=Paper=
{{Paper
|id=Vol-2098/paper20
|storemode=property
|title=One Approach to Fuzzy Matrix Games
|pdfUrl=https://ceur-ws.org/Vol-2098/paper20.pdf
|volume=Vol-2098
|authors=Konstantin N. Kudryavtsev,Irina S. Stabulit,Viktor I. Ukhobotov
}}
==One Approach to Fuzzy Matrix Games==
https://ceur-ws.org/Vol-2098/paper20.pdf
One Approach to Fuzzy Matrix Games
Konstantin N. Kudryavtsev1 , Irina S. Stabulit2 , and Viktor I. Ukhobotov3
1
South Ural State University, Chelyabinsk, Russia
kudrkn@gmail.com
2
South Ural State Agrarian University, Chelyabinsk, Russia
irisku76@mail.ru
3
Chelyabinsk State University, Chelyabinsk, Russia
ukh@csu.ru
Abstract. The paper is concerned with a two-person zero sum matrix
game with fuzzy payoffs, with saddle-point being defined with a clas-
sic definition in matrix game. In order to compare fuzzy numbers some
different ordering operators can be used. The original game can be asso-
ciated with the bimatrix game with crisp payoffs which are the operators
value on a fuzzy payoff. We propose that every player use its ordering
operator. The following statement can be proposed: when the ordering
operators are linear, the same equilibrium strategy profile can be used
for the matrix game, with fuzzy payoffs being the same for the bimatrix
crisp game. We introduce and employ the algorithm of constructing a
saddle-point in a two-person zero sum matrix game with fuzzy payoffs.
In the instances of matrix games we use such ordering operators and
construct the saddle-point.
Keywords: Fuzzy game · Matrix game · Saddle-point · Nash equilib-
rium
1 Introduction
Game theory is well known to take a significant part in decision making, this
theory being often used to model the real world. Applied in real situations the
game theory is difficult to have strict values of payoffs, because it is difficult for
players to analyze some data of game. Thus, the players’ information cannot be
considered complete. Besides the players can have vague targets.
This uncertainty and lack of precision may be modeled as fuzzy games. Fuzzy
sets are known to be initially used in non-cooperative game theory by Butnariu
[3] to prove the belief of each player for strategies of other players. Since then
fuzzy set theory has been applied in cooperative and non-cooperative games.
The results of fuzzy games overview are given in [10]. Recently there were made
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
One Approach to Fuzzy Matrix Games 229
various efforts in fuzzy bi-matrix game theory namely Maeda [11], Nayak [13],
Dutta [6], Seikh [14], Verma and Kumar [16].
In [9], we represented the approach which generalizes some other ideas ([4],
[5],[6] at al.).
2 Fuzzy Numbers
In this part, some basic definitions and results of fuzzy numbers and fuzzy arith-
metic operations will be reminded. Here we will follow to [21].
A fuzzy set can be considered as a subset à of universal set X ⊆ R by its
membership function µÃ (·) with a real number µÃ (x) in the interval [0, 1] and
assigns to each element x ∈ R.
Definition 2.1. A fuzzy subset à defined on R, is said to be a fuzzy number if
its membership function µÃ (x) comply with the following conditions:
(1) µÃ (x) : R → [0, 1] is upper semi-continuous;
(2) µÃ (x) = 0 for ∀ x 6∈ [a, d];
(3) There exist real numbers b, c such that a 6 b 6 c 6 d and
(a) x1 < x2 ⇒ µÃ (x1 ) < µÃ (x2 ) ∀ x1 , x2 ∈ [a, b];
(b) x1 < x2 ⇒ µÃ (x1 ) > µÃ (x2 ) ∀ x1 , x2 ∈ [a, b];
(c) µÃ (x) = 1, ∀x ∈ [b, c].
The α-cut of a fuzzy number à plays an important role in parametric ordering
of fuzzy numbers. The α-cut or α-level set of a fuzzy number Ã, denoted by Ãα ,
is defined as Ãα = {x ∈ R | µÃ (x) > α} for all α ∈ (0, 1]. The support or 0-cut
Ã0 is defined as the closure of the set Ã0 = {x ∈ R | µÃ (x) > 0}. Every α-cut is a
closed interval Ãα = [gà (α), Gà (α)] ⊂ R, where gà (α) = inf{x ∈ R | µÃ (x) > α}
and Gà (α) = sup{x ∈ R | µÃ (x) > α} for any α ∈ [0, 1].
The sets of fuzzy number is denoted as F. Then, two types of fuzzy numbers
are used.
Definition 2.2. Let à be a fuzzy number. If the membership function of à is
given by x−a+l
l , f or x ∈ [a − l, a],
µÃ (x) = a+r−x , f or x ∈ [a, a + r],
r
0, otherwise,
where, a, l and r are all real (crisp) numbers, and l, r are non-negative. Then
à is called a triangular fuzzy number, denoted by à = (a, l, r).
The sets of triangular fuzzy number are denoted as F3 .
Definition 2.3. Let à be a fuzzy number. If the membership function of à is
given by x−a+l
l , f or x ∈ [a − l, a],
1, f or x ∈ [a, b]
µÃ (x) = b+r−x
, f or x ∈ [b, b + r],
r
0, otherwise,
where, a, b, l and r are all real (crisp) numbers, and l, r are non-negative. Then
à is called a trapezoidal fuzzy number, denoted by à = (a, b, l, r). [a, b] is the
core of Ã.
230 K. Kudryavtsev et al.
The sets of trapezoidal fuzzy number are denoted as F4 .
If à = (a1 , l1 , r1 ) and B̃ = (a2 , l2 , r2 ) are two triangular fuzzy numbers,
arithmetic operations on à and B̃ are defined as follows:
Addition: Ã + B̃ = C̃ = (a1 + a2 , l1 + l2 , r1 + r2 ), C̃ ∈ F3 .
Scalar multiplication: ∀ k > 0, k ∈ R,
k à = (ka1 , kl1 , kr1 ), k à ∈ F3 .
If à = (a1 , b1 , l1 , r1 ) and B̃ = (a2 , b2 , l2 , r2 ) are two trapezoidal fuzzy num-
bers, arithmetic operations on à and B̃ are defined as follows:
Addition: Ã + B̃ = C̃ = (a1 + a2 , b1 + b2 , l1 + l2 , r1 + r2 ), C̃ ∈ F4 .
Scalar multiplication: ∀ k > 0, k ∈ R,
k à = (ka1 , kb1 , kl1 , kr1 ), k à ∈ F4 .
Generaly, if à and B̃ are two fuzzy numbers and à + B̃ = C̃, λà = D̃ and
λ = const > 0, then
C̃α = [gà (α) + gB̃ (α), Gà (α) + GB̃ (α)],
and
D̃α = [λgà (α), λGà (α)]
for any α ∈ [0, 1].
To compare fuzzy numbers is crucial issue. There are plenty of various meth-
ods for comparing fuzzy numbers. For instance, fuzzy numbers can be ranked
with the help of defuzzification methods.Defuzzification is the process of produc-
ing a real (crisp) value which correspond to a fuzzy number, the defuzzification
approach being used for ranking fuzzy numbers. Fuzzy numbers are initially
defuzzified and then the obtained crisp numbers are organised using the order
relation of real numbers.
A function for ranking fuzzy subsets in unit interval was introduced by Yager
in [17]. It was based on the integral of mean of the α-cuts. Yager index is
Z1
1
Y (Ã) = (gà (α) + Gà (α)) dα.
2
0
Another methods for ordering fuzzy subsets in the unit interval were sug-
gested by Jain in [8], Baldwin and Guild in [1].
The subjective approach for ranking fuzzy numbers was developed by Ibanez
and Munoz in [7], the following number as the average index for fuzzy number
à beind defined in [7] Z
VP (Ã) = fà (α)dP (α),
Y
with Y is a subset of the unit interval and P is a probability distribution on Y ,
the definition of fà being subjective for decision maker.
One Approach to Fuzzy Matrix Games 231
The ordering operator was proposed by Ukhobotov in [15]
Z1
U (Ã, ν) = ((1 − ν)gà (α) + νGà (α)) dα,
0
with crisp parameter ν ∈ [0, 1], different behavior of the decision maker being
corresponded with different ν.
Other defuzzification operators are given in [2].
Definition 2.4. Let à and B̃ are a fuzzy numbers, T : F → R is the operator of
defuzzification (T (·) = Y (·), VP (·), U (·, ν) etc.).
We say that B̃ is preferable to à by the defuzzification operator T (à T B̃)
if and only if
T (Ã) 6 T (B̃).
The defuzzification operator T is dependant on the order relation T . Then,
we give the example.
Example 2.1. Let T (·) = U (·, ν) and Ã, B̃, C̃ ∈ F3 , Ã = (40, 8, 10), B̃ =
= (45, 20, 10), C̃ = (42, 6, 4).
If X̃ = (a, l, r) ∈ F3 , then
νr − (1 − ν)l
U (X̃, ν) = a + .
2
Next, if ν = 0, then U (Ã, 0) = 36, U (B̃, 0) = 35, U (C̃, 0) = 39. If ν = 21 , then
U (Ã, 12 ) = 40, 5, U (B̃, 12 ) = 42, 5, U (C̃, 21 ) = 41, 5. If ν = 1, then U (Ã, 1) = 45,
U (B̃, 1) = 50, U (C̃, 1) = 44.
Therefore,
B̃ U (·,0) Ã U (·,0) C̃,
à U (·, 12 ) C̃ U (·, 21 ) B̃,
C̃ U (·,1) Ã U (·,1) B̃.
Definition 2.5. If ∀ Ã, B̃ ∈ F, ∀ α, β = const
T (αà + β B̃) = αT (Ã) + βT (B̃),
then the defuzzification operator T (·) is the linear defuzzification operator.
Clear, Yager index Y (·) and operator U (·, ν) is linear.
3 Crisp Games
In this part, some basic definitions of non-cooperative game theory are presented.
232 K. Kudryavtsev et al.
3.1 Noncooperative N-Person Games
Let us consider a non-cooperative N -players game in the class of pure strategies
Γ = hN, {Xi }i∈N , {fi (x)}i∈N i, (1)
where N = 1, ..., N is the set of players’ serial numbers; each player i chooses
and applies its own pure strategy xi ∈ Xi ⊆ Rni , with no coalition with the
others, which induces a strategy profile being formed
Y X
x = (x1 , ..., xN ) ∈ X = Xi ⊂ Rn (n = ni );
i∈N i∈N
for each i ∈ N, a payoff function fi (x) is defined on the strategy profile set X,
which gives the payoff of player i. fi (x) is payoff function of player i (i ∈ N). In
addition, denote (xkzi ) = (x1 , ..., xi−1 , zi , xi+1 , ...xN ), f = (f1 , ..., fN ).
Definition 3.1. A strategy profile xe = (xe1 , ..., xeN ) ∈ X is called a Nash equi-
librium in the game (1) if
max fi (xe kxi ) = fi (xe ) (i ∈ N). (2)
xi ∈Xi
The set of all {xe } in the game (1) will be designated by X e .
3.2 Bimatrix Games
A bimatrix game defined by a pair (A, B) of real m×n matrices being considered,
matrices A and B are payoffs to play I and II, respectively. M denotes the set
of pure strategies of player I (matrix rows) and N stands for the set of pure
strategies of player II (columns).
M = (1, . . . , m), N = (1, . . . , n).
The sets of mixed strategies of the two players being called X and Y , we want
to write expected payoffs as matrix products xAy and xBy, for mixed strategies
x and y, so that x is a row vector and y is a column vector. That is,
X
X = {(x1 , . . . , xm ) | xi > 0 (∀i ∈ M ), xi = 1}
i∈M
and X
Y = {(y1 , . . . , yn ) | yj > 0 (∀j ∈ N ), yj = 1}.
j∈N
Definition 3.2. A pair (xe , y e ) ∈ X × Y is called a Nash equilibrium for the
game (A, B) if
xe Ay e > xAy e ∀x ∈ X,
xe By e > xe By ∀y ∈ Y.
The set of Nash equilibrium for a game (A, B) is proved to be non-empty in [12].
If matrix B = −A, a bimatrix game is considered as a zero-sum matrix game.
A solve of a zero-sum matrix game being a saddle-point.
One Approach to Fuzzy Matrix Games 233
3.3 Internally Instable Set of Nash Equilibrium
Now, consider internal instability of X e . A subset X ∗ ⊂ Rn is internally instable,
if there exist at least two strategy profiles x(j) ∈ X ∗ (j = 1, 2) such that
h i h i
f (x(1) ) < f (x(2) ) ⇔ fi (x(1) ) < fi (x(2) ) (i ∈ N) , (3)
and internally stable otherwise.
Example 3.1. Consider a bimatrix game of the form (A, B), where
2 5 1 1 0 8
A = 3 4 6, B = 9 3 5.
6 7 2 2 7 6
(i) (i)
There are 3 Nash equilibrium strategy profiles x(i) = (x1 , x2 ) (i = 1, 2, 3):
(1) 1 2 4 3
x = 0, , , 0, , ,
3 3 7 7
1 1 4 3
x(2) = 0, , , , 0, , x(3) = ((0, 0, 1), (0, 1, 0)).
2 2 7 7
Consequently, the set X e is internally instable in the game (A, B), as for x(3) it
follows that
(3) (3) (1) (1)
f1 (x(3) ) = x1 Ax2 = 7 > f1 (x(1) ) = x1 Ax2 = 34
7 ,
(3) (3) (1) (1)
f2 (x(3) ) = x1 Bx2 = 7 > f2 (x(1) ) = x1 Bx2 = 17
3 ,
and
(3) (3) (2) (2)
f1 (x(3) ) = x1 Ax2 = 7 > f1 (x(2) ) = x1 Ax2 = 30
7 ,
(3) (3) (3) (2) (2) (2) 11
f2 (x ) = x1 Bx2 = 7 > f2 (x ) = x1 Bx2 = 2 .
We note that in the non-antagonistic setting of the game (1), the internal in-
stability effect vanishes if there exists a unique Nash equilibrium strategy profile
in (1).
Associate the following auxiliary N -criterion problem with the game (1):
Γv = hX e , {fi (x)}i∈N i, (4)
where the set X e of alternatives x coincides with the set of Nash equilibrium
strategy profiles xe in the game (1), and the ith criterion fi (x) is the payoff
function of player i.
Definition 3.3. An alternative xP ∈ X e is Pareto optimal (efficient) in the
problem (4), if ∀x ∈ X e the system of inequalities
fi (x) > fi (xP ) (i ∈ N),
is infeasible, with at least one being a strict inequality. {xP } XP .
Designate by X P the set of all {xP }.
234 K. Kudryavtsev et al.
According to Definition 3.3, the set X P satisfies the inclusionX P ⊆ X e and
is internally stable.
The following statement is obvious: if for all x ∈ X e we have
X X
fi (x) 6 fi (xP ), (5)
i∈N i∈N
then xP gives the Pareto optimal alternative in the problem (4).
Definition 3.4. [18] A strategy profile x∗ ∈ X is called a Pareto-optimal Nash
equilibrium (P ON E) for the game (1) if x∗ is
a) a Nash equilibrium in (1) (Definition 3.1), and
b) a Pareto optimum in (5) (Definition 3.3).
4 Game with Fuzzy Payoffs
Further, a non-cooperative N -person game is considered
Γe = hN, {Xi }i∈N , {fei (x)}i∈N i. (6)
A game (6) differs from (1) according to payoffs functions. In (6), a payoff func-
tion of player i is fei (x) : X → F. In addition, Xi involves only a finite number
of elements, Γe being a finite game with fuzzy payoffs. A game(6) is a bimatrix
game with fuzzy payoffs when N = {1, 2}.
Determining the concept of optimality, we have to compare payoffs. We can
used some defuzzification operator T (T (·) = Y (·), VP (·), U (·, ν) etc.). In [9] we
proposed the following definition.
Definition 4.1. [9] A strategy profile xe = (xe1 , ..., xeN ) ∈ X is called a T (·)-Nash
equilibrium in the game (6) if
fi (xe kxi ) T fi (xe ) (i ∈ N).
We note that the solutions, which defined in [11], [5] and [6], are particular cases
of Definition 4.1.
Next, we consider the associated crisp game for (6)
Γec = hN, {Xi }i∈N , {T (fei (x))}i∈N i. (7)
Theorem 4.1. [9] Let xe is a Nash equilibrium in (7) and T (·) is a linear
defuzzification operator, then xe is T (·)-Nash equilibrium in a game (6).
For example, we consider one bimatrix game with a triangular fuzzy payoffs.
Example 4.1. Consider a bimatrix game with fuzzy payoffs Γe of the form
(A,
e B),
e where A e and Be are the triangular fuzzy matrixes:
(30, 6, 12) (10, 8, 6) (15, 10, 5)
Ae = (20, 10, 5) (22, 6, 10) (30, 5, 10)
(10, 8, 12) (30, 20, 4) (20, 8, 16)
One Approach to Fuzzy Matrix Games 235
and
(10, 4, 6) (15, 10, 5) (20, 15, 4)
e = (20, 10, 10) (10, 6, 14) (15, 10, 5) .
B
(12, 6, 10) (15, 10, 10) (10, 5, 15)
The operator U (·, ν) is used. As a result, we obtain the associated crisp game
(4).
If ν = 0, then the game (4) given as follows
27 6 10 8 10 12, 5
A = 15 19 27, 5 , B = 15 7 10 .
6 20 16 9 10 7, 5
There are 3 pure and mixed U (·, 0)-Nash equilibrium strategy profiles. It is
xe = (xe1 , xe2 ), where
10 ) xe1 = (0, 0, 1), xe2 = (0, 1, 0),
10 9
20 ) xe1 = ( 19 , 19 , 0), xe2 = ( 35 24
59 , 0, 59 ),
0 e 1 8 e 1 9
3 ) x1 = (0, 9 , 9 ), x2 = ( 10 , 10 , 0).
If ν = 12 , then the associated crisp game (4) given as follows
31, 5 9, 5 13, 75 10, 5 13, 75 17, 25
A = 18, 75 23 31, 25 , B = 20 12 13, 75 .
11 26 22 13 15 12, 5
There are 3 U (·, 21 )-Nash equilibrium strategy profiles. It is xe = (xe1 , xe2 ), where
10 ) xe1 = (0, 0, 1), xe2 = (0, 1, 0),
25 27 70 51
20 ) xe1 = ( 52 , 52 , 0), xe2 = ( 121 , 0, 121 ),
0 e 1 4 e 12 31
3 ) x1 = (0, 5 , 5 ), x2 = ( 43 , 43 , 0).
If ν = 1, then the game (4) given as follows
36 13 17, 5 13 17, 5 22
A = 22, 5 27 35 , B = 25 17 17, 5 .
16 32 28 17 20 17, 5
There are 3 U (·, 1)-Nash equilibrium strategy profiles. It is xe = (xe1 , xe2 ), where
10 ) xe1 = (0, 0, 1), xe2 = (0, 1, 0),
5 6
20 ) xe1 = ( 11 , 11 , 0), xe2 = ( 35 27
62 , 0, 62 ),
0 e 3 8 e 10 13
3 ) x1 = (0, 11 , 11 ), x2 = ( 23 , 23 , 0).
Another example: one zero-sum matrix game with a trapezoidal fuzzy payoffs
is considered .
Example 4.2. Let A e be the trapezoidal fuzzy payoff matrixes of the fuzzy zero-
sum matrix game Γ , which is given as follows:
e
Ae = (20, 30, 12, 8) (1, 5, 8, 4) .
(5, 9, 20, 4) (10, 26, 8, 12)
The operator Y (·) is used. As a result, we obtain the associated crisp game (4)
24 2
A= .
3 19
236 K. Kudryavtsev et al.
8 11
The mixed Y (·)-Nash equilibrium is xe = (xe1 , xe2 ), where xe1 = ( 19 , 19 ), xe2 =
17 21
( 38 , 38 ).
5 Fuzzy Matrix Game with Different Preferences
In this section, we consider a two-person zero sum game
Γea = h{1, 2}, {Xi }i=1,2 , fe1 (x)i, (8)
where {1, 2} is the set of players’ serial numbers; each player i chooses and
applies his own pure strategy xi ∈ Xi ⊆ Rni (i = 1, 2), a strategy profile is
x = (x1 , x2 ) ∈ X = X1 × X2 ⊂ Rn (n = n1 + n2 ); a payoff function of player 1
is fe1 (x) : X → F. A payoff function of player 2 is fe2 (x) = −fe1 (x).
In addition, let Xi contains only a finite number of elements. In this case, Γea is
a two-person zero sum matrix game with fuzzy payoffs. This game is determined
by a fuzzy matrix A. e
In the last section, we considered that the players prefer the same defuzzi-
fication operator. However, the players can have the different preferences. For
example, it can be caused by the various attitude to the risk. In this case, the
players can use the different defuzzification operators. The main idea of this
paper is following:
Suppose that the player 1 has decided to use a defuzzification operator T1 (·).
And the player 2 chose to use a defuzzification operator T2 (·) (T1 (·) 6= T2 (·)).
Definition 5.1. A strategy profile x∗ = (x∗1 , x∗2 ) ∈ X is called a T1 (·)T2 (·)-
saddle-point in the game (8) if
fe1 (x1 , x∗2 ) T1 fe1 (x∗1 , x∗2 ) T2 fe1 (x∗1 , x2 ) ∀ x1 ∈ X1 , x2 ∈ X2 .
Next, we consider the associated crisp game for (8)
Γa = h{1, 2}, {Xi }i=1,2 , {T1 (fe1 (x)), −T2 (fe1 (x))}i. (9)
In contrast to (8), a crisp game (9) is a bimatrix game. Usually, the solution
of crisp game (9) is Nash equilibrium. But, a set of Nash equilibriums X e is
internally instable. We will use PONE in this game.
Theorem 5.1. Let x∗ is a Pareto-optimal Nash equilibrium in (9) and T1 (·),
T2 (·) are a linear defuzzification operators, then x∗ is T1 (·)T2 (·)-saddle-point in
a game (8).
For example, we consider zero-sum matrix game with a trapezoidal fuzzy
payoffs from Example 4.2..
Example 5.1. Let A e be the trapezoidal fuzzy payoff matrixes of the fuzzy zero-
sum matrix game Γ , which is given as follows:
e
(20, 30, 12, 8) (1, 5, 8, 4)
.
(5, 9, 20, 4) (10, 26, 8, 12)
The operator Y (·) is used for player 1, and the operator U (·, 0) is used for player
2.
One Approach to Fuzzy Matrix Games 237
As a result, we obtain the associated crisp bimatrix game (9) given as follows
24 2 −14 3
A= , B= .
3 19 5 −6
The Y (·)U (·, 0)-saddle-point is x∗ = (x∗1 , x∗2 ), where x∗1 = ( 11 17 ∗ 17 21
28 , 28 ), x2 = ( 38 , 38 ).
6 Conclusion
In this paper we proposed a method for formalizing and constructing equilib-
rium in fuzzy matrix games to generalize some already known methods. In the
future, we will apply it for formalizing a Berge equilibrium [19] and a coalition
equilibrium [20] in n-person games with fuzzy payoffs. The case of continuous
game is also if great interest. In the case, when we construct equilibrium in pure
strategies, the linearity condition of a defuzzification operator is not required.
We plan to study a continuous games case.
Acknowledgement. The work was supported by Act 211 Government of the
Russian Federation, contract N 02.A03.21.0011 and by Grant of the Foundation
for perspective scientific researches of Chelyabinsk State University (2018).
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