1

Abstract—In this paper, we introduce Fuzzy Nash Equilibrium

to determine a graded representation of Nash equilibriums in

crisp and fuzzy games. This interpretation shows the distribution

of equilibriums in the matrix form of a game and handles

uncertainties in payoffs. Also a new method for ranking fuzzy

values with the user's viewpoint is investigated. By this mean the

definition of Satisfaction Function, which provides the result of

comparison in the form of real value, is developed when users

have preferences regarding the payoffs.

Index Terms—Fuzzy game, fuzzy Nash equilibrium, fuzzy

preference relation, satisfaction function, fuzzy value.

I. INTRODUCTION

game is a decision making system that involves more

than one decision maker each having profits that conflict

with each other. A strategic game first defines each player's

actions (strategy). The combination of all the players’

strategies will determine an outcome to the game and the

payoffs to all players in which each player tries to maximize

his own payoff. The traditional game theory assumes 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, imprecision in the available

information of the environment or the behavior of the other

players. Initially, fuzzy sets were used by Butnariu [1] in non-

cooperative game theory. He used fuzzy sets to represent the

belief of each player for strategies of other players. There have

been several approaches to extend fuzzy cooperative games;

Butnariu in [2] introduced core and stable sets in fuzzy

coalition games where a degree of participation of players in a

coalition is assigned. Moreover, Mares in [3] considered fuzzy

core in fuzzy cooperative game where possibility of each

fuzzy coalition is fuzzy interval as an extension of core in

classic TU games. Also he discussed Shapely value in

cooperative game with deterministic characteristic and fuzzy

coalition. Fuzzy game theory has been applied to many

competitive decision-making situations [4-18]. Vijay et al. [4]

Manuscript received April 11, 2010; revised September 25, 2010 and

October 10, 2011; accepted April 13, 2012. A. Chakeri is with the Electrical and Computer Engineering Department,

Isfahan University of Technology, Isfahan, Iran (e-mail:

alireza.chakeri@ec.iut.ac.ir). F. Sheikholeslam is with the Electrical and Computer Engineering

Department, Isfahan University of Technology, Isfahan, Iran (e-mail:

sheikh@cc.iut.ac.ir). Digital Object Identifier 10.1109/TFUZZ.2012.2203308

considered a game with fuzzy goals and fuzzy parameters and

proved that such a game is equivalent to a primal–dual pair of

certain fuzzy linear programming (FLP) problems with fuzzy

goals and parameters. Chen and Larbani in [6] discussed

multiple attribute decision making with a two person zero-sum

game and achieved simpler criteria for solving the

corresponding FLP. Liu and Kao in [7] defined the value of

games in fuzzy form. Nishizaki and Sakawa [8] discussed

fuzzy bimatrix game and by using a nonlinear programming

the equilibriums were searched. Maeda [9] studied zero-sum

bi-matrix games with fuzzy payoffs. In [10], a fuzzy

differential game approach was proposed to solve the N-

person quadratic differential non-cooperative and cooperative

game. Kima and Leeb in [13] considered fuzzy constraints as

well as fuzzy preference and proved some theorems on the

existence of equilibrium. Song and Kandel in [14] used a

multi-goal problem, where the degree of satisfaction for each

goal was a fuzzy one and the overall payoff is a weighted sum

of the satisfaction of all goals. They assumed that each player

has a fuzzy knowledge about his opponents mixed strategies,

i.e. each player assigns a membership function to the

probability distribution of his opponents’ strategies. Garagic

and Cruz in [15] transformed a game with fuzzy strategies and

fuzzy payoffs to a crisp game using Fuzzy IF-Then rules.

Subsequently, they discussed the Nash equilibriums in the

equivalent crisp game; they proved that this crisp game has at

least one pure strategy Nash equilibrium. Li et al. in [16]

employed two fuzzy approaches including fuzzy multicriteria

decision making method and the theory of fuzzy moves to

investigate the game of chicken. Their model incorporates the

player's subjective manner and imprecise knowledge to the

game model. In [17], [18] a mathematical programming

approach of fuzzy matrix games with intuitionistic fuzzy

payoffs and interval-valued intuitionistic fuzzy (IVIF) payoffs

[19] was developed. It is proven that each matrix game with

IVIF payoffs has a solution.

In this paper, we develop a new approach to N-person crisp

and fuzzy non-cooperative games to obtain Nash equilibriums

for these kinds of problems. The most significant advantages

of using the proposed method are the range of game-theoretic

problems that can be analyzed and the information about

equilibriums that can be obtainable to players. In the proposed

approach, the definition of equilibrium in crisp and fuzzy

games has been generalized to show distribution of Nash

equilibriums in matrix games and also present the amount of

optimality of the players’ strategies by a degree. In this regard,

Fuzzy Nash Equilibriums in Crisp and Fuzzy

Games

Alireza Chakeri, Farid Sheikholeslam, Member, IEEE

A

1063-6706/$31.00 © 2012 IEEE

2

we do not need to determine whether a pure strategy is Nash

equilibrium. Instead, we assign a graded membership to any

pure strategy that describes to what possibility it is Nash

equilibrium. Hence, we can consider strategies with high

degrees of equilibrium which are not necessarily the

equilibrium points. In fuzzy games, fuzzy Nash equilibrium

approach is more appropriate for real world problems which

are modeled by game theory. The proposed approach avoids

loss of any information that happens by defuzzification

method in games and handles uncertainty of payoffs through

all steps of finding Nash equilibrium. It shall be noted that in

this approach the existence of theorem for equilibrium was not

established since the focus is not on existence of equilibrium

but in the degree of equilibrium.

The paper is structured as follows. Section II introduces the

concept of degree of being Nash equilibrium in games with

crisp payoffs using the fuzzy preference relation. In Section

III, a new function modeling the SF is defined. In this

definition the weights of the domain in the fuzzy values are

considered directly in the formula. Also, Section III explains a

new approach in fuzzy games using the satisfaction function.

Moreover, the consequence of player's viewpoints in Nash

degrees of cells is discussed. The conclusion highlights the

main findings of the paper.

II. GAMES WITH CRISP PAYOFFS

A. Non-cooperative N-person Games

This subsection contains the background on game theory,

which are needed to develop the fuzzy Nash equilibrium.

Games have been classified by the number of players, the

number of strategies, the nature of the payoffs function, and

cooperativeness. A normal game consists of a set of players,

their strategies and the payoffs available for all combinations

of players’ strategies.

Non-cooperative N-person strategic game can be

formulated as follows [20]:

1) There are N players to be denoted by NPPP ,...,, 21 .

2) There are a finite number of alternatives for each player

to choose from. Let is denote the number of alternatives

available to iP , and further denote the index set

is,...,2,1 by iX , with a typical element of iX ,

designated as ix .

3) If iPi, chooses a strategy ii Xx , then the payoff

for iP is a single number Ni xxx ,....,, 21 . Also

Niiiiiiii XXXXxXxxx ......,, 111

is the set of all payoffs for iP where ix denotes the

strategies chosen by other players.

4) Players play with a policy of maximizing their payoff,

they take into account the possible rational choices of

the other players and they also make decision

independently.

Unlike one-player decision making, where optimality has an

explicit meaning, in multi person decision making the

optimality is in the form of Nash equilibrium. A pure strategy

Nash Equilibrium is a strategy set wherein, if a player knows

his opponent’s strategy, he is totally satisfied with his

equilibrium strategy and is reluctant to change his strategy

[21].

Definition 1: Classical game theory, determines a cell as a

Nash Equilibrium if and only if it maximizes all players'

payoffs when other players refrain from changing their

strategies, **

1 ,, Nxx is the pure strategy Nash equilibrium

if and only if [22]

iii

Nii

iiiiii

Nii

XxxNi

xxx

xxxx

xxx

*

**

1

***

***

1

,,...,1

,,,,

,,

,,,,

(1)

One can rephrase the classical game theory solution to a

simple algorithm in two steps:

1) Replace iii xx , with 1 if ii xx , is

maximized by ixx , otherwise replace it with 0.

2) Find the minimum of the elements of each cell: if this

value equals 1, the cell is Nash equilibrium and if this

value gets zero, the cell is not Nash equilibrium.

For instance consider the simple game of Table I where

each cell includes two crisp payoffs; the first for player 1 and

the second for player 2. Player 1 and 2 have three strategies,

namely J1, J2, J3 and K1, K2, K3, respectively. Table II shows

the Nash equilibrium cells.

TABLE I A SAMPLE GAME WITH CRISP PAYOFFS

Π1,Π2 K1 K2 K3

J1 (0.85,0.322) (0.97,0.324) (0.81,0.323)

J2 (0.29,0.04) (0.3,0.52) (0.29,0.322)

J3 (0.58,0.51) (0.81,0.322) (0.99,0.88)

TABLE II

RESULTS OF THE GAME OF TABLE I

K1 K2 K3

J1 0 1 0

J2 0 0 0

J3 0 0 1

For each pair of strategies, the numbers 0 and 1 indicate not being

and being a Nash equilibrium, respectively

B. Fuzzy Nash Equilibriums in Games with Crisp Payoffs

According to classical game theory, if a player knowing

3

others' strategies chooses a strategy whereby he cannot get

maximum payoff, he will completely regret his choice.

However, just as the Table I depicts, in real world problems,

there may be situations where the difference between payoffs

are negligible, e.g. payoffs 0.323 and 0.324 in Table I. If a

player chooses 0.323, he/she will be quite satisfied with

his/her choice. Therefore, one can consider 0.322, 0.323 and

0.324 to have approximately same value in Table I. This

approximation prompted the employment of fuzzy logic to

make a soft measurement between payoffs. In this paper,

instead of the logic "greater than" relation, a new measure is

defined, that of the amount being greater; for instance, the

potential amount of being greater between 1000 and 1 is

greater than that between 2 and 1.The amount of being greater

may be perceived differently by each particular player. In

other words, it depends on the mental state and beliefs of that

player. The more meticulous and greedy a player is, the

greater a difference he/she perceives between slightly different

payoffs.

A new term is defined in this paper to determine the amount

of being greater between two payoffs in (2); this value is

interpreted as the preference between two payoffs. Then using

fuzzy preference relation matrix [23], priorities are calculated

by the Least Deviation Method [24], in which the priority

vector is a vector which determines the degree of importance

of alternatives. Here, the grades of being Nash equilibriums

are modeled according to the priority that players feel for their

strategies. This definition for the grade of being Nash

equilibrium seems meaningful because, if a player knows the

opponent’s strategy, he is satisfied with his strategy to the

degree that this strategy has priority for him. The greater

priority the players feel for each cell, the greater the possibility

that the cell is the game’s equilibrium.

Algorithm: First, it is necessary to normalize all payoffs

and make the range between 0 and 1. Let ba, be the payoffs of

selecting strategies ji, , respectively for a player if the

opponents' strategies are fixed. The amount of being greater

between ba, can be calculated as follows:

baa

abab

ba

bab

baba

pij

1,5.))(1()(min1

5.0

1,5.))(1()(min

(2)

The algorithm shown in Fig. 1 is designed to determine to

what degree a cell belongs to the fuzzy set of Nash

Equilibriums. This algorithm assigns each cell the minimum

priority of players as the degree of being Nash equilibrium.

However, the minimum operator can be replaced by any other

T-norm; Example 1: The algorithm described in Fig. 1 is

implemented in the game of Table I assuming 5.0 . The

results of the first two steps are shown in Table III.

In the first row of the game (Table I), player 2 has

approximately the same payoffs and player 1 has relatively

high payoffs for all cells. Therefore, one may expect they have

approximately the same degrees of being Nash equilibrium.

This is exactly what the proposed algorithm has calculated and

is obvious in the first row of Table IV. In other words, a

relatively small difference in player 2’s payoff has resulted in

the distribution of an equilibrium degree in the first row.

TABLE III

PRIORITY MATRIX OF THE GAME OF TABLE I

K1 K2 K3

J1 (0.7,0.332) (0.61,0.335) (0.38,0.333)

J2 (0.05,0.03) (0.05,0.69) (0.03,0.28)

J3 (0.25,0.16) (0.34,0.06) (0.59,0.78)

TABLE IV

DEGREE OF BEING NASH EQUILIBRIUM FOR THE GAME OF TABLE I USING

MINIMUM AS T-NORM OPERATOR

K1 K2 K3

J1 Nash of deg 0.332 Nash of deg 0.335 Nash of deg 0.333

J2 Nash of deg 0.03 Nash of deg 0.05 Nash of deg 0.03

J3 Nash of deg 0.16 Nash of deg 0.06 Nash of deg 0.59

The effect of δ on the priorities of payoffs (0.97, 0.3, 0.81)

and (0.85, 0.29, 0.58) is shown in Fig. 2 and Fig. 3,

respectively. The only correlation that can be identified in

these figures is that as δ increases, the priority of the highest

payoff decreases. As the figures shows, there is no linear

relationship between and the priority.

Fig. 1. The algorithm for finding the degree of belonging to Nash

Equilibrium fuzzy set for each cell of the crisp payoffs game

Step 1: make a matrix with the size of payoffs matrix and

initialize all items to zero (this matrix is called priority matrix in

this algorithm)

Step 2: for all players Pi

- Fix the strategy of other players(x-i)

- Calculate the priority of all payoffs of Pi using fuzzy preference relation

- For all strategies of Pi

o Put priority of strategy xi in the ith element of cell

(xi, x-i)

Step 3: determine the minimum of the elements of each cell as

the degree of being Nash Equilibrium. (Find the graded

membership)

4

Fig. 2. Priority of payoffs (0.97, 0.3, 0.81) versus δ.

Fig. 3. Priority of payoffs (0.85, 0.29, 0.58) versus δ.

III. GAMES WITH FUZZY PAYOFFS

A. Possibility of Being Greater Between Fuzzy Values

In the fuzzy decision making and fuzzy game theory with

fuzzy payoffs, ranking the fuzzy value is a necessary

procedure. Various methods for ranking fuzzy subsets have

been planned [25-32]. Though most methods can only rank

fuzzy values, in [25] the credibility measure as the summation

of possibility and necessity measure is used to show the

degree of greatness. However, their method can not consider

the possibility distribution of fuzzy values. In [26-28], the

satisfaction function (SF) as the truth value of an arithmetic

comparison between fuzzy values was introduced. However,

method in [28] can only rank the fuzzy values when there are

viewpoints but cannot show the degree of being greater. In

addition, it is unclear how a fuzzy number with indefinite

substance can compare with viewpoint. In other words, the

fuzzy sets have nature of possibility but viewpoints are

constructed as user's preferences and interests.

This paper introduces a new method for calculating SF

when users have a viewpoint. The user's viewpoint is

incorporated in the domain of value. By this means, each

element x is extended to xxV , where xV is the user

viewpoint. Hence, the membership function of fuzzy values is

modified dependent on viewpoint, as follows:

))((max)()(

' xhx Axh

A

(3)

where h is a multi-valued mapping, i.e. xxxVh ))(( .

In the proposed method, the length of -cuts of the fuzzy

value is increased or decreased according user viewpoint. We

suggest the following formula for comparison between fuzzy

values A and B where their modified membership function is

used

dydxyx

dydxyxBASF

BA

y

BA

V

)()(

)()()(

''

''

(4)

dydxyx

dydxyxABSF

BA

yBA

V

)()(

)()()(

''

''

(5)

where the operator is a T-norm, e.g., it can be min or the

multiplication operator.

)( BASFV and

)( BASFV determine the possibility of

truth of the BA and BA , i.e. they represent the

possibility that fuzzy number A is smaller than B and A is

larger than B respectively. It is obvious that

1)()( BASFBASF VV .

Since'A and

'B in definition (3) are fuzzy values, all

properties which confirmed in [28] holds for definitions (4)

and (5) in this paper.

Example 2: The following example is presented to show

how the user viewpoint affects the value of SF. It consists of

the categories of optimism, pessimism and neutral user.

Consider two fuzzy values A and B , and three viewpoints 1V ,

2V and 3V shown in Fig. 4 and Fig. 5 respectively. Table V

shows the a comparison between fuzzy values A and B using

the approach in [28] and Table VI shows the results of a

comparison between fuzzy values A and B using (4) and (5).

As shown in Table V, the method for ranking fuzzy values

in [28] is insensitive to the user viewpoints and it is unable to

determine the effect of viewpoints since it can only rank fuzzy

values when user interests exist. However, it seems that the

result for pessimistic players ( 1V ) gives A larger than B , the

result of proposed method is congruous as expected. If the

user is a pessimistic one, he is satisfied with the lower value.

Hence the low value has great importance to the user and

)( BASFV is less than 0.5, because A is closer to zero.

Additionally, if a user is optimistic, he/she prefers to choose

the high value, and the high value is more important to

him/her than a low value. Table VI shows that )( BASFV

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Priority

payoff

0.3

payoff

0.81

payoff

0.97

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Priority

payoff

0.29

payoff

0.58

payoff

0.85

5

for an optimistic user is bigger than )( BASFV for a neutral

user, since in optimism one the high value has more weights

and fuzzy value B has value of domain in high value.

Fig. 4. Two simple fuzzy values A and B.

Fig. 5. Three ViewpointsV1, V2 and V3 show pessimistic, neutral and

optimistic player, respectively.

TABLE V RANKING FUZZY VALUES A AND B USING METHOD IN [28]

V1 V2 V3

B > A B > A B > A

TABLE VI

RANKING FUZZY VALUES A AND B USING (4) AND (5)

V1 V2 V3

SFV (A < B) 0.3 0.9895 0.9914

SFV (A > B) 0.7 0.0105 0.0086

B. Fuzzy Nash Equilibriums in Games with Fuzzy Payoffs

There have been many studies for defining a game in fuzzy

parameters. As discussed earlier, a game has four main

components: a set of players, a set of strategies for each

player, a set of payoffs and preference relationship. Defining

each of these components as a fuzzy component would lead to

a fuzzy game. Most of the previous works on fuzzy games is

concerned with defining fuzzy payoffs and as a result defining

a preference on these fuzzy payoffs. An example is in an

election, where the candidates may select different campaign

issues on which to focus. Different issues may bring different

votes, and the number of votes can only be estimated. For

instance, candidates may think that if they concentrate on a

specific issue for each of the number of votes there is

possibility. Fuzzy sets theory is shown to be an appropriate

means to model these uncertainties.

In this paper, a new method is proposed for finding degree

of being Nash equilibrium of each cell. This explanation

determines the distribution of the degree of being Nash

equilibrium in the matrix game. In fact, the algorithm in

classical game theory, mentioned in section II, is modified in

the case of having uncertainty in payoffs, i.e. it is a

generalization of the classical game theory algorithm. In

classical game theory a crisp payoff is clearly greater than

another one or not, but in the case of a fuzzy payoff, there is

uncertainty in ranking fuzzy values. These uncertainties are

shown by the degree of truth of arithmetic comparison, i.e. the

satisfaction function.

Definition 2: if each player has viewpoint iV , every N-tuple

strategy Nxx ,,1 has a possibility of being pure strategy

Nash equilibrium with the degree of

Ni

Nii

VXxx

Ni

NNash

xxx

xxxSF

xx

iii ,,,,

,,,,min

min

,,

1

1

},...,1{

1

(6)

Also, in matrix games it can be stated in two steps:

1) Replace iii xx , with

iiiiiVXxx

xxxxSFi

ii

,,min .

2) Find the minimum of the elements of each cell. This

value is the degree of being Nash equilibrium for that

cell.

Example 3: Regarding crisp games, two person games are

discussed because they are easier to consider, but they can be

generalized to more than two players. For instance, consider

the fuzzy game described in Table VII where baT , denotes

a fuzzy triangular number with a center on a and boundaries

on ba such as Fig. 6. The results of the proposed algorithm

on the game are shown in Tables VIII and IX for neutral

players.

Fig. 6. A Simple fuzzy triangular number A (T(5,1)).

TABLE VII

A SIMPLE GAME WITH FUZZY PAYOFFS

Π1,Π2 K1 K2 K3

J1 T(5,1),T(3,2) T(6,1),T(3,1) T(5,2),T(3,1.5)

J2 T(3,1),T(1,1) T(3,2),T(4,1) T(3,1),T(3,2)

J3 T(4,1),T(4,2) T(5,2),T(3,2) T(7,1),T(6,2)

1

x

V1(x)

V2(x)

V3(x)

8

A B

2 3 4 5 7

x

1

μA(x)

4 5 6

x

A 1

6

TABLE VIII FINDING NASH EQUILIBRIUM DEGREE FOR EACH NEUTRAL PLAYERS

K1 K2 K3

J1 (0.958,0.5) (0.854,0.5) (0.011,0.5)

J2 (0,0) (0,0.854) (0,0.146)

J3 (0.042,0.041) (0.146,0.002) (0.989,0.958)

TABLE IX

FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR NEUTRAL PLAYERS

K1 K2 K3

J1 Nash of deg 0.5 Nash of deg 0.5 Nash of deg 0.011

J2 Nash of deg 0 Nash of deg 0 Nash of deg 0

J3 Nash of deg 0.041 Nash of deg 0.002 Nash of deg 0.958

For analyzing the effect of user viewpoint in the distribution

of Nash degrees, consider 1V and 3V in Fig. 5 for both players,

in which 1V shows a pessimistic player and 3V shows an

optimistic one. Tables X and XI determine the graded Nash of

each cell for pessimism and optimism players, respectively:

TABLE X

FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR PESSIMISM

PLAYERS

K1 K2 K3

J1 Nash of deg 0.066 Nash of deg 0.262 Nash of deg 0.3

J2 Nash of deg 0 Nash of deg 0.5 Nash of deg 0.052

J3 Nash of deg 0.762 Nash of deg 0.238 Nash of deg 0

TABLE XI

FINDING NASH EQUILIBRIUM DEGREE OF EACH CELL FOR OPTIMISM

PLAYERS

K1 K2 K3

J1 Nash of deg 0.54 Nash of deg 0.424 Nash of deg 0.013

J2 Nash of deg 0 Nash of deg 0 Nash of deg 0

J3 Nash of deg 0.043 Nash of deg 0.003 Nash of deg 0.956

Comparing the results of Tables X and XI, one can

conclude that as players become less greedy, i.e. experiencing

higher degrees of satisfaction from lower payoffs, degrees of

being Nash equilibrium become more widely distributed in the

games matrix and the degrees grow closer to each other. This

effect occurs because when players become less greedy the

priority of different payoffs will increase and become more

similar to others. In the first row of the game, when players

are optimistic, they prefer to choose 2,3T and, hence, this

payoff yields more SF than when players are neutral.

Moreover, when players are pessimistic, cells with a high

Nash degree in neutral and optimistic cases change to cells

with a low Nash degree, and this is because the of inversion of

the user viewpoint.

IV. CONCLUSION

In this paper, a new approach is introduced for analyzing

games more realistically than previous models. In the first

part, only the preference relationship is generalized to a fuzzy

one, i.e., the relationship of "greater than or equal" is extended

to a fuzzy one, which describes how much a crisp number is

greater than or equal to another number. In crisp games, a

fuzzy preference relation was employed for comparing payoffs

and calculating the priority of each payoff using the Least

Deviation method. Using this priority, a value of being

equilibrium is computed and it is shown that this value yields

more realistic results.

In the case of having fuzzy payoffs, the definition of

satisfaction function when players have viewpoints is

improved. The proposed method incorporates player

viewpoints in the domain of fuzzy value and transforms it to

another fuzzy value. The algorithm for finding the Nash

degree of each cell is proposed. Finally, the effect of different

viewpoints on the result of the game is studied. Comparing the

results to the fuzzy Nash equilibrium, the results obtained

through this strategy were more sensitive to the payoffs.

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