(3.32) Let 𝛼 incline to 1 and set 𝛼=0, respectively, it holds 𝑥𝑇𝐴𝐿0,𝐴𝑅0𝑦𝑥𝑇𝐴𝐿0,𝐴𝑅0𝑦,𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦.(3.33) This is a contradiction.
By the similar way, we have there exists no 𝑦𝑆𝐽 such that 𝑥𝑇𝐵𝑦𝑥𝑇𝐵𝑦.

Theorem 3.13. Let Γ({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴,𝐵) be a fuzzy bimatrix game, a pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is the weak Pareto equilibrium strategy of game Γ if and only if(i) there exists no 𝑥𝑆𝐼 such that𝑥𝑇𝐴𝐿0,𝐴,𝐴𝑅0𝑦<𝑥𝑇𝐴𝐿0,𝐴,𝐴𝑅0𝑦,(3.34)(ii) there exists no 𝑦𝑆𝐽 such that𝑥𝑇𝐵𝐿0,𝐵,𝐵𝑅0𝑦<𝑥𝑇𝐵𝐿0,𝐵,𝐵𝑅0𝑦.(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},𝑆𝐼,𝑆𝐽,𝐴,𝐵) be a fuzzy bimatrix game, while Player 𝐼 chooses pure strategy 𝑖 and Player 𝐽 chooses pure strategy 𝑗, we set (1𝜆)(𝑎𝑖𝑗𝑙𝑖𝑗)+𝜆(𝑎𝑖𝑗+𝑟𝑖𝑗) to be the payoffs of player 𝐼 and [(1𝜇)(𝑏𝑖𝑗𝑖𝑗)+𝜇(𝑏𝑖𝑗+𝑧𝑖𝑗)] to be the payoffs of player 𝐽, where 𝜆,𝜇[0,1]. The payoff matrixes of player 𝐼 and 𝐽 are 𝐴(𝜆)=(1𝜆)(𝐴𝐿)+𝜆(𝐴+𝑅),𝐵(𝜇)=(1𝜇)(𝐵𝐻)+𝜇(𝐵+𝑍).(3.36) Then, we have the crisp parametric matrix game Γ(𝜆,𝜇)({𝐼,𝐽},𝑆𝐼,𝑆𝐽,𝐴(𝜆),𝐵(𝜇)), where 𝜆 and 𝜇 are parameters.

Definition 3.14. For 𝜆,𝜇[0,1], the pair (𝑥,𝑦)𝑆𝐼×𝑆𝐽 is called to be the Nash equilibrium of the crisp parametric bimatrix game Γ(𝜆,𝜇), if it holds that(i)𝑥𝑇𝐴(𝜆)𝑦𝑥𝑇𝐴(𝜆)𝑦, 𝑥𝑆𝐼,(ii)𝑥𝑇𝐵(𝜇)𝑦𝑥𝑇𝐵(𝜇)𝑦, 𝑦𝑆𝐽.

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)𝑆𝐼×𝑆𝐽 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 𝑥𝑆𝐼, from Definition 3.14(i), it holds 1𝜆0𝑥𝑇𝐴𝐿0𝑦+𝜆0𝑥𝑇𝐴𝑅0𝑦1𝜆0𝑥𝑇𝐴𝐿0𝑦+𝜆0𝑥𝑇𝐴𝑅0𝑦.(3.37) For 𝑦𝑆𝐽, from Definition 3.14(ii), it holds 1𝜇0𝑥𝑇𝐵𝐿0𝑦+𝜇0𝑥𝑇𝐵𝑅0𝑦1𝜇0𝑥𝑇𝐵𝐿0𝑦+𝜇0𝑥𝑇𝐵𝑅0𝑦.(3.38)
First, we assume there exists 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝑦𝑥𝑇𝐴𝑦 holds. From Definition 2.10, it holds 𝑥𝑇𝐴𝐿0𝑦,𝑥𝑇𝐴𝑅0𝑦𝑥𝑇𝐴𝐿0𝑦,𝑥𝑇𝐴𝑅0𝑦.(3.39) Because 𝑥𝑇𝐴𝐿0𝑦=𝑥𝑇𝐴𝐿0𝑦 and 𝑥𝑇𝐴𝑅0𝑦=𝑥𝑇𝐴𝑅0𝑦 do not occur simultaneously, we have 1𝜆0𝑥𝑇𝐴𝐿0𝑦+𝜆0𝑥𝑇𝐴𝑅0𝑦<1𝜆0𝑥𝑇𝐴𝐿0𝑦+𝜆0𝑥𝑇𝐴𝑅0𝑦,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 𝑥𝑆𝐼 such that 𝑥𝑇𝐴𝐿0𝑦,𝑥𝑇𝐴𝑅0𝑦𝑥𝑇𝐴𝐿0𝑦,𝑥𝑇𝐴𝑅0𝑦(3.41) holds. Then, one of the following cases happens,
Case 1. 𝑥𝑇(𝐴𝐿)𝑦>𝑥𝑇(𝐴𝐿)𝑦or𝑥𝑇(𝐴+𝑅)𝑦>𝑥𝑇(𝐴+𝑅)𝑦;(3.42)Case 2. 𝑥𝑇(𝐴𝐿)𝑦=𝑥𝑇(𝐴𝐿)𝑦,𝑥𝑇(𝐴+𝑅)𝑦=𝑥𝑇(𝐴+𝑅)𝑦.(3.43)
If Case 1 happens, without loss of generality, we set 𝑥𝑇(𝐴𝐿)𝑦>𝑥𝑇(𝐴𝐿)𝑦. It is obviously that there exist 𝜆0(0,1) tended to zero such that 1𝜆0𝑥𝑇(𝐴𝐿)𝑦+𝜆0𝑥𝑇(𝐴+𝑅)𝑦>1𝜆0𝑥𝑇(𝐴𝐿)𝑦+𝜆0𝑥𝑇(𝐴+𝑅)𝑦,(3.44) If Case 2 happens, for any 𝜆(0,1)(1𝜆)𝑥𝑇(𝐴𝐿)𝑦+𝜆𝑥𝑇(𝐴+𝑅)𝑦(1𝜆)𝑥𝑇(𝐴𝐿)𝑦+𝜆𝑥𝑇(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)𝑆𝐼×𝑆𝐽 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).

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