saqarT velos
mecn ierebaTa
erovnu li
ak ad em iis
moam be,
t. 9, #1, 2015
BULL ET IN OF THE GEORGIAN NATIONAL ACADEM Y OF SCIENCE S, vols. 9, no. 1, 2015
Equilibrium Solution of Non-cooperative Bimatrix Game of
Z-Numbers
Mahdieh Akhbari§*, Soheil Sadi-Nezhad **
*§
Department of Industrial Engineering, Science and Research Branch,Islamic Azad University, Tehran, Iran
(Corresponding Author) [email protected]
**
Department of Industrial Engineering, Science and Research Branch,Islamic Azad University, Tehran, Iran
ABSTRACT- In this paper we introduce a concept of equilibrium for a non-cooperative game with payoff
matrices and goals of Z-Numbers. In most related studies, payoffs are considered as fuzzy numbers. In the
present study, another level of uncertainty related to trust or confidence in fuzzy payoffs is also considered.
This type of uncertainty is related to a new kind of numbers in a state of uncertainty. These numbers are
called z-numbers and have been defined by Professor Zadeh in 2011. In order to find the equilibrium solution
in such circumstances, only the payoff maximization is not considered, since considering the uncertainty or
distrust in fuzzy numbers of game matrices, an aspiration level of confidence in the equilibrium solution is
also considered and finally a method is presented to determine the equilibrium solution with respect to the
level of achievement to goals of z-numbers. © 2015 Bull. Georg. Natl.Acad. Sci.
Key words: non-cooperative game theory, z-number, z-number type payoffs, z-number type goals, Equilibrium solution
1. INTRODUCTION
Using game theory as a problem-solving method, we are often asked to express values of payoffs exactly, while in
reality, it is difficult to evaluate the exact values of payoffs due to uncertainty and inadequate information and we
could only know the values of payoffs approximately. Fuzzy set theory is one of the suitable tools in such
circumstances. The first study conducted in fuzzy payoffs matrix game was proposed by Compos in 1989. He
changed the problem of finding a solution to the game using zero-sum fuzzy matrix to the solution of a pair of fuzzy
linear programming problem. This problem was solved using Yager's method for solving fuzzy linear programming.
Then Campos and Gonzalez developed their model in the state of using non-linear ranking functions. Presenting a
definition of feasible solutions and the fuzzy optimal solution, Li (1999). and Li and Yang (2003) could develop a
multi-objective linear programming model and suggest a two-level programming method to solve this problem[5,6].
Bector et al. (2004) also presented a similar model. They defined a defuzzifier function to defuzzify payoffs and
game value. This model replaces fuzzy numbers in dual fuzzy linear models and changes the problem to two crisp
linear programming problems[7]. In the model of Vejay et al. (2004), constraints required for feasible solutions are
first defined, then considering the ranking function, a fuzzy non-linear programming model is developed and like the
model of Bector, this model is also changed to an crisp non-linear programming model[8]. Then Vejay et al.
developed their model to solve two-person finite games using fuzzy goals and payoffs[9,10]. Maeda (2003)
suggested another method in which he changed the problem to an crisp parametric game considering payoffs as
symmetric triangular fuzzy numbers , dividing fuzzy numbers into two sections (crisp and fuzzy) and
© 2015 Bull. Georg. Natl. Acad. Sci.
M.Akhbari
34
considering the concept of α-cuts. Identifying the Nash equilibrium of this problem using certain values of
parameters, the dominant minimax equilibrium strategy is achieved[11]. In 2011 Cunlin and Qiang developed
Maeda's model in a state in which payoffs as asymmetric triangular fuzzy numbers are assumed[12]. Lui and Kao
(2007) developed a method for zero-sum games based on the extension principle and α-cuts[13]. In addition,
Buckley and Jowers (2008) suggested a model similar to Lui and Kao's model. In their model, the optimal strategy is
obtained using Monte Carlo simulation where there is no saddle point [14]. In the model of Xu and Zhao (2005),
payoffs are fuzzy variables. In their first study, they defined the measures of possibility, credibility and fuzzy
expected value and based on them, three types of minimax equilibrium strategies, r-possible, r-credible, and
expected are defined. These equilibrium strategies are obtained using an iterative algorithm based on fuzzy
simulation [15]. In another study conducted in 2006, they considered payoffs as random fuzzy variables. This means
that all payoffs have a membership function, parameters of which are randomly characterized by the distribution
function. Then they defined an algorithm to estimate the fuzzy expected value and to identify optimal strategies [16].
In another study investigating a two-person zero-sum game with payoffs in the form of fuzzy variables, Xu and
Wang (2009) defined the pessimistic and optimistic value of minimax equilibrium strategy in the confidence level of
α, modeled it using fuzzy linear programming model and obtained the optimal equilibrium strategy using particle
swarm optimization (PSO) algorithm and fuzzy simulation[17]. In another study, Xu and Li (2010) considered
payoffs as fuzzy random variables. This means that each payoff is a random variable, the probability density
function parameters of which are fuzzy. In addition, they defined an algorithm to estimate the fuzzy expected value
and to identify optimal strategies [18]. In the approach of Gao-Sheng et al. (2011), each player's expected payoff (in
a state that payoffs are considered as triangular fuzzy numbers) is calculated and explicitly solved using the
definition of fuzzy expected value defined based on credibility measures [19].Nishizaki and Sakawa (2000) studied
multi-objective bimatrix games with fuzzy goals and payoffs. In their approach, the level of achievement to each
objective is defined based on goals for each mixed strategy, and a model based on nonlinear programming is
suggested to achieve the highest degree of achieving the aggregated goal (resulting from aggregating different goals
of the problem) as the objective function of the problem [20].In mentioned studies, uncertainty in strategies and
payoffs is investigated using the concepts of decision making in fuzzy environment or fuzzy mathematics. However,
the effect of the level of confidence or trust in the expression of numbers or fuzzy sets is not considered. In fact, two
levels of uncertainty are discussed in this study. As in previous studies, the first level considers fuzzy numbers and
sets in the definition of payoffs, and the level of trust in these ambiguous information is expressed in the upper level.
In this study, these two levels are investigated with the concept of Z -number defined by professor Lotfi Zadeh
(2011) for the first time[21], in the process of analyzing non-cooperative games and a method similar to that of
Nishizaki and Sakawa( 2000) is used to model the problem and find the equilibrium strategies. The paper is
organized as follows, In Section 2, we describe some basic concepts of non-cooperative bimatrix game and ZNumbers and introduce some basic definitions and notations on bimatrix games with payoffs and goals of znumbers. In Section 3, we focus on developing an optimization model that gives the Equilibrium Solution for this
game. In Section 4, two examples are given to illustrate the solution procedure developed here for solving such
games.
2. NON-COOPERATIVE GAME WITH PAYOFFS AND GOALS OF Z-NUMBERS
This section briefly describes the concept of equilibrium in non-cooperative bimatrix games and then introduce
some basic definitions and notations on bimatrix games with payoffs and goals of z-numbers.
Let I = {1, … , m} and J = {1, … , n} sets are pure strategies of players I and II, respectively. Mixed strategies of
these two players showing the probability distributions of all pure strategies are defined as follows:
x = x1 , … , xm
T
∈ X ≜ {x ∈ Rm
+
y = y1 , … , yn
T
∈ Y ≜ {y ∈ Rn+
i∈I x i
j∈J yi
= 1},
(1)
= 1}
(2)
m
Where x and y are mixed strategies of players I and II, respectively and Rm
+ = {r ∈ R ri ≥ 0, i = 1, … , m}.
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
35
If players I and II choose pure strategies of i ∈ I and j ∈ J, respectively, their payoffs will be a ij and bij ,
respectively. Therefore, two-person non-zero-sum game in normal form can be represented with two following
m × n matrices:
A=
a11
⋮
am1
…
⋱
…
a1n
⋮ ,
a mn
B=
b11
⋮
bm1
…
⋱
…
b1n
⋮ ;
bmn
(3)
(4)
If players I and II choose mixed strategies of x ∈ X and y ∈ Y, respectively, their expected payoffs will be as
follows:
m
n
T
E1 x, y = x Ay =
a ij xi yj ,
i=1 j=1
m
(5)
n
T
E2 x, y = x By =
bij xi yj .
(6)
i=1 j=1
In the non-cooperative bimatrix game (A, B), Nash equilibrium is used for a pair of x ∗ and y ∗ strategies in which for
all other mixed strategies x and y we have:
x ∗T Ay ∗ ≥ x T Ay ∗ ,
(7)
x ∗T By ∗ ≥ x ∗T By.
(8)
Now let us consider the concept of the z-number and then investigate a non-cooperative bimatrix game with payoffs
of z-numbers.
A z-number, Z = (A, B), is an order pair of two fuzzy numbers. The number, A, is a restriction on the values which
the real -valued variable, X, can allocate to itself. The second number, B, is a restriction on degree of trust that X is
A. A and B are usually described in natural language e.g. (about 45 dollars, very sure) [21].
In the bimatrix games with payoffs of z-numbers, each player achieves a payoff in the form of z-number for each
pair of pure strategy. If players I and II choose pure strategies of 𝑖 ∈ 𝐼 and 𝑗 ∈ 𝐽, respectively, their payoffs will be
𝑎𝑖𝑗 , 𝑐𝑖𝑗 and 𝑏𝑖𝑗 , 𝑒𝑖𝑗 , respectively. The first and the second numbers show payoff amount and the level of
confidence, respectively. Therefore, bimatrix game with z-numbers in normal form can be represented with two
following 𝑚 × 𝑛 matrices:
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
M.Akhbari
36
𝐴=
𝑎11 , 𝑐11
⋮
𝑎𝑚1 , 𝑐𝑚1
…
⋱
…
𝑎1𝑛 , 𝑐1𝑛
⋮
𝑎𝑖𝑗 , 𝑐𝑚𝑛
𝐵=
𝑏11 , 𝑒11
⋮
𝑏𝑚1 , 𝑒𝑚1
…
⋱
…
𝑏1𝑛 , 𝑒1𝑛
⋮
.
𝑏𝑖𝑗 , 𝑒𝑚𝑛
,
(9)
(10)
It is assumed that the fuzzy numbers of 𝑎𝑖𝑗 , 𝑏𝑖𝑗 , 𝑐𝑖𝑗 and 𝑒𝑖𝑗 are of LR-type fuzzy numbers and are represented as
follows:
𝑎𝑖𝑗 = 𝑎𝑖𝑗 , 𝛼𝑖𝑗 , 𝛽𝑖𝑗 , 𝛼𝑖𝑗 , 𝛽𝑖𝑗 ≥ 0,
𝑏𝑖𝑗 = 𝑏𝑖𝑗 , 𝜈𝑖𝑗 , 𝜔𝑖𝑗 ,𝜈𝑖𝑗 , 𝜔𝑖𝑗 ≥ 0,
𝑐𝑖𝑗 = 𝑐𝑖𝑗 , 𝛾𝑖𝑗 , 𝛿𝑖𝑗 , 0 ≤ 𝑐𝑖𝑗 , 𝛾𝑖𝑗 , 𝛿𝑖𝑗 ≤ 1, 0 ≤ 𝑐𝑖𝑗 − 𝛾𝑖𝑗 ≤ 1,0 ≤ 𝑐𝑖𝑗 + 𝛿𝑖𝑗 ≤ 1 .
𝑒𝑖𝑗 = 𝑐𝑖𝑗 , 𝜀𝑖𝑗 , 𝜎𝑖𝑗 , 0 ≤ 𝑒𝑖𝑗 , 𝜀𝑖𝑗 , 𝜎𝑖𝑗 ≤ 1, 0 ≤ 𝑒𝑖𝑗 − 𝜀𝑖𝑗 ≤ 1,0 ≤ 𝑒𝑖𝑗 + 𝜎𝑖𝑗 ≤ 1 .
And their membership functions are as follows:
𝜇𝑎𝑖𝑗 𝑝 =
0
𝑝 − 𝑎𝑖𝑗 + 𝛼𝑖𝑗 /𝛼𝑖𝑗
𝑎𝑖𝑗 + 𝛽𝑖𝑗 − 𝑝 /𝛽𝑖𝑗
0
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑝 < 𝑎𝑖𝑗 − 𝛼𝑖𝑗
𝑎𝑖𝑗 − 𝛼𝑖𝑗 ≤ 𝑝 < 𝑎𝑖𝑗
𝑎𝑖𝑗 ≤ 𝑝 ≤ 𝑎𝑖𝑗 + 𝛽𝑖𝑗
𝑎𝑖𝑗 + 𝛽𝑖𝑗 < 𝑝
𝜇𝑏𝑖𝑗 𝑝 =
0
𝑝 − 𝑏𝑖𝑗 + 𝜈𝑖𝑗 /𝜈𝑖𝑗
𝑏𝑖𝑗 + 𝜔𝑖𝑗 − 𝑝 /𝜔𝑖𝑗
0
𝑖𝑓 𝑝 < 𝑏𝑖𝑗 − 𝜈𝑖𝑗
𝑖𝑓 𝑏𝑖𝑗 − 𝜈𝑖𝑗 ≤ 𝑝 < 𝑏𝑖𝑗
𝑖𝑓 𝑏𝑖𝑗 ≤ 𝑝 ≤ 𝑏𝑖𝑗 + 𝜔𝑖𝑗
𝑖𝑓 𝑏𝑖𝑗 + 𝜔𝑖𝑗 < 𝑝
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑝 < 𝑐𝑖𝑗 − 𝛾𝑖𝑗
𝑐𝑖𝑗 − 𝛾𝑖𝑗 ≤ 𝑝 < 𝑐𝑖𝑗
𝑐𝑖𝑗 ≤ 𝑝 ≤ 𝑐𝑖𝑗 + 𝛿𝑖𝑗
𝑐𝑖𝑗 + 𝛿𝑖𝑗 < 𝑝
(13)
𝜇𝑐𝑖𝑗 𝑝 =
0
𝑝 − 𝑐𝑖𝑗 + 𝛾𝑖𝑗 /𝛾𝑖𝑗
𝑐𝑖𝑗 + 𝛿𝑖𝑗 − 𝑝 /𝛿𝑖𝑗
0
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑝 < 𝑒𝑖𝑗 − 𝜀𝑖𝑗
𝑒𝑖𝑗 − 𝜀𝑖𝑗 ≤ 𝑝 < 𝑒𝑖𝑗
𝑒𝑖𝑗 ≤ 𝑝 ≤ 𝑒𝑖𝑗 + 𝜎𝑖𝑗
𝑒𝑖𝑗 + 𝜎𝑖𝑗 < 𝑝
(14)
𝜇𝑒𝑖𝑗 𝑝 =
0
𝑝 − 𝑒𝑖𝑗 + 𝜀𝑖𝑗 /𝜀𝑖𝑗
𝑒𝑖𝑗 + 𝜎𝑖𝑗 − 𝑝 /𝜎𝑖𝑗
0
(11)
(12)
The expected value of the game for player I can be expressed as a z-number of 𝐸1 𝑥, 𝑦 , 𝐶𝑟1 𝑥, 𝑦 in which the
first and the second components show the fuzzy expected value of the game and the level of confidence in the first
component, respectively. The values of these two elements for player I are obtained as follows:
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
37
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
𝐸1 𝑥, 𝑦 = 𝑥 𝑇 𝐴𝑦
(15)
𝑚
𝑛
𝑇
𝐶𝑟1 𝑥, 𝑦 = 𝑥 𝐶 𝑦 =
(16)
𝑐𝑖𝑗 𝑥𝑖 𝑦𝑗
𝑖=1 𝑗=1
Using the fuzzy extension principle, fuzzy membership function of the expected value for player I can be obtained
as follows:
𝜇𝐸1
𝑥,𝑦
𝑝 = 𝑠𝑢𝑝 𝑚𝑖𝑛 𝜇𝑎𝑖𝑗 𝑝
(17)
𝑝=𝑥 𝑇 𝐴𝑦 𝑖,𝑗
Accordingly, the fuzzy expected value for player I is represented as a L_R fuzzy number as follows:
𝐸1 𝑥, 𝑦 = 𝑥 𝑇 𝐴𝑦, 𝑥 𝑇 𝐴𝑦, 𝑥 𝑇 𝐴𝑦
(18)
Where 𝐴 and 𝐴 are 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠, the elements of which are 𝛼𝑖𝑗 and 𝛽𝑖𝑗 , respectively, and their membership
function is as follows:
𝜇𝐸1
𝑥,𝑦
𝑝 =
𝜇𝐸1
𝑥,𝑦
𝑝
0
𝑇
𝑝 − 𝑥 𝐴 − 𝐴 𝑦 /𝑥 𝑇 𝐴𝑦
𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑝 /𝑥 𝑇 𝐴𝑦
0
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑖𝑓
𝑝 < 𝑥𝑇 𝐴 − 𝐴 𝑦
𝑥 𝑇 𝐴 − 𝐴 𝑦 ≤ 𝑝 < 𝑥 𝑇 𝐴𝑦
𝑥 𝑇 𝐴𝑦 ≤ 𝑝 ≤ 𝑥 𝑇 𝐴 + 𝐴 𝑦
𝑥𝑇 𝐴 + 𝐴 𝑦 < 𝑝
(19)
Or,
𝑚
0
𝑚
𝑛
𝑝−
=
𝑚
𝑛
𝑖𝑓
𝑚
𝑛
𝑚
𝑎𝑖𝑗 −𝛼𝑖𝑗 𝑥𝑖 𝑦𝑗 /
𝑖=1 𝑗=1
𝑎𝑖𝑗 + 𝛽𝑖𝑗 𝑥𝑖 𝑦𝑗 − 𝑝 /
𝑖=1 𝑗=1
𝛽𝑖𝑗 𝑥𝑖 𝑦𝑗
𝑖𝑓
𝑚
𝑚
𝑎𝑖𝑗 𝑥𝑖 𝑦𝑗
𝑖=1 𝑗=1
𝑛
𝑎𝑖𝑗 𝑥𝑖 𝑦𝑗 ≤ 𝑝 ≤
𝑖=1 𝑗=1
𝑚 𝑛
𝑖𝑓
(20)
𝑎𝑖𝑗 + 𝛽𝑖𝑗 𝑥𝑖 𝑦𝑗
𝑖=1 𝑗=1
𝑎𝑖𝑗 + 𝛽𝑖𝑗 𝑥𝑖 𝑦𝑗 < 𝑝
𝑖=1 𝑗=1
The expected value of the game for player II, 𝐸2 𝑥, 𝑦 , 𝐶𝑟2 𝑥, 𝑦 can be similarly obtained.
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
𝑛
𝑎𝑖𝑗 −𝛼𝑖𝑗 𝑥𝑖 𝑦𝑗 ≤ 𝑝 <
𝑖=1 𝑗=1
𝑚 𝑛
𝑖=1 𝑗=1
0
𝑛
𝑎𝑖𝑗 −𝛼𝑖𝑗 𝑥𝑖 𝑦𝑗
𝑖=1 𝑗=1
𝛼𝑖𝑗 𝑥𝑖 𝑦𝑗 𝑖𝑓
𝑖=1 𝑗=1
𝑚 𝑛
𝑛
𝑝<
M.Akhbari
38
3. AN EQUILIBRIUM SOLUTION FOR A BIMATRIX GAME WITH PAYOFFS AND GOALS OF ZNUMBERS
In the present study, since the expected value of each player's is of z-numbers type, we consider a goal of z-numbers
type for them. Assume that for player I, we consider a goal of z-number, 𝐺1 , 𝑅1 ,in which 𝐺1 , 𝑅1 are fuzzy numbers
indicating the desired payoff of the game and the acceptable level of trust or confidence in the solution.
If the payoff membership degree of player I in 𝐺1 is higher, it will be more desirable in his opinion. Membership
function 𝜇𝐺1 defined on the set of real numbers is as follows
𝜇𝐺1
0
𝑝−𝑎
𝑝 =
𝑎−𝑎
1
𝑝≤𝑎
𝑎 ≤ 𝑝 ≤𝑎
(21)
𝑝≥𝑎
Accordingly, if the payoff of player I is less than 𝑎, its membership degree in 𝐺1 will be equal to zero and not
desirable and will be linearly increased in the distance between 𝑎 and 𝑎 , and the membership degree or the
maximum desirability is obtained in amounts higher than maximum 𝑎. Therefore, membership degree in 𝐺1
indicates the level of achievement to the goal.
Here, the level of achievement to the goal is obtained using the concept of Bellman-Zadeh fuzzy decision-making
[22]. Membership degree of fuzzy expected value 𝐸1 𝑥, 𝑦 in the 𝐺1 is defined as the following for player I,
𝑑1 𝑥, 𝑦 = 𝑚𝑎𝑥 𝑚𝑖𝑛{ 𝜇𝐸1
𝑝
𝑥,𝑦
𝑝 , 𝜇𝐺1 𝑝 }
(22)
In fact, 𝑑1 𝑥, 𝑦 is the maximum points of intersection between membership functions of 𝐸1 𝑥, 𝑦 and 𝐺1 . 𝑑1 𝑥, 𝑦
can be calculated in the following states:
𝑑1 𝑥, 𝑦 =
0
𝑋𝑇 𝐴 + 𝐴 𝑌 − 𝑎
𝑎 − 𝑎 + 𝑋 𝑇 𝐴𝑌
1
𝑥𝑇 𝐴 + 𝐴 𝑦 ≤ 𝑎
𝑖𝑓
𝑖𝑓
𝑎 < 𝑥𝑇 𝐴 + 𝐴 𝑦 ≤ 𝑎
𝑖𝑓
(23)
𝑎 < 𝑥𝑇 𝐴 + 𝐴 𝑦
Briefly:
𝑑1 𝑥, 𝑦 = 𝑚𝑖𝑛{𝑚𝑎𝑥 0,
𝑋𝑇 𝐴 + 𝐴 𝑌 − 𝑎
d1 x, y is shown in Figure (1).
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
𝑎 − 𝑎 + 𝑋 𝑇 𝐴𝑌
, 1}
(24)
39
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
𝜇𝐸1
𝑥,𝑦
𝑝
𝜇𝐺1 𝑝
1
𝑑1 𝑥, 𝑦
0
𝑎
𝑎
p
Fig.1 the degree of achievement to the goal (Nishizaki and.Sakawa 2000)
𝑑2 𝑥, 𝑦 is calculated similarly.
Since we do not want to limit the improvement of players' payoffs, we identify the values of parameters 𝑎 and 𝑎 so
that the following equation is always true:
0<
𝑋𝑇 𝐴 + 𝐴 𝑌 − 𝑎
𝑎 − 𝑎 + 𝑋 𝑇 𝐴𝑌
<1
(25)
Therefore, we can define the values of parameters 𝑎 and 𝑎 as follows:
𝑎 = 𝑚𝑖𝑛 𝑎𝑖𝑗 − 𝛼𝑖𝑗
(26)
𝑎 = 𝑚𝑎𝑥 𝑎𝑖𝑗 + 𝛽𝑖𝑗
(27)
𝑖,𝑗
𝑖,𝑗
In fact, the second part of the goal or 𝑅1 shows the acceptable level of trust or confidence of player I in the solution
which means player I accepts only solution that the level of the confidence of it, 𝐶𝑟1 , is greater than 𝑅1. 𝑅1 is a fuzzy
set defined as a L-R fuzzy number on the set of real numbers in interval 0,1 . Since confidence in the solution is
resulted from fuzzy numbers of payoff confidence, at least one of these values or 𝑐𝑖𝑗 s should be higher than or equal
𝑅1, or in other words 𝑅1should be smaller than the maximum 𝑐𝒊𝒋 s to have at least one feasible solution. The set of
possible fuzzy values of 𝑅1 is called 𝐻1 . Here, the greater relationship is defined through the comparison of parties'
desuzzified values calculated by center of gravity method.
𝑅1 = 𝑅1 , 𝑙1 , 𝑟1 ∈ 𝐻1 = ℎ = ℎ, 𝑙, 𝑟 ℎ ≤ 𝑚𝑎𝑥 𝑐𝑖𝑗 , ℎ − 𝑙 ≥ 0, ℎ + 𝑟 ≤ 1
(28)
Membership function 𝑅1 is defined as follows:
𝜇𝑅1 𝑝 =
0
𝑝 − 𝑅1 + 𝑙1 /𝑙1
𝑅1 + 𝑟1 − 𝑝 /𝑟1
0
𝑖𝑓 𝑝 < 𝑅1 − 𝑙1
𝑖𝑓 𝑅1 − 𝑙 ≤ 𝑝 < 𝑅1
𝑖𝑓 𝑅1 ≤ 𝑝 ≤ 𝑅1 + 𝑟1
𝑖𝑓 𝑅1 + 𝑟1 < 𝑝
The minimum level of trust or confidence of player II (𝑅2 = 𝑅2 , 𝑙2 , 𝑟2 ) can be similarly obtained.
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
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M.Akhbari
40
Accordingly, the set of feasible solutions (S) can be defined as follows:
𝑆 = { 𝑥, 𝑦 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌, 𝐶𝑟1 𝑥, 𝑦 ≥ 𝑅1 , 𝐶𝑟2 𝑥, 𝑦 ≥ 𝑅2 }
(30)
According to the definition of the set of feasible solutions (S), the level of achievement to the first player's goal or
𝑑1 𝑥, 𝑦 can be defined as follows:
𝑥𝑇 𝐴 + 𝐴 𝑦 − 𝑎
𝑑1 𝑥, 𝑦 =
𝑎 − 𝑎 + 𝑦 𝑇 𝐴𝑦
0
𝑖𝑓 𝑥, 𝑦 ∈ 𝑆
𝑖𝑓
(31)
𝑥, 𝑦 ∉ 𝑆
The level of achievement to the second player's goal or 𝑑2 𝑥, 𝑦 can be similarly defined.
Therefore, considering the level of achievement to the goal with a pair of mixed strategies (𝑥 ∗ and 𝑦 ∗ ), an
equilibrium solution is defined so that for other mixed strategies x and y,
𝑑1 𝑥 ∗ , 𝑦 ∗ ≥ 𝑑1 𝑥, 𝑦 ∗ ,
𝑑2 𝑥 ∗ , 𝑦 ∗ ≥ 𝑑2 𝑥 ∗ , 𝑦 .
(32)
Optimal solutions 𝑥 ∗ and 𝑦 ∗ of both following optimization problems are strategies that satisfy above conditions
(32):
𝑚𝑎𝑥 𝑑1 𝑥, 𝑦 ∗ ,
Subject to
𝐶𝑟1 𝑥, 𝑦 ∗ ≥ 𝑅1 ,
𝑒 𝑚𝑇 𝑥 = 1,
(33)
𝑥 ≥ 0𝑚 ,
and
𝑚𝑎𝑥 𝑑2 𝑥 ∗ , 𝑦 ,
Subject to
𝐶𝑟2 𝑥 ∗ , 𝑦 ≥ 𝑅2 ,
𝑒
𝑛𝑇
(34)
𝑦 = 1,
𝑦 ≥ 0𝑛 .
where em and en are m-dimensional and n-dimensional column vectors every entry of which is a unit, and 0m and
0n are m-dimensional and n-dimensional column vectors every entry of which is a zero.
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
41
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
According to the equations (16) and (31), the problems (33) and (34) can be rewritten as the following:
𝑚𝑎𝑥
𝑥𝑇 𝐴 + 𝐴 𝑦∗ − 𝑎
𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦 ∗
Subject to
𝑅1 − 𝑥 𝑇 𝐶 𝑦 ∗ ≤ 0,
𝑒
𝑚𝑇
(35)
𝑥=1,
𝑥 ≥ 0𝑚 ,
And,
𝑚𝑎𝑥
𝑥 ∗𝑇 𝐵 + 𝐵 𝑦 − 𝑏
𝑏 − 𝑏 + 𝑥 ∗𝑇 𝐵 𝑦
Subject to
𝑅2 − 𝑥 ∗𝑇 𝐸 𝑦 ≤ 0,
𝑒
𝑚𝑇
(36)
𝑦 = 1,
𝑦 ≥ 0𝑛 .
Considering the method of ranking numbers using the center of gravity, the first constraint of the problem (35) can
be modified as follows:
𝑚
𝑛
3𝑐𝑖𝑗 −𝛿𝑖𝑗 + 𝛾𝑖𝑗 𝑥𝑖 𝑦𝑗∗ ≤ 0 →
3𝑅1 − 𝑙1 + 𝑟1 −
𝑖=1 𝑗=1
𝑚
𝑛
1−
𝑖=1 𝑗=1
3𝑐𝑖𝑗 − 𝛿𝑖𝑗 + 𝛾𝑖𝑗
𝑥 𝑦 ∗ ≤ 0 → 1 − 𝑥 𝑇 𝐾𝑦 ∗ ≤ 0
3𝑅1 − 𝑙1 + 𝑟1 𝑖 𝑗
Where 𝐾 are 𝑚 × 𝑛 𝑚𝑎𝑡𝑟𝑖𝑥, the elements of which, 𝑘𝑖𝑗 is obtained as follows:
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
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M.Akhbari
42
𝑘𝑖𝑗 =
3𝑐𝑖𝑗 − 𝛿𝑖𝑗 + 𝛾𝑖𝑗
.
3𝑅1 − 𝑙1 + 𝑟1
Similarly, in the problem (36), we have 1 − 𝑥 ∗𝑇 𝑀𝑦 ≤ 0.
Accordingly, the two problems can be rewritten as follows:
𝑚𝑎𝑥
𝑥𝑇 𝐴 + 𝐴 𝑦∗ − 𝑎
𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦 ∗
Subject to
1 − 𝑥 𝑇 𝐾𝑦 ∗ ≤ 0,
(38)
𝑒 𝑚𝑇 𝑥 = 1 ,
𝑥 ≥ 0𝑚 ,
And,
𝑚𝑎𝑥
𝑥 ∗𝑇 𝐵 + 𝐵 𝑦 − 𝑏
𝑏 − 𝑏 + 𝑥 ∗𝑇 𝐵 𝑦
Subject to
1 − 𝑥 ∗𝑇 𝑀𝑦 ≤ 0,
𝑒
𝑚𝑇
(39)
𝑦 = 1,
𝑦 ≥ 0𝑛 .
By applying the necessary Kuhn-Tucker Conditions for the problems (38) and (39), we have:
𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴 𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦 + 𝜆 + 𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦
2
2
= 0,
= 0,
𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
2
𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦 𝑥 𝑇 𝐵 + 𝑏 − 𝑥 𝑇 𝐵𝑦 𝑥 𝑇 𝐵 + 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
(40)
(41)
𝜏𝐾𝑦 + 𝜆𝑒 𝑚 ≤ 0,
2
𝜏 𝑥 𝑇 𝑀 + 𝜆 𝑒 𝑛 ≤ 0,
(42)
(43)
43
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
1 − 𝑥 𝑇 𝐾𝑦 ≤ 0,
(44)
𝑇
1 − 𝑥 𝑀𝑦 ≤ 0,
(45)
𝑒 𝑚𝑇 𝑥 = 1,
(46)
𝑒
𝑛𝑇
𝑦 = 1,
(47)
𝜏 1 − 𝑥 𝑇 𝐾𝑦 = 0,
(48)
𝑇
𝜏 1 − 𝑥 𝑀𝑦 = 0,
(49)
𝑥 ≥ 0𝑚 ,
(50)
𝑛
𝑦≥ 0 .
(51)
Where 𝜏 𝑎𝑛𝑑 𝜏 are positive and 𝜆 𝑎𝑛𝑑 𝜆 are scalar variables.
Lemma 1. 𝑥 and 𝑦 satisfy the Kuhn-Tucker Conditions (40)-(51) If and only if there exists an optimal solution to
the mathematical programming problem and 𝑥 and 𝑦 are components of the optimal solution,
𝑚𝑎𝑥 𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
2
+ 𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦
2
+ 𝜆+𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦 ,
Subject to
𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
2
𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦 𝑥 𝑇 𝐵 + 𝑏 − 𝑥 𝑇 𝐵𝑦 𝑥 𝑇 𝐵 + 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦
1 − 𝑥 𝑇 𝐾𝑦 ≤ 0,
1 − 𝑥 𝑇 𝑀𝑦 ≤ 0
𝑒 𝑚𝑇 𝑥 = 1,
𝑒 𝑛𝑇 𝑦 = 1,
𝜏 1 − 𝑥 𝑇 𝐾𝑦 = 0,
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
𝜏𝐾𝑦 + 𝜆𝑒 𝑚 ≤ 0,
2
𝜏 𝑥 𝑇 𝑀 + 𝜆 𝑒 𝑛 ≤ 0,
(52)
M.Akhbari
44
𝜏 1 − 𝑥 𝑇 𝑀𝑦 = 0.
Proof: The constraints of the optimization problem (52) are the constraints (42)-(51) of Kuhn-Tucker conditions.
Assume that the feasible region of the above-mentioned problem is called S. In this region, according to (42) and
(43), we have:
𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
= 𝑥𝑇
2
+ 𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦 + 𝜆 + 𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦
2
𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑥 𝑇 𝐴𝑦 𝐴𝑦 + 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
+ 𝑏 − 𝑥 𝑇 𝐵𝑦 𝑥 𝑇 𝐵 + 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦
2
𝜏𝐾𝑦 + 𝜆𝑒 𝑚
2
+ 𝑦{ 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦 𝑥 𝑇 𝐵
𝜏𝑥 𝑇 𝑀 + 𝜆𝑒 𝑛 } ≤ 0
(53)
For any
𝑥, 𝑦, 𝜆, 𝜆, 𝜏, 𝜏 ∈ 𝑆
Therefore,
𝑚𝑎𝑥𝑥,𝑦,𝜆,𝜆,𝜏,𝜏 𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦
2
2
+ 𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦 + 𝜆 +
(54)
≤ 0,
Let 𝑥 ∗ , 𝑦 ∗ , 𝜆∗ , 𝜆∗ , 𝜏 ∗ , 𝜏 ∗ satisfy Kuhn-Tucker conditions (40)-(51). Therefore, according to the (40) and (41), we have:
𝑎𝑥 ∗ 𝑇 𝐴 + 𝐴 𝑦 ∗ − 𝑎𝑥 ∗ 𝑇 𝐴 𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑎 − 𝑎 + 𝑥 ∗ 𝑇 𝐴𝑦 ∗
+ 𝜏 ∗ 𝑏 − 𝑏 + 𝑥 ∗ 𝑇 𝐵𝑦 ∗
2
2
+ 𝑏𝑥 ∗ 𝑇 𝐵 + 𝐵 𝑦 ∗ − 𝑏𝑥 ∗ 𝑇 𝐵𝑦 ∗ + 𝜆∗
(55)
= 0.
Based on (54), (55) and the fact 𝑥 ∗ , 𝑦 ∗ , 𝜆∗ , 𝜆∗ , 𝜏 ∗ , 𝜏 ∗ ∈ 𝑆 ,we have
𝑎𝑥 ∗ 𝑇 𝐴 + 𝐴 𝑦 ∗ − 𝑎𝑥 ∗ 𝑇 𝐴 𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑎 − 𝑎 + 𝑥 ∗ 𝑇 𝐴𝑦 ∗
+ 𝜏 ∗ 𝑏 − 𝑏 + 𝑥 ∗𝑇 𝐵𝑦 ∗
2
+ 𝑏𝑥 ∗ 𝑇 𝐵 + 𝐵 𝑦 ∗ − 𝑏𝑥 ∗ 𝑇 𝐵𝑦 ∗ + 𝜆∗
2
= 𝑚𝑎𝑥 𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
𝑥,𝑦,𝜆,𝜆,𝜏,𝜏
+ 𝜆 + 𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵𝑦
2
+ 𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦
2
Conversely, let 𝑥 ∗ , 𝑦 ∗ , 𝜆∗ , 𝜆∗ , 𝜏 ∗ , 𝜏 ∗ ( is an optimal solution of the problem (52). Therefore, we have
Bull. Georg. Natl. Acad. Sci., vols. 9, no.1, 2015
(56)
45
Equilibrium Solution of Non-cooperative Bimatrix Game of Z-Numbers
𝑎𝑥 ∗ 𝑇 𝐴 + 𝐴 𝑦 ∗ − 𝑎𝑥 ∗ 𝑇 𝐴 𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑎 − 𝑎 + 𝑥 ∗ 𝑇 𝐴𝑦 ∗
+ 𝜏 ∗ 𝑏 − 𝑏 + 𝑥 ∗𝑇 𝐵𝑦 ∗
2
2
+ 𝑏𝑥 ∗ 𝑇 𝐵 + 𝐵 𝑦 ∗ − 𝑏𝑥 ∗ 𝑇 𝐵𝑦 ∗ + 𝜆∗
≤ 0.
Considering the existence of the equilibrium solution and Kuhn-Tucker conditions (34)-(45), there exists at least one
solution 𝑥, 𝑦, 𝜆, 𝜆, 𝜏, 𝜏 which satisfies
𝑎𝑥 𝑇 𝐴 + 𝐴 𝑦 − 𝑎𝑥 𝑇 𝐴 𝑦 + 𝜆 + 𝜏 𝑎 − 𝑎 + 𝑥 𝑇 𝐴𝑦
=0
2
+ 𝑏𝑥 𝑇 𝐵 + 𝐵 𝑦 − 𝑏𝑥 𝑇 𝐵𝑦 + 𝜆 + 𝜏 𝑏 − 𝑏 + 𝑥 𝑇 𝐵 𝑦
2
Therefore, if 𝑥 ∗ , 𝑦 ∗ , 𝜆∗ , 𝜆∗ , 𝜏 ∗ , 𝜏 ∗ ( is the optimal solution of problem (52), there must be
𝑎𝑥 ∗ 𝑇 𝐴 + 𝐴 𝑦 ∗ − 𝑎𝑥 ∗ 𝑇 𝐴 𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑎 − 𝑎 + 𝑥 ∗ 𝑇 𝐴𝑦 ∗
+ 𝜏 ∗ 𝑏 − 𝑏 + 𝑥 ∗𝑇 𝐵𝑦 ∗
2
2
+ 𝑏𝑥 ∗ 𝑇 𝐵 + 𝐵 𝑦 ∗ − 𝑏𝑥 ∗ 𝑇 𝐵𝑦 ∗ + 𝜆∗
= 0.
According to the first and the second constraints of problem (52),
𝑎𝑥 ∗ 𝑇 𝐴 + 𝐴 𝑦 ∗ − 𝑎𝑥 ∗ 𝑇 𝐴 𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑎 − 𝑎 + 𝑥 ∗ 𝑇 𝐴𝑦 ∗
𝑏𝑥 ∗ 𝑇 𝐵 + 𝐵 𝑦 ∗ − 𝑏𝑥 ∗ 𝑇 𝐵𝑦 ∗ + 𝜆∗ + 𝜏 ∗ 𝑏 − 𝑏 + 𝑥 ∗ 𝑇 𝐵𝑦 ∗
2
2
= 0,
= 0..
Hence 𝑥 ∗ , 𝑦 ∗ , 𝜆∗ , 𝜆∗ , 𝜏 ∗ , 𝜏 ∗ (satisfies Kuhn-Tucker conditions.
Theorem 1. For a bimatrix game 𝐴, 𝐵 ) with payoffs of Z numbers, let membership functions of the first part of
payoffs and goals of Players I and II be linear. The necessary conditions for x and y to be an equilibrium solution
with respect to the degree of attainment of the goal is that x and y are components of an optimal solution to the
mathematical programming problem (52).
Since the constraints of the problems (38) and (39) are linear and convex and the objective functions of problems
(38) and (39) are concave, the necessary Kuhn-Tucker conditions (40)-(51) are also sufficient. Therefore,
𝑥 ∗ , 𝑦 ∗ (obtained from the solution of problem (52) will be the optimal solution of problems (38) and (39) and
consequently will be the equilibrium solution of bimatrix game with payoffs of z-numbers.
Numerical Example –Consider the bimatrix game with payoffs of z-numbers:
𝐴=
100,10,15 , 0.7,0.05,0.05
80,5,5 , 0.8,0.03,0.02
85,5,10 , 0.9,0.04,0.05
185,15,20 , 0.75,0.1,0.08
,
𝐵=
90,5,15 , 0.75,0.05,0.1
110,10,5 , 0.65,0.03,0.05
100,10,10 , 0.9,0.1,0.05
135,5,10 , 0.85,0.05,0.05
,
The level of expected confidence for the first and the second players is considered about 80%, 𝑅1 = 0.8,0.05,0.05
and 75%, 𝑅2 = 0.75,0.05,0.05 , respectively. Goal parameters including 𝑎 and 𝑎 are calculated based on (26) and
Bull. Georg. Natl. Acad. Sci., vols 9, no. 1, 2015
M.Akhbari
46
(27) and their values are equal to 75 and 205, respectively. Similarly, the values of 𝑏 𝑎𝑛𝑑 𝑏 are 145 and 85,
respectively.
The equilibrium solution of the above-mentioned problem is obtained through solving the mathematical
programming problem (50):
𝑥 = 0.354, 0.645
𝑦 = 0,1
And according to (15) fuzzy values of the game for the two players are:
𝐸1 𝑥, 𝑦 = 149.58,11.458,16.458
𝐸2 𝑥, 𝑦 = 122.6,6.77,10
In addition, , based on (17) the level of achievement to the goals for the first and the second players is 60% and
68%, respectively, and level of confidence of them to the solution are equal to
𝐶𝑟1 = 0.8,0.078,0.069 ,
𝐶𝑟2 = 0.86,0.067,0.05 ,
4. CONCLUSIONS
The present study investigated the equilibrium solution for non-cooperative bimatrix game with payoffs and goals of
z-numbers. The equilibrium solution was defined based on the level of achievement to the goal. Then two nonlinear
programming problems were developed using the objective function of the level of achievement to fuzzy goal and
the constraint of the confidence level for each player. We have shown the sufficient and necessary conditions that
pair of strategies be the equilibrium solution and an optimization model was developed to find them. Finally, an
illustrative examples is given in order to show the detailed calculation process.
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