1 Introduction
In the real world, many scientific and engineering issues can be seen as or converted to decision making problems. Generally, decision making is a complicated process that aims to select an alternative which gives decision makers the highest interests among a variety of options based on the performances or evaluations of alternatives on various criteria including benefit criteria and cost criteria tzeng2011multiple2011 ; Bouyssou2013 ; XDWJIJIS21929 . So far, many technologies and theories have been developed to help decision makers to achieve the optimal decision making, for example analytic hierarchy process (AHP) Saaty1980 ; Emrouznejad20175522 , analytic network process (ANP) saaty2001analyticRWS , technique for order preference by similarity to ideal solution (TOPSIS) hwang1981multipleSpringer ; Zavadskas1503 . Two important aspects are worthy of particular concern in almost every decision making process. One is the representation of evaluations to alternatives, the other is the method to find the optimal alternative. These two aspects are given special attention in this paper.
Regarding the representation of alternatives’ evaluations or performances, essentially it is the issue of information modeling, and there have already many existing means TverskyA54 ; HerreraF84337 ; HerreraF86 . One of the simplest and most straightforward ways is using real numbers to express the performances of alternatives, which is founded on the classical expected utility theory rabin2013risk241 ; levy2016expected and has the most solid theoretical basis. But that means is deficient in specially representing uncertain evaluations, where the uncertainty does not only include randomness but also involve vagueness, imprecision, ambiguity, and so on kruse2012uncertainty ; mendel2017sources245 . With the advance of uncertainty reasoning technologies, many theories have developed to tackle uncertain information and widely used in decision making field, such as theory of interval numbers moore2009introduction , possibility theory dubois2012possibility , soft set theory maji2003soft45 , rough set theory pawlak2012rough , fuzzy set theory zadeh1965fuzzy ; zimmermann2011fuzzy , etc. Among them, fuzzy set theory, aiming to cope with the uncertainty of fuzziness, has received much attention and is increasingly flourishing since its intuitive physical meaning and profound philosophical foundation chan2007global354 ; tang2017analysis387 ; liu2013fuzzy621 ; reza2013fuzzy59 ; zhang2017novelfuzzy . On the basis of fuzzy set theory, many branches, such as intuitionistic fuzzy set Atanassov2012Book and hesitant fuzzy set torra2010hesitant256 , constantly emerge, which provide diverse tools to express complicated evaluations in decision making. However, when human beings are involved in the decision making process, overcomplicated data structures will make it hard to get people’s authentic evaluations about the alternatives. In balancing the effectiveness and conciseness, fuzzy set theory is a feasible and preferable solution to model the uncertainty in decision making. Therefore, in this paper the decision making under fuzzy environment is of concern.
With respect to the methods of selecting the best alternative in decision making, there are much many theories for various type of decision making problems such as multicriteria decision making (MCDM), group decision making (GDM), and so on. Among these theories of decision making, many of them consider the situation that the interests of decision makers are basically consistent, but do not pay enough attention on the conflict of decision makers’ interests. In many situations the gain a decision maker obtains is not only dependent on his decisions, but also affected by the actions taken by other decision makers aplakfuzzy2013 ; xiong2014ambiguous70 ; deng2014evidential244 . What’s more important, these decision makers are conflict of interests. This type of decision making issue is generically referred to as adversarial decision making yager2008ICCCDadversarial . In the adversarial decision making, the alternatives of decision makers are changed to strategies that they can choose to achieve their benefits, and a decision maker aims to determine optimal strategy against adversarial opponents. Some researches have been done for the adversarial decision making. For example, Yager yager2008knowledge231 proposed a knowledgebased approach to adversarial decision making which mainly studies the use of a decision maker’s expertise, knowledge, and perceptions, about his adversary to construct a knowledge base about the action his believes his adversary will take. In pelta2009conflict179 , Pelta and Yager presented a mathematical framework to investigate the balance between inducing confusion and attaining payoff in adversarial decision making. Recent other work, to name but a few, can be found in references rege2014criminological114 ; villacorta2011ant ; villacorta2012theoretical186 ; razuri2013adversarial163 ; froeb2016adversarial593 .
In the most basic form of adversarial decision making, it involves two competitive participants and each of both sides chooses an strategy without knowing the choice of the other. In a perspective of game theory, it constitutes a twoperson zero sum or nonconstant sum game washburn2013two201 . Therefore, game theory is feasibly used in researching the adversarial decision making problem aplakfuzzy2013 ; xiong2014ambiguous70 ; yager1999game82 . Based on two aspects of considerations, the representation of human being’s uncertain evaluations and methods for decision making, in this paper we suggest a D numbers theory (DNT) based gametheoretic framework for two persons’ adversarial decision making under fuzzy environment, where DNT deng2012DJICS99 ; xydeng2017DNCR is a new uncertainty reasoning theory which generalizes DempsterShafer theory Dempster1967 ; Shafer1976 and already has some applications fan2016hybrid44 ; liu2014failure4110 ; xiao2016intelligent3713518 ; wang2016route6651 ; deng2014environmental412 . Within the presented framework, the uncertainty involved in the evaluations to decision makers’ strategies is expressed by fuzzy linguistic variables, and the conflict relationship between participants is modelled by a twoperson nonconstant sum game. Especially, DNT is used to handle and integrate the uncertain evaluations, which fully considers the nonexclusiveness among evaluations expressed fuzzy linguistic variables that is ignored by many previous studies. An illustrative application is given to show the effectiveness of the proposed framework.
The contribution of this paper is two folds. At first, we further improve the basis of DNT by presenting a uniform combination rule for D numbers with complete information or incomplete information, and defining the belief measure and plausibility measure for D numbers, and further clarifying the concept of nonexclusiveness in DNT, and providing a new form for the definition of D numbers. At second, by combining fuzzy set theory, game theory and DNT, we present a new gametheoretic framework for adversarial decision making under fuzzy environment, where the the nonexclusiveness among fuzzy evaluations are fully taken into consideration by exploiting DNT. The rest of this paper is organized as follows. Section 2 gives a brief introduction about fuzzy set theory, DempsterShafer theory, and twoperson nonconstant sum game. In Section 3, the DNT is presented as a generalization of DempsterShafer theory on a set with nonexclusive elements. Section 4 proposes a DNT based gametheoretic framework for adversarial decision making under fuzzy environment, and an illustrative application is given in Section 5 to show the effectiveness of the proposed framework. Finally, conclusions are given in Section 6.
2 Preliminaries
2.1 Fuzzy set theory
Fuzzy set theory was first introduced by Zadeh zadeh1965fuzzy in 1965 to deal with the uncertainty information. In some real application environments, the states are subjective concepts which are too complex or too illdefined to be reasonably described in conventional quantitative expressions. In those situation, fuzzy set theory provides an efficiently simple way to express the vagueness or imprecise information dabbaghian2014sustainability54 ; jiang2018intuitionistic ; song2017uncertainty464 ; zheng2018evaluation .
Let be the universe of discourse, a fuzzy set is characterized by a membership function satisfying
(1) 
where is called the membership degree of belonging to fuzzy set .
For a finite set , the fuzzy set is often denoted by . It is easily found that a fuzzy set is described entirely by its membership function. When takes value from , fuzzy set degenerates into a classical set.
A fuzzy number is a fuzzy subset of the real number , and its membership function is
(2) 
where is a real number and there definitely exists an element such that . Triangular fuzzy numbers are the most widely used fuzzy numbers. A triangular fuzzy number is usually denoted as , as graphically shown in Figure 1, which has the following membership function
(3) 
where .
In theory and practice, fuzzy numbers are usually associated with linguistic variables to express the fuzzy evaluation to objects. A linguistic variable is a variable whose values are represented by words or sentences in a natural or artificial language, for example “Very Low”, “Low”, “Medium”, “High”, “Very High”, where there values are usually expressed by fuzzy numbers.
In some applications, it may need to transform a fuzzy number to a real number. There are many approaches proposed for this task. A representative approach is the graded mean integration representation method developed Chou Chou2003The4510 . Based on that method, given a triangular fuzzy number , its graded mean integration representation is defined as
(4) 
2.2 DempsterShafer theory
DempsterShafer theory (DST) Dempster1967 ; Shafer1976 , also called belief function theory or evidence theory, is a popular tool for uncertainty reasoning because of its advantages in expressing uncertainty AnImprovedAPIN2017 ; jirouvsek2017new ; jiang2017uncertainty and handling decision making yager2015dempster80 ; han2016evaluation461 , clustering and classification denoeux2016evidential ; xu2017data116 , failure mode and effects analysis gong2018research112 ; jiang2017failure57 , and other issues Fontani84593 ; Elmore476 ; zheng2018dependence
with uncertainty. This theory needs weaker conditions than the Bayesian theory of probability, so it is often regarded as an extension of the Bayesian theory. As a theory of reasoning under the uncertain environment, DST has an advantage of directly expressing the “uncertainty” by assigning the basic probability to a set composed of multiple objects, rather than to each of the individual objects. In DST, the information from each information source is seen as a piece of evidence. When there are more than two pieces of evidence, a combination rule is provided to fuse them. For completeness of the explanation, a few basic concepts in DST are introduced as follows.
Let be a set of mutually exclusive and collectively exhaustive events, indicated by
(5) 
where set is called a frame of discernment (FOD). The power set of is indicated by , namely
(6) 
The elements of or subsets of are called propositions.
Let a FOD be , a mass function defined on is a mapping from to , formally defined by:
(7) 
which satisfies the following condition:
(8) 
In DST, a mass function is also called a basic probability assignment (BPA). The assigned basic probability measures the belief exactly assigned to and represents how strongly the evidence supports . If , is called a focal element, and the union of all focal elements is called the core of the mass function.
Given a BPA, its associated belief measure and plausibility measure express the lower bound and upper bound of the support degree to each proposition in that BPA, respectively. They are defined as
(9) 
(10) 
where . Obviously, for each , and is called the belief interval of .
If there are two pieces of evidence, a Dempster’s rule of combination is usually used to combine them. This rule assumes that the BPAs are mutually independent.
Let and be two BPAs defined on FOD , the Dempster’s rule to combine them, denoted by , is defined as follows:
(11) 
with
(12) 
where is a normalization constant, called conflict coefficient between and . Note that the Dempster’s rule is only applicable to such two BPAs which satisfy the condition .
The Dempster’s rule plays a very important role in DST, and satisfies commutative and associative properties, i.e., (i) and (ii) . Thus if there exist multiple BPAs, the combination of them can be carried out in a pairwise way with any order.
In order to make decision in terms of a BPA, an approach, called pignistic probability transformation (PPT), is proposed by Smets and Kennes smets1994transferable662 to derive a distribution of probabilities from the BPA. The PPT function is defined as follows. Let be a BPA on FOD , a PPT function associated to is defined by
(13) 
where and is the cardinality of proposition .
2.3 Twoperson nonconstant sum game
Game theory provides a mathematical framework to explain and address the interactive decision situations where the aims, goals and preferences of the participating agents are potentially in conflict and it is extensively applied from economics to biology, and to other disciplines Colman2013 ; Coupled151 ; liu2017evidentialgame ; WangSAe1601444 ; CSF103177 . A strategic game consists of three components which are a finite set of players, a nonempty set of strategies for each player, and payoff function for each player in every strategy combination, respectively. The most commonly used solution concept in game theory is that of Nash equilibrium which captures a steady state of the play of a strategic game in which each player holds the correct expectation about the other players’ behavior and acts rationally Osborne1994 .
Twoperson nonconstant sum game is a kind of widely addressed game, such as iterated prisoner’s dilemma game. In this game there are two players, player 1 and player 2. Assuming player 1 has a finite strategy set including strategies, and layer 2 has a finite strategy set including strategies. The payoffs of player 1 and player 2 are determined by functions and , respectively, where and . A combination of players’ strategies, denoted as , is a Nash equilibrium of this twoperson nonconstant sum game if
(14) 
(15) 
The Nash equilibrium of a twoperson nonconstant sum game is a selfenforcing combination of strategies in which no player can gain more by unilaterally deviating from it.
3 D numbers theory: A generalization of DST on a set with nonexclusive elements
Although DST has many merits in dealing with uncertain information involving imprecision and ignorance, but it is limited by some hypotheses and constraints that are often hardly satisfied. Related discussions have been detailed in some previous studies deng2012DJICS99 ; xydeng2017DNCR ; deng2014environmental412 . In summary, there two mainly two aspects. At first, in DST the FOD denoted as must be composed by mutually exclusive elements. Formally, for any they have to meet . It is called FOD’s exclusiveness hypothesis. By following the hypothesis, if DST is used in fuzzy linguistic environment, any pair of linguistic variables such as “Good” and “Very Good” must be exclusive strictly, which however is questionable. Secondly, in DST the sum of basic probabilities or beliefs in a BPA must be 1, which is called BPA’s completeness constraint. However, in some cases, due to lack of knowledge and information, it is possible to obtain an incomplete BPA whose sum of basic probabilities is less than 1. This incompleteness may be caused by an incomplete FOD which corresponds to the case of open world environment, or may be from the lack of information in a closed world environment.
D numbers theory (DNT) is proposed to overcome these limitations mentioned above in DST. This theory is a developing theory in which some key issues still remain unsolved. In deng2012DJICS99 , Deng first given the definition of D numbers and a combination rule for a special form of D numbers, but that rule is not universal. Besides, in contrast to DST, there are not belief measure and plausibility measure for D numbers so far, so that the lower bound and upper bound of the support in a D number to each proposition can not be derived. In order to let DNT really be a generalization of DST, in this paper a general rule for combining D numbers is proposed, and the belief measure and plausibility measure for D numbers are also developed.
3.1 Definition of D numbers
At first, the definition of D numbers is given as follows. Let be a nonempty finite set , a D number is a mapping formulated by
(16) 
with
(17) 
where is the empty set and is a subset of .
From the above definition, a D number is defined on a set with nonexclusive elements, which means that any pair of elements in , for example , are not required to be strictly exclusive, i.e. . Here, we still call as a FOD, but should note that a FOD in DNT is a set consisting of nonexclusive elements. Besides, according to Definition 3.1, in a D number the information is not required to be complete. If , we say that the D number is informationcomplete. By contrast, if the D number is informationincomplete. The degree of information’s completeness in a D number can be simply expressed by its value . In previous studies, we find that a D number with incomplete information is hard to handle mathematically. Facing that, from the view of math a new nonempty set can be imported to transform a D number with incomplete information to the informationcomplete case by letting . As a result, a new definition about D numbers is obtained below. A D number defined on a nonempty finite set is a mapping satisfying
(18) 
and
(19) 
where is the empty set, is a subset of , and is a nonempty set.
If and becomes a set of mutually exclusive elements, the D number will be completely reduced to a BPA in DST. Therefore, D number is a generalization of BPA. In addition, since in DNT a D number allows , it is very similar with the open world assumption in the transferable belief model (TBM) of DST Smets1994 . But DNT is essentially different from TBM. At first, in contrast to TBM which implements the open world assumption by letting , DNT holds and lets instead. Secondly, in TBM represents all elements that are not included in the FOD, thus the exclusiveness is still hold between FOD and . However, in DNT can have an intersection with FOD , even becoming the subset of . Only if and , the open world assumption is hold. Hence, a D number with incomplete information (i.e. ) does not absolutely correspond to the open world environment.
3.2 FOD’s nonexclusiveness in DNT
As mentioned above, in DNT the elements of FOD are not required to be mutually exclusive, which means that may be not completely exclusive to for any and . As a natural generalization, the concept of nonexclusiveness can be extended to the subsets of from the elements of : for two nonempty sets and , may be also not completely exclusive to . Further, the nonexclusiveness can be applied to where expresses the incomplete information in DNT as shown in Definition 3.1. In order to quantitatively represent the nonexclusiveness in , a membership function is developed to measure the nonexclusive degrees. Given , the nonexclusive degree between and is characterized by a mapping :
(20) 
with
(21) 
and
(22) 
where . If letting the exclusive degree between and be denoted as , then .
According to Definition 3.2, the nonexclusive degree between and is 1 if and have intersections, otherwise is taking a value from . Obviously, if for any , the FOD in DNT is degenerated to classical FOD in DST. As expressed above, the nonexclusiveness of FOD is one of the most important properties in DNT. In our previous studies xydeng2017DNCR ; Deng2017Fuzzy2086 , a simple approach is developped to calculate all nonexclusive degrees in power set space if we have the nonexclusive degrees of any pair of elements in , which is presented as follows
(23) 
where . A numerical example is given below to illustrate the approach shown in Eq. (23).
Supposing there is a set of linguistic variables in which every linguistic variable is represented by a triangular fuzzy number given in Table 1 and graphically presented as Figure 2. The set is seen as a FOD, and let represent the possible incomplete information in DNT. For simplicity, it is assumed that for any . Now the nonexclusive degrees between elements in .
Linguistic variable  Fuzzy number 

Very Poor (VP)  (0.00, 0.00, 0.25) 
Poor (P)  (0.10, 0.25, 0.39) 
Medium Poor (MP)  (0.25, 0.39, 0.53) 
Medium (M)  (0.39, 0.53, 0.68) 
Medium Good (MG)  (0.53, 0.68, 0.86) 
Good (G)  (0.68, 0.86, 0.97) 
Very Good (VG)  (0.86, 1.00, 1.00) 
At first, we calculate the nonexclusive degrees between elements in FOD . For that purpose, in this paper the areas of fuzzy numbers are utilized. Let , be two fuzzy numbers with membership functions and , respectively. And the intersection of and is denoted as with membership function . Correspondingly, indicates the union of and whose membership function is . Then, we define the nonexclusive degree between and as
(24) 
where and represent the areas of fuzzy numbers and , respectively. Based on Eq. (24), and Figure 2, and the assumed , each nonexclusive degree between elements in , therefore, can be obtained as shown in the following matrix
At second, once having the above nonexclusive degree matrix of between elements in , according to Eq. (23) we can calculate the nonexclusive degree of any pair of elements in . For example, as for and , we have
The other nonexclusive degrees between elements in can also be derived by the means.
3.3 Belief measure and plausibility measure for D numbers
In previous studies, the belief and plausibility for D numbers are not addressed. By considering the nonexclusiveness of FOD in DNT, in this paper we propose a belief measure and a plausibility measure for D numbers as follows. Let represent a D number defined on and express the incomplete information in , for any proposition , its belief measure is defined as
(25) 
and its plausibility measure is defined as
(26) 
where .
For the above definition, because for , the plausibility measure can be rewritten as
(27) 
As same as DST, is called the belief interval of in DNT, which expresses the lower bound and upper bound of support degree to proposition . And it is easy to find that the and for D numbers will degenerate to classical belief measure and plausibility measure in DST if the associated D number is a BPA in fact.
3.4 Combination rule for D numbers
How to combine pieces of information is a crucial issue in the theory of information fusion. Ideally, a combination rule for D numbers should be degenerated to the Dempster’s rule of combination under a certain conditions, since DNT is designed as a generalization of DST. The rule given in deng2012DJICS99 is obviously not satisfactory, and its application is limited since that rule is not universal. In a very recent study xydeng2017DNCR , we given new combination rules for D numbers with complete information and incomplete information, respectively, from a perspective of conflict redistribution Dempster1967 ; yager1987dempster412 ; dubois1988representation43 . By reexamining the rules given in xydeng2017DNCR , we find that they do not well address the case of informationincompleteness and import a D numbers’ values related function which is difficulty determined in practice. In this paper, by inheriting the idea in xydeng2017DNCR , we present a uniform combination rule for D numbers with complete information and incomplete information. The proposed uniform rule is universal for any forms of D numbers and can be totally reduced to Dempster’s rule of DST.
Let , be two D numbers defined on , and , express the incomplete information in and respectively. The combination of and , indicated by , is defined by
(28) 
with
(29) 
where .
From Definition 3.4, the presented rule essentially is a redistribution of exclusive conflict, therefore it is called the exclusive conflict’s redistribution (ECR) rule. The proposed ECR rule simultaneously considers the nonexclusiveness of FOD and possible informationincompleteness in D numbers. In this rule, the conflict and incomplete information are represented by and , respectively. And it can be completely degenerated to classical Dempster’s rule of DST if for any and , . A numerical example is given below to show the combination process of D numbers by means of the ECR rule.
Assume there are two D numbers defined on :
, , , ;
, , , .
And the nonexclusive degrees between subsets of are assumed in the following matrix:
The combination result of and can be obtained through the following steps. At first, an intersection/union table is calculated as shown Table 2, where each item is derived by either assigned to if or assigned to if .
(0.2)  (0.015)  (0.1)  (0.01)  
(0.008)  (0.06)  (0.04)  (0.002)  
(0.04)  (0.03)  (0.02)  (0.002)  
(0.016)  (0.006)  (0.008)  (0.02) 
Then according to Eq. (29), we can have the conflict coefficient . Therefore, in terms of Eq. (28) the final result of combining and is obtained
It must be pointed out that the proposed ECR rule in Definition 3.4 satisfies the commutative property, i.e. , but does not preserve the associative property, namely . In the theory of information fusion, there are two main schemes when combining pieces of information florea2009robust102 . One is the aggregating scheme where evidences represent different opinions about the same event, the other is the updating scheme in which evidences express sequential opinions about a dynamic event. For the commutative and associative rules, such as the Dempster’s rule, there is no difference between these two schemes. But for the nonassociative rules, the two schemes provide different results. In DNT, the ECR rule is naturally suitable for the updating scheme, but is not appropriate to be used in the aggregating scheme. Facing that, a weighted average combination (WAC) method is suggested to combine multiple D numbers about the same event based on the proposed ECR rule.
Suppose there are D numbers indicated by , , , , and every D number is given a weighting factor , , satisfying . At first, the averaging D number among , , , is defined as
(30) 
Then, the result of combining , , , is obtained by
(31) 
where , , and is the ECR rule given in Definition 3.4.
4 Proposed D numbers theory based gametheoretic framework
In this section, a DNT based gametheoretic framework is proposed for adversarial decision making under uncertain environment, whose flow diagram is graphically shown in Figure 3. Underlying this framework, a twoperson nonconstant sum game is considered. The framework mainly consists of four phases including “Game analysis”, “Strategy assessment”, “Payoff matrix construction”, and “Equilibrium calculation”, respectively. In the following text, we briefly describe these phases, leaving the details in the next section to explain via an illustrative application.

By means of a formed decision making group (DMG), in the phase of game analysis the players in the game are easily determined at first. Then, the DMG will define a strategy set for each player by analyzing the current decision making situation.

In the phase of strategy assessment, the main task is to evaluate the strategies of each player. At first, the decision making criteria are identified as the basis of assessment. Secondly, by considering an uncertain decision making environment, we define fuzzy linguistic variables to express the uncertainty in the evaluations. Based on the criteria and linguistic variables determined previously, each strategy for each player is evaluated by the DMG. As a result, a multiexperts and multicriteria decision making matrix with fuzzy evaluations is formed for every strategy of each player.

The third phase is to construct the payoff matrix of the game. For the decision making matrix of each strategy, firstly we collect the evaluations from multiexperts on every criterion and express them in a D number, which is to construct a D numbersvalued MCDM matrix for every strategy. Then, we combine these D numbers on different criteria for every strategy to generate a D numbersvalued payoff matrix, where the fusion of evaluations on multicriteria is implemented. At last, in order to calculate the equilibria of the game, the generated D numbersvalued payoff matrix is converted to a realvalued payoff matrix via the defuzzification of D numbers.

In the phase of equilibrium calculation, all equilibria of the game will be found at first according to the realvalued payoff matrix obtained above. Evidently, there are two cases about the found equilibria. One is that there is only an equilibrium point, in this case the unique equilibrium naturally become the outcome of the game. The other is that there are at least two equilibrium points, in the case we must choose an optimal equilibrium point as the outcome of the game. In game theory, different approach may yield different optimal equilibrium point, where each optimal equilibrium is associated with different point of view on “optimal”. The determination of the optimal equilibrium is on the basis of practical applications and specific demands, therefore it is not the focus of the study. By some means an optimal equilibrium is assumed to be determined, the DMG will reexamine the equilibrium prudently. If the optimal equilibrium is not satisfactory, a new equilibrium point will be required; Otherwise, if it is approved, the optimal equilibrium will be used to generate the action plan for the adversarial decision making.
5 An illustrative application
In this section, an example adapted from reference aplakfuzzy2013 is given to illustrate the process of applying the proposed DNT based gametheoretic framework and verify its effectiveness. For more details about the application, please refer to aplakfuzzy2013 .
In this application, it is assumed that there is a territorial dispute between two countries Alpha and Beta about the belonging of a buffer zone (BZ), as shown in Figure 4. Recently, Beta unilaterally claims that the BZ belongs to it. For this reason, Alpha call for an international aid against Beta’s demand. United Nations send and located international peace support force to control the area. Regarding to this situation, the conflict between countries is formalized in twoperson nonconstant game theory and analyzed by the proposed DNT based gametheoretic framework. Before the analysis, it is assumed a DMG consisting of ten decision makers (DM) is established in advance.
5.1 Game analysis
In this phase, the DMG is to determine the players in the game and define possible strategies for each player. Regarding the players, obviously there are two players, namely Alpha and Beta. Regarding the strategies, by following reference aplakfuzzy2013 it is assumed that there are five strategies for Alpha and four strategies for Beta.
The strategies of Alpha are

: Main and rear area control

: Sector control

: Area control as whole

: Strong area control as whole

: Area control with local forces
The strategies of Beta are

: Attack by itself

: Corporate attack

: Passive attitude

: Ownership of conflict zone (BZ)
5.2 Strategy assessment
In the phase of strategy assessment, at first we identify key criteria so as to evaluate the strategies for players. In previous study aplakfuzzy2013 , six key criteria have been identified, including “Management”, “Protection”, “Mobility”, “Logistic”, “Flexibility”, “Simplicity”, and a fuzzy weigh for every criterion has been also determined, as shown in Table 3.
Criterion  Fuzzy weight 

Management ()  (0.53, 0.91, 1.00) 
Protection ()  (0.39, 0.66, 1.00) 
Mobility ()  (0.10, 0.41, 0.68) 
Logistic ()  (0.25, 0.60, 1.00) 
Flexibility ()  (0.39, 0.82, 1.00) 
Simplicity ()  (0.10, 0.49, 1.00) 
Secondly, a group of fuzzy linguistic terms should be defined for the sake of expressing DM’s uncertain evaluations to the strategies. In the paper, the linguistic variables given in Table 1 are employed directly. These linguistic variables, ranked from “Very Poor” (VP) to “Very Good” (VG), constitute a scale of seven grades, as graphically shown in Figure 2.
Thirdly, according to the criteria identified above, the DMG is to evaluate these strategies for each player by using the defined fuzzy linguistic variables. Different from conventional MCDM process, each evaluation matrix is established by given opposite player’s strategies from a game theory perspective since the application is in an adversarial decision making environment. For example, Table 4 gives the evaluations to Alpha’s all strategies in the case of Beta choosing as its strategy, which is abbreviated as Case for short. And since there are ten decision makers in the DMG, Table 4 is also an evaluation matrix of a group decision making. By means of this way, we can obtain the other evaluation matrices, namely Case, Case, Case for Alpha, and Case, Case, Case, Case, Case for Beta. In this paper, all of the evaluation matrices are from literature aplakfuzzy2013 .
Criterion  Strategy  DM 1  DM 2  DM 3  DM 4  DM 5  DM 6  DM 7  DM 8  DM 9  DM 10 

MG  MG  M  MG  G  G  G  MG  G  G  
G  G  MG  M  VG  G  MG  MG  VG  M  
P  VP  VP  VP  VP  VP  VP  P  VP  VP  
VP  P  P  VP  MP  P  P  P  P  MP  
P  VP  VP  P  VP  VP  VP  P  P  VP  
M  MP  M  M  MP  P  G  MG  MG  MG  
G  M  M  G  M  M  MG  MP  M  MP  
MP  VP  VP  P  VP  P  VP  P  MP  P  
MP  MP  VP  P  MP  M  MP  P  M  VP  
VP  MP  MP  VP  MP  VP  VP  VP  VP  MP  
VG  VG  MG  VG  VG  G  VG  MG  MG  VG  
VG  MG  G  VG  MG  VG  VG  VG  MG  MG  
MP  M  P  P  M  P  P  P  P  P  
MG  MG  M  MG  MG  MG  M  M  MP  G  
MP  VP  P  P  VP  M  VP  VP  P  M  
VG  VG  VG  MG  VG  VG  VG  VG  G  VG  
G  MG  G  MG  M  M  MG  G  G  P  
G  MP  MP  P  M  P  MP  MP  M  MP  
M  M  M  MP  P  MP  M  P  M  P  
MP  VP  MP  P  M  P  P  MP  P  MP  
MP  VG  MP  M  VP  G  M  MG  G  MP  
G  G  G  MG  M  MG  M  M  MG  MG  
MP  MP  VP  P  VP  M  VP  P  P  MP  
MP  M  P  P  P  P  P  MP  VP  P  
VP  MP  P  VP  MP  MG  VP  P  VP  M  
MG  M  G  MG  G  MG  G  G  MG  G  
MP  MG  M  MP  M  MG  MG  MG  MG  MG  
M  M  MG  M  MG  M  M  MG  M  MG  
MP  MP  M  MG  M  MP  MP  M  M  MP  
P  P  MP  MG  MP  MP  P  MG  M  M 
5.3 Payoff matrix construction
Through the above two phases, nine gametheoretic evaluation matrices have been established for each player’s strategies. But it is difficult to find the solution (or equilibrium point) of the adversarial decision making problem directly based on these evaluation matrices. We must construct a payoff matrix for this game in terms of the evaluation matrices. During the process, it will implement the fusion of evaluations from multiple decision makers and multiple criteria, where DNT is used to deal with these nonexclusive fuzzy evaluations.
Firstly, for every strategy we transform the fuzzy evaluations on the same criterion given by ten decision makers into a D number to realize the fusion of multiple experts. For example, in Table 4, for Alpha’s strategy , on four decision makers give the evaluation of MG, one gives M, and the other five give G. By assuming these decision makers have the same importance, hence a D number is generated as follows
Here, the above D number is simply denoted as ({MG}, 0.4; {M}, 0.1; {G}, 0.5). Via this means, all fuzzy evaluations can be integrated and reexpressed by D numbers. In this application, all the obtained D numbers are informationcomplete. As a result, every fuzzy evaluation matrix, like Case in Table 4, is transformed to a D numbersvalued MCDM matrix. Table 5 shows the D numbersvalued MCDM matrix derived from Table 4.
Criterion  Strategy  Evaluation 

({M}, 0.1; {MG}, 0.4; {G}, 0.5)  
({M}, 0.2; {MG}, 0.3; {G}, 0.3; {VG}, 0.2)  
({VP}, 0.8; {P}, 0.2)  
({VP}, 0.2; {P}, 0.6; {MP}, 0.2)  
({VP}, 0.6; {P}, 0.4)  
({P}, 0.1; {MP}, 0.2; {M}, 0.3; {MG}, 0.3; {G}, 0.1)  
({MP}, 0.2; {M}, 0.5; {MG}, 0.1; {G}, 0.2)  
({VP}, 0.4; {P}, 0.4; {MP}, 0.2)  
({VP}, 0.2; {P}, 0.2; {MP}, 0.4; {M}, 0.2)  
({VP}, 0.6; {MP}, 0.4)  
({MG}, 0.3; {G}, 0.1; {VG}, 0.6)  
({MG}, 0.4; {G}, 0.1; {VG}, 0.5)  
({P}, 0.7; {MP}, 0.1; {M}, 0.2)  
({MP}, 0.1; {M}, 0.3; {MG}, 0.5; {G}, 0.1)  
({VP}, 0.4; {P}, 0.3; {MP}, 0.1; {M}, 0.2)  
({MG}, 0.1; {G}, 0.1; {VG}, 0.8)  
({P}, 0.1; {M}, 0.2; {MG}, 0.3; {G}, 0.4)  
({P}, 0.2; {MP}, 0.5; {M}, 0.2; {G}, 0.1)  
({P}, 0.3; {MP}, 0.2; {M}, 0.5)  
({VP}, 0.1; {P}, 0.4; {MP}, 0.4; {M}, 0.1)  
({VP}, 0.1; {MP}, 0.3; {M}, 0.2; {MG}, 0.1; {G}, 0.2; {VG}, 0.1)  
({M}, 0.3; {MG}, 0.4; {G}, 0.3)  
({VP}, 0.3; {P}, 0.3; {MP}, 0.3; {M}, 0.1)  
({VP}, 0.1; {P}, 0.6; {MP}, 0.2; {M}, 0.1)  
({VP}, 0.4; {P}, 0.2; {MP}, 0.2; {M}, 0.1; {MG}, 0.1)  
({M}, 0.1; {MG}, 0.4; {G}, 0.5)  
({MP}, 0.2; {M}, 0.2; {MG}, 0.6)  
({M}, 0.6; {MG}, 0.4)  
({MP}, 0.5; {M}, 0.4; {MG}, 0.1)  
({P}, 0.3; {MP}, 0.3; {M}, 0.2; {MG}, 0.2) 
Secondly, the evaluations to every strategy on different criteria, now expressed by D numbers, are fused to derive an integrated evaluation to each strategy. This is a typical fusion process of evaluations on multiple criteria, but the data is expressed by D numbers. Within the process, the ECR based WAC method, shown in Eqs. (30) and (31), is used to combine multiple D numbers, where the weight factors of D numbers are derived from the fuzzy weights of criteria given in Table 3. At the firs step, the fuzzy weights given in Table 3 are transformed to crisp values by using Eq. (4), we have , , , , , . At the second step, these crisp values are normalized as the weight factors of the criteria, namely
Now we can fuse the evaluations on all criteria for every strategy. Let’s use strategy as the example. From Table 5, given Beta’s strategy the evaluations to are

On : ;

On : ;

On : ;

On : ;

On : ;

On : .
Based on the weight factors calculated above, the averaging D number of , , , , , , denoted as , therefore, is
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