=Paper=
{{Paper
|id=Vol-1987/paper50
|storemode=property
|title=A Bimatrix Game with Fuzzy Payoffs and Crisp Game
|pdfUrl=https://ceur-ws.org/Vol-1987/paper50.pdf
|volume=Vol-1987
|authors=Konstantin N. Kudryavtsev,Irina S. Stabulit,Viktor I. Ukhobotov
}}
==A Bimatrix Game with Fuzzy Payoffs and Crisp Game==
https://ceur-ws.org/Vol-1987/paper50.pdf
A Bimatrix Game with Fuzzy Payoffs and Crisp Game
Konstantin N. Kudryavtsev Irina S. Stabulit
South Ural State University Chelyabinsk State University
Lenin prospekt, 76, Bratiev Kashirinykh st., 129
454080 Chelyabinsk, Russia 454001 Chelyabinsk, Russia
kudrkn@gmail.com irisku76@mail.ru
Viktor I. Ukhobotov
Chelyabinsk State University
Bratiev Kashirinykh st., 129
454001 Chelyabinsk, Russia
ukh@csu.ru
Abstract
In this paper we consider a bimatrix game with fuzzy payoffs. To
compare a fuzzy numbers, some different ordering operators can be
used. We define a Nash equilibrium in fuzzy game using the ordering
operator. The game with crisp payoffs is associated with the original
game. Here, a crisp payoffs are the operator’s value on a fuzzy payoff.
We propose the following statement: if the ordering operator is linear,
then the game with payoffs have same Nash equilibrium strategy profile
as the crisp game. We present an algorithm for constructing a Nash
equilibriumin in a bimatrix game with fuzzy payoffs and we are using
this fact. We use such ordering operators, and construct the Nash
equilibrium in examples of bimatrix games.
1 Introduction
Game theory takes an important role in decision making and actively is used for modeling the real world.
Applying game theory in real situations, it is difficult to have strict value of payoffs, because players are not able
to analyze some data of game and as a result, their information isn‘t complete. This lack of precision and certainty
may be modeled by different ways such as fuzzy games. Initially, fuzzy sets were used by [Butnariu, 1978] 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 and cooperative games. Overview
of the results of fuzzy games are in [Larbani, 2009]. In recent past, various attempt have been made in fuzzy
bi-matrix game theory namely [Maeda, 2003], [Nayak, 2009], [Dutta, 2014], [Seikh et al., 2015]
In this paper, we present an approach that generalizes some other ideas ([Campos, 1989],
[Cunlin, 2011],[Dutta, 2014] at al.).
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org
343
2 Fuzzy Numbers
In this section, we recall some basic concepts and result of fuzzy numbers and fuzzy arithmetic operations. Here,
we follow [Zimmermann, 1996].
A fuzzy set is defined as a subset à of universal set X ⊆ R by its membership function µÃ (·) with assigns to
each element x ∈ R, a real number µÃ (x) in the interval [0, 1].
Definition 2.1. A fuzzy subset à defined on R, is said to be a fuzzy number if its membership function µÃ (x)
satisfies the following conditions:
(1) µÃ (x) : R → [0, 1] is upper semi-continuous;
(2) µÃ (x) = 0 outside some interval [a, d];
(3) There exist real numbers b, c such that a 6 b 6 c 6 d and
(a) µÃ (x) is monotonic increasing on [a, b];
(b) µÃ (x) is monotonic decreasing on [c, d];
(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].
We denote the sets of fuzzy number as F. Next, we use two types of fuzzy numbers.
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).
We denote the sets of triangular fuzzy number as F3 .
Definition 2.3. Let à be a fuzzy number. If the membership function of à is given by
x−a+l
, f or x ∈ [a − l, a],
l
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.
We denote the sets of trapezoidal fuzzy number as F4 .
Let à = (a1 , l1 , r1 ) and B̃ = (a2 , l2 , r2 ) be two triangular fuzzy numbers. Then arithmetic operations on Ã
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 .
Let à = (a1 , b1 , l1 , r1 ) and B̃ = (a2 , b2 , l2 , r2 ) be two trapezoidal fuzzy numbers. Then 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 .
In general, let à and B̃ be two fuzzy numbers. If à + B̃ = C̃, λà = D̃ and λ = const > 0, then
C̃α = [gà (α) + gB̃ (α), Gà (α) + GB̃ (α)],
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and
D̃α = [λgà (α), λGà (α)]
for any α ∈ [0, 1].
Comparing of fuzzy numbers is a very important question. Various methods for comparing fuzzy numbers have
been proposed. For example, fuzzy numbers can be ranked using the defuzzification methods. A defuzzification is
the process of producing a real (crisp) value corresponding to a fuzzy number. In order to rank fuzzy numbers are
using the defuzzification approach, the fuzzy numbers are first defuzzified and then, the obtained crisp numbers
are ordered using the order relation of real numbers.
Yager in [Yager, 1981] introduced a function for ranking fuzzy subsets in unit interval, which is based on the
integral of mean of the α-cuts. Yager index is
∫1
1
Y (Ã) = (gà (α) + Gà (α)) dα.
2
0
Jain in [Jain, 1977], Baldwin and Guild in [Baldwin, 1979] were also suggested methods for ordering fuzzy
subsets in the unit interval.
Ibanez and Munoz in [Ibanez, 1989] have developed a subjective approach for ranking fuzzy numbers. In
[Ibanez, 1989], Ibanez and Munoz defined the following number as the average index for fuzzy number Ã
∫
VP (Ã) = fà (α)dP (α),
Y
where Y is a subset of the unit interval and P is a probability distribution on Y . The definition of fà could be
subjective for decision maker.
Ukhobotov in [Ukhobotov, 2016] proposed the ordering operator
∫1
U (Ã, ν) = ((1 − ν)gà (α) + νGà (α)) dα,
0
where crisp parameter ν ∈ [0, 1]. Different ν correspond to different behavior of the decision maker.
Some other defuzzification operators were given in [Basiura et al., 2015].
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 order relation ≼T depends on the defuzzification operator T . Next we give the example.
Example 2.1. Let Ã, B̃, C̃ ∈ F3 , Ã = (50, 10, 20), B̃ = (55, 30, 12), C̃ = (52, 30, 30), T (·) = U (·, ν).
If X̃ = (a, l, r) ∈ F3 , then
νr − (1 − ν)l
U (X̃, ν) = a + .
2
Next, if ν = 0, then U (Ã, 0) = 45, U (B̃, 0) = 40, U (C̃, 0) = 37. If ν = 12 , then U (Ã, 12 ) = 52, 5, U (B̃, 12 ) = 51, 5,
U (C̃, 21 ) = 52. If ν = 1, then U (Ã, 1) = 60, U (B̃, 1) = 61, U (C̃, 1) = 67.
Therefore,
C̃ ≼U (·,0) B̃ ≼U (·,0) Ã,
B̃ ≼U (·, 21 ) C̃ ≼U (·, 21 ) Ã,
à ≼U (·,1) B̃ ≼U (·,1) C̃.
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.
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3 Crisp Games
In this section, we present some basic definitions of non-cooperative game theory.
3.1 Noncooperative N-Person Games
Consider a non-cooperative game of N players in the class of pure strategies
Γ = ⟨N, {Xi }i∈N , {fi (x)}i∈N ⟩, (1)
where N = 1, ..., N is the set of players’ serial numbers; each player i chooses and applies his own pure strategy
xi ∈ Xi ⊆ Rni , forming no coalition with the others, which induces a strategy profile
∏ ∑
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 (x∥zi ) = (x1 , ..., xi−1 , zi , xi+1 , ...xN ),
f = (f1 , ..., fN ).
Definition 3.1. A strategy profile xe = (xe1 , ..., xeN ) ∈ X is called a Nash equilibrium in the game (1) if
max fi (xe ∥xi ) = 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
We consider a bimatrix game defined by a pair (A, B) of real m × n matrices. Matrices A and B are payoffs to
play I and II, respectively. The set of pure strategies of player I (matrix rows) is denoted by M and the set of
pure strategies of player II (columns) is denoted by N .
M = (1, . . . , m), N = (1, . . . , n).
The sets of mixed strategies of the two players are called X and Y . For mixed strategies x and y, we want to
write expected payoffs as matrix products xAy and xBy, so that x should be a row vector and y should be a
column vector. That is, ∑
X = {(x1 , . . . , xm ) | xi > 0 (∀i ∈ M ), xi = 1}
i∈M
and ∑
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.
From [Nash, 1950] implies that the set of Nash equilibrium for a game (A, B) is non-empty.
We recall, a bimatrix game is a zero-sum bimatrix game if matrix B = −A. A solve of a zero-sum bimatrix
game is sadle-point.
4 Game with Fuzzy Payoffs
Further, we consider a non-cooperative N -person game
e = ⟨N, {Xi }i∈N , {fei (x)}i∈N ⟩,
Γ (3)
which differs from (1) only payoffs functions. In (3), a payoff function of player i is fei (x) : X → F. In addition,
Xi contains only a finite number of elements. Γe is a finite game with fuzzy payoffs. If N = {1, 2}, then (3) is a
bimatrix game with fuzzy payoffs.
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To determine the concept of optimality, we must compare payoffs. We use some defuzzification operator T
(T (·) = Y (·), VP (·), U (·, ν) etc.). We propose the following definition.
Definition 4.1. A strategy profile xe = (xe1 , ..., xeN ) ∈ X is called a T (·)-Nash equilibrium in the game (3) if
fi (xe ∥xi ) ≼T fi (xe ) (i ∈ N).
We note that the solutions, which defined in [Maeda, 2003], [Cunlin, 2011] and [Dutta, 2014], are particular cases
of Definition 4.1.
Next, we consider the associated crisp game for (3)
e c = ⟨N, {Xi }i∈N , {T (fei (x))}i∈N ⟩.
Γ (4)
Theorem 4.1. Let xe is a Nash equilibrium in (4) and T (·) is a linear defuzzification operator, then xe is
T (·)-Nash equilibrium in a game (3).
For example, we consider one bimatrix game with a triangular fuzzy payoffs.
Example 4.1. Let A e and Be are the triangular fuzzy payoff matrixes of the fuzzy bimatrix game Γ,
e given as
follows: ( ) ( )
e= (20, 5, 10) (5, 10, 5) e= (10, 10, 5) (15, 5, 10)
A , B .
(5, 10, 5) (10, 5, 5) (10, 10, 20) (5, 10, 5)
We use the operator U (·, ν).
If ν = 0, then the associated crisp game (4) given as follows
( ) ( )
17, 5 0 5 12, 5
A= , B= .
0 7, 5 5 0
The mixed U (·, 0)-Nash equilibrium is xe = (xe1 , xe2 ), where xe1 = ( 25 , 53 ), xe2 = ( 10
3 7
, 10 ).
1
If ν = 2 , then the associated crisp game (4) given as follows
( ) ( )
21, 25 3, 75 8, 75 16, 25
A= , B= .
3, 75 10 12, 5 3, 75
7
The mixed U (·, 12 )-Nash equilibrium is xe = (xe1 , xe2 ), where xe1 = ( 13 6
, 13 5 14
), xe2 = ( 19 , 19 ).
If ν = 1, then the associated crisp game (4) given as follows
( ) ( )
25 7, 5 12, 5 20
A= , B= .
7, 5 12, 5 20 7, 5
The mixed U (·, 1)-Nash equilibrium is xe = (xe1 , xe2 ), where xe1 = ( 58 , 83 ), xe2 = ( 29 , 79 ).
To give another example: we consider one zero-sum bimatrix game with a trapezoidal fuzzy payoffs.
Example 4.2. Let A e is the triangular fuzzy payoff matrixes of the fuzzy zero-sum bimatrix game Γ, e given as
follows: ( )
(20, 30, 12, 8) (1, 5, 8, 4)
.
(5, 9, 20, 4) (10, 30, 8, 12)
We use the operator Y (·). The associated crisp game (4) given as follows
( )
24 2
A= .
3 19
8 11
The mixed Y (·)-Nash equilibrium is xe = (xe1 , xe2 ), where xe1 = ( 19 , 19 ), xe2 = ( 17 21
38 , 38 ).
5 Conclusion
In this article, we proposed a method for formalizing and constructing equilibria in fuzzy games. This method
generalized some already known methods. In the future, we plan to apply it for formalization a Berge equi-
librium [Zhukovskiy, 2017 a], a Pareto optimal Nash equilibrium [Zhukovskiy, 2016], [Kudryavtsev et al., 2016],
a coalition equilibrium [Zhukovskiy, 2017 b] in n-person games with fuzzy payoffs. Also, we will study games
under uncertainty [Zhukovskiy, 2013] with fuzzy payoffs.
347
Acknowledgements
The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0011 and Grant
of the Foundation for perspective scientific researches of Chelyabinsk State University (2017)
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