A conflict analysis framework considering fuzzy preferences

2.1Fuzzy preference and stability definitions

Definition 1. (Fuzzy Preference): A fuzzy preference of DM i over S is a fuzzy relation on S, represented by a matrix R _i  = (r _kt )  _w×w , with membership function μ _R  : S × S → [0,  1], where μ _R  (s _k ,  s _t ) = r _kt denotes the preference degree of state s _k over s _t , satisfying r _kt  + r _tk  = 1 and r _kk  = 0.5.

The preference degree is the level of certainty that a DM will prefer one state over the other. Note that the fuzzy preference model allows for both crisp and fuzzy preferences between two states. When a DM holds a crisp preference, the preference degrees over all states are 0, 0.5, or 1.

Definition 2. (Fuzzy Relative Strength of Preference (FRSP)): Let rkti denote the preference degree of state s _k over s _t for DM i. Then, DM i’s FRSP of state s _k over s _t is defined as αi(sk, st)=rkti-rtki, where -1 ≤ α ⁱ  (s _k ,  s _t ) ≤1.

The concept of FRSP is used to measure how strongly a DM prefers one state to the other. In stability analysis, each DM with fuzzy preferences may select a level of FRSP to decide whether to maintain or leave a focal state. The FRSP level represents the DM’s fuzzy satisficing threshold.

Definition 3. (Fuzzy Satisficing Threshold (FST)): Let DM i’s FST be denoted as γ _i (0 < γ _i  ≤ 1). Then, DM i would prefer to move from state s to s _k if and only if (iff) α ⁱ  (s _k ,  s) ≥ γ _i .

Notice that a DM may have different FSTs at different times, and different DMs in a conflict may hold different FSTs as well.

Bashar et al. [10] provides four fuzzy stability definitions to identify possible equilibriums for the conflict. The fuzzy stability definitions vary predictably according to the DMs’ FSTs. Generally, a state is fuzzy stable for a DM iffleaving a focal state does not meet the DM’s FST. It is necessary to define the movements for a single DM and a coalition of DMs before defining the fuzzy stability concepts.

Definition 4. (Fuzzy Unilateral Improvement List (FUIL)): Let γ be DM i’s FST, and R _i  (s) be the set of reachable states from state s for DM i. Then, state s _k  ∈ R _i  (s) is called a Fuzzy Unilateral Improvement (FUI) from s for DM i iff α ⁱ  (s _k ,  s) ≥ γ _i . The set of all FUIs from s for DM i is called the FUIL, which is denoted by R˜i,γ+(s), where R˜i,γ+(s)={sk∈Ri(s):αi(sk, s)≥γi}.

Definition 5. (FUIL for a Coalition of DMs): For H ∈ N, let H = {1,  2, …,  m}, γ _H  = {γ ₁,  γ ₂, …,  γ _m }, and Ω˜H,γH+(s, s1) denote the set of all last or final DMs who can move in one step from state s to s ₁. The set R˜H,γH+(s) satisfies the following criteria: (i) if i ∈ H and s1∈R˜i+(s), then s1∈R˜H,γH+(s) and i∈Ω˜H,γH+(s, s1); (ii) if i ∈ H, s1∈R˜H,γH+(s), s2∈R˜H,γH+(s1), and Ω˜H,γH+(s, s1)≠{i}, then s2∈R˜H,γH+(s) and i∈Ω˜H,γH+(s, s2).

In addition to the above definitions, the fuzzy stability concepts are shown in Table 1.

A state that is fuzzy stable for all DMs under a specific fuzzy stability definition is called a fuzzy equilibrium (FE) under that definition.

2.2Fuzzy option prioritization

In the option prioritization method set forth in [8], each DM i possesses an ordered list of preference statements P _i  = [Ω ₁,  Ω ₂, …,  Ω _j , …,  Ω _q ] in which the preference statements that are more important for DM i appear earlier in the list. Each preference statement, which is expressed in terms of options and logical connectives, takes a truth-value of either “True” (T) or “False” (F) at each state.

Denote Ω _j  (s) as the truth-value of preference statement Ω _j at state s, and let ψ _j  (s) be the score to state s based upon preference statement Ω _j . Define ψj(s)={2q-j,if Ωj(s)=T0,otherwise and ψ(s)=∑j-1qψj(s); then, the states can be sorted based on their scores. Specifically, s _k  ≻ i s _t iff ψ (s _k ) > ψ (s _t ), and s _k  ∼ i s _t iff ψ (s _k ) = ψ (s _t ).

In some situations, however, preference statements cannot be judged as simply “true” or “false” at some feasible states. Under these circumstances, Bashar et al. [11] used fuzzy truth-values represented by numerical values in the interval [0,  1] to judge the truthfulness of a preference statement at a feasible state. Let σ _j  (s) = σ (Ω _j ,  s) denote the fuzzy truth-value of preference statement Ω _j at state s ∈ S. Then, σ _j  (s) =0 is equivalent to Ω _j  = F, while σ _j  (s) =1 is equivalent to Ω _j  = T.

Let a lower transformation function be l (x) = x ^p and an upper transformation function be u (x) =2x - x ^p , where 1 ≤ p ≤ 2, 0 ≤ x = σ _j  (s) ≤1, and 0 ≤ l (x) ≤ u (x) ≤1. The value of p depends on a DM’s level of confidence about his judgment with respect to the truth degree of a preference statement at astate.

Definition 6. (Fuzzy Truth Value Interval): Define σjL(s)=l(σj(s)), and σjU(s)=u(σj(s)); then, [σjL(s), σjU(s)]⊂[0, 1] is called the DM’s fuzzy truth value interval of Ω _j at state s.

Definition 7. (Fuzzy Score Interval): For σj(s)=[σjL(s), σjU(s)], choose a real number α > 0 satisfying α > α ² + α ² + … + α ^q , such as 0<α≤12. Let ψ˜jL(s)=αjσjL(s) and ψ˜jU(s)=αjσjU(s); then, ψ˜j(s)=[ψ˜jL(s), ψ˜jU(s)] is called the DM’s incremental fuzzy score intervalof s for Ω _j . Let ψ˜L(s)=∑j=1qψ˜jL(s) and ψ˜U(s)=∑j=1qψ˜jU(s); then, ψ˜(s)=[ψ˜L(s), ψ˜U(s)] is called the DM’s fuzzy score interval of state s.

Notice that the value of α depends on how much truth for a preference statement at a particular state exactly balances truth-value 1 of the next less important preference statement at that state.

The fuzzy score intervals of states can be compared in pairs to calculate the preference degree over two states using Definition 8.

Definition 8. For Lk=ψ˜U(sk)-ψ˜L(sk) and Lt=ψ˜U(st)-ψ˜L(st), one can use the following function (1) to calculate r (s _k ,  s _t ) = r _kt :

When the truth-values are either 0 or 1, the preference is crisp.

3.1DMs and options

In this conflict, the central government’s objective is to take measures to promote cooperation between upstream and downstream stakeholders. Therefore, three DMs are involved in the conflict: the upstream areas (DM 1), the downstream regions (DM 2), and the central government (DM 3). Table 2 gives the options of the three DMs.

3.2Feasible states and graph model

Since states in which DM 1 or DM 2 select no option or more than one option are infeasible and should be emitted, and states where DM 1 selects option A₃ are indistinguishable and should be combined into one, thirteen feasible states remain. These are given in Table 3, in which “Y” and “N” indicate that an option is taken or not taken, respectively, by its DM, while a dash (“-”) means either Y or N.

Figure 1 shows the integrated graph model of the conflict, in which circles represent the feasible states, while directed arcs represent state transitions controlled by different DMs. Arc tails indicate the initial states, while arrowheads indicate the terminal states after moving from the initial states.

3.3Fuzzy preferences

Table 4 furnishes explanations of the prioritized preference statements for each DM.

DM 1’s preferences regarding some states are fuzzy. For example, the truth-value of DM 1’s preference statement Ω ₁ (-A₃) is generally “true” at states s ₁₀ and s ₁₂ and “false” at s ₁₃; that is, σ ₁ (10) =1, σ ₁ (12) =1, σ ₁ (13) =0. However, when DM 2 selects B₁ and DM 3 chooses C₂, DM 1 might prefer state s ₁₃ to s ₁₀ or s ₁₂. In other words, DM 1 may not prefer to select “-A₃” with 100% truth (that is, a truth degree of 1), and the truth-value of DM 1’s preference statement Ω ₁ at states s ₁₀, s ₁₂, and s ₁₃ might be σ ₁ (10) =0.85, σ ₁ (12) =0.65, and σ ₁ (13) =0.15, respectively.

Since DM 3’s decision whether or not to select option C₂ will not have any significant influence on DM 1 when DM 1 chooses A₂, DM 1 might assign a non-zero truth degree to Ω ₂ (-C₂) at s ₇ and s ₇, and might assign a non-one truth degree to Ω ₂ at s ₃ and s ₄. In this paper, the truth value of DM 1’s preference statement Ω ₂ at states s ₃, s ₄, s ₇, and s ₈ are assumed to be σ ₂ (3) =0.4, σ ₂ (4) =0.35, σ ₂ (7) =0.5, and σ ₂ (8) =0.6. Table 5 presents the fuzzy truth-values of DM 1 considering all similar circumstances.

In this conflict, both DM 2 and DM 3 have crisp preferences. Therefore, the truth degrees of DM 2 and DM 3 are either “0” or “1,” as shown in Table 5.

DM 1, DM 2, and DM 3 have six, six, and five preference statements, respectively, in Table 4. Accordingly, Table 5 lists a total of six, six, and five truth degrees at each state for DM 1, DM 2 and DM 3, respectively, appearing in decreasing order of importance of the preference statements. For instance, the second entry in the fourth row and second column of Table 5, 0.4, represents the truth degree of the second most important preference statement “-C₂” of DM 1 at state s ₄.

Based on the fuzzy truth-values in Table 5, a fuzzy score interval for each state for each DM can be calculated employing Definition 8. With the fuzzy score intervals and set parameters p = 1 and α=13, Equation (1) allows the fuzzy preference degrees for each DM to be calculated. DM 1’s fuzzy preferences are represented by the matrix R _DM1 in Table 6. Table 7 shows the crisp preferences of DM 2 and DM 3.

3.4Fuzzy stability analysis

DMs’ FSTs usually play a vital role in fuzzy stability analysis. This paper considers two different FSTs for DM 1: (i) γ ₁ = 0.25 and (ii) γ ₁ = 0.55. Since both DM 2 and DM 3 hold crisp preferences, their FSTs are the same (γ ₂ = γ ₃ = 1.0). After setting each DM’s FST and employing the four fuzzy stability concepts presented in Table 1, the fuzzy stability analysis results of the water pollution conflict can be determined as shown in Table 8.

Table 8 shows that when weaker satisficing criteria for DM 1 (γ ₁ = 0.25) are considered, states s ₁₀, s ₁₁, and s ₁₃ are FNash stable states, while states s ₃, s ₄, s ₇, and s ₈ satisfy FGMR and FSMR stability. Under these conditions, the conflict will have more opportunities to stay at state s ₁₀, where it cannot be solved effectively. However, under stronger satisficing criteria for DM 1 (γ ₁ = 0.55), state s ₁₀ disappears from the stable state list, state s ₁₂ joins the FGMR and FSMR stable state list, and state s ₁₁ has a greater chance to become an equilibrium for the conflict, which is favorable for all DMs. These results illustrate that the fuzziness of DM 1’s preferences is strong and will influence the conflict’s development and solutions.

Note that the distinguishable state s ₁₃ is a stable state that did not occur in reality. In reality, the final outcome of the conflict was state s ₁₁, where under the pressure of DM 3’s both Encourage and Punish options, DM 1 reduced pollution emissions and DM 2 did not take option Sanction. The above analysis is consistent with the actual trajectory of the conflict, a result that demonstrates the feasibility and applicability of this conflict model.