Matrix Game with Znumbers
 Author: Bandyopadhyay Sibasis, Raha Swapan, Nayak Prasun Kumar
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 15, Issue1, p60~71, 25 March 2015

ABSTRACT
In this paper, a matrix game is considered in which the elements are represented as Znumbers. The objective is to formalize the human capability for solving decisionmaking problems in uncertain situations. A ranking method of Znumbers is proposed and used to define pure and mixed strategies. These strategies are then applied to find the optimal solution to the game problem with an induced pay off matrix using a min max, max min algorithm and the multisection technique. Numerical examples are given in support of the proposed method.

KEYWORD
Znumber , Interval number , Matrix game , Multisection technique

1. Introduction
In the modern world, one is faced with innumerable problems arising from existing socioeconomic conditions that involve varying degrees of imprecision and uncertainty. We often need to take decisions in conflicting situations based on uncertain, ambiguous, or incomplete information. Human beings have a tremendous capability to form rational decisions based on such imprecise information. It is hard to formalize these human capabilities. This challenge has motivated us to consider a game theoretic model under conditions of uncertainty.
Game theory problems under uncertainty have been considered by many researchers, e.g., Nayak and Pal [1–3], Narayanan [4], and Nishizaki [5]. Narayanan [4] solved a 2 × 2 interval game using the probability and possibility approach but no certain distribution function has been used. Nayak and Pal [3] established a method of solution of a matrix game using interval numbers. But the solution has been considered under a certain condition which has been obviated by the authors in their work [6]. Biswas and Bose [7] constructed a quadratic programming model under fuzzily described system constraints on the basis of degree of satisfaction. Veeramani and Duraisamy [8] suggested a new approach for solving fuzzy linear programming problem using the concept of nearest symmetric triangular fuzzy number approximations with preserve expected interval. But this approach is not efficient when a primal basic feasible solution is not in hand. Ebrahimnejad and Nasseri [9] overcame this shortcoming using a new algorithm. Iskander [10] proposed a new approach for solving stochastic fuzzy linear programming problem using triangular fuzzy probabilities. Apart from this Kumar and Kar [11], Marbini and Tavana [12], Ebrahemnijad and Nasseri [13] have contributed substantially to the application of fuzzy mathematics in operations research. However, there are many shortcomings in the abovementioned techniques for solving game problems in uncertain situations. These can be categorized as follows:
We consider the theory of fuzzy sets to address the uncertainty that occurs in industrial problems or machine learning. It is unlikely that we always have complete knowledge about the domain set. For example, when we try to address an object as ‘beautiful,’ we may not have complete knowledge about the parameters by which the beauty of an object can be explained, or the parameters, if assumed, may not match other people’s choices, i.e., they may not be unique. However, the reliability of the information must be taken into consideration, and this aspect is lacking in the fuzzy set description. We use soft set theory or rough set theory to describe uncertain situations. However, we may not know the complete set of parameters E in the case of a soft set, i.e., there is an issue regarding the reliability of the information concerned. In the rough set description, the lower or upper approximation or boundary region may not always be described, because knowledge or information about the equivalence relation R or domain U are only approximations or assumptions, and may not be known in advance. Additionally, it is not guaranteed that the desired optimal solution will be compatible with industrial applications. Different optimization techniques give rise to different optimal results, and we can only compare these results with those obtained by existing techniques. We cannot, however, ensure that a technique gives an actual optimal solution that will be universally accepted. The main reason behind this is that we cannot address the uncertainty properly, and thus use a number of assumptions.
Thus, we need to develop the mathematical structures that provide a generalization of uncertain situations and that consider the reliability of available information. Zadeh [14] has made an attempt towards such a generalization by proposing Znumbers. A Znumber is an ordered pair (
A ,B ) in whichA represents the restriction on a realvalued uncertain variableX andB is the measure of sureness, reliability, or certainty aboutA . However, the Znumber lacks informativeness when first introduced. Their composition in a summation has been defined [14], but the ranking or ordering of Znumbers has not yet been considered. In this paper, we introduce a ranking of Znumbers in the specific case whereA represents a restriction on the measure of the possibility distribution with a Gaussian membership function, andB is a restriction on the probability measure with a normal density function. This ranking method is then used to solve a twoperson zerosum game using min − max and max − min principles [6] and the multisection technique.The remainder of this paper is organized as follows. In Section 2, we discuss the concept of Znumbers, before presenting some basic definitions, notation, and comparisons related to interval numbers in Section 3. In Section 4, a matrix game with Znumbers is proposed, and then Section 5 discusses the interval approximation of Znumbers, and introduces some definitions, explanations, theorems about matrix games with Znumbers, pure and mixed strategies, and saddle points. In section 6, a computational procedure is proposed along with a minmax algorithm, maxmin algorithm, and multisection technique. Section 7 presents an example in support of the proposed method, along with discussions related to our algorithm and the results of the numerical example. Brief conclusions are given in Section 8.
2. Znumber
Let us consider a fuzzy set defined over a universe of discourse
X . How will we define ifX is not known in advance? Again, if it is assumed thatX is known with certain parameters, then what is the reliability of such an information? We may say that we can define as type−2 fuzzy set where,X is a type−1 fuzzy set. In that case,X is also defined over some domain set . However, what will happen if is also not known? In Znumber representation, we may always constructX with some arbitrary elementx ∈X with some probability or possibilityp . In that case, we can always associate a statementx with some statementG asx isG isλ where,
λ acts as a fuzzy quantifier and we consider it as probability induced by some possibility membership and we writeprob(
x isG ) =λ (fuzzy granularity [15]).This situation, inspires us to define a
Z number formally. A ZnumberZ is defined [14] as an ordered pair (A ,B ) whereA andB are two fuzzy numbers and a Zvaluation is defined aswhere,
X is a realvalued uncertain variable,A is a restriction on the real values of the uncertain variableX andB is the reliability or certainty ofA . Here, we considerX to be a random variable,A_{X} to be a restriction on the measure of a possibility distribution with membership functionµ_{AX} andp_{X} is the restriction on probability distribution(density) function ofX . The scalar productµ_{AX} .p_{X} gives the probability measure,P_{AX} ofA_{X} and it is given asHere, we recall that
B is a restriction on the probability measure ofA and it is not a restriction on the probability ofA . Here, we attempt to ordering of the Znumbers. For that purpose, here we consider the membership function as gaussian membership function. The reason behind taking such a function as membership function is that it is nonlinear in nature and assume the situation which occurs generally in industrial applications. Unless otherwise stated by Znumber we here understand Zvaluation. For the sake of computation here we will consider a Znumber as (X ;A_{X} ,P_{AX} ) or (X ;µ_{AX} ,P_{AX} ) or(X ; <c ,σ >,P_{AX} ) wherec andσ are parameters of the Gaussian membership function. Here we may consider some examples like (Population of India, about 1200 million, very likely ),(degree of satisfaction, very high, not sure) which are considered as Znumber.To consider an arithmetic operation, let
Z_{X} = (X ;µ_{AX} ,P_{AX} ) andZ_{Y} = (Y ;µ_{AY} ,P_{AY} ) be two Znumbers. Using the extension principle as described by [14], we obtainZ_{X} +Z_{Y} =Z _{X+Y}3. Intervalnumber
Let us consider that ℜ represents the set of all real numbers. We define an interval, Moore [16], as
where
a_{L} anda_{R} are said to be the lower and upper limits of the interval , respectively. Ifa_{L} =a_{R} then is reduced to a real numbera , wherea =a_{L} =a_{R} . Corresponding interval arithmetic is given by 3.For,
The order relation of interval numbers is discussed in several literature [16, 17]. Recently Chakrabortty et al. [18] proposed a revised definition of order relations between interval costs(or times) for minimization problems and interval profits for maximization problems for optimistic and pessimistic decision making. Let us suppose, the intervals and represent the uncertain interval costs (or times) or profits in centerradius form.
For minimization problems the order relation ‘ ≤_{o min}’ between the intervals and is
This implies that is superior to and is accepted. This order relation is not symmetric.
In pessimistic decision making, the decision maker expects the minimum cost/time for minimization problems according to the principle ‘Less uncertainty is better than more uncertainty’.
For minimization problems, the order relation ‘ <_{p min}’ between the intervals and is
(i) iff , for typeI and typeII intervals, (ii) iff and , for typeIII intervals.
which is said to be matrix game with Znumbers.
In this paper, arithmetic operations on interval [
a ,b ] and their ranking as proposed in [6] serves as a level−1 computation [14] and the same is used in ranking of Znumbers. Gregorzewski [19] proposed a method for interval approximation of fuzzy number. Here, in same way, we will approximate a Gaussian fuzzy number to an interval number. For that, let us consider a Gaussian fuzzy number <x ,µ (x ;c ,σ )x ∈X > where the membership function is given asNow, we define an
α cut setA_{α} asA_{α} = {x :µ (x ;c ,σ ) ≥α }. Then,and let us consider that and . Let [
a ,b ] be the corresponding interval approximation. ThenTherefore, if
Z = (X ;A_{X} ,P_{AX} ) be aZ number with Gaussian membership functionµ (x ;c ,σ ) then the corresponding interval approximation is andHere, we get the probability density function
p_{Xij} as the normal density functionN (c_{ij} ,σ ). Hence,where . Then, we obtain from Eq. (9) that
Now, let us consider two
Z numbersZ _{1},Z _{2} and corresponding interval approximations as and Using the interval arithmetic, we propose the ranking ofZ numbers as4. Solution of Matrix Game
Suppose, the payoff for player
A in a matrix game with Znumber be represented as (X_{ij} ;A_{Xij} ,P_{AXij} ). Then, the corresponding interval approximation will be given by . The payoff matrix with elements as interval approximation of Znumber can then be represented aswhere ,
i = 1, 2 · · ·m ,j = 1, 2, · · ·n andTheorem 4.1. If Z _{1}(X _{1};A _{X1} ,P _{AX1} ) ≤Z _{2}(X _{2};A _{X2} ,P _{AX2})then P _{AX1} ≥P _{AX2}for optimistic decision maker and P _{AX1} ≤P _{AX2}for pessimistic decision maker .Proof . From the expression in (13) we see thatP_{AXij} depend only onσ_{ij} and not onc_{ij} . Using this fact and combining the relation in (14) we can easily construct the proof of the theorem.Notes : Here it should be noted that for pessimistic decision maker the degree of certaintyP_{AX} is lesser iff the membership value or the interval approximation is lesser and for optimistic decision maker the degree of certainty does not matter at all, it only gives the degree of reliability of the information.4.1 Pure Strategy
In the context of Znumber, a pure strategy may be considered as a decision making rule in which one particular course of action is selected with some degree of reliability or certainty for the pay off considered. Actually, Znumber gives higher level of generality compared to interval numbers where length of the interval actually measures the certainty. For lack of informativeness of Znumber, we develop the concept of pure strategy in the domain of interval numbers with parallel computation. For matrix game with Znumber, we define the min max and max min as
where ‘∨
' and ‘∧' the max and min operators for two Znumber in the domain of Znumbers Z respectively. In accordance with ranking of Znumbers in (14), for games such as with pure strategy, we define the concept of saddle point solution.Definition 4.1. (Saddle Point) The concept of saddle point in classical form was proposed by Von Neumann and Morgenstern [20]. The (k ,r )th position of the payoff matrix with Znumbers is said to be a saddle point of the matrix game , if and only if,The position (
k ,r ) is said to be a saddle point, the entry itself [a_{kr} ,b_{kr} ] represents the value of the game (denoted by ) and the pair of pure strategies leading to it are optimal pure strategies. Now we have to confirm that the relation as defined here for the saddle point solution actually exists for the matrix game with Znumbers. For that purpose we must consider the following theorems.Theorem 4.2. Let ;i = 1, 2, ...,m ;j = 1, 2, ...,n be the m ×n payoff matrix for a twoperson matrix game Γwith Znumbers .Suppose and both exist .Then Proof . For some fixedi , we have, by using the order relation onZ ,From (19) and (20) we have,
Here, we see that is independent of
j , since (X_{ij} ;A_{Xij} ,P_{Xij} ) has obtained minimum value for some fixed value ofj . Hence we writeAgain, the righthand side of (21) is independent of
i , hence, we obtainHence the theorem.
Theorem 4.3. Let both and exist .Then a necessary and sufficient condition that (X_{ij} ;A_{Xij} ,P_{Xij} )will be a saddle point at i =k ,j =r is and Proof .Condition is necessary : LetLet
i =k make a maximum and letj =r make a minimum. Then, we writeAs, , we have . Also, using the order relation over Znumbers in
Z ,This is one of the conditions for (
X_{ij} ;A_{Xij} ,P_{AXij} ) to have a saddle point. The other condition can similarly be deduced.Condition is sufficient : Let (X_{kr} ;A_{Xkr} ,P_{AXkr} ) be the saddle point of the payoff matrix , then fori =k ,j =r we have, by definition of saddle pointor,
or,
or,
as
and
Using the above Theorem 4.2 we have,
Hence the necessary and sufficient condition for the existence of a saddle point is proved.
Example 4.1. Let us consider the 2 × 2 matrix game with Znumber having the payoff matrix as in the following:It can be easily verified that Therefore, the matrix game with Znumber has a saddle point at (1, 1) and the optimal strategies for players
A andB are the pure strategiesA _{1} andB _{1}, respectively. The value of the matrix game is4.2 Mixed Strategy
In a situation where the saddle point of a pay off matrix does not exist we allow mixed strategies to get a solution. In mixed strategies, the probability with which a player chooses a particular strategy is considered. In the context of Znumber, we can say that in mixed strategy game we find an expected pay off with some reliability or certainty of the pay off obtained. Suppose and be the
m andn dimensional vector spaces, respectively. We denote x = (x _{1},x _{2}, ··· ,x_{m} )^{T} and y = (y _{1},y _{2}, ...,y_{n} )^{T} , respectively, where the symbol ‘T' denotes the transpose of a vector. The strategy spaces for playersA andB are denoted asrespectively. Vectors x ∈
S_{A} , y ∈S_{B} are called mixed strategies of playersA andB , respectively. Now, we should remember that Znumber is a higher(level 3) level of generality [14] and all the operational rules like multiplication, division are not known. In that case, it is better idea to find the mixed strategy solution using the interval approximation. Interval is a particular case of a Znumber and it is level−1 domain of computation. Now, the question may arise: Does an optimal mixed strategy solution with interval numbers actually correspond to an optimal mixed strategy solution with Znumbers? To get an answer to this question we must consider the following theorem where, a Znumber is modelled with a gaussian membership function and normal probability density function.Theorem 4.4. An optimal solution of the matrix game with payoff elements as interval approximation of some Znumber corresponds to the optimal solution of the matrix game with payoff elements the concerned Znumber .Proof . Let us consider a maximization problem where, the elements of the pay off matrix are interval approximation of Znumbers. Let us construct an interval approximation functionϕ :Z →I (ℜ). Now, the author’s [19] approach of interval approximation assures that the set of such functions is nonempty. We first establish thatϕ is a bijective mapping. LetZ _{1},Z _{2} ∈Z . We can then findc _{1},c _{2},σ _{1},σ _{2} ∈ ℜ such thatUsing interval arithmetic, we can easily verify that
ϕ (Z _{1}) =ϕ (Z _{2}) ⇒c _{1} =c _{2} andσ _{1} =σ _{2}. HenceZ _{1} =Z _{2}. Therefore,ϕ is injective.Similarly, using interval arithmetic, we can easily verify that for every interval
c _{1},c _{2},σ _{1},σ _{2} ∈ ℜ we can findZ _{1},Z _{2} ∈Z , the set of solutions withZ numbers, assuring thatϕ is surjective. Hence,ϕ is bijective. This property assures that every solution inI (ℜ) corresponds exactly to one solution inZ . LetI ^{∗} be the optimal solution of the maximization problem. Then, for every solutionI , we must have,Now, let us construct
I andI ^{∗} asUsing equation (4.2) we have either
Since,
ϕ is bijective we findZ ^{∗} such thatZ ≤Z ^{∗}∀Z ∈Z . Therefore,Z ^{∗} is the optimal solution of the maximization problem modelled by Znumbers. Similar approach can be made for minimization problem. Hence the theorem is proved.Note: We compute the inverse function , a Znumber ∀[a ,b ] ∈ ℜ which gives the optimal solution of the matrix game with Znumber corresponding to the optimal solution with interval number.Definition 4.2. (Interval expected payoff ): If the mixed strategies x = (x _{1},x _{2}, ...,x_{m} ) and y = (y _{1},y _{2}, ...,y_{n} ) are proposed by playersA andB respectively, then the expected payoff of the playerA by playerB is defined bywhere,
CL = (a11 + a22 − b12 − b21)x1y1 + (a12 − b22)x1 + (a21 − b22)y1 + a22 CU = (b11 + b22 − a12 − a21)x1y1 + (b12 − a22)x1 + (b21 − a22)y1 + b22
The composition rules on interval numbers [6] are used in this definition (3) of expected payoffs.
Definition 4.3. Suppose, and be two intervals defined over ℜ. Let us consider that there exist strategies x^{∗} ∈S _{A}, y^{∗} ∈S_{B} . If, for any strategy x ∈S _{A}, y ∈S_{B} , satisfies boththen, x ∈
S _{A}, y ∈S_{B} , is said to be a reasonable solution to the interval matrix game and are called reasonable values for playersA andB , respectively; x^{∗} and y^{∗} are called reasonable strategies for playersA andB , respectively.Let
U andW be the sets of reasonable values for playersA andB , respectively.Definition 4.4. Let us consider that there exist two reasonable values and If there do not exist reasonable values and such that they satisfy both and , then is said to be a solution of the interval matrix game ; x^{∗} is called an optimal (or a maximin) strategy for playerA and y^{∗} is called an optimal (or a minimax) strategy for player and are called PlayerA ’s gainfloor andB ’s losscelling, respectively.5. Computational Methods
In this section, we discuss the computation procedure to find out the solution to a matrix game with Znumber. We first transform (through approximation) the pay off matrix with Znumbers to a pay off matrix with corresponding intervalnumber. We then use multisection technique and min max algorithm [6] to solve the matrix game. The multisection algorithm is formulated according to the approach given in the work of Chakrabortty et al. [18]. The concept of multisection is inspired by the concept of multiple bisection, where more than one bisection is made at a single iteration cycle. The basis of this method is the comparison of intervals (as described in Section 3 of this paper) according to the decision makers point of view.
Algorithm for multisection technique Input:
λ (number of divisions),y ,l (lower bound)andu (upper bound) ofx .Output: Probability
x ^{∗}Step 1://calculation of step lengths//calculate step length
h = (u −l )/λ end for
Step 2://Division of concerned region into equal subregions //
Step 2.1: For
j = 0 toλ − 1 Calculatel _{0} =l +j ∗h Step 2.2: //Call the function
C_{L} andC_{U} //.Section 3,
Calculate
C_{L} = lower value of the interval number ,obtained by as in Eq.
(23)
Calculate
C_{U} = upper value of the interval number ,obtained by as in Eq.
(23)
Step 2.3: For
j 1 = 0 toλ − 1 Calculatel _{1} =l +j 1 ∗h Step 2.4: Calculate
l_{min} = lower value of the interval number ,obtained by as in Eq.
(23) at
l _{0}Calculate
u_{min} upper value of the interval number ,obtained by as in Eq.
(23) at
l _{0}Calculate
C_{l} = lower value of the interval number ,obtained by as in Eq.
(23) at
l _{1}Calculate
C_{u} = upper value of the interval number ,obtained by as in Eq.
(23) at
l _{1}Step 2.5: Applying required order relation (defined in Section 3)
between any two interval numbers [
C_{L} ,C_{L} ] and [l _{min},u _{min}] choose the optimal interval number.end
j 1 loopStep 2.6: Choose the subregion
E^{opt} amongE_{j} obtained in step 2.5 which has a better objective function value by comparing the interval valuesE_{j} to each other.Step 3: //calculation of widths//.
Step 3.1: Calculate widths
w_{j} =u_{j} −l_{j} ofE_{j} whereu_{j} andl_{j} are upper bounds and lower bounds ofE_{j} Step 3.2: While
w_{j} >ε break
Step 3.3: Set
E^{opt} ←E_{j} Return to step 2.1
end for
endwhile.
end
j loopOutput
END MULTISECTION
On the basis of this technique we have developed an algorithm for max min and min max solutions of a single objective interval game.
5.1 MinMax Principle
Algorithm for min max principle We conduct operations not on the degree of certainty
P_{AXi j} but on interval numbers [6] using the following steps:Step 1: Put
y _{1} =nh ,whereh = 1/M andn = 0, 1, 2, 3 ···M = Number of divisions of the interval [0, 1]Step 2: For
n =i Find max/optimistic order relation of ,
where 0 ≤
x _{1} ≤ 1 by using multisection algorithm 4.1.Step 3: Let the solution set for
x isUsing pessimistic order relation find minimum of
Suppose it occurs at
x _{1} ^{∗}, which is a crisp number.Step 4: Calculate
Step 5: Using pessimistic order relation calculate for 0 ≤
y _{1} ≤ 1by multisection technique. Suppose, the minimum value is (say).
Then (by Theorem 4.2) and
Therefore (
x _{1}^{∗},y _{1}^{∗}) is the optimal solution.5.2 MaxMin Principle
Algorithm for max min principle 4.2.1 Step 1: Put
x _{1} =nk , wherek = 1/N andn = 0, 1, 2, 3, ···N = number of divisions of the interval [0, 1]Step 2: For
n =i Find min/pessimistic order relation of
where 0 ≤
y _{1} ≤ 1 by using multisection technique 4.1.Step 3: Let the solution set for
y isUsing optimistic order relation find maximum of
Suppose it occurs at
y _{1} ^{∗} , which is a crisp number.Step 4: Calculate .
Step 5: Using optimistic order relation calculate
by multisection technique. Suppose the maximum value is (say).
Then (by Theorem 4.2) and
Therefore e (
x _{1}^{∗},y _{1}^{∗}) is the optimal solution.Thus is a reasonable solution of the interval matrix game , is player
A ’s gainfloor and is playerB ’s lossceiling.6. Demonstration ― An Example
Suppose, a company conducts an opinion pole about an election. They place some questions in front of the voters and get answers as ‘We are not very sure that the candidate A’s honesty is high’ or ‘It is very likely that inflation during the period of the present government is high’. In such cases, we can consider the statements as (A’s degree of honesty, high, not sure) or (price hike, high, very likely). These conditions are representation of Znumbers. When this happens between two candidates in an election, then it forms a matrix game with Znumber. Suppose, the payoff matrix is given by
What are the optimal strategies and what is the value of game?
This is an example of 2 × 2 matrix game with Znumber which has no saddle point because
Using the definition of interval expected pay off (4.2) When we run min max and max min programmes in TURBOC we get the reasonable solutions as
x ^{∗} = (0.0.9375, 0.0625),y ^{∗} = (0.5, 0.5),and
6.1 Results and Discussions
In this example, we obtain a reasonable solution
and
as respectively gainfloor and lossceiling of players
A andB with the probabilitiesx ^{∗} = (0.0.9375, 0.0625),y ^{∗} = (0.5, 0.5). There is a significance behind the result obtained and the technique adopted.(i) Here the pay off actually means some restriction on the measure of possibility that one can gain or loose with some degree of certainty and value of the game actually means measure of possibility of a solution to be an optimum solution. (ii) We have tried to arrive at a reasonable solution with some degree of certainty which is compatible with the real world situation as most of the optimization results obtained with other numbers or intervals lack some compatibility with the real world situation. For example, there are several kinds of imprecisions. What imprecision will then be modelled using a particular approach is a matter of concern as compared to the existing techniques [6]. In our approach, we have tried to consider a typical imprecision by using Znumber. (iii) We have tried to arrive at a higher degree of generality in the decision making process by using Znumber.
7. Conclusions
In a decisionmaking process, the information is often found to be imprecise, incomplete, e.g., ‘about 5%’, ‘high price” etc. In such a situation it is unlikely that usual approach would give a desired result. Again, formalization of the imprecision hardly occurs in our optimization models and there is no universal model which can consider all types of imprecisions. We often model certain types of imprecise data with certain type of membership function or interval numbers. It does not, however, ensure the optimization universally, i.e., there may be a chance to arrive at a better optimal solution if we model it otherwise. So, there is a need for formalization of imprecision and for that purpose we have used Znumber as a pay off which actually gives the degree of certainty. On the restriction of the measure of possibility of pay off one would gain or loose. Though we have modelled a Znumber with a particular type of membership and density function there is scope of further generalization. There is also scope for using this procedure to solve multiobjective decisionmaking problems.

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