Conflict Analysis Based on Three-Way Decision Theoretic Fuzzy Rough Set over Two Universes

Conflict is inevitable in many fields of human endeavor, namely, business and governmental negotiations, political and legal disputes, labor management, military operations, service and physical resources, commercial security policies, information decisions, water resource allocation, and risk control [1-14]. Therefore, it is very important to analyze and resolve conflicts in an accurate manner. So far, many models and methods have been developed for conflict analysis [2-7, 15, 16, 19-23].

Since the late 1990s, the rough set (RS) has been applied to conflict resolution and decision-making. Based on the RST, Z. Pawlak [4, 5] outlined a novel approach for conflict analysis. Later, various scholars have identified the limitations of Pawlak’s conflict analysis model, and made many attempts to improve the model. First, Ramanna et al. [2] extended the RS-based conflict model to a complex conflict model. Next, Zhu and Wang [6] proposed the concept of associated conflict, and developed an approach for associated conflict analysis, using covering-based granular computing. Soon, Liu et al. [7] introduced a comprehensive strategy for associated conflict analysis through covering-based granular computing. Deja and Ślęak [16] described the nature of conflicts and several solutions to the fundamental problems of conflicts, and defined conflict situation models that clearly illustrate conflict components.

Pawlak’s RS theory might be invalid if there are two or more correlated universes, such as investment plan [17] and medical diagnosis [18]. Despite their correlations, investor (sufferer) and project (symptom) belong to different universes. Therefore, it is necessary to promote RS-based conflict model to multiple universes. Drawing on the RS theory, B.Z. Sun et al. [19, 20] extended Pawlak’s conflict analysis model to two universes, and also combined probabilistic rough set with three-way decision, creating a new conflict analysis model.

The RS-based conflict analysis is critical to decision-making. Lang et al. [21] presented the concepts of probabilistic conflict set, neutral set and allied conflict set, calculated the thresholds of conflict analysis with the decision theoretic RST, and designed incremental algorithms for the said sets in dynamic information system. With the aid of group decision theory, Lang et al. [22] also introduced methods to compute positive, neutral and negative alliances for conflicts in Pythagorean fuzzy information systems. Later, Lang et al. [23] proposed the concepts of agreement, disagreement and neutral subsets of a strategy with two evaluation functions, put forward a three-way decisions-based conflict analysis models to trisect the universe of agents, and employed a pair of two-way decisions models to interpret the mechanism of the three-way decision rules for an agent.

The fuzzy set (FS) and RS can effectively handle conflicts under uncertainty. So far, many have analyzed conflicts based on the RS or FS, but few have tackled conflict analysis from the perspective of FRS (FRS). To make up for this gap, this paper mainly investigates conflict analysis based on three-way decision FRS over two universes. Inspired by the RS-based decision rules proposed by Sun et al. [20] for conflict over two universes, the authors presented the decision rules for conflict based on the FRS over two universes, and then developed a three-way decision model of (0,1)-probabilistic FRS and 0.5-probabilistic FRS.

The reminder of this paper is organized as follows: Section 2 reviews the CIS over two universes and introduces the FRS-CIS over two universes; Section 3 presents the three-way decision theoretic RS under the CIS over two universes, and illustrates the three-way decision models of (0,1)-probabilistic FRS and 0.5-probabilistic FRS with examples; Section 4 sums up the research findings.

This section introduces the three-way decision theoretic RS under the CIS over two universes, and then sets up the three-way decision models of (0,1)-probabilistic FRS and 0.5-probabilistic FRS.

3.1 The three-way decision theoretic RS under the CIS over two universes

The decision rules for probabilistic RS of the CIS over two universes were presented, according to the decision-theoretic framework for probabilistic RS proposed by Yao et al. [26, 27] based on the Bayesian decision procedure.

Suppose U and V are the agent set and issue set of a conflict, respectively, and XÎ2^V is a feasible consensus strategy of the conflict. Let A={d₁, d₂, d₃} be the three decisions of agent u_iÎU over the feasible consensus strategy X: agreement, disagreement and neutrality. In the case of X, the cost or risk to agree, disagree or stay neutral can be respectively denoted as λ₁₁=λ(d₁|X), λ₂₁=λ(d₂|X) and λ₃₁=λ(d₃|X)  ; in the case of X^c, the cost or risk to agree, disagree or stay neutral can be respectively denoted as  λ₁₂=λ(d₁|X^c), λ₂₂=λ(d₂|X^c) and λ₃₂=λ(d₃|X^c).

Based on the rough set under the CIS over two universes, the expected risk $R^{\circ}\left(d_{t} | u_{i}\right)(\circ \in\{+,-\}, t=1,2,3)$ of any agent u_iÎU to agree with, disagree with or stay neutral to the feasible consensus strategy X can be expressed as:

$R^{+}\left(d_{1} | u_{i}\right)=R\left(d_{1} | F^{+}\left(u_{i}\right)\right)=$ $\lambda\left(d_{1} | X\right) P\left(X | F^{+}\left(u_{i}\right)\right)+\lambda\left(d_{1} | X^{c}\right) P\left(X^{c} | F^{+}\left(u_{i}\right)\right)$   (7)

$R^{+}\left(d_{2} | u_{i}\right)=R\left(d_{2} | F^{+}\left(u_{i}\right)\right)=$ $\lambda\left(d_{2} | X\right) P\left(X | F^{+}\left(u_{i}\right)\right)+\lambda\left(d_{2} | X^{c}\right) P\left(X^{c} | F^{+}\left(u_{i}\right)\right)$ (8)

$\begin{aligned} R^{+}\left(d_{3} | u_{i}\right)=R\left(d_{3}\right.&\left.| F^{+}\left(u_{i}\right)\right) \\ &=\lambda\left(d_{3} | X\right) P\left(X | F^{+}\left(u_{i}\right)\right) \\ &+\lambda\left(d_{3} | X^{c}\right) P\left(X^{c} | F^{+}\left(u_{i}\right)\right) \end{aligned}$ (9)

and

$R^{-}\left(d_{1} | u_{i}\right)=R\left(d_{1} | F^{-}\left(u_{i}\right)\right)=$ $\lambda\left(d_{1} | X\right) P\left(X | F^{-}\left(u_{i}\right)\right)+\lambda\left(d_{1} | X^{c}\right) P\left(X^{c} | F^{-}\left(u_{i}\right)\right)$ (10)

$\begin{aligned} R^{-}\left(d_{2} | u_{i}\right)=R\left(d_{2} |\right.&\left.F^{-}\left(u_{i}\right)\right) \\ &=\lambda\left(d_{2} | X\right) P\left(X | F^{-}\left(u_{i}\right)\right) \\ &+\lambda\left(d_{2} | X^{c}\right) P\left(X^{c} | F^{-}\left(u_{i}\right)\right) \end{aligned}$   (11)

$\begin{aligned} R^{-}\left(d_{3} | u_{i}\right)=R\left(d_{3}\right.&\left.| F^{-}\left(u_{i}\right)\right) \\ &=\lambda\left(d_{3} | X\right) P\left(X | F^{-}\left(u_{i}\right)\right) \\ &+\lambda\left(d_{3} | X^{c}\right) P\left(X^{c} | F^{-}\left(u_{i}\right)\right) \end{aligned}$   (12)

Here, the relation for λ_ij=λ_ij(i=1,2,3, j=1,2)  is assumed as follows:

$\lambda_{11} \leq \lambda_{31}<\lambda_{21}, \lambda_{12}>\lambda_{32} \geq \lambda_{22}$    (13)

For any agent u_i∈U, the minimum risk decision rules for F⁺(u_i)={a_j|a_j(u_i)=+, a_j∈V}  u_i∈U  based on the RS under the CIS over two universes can be derived from the Bayesian decision procedure over two universes[28, 29]:

(P1) Making the decision $\operatorname{Pos}_{p}^{+}(X)$ if R(d₁|F⁺(u_i))<R(d₂|F⁺(u_i)) and R(d₁|F⁺(u_i))<R(d₃|F⁺(u_i));

(N1) Making the decision $\operatorname{Neg}_{P}^{+}(X)$ if R(d₂|F⁺(u_i))<R(d₁|F⁺(u_i))  and R(d₂|F⁺(u_i))<R(d₃|F⁺(u_i));

(B1) Making the decision $\mathrm{Bn}_{p}^{+}(X)$ if R(d₃|F⁺(u_i))<R(d₁|F⁺(u_i)) and R(d₃|F⁺(u_i))<R(d₂|F⁺(u_i));

Based on the RS under the CIS over two universes, the minimum risk decision rules (P1), (N1) and (B1) can be written as follows for any agent u_i∈U:

(P2) Making the decision $\operatorname{Pos}_{P}^{+}(X)$ if P(X|F⁺(u_i))≥α  and P(X|F⁺(u_i))≥γ; (N2) Making the decision $\operatorname{Neg}_{P}^{+}(X)$ if P(X|F⁺(u_i))≤β and P(X|F⁺(u_i))<γ; (B2) Making the decision $\mathrm{Bn}_{P}^{+}(X)$ if P(X|F⁺(u_i))<αand P(X|F⁺(u_i))>β.

where,

$\alpha=\frac{\lambda_{12}-\lambda_{32}}{\left(\lambda_{12}-\lambda_{32}\right)+\left(\lambda_{31}-\lambda_{11}\right)}$    (14)

$\beta=\frac{\lambda_{32}-\lambda_{22}}{\left(\lambda_{32}-\lambda_{22}\right)+\left(\lambda_{21}-\lambda_{31}\right)}$   (15)

$\gamma=\frac{\lambda_{12}-\lambda_{22}}{\left(\lambda_{12}-\lambda_{22}\right)+\left(\lambda_{21}-\lambda_{11}\right)}$    (16)

and satisfies $0 \leqslant \beta<\alpha \leqslant 1$.

By the principle of three-way decision $[4,28],$ the following equations can be derived from the RS under the CIS:

(P3) Agreement (Decision positive region):

$\operatorname{Pos}_{P}^{+}(X)=\operatorname{Apr}_{P}^{+}(X)=\left\{\mathrm{u}_{\mathrm{i}} \in U\left|P\left(X | F^{+}\left(u_{i}\right)\right) \geq \alpha\right\}\right.$

(B3) Neutrality (Decision boundary region):

$\operatorname{Bn}_{P}^{+}(X)=\left\{\mathrm{u}_{\mathrm{i}} \in U\left|\beta<P\left(X | F^{+}\left(u_{i}\right)\right)<\alpha\right\}\right.$

(N3) Disagreement (Decision negative region):

$\operatorname{Neg}_{P}^{+}(X)=\left\{\mathrm{u}_{\mathrm{i}} \in U\left|P\left(X | F^{+}\left(u_{i}\right)\right)<\beta\right\}\right.$

Based on the RS under the CIS, the following equations can be derived for $R^{-}\left(d_{1} | u_{i}\right), R^{-}\left(d_{2} | u_{i}\right)$ and $R^{-}\left(d_{3} | u_{i}\right)$

(P3') Agreement (Decision positive region):

$\operatorname{Pos}_{P}^{-}(X)=\operatorname{Apr}_{P}^{-}(X)=\left\{u_{i} \in U\left|P\left(X | F^{-}\left(u_{i}\right)\right) \geq \alpha\right\}\right.$

(B3') Neutrality (Decision boundary region):

$\operatorname{Bn}_{P}^{-}(X)=\left\{u_{i} \in U\left|\beta<P\left(X | F^{-}\left(u_{i}\right)\right)<\alpha\right\}\right.$

(N3') Disagreement (Decision negative region):

$\operatorname{Neg}_{P}^{-}(X)=\left\{u_{i} \in U\left|P\left(X | F^{-}\left(u_{i}\right)\right) \leq \beta\right\}\right.$

Both the set of agreed issues $F^{+}\left(u_{i}\right)$ and the set of disagreed issues $F^{-}\left(u_{i}\right)$ contain decision rules in the CIS over two universes. Therefore, the decision rules in the two set-valued mappings should be considered. For $0 \leq \beta<\alpha \leq 1,$ any agent $u_{i} \in U$ and any feasible consensus strategy $X \in 2^{V},$ the following decision rules are available for the RS under the CIS:

(P4) If $P\left(X | F^{+}\left(u_{i}\right)\right) \geq \alpha$ and $P\left(X | F^{-}\left(u_{i}\right)\right) \leq \beta,$ then

agent $u_{i}$ agrees with the feasible consensus strategy $X$

(N4) If $P\left(X | F^{+}\left(u_{i}\right)\right) \leq \beta$ and $P\left(X | F^{-}\left(u_{i}\right)\right) \geq \alpha,$ then

agent $u_{i}$ disagrees with the feasible consensus strategy $X$

(B4) If $\beta<P\left(X | F^{+}\left(u_{i}\right)\right)<\alpha$ and $\beta<P\left(X | F^{-}\left(u_{i}\right)\right)<\alpha$

then agent $u_{i}$ remains neutral to the feasible consensus strategy $X$

3.2 Three-way decision model of (0,1)-probabilistic FRS under FRS-CIS over two universes

This sub-section discusses the principle and framework of three-way decision model of (0, 1)-probabilistic FRS in the FRS-CIS.

Suppose $U$ and $V$ are the investor set and project set of an FRS conflict, respectively, and $Y \in 2^{V}$ is a feasible consensus strategy of the conflict. Let $A=\left\{d_{1}, d_{2}, d_{3}\right\}$ be the three decisions of agent $u_{i} \in U$ over the feasible consensus strategy $Y:$ agreement, disagreement and neutrality. In the case of $f^{+}$ the cost or risk to agree, disagree or stay neutral can be respectively denoted as $\lambda_{11}=\lambda\left(d_{1} | f^{+}\right), \lambda_{21}=\lambda\left(d_{2} | f^{+}\right)$ and $\lambda_{31}=\lambda\left(d_{3} | f^{+}\right) ;$ in the case of $\left(f^{+}\right)^{c},$ the cost or risk to agree, disagree or stay neutral can be respectively denoted as $\lambda_{12}=$ $\lambda\left(d_{1} |\left(f^{+}\right)^{c}\right), \lambda_{22}=\lambda\left(d_{2} |\left(f^{+}\right)^{c}\right)$ and $\lambda_{32}=\lambda\left(d_{3} |\left(f^{+}\right)^{c}\right)$.

Based on the (0,1) -probabilistic FRS under FRS-CIS over two universes, the expected risk $R^{\circ}\left(d_{t} | u_{i}\right)(\circ \in\{+,-\}, t=$ 1,2,3) of any investor $u_{i} \in U$ to agree with, disagree with or stay neutral to the feasible consensus strategy $Y$ can be expressed as:

$R^{+}\left(d_{1} | u_{i}\right)$ $=\lambda\left(d_{1} | f^{+}\left(u_{i}\right)\right) P\left(f^{+}\left(u_{i}\right) | Y\right)$ $+\lambda\left(d_{1} |\left(f^{+}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{+}\left(u_{i}\right)\right)^{c} | Y\right)$   (17)

$R^{+}\left(d_{2} | u_{i}\right)=\lambda\left(d_{2} | f^{+}\left(u_{i}\right)\right) P\left(f^{+}\left(u_{i}\right) | Y\right)+$

$\lambda\left(d_{2} |\left(f^{+}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{+}\left(u_{i}\right)\right)^{c} | Y\right)$   (18)

$R^{+}\left(d_{3} | u_{i}\right)$ $=\lambda\left(d_{3} | f^{+}\left(u_{i}\right)\right) P\left(f^{+}\left(u_{i}\right) | Y\right)$ $+\lambda\left(d_{3} |\left(f^{+}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{+}\left(u_{i}\right)\right)^{c} | Y\right)$ (19)

and

$R^{-}\left(d_{1} | u_{i}\right)$ $=\lambda\left(d_{1} | f^{-}\left(u_{i}\right)\right) P\left(f^{-}\left(u_{i}\right) | Y\right)$ $+\lambda\left(d_{1} |\left(f^{-}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{-}\left(u_{i}\right)\right)^{c} | Y\right)$   (20)

$R^{-}\left(d_{2} | u_{i}\right)$ $=\lambda\left(d_{2} | f^{-}\left(u_{i}\right)\right) P\left(f^{-}\left(u_{i}\right) | Y\right)$ $+\lambda\left(d_{2} |\left(f^{-}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{-}\left(u_{i}\right)\right)^{c} | Y\right)$  (21)

$R^{-}\left(d_{3} | u_{i}\right)$ $=\lambda\left(d_{3} | f^{-}\left(u_{i}\right)\right) P\left(f^{-}\left(u_{i}\right) | Y\right)$

$+\lambda\left(d_{3} |\left(f^{-}\left(u_{i}\right)\right)^{c}\right) P\left(\left(f^{-}\left(u_{i}\right)\right)^{c} | Y\right)$   (22)

Here, the relation for $\lambda_{i j}(i=1,2,3, j=1,2)$ is assumed as follows:

$\lambda_{11} \leq \lambda_{31}<\lambda_{21}, \lambda_{12}>\lambda_{32} \geq \lambda_{22}$    (23)

For any investor $u_{i} \in U,$ the minimum risk decision rules for $f^{+}\left(u_{i}\right)=\left\{a_{j} | a_{j}\left(u_{i}\right)=1, a_{j} \in V\right\}, u_{i} \in U$ based on the (0,1) -probabilistic FRS under FRS-CIS over two universes can be derived from the Bayesian decision procedure over two universes [28,29]:

(P1) Making the decision $\operatorname{Pos}_{P}^{+}(X)$ if $R^{+}\left(d_{1} | u_{i}\right)<$ $R^{+}\left(d_{2} | u_{i}\right)$ and $R^{+}\left(d_{1} | u_{i}\right)<R^{+}\left(d_{3} | u_{i}\right)$;

(N1) Making the decision $\operatorname{Neg}_{P}^{+}(X)$ if $R^{+}\left(d_{2} | u_{i}\right)<$ $R^{+}\left(d_{1} | u_{i}\right)$ and $R^{+}\left(d_{2} | u_{i}\right)<R^{+}\left(d_{3} | u_{i}\right)$;

(B1) Making the decision Bn $_{P}^{+}(X)$ if $R^{+}\left(d_{3} | u_{i}\right)<$ $R^{+}\left(d_{1} | u_{i}\right)$ and $R^{+}\left(d_{3} | u_{i}\right)<R^{+}\left(d_{2} | u_{i}\right)$ ;

Based on the (0,1) -probabilistic FRS under FRS-CIS over two universes, the minimum risk decision rules (P1), (N1) and (B1) can be written as follows for any agent $u_{i} \in U$;

(P2) Making the decision $\operatorname{Pos}_{P}^{+}(X)$ if $P\left(f^{+}\left(u_{i}\right) | Y\right) \geq \alpha$ and $P\left(f^{+}\left(u_{i}\right) | Y\right) \geq \gamma$;

(N2) Making the decision $\operatorname{Neg}_{P}^{+}(X)$ if $P\left(f^{+}\left(u_{i}\right) | Y\right) \leq \beta$ and $P\left(f^{+}\left(u_{i}\right) | Y\right)<\gamma$;

(B2) Making the decision $\operatorname{Bn}_{P}^{+}(X)$ if $P\left(f^{+}\left(u_{i}\right) | Y\right)<\alpha$ and $P\left(f^{+}\left(u_{i}\right) | Y\right)>\beta$.

where,

$\alpha=\frac{\lambda_{12}-\lambda_{32}}{\left(\lambda_{12}-\lambda_{32}\right)+\left(\lambda_{31}-\lambda_{11}\right)}$    (24)

$\beta=\frac{\lambda_{32}-\lambda_{22}}{\left(\lambda_{32}-\lambda_{22}\right)+\left(\lambda_{21}-\lambda_{31}\right)}$   (25)

$\gamma=\frac{\lambda_{12}-\lambda_{22}}{\left(\lambda_{12}-\lambda_{22}\right)+\left(\lambda_{21}-\lambda_{11}\right)}$   (26)

and satisfies the condition of $0 \leq \beta<\alpha \leq 1$.

By the principle of three-way decision $[4,28],$ the following equations can be derived from the (0,1) -probabilistic FRS under FRS-CIS over two universes:

(P3) Agreement (Decision positive region): $\operatorname{Pos}_{P}^{+}(Y)=\left\{u_{i} \in U\left|P\left(f^{+}\left(u_{i}\right) | Y\right) \geq \alpha\right\}\right.$,

(B3) Neutrality (Decision boundary region): $\operatorname{Bn}_{P}^{+}(Y)=\left\{u_{i} \in U\left|\beta<P\left(f^{+}\left(u_{i}\right) | Y\right)<\alpha\right\}\right.$,

(N3) Disagreement (Decision negative region): $\operatorname{Neg}_{P}^{+}(Y)=\left\{u_{i} \in U\left|P\left(f^{+}\left(u_{i}\right) | Y\right) \leq \beta\right\}\right.$.

Based on the ( 0,1 )-probabilistic FRS under FRS-CIS over two universes, the following equations can be derived for $R^{-}\left(d_{1} | u_{i}\right), R^{-}\left(d_{2} | u_{i}\right)$ and $R^{-}\left(d_{3} | u_{i}\right)$:

(P3') Agreement (Decision positive region): $\operatorname{Pos}_{P}^{-}(Y)=\left\{u_{i} \in U\left|P\left(f^{-}\left(u_{i}\right) | Y\right) \geq \alpha\right\}\right.$,

(B3') Neutrality (Decision boundary region): $\operatorname{Bn}_{P}^{-}(Y)=\left\{u_{i} \in U\left|\beta<P\left(f^{-}\left(u_{i}\right) | Y\right)<\alpha\right\}\right.$,

(N3') Disagreement (Decision negative region): $\operatorname{Neg}_{P}^{-}(Y)=\left\{u_{i} \in U\left|P\left(f^{-}\left(u_{i}\right) | Y\right) \leq \beta\right\}\right.$.

Both the set of agreed issues $f^{+}\left(u_{i}\right)$ and the set of disagreed issues $f^{-}\left(u_{i}\right)$ contain decision rules in the FRS-CIS over two universes. Therefore, the decision rules in the two set-valued mappings should be considered. For $0 \leq \beta<\alpha \leq 1,$ any investor $u_{i} \in U$ and any feasible consensus strategy $Y \in 2^{V}$ the following decision rules are available for the (0,1) probabilistic FRS under FRS-CIS over two universes:

(P4) If $P\left(f^{+}\left(u_{i}\right) | Y\right) \geq \alpha$ and $P\left(f^{-}\left(u_{i}\right) | Y\right) \leq \beta,$ then the investor $u_{i}$ agrees with the feasible consensus strategy $Y$;

(N4) If $P\left(f^{+}\left(u_{i}\right) | Y\right) \leq \beta$ and $P\left(f^{-}\left(u_{i}\right) | Y\right) \geq \alpha,$ then the investor $u_{i}$ disagrees with the feasible consensus strategy $Y$;

(B4) For $u_{i} \in U$ satisfying neither (P4) nor (N4), then the investor $u_{i}$ remains neutral to the feasible consensus strategy $Y$.

Example 3 (Continued from Example 1 ) The risks of making different decisions can be empirically determined as: $\lambda_{11}=0.35, \lambda_{21}=0.75, \lambda_{31}=0.65, \lambda_{12}=0.8, \lambda_{22}=$ $0.12, \lambda_{32}=0.2$.

The loss function satisfies formula (23). By formulas ( 24 ) and $(25),$ we have $\alpha=0.6667$ and $\beta=0.4444$.

The following results can be derived from minimum risk decision rules and the principle of three-way decision based on (0,1) -probabilistic FRS under the FRS-CIS over two universes:

$\operatorname{Pos}_{P}^{+}(Y)=\emptyset, \operatorname{Bn}_{P}^{+}(Y)=\left\{u_{9}\right\}, \operatorname{Neg}_{P}^{+}(Y)=$ $\left\{u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7}, u_{8}, u_{10}\right\}$

$\operatorname{Pos}_{p}^{-}(Y)=\left\{u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{9}\right\}, \operatorname{Bn}_{P}(Y)=$ $\emptyset, \operatorname{Neg}_{P}^{-}(Y)=\left\{u_{1}, u_{7}, u_{8}, u_{10}\right\}$

According to the above decision rules, no sufferer agrees with the feasible consensus strategy $Y,$ sufferers $u_{2}, u_{3}, u_{4}, u_{5}$ and $u_{6}$ disagree with the feasible consensus strategy $Y,$ and sufferers $u_{1}, u_{7}, u_{8}, u_{9}$ and $u_{10}$ remain neutral to the feasible consensus strategy $Y$.

3.3 Three-way decision model of 0.5-probabilistic FRS under FRS-CIS over two universes

Considering all the information for the FRS-CIS over two universes, 0.5 -probabilistic FRS is more accurate than the (0,1)-probabilistic FRS in classification. The expected risks and decision rules of 0.5 -probabilistic FRS are similar to those of (0,1)-probabilistic FRS. For $0 \leq \beta<\alpha \leq 1,$ any investor $u_{i} \in U$ and any feasible consensus strategy $Y \in 2^{V},$ the following decision rules are available for 0.5 -probabilistic FRS under FRS-CIS over two universes:

For $0 \leq \beta<\alpha \leq 1,$ any investor $u_{i} \in U$ and any feasible consensus strategy $Y \in 2^{V},$ the following decision rules are available for 0.5 -probabilistic FRS under FRS-CIS over two universes:

(P5) If $P\left(f^{+}\left(u_{i}\right) | Y\right) \geq \alpha$ and $P\left(f^{-}\left(u_{i}\right) | Y\right) \leq \beta,$ then then agent $u_{i}$ agrees with the feasible consensus strategy $Y$;

(N5) If $P\left(f^{+}\left(u_{i}\right) | Y\right) \leq \beta$ and $P\left(f^{-}\left(u_{i}\right) | Y\right) \geq \alpha,$ then then agent $u_{i}$ disagrees with the feasible consensus strategy$Y$;

(B5) For $u_{i} \in U$ satisfying neither (P5) nor (N5), then then agent $u_{i}$ remains neutral to the feasible consensus strategy $Y$.

Example 4 (Continued from Example 2 ) The risks of making different decisions are similar to those in example 3 Therefore, we have $\alpha=0.6667$ and $\beta=0.4444$

The following results can be derived from minimum risk decision rules and the principle of three-way decision based on 0.5 -probabilistic FRS under the FRS-CIS over two universes:

$\operatorname{Pos}_{P}^{+}(Y)=\left\{u_{1}, u_{2}, u_{4}, u_{5}, u_{6}, u_{7}, u_{9}, u_{10}\right\}, \operatorname{Bn}_{P}^{+}(Y)=$ $\left\{u_{\mathrm{g}}\right\}, \operatorname{Neg}_{P}^{+}(Y)=\left\{u_{3}\right\}$.

$\operatorname{Pos}_{P}^{-}(Y)=\left\{u_{3}\right\}, \operatorname{Bn}_{P}^{-}(Y)=\emptyset, \operatorname{Neg}_{P}^{-}(Y)=$ $\left\{u_{1}, u_{2}, u_{4}, u_{5}, u_{6}, u_{7}, u_{8}, u_{9}, u_{10}\right\}$.

According to the above decision rules, sufferers $u_{1}, u_{2}, u_{4}, u_{5}, u_{6}, u_{7}, u_{9}$ and $u_{10}$ agree with the feasible consensus strategy $Y,$ sufferer $u_{3}$ disagrees with feasible consensus strategy $Y,$ and sufferer $u_{8}$ remains neutral to the feasible consensus strategy $Y$.