Personal Model of Conflict Resolution

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Conflict analysis based on the graph model for conflict

resolution and grey matrix Xueshan Han, Thi Dieu Linh Nguyen and Haiyan Xu

College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, China

Abstract

Purpose – The purpose of this paper is to propose a complete theory of grey conflict analysis model based on grey game and the graph model for conflict resolution and also, to illustrate a case of “prisoner’s dilemma” in the traditional grey game as an example.

Design/methodology/approach – Based on the theories of grey game and graph model for conflict resolution, this paper concentrates on the model of grey conflict analysis in a case of two players under the condition of symmetrical loss information. By analyzing decision makers, strategies, states, graph model and grey potential, and the number of decision makers’ steps, the pure strategy Nash equilibrium is extended to grey potential-general metarationality, grey potential-symmetrical metarationality, and grey potential-sequential stability. Meanwhile, the logical relationships between solutions are discussed. A specific case study is carried out to illustrate how the proposed grey conflict analysis model is used in practice.

Findings – The results in this paper indicate that more stable solutions are found when one considers the grey potential-general metarationality, the grey potential-symmetrical metarationality, and the grey potential-sequential stability, and then solve the paradox of “prisoner’s dilemma”.

Practical implications – This new grey conflict analysis model could be used to provide useful information for policy makers during existing conflicts or negotiations among parties or enterprises.

Originality/value – The paper succeeds in constructing a new grey conflict analysis model, in which the solution concepts are studied; and the two-player grey game will be extended to n-players in the near future.

Keywords Grey systems, Conflict resolution, Game theory, Conflict analysis, Grey game, Grey potential, Pure strategy, Equilibrium

Paper type Research paper

1. Introduction The classical game theory ( John and Osakar, 1944) was built on the precise mathematical fundamental. However, sometimes it cannot be used in practice due to the requirement for complete information. For example, in the zero-sum games of two-players, the profit and loss matrix that two game players depend on is from the judgment information. Due to the cognitive level, incomplete information, grey information (Deng, 1982) and the uncertainty of the future, game players may not have significant understanding for their strategic characteristics, space and payoff function, even if the information involved parties are symmetrical. Many researches are focusing their study on the game containing incomplete information. For example, Professor Fang and Liu (2003a, b, c) proposed the grey matrix game model based on pure strategy. Sequentially, Fang and Liu (2006) developed the matrix game of grey situation-pure-tactical Nash equilibrium in symmetrical loss information of game-equilibrium in n-people. They applied the ideology and theory of grey system to establish the standard grey matrix game and thus result in non-classical grey

The current issue and full text archive of this journal is available at

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Conflict analysis

Grey Systems: Theory and Application

Vol. 3 No. 1, 2013 pp. 95-106

q Emerald Group Publishing Limited 2043-9377

DOI 10.1108/20439371311293723

number areas. Based on the zero-sum finite games of two-players and the most conservative game decision problem, Fang et al. proposed the necessary and sufficient conditions, as well as mentioned the concept of grey saddle point. Tao et al. (2004) introduced the definition of grey saddle point of grey mixed strategy based on mixed strategy of the grey matrix games. Also working on grey matrix and grey matrix game theory, Luo and Wu (2005) proposed the upper and lower equilibrium solution, ideal equilibrium solution, u positioned equilibrium solution and average positioned equilibrium solution. Luo and Wu (2006) focused on the concept of equilibrium solution and then established a method to rank alternatives in equilibrium solutions. By building the size relations of the potential of interval grey number, Mi and Fang (2005) proposed the position dominant strategy of grey potential and defined the pure strategy solution. On this point, Fang and Liu (2006) further introduced a condition of the optimal pure strategy and solution, as well as discussed the risks that game players face. Fang and Liu (2006) also researched the symmetrical loss information of game-equilibrium in a group of N people to define the grey situation-pure-tactical Nash equilibrium to make it easier to find the equilibrium of the pure grey situation up-and-down approach.

It is necessary to mention the size relations of the grey number while studying grey games equilibrium. The main methods to compare the size of the interval grey number include grey position comparison (Fang and Liu, 2003a, b, c; Mi and Fang, 2005), simple grey number comparison (Fang and Liu, 2003a, b, c) and the probability distribution of the grey number (Xie and Liu, 2009).

Grey games is mostly based on the traditional game theory but also considers the profit and loss value of the symmetrical defect by using a strategy analysis method. In cases using conflict analysis of situation analysis of metagame theory (Howard, 1976), the results will be much closer to the result of actual conflict. By simplifying the algorithms and adding more constraints, Fraser and Hipel (1979) proposed the F-H conflict analysis method. This method is more complete and compatible with actual conditions. It is expected to illustrate the essence structure of the objective world and establish a conflict model by extracting large complex and irregular information based on players’ strengths, attitudes and goals which are arranged in the order of preferences. The conflict equilibrium solutions are found by analyzing players’ stability.

Kilgour et al. (1987, 1993) published the graph model for conflict resolution (GMCR) based on the idea of the F-H method. In GMCR, solution concepts, which characterize DM’s possible patterns in conflicts, play an important role to calculate the equilibrium solution. To depict a diversity of decision types, a variety of solution concepts has been put forward such as Nash stability (Nash, 1950, 1951), general metarationality (GMR) (Howard, 1971), symmetric metarationality (SMR) (Howard, 1971), sequential stability (SEQ) (Fraser and Hipel, 1979), limited-move stability (Fang et al., 1993) and non-myopic stability (Kilgour, 1984). The GMCR has been applied widely to environment conflict (Obeidi et al., 2002), business negotiation (Hipel et al., 2001; Xin et al., 2005), engineering negotiation (Moustafa et al., 2006) and so on. However, this method requires decision makers (DMs) to provide preference information which is hard to satisfy in real conflicts.

For that reason, this paper discusses the conflict analysis of game theory and metagame theory in grey system and introduces the grey conflict analysis model based on the game theory framework by transforming from the static game into the expansion of the dynamic game. The computation of grey potential, weakening the preferred conflict analysis and DMs’ state transition, will expand the Nash equilibrium of the original grey

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potential pure strategy (adding one more consideration for equilibrium solution). This renders the game model closer to the reality and allows the intuitive sense of humans to have greater applicability.

2. Basic concepts of grey game 2.1 Basic definitions of grey game with pure strategy Definition 2.1. (Grey matrix) For each element of any certain matrix, if there exists grey number aij(^) (i ¼ 1, 2, . . . , m; j ¼ 1, 2, . . . , n), then this matrix is called grey number matrix and used A0ð^Þ ¼ ðaijð^ÞÞm£n ði ¼ 1; 2; . . . ; m; j ¼ 1; 2 . . . ; nÞ to represent.

Definition 2.2. (Grey matrix game theory) In the two-person zero-sum finite games, players set the profit and loss matrix as grey matrix, then we call this matrix grey matrix game and denote as G0(^) ¼ {S01, S

0 2; A

0(^)}, where S01 ¼ {a 0 1, a

0 2, . . . , a

0 m} is the

set strategies of player 1 and S02 ¼ {b 0 1, b

0 2, . . . , b

0 m} is that of player 2. A

0(^) is players’ grey profit and loss matrix.

Definition 2.3. (Symmetrical incompleteness of profit and loss value information) In a game, we use the interval grey number to represent the profit and loss value information, which is mutual knowledge, thus in this situation the profit and loss value of the game is named defect and listed as U(^) ¼ {u1(^), u2(^), . . . , un(^)}.

Remark. In order to make it easier to understand, in this paper, all grey numbers are limited as interval grey numbers as interval grey numbers in various types of grey numbers are the most representative.

Definition 2.4. (The static game of the profit and loss value information symmetrical defect) In a game, the profit and loss value information U(^) ¼ {u1(^), u2(^), . . . , un(^)} and players’ corresponding set of strategies is called S(^) ¼ {s1(^), s2(^), . . . , sn(^)} then this static game of the profit and loss value symmetrical defects of the game is set as G(^) ¼ {S(^), U(^)}, simplified as G(^).

For example, in a relatively closed color TV market, there are two oligopoly manufacturers. In order to seize more market share, they decide whether to decrease- or not-decrease-price strategy. In the real world, the profit and loss matrix is not clear and accurate but in terms of grey matrix, therefore we can use the interval grey numbers [aij, bij] [cij, dij] to represent this. Where [aij, bij] indicates that for manufacturer 1, because of the income value information of related defect, the corresponding revenue value is interval grey number; aij represents the minimum income that manufacturer 1 can achieve in this case, bij represents the maximum income that can be achieved using this strategy. Similarly, cij, dij are those of manufacturer 2 (Table I).

To be able to conduct the equilibrium analysis on the profit and loss value information symmetrical defect game mentioned above, there must be a method that can determine the size of the interval grey number. In some cases where the interval grey number is not easy to distribute clearly, the size comparison of the interval grey number is a trend. In this paper, we use the grey potential to compare the size of the interval grey number.

Manufacturer 2 Decrease Not-decrease

Manufacturer 1 Decrease [a11, b11], [c11, d11] [a12, b12], [c12, d12] Not-decrease [a21, b21], [c21, d21] [a22, b22], [c22, d22]

Table I. The grey game between

two manufacturers

Conflict analysis

2.2 Compare the size of the interval grey number Given any two interval grey number ^ij [ [aij, bij] and ^st [ [ast, bst], where bst $ bij $ ast (Figure 1). Based on the value of the endpoints position of the two grey numbers, we can split into three intervals, [aij, ast], [ast, bij], [bij, bst], where [ast, bij] is the intersection area of two grey numbers.

(1) Equipollence potential degree. Consider the intersection area of two grey numbers in the equilibrium region, set EPDij!st ¼ ðbij 2 astÞ=ðbij 2 aijÞ as grey number ^ij is the equipollence potential degree of ^st. Similarly, set EPDst!ij ¼ ðbij 2 astÞ=ðbst 2 astÞ as grey number ^st is the equipollence potential degree of ^ij.

(2) Superiority potential degree. Based on the value of the endpoints position of two grey numbers, we set [bij, bst], on the right side of the intersection area, as the superiority region and SPDst!ij ¼ ðbst 2 aijÞ=ðbst 2 astÞ as grey number ^st is the superiority potential degree of ^ij.

(3) Inferior potential degree. Similarly, we set [aij, ast], on the left side of the intersection area, as the inferior region and IPDij!st ¼ 2ðast 2 aijÞ=ðbij 2 aijÞ as grey number ^ij is the inferior potential degree of ^st.

The sum of superiority potential degree and inferior potential degree of any given interval grey numbers ^ij [ [aij, bij] and ^st [ [ast, bst] is called potential difference, simply called potential:

. If SPDij!st þ IPDij!st . 0, then we say grey number ^ij is advantageous to grey number ^st and denoted ^ij s ^st.

. If SPDij!st þ IPDij!st , 0, then we say grey number ^ij is disadvantageous to grey number ^st and denoted ^ij a ^st.

. If SPDij!st þ IPDij!st ¼ 0, then we say grey number ^ij is equivalent to grey number ^st and denoted ^ij , ^st.

The set of size relation of grey number potential is a whole set which was detailed by Fang and Liu (2006). Although the size of grey number potential is not always the real size of grey number, it is helpful for people’s decision making under the conditions of defect information, as well as providing new research tools and ideas to search for the potential equilibrium policy of grey game.

As mentioned above, the size of grey number is expressed in terms of a pair of binary relations ( s , ,). It is assumed to possess the following properties:

. s is asymmetric, i.e. for any grey number ^ij, ^st, ^ij s ^st and ^ij a ^st cannot hold true at the same time.

. , is reflexive, i.e. ^ij , ^ij and symmetric, i.e. if ^ij , ^st then ^st , ^ij.

. ( s , ,) is complete, i.e. for any grey number ^ij, ^st, then one of ^ij s ^st, ^ij a ^st or ^ij , ^st is true.

Figure 1. The relationship of grey interval numbers

Number axis

[aij, bij] [ast, bst]

aij bijast bst

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3. The grey conflict analysis model We use V ¼ {N, S, P, (S, A)} to denote the grey conflict analysis model. N represents for a non-empty set of players. In this paper, there are two-players, so N ¼ {i, j}, each player chooses one strategy, so the group of players’ strategies is a situation (a state) (in order to make it unanimous, in this article, every strategy group is called a state). S, P stand for the non-empty set of all feasible states and the preference of each DM for each state, respectively, (according to the merits degree gained from the size comparison of grey potential). (S, A) indicates the set of all DMs of the state transition.

3.1 Graph model and conflict analysis There are a few ways to define the equilibrium solutions of grey conflict analysis at first. In the conflict analysis, using graph theory to represent the conflict, a directed graph is defined as two-dimension (S, A), where S ¼ {s1, s2, . . . , sn} is vertices set (each point represents for a situation); A ¼ {a1, a2, . . . , an} has an arc that can change the situation causing by the change off DMs. The tail of arc means the original situation; the head of arc denotes the next situation; each DM responds to a graph and reachable matrix (Figure 2). For that, we can define the reachable matrix Ri(n£n) of DM:

Riðs; qÞ ¼ 1; if DM can move from state s to state q in one step

0; others

(

Figure 2. The analysis procedure

of the grey conflict analysis model

Conflict

Decision makers

Options

Feasible states

Profit and loss value

Grey potential

Preference

Individual stabilities

Equilibrium

Information to assist decision

makers

Modeling

Analysis

Conflict analysis

From the above reachable matrix, we can get the set of player i’s reachable Ri(s) from state s:

RiðsÞ ¼ {q [ S : Riðs; qÞ ¼ 1}

A unilateral improvement means that a DM moves from a particular state to a preferred state. To represent unilateral improvements, each player’s reachability matrix Ri can be replaced by R

þ i , defined by:

R þ i ðs; qÞ ¼

1; if Riðs; qÞ ¼ 1; and piðsÞ a piðqÞ

0; others

(

Similarly, the set of player i’s unilateral improvements from state s can be defined by:

R þ i ðsÞ ¼ {q [ S : R

þ i ðs; qÞ ¼ 1}

According to the above several ways, the definitions of equilibrium solution under grey game theory are as follows.

3.2 Equilibrium solutions under grey potential In the grey conflict analysis model, if a DM has no incentive to deviate from a certain state then this state is considered stable. If a state is stable for all DMs, it is an equilibrium state and is regarded as a possible resolution of the conflict.

Definition 3.1. (Grey potential-Nash equilibrium), simplified as GP-Nash. Let i [ N. A state s [ S is GP-Nash stable for DM i, denoted by s [ S

GP-Nash i ,

iff (if and only if ) R þ i ðsÞ ¼ F.

Additionally, s is said to be an equilibrium state of GP-Nash if s [ S GP-Nash i for all

i [ N. From Definition 3.1 we can get: under GP-Nash equilibrium stability, there is no

unilateral improvement for DM i, when it transferred to other state, the rest states must become disadvantages to the initial state, the initial state of the choices is the final state.

Definition 3.2. (Grey potential-general metarationality), simplified as GP-GMR. Let i [ N. A state s [ S is GP-GMR stable for DM i, denoted by s [ S

GP-GMR i , iff for

every s1 [ R þ i ðsÞ, there exists at least one s2 [ Rj (s1), such that pi (s2) # pi (s).

Additionally, s is said to be an equilibrium state of GP-GMR if s [ S GP-GMR i for all

i [ N. In Definition 3.2 “ # ” is equivalent to “ a ” or “ , ”. Player i deems that the

opponent will sanction i’s improvement at any cost. Definition 3.3. (Grey potential-symmeral metarationality), simplified as GP-SMR. Let i [ N. A state s [ S is GP-SMR stable for DM i, denoted by s [ S

GP-SMR i , iff for

every s1 [ R þ i ðsÞ, there exists s2 [ Rj (s1), such that pi (s2) # pi (s) and pi (s3) # pi (s) for

all s3 [ Rj (s2). Additionally, s is said to be an equilibrium state of GP-SMR if s [ S

GP-SMR i for all

i [ N. GP-SMR solution is similarly to GP-GMR, however, for player i, there is still an

opportunity to counter-back and the conflict ends after his or her counter-move. Definition 3.4. (Grey potential-sequential stability), simplified as GP-SEQ.

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100

Let i [ N. A state s [ S is GP-SEQ stable for DM i, denoted by s [ S GP-SEQ i , iff for

every s1 [ R þ i ðsÞ, there exists at least one s2 [ R

þ j ðs1Þ, such that pi (s2) # pi (s).

Additionally, s is said to be an equilibrium state of GP-SEQ if s [ S GP-SEQ i for all

i [ N. GP-SEQ solution is similarly to GP-GMR, the only difference is that the opponent

will sanction i’s improvement, but only using their own improvements. The first column of Table II describes the approaches of different solution concepts

to foresight which measures the maximum number of moves foreseen by a DM; GP-Nash stability has foresight one; the other definitions (GP-GMR, GP-SMR and GP-SEQ) have foresight two or three.

The disimprovement criterion indicates a DM’s willingness to move to a less preferred state. In GP-Nash and GP-SEQ stability, disimprovements cannot be accepted and in GP-GMR and GP-SMR, only disimprovements by opponents with the purpose of sanctioning are allowed.

The last column of Table II shows the knowledge of preferences in the viewpoint of DM’s. Under GP-Nash, GP-GMR and GP-SMR, opponents’ preference rankings are not required. But under GP-SEQ stability, preference rankings for the focal DM and his opponent are required.

3.3 Interrelationships of solution concepts

Theorem 3.1. SGP-Nash i

# SGP-SMR i

; SGP-Nash i

# SGP-GMR i

; SGP-Nash i

# SGP-SEQ i

Proof. For i [ N, if state s [ S GP-Nash i , then R

þ i ðsÞ ¼ F, which implies Definitions

3.2-3.4, are satisfied because no s1 belongs to R þ i ðsÞ. Hence s [ S

GP-GMR i , s [ S

GP-SMR i ,

s [ S GP-SEQ i . This proves that S

GP-Nash i

# SGP-SMR i

; SGP-Nash i

# SGP-GMR i

;

SGP-Nash i

# SGP-SEQ i

. A

Theorem 3.2. SGP-SMR i

# SGP-GMR i

. Proof. For i [ N, if state s [ S

GP-SMR i , two cases may arise. First, R

þ i ðsÞ ¼ F. In

this case, s [ S GP-Nash i , and hence, s [ S

GP-GMR i according to the theorem 3.1. Second,

R þ i ðsÞ – F. Because s [ S

GP-GMR i , according to Definition 3.3, for every s1 [ R

þ i ðsÞ,

there exists s2 [ Rj (s1) such that pi (s2) # pi (s), and pi (s3) # pi (s) for all s3 [ Rj (s2). Discarding the information about s3, it is easy to see that Definition 3.2 is satisfied.

Hence, s [ S GP-SMR i as well. This proves that S

GP-SMR i # S

GP-GMR i . A

Theorem 3.3. S GP-SEQ # S GP-GMR. Proof. For i [ N, if state s [ S

GP-SEQ i , again, two cases may arise. First, R

þ i ðsÞ ¼ F.

In this case, s [ S GP-Nash i , therefore, s [ S

GP-GMR i according to the above argument.

Second, R þ i ðsÞ – F. Because s [ S

GP-SEQ i , for every s1 [ R

þ i ðsÞ, there exists at least one

Foresight Disimprovements Knowledge of preferences

GP-Nash 1 Never Own GP-GMR 2 By opponents Own GP-SMR 3 By opponents Own GP-SEQ 2 Never All

Table II. The differences between

equilibrium solutions

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101

s2 [ R þ j ðs1Þ such that pi (s2) # pi (s). However, s2 [ R

þ j ðs1Þ # Rjðs1Þ, so, according to

Definition 3.2, state s [ S GP-GMR i . This proves that S

GP-SEQ i

# SGP-GMR i

. A The logical relationships among four stability solutions as follows:

S GP-Nash # S GP-SMR # S GP-GMR; S GP-Nash # S GP-SEQ # S GP-GMRand as shows in Figure 3.

4. Example In a relatively closed colored TV market, let’s consider the problem of “prisoner’s dilemma” between two oligopoly manufacturers. Due to several reasons, their profit and loss value information are symmetry detect, the value (unit: million) can be used by interval grey number, as shown in Table III.

4.1 DMs, options and states There are two DMs, as shown in the “prisoner’s dilemma” above. Manufacturers 1 and 2, we simply denote as DM1, DM2, respectively. DM1 has two options: decrease price or not-decrease price, so does DM2. Therefore, this game includes four states: s1 (decrease, decrease), s2 (decrease, not-decrease), s3 (not-decrease, decrease), s4 (not-decrease, not-decrease).

4.2 Grey potential and preference information According to the definition of grey potential in part 2, we can calculate the superiority potential degree and inferior potential degree of four states.

For DM1, We use these following formulas to calculate the superiority potential degree and the inferior potential degree between state s1 (decrease, decrease) and state s3 (not-decrease, decrease):

SPDs1!s3 ¼ 1 2 0

1 2 ð21Þ ¼

2 IPDs1!s3 ¼ 2

ð22Þ 2 ð21Þ

1 2 ð21Þ ¼

Then:

SPDs1!s3 þ IPDs1!s3 ¼ 1

2 þ

2 ¼ 1 . 0

Figure 3. Interrelationships of solution concepts

GP-SEQ

GP-SMR

GP-GMR

GP- Nash

Manufacturer 2 Decrease Not-decrease

Manufacturer 1 Decrease [21, 1], [21, 1] [1, 2.5], [22, 0] Not-decrease [22, 0], [1, 2.5] [0, 2], [0, 2]

Table III. The grey profit and loss matrix between two manufacturers

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102

So, for manufacturer 1, state s1 is advantageous to state s3; similarly, we can conclude that state s2 is better than state s4, and state s4 is advantageous to state s1; according to the property of preference, we can get the preference information of manufacturer 1 as follows:

P1 : s2 s s4 s s1 s s3

For DM2, the calculation of the superiority potential degree and the inferior potential degree between state s3 (not-decrease, decrease) and state s4 (not-decrease, not-decrease) as follows:

SPDs3!s4 ¼ 2:5 2 2

2:5 2 1 ¼

3 IPDs3!s4 ¼ 2

0 2 1

2:5 2 1 ¼

3 Then:

SPDs3!s4 þ IPDs3!s4 ¼ 1

3 þ

3 ¼ 1 . 0

So, for manufacturer 2, state s3 is advantageous to state s4; similarly, we can conclude that state s1 is better than state s2 and state s4 is advantageous to state s1; according to the property of preference, we can get the preference information of manufacturer 2 as follows:

P2 : s3 s s4 s s1 s s2

4.3 Graph model When manufacturer 2 chooses the decrease strategy, for manufacturer 1, he can move from decrease to not-decrease, also, he can move from not-decrease to decrease; he can transform between state s1 and state s3; manufacturer 1 can transform between state s2 and state s4 when manufacturer 2 selects the not-decrease strategy.

Similarly, we can get the state transition of manufacturer 2. The graph model of DM1 and DM2 as shown in Figure 4.

4.4 Equilibrium states and analysis According to Definitions 3.1-3.4, the process to calculate equilibrium is as follows.

Considering state s1 (decrease, decrease): because R þ 1 ðs1Þ ¼ F, so the state s1 is

GP-Nash equilibrium state for DM1; additionally, because R þ 2 ðs1Þ ¼ F, so the state s1 is

GP-Nash equilibrium state for DM2. Therefore, state s1 is GP-Nash equilibrium. From the logical relationships among equilibrium solutions, we can conclude that state s1 is also GP-GMR, GP-SMR and GP-SEQ equilibrium state.

Let’s consider state 4 for DM1 to explain why a state possessing a unilateral improvement can be GP-GMR stable. The diagram is shown in the first graph in Figure 5. DM1 can make a unilateral improvement to state 2 or else stay at state 4.

Figure 4. Graph model for

DM1 and DM2s3

a1 a2 a3 a4

s1 s2

s3 s4

Conflict analysis

103

From state 2, DM2 can move to state 1 or else stay at state 2. However, notice that state 1 is less preferred than state 4 by DM1, so DM1 is willing to stay at the initial state and then state 4 is GP-GMR stable for DM1.

Notice that when DM2 moves from state 2 to state 1, state 1 is more preferred than state 2 for DM2, so the state 4 is also GP-SEQ stable.

The GP-SMR stable as shown in the second graph in Figure 5, is similar to Figure 4(a), the only difference is that the game is not over when DM2 moves to state 1 and the DM1 has the chance to move from state 1 to state 3. While we notice that state 3 is also less preferred than the initial state 4, so DM1 is willing to stay at state 4 and then state 4 is GP-SMR stable for DM1.

The unstable state is shown the third graph in Figure 5. DM1 can make a unilateral improvement move from state 3 to state 1 or else stay at state 3. From state 1, DM2 can move to state 2 or else stay at state 1. However, notice that state 2 is preferred than state 3 by DM1, so DM1 is willing to move, then state 3 is not stable for DM1.

Similarly, we can analysis the equilibrium states of DM2; the final equilibrium states as shown in Table IV.

In Table IV, “1”, “2” indicate DM1 and DM2, respectively; “E” is the abbreviation of “equilibrium”, “U” denotes that the state is a stable state under the equilibrium solution. For example, state s2, it is GP-Nash stable state for DM1, but it’s not GP-Nash stable state for DM2; while, state s1, it’s GP-Nash stable for both DM1 and DM2, then the state is GP-Nash equilibrium.

From the results of equilibrium states in Table IV, we can conclude as follows: when considering the GP-Nash equilibrium, only one state s1 (decrease, decrease) is the final equilibrium state, which is consistent with the classical game theory. All DMs are

Figure 5. The diagram of calculate equilibrium solutions

s2s1

Note: “s” means “stay”

DM1

DM2

DM1

DM2

DM1

DM2

s s

GP-Nash GP-GMR GP-SMR GP-SEQ 1 2 E 1 2 E 1 2 E 1 2 E

s1 U U U U U U U U U U U U s2 U U U U s3 U U U U s4 U U U U U U U U U

Table IV. The final equilibrium states

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104

considering the best of personal, but in fact, the individual interest has conflict with the collective interest. That is the “prisoner’s dilemma”. When we consider the GP-GMR, GP-SMR and GP-SEQ solutions, we can find that except state s1 is the stable state, state s4 (not-decrease, not-decrease) is also the stable state. The main reason is that: the DM considers only one step under GP-Nash equilibrium. The explanation is that the DM is short-sighted and has no long-term vision. In contrast, the DM will consider multiple steps under the GP-GMR, GP-SMR and GP-SEQ stability. Therefore, for a sufficient farsighted DM, he or she will select state s4 (not-decrease, not-decrease) as the last equilibrium state.

5. Conclusions and future work In this paper, we combine the grey game theory with the GMCR in two persons based on pure strategy and construct the grey conflict analysis model. Detailed contributions along this research include:

. From the number of DM’s steps, pure strategy GP-Nash solution was extended to GP-GMR, GP-SMR and GP-SEQ concepts.

. Interrelationships of solution concepts were investigated; theoretical results established interrelationships of the four solution concepts.

. An illustrative example was developed to demonstrate how the grey conflict analysis model and these solution concepts can be applied, the example results of this paper showed that the proposed model is reliable and reasonable.

The proposed method in this paper establishes a hybrid framework for conflict analysis by combining grey game and GMCR together. The hybrid system that extends GMCR by allowing preference to be expressed in grey interval values is more general than existing grey game theory and GMCR.

To allow complex cases such as economic and social applications to be efficiently and expeditiously analyzed, the two-player game will be extended to n-players in the near future.

References

Deng, J.L. (1982), “The control problems of grey systems”, Systems and Control Letters, Vol. 1 No. 5, pp. 288-94.

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