**Nash Equilibrium Strategies in Fuzzy Games**

Alireza Chakeri, Nasser Sadati and Guy A. Dumont

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54677

## **1. Introduction**

16 Will-be-set-by-IN-TECH

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UK, 1998.

Modern description of game theory is generally considered to have been started with the book "*Theory of Games and Economic Behavior*" [1]. Modern game theory has grown extremely well, in particular after the influential results of Nash [2-4]. It has been widely applied to many problems in economics, engineering, politics, etc.

A game is a model of a situation where two or more groups are in dispute over some issues [5]. The participants in a game are called the players. The possible actions available to players are referred to as strategies. When each player selects a strategy, it will determine an outcome to the game and the payoffs to all players, while tries to maximize his own payoff.

Classical game theory uses the extensive form and the strategic form to explain a game. The extensive form is represented by a game tree, in which the players make sequential actions. However, the strategic form is usually used to describe games with two decision makers, in which players' choices are made simultaneously. Unlike one-player decision making, where optimality has a clear sense, in multi person decision making the optimality is in the form of NE. An NE strategy is a strategy wherein, if a player knows his opponent's strategy, he is totally satisfied and is unwilling to change his strategy.

In classical game theory it is assumed that all data of a game are known exactly by players. However, in real games, the players are often not able to evaluate exactly the game due to lack of information and precision in the available information of the situation. Harsanyi [6] treated imprecision in games with a probabilistic method and developed the theory of Bayesian games. This theory could not entirely solve the problem of imprecision in games, because it was limited to only one possible kind of imprecision. However, in reality, imprecision is of different types and can be modeled by fuzzy sets. The notion of fuzzy sets first appeared in the paper written by Zadeh [7]. This notion tries to show that to what degree an element belongs to a set. The degree, to which an element belongs to a set, is an element of the continuous interval [0, 1] rather than the Boolean values. Using the notion of

fuzzy sets, each component in a game (set of players, set of strategies, set of payoffs, etc) can be fuzzified. Initially, fuzzy sets were used by Butnariu [8] in non-cooperative game theory. He used fuzzy sets to represent the belief of each player for strategies of other players. Since then, fuzzy set theory has been used in many non-cooperative [9-15] and cooperative games [16-17].

Nash Equilibrium Strategies in Fuzzy Games 311



(1)

2. ( ) *<sup>M</sup>*

*M x RMx*

*x* is piecewise continuous.

fuzzy number *M* are closed real intervals, that is

membership function of *B* is defined as follows

 

membership function

In definition 3, if *fx x*

the following membership function

1 1

0

, , , ,

 

*y fx x*

*y y*

1

, ,

*B fM M*

 

*n n*

*f M*

*y*

*M* is a fuzzy number with the following membership function

is also called

*M ab*, 0,1

Fuzzy extension principle, which was introduced by Zadeh in [19], is an essential principle in fuzzy set theory to generalize the concepts and structures of classical mathematics to

**Definition 2**: Let 1 , , *U U <sup>n</sup>* are UoDs and *UU U* <sup>1</sup> *<sup>n</sup>* be their Cartesian product. Also assume that 1 , , *<sup>M</sup> Mn* are fuzzy subsets of 1 , , *U U <sup>n</sup>* , respectively. Moreover, let <sup>1</sup> , , *<sup>n</sup> <sup>y</sup> fx x* be a mapping from *<sup>U</sup>* to*<sup>Y</sup>* . Now if *B fM M* <sup>1</sup> , , *<sup>n</sup>* , then the

*x x fy*

*n*

1

1

*f y*

*x fy*

1

*f y*

1

(4)

into fuzzy number

(3)

 

(2)

 

 

1

*M M <sup>n</sup> x x*

on fuzzy numbers can be represented by the following definitions, respectively.

,

0

sup

*xy f x*

sup min , ,

*n*

1

Based on the above definition, one-dimensional operators and two-dimensional operators

**Definition 3**: Assume that *M* is a fuzzy number and *f* : *R R* is a one-dimensional operator. According to fuzzy extension principle, *f M* is a fuzzy set with the following

*M*

, then the scalar product of the real value

*M M <sup>x</sup> x* 

**Definition 4**: Assume that *M* and *N* are fuzzy numbers and *fRR R* : is a twodimensional operator. According to fuzzy extension principle, *M N* is a fuzzy set with

 

 ( ) , 0,1 

**2.1. Fuzzy extension principle** 

fuzzy mathematics as follows [20]

In this chapter, we will extend the NE set to fuzzy set in games with fuzzy numbers as payoffs. In this regard, using ranking fuzzy numbers, a fuzzy preference relation is constructed over payoffs and then the resultant priorities of payoffs are considered as the grades of being NE. Hence, if a player knows the opponent's strategy, he is satisfied with his own strategy by the degree that this strategy has priority for him. The more priority the players feel for each strategy, the more possible the strategy becomes the game's equilibrium. This generalization shows the distribution of Nash grades in the matrix form of the game. In other words, we can consider strategies with high grades of equilibrium which are not necessarily the equilibrium points. In the proposed approach, the effect of different viewpoints (optimism and pessimism) on the result of the game is also studied.

The remainder of this chapter is organized as follows. Section 2 reviews the backgrounds on fuzzy set theory. This section discusses the basic definition of fuzzy numbers, fuzzy extension principle and ranking fuzzy numbers. Section 3 briefly presents the fuzzy preference relation and a way to obtain priorities from fuzzy preference matrix. In this section, preference ordering of the alternatives and its transformation into fuzzy preference relation is also discussed. Section 4 introduces the proposed algorithm to find the Nash grades for pure and mixed strategies in the matrix form of the game. Two examples and their detailed results are presented in Section 5. Finally, Section 6 contains some concluding remarks.

#### **2. Backgrounds on fuzzy set theory**

There are two well-known frameworks for quantifying the lack of knowledge and precision, namely, probabilistic and possibilistic uncertainty type. The probabilistic framework deals with the uncertain events with a probability distribution function (PDF). However, there are some situations in which there is not much information about the PDF of uncertain parameters or they are inherently not repeatable. In possibilistic framework, for each uncertain event *<sup>M</sup>* , a membership function ( ) *<sup>M</sup> x* is defined which describes how much each element *x* , of universe of discourse *U* (the set of all values that *x* can take) belongs to *M* . Different types of membership functions can be used to describe uncertain values. Fuzzy numbers have also been used in many decision making problems. The following definition of fuzzy numbers is most commonly used [18];

**Definition 1:** The fuzzy number *M* is a convex normalized fuzzy set of the real line *R* such that

1. There is exactly one <sup>0</sup> *x* with <sup>0</sup> ()1 *<sup>M</sup>x* . 2. ( ) *<sup>M</sup>x* is piecewise continuous.

310 Game Theory Relaunched

[16-17].

remarks.

that

**2. Backgrounds on fuzzy set theory** 

uncertain event *<sup>M</sup>* , a membership function ( ) *<sup>M</sup>*

1. There is exactly one <sup>0</sup> *x* with <sup>0</sup> ()1 *<sup>M</sup>*

definition of fuzzy numbers is most commonly used [18];

*x* .

fuzzy sets, each component in a game (set of players, set of strategies, set of payoffs, etc) can be fuzzified. Initially, fuzzy sets were used by Butnariu [8] in non-cooperative game theory. He used fuzzy sets to represent the belief of each player for strategies of other players. Since then, fuzzy set theory has been used in many non-cooperative [9-15] and cooperative games

In this chapter, we will extend the NE set to fuzzy set in games with fuzzy numbers as payoffs. In this regard, using ranking fuzzy numbers, a fuzzy preference relation is constructed over payoffs and then the resultant priorities of payoffs are considered as the grades of being NE. Hence, if a player knows the opponent's strategy, he is satisfied with his own strategy by the degree that this strategy has priority for him. The more priority the players feel for each strategy, the more possible the strategy becomes the game's equilibrium. This generalization shows the distribution of Nash grades in the matrix form of the game. In other words, we can consider strategies with high grades of equilibrium which are not necessarily the equilibrium points. In the proposed approach, the effect of different

The remainder of this chapter is organized as follows. Section 2 reviews the backgrounds on fuzzy set theory. This section discusses the basic definition of fuzzy numbers, fuzzy extension principle and ranking fuzzy numbers. Section 3 briefly presents the fuzzy preference relation and a way to obtain priorities from fuzzy preference matrix. In this section, preference ordering of the alternatives and its transformation into fuzzy preference relation is also discussed. Section 4 introduces the proposed algorithm to find the Nash grades for pure and mixed strategies in the matrix form of the game. Two examples and their detailed results are presented in Section 5. Finally, Section 6 contains some concluding

There are two well-known frameworks for quantifying the lack of knowledge and precision, namely, probabilistic and possibilistic uncertainty type. The probabilistic framework deals with the uncertain events with a probability distribution function (PDF). However, there are some situations in which there is not much information about the PDF of uncertain parameters or they are inherently not repeatable. In possibilistic framework, for each

each element *x* , of universe of discourse *U* (the set of all values that *x* can take) belongs to *M* . Different types of membership functions can be used to describe uncertain values. Fuzzy numbers have also been used in many decision making problems. The following

**Definition 1:** The fuzzy number *M* is a convex normalized fuzzy set of the real line *R* such

*x* is defined which describes how much

viewpoints (optimism and pessimism) on the result of the game is also studied.

*M x RMx* ( ) , 0,1 is also called -cut of *M* . From definition 1, -cuts of a fuzzy number *M* are closed real intervals, that is

$$M\_{\alpha} = \left[a\_{\alpha}, b\_{\alpha}\right] \quad \forall \alpha \in \left[0, 1\right] \tag{1}$$

#### **2.1. Fuzzy extension principle**

Fuzzy extension principle, which was introduced by Zadeh in [19], is an essential principle in fuzzy set theory to generalize the concepts and structures of classical mathematics to fuzzy mathematics as follows [20]

**Definition 2**: Let 1 , , *U U <sup>n</sup>* are UoDs and *UU U* <sup>1</sup> *<sup>n</sup>* be their Cartesian product. Also assume that 1 , , *<sup>M</sup> Mn* are fuzzy subsets of 1 , , *U U <sup>n</sup>* , respectively. Moreover, let <sup>1</sup> , , *<sup>n</sup> <sup>y</sup> fx x* be a mapping from *<sup>U</sup>* to*<sup>Y</sup>* . Now if *B fM M* <sup>1</sup> , , *<sup>n</sup>* , then the membership function of *B* is defined as follows

$$\begin{aligned} \mu\_{B}\left(y\right) &= \mu\_{f\left(\tilde{M}\_{1},...,\tilde{M}\_{n}\right)}\left(y\right) \\ &= \begin{cases} \sup\limits\_{y=f\left(x\_{1},...,x\_{n}\right)} \min\left\{\mu\_{\tilde{M}\_{1}}\left(x\_{1}\right),...,\mu\_{\tilde{M}\_{n}}\left(x\_{n}\right)\right\} & f^{-1}\left(y\right) \neq \phi \\\ y=f\left(x\_{1},...,x\_{n}\right) & \text{if } f^{-1}\left(y\right) = \phi \end{cases} \end{aligned} \tag{2}$$

Based on the above definition, one-dimensional operators and two-dimensional operators on fuzzy numbers can be represented by the following definitions, respectively.

**Definition 3**: Assume that *M* is a fuzzy number and *f* : *R R* is a one-dimensional operator. According to fuzzy extension principle, *f M* is a fuzzy set with the following membership function

$$\mu\_{f(\check{M})}(y) = \begin{cases} \sup\_{\boldsymbol{x}, \boldsymbol{y} = f(\boldsymbol{x})} \mu\_{\check{M}}(\boldsymbol{x}) & f^{-1}(\boldsymbol{y}) \neq \phi \\ 0 & f^{-1}(\boldsymbol{y}) = \phi \end{cases} \tag{3}$$

In definition 3, if *fx x* , then the scalar product of the real value into fuzzy number *M* is a fuzzy number with the following membership function

$$
\mu\_{\mathcal{A},\tilde{\mathcal{M}}} \left( \boldsymbol{\chi} \right) = \mu\_{\tilde{\mathcal{M}}} \left( \frac{\boldsymbol{\chi}}{\mathcal{X}} \right) \tag{4}
$$

**Definition 4**: Assume that *M* and *N* are fuzzy numbers and *fRR R* : is a twodimensional operator. According to fuzzy extension principle, *M N* is a fuzzy set with the following membership function

$$\mu\_{\tilde{M}\otimes\tilde{N}}(z) = \sup\_{z=x\otimes y} \min\left(\mu\_{\tilde{M}}(x), \mu\_{\tilde{N}}(y)\right) \tag{5}$$

Nash Equilibrium Strategies in Fuzzy Games 313

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For convenience, let *Y* [0,1] and *P* is the Lebesgue measure on [0, 1], as used in [24]. Then,

( ) (1 ) *VM b ad <sup>P</sup>* 

As a special case, the average index for triangular fuzzy numbers can be given as follows

*Vb ab <sup>L</sup>*

where, a triangular fuzzy number with notation ( , , ) *M Tacb* is used as shown in Figure 1.

Preference relation is one of the most regular tools for stating decision maker's preferences. In the process of making decision, individuals are asked to give their preferences over an alternative, which is based on their comparison according to one's desire. Various kinds of preference relations have been developed including multiplicative preference relation [25], fuzzy preference relation [26], linguistic preference relation [27] and intuitionistic preference

For a decision making situation, let *X xx x* 1 2 , , , *<sup>n</sup>* be a discrete set of alternatives. A

domain of representation of preference degrees provided by the decision maker for each pair of alternatives. In many situations, due to lack of information about the problems, the goals, constraints and consequences are not precisely known. Because of these uncertainties, fuzzy set theory allows a more flexible framework to express the preferences. Fuzzy preferences show the fuzziness of the decision maker's preferences. In [26], fuzzy preference

**Definition 5**: A fuzzy preference relation *<sup>R</sup>* on the set *<sup>X</sup>* is defined as a matrix *ij n n R r*

*<sup>P</sup> XX D* , where *D* is the

 

 

1

0

<sup>1</sup> ( ) <sup>2</sup>

   

(11)

the following ( ) *V MP* is derived for a specific

**Figure 1.** Triangular fuzzy number, *M Tacb* (,,) .

preference relation *P* on the set *X* is defined by a function

**3. Preference relations** 

relation is defined as follows.

with some properties given as

relation [28].

**3.1. Fuzzy preference relations** 

As a special case, the result of the summation operator is also a fuzzy number as follows

$$\mu\_{\tilde{M}+\tilde{N}}\left(z\right) = \sup\_{z=x+y} \min\left(\mu\_{\tilde{M}}(x), \mu\_{\tilde{N}}(y)\right) \tag{6}$$

#### **2.2. Ranking fuzzy numbers**

Ranking fuzzy numbers seems a necessary procedure in decision making when alternatives are fuzzy numbers. Various methods for ranking fuzzy subsets have been proposed. Yager in [21] introduced a function for ranking fuzzy subsets in unit interval which is based on the integral of mean of the -cuts. Jain in [22], Baldwin and Guild in [23] were also suggested methods for ordering fuzzy subsets in the unit interval. Ibanez and Munoz in [24] have developed a subjective approach for ranking fuzzy numbers. In this chapter, we use the subjective approach, as introduced in [24].

Ibanez and Munoz in [24] defined the following number as the average index for *M*

$$\forall V\_P(\tilde{M}) = \int\_Y f\_{\tilde{M}}(\alpha)dP(\alpha) \qquad \qquad \forall \tilde{M} \in \mathcal{U} \tag{7}$$

where*Y* is a subset of the unit interval and *P* is a probability distribution on*Y* .

The definition of *<sup>M</sup> f* could be subjective for decision maker. In [24] the following definition for *<sup>M</sup> f* has been suggested, in which the parameter determines the optimism-pessimism degree of the decision maker

$$f\_{\tilde{M}}^{\lambda}: \mathcal{Y} \to \mathbb{R}, \quad f\_{\tilde{M}}^{\lambda}(\alpha) = \lambda b\_{\alpha} + (1 - \lambda)a\_{\alpha} \tag{8}$$

where [0,1] and *M a b*, .

When an optimistic decision maker ( 1 ) wants to choose the greatest value, the upper extreme of the interval ( *b* ) would be chosen, i.e. he prefers to choose the greatest possible value. A pessimism person ( 0 ) on the opposite prefers to decide on the lower extreme of the interval ( *a*).

Using ( ) *VP* in (7), the ordering relations between fuzzy numbers *A* and *B* can be given as

$$
\tilde{A} \le \tilde{B} \iff V\_P(\tilde{A}) \le V\_P(\tilde{B}) \qquad \qquad \forall \tilde{A}, \tilde{B} \in \mathcal{U} \tag{9}
$$

The fuzzy number *A* is not preferred to *B* , if and only if their average index is the same

$$
\tilde{A} \approx \tilde{B} \Longleftrightarrow V\_P(\tilde{A}) = V\_P(\tilde{B}) \quad \forall \tilde{A}, \tilde{B} \in \mathcal{U} \tag{10}
$$

For convenience, let *Y* [0,1] and *P* is the Lebesgue measure on [0, 1], as used in [24]. Then, the following ( ) *V MP* is derived for a specific 

$$W\_P(\tilde{M}) = \prod\_{0}^{1} (\lambda b\_{\alpha} + (1 - \lambda)a\_{\alpha})d\alpha \tag{11}$$

As a special case, the average index for triangular fuzzy numbers can be given as follows

$$V\_L^{\lambda} = b\lambda + (a - \frac{1}{2}b) \tag{12}$$

where, a triangular fuzzy number with notation ( , , ) *M Tacb* is used as shown in Figure 1.

**Figure 1.** Triangular fuzzy number, *M Tacb* (,,) .

#### **3. Preference relations**

312 Game Theory Relaunched

**2.2. Ranking fuzzy numbers** 

integral of mean of the

The definition of *<sup>M</sup>*

degree of the decision maker

extreme of the interval ( *b*

the interval ( *a*

 [0,1] and *M a b*, 

value. A pessimism person ( 0

).

When an optimistic decision maker ( 1

for *<sup>M</sup>*

where  *M N* sup min ( ), ( ) *M N*

*y*

*M N* sup min ( ), ( ) *M N*

*<sup>y</sup>*

Ranking fuzzy numbers seems a necessary procedure in decision making when alternatives are fuzzy numbers. Various methods for ranking fuzzy subsets have been proposed. Yager in [21] introduced a function for ranking fuzzy subsets in unit interval which is based on the

methods for ordering fuzzy subsets in the unit interval. Ibanez and Munoz in [24] have developed a subjective approach for ranking fuzzy numbers. In this chapter, we use the

Ibanez and Munoz in [24] defined the following number as the average index for *M*

*V M f dP* 

 

( ) () () *<sup>P</sup> <sup>M</sup> Y*

where*Y* is a subset of the unit interval and *P* is a probability distribution on*Y* .

: , ( ) (1 ) *M M fY Rf b a*

 

Using ( ) *VP* in (7), the ordering relations between fuzzy numbers *A* and *B* can be given as

The fuzzy number *A* is not preferred to *B* , if and only if their average index is the same

( ) () , *<sup>A</sup> P P B V A V B AB U* (10)

 

*z x*

*z x*

 

  -cuts. Jain in [22], Baldwin and Guild in [23] were also suggested

*f* could be subjective for decision maker. In [24] the following definition

 

) would be chosen, i.e. he prefers to choose the greatest possible

) on the opposite prefers to decide on the lower extreme of

( ) () , *<sup>A</sup> P P B V A V B AB U* (9)

) wants to choose the greatest value, the upper

 

*M U* (7)

determines the optimism-pessimism

(8)

(5)

(6)

*zx y*

As a special case, the result of the summation operator is also a fuzzy number as follows

*zxy*

*f* has been suggested, in which the parameter

 .

subjective approach, as introduced in [24].

#### **3.1. Fuzzy preference relations**

Preference relation is one of the most regular tools for stating decision maker's preferences. In the process of making decision, individuals are asked to give their preferences over an alternative, which is based on their comparison according to one's desire. Various kinds of preference relations have been developed including multiplicative preference relation [25], fuzzy preference relation [26], linguistic preference relation [27] and intuitionistic preference relation [28].

For a decision making situation, let *X xx x* 1 2 , , , *<sup>n</sup>* be a discrete set of alternatives. A preference relation *P* on the set *X* is defined by a function *<sup>P</sup> XX D* , where *D* is the domain of representation of preference degrees provided by the decision maker for each pair of alternatives. In many situations, due to lack of information about the problems, the goals, constraints and consequences are not precisely known. Because of these uncertainties, fuzzy set theory allows a more flexible framework to express the preferences. Fuzzy preferences show the fuzziness of the decision maker's preferences. In [26], fuzzy preference relation is defined as follows.

**Definition 5**: A fuzzy preference relation *<sup>R</sup>* on the set *<sup>X</sup>* is defined as a matrix *ij n n R r*

with some properties given as

$$\begin{cases} r\_{ij} + r\_{ji} = 1, \\ r\_{ij} \ge 0, \\ r\_{il} = 0.5, \end{cases} \qquad \forall i, j = 1, 2, \dots, n \tag{13}$$

Nash Equilibrium Strategies in Fuzzy Games 315

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**Step 3.** ( ) max ( ) <sup>1</sup> *<sup>n</sup>*

**Step 4.** Set 1 *k k* and go to step 2. **Step 5.** Obtain *w* as the priority vector.

**3.2. Preference ordering of the alternatives** 

 

*<sup>m</sup> w k i i w k* , calculate *T k*( ) and then ( ) *<sup>i</sup> x k* using

( ) ( ) () ( ) ( ) ( ) ( )

*w k w k Tk hr h r*

 

*i*

This means that the alternative *x2* is the best and alternative *x3* is the worst.

**3.3. Transforming preference ordering into fuzzy preference relation** 

introduced to achieve fuzzy preference relation from an ordering preference

*v x*

where *ij r* denotes the preference between alternatives *<sup>i</sup> x* and *<sup>j</sup> x* .

*i*

*x k*

*mj jm j m m j j m*

> () () ( ) ( ) *m*

Individuals usually provide their preferences over the alternatives by preference ordering. Let's assume that an expert expresses his preferences on *X* by a preference ordering *O ox ox ox* ( ), ( ), , ( ) 1 2 *<sup>n</sup>* . It is assumed that the lower the position of the alternative in the preference ordering, the more contentment for the individual. For instance, an expert states his preference ordering on *X xxx* <sup>123</sup> , , by the following ordering; 123 *ox ox ox* ( ) 2, ( ) 1, ( ) 3 .

An alternative satisfies the decision maker according to its position in the ordering preference relation. Various mappings to transform the ordering preference relation into fuzzy preference relation have been proposed [32-34]. In [32] a crisp relation was introduced for assessing the fuzzy preference relation, where the preference between alternatives *<sup>i</sup> <sup>x</sup>* and *<sup>j</sup> <sup>x</sup>* depends only on the values of ( )*<sup>i</sup> o x* and ( )*<sup>j</sup> o x* . In [33-34], the following relations was

> ()1 ()1 <sup>1</sup> *i*

In a strategic game, there are *n* players and *<sup>i</sup> ns* strategies for player*i* . Suppose *Xi* is the

*o x*

*n*

(18)

<sup>1</sup> 1 () () <sup>2</sup> *ij i j r vx vx* (19)

*w k w k* 

> *i Tkw k*

*w k*

*j m*

(16)

**Step 6.** End.

**4. Fuzzy games** 

strategy set for player*i* defined as follows

where *ij r* denotes the preference degree of the alternative *<sup>i</sup> x* to *<sup>j</sup> x* .

In particular, 0.5 *ij r* shows indifference between *<sup>i</sup> x* and *<sup>j</sup> x* . The case of 0.5 *ij r* indicates that *<sup>i</sup> x* is preferred to *<sup>j</sup> x* . As *ij r* increases, the degree of preference gets larger. Also 1 *ij r* shows that *<sup>i</sup> x* is absolutely preferred to *<sup>j</sup> x* and vice versa.

Deriving priorities, that is the degrees of importance of alternatives, are the main aspect of preference relations. Quite a number of approaches have been developed to derive priorities from fuzzy preference relations. Lipovetsky and Michael-Conklin [29] introduced an optimization approach and an eigen-problem to produce robust priority estimation of a fuzzy preference relation. Xu [30] developed a weighted least square approach and an eigenvector method for priorities of fuzzy preference relations. Xu and Da [31] proposed the Least Deviation method to obtain the priority vector of a fuzzy preference relation and considered its properties. In this chapter, we use the Least Deviation method presented in [31] to derive priorities from fuzzy preference relation. The algorithm to derive priorities in [31] is mentioned briefly as follows.

Let ( ) *R rij n n* be a fuzzy preference relation and 1 2 , ,, *W ww wn* be the priority vector which is going to be calculated such that *wi* shows the degree of priority of alternative *<sup>i</sup> x* , with the following properties

$$\sum\_{i=1}^{n} w\_i = 1; \ w\_i \ge 0 \tag{14}$$

Let *k* be the number of iterations:

**Step 1.** Initialize the weight 1 2, (0) ( , , , ) *W ww wn* , specify parameter 0 1 and let *k* 0 .

**Step 2.** Calculate the following term, where (2 1) () 9 *ij <sup>r</sup> ij h r*

$$\mu\_i(w(k)) = \sum\_{j=1}^n \left[ h(r\_{ij}) \left( \frac{w\_j(k)}{w\_i(k)} \right) - h(r\_{ji}) \left( \frac{w\_i(k)}{w\_j(k)} \right) \right] \,\forall i \tag{15}$$

If ( ) *<sup>i</sup> w k* for all *i* , then update *w* b*y w k*( ) and go to step 5, otherwise continue with step 3.

**Step 3.** ( ) max ( ) <sup>1</sup> *<sup>n</sup> <sup>m</sup> w k i i w k* , calculate *T k*( ) and then ( ) *<sup>i</sup> x k* using

$$T(k) = \sqrt{\left(\sum\_{j \neq m} h(r\_{mj}) \left(\frac{w\_j(k)}{w\_m(k)}\right)\right) \bigg/ \left(\sum\_{j \neq m} h(r\_{jm}) \left(\frac{w\_m(k)}{w\_j(k)}\right)\right)}\tag{16}$$

$$\mathbf{x}\_i(k) = \begin{cases} T(k)w\_m(k) \\ w\_i(k) \end{cases} \tag{17}$$


**Step 6.** End.

314 Game Theory Relaunched

where *ij*

In particular, 0.5 *ij*

*<sup>i</sup> x* is preferred to *<sup>j</sup> x* . As *ij*

[31] is mentioned briefly as follows.

with the following properties

Let *k* be the number of iterations:

*k* 0 .

If ( ) 

step 3.

*<sup>i</sup> w k*

1,

0.5,

*r* denotes the preference degree of the alternative *<sup>i</sup> x* to *<sup>j</sup> x* .

*ij ji ij ii*

*r r*

*r*

 

that *<sup>i</sup> x* is absolutely preferred to *<sup>j</sup> x* and vice versa.

0, , 1,2,...,

(13)

*r* indicates that

*r* shows

*r ij n*

*r* shows indifference between *<sup>i</sup> x* and *<sup>j</sup> x* . The case of 0.5 *ij*

Deriving priorities, that is the degrees of importance of alternatives, are the main aspect of preference relations. Quite a number of approaches have been developed to derive priorities from fuzzy preference relations. Lipovetsky and Michael-Conklin [29] introduced an optimization approach and an eigen-problem to produce robust priority estimation of a fuzzy preference relation. Xu [30] developed a weighted least square approach and an eigenvector method for priorities of fuzzy preference relations. Xu and Da [31] proposed the Least Deviation method to obtain the priority vector of a fuzzy preference relation and considered its properties. In this chapter, we use the Least Deviation method presented in [31] to derive priorities from fuzzy preference relation. The algorithm to derive priorities in

Let ( ) *R rij n n* be a fuzzy preference relation and 1 2 , ,, *W ww wn* be the priority vector which is going to be calculated such that *wi* shows the degree of priority of alternative *<sup>i</sup> x* ,

1; 0

*r ij h r*

*wk wk*

for all *i* , then update *w* b*y w k*( ) and go to step 5, otherwise continue with

(14)

(15)

and let

*i i*

( ) ( ) () ( ) ( ) , () () *<sup>n</sup> <sup>j</sup> <sup>i</sup>*

*j i j w k w k wk hr hr <sup>i</sup>*

*w w*

1

**Step 1.** Initialize the weight 1 2, (0) ( , , , ) *W ww wn* , specify parameter 0 1

*i*

**Step 2.** Calculate the following term, where (2 1) () 9 *ij*

1

*i ij ji*

*n*

*r* increases, the degree of preference gets larger. Also 1 *ij*

#### **3.2. Preference ordering of the alternatives**

Individuals usually provide their preferences over the alternatives by preference ordering. Let's assume that an expert expresses his preferences on *X* by a preference ordering *O ox ox ox* ( ), ( ), , ( ) 1 2 *<sup>n</sup>* . It is assumed that the lower the position of the alternative in the preference ordering, the more contentment for the individual. For instance, an expert states his preference ordering on *X xxx* <sup>123</sup> , , by the following ordering; 123 *ox ox ox* ( ) 2, ( ) 1, ( ) 3 . This means that the alternative *x2* is the best and alternative *x3* is the worst.

#### **3.3. Transforming preference ordering into fuzzy preference relation**

An alternative satisfies the decision maker according to its position in the ordering preference relation. Various mappings to transform the ordering preference relation into fuzzy preference relation have been proposed [32-34]. In [32] a crisp relation was introduced for assessing the fuzzy preference relation, where the preference between alternatives *<sup>i</sup> <sup>x</sup>* and *<sup>j</sup> <sup>x</sup>* depends only on the values of ( )*<sup>i</sup> o x* and ( )*<sup>j</sup> o x* . In [33-34], the following relations was introduced to achieve fuzzy preference relation from an ordering preference

$$\upsilon(\mathbf{x}\_i) = 1 - \frac{o(\mathbf{x}\_i) - 1}{n - 1} \tag{18}$$

$$r\_{ij} = \frac{1}{2} \left( 1 + \upsilon(\mathbf{x}\_i) - \upsilon(\mathbf{x}\_j) \right) \tag{19}$$

where *ij r* denotes the preference between alternatives *<sup>i</sup> x* and *<sup>j</sup> x* .

#### **4. Fuzzy games**

In a strategic game, there are *n* players and *<sup>i</sup> ns* strategies for player*i* . Suppose *Xi* is the strategy set for player*i* defined as follows

$$X\_i = \left\langle \mathbf{x}\_{1i}, \mathbf{x}\_{2i}, \dots, \mathbf{x}\_{ns\_i i} \right\rangle \tag{20}$$

Nash Equilibrium Strategies in Fuzzy Games 317

1. When there is not sufficient information about the payoffs and they are not inherently

2. When calculation of payoffs is difficult or time-consuming. Hence, they are usually

In this chapter, we consider pure and mixed strategies in games with fuzzy numbers as payoffs. In this regard, in pure strategies, ( , ) *ii i s s* is considered as a fuzzy number. Also, in mixed strategies, the expected values are calculated based on the fuzzy extension principle

In classical game theory, a crisp payoff is either greater than or less than others. However, in fuzzy ones, there are uncertainties in comparing fuzzy payoffs. We model these uncertainties using fuzzy preference relation on the preference ordering of the expected values. In this regard, using ranking fuzzy numbers, a fuzzy preference relation reflecting the uncertainties in payoffs is constructed and then, the resultant priorities of payoffs are considered as the grades of being NE. This definition for the grade of being NE seems meaningful because if a player knows the opponent's strategy, he is satisfied with his strategy by the degree that this strategy has priority for him. The more priority the players get for each strategy, the more possible the strategy is the game's equilibrium. The proposed algorithm to find the grade of being NE for

> .

, set its grade of being NE equal to its corresponding

, set its overall grade of being NE equal to the

repeatable.

defined by fuzzy if-then rules.

every mixed strategy is shown in Table 1.

4. Fix the opponent's mixed strategy *<sup>i</sup>*

1. *i* , do steps 2-9.

according to (9).

, do steps 4-9.

the least deviation method.

9. For every mixed strategy , *i i*

10. For every mixed strategy , *i i*

3. *<sup>i</sup>* 

5. *<sup>i</sup>* 

(18-19).

priorities.

and definitions 3 and 4, because the payoffs are fuzzy numbers.

**5. Playing games with fuzzy numbers as payoffs** 

2. For player*i* , determine the optimism-pessimism degree

 

> 

**Table 1.** Algorithm to find the grade of being NE for each mixed strategy

minimum of its grade of being NE for each player.

.

, calculate the average index of the resultant expected value according to (11).

7. Transform the ordering preference relation into fuzzy preference relation according to

8. Derive priorities of expected values from the resultant fuzzy preference relation based on

6. Order the expected values according to their average index in descending order,

In pure strategy, let *is* denotes the strategy chosen by player*i* and *<sup>i</sup> s* denotes the strategies chosen by the other players. Then 1 (, ,, , ) (, ) *i i n ii i s s s ss* is the payoff achieved by player*i* . By definition of classical game theory, \* \* <sup>1</sup> (, ,) *<sup>n</sup> s s* is the pure strategy Nash equilibrium, if and only if [35]

$$\Pi\_i(\mathbf{s}\_1^\*, \dots, \mathbf{s}\_i^\*, \dots, \mathbf{s}\_n^\*) \ge \Pi\_i(\mathbf{s}\_1^\*, \dots, \mathbf{s}\_{i'}, \dots, \mathbf{s}\_n^\*) \quad \forall i \in \{1, \dots, n\}, \forall \mathbf{s}\_i \in \mathbf{X}\_i \tag{21}$$

In mixed strategy, each player assigns a probability distribution <sup>1</sup> , , *<sup>i</sup> i i ns i i* over his strategies, where *<sup>i</sup>* determines all possible probability distributions for player *i* . Then, the expected payoff for player *i* is the real value <sup>1</sup> , , *i n E* , where *E* denotes the expectation operator. In this regard, mixed strategy Nash equilibrium \* \* <sup>1</sup> ( ,, ) *<sup>n</sup>* is defined as follows

$$E\_i(\stackrel{\*}{\sigma\_1}, \dots, \stackrel{\*}{\sigma\_{i'}}, \dots, \stackrel{\*}{\sigma\_n}) \ge E\_i(\stackrel{\*}{\sigma\_1}, \dots, \stackrel{\*}{\sigma\_{i'}}, \dots, \stackrel{\*}{\sigma\_n}) \quad \forall i \in \{1, \dots, n\}, \forall \sigma\_i \in \Sigma\_i \tag{22}$$

In real games, players must often make their decisions under unclear or fuzzy information. In this regard, there are several approaches for explaining games with fuzzy set theory. As discussed earlier, a game has three main components: a set of players, a set of strategies and a set of payoffs. The set of players is defined as fuzzy set when the concept of coalition in cooperative games is fuzzified. Butnariu in [17] proposed core and stable sets in fuzzy coalition games, and introduced a degree of participation for players in a coalition. Mares in [16] considered fuzzy core in fuzzy cooperative games, where possibility for each fuzzy coalition was considered as fuzzy interval, and an extension of the core in classic TU games. He discussed Shapely value in cooperative games with deterministic characteristics and fuzzy coalitions. In [9], the concept of fuzzy strategies has been introduced. It defines a strategy set consisting of fuzzy subspaces of strategy spaces and assigned a fuzzy payoff for each set of player's strategies. Hence, they have defined a fuzzy inference system (fuzzy If-Then rules). However, they have assumed some real values as strategies and have solved the games with common crisp methods. The first two steps seem rational for modeling any system according to its specifications, however the final step is not a reasonable fuzzy decision making approach to find NEs. Generally, since the final decision of a player is a number in real world problems, they can not adopt fuzzy strategies except when meaningful interpretations exist. However, we should remark that fuzzy strategies are constructive to model the games and calculate the payoffs.

Defining payoffs as fuzzy sets is reasonable in the following two main situations, and lead to fuzzy numbers as payoffs.


In this chapter, we consider pure and mixed strategies in games with fuzzy numbers as payoffs. In this regard, in pure strategies, ( , ) *ii i s s* is considered as a fuzzy number. Also, in mixed strategies, the expected values are calculated based on the fuzzy extension principle and definitions 3 and 4, because the payoffs are fuzzy numbers.

### **5. Playing games with fuzzy numbers as payoffs**

In classical game theory, a crisp payoff is either greater than or less than others. However, in fuzzy ones, there are uncertainties in comparing fuzzy payoffs. We model these uncertainties using fuzzy preference relation on the preference ordering of the expected values. In this regard, using ranking fuzzy numbers, a fuzzy preference relation reflecting the uncertainties in payoffs is constructed and then, the resultant priorities of payoffs are considered as the grades of being NE. This definition for the grade of being NE seems meaningful because if a player knows the opponent's strategy, he is satisfied with his strategy by the degree that this strategy has priority for him. The more priority the players get for each strategy, the more possible the strategy is the game's equilibrium. The proposed algorithm to find the grade of being NE for every mixed strategy is shown in Table 1.

1. *i* , do steps 2-9.

316 Game Theory Relaunched

as follows

equilibrium, if and only if [35]

   

constructive to model the games and calculate the payoffs.

to fuzzy numbers as payoffs.

*X xx x i i i ns i* 1 2 , ,..., *<sup>i</sup>* (20)

*i i n ii i s s s ss* is the payoff achieved by

<sup>1</sup> (, ,) *<sup>n</sup> s s* is the pure strategy Nash

 over

, where *E* denotes the

is defined

(22)

<sup>1</sup> ( ,, ) *<sup>n</sup>* 

In pure strategy, let *is* denotes the strategy chosen by player*i* and *<sup>i</sup> s* denotes the strategies

 \*\*\* \* \* 1 1 ( , , , , ) ( , , , , ) 1, , ,

In mixed strategy, each player assigns a probability distribution <sup>1</sup> , , *<sup>i</sup> i i ns i i*

expectation operator. In this regard, mixed strategy Nash equilibrium \* \*

 

 \*\*\* \* \* 1 1 , , , , , , , , 1, , , *i i ni i n i i EE i*

 *n*

In real games, players must often make their decisions under unclear or fuzzy information. In this regard, there are several approaches for explaining games with fuzzy set theory. As discussed earlier, a game has three main components: a set of players, a set of strategies and a set of payoffs. The set of players is defined as fuzzy set when the concept of coalition in cooperative games is fuzzified. Butnariu in [17] proposed core and stable sets in fuzzy coalition games, and introduced a degree of participation for players in a coalition. Mares in [16] considered fuzzy core in fuzzy cooperative games, where possibility for each fuzzy coalition was considered as fuzzy interval, and an extension of the core in classic TU games. He discussed Shapely value in cooperative games with deterministic characteristics and fuzzy coalitions. In [9], the concept of fuzzy strategies has been introduced. It defines a strategy set consisting of fuzzy subspaces of strategy spaces and assigned a fuzzy payoff for each set of player's strategies. Hence, they have defined a fuzzy inference system (fuzzy If-Then rules). However, they have assumed some real values as strategies and have solved the games with common crisp methods. The first two steps seem rational for modeling any system according to its specifications, however the final step is not a reasonable fuzzy decision making approach to find NEs. Generally, since the final decision of a player is a number in real world problems, they can not adopt fuzzy strategies except when meaningful interpretations exist. However, we should remark that fuzzy strategies are

Defining payoffs as fuzzy sets is reasonable in the following two main situations, and lead

his strategies, where *<sup>i</sup>* determines all possible probability distributions for player *i* . Then,

*i i ni in i i s s s s s s i n sX* (21)

 

chosen by the other players. Then 1 (, ,, , ) (, )

the expected payoff for player *i* is the real value <sup>1</sup> , , *i n E*

player*i* . By definition of classical game theory, \* \*

2. For player*i* , determine the optimism-pessimism degree .

3. *<sup>i</sup>* , do steps 4-9.

4. Fix the opponent's mixed strategy *<sup>i</sup>* .

5. *<sup>i</sup>* , calculate the average index of the resultant expected value according to (11).

6. Order the expected values according to their average index in descending order, according to (9).

7. Transform the ordering preference relation into fuzzy preference relation according to (18-19).

8. Derive priorities of expected values from the resultant fuzzy preference relation based on the least deviation method.

9. For every mixed strategy , *i i* , set its grade of being NE equal to its corresponding priorities.

10. For every mixed strategy , *i i* , set its overall grade of being NE equal to the minimum of its grade of being NE for each player.

**Table 1.** Algorithm to find the grade of being NE for each mixed strategy

### **6. Simulation results**

#### **6.1. Application in pure strategy**

In this subsection, games with two players are considered because of their easier understanding. However, our algorithm can be implemented to more than two players. Consider a fuzzy game, as defined in Table 2. Player one and player two have three strategies, namely a1, a2, a3 and b1, b2, b3, respectively. Each cell includes two fuzzy triangular numbers as payoffs; the first for player one and the second for player two.

Nash Equilibrium Strategies in Fuzzy Games 319

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

 b1 b2 b3 a1 0.08 0.7 0.22 a2 0.08 0.08 0.08 a3 0.22 0.08 0.7

a1 0.7 0.08 0.22 a2 0.08 0.08 0.08 a3 0.22 0.08 0.7

The difference between the results of Tables 5 and 6 is meaningful. As the player 2 becomes optimistic, he prefers to choose the greatest possible outcome. Hence, between three alternatives*TTT* (3,2), (3,1), (3,1.5) , *T*(3,2) has the most priority. But a pessimistic player

In this subsection, mixed strategies in bi-matrix games are considered. Consider a fuzzy bi-

a1 T(10,11,12),T(3,4,5) T(3,4,5),T(0.5,2,3.5) a2 T(2,4,6),T(0,1,2) T(6,7,8),T(2,4,6)

> ).

Using the proposed approach, every mixed strategy has a grade of being NE as shown in

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

p2

1 2 , b1 b2

b1 b2 b3

**Table 5.** Grades of being NE for λ ≤ 1/4

**Table 6.** Grades of being NE for λ > 1/4

prefers to choose alternative*T*(3,1) .

**6.2. Application in mixed strategy** 

**Table 7.** A sample game with fuzzy payoffs

**Figure 2.** Grades of being NE for Player 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 1

Figures 2 and 3 for each neutral player ( 0.5

matrix game as defined in Table 7.


#### **Table 2.** A sample game with fuzzy payoffs

Implementing the algorithm presented in Table 1, there can be several grades of being NE for the game corresponding to different s. Tables 3 and 4 present the priority of each payoff according to possible choices of .


**Table 3.** Priorities of payoffs for λ ≤ ¼


#### **Table 4.** Priorities of payoffs for λ > 1/4

The minimum priority of all payoffs in a cell is interpreted now as the grade of being Nash equilibrium of that cell. The results of the mentioned game is tabulated in Tables 5 and 6 for the possible choice of , respectively


**Table 6.** Grades of being NE for λ > 1/4

318 Game Theory Relaunched

**6. Simulation results** 

**6.1. Application in pure strategy** 

**Table 2.** A sample game with fuzzy payoffs

for the game corresponding to different

payoff according to possible choices of

**Table 3.** Priorities of payoffs for λ ≤ ¼

**Table 4.** Priorities of payoffs for λ > 1/4

, respectively

the possible choice of

In this subsection, games with two players are considered because of their easier understanding. However, our algorithm can be implemented to more than two players. Consider a fuzzy game, as defined in Table 2. Player one and player two have three strategies, namely a1, a2, a3 and b1, b2, b3, respectively. Each cell includes two fuzzy

triangular numbers as payoffs; the first for player one and the second for player two.

1 2 , b1 b2 b3

a1 T(4,5,6),T(1,3,5) T(5,6,7),T(2,3,4) T(4,5,6),T(1.5,3,4.5)

a2 T(2,3,4),T(0,1,2) T(1,3,5),T(3,4,5) T(2,3,4),T(1,3,5) a3 T(3,4,5),T(2,4,6) T(3,5,7),T(1,3,5) T(6,7,8),T(4,6,8)

Implementing the algorithm presented in Table 1, there can be several grades of being NE

b1 b2 b3

a1 (0.7,0.08) (0.7,0.7) (0.22,0.22) a2 (0.08,0.08) (0.08,0.7) (0.08,0.22)

a3 (0.22,0.22) (0.22,0.08) (0.7,0.7)

 b1 b2 b3 a1 (0.7,0.7) (0.7,0.08) (0.22,0.22) a2 (0.08,0.08) (0.08,0.7) (0.08,0.22) a3 (0.22,0.22) (0.22,0.08) (0.7,0.7)

The minimum priority of all payoffs in a cell is interpreted now as the grade of being Nash equilibrium of that cell. The results of the mentioned game is tabulated in Tables 5 and 6 for

s. Tables 3 and 4 present the priority of each

. The difference between the results of Tables 5 and 6 is meaningful. As the player 2 becomes optimistic, he prefers to choose the greatest possible outcome. Hence, between three alternatives*TTT* (3,2), (3,1), (3,1.5) , *T*(3,2) has the most priority. But a pessimistic player prefers to choose alternative*T*(3,1) .

#### **6.2. Application in mixed strategy**

In this subsection, mixed strategies in bi-matrix games are considered. Consider a fuzzy bimatrix game as defined in Table 7.


**Table 7.** A sample game with fuzzy payoffs

Using the proposed approach, every mixed strategy has a grade of being NE as shown in Figures 2 and 3 for each neutral player ( 0.5 ).

**Figure 2.** Grades of being NE for Player 1

Nash Equilibrium Strategies in Fuzzy Games 321

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

**Figure 5.** Grades of being NE for optimistic players

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p1

**Figure 6.** Grades of being NE for pessimistic players

In this chapter, some concepts in fuzzy sets including fuzzy numbers, fuzzy extension principle, ranking of fuzzy numbers and fuzzy preference relations were briefly introduced and consequently used to develop a new approach for practically analyzing the games with fuzzy numbers as payoffs. In this regard, a fuzzy preference relation was constructed on the preference ordering of payoffs using ranking fuzzy numbers. The priority of each payoff then was derived using the least deviation method. The priorities of payoffs were interpreted as the grades of being NE. We should remark that, in this chapter, we were not looking to show the pure and mixed strategy NEs, but rather we tried to assign a graded membership to each

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

p2

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

p2

strategy to determine how much it is NE, i.e. it has the possibility for being NE.

**7. Conclusion** 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p1

**Figure 3.** Grades of being NE for Player 2

Now, the minimum priorities for players can be interpreted as the overall grades of being NE. We remark that the minimum operator can be replaced by any other T-norms. The results of the game presented in Table 7, is shown in Figure 4.

**Figure 4.** Grades of being NE for neutral players

Grade of being NE is high around two pure strategies, that is 1 2 *p p* (0,1), (0,1) and 1 2 *p p* (1,0), (1,0) . The mixed strategy 1 2 *p p* (0.7,0.3), (0.3,0.7) has also high grade of being NE.

In addition, the effect of optimism-pessimism degree of players in distribution of Nash grades is studied. For instance, if two players are optimistic ( 1 ), the Nash grades are the same as Figure 5. If two players are pessimistic ( 0 ), the Nash grades are the same as Figure 6.

The difference between the results of Figures 4 and 5 is important. As the player becomes optimistic, he prefers to choose the greatest possible outcome. Hence, payoff *T*(4,1.5) gets more priority than payoff *T*(4,1) . In addition, Nash grades for neutral players are approximately the combination of Nash grades for optimistic and pessimistic players.

**Figure 5.** Grades of being NE for optimistic players

**Figure 6.** Grades of being NE for pessimistic players

#### **7. Conclusion**

320 Game Theory Relaunched

**Figure 3.** Grades of being NE for Player 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 1

**Figure 4.** Grades of being NE for neutral players

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p1

of being NE.

Figure 6.

results of the game presented in Table 7, is shown in Figure 4.

Now, the minimum priorities for players can be interpreted as the overall grades of being NE. We remark that the minimum operator can be replaced by any other T-norms. The

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

p2

Grade of being NE is high around two pure strategies, that is 1 2 *p p* (0,1), (0,1) and 1 2 *p p* (1,0), (1,0) . The mixed strategy 1 2 *p p* (0.7,0.3), (0.3,0.7) has also high grade

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> <sup>0</sup>

p2

In addition, the effect of optimism-pessimism degree of players in distribution of Nash

The difference between the results of Figures 4 and 5 is important. As the player becomes optimistic, he prefers to choose the greatest possible outcome. Hence, payoff *T*(4,1.5) gets more priority than payoff *T*(4,1) . In addition, Nash grades for neutral players are

approximately the combination of Nash grades for optimistic and pessimistic players.

), the Nash grades are the

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

), the Nash grades are the same as

grades is studied. For instance, if two players are optimistic ( 1

same as Figure 5. If two players are pessimistic ( 0

In this chapter, some concepts in fuzzy sets including fuzzy numbers, fuzzy extension principle, ranking of fuzzy numbers and fuzzy preference relations were briefly introduced and consequently used to develop a new approach for practically analyzing the games with fuzzy numbers as payoffs. In this regard, a fuzzy preference relation was constructed on the preference ordering of payoffs using ranking fuzzy numbers. The priority of each payoff then was derived using the least deviation method. The priorities of payoffs were interpreted as the grades of being NE. We should remark that, in this chapter, we were not looking to show the pure and mixed strategy NEs, but rather we tried to assign a graded membership to each strategy to determine how much it is NE, i.e. it has the possibility for being NE.

## **Author details**

#### Alireza Chakeri

*Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran* 

Nash Equilibrium Strategies in Fuzzy Games 323

[15] Li K. W, Karray F, Hipel K. W, Kilgour D. M (2001) Fuzzy Approach to the Game of

[17] Butnariu D (1979) Solution Concepts for n-Person Fuzzy Games. Advances in Fuzzy Set

[18] Zimmermann H. J (1996) Fuzzy Set Theory and its Applications. Kluwer Academic

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Nasser Sadati

*Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran* 

*Electrical and Computer Engineering Department, The University of British Columbia, Vancouver, BC, Canada* 

Guy A. Dumont

*Electrical and Computer Engineering Department, The University of British Columbia, Vancouver, BC, Canada* 

## **8. References**


**Author details** 

*Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology,* 

*Intelligent Systems Laboratory, Electrical Engineering Department, Sharif University of Technology,* 

*Electrical and Computer Engineering Department, The University of British Columbia, Vancouver,* 

*Electrical and Computer Engineering Department, The University of British Columbia, Vancouver,* 

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Alireza Chakeri

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*Tehran, Iran* 

*BC, Canada* 

*BC, Canada* 

Guy A. Dumont

**8. References** 

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Nasser Sadati


[35] Fudenberg D, Tirole J (1991) Game Theory. MIT Press, Cambridge, MA.

**Chapter 16** 

© 2013 Wagner, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Wagner; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Game Theory as Psychological Investigation** 

Over the course of history mathematics and science have become increasingly entangled with one another. This has been especially true in the physical science wherein mathematical derivations have resulted in subsequent experimental pursuits. While the social sciences have relied on mathematical descriptions in recent generations it was not until the more general employment of game theory that mathematical modeling became itself a directive tool for subsequent evaluation as it had so become in the physical sciences, most prominently perhaps in high energy physics. This chapter shows how game theory forced upon the social sciences new avenues of investigation. It also shows that once those investigations were fully underway they in turn forced new considerations on the practice of game theory modeling. This boot-strapping dynamic between mathematical game theory and social science represents a novel turn in the relationship between the two disciplines bringing their relationship more into parallel alignment with what has long existed between

Science and mathematics travel along coincidental paths. Just how coincidental is a source of perennial speculation and argument among sociologists, philosophers and historians of both science and mathematics. If the paths math and science travel are only coincidentally linked then neither serves as foundation or guide for the other. On the other hand, if the paths are destined to be linked in some fashion then the affiliation between the two is more than merely coincidental. Any answer to this query seems necessarily to raise the question of whether either, at least in its modern form, can exist without the contemporary development of the other. Any effort to address this issue inevitably prompts consideration of the ageless

Early explanations of the natural world can be put forward in strictly naturalistic terms common to the researcher's native tongue or in some cases scholarly language such as Latin or Greek in the West and Mandarin in the East. The science of antiquity was largely free of any necessary bondage to mathematics. Admittedly as soon as early cosmic observers in

Paul A. Wagner

**1. Introduction** 

http://dx.doi.org/10.5772/53932

physical science and mathematics.

question of whether mathematics is invented or discovered.

Additional information is available at the end of the chapter
