The use of three ratings and three types of agent relations naturally links conflict analysis to the idea of thinking in threes that underlies the theory of three-way decision introduced by Yao [34,35,37]. In a broad sense, threeway decision involves a philosophy of thinking in threes, a methodology of working in threes, and a mechanism of processing in threes [37]. A popular concrete model is the Trisecting-Acting-Outcome (TAO) model [35], which involves dividing a whole into three parts, devising and applying strategies to process the three parts, and evaluating and optimizing the outcome. Three-way decision and its underlying philosophy of thinking in threes have been widely applied to many disciplines and fields [2,7,19,38,43]. Three-way classification [6,8,17,20,33,41,44] is one of the earliest studied specific topics of three-way decision. It focuses on trisecting a universe into positive instances, negative instances, and those in the boundary that cannot be determined to be positive or negative. Such a trisection is commonly based on the evaluation of objects. Yao [34] summarizes two evaluation-based three-way classification models. One uses a single evaluation function to perform both the positive and negative evaluations, and the other uses two separate evaluation functions. The idea of computing with threes also appears in a few other topics, such as shadowed sets and bipolar representations. Pedrycz [24][25][26] proposes the concept of a shadowed set that transforms a fuzzy set into three disjoint elevated, reduced, and shadowed areas by applying two opposite operations of elevation and reduction. The idea of using three areas indicates a close connection between shadowed sets and three-way decision, which has been studied by a few researchers [39,40,42,45,47]. Dubois and Prade [3,4] propose the concept of bipolarity, which refers to the propensity of the human mind for reasoning and decision-making based on the two opposite positive and negative effects. Although bipolarity shares many common features with three-way decision, their connections are not well explored in the literature.

Conflict analysis is one of the many topics that share a common idea of using threes with three-way decision. A combination of the two topics leads to three-way conflict analysis that studies a conflict situation via trisecting agents, issues, and agent pairs. Specifically, most research on three-way conflict analysis [5, 11-14, 16, 30-32, 36] follows the three-way classification model based on a single evaluation function. Sun, Ma, and Zhao [32] introduce a model of trisecting agents and issues by generalizing the concepts of lower and upper approximations over two universes in rough set theory. This work is further extended by Sun et al. [30] based on the probabilistic rough set approximations over two universes. Lang, Miao, and Cai [13] apply a pair of thresholds (α, β) on an evaluation function to trisect agent pairs instead of a single threshold of 0.5 in Pawlak’s model [21][22][23]. Fan, Qi, and Wei [5] propose a quantitative model of three-way conflict analysis based on an evaluation function generalized from derivation operators in formal concept analysis. Yao [36] argues about the inconsistency of interpreting the neutral rating 0 in Pawlak’s model and discusses trisections of agents and agent pairs based on a single evaluation function.

A potential issue in most existing models is that they use a single evaluation function to reflect two opposite aspects. For instance, an auxiliary function commonly uses the positive value +1 for an alliance relation between two agents on a specific issue, the negative value -1 for a conflict relation, and the value 0 for a neutrality relation. This may lead to some difficulties in understanding and interpreting the computation of values from an auxiliary function. In particular, when investigating the relationships between agents on a set of issues, the average-based method is commonly used to aggregate the values from an auxiliary function on individual issues. Therefore, a pair of +1 and -1 gives the same result as a pair of 0 and 0. Consequently, two agents allied on one issue and in conflict on another have the same relation as two other agents that are neutral to each other on both issues.

To solve the above semantic issues, we use two separate functions to manage the two opposite aspects represented by an auxiliary function. Specifically, we use an alliance function and a conflict function, respectively, to evaluate the alliance and conflict degrees between two agents on a single issue. When it comes to a set of issues, the alliance and conflict degrees on individual issues are aggregated separately into an aggregated alliance degree and an aggregated conflict degree. These two aggregated degrees are then considered together to define the final alliance, neutrality, and conflict relations between two agents. Intuitively, we define the alliance relation if the alliance degree suggests strong alliance and the conflict degree suggests non-conflict, the conflict relation if it is the other way around, and the neutrality relation otherwise. Many other crucial topics in conflict analysis could also be investigated via the two proposed functions. Particularly, we explore the definition of alliance sets and the description of their decisions based on the trisections of agent pairs. With the trisections of issues and agents, we present to formulate a strategy as a conjunction of issue-rating pairs and further explore the relationships between strategies and agents. With a separation of alliance and conflict functions, we can gain a more in-depth understanding of the relations regarding agents and issues in conflict analysis.

The remainder of this paper is organized as follows. Section 2 reviews basic concepts related to existing three-way conflict analysis with a single evaluation function, including a rating function and an auxiliary function. Section 3 presents the alliance and conflict functions that separate the two opposite aspects in an auxiliary function. Based on these two functions, we discuss the trisections of agents, issues, and agent pairs in Sections 4 and 5. Particularly, Section 4 proposes the concept of alliance sets by trisecting agent pairs and investigates the representation of their decisions. Section 5 presents a formal description of a strategy (i.e., a logic conjunction of issue-rating pairs) and further investigates the trisection of agents towards a specific strategy. In Section 6, we compare the proposed models with two evaluation functions and the existing models with a single evaluation function. An application with a realworld conflict situation is also given in Section 6 to illustrate the proposed models. In the end, Section 7 concludes this work and discusses possible directions for future work.

This section reviews and summarizes basic concepts in three-way conflict analysis. Particularly, we discuss trisections with a single evaluation function, including trisections of agents, issues, and agent pairs. Examples are given at the end to illustrate the concepts.

A conflict problem is usually formulated by a three-valued situation table [22,36], which gives the opinions, attitudes, or ratings of agents on issues. The attitudes are commonly considered to have three cases, namely, positive, negative, and neutral, represented by three numeric values of +1, -1, and 0, respectively. A three-valued situation table is formally defined as follows.

Definition 1. A three-valued situation table is a triplet S = (A, I, r), where A is a finite nonempty set of agents, I is a finite nonempty set of issues, and r : A × I → {+1, -1, 0} is a rating function. The value r(x, i) is called the rating of an agent x ∈ A on an issue i ∈ I and interpreted as follows:

Yao [36] extends a three-valued situation table to a many-valued situation table, which considers ratings from the interval [-1, +1]. We restrict our discussion to three-valued situation tables in this work. The generalization to many-valued situation tables might be a direction for our future work. According to the three ratings of agents on a specific issue, we can immediately construct a trisection of agents as follows.

Definition 2. For a subset of agents X ⊆ A and an issue i ∈ I, the trisection of X with respect to i is defined as [36]:

The three sets X + i , X - i , and X 0 i include the agents with a positive, negative, and neutral rating on a single issue i, respectively. When it comes to a set of issues, we need to aggregate the ratings of an agent on every single issue in the set. A commonly used method is to take the average [36]. Formally, the rating of an agent x ∈ A on a nonempty subset of issues ∅ J ⊆ I is given by a rating function r : A × (2 I -{∅}) → [-1, +1] as follows:

where 2 I is the power set of I and |J| is the cardinality of J. It should be noted that we use the same symbol r to represent different rating functions for simplicity. One may simply differentiate the functions by their parameters.

The aggregated rating r(x, J) can be any value in the interval [-1, +1]. Accordingly, we may apply two thresholds on the aggregated ratings to construct a trisection of agents with respect to J. Definition 3. For a subset of agents X ⊆ A and a nonempty subset of issues ∅ J ⊆ I, the trisection of X with respect to J is defined as [36]:

where (l, h) is a pair of thresholds satisfying -1 ≤ l < 0 < h ≤ +1.

Intuitively, if the aggregated rating r(x, J) is high enough (i.e., r(x, J) ≥ h), the agent x is considered to have a positive attitude on J as a whole. Similarly, if r(x, J) is low enough (i.e., r(x, J) ≤ l), x is considered to have a negative attitude on J. Otherwise, r(x, J) is neither strong enough to suggest a positive nor a negative attitude and thus, x is considered to have a neutral attitude on J.

One can investigate the trisections of issues by simply exchanging the roles of agents and issues. Due to this reason, the trisections of issues are usually not explicitly explored in the literature. Although the trisections of issues follow the same formulations as the trisections of agents, they imply different meanings and views in understanding the conflict situation. Furthermore, the trisections of issues will also be investigated in our proposed models in the following sections. Thus, we briefly discuss the trisections of issues with respect to a single agent and a subset of agents. The trisection of issues with respect to a single agent can be obtained based on the ratings of agents. Definition 4. For a subset of issues J ⊆ I and an agent x ∈ A, the trisection of J with respect to x is defined as [36]:

With respect to a nonempty set of agents, we aggregate their ratings on a single issue as the average. Formally, the aggregated rating function r :

Accordingly, one can trisect a set of issues as given in the following definition.

Definition 5. For a nonempty subset of agents ∅ X ⊆ A and a subset of issues J ⊆ I, the trisection of J with respect to X is defined as:

where (l, h) is a pair of thresholds satisfying -1 ≤ l < 0 < h ≤ +1.

Intuitively, if the aggregated rating from a group of agents X on an issue i is high enough (i.e., r(X, i) ≥ h), then the group X as a whole is considered to have a positive attitude on i. Similarly, if the aggregated rating is low enough (i.e., r(X, i) ≤ l), then the group has a negative attitude on i. Otherwise, the group has a neutral attitude on i.

Another essential topic of three-way conflict analysis is investigating the relationships between agents. Basically, there are three relationships between agents, namely, the alliance, conflict, and neutrality relations [21]. These relations are defined based on the ratings of agents on either a single issue or a subset of issues. An auxiliary function Φ is commonly used to help formulate the relations. Intuitively, a positive value of Φ represents the tendency of alliance, and a negative value represents the tendency of conflict.

Formally, an auxiliary function with respect to a single issue i ∈ I is a three-valued mapping Φ i : A × A → {+1, -1, 0}, where the three values are interpreted as follows:

x and y are allied on i, iff Φ i (x, y) = +1; x and y are in conflict on i, iff Φ i (x, y) = -1; x and y are neutral on i, iff Φ i (x, y) = 0.

The neutrality relation is simply interpreted as the two agents being neither allied nor in conflict. Accordingly, one can formally define a trisection of agent pairs with respect to a single issue.

Definition 6. Given an auxiliary function Φ i with respect to an issue i ∈ I, the alliance relation R = i , conflict relation R ≍ i , and neutrality relation R ≈ i between agents with respect to i are defined as:

There are different specific definitions of Φ i as will be illustrated in Example 1. Commonly, the following properties should be satisfied by a meaningful auxiliary function Φ i :

(1)

That is, an agent is always self-allied, and the value for x and y is equal to the value for y and x. The latter implies that the alliance, conflict, and neutrality relations are symmetric. In addition, it is reasonable to consider two agents to be allied if they both have a positive or negative attitude on the issue i. Similarly, it is reasonable to consider two agents to be in conflict if one of them is positive and the other is negative. In the case where at least one agent holds a neutral rating, there are different views of defining the relations based on different understanding and interpretations of neutral ratings. From the above analysis, we define a general template of Φ i as follows: for x, y ∈ A,

where the notation * represents a value in {+1, -1, 0}. Since Φ i (x, y) is actually defined with respect to the ratings from x and y, we may alternatively use the notation Φ(r(x, i), r(y, i)) when we do not concern about the specific agents who hold the ratings. Table 1 shows the values of Φ i in the general template for two different agents. We can generalize the auxiliary function in Equation ( 8) with respect to a set of issues by simply taking the average. Formally, an auxiliary function Φ J : A × A → [-1, +1] with respect to a subset of issues J ⊆ I can be defined as follows [22,36]:

While Φ i is a three-valued function, Φ J is a many-valued function that takes values in the interval [-1, +1]. Applying two thresholds on Φ J , one can immediately construct a trisection of agent pairs and accordingly, define the alliance, conflict, and neutrality relations.

Definition 7. Given an auxiliary function Φ J with respect to a subset of issues J ⊆ I, the alliance relation R = J , conflict relation R ≍ J , and neutrality relation R ≈ J with respect to J are defined as:

where (l, h) is a pair of thresholds satisfying -1 ≤ l < 0 < h ≤ +1.

Intuitively, two agents x and y are allied on J if they have a high tendency for alliance (i.e., Φ J (x, y) ≥ h). They are in conflict if their tendency for conflict is strong enough (i.e., Φ J (x, y) ≤ l). Otherwise, they are neutral on J.

Aggregating ratings with respect to a group of issues or agents leads to continuous values from the interval [-1, +1] instead of the three discrete values +1, -1, and 0. Accordingly, the trisections with respect to a group of issues or agents need a pair of thresholds (l, h) to cut the continuous aggregated values, as in Definitions 3, 5, and 7. This idea is also commonly used in other topics to transform a set of continuous quantitative values into several qualitative parts or levels. For example, a shadowed set [24][25][26] adopts a similar idea to transform a fuzzy set with continuous membership degrees in the unit interval [0, 1] into three qualitative levels of membership degrees. The transformation is performed through two operations of elevation and reduction with a pair of thresholds. Particularly, the elevation operation elevates the membership degrees at or above one threshold to 1, and the reduction operation reduces the membership degrees at or below another threshold to 0. The membership degrees between the two thresholds are mapped to the whole unit interval [0, 1]. A shadowed set can also be considered as a three-way approximation of fuzzy sets [39]. The trisections in Definitions 3, 5, and 7 can also be explained through similar elevation and reduction operations. We elevate an evaluation value from a rating function (or an auxiliary function) to +1 that represents a positive attitude (or an alliance relation) if it is higher than or equal to a threshold h. Similarly, we reduce an evaluation value from a rating function (or an auxiliary function) to -1 that represents a negative attitude (or a conflict relation) if it is lower than or equal to another threshold l. For a value in-between the two thresholds, we either elevate or reduce it to the neutral value 0. A similar idea of dealing with values between two thresholds is also discussed for shadowed sets by Yao, Wang, and Deng [39], where they either elevate or reduce the values to a middle membership degree of 0.5. It is an interesting direction of our future work to further investigate the relations between conflict analysis and shadowed sets in this regard.

We give two examples to illustrate the concepts discussed in this section. Particularly, Example 1 illustrates specific auxiliary functions that fit into our general template in Equation ( 8) and Example 2 illustrates the trisections.

Example 1 (Pawlak’s and Yao’s auxiliary functions). We have proposed a general definition of an auxiliary function in Equation (8). Here, we review two specific auxiliary functions proposed by Pawlak [22] and Yao [36] in terms of our general definition. Concretely, for any two agents x, y ∈ A and an issue i ∈ I, Pawlak defines the auxiliary function Φ P i : A × A -→ {+1, -1, 0} as:

Instead of defining an auxiliary function, Yao proposes a distance function regarding two agents on a single issue. An auxiliary function can be easily defined based on the distance function. For x, y ∈ A and i ∈ I, the Yao’s auxiliary function Φ Y i : A × A -→ {+1, -1, 0} can be defined as:

To clearly show the difference between Pawlak’s and Yao’s auxiliary functions, Table 2 gives the values of Φ P i and Φ Y i for two different agents. These two auxiliary functions differ in the understanding of the relations between two different agents with both neutral ratings on an issue i. Concretely, for two different agents x, y ∈ A with r(x, i) = r(y, i) = 0, in Pawlak’s opinion, they are neutral on i, that is, Φ P (r(x, i), r(x, i)) = 0; in Yao’s opinion, they are allied on i, that is, Φ Y (r(x, i), r(x, i)) = +1.

Table 2: The values of Φ P (r(x, i), r(y, i)) and Φ Y (r(x, i), r(y, i)) with x y With respect to a set of issues J ⊆ I, we could apply the general template in Equation (9) with Pawlak’s auxiliary function in Equation (11) and get the following aggregated auxiliary function:

Similarly, we can define Yao’s auxiliary function with respect to J as:

Example 2 (Trisections of agents, issues, and agent pairs). We illustrate the trisections of agents and issues using the rating functions r defined in Section 2.1, and the trisection of agent pairs using Pawlak’s and Yao’s auxiliary functions given in the above Example 1. Consider the three-valued situation table given in Table 3. For simplicity, we consider a set of agents X = A = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } and a set of issues J = I = {i 1 , i 2 , i 3 , i 4 , i 5 } when constructing the trisections. Firstly, we construct the trisection of agents X with respect to J. According to the aggregated rating function in Equation (2), we compute the ratings of every agent in X on J as:

By applying a pair of thresholds (-3 5 , + 3 5 ), we get the trisection of X according to Definition 3 as:

Secondly, we construct the trisection of issues J with respect to X. According to the aggregated rating function in Equation (5), we compute the ratings of agents in X as a whole on every issue in J as:

By applying a pair of thresholds (-1 3 , + 1 3 ), we get the trisection of J according to Definition 5 as:

Finally, we construct the trisection of agent pairs using Pawlak’s and Yao’s auxiliary functions. According to Equations (13) and (14), we compute Pawlak’s and Yao’s auxiliary functions with respect to J as given in Table 4. For example, for the two agents x 1 and x 2 , we have:

;

By applying a pair of thresholds (-1 2 , + 1 2 ), we construct the trisections of agent pairs according to Definition 7.

x 1 The trisection based on Pawlak’s auxiliary function in Table 4 is constructed as:

x 2 ), (x 3 , x 3 ), (x 3 , x 4 ), (x 4 , x 3 ), (x 4 , x 4 ), (x 5 , x 5 ), (x 5 , x 6 ), (x 6 , x 5 ), (x 6 , x 6 )},

x 3 ), (x 2 , x 4 ), (x 2 , x 5 ), (x 2 , x 6 ), (x 3 , x 1 ), (x 3 , x 2 ), (x 3 , x 5 ), (x 3 , x 6 ), (x 4 , x 1 ), (x 4 , x 2 ), (x 4 , x 5 ), (x 5 , x 1 ), (x 5 , x 2 ), (x 5 , x 3 ), (x 5 , x 4 ), (x 6 , x 1 ), (x 6 , x 2 ), (x 6 , x 3 )}.

(

The trisection based on Yao’s auxiliary function in Table 4 is constructed as:

x 2 ), (x 3 , x 3 ), (x 3 , x 4 ), (x 4 , x 3 ), (x 4 , x 4 ), (x 5 , x 5 ), (x 5 , x 6 ), (x 6 , x 5 ), (x 6 , x 6 )},

x 3 ), (x 2 , x 4 ), (x 2 , x 5 ), (x 2 , x 6 ), (x 3 , x 1 ), (x 3 , x 2 ), (x 3 , x 5 ), (x 3 , x 6 ), (x 4 , x 1 ), (x 4 , x 2 ), (x 4 , x 5 ), (x 5 , x 1 ), (x 5 , x 2 ), (x 5 , x 3 ), (x 5 , x 4 ), (x 6 , x 1 ), (x 6 , x 2 ), (x 6 , x 3 )}.

(21)

We present the alliance and conflict functions that separate the two opposite semantics measured in one auxiliary function, including both unaggregated and aggregated formulations. Examples are given at the end to illustrate the presented concepts.

As discussed in the previous section, the existing trisections of agents, issues, and agent pairs in three-way conflict analysis are commonly based on a single evaluation function, either a rating or an auxiliary function. A common feature of these functions is that they measure two opposite attitudes or relations in one formula. The rating function gives both positive and negative attitudes, and the auxiliary function defines both alliance and conflict relations. This may induce certain difficulties in understanding, interpreting, and applying these functions in analyzing conflict problems. For example, when using the average function to aggregate the ratings of an agent on a set of issues, two opposite ratings +1 and -1 on two issues have the same effect as two neutral ratings 0 and 0. This issue is actually illustrated in Example 2, where the aggregated ratings of agents x 1 and x 5 on

5 (see Equation ( 15)). As a result, x 1 and x 5 are determined to have the same attitude toward the set of issues J, even though x 1 clearly expresses a non-neutral attitude on issue i 1 only, and x 5 has non-neutral attitudes on all issues in J. As a mirror image, aggregating the ratings of a set of agents on a single issue also faces the same challenge.

There is a similar difficulty in aggregating an auxiliary function Φ i into Φ J . A pair of alliance and conflict relations on two issues has the same effect as a pair of neutrality relations. Consider a subset of issues J = {i 2 , i 3 , i 4 , i 5 } from Table 3. We first use Yao’s auxiliary function for a single issue (in Table 2(b)) to compute the relations between agents x 1 and x 3 on every single issue in J, as given in the first line of Table 5. Then we take their average as the aggregated value of Φ Y J (x 1 , x 3 ) which is 0. We perform the same computation for agents x 3 and x 5 which is given in the second line of Table 5. Clearly, we have

Therefore, the agent x 3 is determined to have the same relation with x 1 and x 5 on J (i.e., neutrality), even though the agent x 3 is in conflict with x 5 on two issues but not with x 1 for any issue. A conflict relation on an issue is dissolved by an alliance relation on another. The same problem exists in the auxiliary function Φ P J . For example, we have Φ P J (x 1 , x 2 ) = Φ P J (x 3 , x 5 ) = 0 as given in Table 5.

Computing some values of Φ Y J and Φ P J in Table 3 The reason for the above issues is that three-way conflict models with a single evaluation function do not distinguish between two opposite impacts, that is, either positive/negative attitudes or alliance/conflict relations. Intuitively in decision-making, people usually weigh the positive/good aspect and the negative/bad aspect separately and then combine them to make a final decision. A similar idea has been used in the three-way classification model based on two evaluation functions of positive and negative [34], as well as in the concept of bipolarity [3,4]. Following the same idea, we propose to split the auxiliary function, which describes both alliance and conflict, into two separate functions. An alliance function evaluates the alliance degree, and a conflict function evaluates the conflict degree. To clearly show their connections to the existing works, we define the alliance and conflict functions in terms of a given auxiliary function. Since auxiliary functions are defined through ratings, one may equivalently define these two functions in terms of ratings. Definition 8. Given an auxiliary function Φ i regarding a single issue i ∈ I, the alliance function

and the conflict function Φ ≍ i : A × A → {0, 1} with respect to i are defined as:

The function Φ i can be any specific auxiliary function that satisfies the general formulation in Equation (8).

Intuitively, we use the values 1 and 0 to represent an answer of “yes” and “no” to the alliance and conflict relations. Accordingly, the alliance function describes the alliance and non-alliance relations between agents. If Φ = i (x, y) = 1, then x and y are allied; otherwise, x and y are not allied, which does not necessarily mean that they are in conflict. Similarly, the conflict function gives the conflict and non-conflict relations between agents. If Φ ≍ i (x, y) = 1, then x and y are in conflict; otherwise, x and y are not in conflict, which does not necessarily mean that they are allied.

As mentioned above, one may also define the alliance and conflict functions through ratings rather than a given auxiliary function. In that case, an auxiliary function and the pair of alliance and conflict functions can be interchangeably defined. Specifically, we may define an auxiliary function through a pair of alliance and conflict functions as:

Equivalently, we have:

In our following discussion, we may alternatively denote the alliance and conflict functions with ratings as their parameters instead of the agents, particularly when we emphasize the ratings rather than the agents who give the ratings. Formally, for two agents x, y ∈ A, we have:

The alliance Φ = i and conflict Φ ≍ i functions can be aggregated with respect to both issues and agents. Specifically, we may consider the following four cases of aggregation: (C1) a set of issues only;

(C2) a set of agents as the first parameter only;

(C3) a set of agents as the second parameter only;

(C4) any combination of the above three.

We propose a general definition of the aggregated alliance and conflict functions that could cover all these four cases. Definition 9. A general definition of the aggregated alliance and conflict functions is given as follows: for J ⊆ I and X, Y ⊆ A,

From the general definition, one may easily define other cases of aggregation mentioned above by degenerating one or more sets of J, X, and Y into a singleton set. For example, for the first case of aggregation mentioned above (i.e., a set of issues only), we may simply degenerate the set X into {x} and Y into {y}. Furthermore, for simplicity, we may use x and y instead of the singleton sets in the notations. Thus, we get the following aggregated functions:

Similarly, we may also easily define the other cases of aggregated functions. Corresponding to all the cases of aggregation mentioned above, we arrive at the following 7 cases in total:

(C1) a set of issues only: Φ = J (x, y) and Φ ≍ J (x, y);

(C2) a set of agents as the first parameter only: Φ = i (X, y) and Φ ≍ i (X, y);

(C3) a set of agents as the second parameter only:

(C4) any combination of the above three:

We omit their definitions for simplicity. Following the two functions defined in Equation ( 26), all these aggregated functions are defined in terms of the unaggregated functions Φ = i (x, y) and Φ ≍ i (x, y). Alternatively, one can also define an aggregated function in the case (C4) in terms of those in the cases from (C1) to (C3). For example, we can define Φ = i (X, Y) by aggregating Φ = i (x, Y) or Φ = i (X, y) as follows:

Similarly, we can define Φ = J (X, Y) by aggregating Φ = i (X, Y) or Φ = J (x, Y) as follows:

As a result, the relationships among alliance functions and those among conflict functions can be shown by the lattices in Figure 1. A function can be defined in terms of any function below it as long as a path connects them. We get more general functions when going upwards and special cases when going downwards. An essential issue in analyzing the agent relations is the choice of a subset of issues J ⊆ I. Two agents (or two groups of agents) may have different relations with respect to different subsets of issues. For a single agent x ∈ A, a subset of issues J can be trisected into three disjoint parts J + x , J - x , and J 0 x (as explained in Definition 4). The agent has a neutral attitude towards issues in J 0

x , which can be interpreted as an unclear, uncertain, undetermined, or changeable attitude. In analyzing the relations between agents, it is reasonable to focus on the issues where the agents have clear, definite, and non-neutral attitudes. For an agent x ∈ A, we denote the set of issues on which x has non-neutral attitudes as:

which is called the set of non-neutral issues of x. One can define the alliance and conflict functions with respect to the non-neutral issues of x by using J +- x instead of J in Equation ( 27):

The function Φ = J +-

x (x, y) represents the proportion of issues where x and y are allied out of the non-neutral issues in J +-

x , which can be considered as the degree of y supporting x with respect to J +- x . Similarly, the function Φ ≍ J +-

x (x, y) gives the proportion of issues where x and y are in conflict out of J +-

x , which can be considered as the degree of y opposing x with respect to J +-

x . The two functions in Equation ( 31) are not symmetric since different denominators are used in the computation, that is,

x (x, y) = 0 for any other agent y ∈ A. In other words, an agent with all neutral ratings neither supports nor opposes other agents.

We give two examples to illustrate the concepts of alliance and conflict functions introduced in this section. Particularly, Example 3 defines three pairs of alliance and conflict functions based on Pawlak’s and Yao’s auxiliary functions introduced in Example 1, including both unaggregated and aggregated functions. Example 4 further illustrates these three pairs of functions by showing the computation of their values with a specific situation table, including with respect to a set of issues J and a set of non-neutral issues in J regarding a specific agent.

Example 3 (Three pairs of alliance and conflict functions based on Pawlak’s and Yao’s auxiliary functions). We first illustrate the unaggregated alliance and conflict functions. By applying Pawlak’s and Yao’s auxiliary functions in Equations (11) and (12) to Definition 8, we get the alliance and conflict functions for individual agents with respect to a single issue i ∈ I as follows:

Tables 6 and7 give the values of these four functions in terms of ratings as parameters instead of agents. The value 0 * means that the value is 1 instead of 0 if x = y. Apparently, Pawlak’s and Yao’s auxiliary functions induce the same conflict function, that is, we have Φ ≍P i (x, y) = Φ ≍Y i (x, y). As discussed in Example 1, Pawlak’s and Yao’s auxiliary functions express different opinions on the relations between two different agents with both neutral ratings. Thus, the two alliance functions only differ in the case of two different agents with both neutral ratings. While Pawlak considers them to be neutral to each other, Yao considers them to be allied.

Table 6: The values of Φ =P (r(x, i), r(y, i)) and Φ ≍P (r(x, i), r(y, i))

Table 7: The values of Φ =Y (r(x, i), r(y, i)) and Φ ≍Y (r(x, i), r(y, i))

We then illustrate the aggregated alliance and conflict functions. Based on the functions in Equation (32), we can define the aggregated alliance and conflict functions induced from Pawlak’s and Yao’s auxiliary functions. As an illustration, we consider the case (C1) of aggregation, that is, a set of issues. We have given the general definition of Φ = J (x, y) and Φ ≍ J (x, y) in Equation (27). By simply replacing the general alliance and conflict functions Φ = i (x, y) and Φ ≍ i (x, y) with those in Equation (32), we arrive at the following aggregated alliance and conflict functions with respect to a subset of issues J ⊆ I:

At last, we illustrate the aggregated alliance and conflict functions with respect to non-neutral issues. By replacing the general unaggregated alliance and conflict functions used in Equation (31) with those in Equation (32), we can easily get the following aggregated functions with respect to the non-neutral issues of the first agent:

These two pairs of aggregated alliance and conflict functions are actually equivalent, that is, we have Φ =P

x (x, y). Recall that the unaggregated alliance functions induced from Pawlak’s and Yao’s functions only differ in the case where two different agents both hold a neutral rating on a single issue, as shown in Tables 6(a) and 7(a). Such an issue will not appear in J +-

x . The unaggregated conflict functions induced from Pawlak’s and Yao’s function are exactly the same as shown in Tables 6(b) and 7(b). Thus, we must have Φ ≍P

Example 4 (Computing the three pairs of alliance and conflict functions in Example 3). We further illustrate the three pairs of alliance and conflict functions constructed in Example 3 with the situation table given in Table 8. For simplicity, let us consider a set of issues J = I = {i 1 , i 2 , i 3 , i 4 }. Based on the functions defined in Equation (33), one can easily compute the alliance and conflict degrees between agents with respect to J, which are shown in Tables 9 and10. We explicitly label the agents as the first or second parameter in the functions, which is important in those with respect to the non-neutral issues. As discussed in Example 3, with respect to the non-neutral issues, the alliance and conflict functions are equivalent in Pawlak’s and Yao’s cases. Thus, we give the values in both cases in Table 11 where we omit the superscripts of P and Y. We take the two agents x 4 and x 10 to illustrate the computation. The alliance and conflict degrees from the three pairs of functions are computed as:

;

;

x 4 0 0

x 9 0 0 x 10 0 0 0

x 11 0 0 0

x 4 x 6 0 0

x 9 0 0

Based on the alliance and conflict functions, we present the concept of alliance sets and investigate the description of their decisions. As shown in Figure 2, our discussion includes two streams, one with respect to a set of issues J ⊆ I and another with respect to the non-neutral issues J +- x ⊆ J regarding an agent x. Intuitively, an alliance set for

Table 11: The values of Φ = J +-and Φ ≍ J +-with respect to Table 8 a specific agent is the group of agents that are in the alliance relation with it. In a special case that the agents in an alliance set are allied with each other, we can derive a concept of the maximal consistent alliance set. The agents in an alliance set are in close co-operation. Thus, it is useful to analyze their ratings of issues as a whole, which is called the decision of the alliance set. Examples are given at the end to illustrate the presented concepts.

We study the relationships between agents based on the proposed alliance and conflict functions. By combining the two opposite viewpoints represented in the two functions, we define the alliance, conflict, and neutrality relations as follows.

Definition 10. Suppose Φ = J and Φ ≍ J are the alliance and conflict functions with respect to a subset of issues J ⊆ I. Given two pairs of thresholds (l a , h a ) and (l c , h c ) with 0 ≤ l a < h a ≤ 1 and 0 ≤ l c < h c ≤ 1, the alliance relation R = J (Φ = , Φ ≍ ), conflict relation R ≍ J (Φ = , Φ ≍ ), and neutrality relation R ≈ J (Φ = , Φ ≍ ) are defined as:

The above trisection of agent pairs can also be explained through the elevation and reduction operations in shadowed sets that are discussed at the end of Section 2.2. If the alliance degree is high enough and at the same time, the conflict degree is low enough, that is, Φ = J (x, y) ≥ h a ∧ Φ ≍ J (x, y) ≤ l c , we elevate the alliance degree to 1 and reduce the conflict degree to 0. Accordingly, the two agents x and y are determined to be allied. One can similarly explain the conflict and neutrality relations. A difference from the existing definitions of these relations is that we apply both elevation and reduction operations in constructing one relation.

Definition 10 actually gives a trisection of agent pairs in A×A with respect to J. One can easily define the trisection with respect to a single issue i ∈ I by degenerating J to {i}. Such a trisection is actually equivalent to the trisection induced from an auxiliary function (i.e., Definition 6).

Theorem 1. With respect to an issue i ∈ I, the trisections of agent pairs with an auxiliary function and with a pair of AS J (x) AS J +-(x)

Trisecting issues (Definition 14)

Decision of an alliance set X (Definition 15) Des J (X) Des J +-(X)

1 with respect to J ⊆ I 2 with respect to alliance and conflict functions are equivalent, that is:

Proof. By Definition 10, we get the trisection of agent pairs with respect to a single issue i ∈ I as:

where we use i to denote the singleton set {i} for simplicity. Since 0 ≤ l a < h a ≤ 1 and 0 ≤ l c < h c ≤ 1, we have l a < 1, h a > 0, l c < 1, and h c > 0. According to Definition 8, the values of alliance and conflict functions Φ = i and Φ ≍ i 20 can only take 1 or 0. Therefore, we have:

The alliance and conflict functions enable us to measure the two opposite views of alliance and conflict separately when trisecting agent pairs. These two views are combined to define the three relations of alliance, conflict, and neutrality. Such a separation and combination of the two views may provide an in-depth understanding of the agent relations. As shown in Table 12, from the alliance view, if the alliance degree of two agents x and y is high enough, i.e., Φ = J (x, y) ≥ h a , then they are strongly allied; if the alliance degree is low enough, i.e., Φ = J (x, y) ≤ l a , then they are considered not to be allied; otherwise, they are weakly allied. Similarly, from the conflict view, if the conflict degree of x and y is high enough, i.e., Φ ≍ J (x, y) ≥ h c , then they are in strong conflict; if the conflict degree is low enough, i.e., Φ ≍ J (x, y) ≤ l c , then they are not in conflict; otherwise, they are in weak conflict. The combinations of two views give the final definition of the relationship between the two agents. A pair of two agents belongs to the alliance relation if they are strongly allied and not in conflict, and a pair of agents belongs to the conflict relation if they are in strong conflict and not allied. One may also use less restrictive conditions in defining the relations if reasonable in specific situations. Based on Definition 10, we can easily define the alliance, conflict, and neutrality relations with respect to the non-neutral issues as:

By grouping the agents allied with a specific agent x, one can get the alliance set of x.

Definition 11. The alliance sets of an agent x ∈ A with respect to a subset of issues J ⊆ I and the non-neutral issues J +- x ⊆ J are respectively defined as:

By Equation ( 36), the alliance relation R = J (Φ = , Φ ≍ ) is reflexive and symmetric, but not transitive. In other words, if (x, y) ∈ R = J (Φ = , Φ ≍ ) and (y, z) ∈ R = J (Φ = , Φ ≍ ), we may not have (x, z) ∈ R = J (Φ = , Φ ≍ ). By Equation ( 40), the alliance relation

As a result, although the agents in an alliance set AS J (x) are all allied with x, they are not necessarily allied with each other, which may induce some difficulties in analyzing the alliance sets and agent relations. To solve this problem, we present the definition of maximal consistent alliance sets by adopting the ideas in the concept of maximal consistent blocks proposed by Leung [15]. Definition 12. A subset of agents M ⊆ A is a maximal consistent alliance set with respect to a subset of issues J ⊆ I if it satisfies the following conditions:

The family of maximal consistent alliance sets with respect to J is denoted as MA J .

Intuitively, a maximal consistent alliance set is a maximal set where every two agents are allied with each other with respect to a set of issues. By taking Definition 12, one can easily define the concept of a maximal consistent alliance set with respect to non-neutral issues. Accordingly, the family of maximal consistent alliance sets with respect to non-neutral issues is denoted as MA J +-.

Intuitively, the agents in an alliance set share approximately the same attitudes towards a set of issues. These shared attitudes represent the decision or description of these agents as a whole. An agent in an alliance set supports or at least does not oppose this decision. To represent such decisions, we first define the formal description of an agent x ∈ A regarding a subset of issues J ⊆ I in terms of the ratings, which is also called the decision of x on J. Recall that a subset of issues J can be trisected with respect to an agent x into three parts, namely, positive issues J +

x , negative issues J -

x , and neutral issues J 0 x (see Definition 4). The description or decision of an agent on a set of issues is formally defined as follows.

Definition 13. For a subset of issues J ⊆ I, the description or decision of an agent x ∈ A on J is defined as:

In the alliance set AS J (x) of an agent x ∈ A, all the agents are allied with x by sharing attitudes with x towards the subset of issues J. In other words, the decision of the alliance set AS J (x) can be represented by the description or decision of x:

Similarly, the decision of alliance set AS J +-(x) is the description of an agent x ∈ A with respect to the non-neutral issues J +- x :

It should be noted that an agent with a neutral rating on every issue under consideration does not have a description with respect to the non-neutral issues. Therefore, there is no valid decision of alliance set AS J +-(x) if J +- x = ∅. When it comes to a maximal consistent alliance set, there is no such “primary” agent that can represent the decision of the whole set. Nevertheless, we may follow the format in Equation ( 42) by trisecting a set of issues J ⊆ I into three subsets of positive, negative, and neutral issues. This requires an aggregation of the ratings from the agents in the maximal consistent alliance set on J. Under the framework of our study, we obtain the aggregation of ratings and the trisection of J based on the proposed alliance and conflict functions. Instead of simply taking average to aggregate the ratings, we propose another approach by introducing two imaginary agents. A positive imaginary agent x + has a positive rating +1 on all issues in I and a negative imaginary agent x -has a negative rating -1 on all issues in I. These two agents represent two extremes of positive and negative attitudes on I. Formally, the two agents x + and x - are described over I as:

Then for a given set of agents X, we compare it with x + and x -from both views of alliance and conflict. More specifically, for i ∈ I, we compute the following aggregated alliance and conflict functions regarding x + /x -and X by Definition 9 as follows:

|X| ,

|X| ;

|X| ,

x + and X

x -and X

Table 14: The cases of alliance and conflict between x + /x -and X Thus, X can be reasonably considered to have a positive rating +1 on these issues. Similarly, J - X (Φ = , Φ ≍ ) includes the issues where x + and X are strongly in conflict but not allied. Thus, X can be reasonably considered to have a negative rating -1 on these issues. Otherwise, X has a neutral rating 0. One can easily make a similar analysis about the other three trisections in Definition 14.

According to Definition 14, one can easily define the corresponding four trisections of issues with respect to a single agent x ∈ A by degenerating X to {x}. These trisections are equivalent with each other and with the corresponding trisection based on a rating function (i.e., Definition 4).

Theorem 2. With respect to a single agent x ∈ A, the trisections of a set of issues J ⊆ I with a rating function and with any pair of alliance and conflict functions in Table 13 are equivalent, that is:

where we use x to represent the singleton set {x} for simplicity.

Proof. Since 0 ≤ l p < h p ≤ 1 and 0 ≤ l n < h n ≤ 1, we have l p < 1, h p > 0, l n < 1, and h n > 0. By Definitions 14 and 8, we get the first trisection of issues with respect to x as:

Similarly, one can prove that the other three trisections are also equivalent with the trisection of J + x , J - x , and J 0 x .

Using a maximal consistent alliance set M ∈ MA J as the set X in Definition 14, a subset of issues J ⊆ I can be trisected into three pair-wise disjoint parts. Accordingly, we can formulate the decision of a maximal consistent alliance set as follows.

Definition 15. Given a maximal consistent alliance set M ∈ MA J , let J + M (Φ * , Φ • ), J - M (Φ * , Φ • ), and J 0 M (Φ * , Φ • ) denote a trisection of a set of issues J ⊆ I where * and • represent either = or ≍. Then the decision of M is represented as:

Similarly, with respect to the non-neutral issues

, one can easily get the decision of M as:

We give two examples to illustrate the concepts introduced in this section. Particularly, Example 5 uses Pawlak’s and Yao’s auxiliary functions to construct the alliance and conflict functions regarding x + and x -defined in Equation (46). Based on these functions, we continue with our discussion in Example 4 in Section 3 to illustrate the concepts of alliance sets, maximal consistent alliance sets, and the representation of their decisions, which is given in Example 6.

Example 5 (Alliance and conflict functions regarding x + and x -based on Pawlak’s and Yao’s auxiliary functions). We illustrate the alliance and conflict functions defined in Equation (46) with Pawlak’s and Yao’s auxiliary functions. We have defined the unaggregated alliance and conflict functions with both Pawlak’s and Yao’s auxiliary functions in Example 3 in Equation (32). By using either x + or x -as the first parameter, we can compute the following unaggregated functions: for i ∈ I and y ∈ A,

We use the rating r(y, i) as the conditions, which can be easily induced from the original conditions regarding Φ P i or Φ Y i in Equation (32). Then we aggregate these functions as defined in Equation (46), which leads to the following alliance and conflict functions: for X ⊆ A,

where X + i and X - i are as defined in Definition 2 (i.e., X +

) are the proportions of the agents in X with a positive rating +1 on the issue i; and Φ = i (x -, X) and Φ ≍ i (x + , X) are the proportions of those with a negative rating -1. Accordingly, based on Pawlak’s and Yao’s auxiliary functions, we may get a combination of the positive and negative views by using

Example 6 (Alliance sets, maximal consistent alliance sets, and their decisions). Using the alliance and conflict functions defined in Example 5, we continue with our discussion in Example 4 to illustrate the concepts of alliance sets, maximal consistent alliance sets, and their decisions. For simplicity, we illustrate these concepts with respect to the non-neutral issues. Let us consider a set of issues J = I = {i 1 , i 2 , i 3 , i 4 }. Firstly, we illustrate the concept of alliance sets and their decisions. We apply two thresholds h a = 1 2 and l c = 1 3 in Equation (40) to construct the alliance relation and accordingly, compute the alliance sets by Definition 11. We get the following alliance sets with respect to the non-neutral issues:

Then we can simply represent the decision of an alliance set AS J +-(x i ) as the description of x i with respect to its non-neutral issues J +- x i . For example, we have:

Secondly, we illustrate the concept of maximal consistent alliance sets. By Definition 12 and Equation (57), we compute the maximal consistent alliance sets with respect to the non-neutral issues as:

Thirdly, we illustrate the decisions of maximal consistent alliance sets. Let us consider the alliance and conflict functions induced from Pawlak’s auxiliary function, and the first case of combination in Definition 14. It should be noted that according to Equation (56), one can actually get the same results using any other combination in Definition 14. By applying a pair of thresholds

2 ) (k = 1, 2, 3, 4), we get the trisections of J regarding each maximal consistent alliance set in Equation (59) as follows:

(

Then according to Definition 15, the decisions of the maximal consistent alliance sets in Equation (59) with respect to the non-neutral issues are given as:

We use the empty set ∅ to denote a representation with no condition. This can be justified by the fact that a description can be easily transformed into an equivalent set representation, in particular, a set of issue-rating pairs with the conjunctive relation assumed.

In this section, we apply the proposed alliance and conflict functions in studying another essential topic in conflict analysis, that is, strategy. Specifically, we present a formal representation of a strategy as a logic conjunction of issuerating pairs. Based on it, we investigate the families of strategies with respect to a subset of issues and the non-neutral issues, as shown in Figure 3. For each strategy, we discuss its relationship with the agents in a given subset X ⊆ A. Particularly, an agent may support, oppose, or be neutral to the strategy, which leads to a trisection of X. These attitudes from agents in X on the strategy indicate the feasibility of a strategy in resolving the conflict. Examples are given at the end to illustrate the presented concepts.

The family of strategies with respect to a subset of issues J ⊆ I the non-neutral issues J +-⊆ J Trisecting agents X ⊆ A based on a strategy X Neutral Supporting Opposing Figure 3: A framework of the discussion in Section 5

A strategy is usually defined as a set of issues in the existing works. The issues in the strategy are implicitly associated with a positive rating, and those not included in the strategy are associated with a non-positive rating that could be either negative or neutral. To further clarify these non-positive ratings, we present to list all the three types of positive, negative, and neutral ratings in a formal representation of a strategy. Particularly, we discuss two equivalent formats: a set representation and a logic representation. A set representation uses a set of issue-rating pairs to formulate a strategy S as follows:

where J S is the set of issues involved in S and r i is a rating on the issue i ∈ J S . The issue-rating pairs in the set are assumed to have a logic conjunction relation. Accordingly, we can alternatively adopt the following logic representation:

Furthermore, according to the ratings, the issues in J S can be trisected into positive J + S , negative J - S , and neutral J 0 S sets. Thus, the strategy S can also be represented in the following format:

The family of strategies is denoted as S. For a given subset J ⊆ I, one can formulate 3 |J| strategies. Thus, for |I| = n, one can get the cardinality of S as:

where the empty strategy is also considered as valid.

Both the set and logic representations can also be recursively formulated through the concept of atomic strategies. An atomic strategy is defined as a strategy that involves only one issue. We can formulate three atomic strategies of positive, negative, and neutral ratings for every single issue. Any other strategy can be formulated through atomic strategies regarding the involved issues. We formally define strategies following this idea in terms of logic representations. One can easily formulate the equivalent set representations.

Definition 16. The logic representation of a strategy can be defined by the following cases:

(1) The empty strategy is represented as ∅.

(2) Atomic strategies: ∀i ∈ I, we can formulate three atomic strategies, (+)

A positive atomic strategy :

A negative atomic strategy : S - i = ⟨i, -1⟩; (0) A neutral atomic strategy : S 0 i = ⟨i, 0⟩.

(3) Composite strategies: ∀J ⊆ I, one can formulate a composite strategy on J as,

where * represent +, -, or 0.

According to Definition 16, we can further represent the strategy in Equation (64) as:

For example, suppose S = ⟨i 1 , +1⟩∧⟨i 2 , -1⟩∧⟨i 3 , 0⟩. Then S is a composition of three atomic strategies S + i 1 = ⟨i 1 , +1⟩, S - i 2 = ⟨i 2 , -1⟩, and S 0 i 3 = ⟨i 3 , 0⟩, and can be represented as:

By Definition 16, one can compose a set of atomic strategies on different issues into a composite strategy and decompose a given strategy into a set of atomic strategies. The decomposition enables us to study a strategy through a set of atomic strategies. We arrive at a uniform formulation of describing a strategy in Equation ( 64) and an agent in Equation (42), that is, a conjunction of issue-rating pairs. Although taking the same format, these two descriptions have different semantics. The construction of a strategy may not involve any specific agent. The uniform representation suggests that we may consider a strategy as an imaginary “agent”. Following this idea, we may explore the relationships between a strategy and an agent by using the same approaches to the relationships between agents that are discussed in Sections 3 and 4. By modifying Definition 8, we define the unaggregated alliance and conflict functions regarding a strategy and an agent as given in the following definition.

Definition 17. Given an auxiliary function Φ i regarding a single issue i ∈ I, the alliance function Φ = i : S × A → {0, 1} and the conflict function Φ ≍ i : S × A → {0, 1} are defined as:

where Φ i (S, x) is computed as if S were an agent.

One can apply the same approaches discussed in Section 3 to define the aggregated alliance and conflict functions regarding strategies and agents. In particular, we give the following aggregated functions that will be used in our following discussion:

Accordingly, we may trisect a subset of agents as given in the following definition.

Definition 18. For a subset of agents X ⊆ A and a strategy S ∈ S, given two pairs of supporting and opposing thresholds (l s , h s ) and (l o , h o ) with 0 ≤ l s < h s ≤ 1 and 0 ≤ l o < h o ≤ 1, X can be trisected into the supporting X + J S (Φ = , Φ ≍ ), opposing X - J S (Φ = , Φ ≍ ), and neutral X 0 J S (Φ = , Φ ≍ ) sets regarding S as follows:

One may also apply Definition 11 of alliance sets by considering a strategy S as an imaginary agent. Then the supporting set X + J S (Φ = , Φ ≍ ) will induce the alliance set of S with respect to J S . The alliance and conflict functions regarding a strategy and an agent in Equations ( 69) and (70) can also be computed through the functions regarding the involved atomic strategies.

Theorem 3. The alliance function Φ = J S : S× A → {0, 1} and the conflict function Φ ≍ J S : S× A → {0, 1} can be computed as: for S ∈ S and x ∈ A,

where * denotes +, -, or 0.

For instance, for the strategy S = S + i 1 ∧ S - i 2 ∧ S 0 i 3 , we have:

Our above discussion can be easily applied with respect to the non-neutral issues. For a given strategy S ∈ S, we have the set of its non-neutral issue as:

that is, the set of issues with a non-neutral rating involved in S. Then one can induce a non-neutral strategy S from S as:

which removes all the issue-rating pairs with a neutral rating from S. The family of non-neutral strategies is denoted as S. For a given subset J ⊆ I, one can formulate 2 |J| non-neutral strategies. Thus, for |I| = n, one can get the cardinality of S as:

where the empty strategy is also considered as valid.

One may analyze the relationships between strategies and agents with respect to the non-neutral issues. In particular, we apply Definition 18 with J S = J +- S = J + S ∪ J - S instead of J S to trisect a subset of agents X ⊆ A as:

where the alliance and conflict functions are computed as:

By Theorem 3, the two functions Φ = J S ( S, x) and Φ ≍ J S ( S, x) can be computed as the sum of the functions regarding the atomic non-neutral strategies involved in S.

We give two examples to illustrate the concepts introduced in this section. Particularly, Example 7 uses Pawlak’s and Yao’s auxiliary functions to construct the alliance and conflict functions regarding strategies and agents. Based on these functions, Example 8 continues with our discussion in Example 6 in Section 4 to illustrate the relationships between strategies and agents with respect to non-neutral issues.

Example 7 (Alliance and conflict functions regarding strategies and agents based on Pawlak’s and Yao’s auxiliary functions). Based on Pawlak’s and Yao’s auxiliary functions, the alliance and conflict functions for a strategy S ∈ S and an agent x ∈ A with respect to a subset of issues J ⊆ I are respectively defined as:

Similarly, the alliance and conflict functions with respect to a non-neutral strategy S ∈ S and an agent x ∈ A are defined as:

where

Furthermore, by the definition of Pawlak’s auxiliary function given in Equation (11), for an agent x ∈ A, we have the following duality:

Accordingly, for any two strategies, if the set of issues with a positive rating +1 in one strategy is the same as the set of issues with a negative rating -1 in another and vice versa, they can be considered as two dual strategies. Formally, two dual strategies can be represented as:

Then by Equations (80) and (82), we have:

which also holds for those induced from Yao’s auxiliary function. Therefore, based on Pawlak’s and Yao’s auxiliary functions, one can trisect a subset of agents X ⊆ A with respect to the non-neutral issues J S by using the values of alliance functions only, which will significantly simplify our computation in the following Example 8.

Example 8 (Relationships between strategies and agents). We continue with Example 6 to illustrate the nonneutral strategies and their relationships with agents based on the alliance and conflict functions defined in the above Example 7. Firstly, Table 15 lists the family of non-neutral strategies S = { S 0 , S 1 , S 2 , S 3 , • • • , S 80 }, where S 0 = ∅. We omit the empty strategy S 0 in Table 15 for simplicity. Secondly, by using the values computed from Equation (84), one can get a trisection of X = A with respect to each strategy in Table 15. The values of the alliance functions regarding each non-neutral strategy and each agent are shown in Table 16. We take S 11 = S 1 ∧ S 4 as an illustration of the computation:

Finally, we illustrate the relationships between strategies and agents. By taking the thresholds l s = 0, h s =

The family of nonempty non-neutral strategies S l o = 0, and h o = 1 2 , we get the trisection of X with respect to S 10 as:

where S 9 and S 10 are dual strategies. By using the thresholds l s = 0, h s = 1 4 , l o = 0, and h o = 1 4 , we get the trisection of X with respect to S 65 as:

an auxiliary function Φ i (x, y) in Equation ( 8) and its aggregation Φ J (x, y) in Equation ( 9), respectively. In the proposed model with two functions, we define a pair of alliance and conflict functions Φ = i (x, y) and Φ ≍ i (x, y) (i.e., Definition 8) by splitting the auxiliary function Φ i (x, y). Furthermore, with respect to an arbitrary subset of issues and the set of non-neutral issues, we respectively define two pairs of alliance and conflict functions in Equations ( 27) and (31) to trisect the agent pairs (i.e., Equations ( 36) and ( 40)). According to Theorem 1, with respect to a single issue, the trisections of agent pairs with either a single auxiliary function or a pair of alliance and conflict functions are equivalent.

Secondly, for semantics, we define the alliance and conflict functions to reduce the difficulty of applying a rating function to trisect agent pairs. The proposed model with two functions and the existing model with a single function serve the same purpose. They both divide agent relationships into three pair-wise disjoint parts: alliance, conflict, and neutrality relations.

We discuss a real-world application to illustrate the proposed three-way conflict analysis model with alliance and conflict functions. Specifically, we use the proposed model to help the government of Gansu province in China to address conflict problems. The same application is also used by Sun et al. in [30] as a case study.

The three-valued situation table is given in Table 18. A = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 , x 12 , x 13 , x 14 } is a set of fourteen cities in Gansu province, namely, Lanzhou, Jinchang, Baiyin, Tianshui, Jiayuguan, Wuwei, Zhangye, Pingliang, Jiuquan, Qingyang, Dingxi, Longnan, Linxia, and Gannan. I = {i 1 , i 2 , i 3 , i 4 , i 5 , i 6 , i 7 , i 8 , i 9 , i 10 , i 11 } is a set of issues involved in the development plans, namely, the construction of roads, factories, entertainment, educational institutions, the total population of residence, ecology environment, the number of senior intellectuals, the traffic capacity, mineral resources, sustainable development capacity, and water resources carrying capacity.

Table 18: A three-valued situation table [30] We consider the non-neutral issues and compute the alliance and conflict degrees between cities according to Equation (34), which is given in Table 19. Then by applying a pair of thresholds ( x 2 Φ = J +- x 13 Φ = J +-

Accordingly, Lanzhou (i.e., x 1 ), the capital city of Gansu, only allies with itself. Jinchang (i.e., x 2 ) has the biggest alliance set. We consider the non-neutral issues again to formally represent the decisions of the alliance sets. For example, the decisions of the alliance sets regarding Lanzhou and Jinchang are: topics of three-way conflict analysis. The first topic is about the relationship between agents. Particularly, we define the alliance sets and the maximal consistent alliance sets. Furthermore, we present the conjunction of issue-rating pairs as a formal representation of the decision or description of an alliance set. In the second topic, we formally define the strategies through issue-rating pairs and investigate their relationships with agents. To verify the effectiveness of the trisections with the proposed alliance and conflict functions, we compare them with the trisections in existing models based on a single evaluation function. Finally, we apply the proposed model in a real-world application to help the government of Gansu province make the development plan.

Determining the thresholds used in the trisections is a fundamental issue that needs further discussion in future work. Moreover, our approaches can be generalized with respect to a few aspects, such as weighted agents, conflict sets of agents instead of alliance sets, incomplete situation tables, and dynamic situation tables. In addition, it is worth investigating the connections and differences between three-way conflict analysis and the concept of bipolarity and shadowed sets, which may inspire exciting results in these closely related topics.

The values of Φ(r(x, i), r(y, i)) with x y