COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.4, October 2015
DOI : 10.5121/ijfls.2015.5402 11
COMPARISON OF DIFFERENT APPROXIMATIONS OF
FUZZY NUMBERS
D. Stephen Dinagar1
and K.Jivagan2
1,2
PG and Research Department of Mathematics, T.B.M.L. College, Porayar, India.
ABSTRACT
The notions of interval approximations of fuzzy numbers and trapezoidal approximations of fuzzy numbers
have been discussed. Comparisons have been made between the close-interval approximation, value-
ambiguity interval approximation and distinct approximation with the corresponding crisp and trapezoidal
fuzzy numbers. A numerical example is included to justify the above mentioned notions.
KEYWORDS
Close interval approximations, value ambiguity interval approximations, Distinct approximations.
1. INTRODUCTION
The game theory is widely applied to military affairs, physical training, and commercial
production and so on. But the classical game theory is based on the double value logic theory lose
sight of much fuzzy information and grey information.
A game is a description of a decision-making situation involving more than one decision maker.
Game theory is generally considered to have begun with the publication of von Neumann and
Morgenstern’s The Theory of Games and Economic Behavior [1,13] in 1944. The development
of game theory accelerated in papers by Nash [5,6], Shapley [7], [8], and Gillies [4]. The
behavior of players in a game is assumed to be rational and influenced by other rational players’
behaviors, which distinguishes a game from the general decision-making problem.
In order to analyze the behaviors of players and construct a method for each player to choose his
action, a strategic game first defines each individual’s alternative actions. The combination of all
the players’ strategies will determine a unique outcome to the game and the payoffs to all players.
A solution for a player in a game should allow that player to win or satisfy his objectives for the
game. For example, he can maximize his own payoff and/or minimize his opponent’s payoff.
Formally, the solution of a game is a situation in which each player plays a best response to the
other players’ actual strategy choices. This is the concept of equilibrium.
There are several methods for obtaining the equilibrium of some special kinds of games such as
the dominant strategy for dominant strategy equilibrium or a mixed strategy for mixed strategy
equilibrium. A linear programming method is used in matrix games. Most of these methods are
based on the maximin principle for selecting optimal strategies. But in 1978, Butnariu [2] pointed
out that one of the major assumptions of classical game theory is that all possible choosing
strategies are equally possible choices for a player.

12
The notion of fuzzy sets first appeared in the papers written by Zadeh [14,15]. This notion tries to
show that an object more or less corresponds to a particular category. The degree to which an
element belongs to a category is an element of the continuous interval [0,1] rather than the
Boolean pair {0,1} [15]. Using the notion of fuzzy sets, the payoff function in a game can be
fuzzified. Furthermore, the solution to the game may also be a fuzzy set.
The paper is organized as follows: The section 1 gives introduction is introductory in nature.
Section 2 gives the basic definitions and concepts needed for this work. In section 3, the
comparisons of different approximations in ‘fuzzy game notion’ have been presented. In section
4, the comparisons of different approximations in two person zero-sum method in fuzzy
environment have been discussed. In section 5, a relevant numerical example to compare the
different approximations of fuzzy numbers to justify the above discussed notions are included.
The concluding remarks are also added in the last section.
2. PRELIMINARIES
2.1. Definition (Fuzzy Set [FS])
Let X denote a universal set i.e., X={x}; then the characteristic function which assigns certain
values or a membership grade to the elements of this universal set within a specified range {0,1}
is known as the membership function and the set thus defined is called a fuzzy set. The
membership graders correspond to the degree to which an element is compatible with the concept
represented by the fuzzy set. If A
~µ is the membership function defining fuzzy set Ã, then,
]1,0[:µA
~ →X , where [0, 1] developed the interval of real numbers from 0 to 1.
2.2. Definition (Fuzzy Number [FN])
A convex and normalized fuzzy set defined on R. whose membership function is piecewise
continuous is called fuzzy number. A fuzzy set is called normal when at least one of its elements
attains the maximum possible membership grade.
i.e., 1)(max =xA
x
µ ,∀ x∈R
2.3. Definition ((((αααα-CUT)
An α -cut of a fuzzy set Ã is a crisp set αA
~
that contains all the elements of the universal set X
that have a membership grade in A greater than or equal to the specified value of a. Thus,
}10,)(µ;{xA
~
~ ≤≤≥∈= xxX A
αα
2.4. Definition (Close Interval Approximation of Trapezoidal Fuzzy Numbers
[CIATFN]) [11]
Let ‫ܣ‬ሚ =(ܽଵ, ܽଶ, ܽଷ, ܽସ) be a trapezoidal fuzzy number, and its interval approximation of fuzzy
number ]
~
[A ],[ UL
aa αα= is said to be close interval approximation if
2
}5.0)(/inf{ 21 aa
xAxa AL
+
=≥∈= µ
2
}5.0)(/{ 43 aa
xAxSupa AU
+
=≥∈= µ

13
2.5. Definition (Value of a Fuzzy Number [VFN]) [9]
Let Ã be a fuzzy number with α-cut representation (‫ܣ‬௅ሺߙሻ, ‫ܣ‬௎ሺߙሻ), then the value of Ã is defined
as
ܸ݈ܽሺÃሻ = න ߙሾ‫ܣ‬௎ሺߙሻ + ‫ܣ‬௅ሺߙሻሿ
ଵ
଴
݀ߙ
2.6. Definition (Ambiguity of a Fuzzy Number [AFN]) [9]
Let Ã be a fuzzy number with α-cut representation (‫ܣ‬௅ሺߙሻ, ‫ܣ‬௎ሺߙሻ), then the ambiguity of Ã is
defined as
‫ܾ݉ܣ‬ሺÃሻ = න ߙሾ‫ܣ‬௎ሺߙሻ − ‫ܣ‬௅ሺߙሻሿ
ଵ
଴
݀ߙ
2.7. Definition (Value-Ambiguity Interval Approximation of Fuzzy Number
[VAIA])[10]
An interval approximation operator ܸ஺ is defined as
ܸ஺: F(R) → ‫ܨ‬ூሺܴሻ in a way that ܸ஺: Ã→ܸ஺ሺÃሻ= [ܽି, ܽା]
where ܽି
= 2‫׬‬ ߙሺ‫ܣ‬௅ሺߙሻ
ଵ
଴
ሻ݀ߙ
ܽା
=2‫׬‬ ߙሺ‫ܣ‬௎ሺߙሻ
ଵ
଴
ሻ݀ߙ
Then the operator ܸ஺ is called a value-ambiguity interval approximation operator, Also the
interval ሾܽି
, ܽା
] is called the value – ambiguity interval approximation of Ã.
2.8. Definition (Distinct Approximation of Trapezoidal Fuzzy Number [DATrFN]) [12]
Let ‫ܣ‬ሚᇱ
= (ܽଵ
ᇱ
, ܽଶ
ᇱ
, ܽଷ
ᇱ
, ܽସ
ᇱ
) be a distinct approximation of ‫ܣ‬ሚ =(ܽଵ, ܽଶ, ܽଷ, ܽସ). where ܽଵ
ᇱ
=
ଷ௔భି௔మ
ଶ
,
ܽଶ
ᇱ
=
௔భା௔మ
ଶ
, ܽଷ
ᇱ
=
௔రା௔య
ଶ
, ܽସ
ᇱ
=
ଷ௔రି௔య
ଶ
Then the membership function of DATrFN is











+
≤<
−
−
−+
+
≤≤
+
−
<≤
+
−
+−
=
otherwise
aa
x
aa
aa
xaa
aa
x
aa
a
x
aa
aa
aax
xA
0
22
3
if
)(2
2)(
22
if1
2
3a
2
if
)(2
)(2
)(
4334
34
43
2143
2121
12
12
'
~µ

14
2.9. Definition (Associated Real Value of Approximation of Interval Fuzzy Number)
If 1 2[ ] [ , ]A t t=% is an close interval approximation of trapezoidal fuzzy number
),,,(
~
4321 aaaaA = , then its associated real value is given by
1 2[ ]ˆ
2
t t
A
+
=
2.10. Definition (Associated Real Value of Trapezoidal Fuzzy Number)
If ),,,(
~
4321 aaaaA = is a trapezoidal fuzzy number, then its associated real value is given by
4
)(ˆ 4321 aaaa
A
+++
=
2.11. Definition (Ranking of Fuzzy Numbers [RFN])
Let A
~
and B
~
be any two fuzzy numbers. Then A
~
is greater than B
~
if and only if BA ˆˆ f and
denoted by BA
~~
f .
2.12. Definition (Maximum of Fuzzy Numbers)
Let A
~
and B
~
be any two fuzzy numbers. Then the maximum value of A
~
and B
~
denoted by
( )BAMax
~
,
~
and is defined by ( ),Max A B A=% %% , if BA
~~
f .
2.13. Definition (Minimum of Fuzzy Numbers)
Let A
~
and B
~
be any two fuzzy numbers. Then the minimum value of A
~
and B
~
denoted by
( )BAMin
~
,
~
and is defined by ( ),Min A B A=% %% , if A B% %p .
2.14. Definition (Arithmetic Operations of Interval Fuzzy Numbers)
Let [ ],,
~
21 aaA = [ ]21,
~
bbB = ,∀ Rbbaa ∈2121 ,,, , be any two interval fuzzy numbers. Then
Addition:
[ ]21,aa ( )[ ] [ ]221121 ,, bababb ++=+
Subtraction:
[ ]21,aa ( )[ ] [ ]122121 ,, bababb −−=−
Multiplication:
[ ]1 2,a a ( )[ ] [ ]1 2 1 2 2 1, ,b b a b a b• = • •

15
2.15. Arithmetic Operations of Trapezoidal Fuzzy Numbers
Let ),,,(
~
4321 aaaaA = and ),,,(
~
4321 bbbbB = be any two trapezoidal fuzzy numbers.
Then
Addition:
=+ BA )( ),,,( 44332211 babababa ++++
Subtraction:
),,,()( 14233241 babababaBA −−−−=−
Multiplication:
),,,)()(,,,()( 43214321 bbbbaaaaBA •=•
),,,( 44332211 babababa=
3. FUZZY APPROACH TO STRATEGIC GAMES
A strategy is a complete plan for a player to decide how to play the game [3]. Without detailed
knowledge of the sequence of moves in a strategic game, we can still analyze the behaviors of
players; all that is needed for a solution to a game is to indicate what the player would do in the
situation when he must make a move. So it is sufficient if we list each individual’s strategies, the
combination of which will determine a unique outcome (payoff to each player) of a game. A
game is in strategic form if only the set of players I, the set of strategies iS and the payoff
functions )(sPi for each player Ii ∈ are given, where ( )ISSSS ,....., 21= . The concepts of
games in strategic form and mixed strategies used in this paper, as well as the notations, are as
defined and described in [3].
3.1. Methodology of Comparison of Different Approximations by Using Two-
Person Zero Sum in Existing Saddle Point Method in Fuzzy Environment
The fuzzy game with two players were the gain of one player is equal to the loss of the other
player is called two person ‘zero’ sum game. The resulting gain or loss for the player ‘A’ in a
game between players ‘A’ and ‘B’ can be represented in the form of a matrix which is called the
payoff matrix. In each entry are fuzzy numbers. To obtain the solution of the game we can use the
Minimax- Maximin principle. Using which we find the fuzzy saddle point.
Using this fuzzy saddle point, the value of the game is obtained and this saddle point has a
corresponding strategy for player ‘A’ and ‘B’. If a saddle point exists then the strategy follows by
pure-strategy. (If fuzzy saddle point does not exist then we have mixed strategy). The value of the
game which corresponding to the saddle point represents the Maximum guaranteed gain for
player ‘A’.
(i.e.,) Guaranteed
(i.e.,) Maximum possible loss for player ‘B’

16
The value of the fuzzy game is used to represent ‘V
~
’moreover if a saddle point exists then the
game is said to be strictly determined.
In general for any given fuzzy game the maximin value is denoted by ‘V
~
’ and minimax value is
denoted by ‘V
~
’ for any fuzzy game the following inequalities is
VVV
~~~
≤≤ if and only if
However if a fuzzy saddle point exists.
VVV
~~~
==
If the value of the game 0
~~
=V that such a fuzzy game is said to be fair game.
Steps to find the fuzzy saddle point
Step: 1
Find the minimum variable of each row of the payoff matrix and mark them.
Step: 2
Find the Maximum variable of each column of the payoff matrix and mark them.
Step: 3
If in this process then there exists a point in the payoff matrix with the both above marks. Then
this point corresponds to the saddle point. Using this value of saddle point is found. Using which
corresponding pure strategies for players ‘A’ and ‘B’ are found.
3.2. Close-Interval Approximation of Trapezoidal Fuzzy Numbers
(CIATrFN)
Let ),,,(
~
),,,,(
~
),,,,(
~
432143214321 ccccCbbbbBaaaaA === and ),,,(
~
4321 ddddD = be a
trapezoidal fuzzy numbers. Then the close-interval approximations of them given by
=)
~
(ACD 




 ++
2
,
2
4321 aaaa
, =)
~
(BCD 




 ++
2
,
2
4321 bbbb
, =)
~
(CCD 




 ++
2
,
2
4321 cccc
and
=)
~
(DCD 




 ++
2
,
2
4321 dddd
Now, solve the fuzzy game whose payoff matrix is given by
Player B Min
Player A
( )
( ) 2221
1211
43214321
43214321
2
,
22
,
2
2
,
22
,
2
gorg
gorg
ddddcccc
bbbbaaaa

















 ++





 ++





 ++





 ++
Max ( ) 2111 gorg ( ) 2212 gorg
If Minimax = Maximin,
Therefore the saddle point is exists. Then we can find the value of the fuzzy game( )V% . And then
we can find the Strategy of player A. i.e., AS and Strategy of player B. i.e., BS

17
3.3. Value-Ambiguity Interval Approximation of Trapezoidal Fuzzy Numbers
(VAIA)
Let ),,,(
~
4321 aaaaA = ; ),,,(
~
4321 bbbbB = ; ),,,(
~
4321 ccccC = and ),,,(
~
4321 ddddD = are the
trapezoidal fuzzy numbers. Then the value of the fuzzy number A
~
is
i.e., ;
6
)(2
)
~
( 1234 aaaa
AVal
+++
=
and similarly for
;
6
)(2
)
~
( 1234 bbbb
BVal
+++
= ;
6
)(2
)
~
( 1234 cccc
CVal
+++
=
6
)(2
)
~
( 1234 dddd
DVal
+++
=
and the Ambiguity of fuzzy number A
~
is
i.e., ;
6
)(2
)
~
( 1234 aaaa
AAmb
−−+
=
and similarly for
;
6
)(2
)
~
( 1234 bbbb
BAmb
−−+
= ;
6
)(2
)
~
( 1234 cccc
CAmb
−−+
=
6
)(2
)
~
( 1234 dddd
DAmb
−−+
=
Therefore the Value-Ambiguity interval approximations of trapezoidal fuzzy number A
~
is denoted
by ).
~
(AVA
[ ] 




 ++
=+−=
3
2
,
3
2
)
~
()
~
(),
~
()
~
()
~
( 3421 aaaa
AAmbAValAAmbAValAVA
,
Similarly
[ ] 




 ++
=/+//−/=/
3
2
,
3
2
)
~
()
~
(),
~
()
~
()
~
( 3421 bbbb
BAmbBValBAmbBValBVA
,
[ ] 




 ++
=+−=
3
2
,
3
2
)
~
()
~
(),
~
()
~
()
~
( 3421 cccc
CAmbCValCAmbCValCVA
,
[ ] 




 ++
=+−=
3
2
,
3
2
)
~
()
~
(),
~
()
~
()
~
( 3421 dddd
DAmbDValDAmbDValDVA
.
Now, solve the above fuzzy game whose payoff matrix is given by

18
Player B Min
Player A
( )
( ) 2221
1211
34213421
34213421
3
2
,
3
2
3
2
,
3
2
2
2
,
2
2
2
2
,
3
2
gorg
gorg
ddddcccc
bbbbaaaa

















 ++





 ++





 ++





 ++
Max
( ) 2111 gorg ( ) 2212 gorg
the saddle point is exists. Then we can find the value of the fuzzy game ( )V% . And then we can
find Strategy of player A. i.e., AS and Strategy of player B. i.e., BS .
3.4. Distinct Approximations of Trapezoidal Fuzzy Numbers (DATFN)
Let ‫ܣ‬ሚᇱ
= (ܽଵ
ᇱ
, ܽଶ
ᇱ
, ܽଷ
ᇱ
, ܽସ
ᇱ
) be a distinct approximation of trapezoidal fuzzy number ‫ܣ‬ሚ =(ܽଵ,
ܽଶ, ܽଷ, ܽସ), where ܽଵ
ᇱ
=
ଷ௔భି௔మ
ଶ
, ܽଶ
ᇱ
=
௔భା௔మ
ଶ
, ܽଷ
ᇱ
=
௔రା௔య
ଶ
, ܽସ
ᇱ
=
ଷ௔రି௔య
ଶ
. Distinct approximations of A
~
,
( ) 




 −++−
=
2
3
,
2
,
2
,
2
3~ 34432121 aaaaaaaa
AD ,
Similarly for
),,,(
~
4321 bbbbB = , then its distinct approximations
( ) 




 −++−
=
2
3
,
2
,
2
,
2
3~ 34432121 bbbbbbbb
BD
),,,(
~
4321 ccccC = then its distinct approximations
( ) 




 −++−
=
2
3
,
2
,
2
,
2
3~ 34432121 cccccccc
CD
and ),,,(
~
4321 ddddD = then its distinct approximations
( ) 




 −++−
=
2
3
,
2
,
2
,
2
3~ 34432121 dddddddd
DD
Player B Min
Player A
( )
( ) 2221
1211
3443212134432121
3443212134432121
2
3
,
2
,
2
,
2
3
2
3
,
2
,
2
,
2
3
2
3
,
2
,
2
,
2
3
2
3
,
2
,
2
,
2
3
gorg
gorg
ddddddddcccccccc
bbbbbbbbaaaaaaaa

















 −++−





 −++−





 −++−





 −++−
Max
( ) 2111 gorg ( ) 2212 gorg
Therefore the saddle point is exists. Then we can find the value of the fuzzy game( )V% . And then
we can find Strategy of player A i.e., AS and Strategy of player B i.e., BS

19
4. MIXED STRATEGY OF TWO PERSON ZERO-SUM METHOD (WITHOUT
SADDLE POINT) TO COMPARE DIFFERENT APPROXIMATIONS
METHODS IN FUZZY ENVIRONMENT
Consider a fuzzy game whose payoff matrix is of order 2x2 and which does not have saddle
point.
Consider the payoff matrix given by
Player B Min
2
1
2221
1211
APlayer
P
P
aa
aa






Max
1q 2q
If there is no saddle point for the above payoff matrix then,
We know that, mixed strategy is followed. Moreover the value of the game is
( ) ( )21122211
21122211~
aaaa
aaaa
V
+−+
−
=
( ) ( )21122211
2122
1
aaaa
aa
P
+−+
−
=
We know that,
121 =+ PP 12 1 PP −=⇒ ,
Also,
( ) ( )21122211
1222
1
aaaa
aa
q
+−+
−
= and 121 =+ qq 12 1 qq −=⇒
4.1. Example: Game with and without Saddle Point Method in Crisp
Consider the game
Player B
Player A








2.72.5
57.4

20
Player B Min
Player A
2.5
7.4
2.72.5
57.4








Max
2.5 2.7
If Minmax = Maximin
Therefore the saddle point is exist. Therefore the value of the fuzzy game 2.5)
~
( =V
The Strategy for player A. i.e., 5=AS
Strategy for player B. i.e., 2.7=BS
4.2. Fuzzy Game with and without Saddle Point Method
Let )7,5,4,3(
~
=A , )7,6,5,2(
~
=B , )9,6,4,2(
~
=C , )11,8,7,4(
~
=D are the trapezoidal fuzzy
numbers. Now, solve the above fuzzy game whose payoff matrix is given by
Player B Min
Player A
( )
( ) ( )
( )
( )9,6,4,2
7,5,4,3
11,8,7,49,6,4,2
7,6,5,2)7,5,4,3(








Max
( )9,6,4,2 ( )11,8,7,4
If Minmax = Maximin
Therefore the saddle point is exist.
Therefore the value of the fuzzy game 2.5)
~
( =V .
The Strategy for player A. i.e., 7.4=AS
Strategy for player B. i.e., 2.7=BS
4.3. Close Interval Approximation of Fuzzy Game without Saddle Point
Method
Let )7,5,4,3(
~
=A be a trapezoidal fuzzy number and its close interval approximation is
[ ] [ ]6,5.3
~
=A (by def 2.4).
Similarly for )7,6,5,2(
~
=B , then its close interval approximation is [ ]5.6,5.3]
~
[ =B ,
(2,4,6,9)C =% , then the close interval approximation is [ ][ ] 3,6.5C =% ,
and )11,8,7,4(
~
=D , then the close interval approximation of [ ]5.9,5.5]
~
[ =D .
Player B Min
Player A
[ ]
[ ] [ ] [ ]5.6,3
]6,5.3[
5.9,5.55.6,3
]5.6,5.3[6,5.3








Max
[ ]5.6,3 [ ]5.9,5.5

21
If Minmax = Maximin
Therefore the saddle point is exist.
Therefore the value of the fuzzy game 7.4
~
=V .
The Strategy for player A. i.e., 75.4]6,5.3[ ==AS
Strategy for player B. i.e., [ ] 5.75.9,5.5 ==BS
4.4. Value-Ambiguity Interval Approximation of Fuzzy Game without Saddle
Point Method
Let )7,5,4,3(
~
=A , )7,6,5,2(
~
=B , )7,6,4,2(
~
=C , )11,8,7,4(
~
numbers and its value ambiguity interval approximation of A
~
is
,
6
)(2
)
~
( 1234 aaaa
AValue
+++
=
6
)(2
)
~
( 1234 aaaa
AValue
+++
= , 16.5)
~
( =BValue , 8.4)
~
( =CValue , and 5.7)
~
( =DValue .
Also,
6
)(2
)
~
( 1234 aaaa
AAmbiguity
−−+
= ,
1)
~
( =AAmbiguity , 16.1)
~
( =BAmbiguity , 5.1)
~
( =CAmbiguity , and 5.1)
~
( =DAmbiguity .
Then , the value –Ambiguity interval approximation of A
~
is
)]
~
()
~
(),
~
()
~
([)
~
( AAmbAValAAmbAValAVA +−= , which implies
]3.6,3.4[)
~
( =AVA ,
Similarly,
]32.6,4[)
~
( =BVA , ]3.6,3.3[)
~
( =CVA , ]9,6[)
~
( =DVA .
Player B Min
Player A
[ ]
[ ] [ ] [ ]3.6,3.3
]32.6,4[
9,63.6,3.3
]32,6.4[3.6,3.4








Max
[ ]3.6,3.4 [ ]9,6
If Minmax ≠ Maximin

22
Therefore the saddle point does not exist. Hence we have
Player B Min
Player A
[ ]
[ ] [ ]
1
2
4.3,6.3 [4.6,32]
3.3,6.3 6,9
P
P
 
 
 
 
 
Max
1q 2q
We know that,
Mixed strategy is followed. Moreover the value of the game is
( ) ( )21122211
21122211~
aaaa
aaaa
V
+−+
−
=
]]3.6,32.3[]32.6,4[[]]9,6[]3.6,3.4[[
]3.6,32.3][32.6,4[]9,6][3.6,3.4[~
+−+
−
=V
Therefore 5.19V =%
We know that,
( ) ( )21122211
2122
1
aaaa
aa
P
+−+
−
=
( )7.2,3
]3.6,3.3[]9,6[ −
= = 94.01 =P
121 =+ PP 12 1 PP −=⇒ 06.0=
Also,
( ) ( )21122211
1222
1
aaaa
aa
q
+−+
−
=
[ ]7.2,3
]32.6,4[]9,6[ −
= 82.0=
and 121 =+ qq 12 1 qq −=⇒ 18.0=
Therefore the value of the fuzzy game 19.5)
~
( =V .
The Strategy for player A. i.e., 5.0)06.0,94.0( ==AS
Strategy for player B. i.e., 5.0)18.0,82.0( ==BS
4.5. Distinct Approximation of Fuzzy Game without Saddle Point Method
Let )7,5,4,3(
~
=A , )7,6,5,2(
~
=B , )7,6,4,2(
~
=C , )11,8,7,4(
~
numbers and its Distinct approximation of Trapezoidal fuzzy number A
~
is
( ) 




 −++−
=
2
3
,
2
,
2
,
2
3~ 34432121 aaaaaaaa
AD . Using this, ( ) ( )8,6,5.3,5.2
~
=AD ,
( ) ( )5.7,5.6,5.3,1
~
=BD , ( ) ( )5.7,5.6,3,1
~
=CD , ( ) ( )5.12,5.9,5.5,5.2
~
=DD .

23
Player B Min
Player A
( ) ( )
( ) ( )
( )
( )5.7,5.6,3,1
5.7,5.6,5.3,1
5.12,5.9,5.5,5.25.7,5.6,3,1
5.7,5.6,5.3,18,6,5.3,5.2








Max
( )8,6,5.3,5.2 ( )5.12,5.9,5.5,5.2
If Minmax ≠ Maximin
Therefore the saddle point does not exist.
Hence we have
Player B Min
Player A
( ) ( )
( ) ( ) 2
1
5.12,5.9,5.5,5.25.7,5.6,3,1
5.7,5.6,5.3,18,6,5.3,5.2
P
P








Max
1q 2q
We know that,
Mixed strategy is followed. Moreover the value of the game is
( ) ( )21122211
21122211~
aaaa
aaaa
V
+−+
−
=
32.1
)]15,13,5.6,2[()]5.20,5.15,9,5[(
)]5.7,6.6,3,1)(5.7.,6,5.3,1[()]5.12,5.9,5.5,5.2(),8,6,5.3,5.2[(~
=
−
−
=V
We know that,
( ) ( )
22.0
21122211
2122
1 =
+−+
−
=
aaaa
aa
P
121 =+ PP 12 1 PP −=⇒ ( )78.0=
Also,
( ) ( )
21.0
21122211
1222
1 =
+−+
−
=
aaaa
aa
q
and 121 =+ qq 12 1 qq −=⇒ )79.0(=
Therefore the value of the fuzzy game )32.1()
~
( =V .
The Strategy for player A. i.e., [ ] 5.0)78.0,22.0( ==AS
Strategy for player B. i.e., ( )[ ] 5.079.0,21.0 ==BS

24
5. COMPARISON OF DIFFERENT APPROXIMATIONS OF FUZZY NUMBERS
IN TWO-PERSON ZERO SUM GAME METHOD
Here we have compared the different approximations of fuzzy numbers with its corresponding
crisp number.
Example
4.1 The crisp number value of the game 2.5)
~
( =V
4.2 The value of the game with the aid of trapezoidal fuzzy numbers 2.5)
~
( =V
4.3 The value of the game with the aid of close interval approximations of fuzzy numbers
7.4)
~
( =V
4.4 The value of the game with the aid of Value-Ambiguity interval Approximations of fuzzy
numbers 19.5)
~
( =V
4.5 The value of the game with the aid of Distinct Approximation of fuzzy numbers 32.1)
~
( =V
From the table it is noted that the value-ambiguity interval approximation is very close to the
crisp number value of the game. Then we can conclude that the value-ambiguity interval
approximation is much better than any other methods.
6.CONCLUSION:
In this paper, comparisons between different approximations of fuzzy numbers have been made
by the aid of fuzzy game theory. The different approximations such as close interval, value-
ambiguity and distinct approximations have been employed in a fuzzy game are found and it is
observed and concluded that the value-ambiguity approximation of fuzzy numbers is the best
approximation than other methods.

25
REFERENCES
[1] Billot .A., (1992) Economic Theory of Fuzzy Equilibria: An Axiomatic Analysis. ew York: Springer-
Verlag.
[2] Butnariu .D.,( 1978) “Fuzzy Games: A Description of the Concept,” Fuzzy Sets Syst., Vol. 1, Pp.
181–192.
[3] Eichberger.J., (1993) Game Theory for Economists. New York: Academic.
[4] Gillies.D.,(1953) “Locations of Solutions,” Informal Conf. Theory of N-Person Games Princeton
Mathematics, Princeton, Nj, Pp. 11–12.
[5] Nash .J., (1950) “Equilibrium Points in N-Person Games,” Proc. Nat. Academy Sci., New York, Ny,
Vol. 36, Pp. 48–49.
[6] Nash .J., (1950) “The Bargaining Problem,” Econometrica, Vol. 18, Pp. 155–162,
[7] Shapley .L., (1953) “A Value for N-Person Games,” Kuhn and Tucker, Eds., in Contributions to the
Theory of Games. Princeton, Nj: Princeton Univ.Press, Pp. 207–317.
[8] Shapley .L., (1953) “Open Question,” Informal Conf. Theory of N-Person Games Princeton
Mathematics, Princeton, Nj, Pp. 11–12.
[9] Stephen Dinagar .D and Jivagan .K., (2014). “A Note on Interval Approximation of Fuzzy
Numbers”, Proceedings of the International Conference on Mathematical Methods and
Computation, ICOMAC.
[10] Stephen Dinagar .D and Jivagan .K., (2015) “Solving Transportation Problems using Value-
Ambiguity interval Approximations of fuzzy numbers. International Journal of Recent Development
in Engineering and Technology. (ISSN-2347- 6435). Volume 2, Issue 5.
[11] Stephen Dinagar .D and Jivagan .K., (2015) Duality in Close Interval Approximation of Fuzzy
Number Linear Programming. "Jamal Academic Research Journal (JARJ)" (ISSN No.:0973-0303).
Proceedings of the International Conference on Mathematical Methods and Computation, ICOMAC,
pp.674-681.
[12] Stephen Dinagar.D and Jivagan.K., (2015) “On Complement of Distinct Approximations of Fuzzy
Number”. International Journal of Scientific Research and Engineering Trends. Volume 1, Issue 2,
March-2015, ISSN (Online):2395- 566X.
[13] Von Neumann.J and Morgenstern.D., (1944). “The Theory of Games in Economic Bahavior. New
York: Wiley.
[14] Zadeh.L.A., (1965) “Fuzzy Sets,” Inform. Contr., Vol. 8, Pp. 338–353.
[15] Zadeh.L.A., (1968),“Probability Measures And Fuzzy Events,” J. Mathematics Anal.Applicat., Vol.
23, No. 2, Pp. 421–427.

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