# Measuring Conflicts Using Cardinal Ranking: An Application to Decision Analytic Conflict Evaluations

Authors: Tobias Fasth; Aron Larsson; Love Ekenberg; Mats Danielson
Journal: Advances in Operations Research
 (2018)

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## Abstract

One of the core complexities involved in evaluating decision alternatives in the area of public decision-making is to deal with conflicts. The stakeholders affected by and involved in the decision often have conflicting preferences regarding the actions under consideration. For an executive authority, these differences of opinion can be problematic, during both implementation and communication, even though the decision is rational with respect to an attribute set perceived to represent social welfare. It is therefore important to involve the stakeholders in the process and to get an understanding of their preferences. Otherwise, the stakeholder disagreement can lead to costly conflicts. One way of approaching this problem is to provide means for comprehensive, yet effective stakeholder preference elicitation methods, where the stakeholders can state their preferences with respect to actions part of the current agenda of a government. In this paper we contribute two supporting methods: (i) an application of the cardinal ranking (CAR) method for preference elicitation for conflict evaluations and (ii) two conflict indices for measuring stakeholder conflicts. The application of the CAR method utilizes ado nothing alternative to differentiate between positive and negative actions. The elicited preferences can then be used as input to the two conflict indices indicating the level of conflict within a stakeholder group or between two stakeholder groups. The contributed methods are demonstrated in a real-life example carried out in the municipality of Upplands Väsby, Sweden. We show how a questionnaire can be used to elicit preferences with CAR and how the indices can be used to semantically describe the level of consensus and conflict regarding a certain attribute. As such, we show how the methods can provide decision aid in the clarification of controversies.


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## Body

## 1. Introduction

One of the core complexities involved in evaluating decision alternatives in the area of public decision-making is to deal with conflicts. The stakeholders affected by and involved in the decision often have conflicting preferences regarding the actions under consideration. For an executive authority, these differences of opinion can be problematic during both implementation and communication, even though the decision is rational with respect to an attribute set perceived to represent social welfare; see, e.g., [1–3].Of particular interest for the decision-making forum are preferential conflicts between stakeholders, since such conflicts may cause delays in the decision process due to obstructions, hassles, and/or locked negotiations [4–6]. For instance, Hansson et al. [6] describe that the development plans of Husby, a suburb to Stockholm, had been on hold for several years due to conflicts. Another example is given in Danielson et al. [5], where three infrastructure decisions in Nacka, a municipality in the Stockholm region, were delayed for several years due to conflicts between stakeholders. Therefore, to avoid stakeholder conflicts it is important for the executive authority to involve the stakeholders in the process and to get an understanding of their preferences [7, 8].This calls for a desire to become better informed with regard to potential controversies, and it has been discussed in contemporary decision analysis literature that interaction with stakeholders using web-based techniques is one feasible approach to obtain stakeholder preferences and then inform decision-makers by utilizing decision analysis techniques. For instance, French et al. [1] suggest that web-based approaches can be used to support and structure the democratic process. Hansson et al. [6] suggest that a solution to the above-mentioned problem of conflicts could be to actively interact with stakeholders early in the process by the use of social media.In this approach, an important first step is to elicit the opinions or attitudes of the citizens regarding a set of “actions”. These are potential actions that in the future may be redefined into more well-defined projects. It has been recognized that the selected methods must be easy to use for less experienced stakeholders and at the same time be sufficiently powerful to enable them to provide meaningful feedback [9]. To facilitate scalable elicitation, many approaches (including ours) promote the use of web-based surveys or questionnaires distributed to stakeholders early in the planning phase [3].In this paper we contribute with an application of the cardinal ranking (CAR) method [10], extended with a feature for conflict evaluations, and a method for measuring conflict within a stakeholder group and between two stakeholder groups. The contributions support the decision process in a public decision-making setting.Cardinal Ranking for Conflict Evaluations. To the best of our knowledge, no method exists that utilizes preferences elicited by a cardinal ranking approach in conflict evaluations. In this application of CAR [10], the respondents, in addition to cardinally ranking the elements, state the performance of the elements relative to ado nothing alternative. For example, supporting statements are like“action 1 is better than action 2, and both actions are negative relative the do nothing alternative and“action 2 is much better than action 3, action 2 is positive whereas Action 3 is negative relative the do nothing alternative”.Conflict Indices. In a group decision analysis setting, it is of interest to investigate whether a certain alternative is conflict-prone with respect to one or more attributes. The preferences elicited by the application of CAR can be used for assessing the level of conflict within or between stakeholder groups. We describe two such indices, the Within-Group Conflict Index and the Between-Group Conflict Index, both utilizing a sum of squares, used in Ward’s clustering method [11, p. 466] and extended with a positive stakeholder scaling constant. The underlying group formation is similar to the collective attractiveness and unattractiveness indices presented by [12]. The approaches differ in how the value functions are created and utilized. Bana e Costa [12] creates one value function per criterion by applying MACBETH and its semantic categories. In our approach the respondents use CAR to create their individual value functions per criterion. In turn, this enables individual value estimates of thedo nothing alternative which is not necessarily viewed as a “neutral” alternative.With this we contribute to the first important step in a participatory group decision process by elucidating both conflict-prone actions and nonconflicting actions. An action is conflict-prone when there are strong opposing preferences regarding the performance of the action, either within a stakeholder group or between two stakeholder groups. An action is a nonconflicting action when the stakeholders have similar preferences.


### 1.1. Case Setting

In this paper, we describe a case conducted in Upplands Väsby municipality slightly north of Stockholm City. In this case, we analyzed the citizens preferences regarding a set of actions that could be implemented in the future. A previous reporting of the case can be found in Chapter 7 of Ekenberg et al. [13], in which a simplified conflict measure was utilized. In the reported method, the stakeholders were divided into two groups, the con- and the pro-group. The conflict index was then measured as the difference between the arithmetic means of the groups part-worth values. The methods presented in this paper are rather based on Ward’s method [11].Typically, the actions are described quite briefly, for example, in short sentences such as “Build residential area near the lake” or “Build apartments in the town centre”. In the paper, we use the term “action” to distinguish from the more traditional term “alternatives” employed within the field of decision analysis. In our setting, an action is a tentative project proposal without an associated cost but rather a line of direction for a future project. This is somewhat similar to the tentative proposals defined as “topics” by [14].In order to understand attitudes, the questionnaire must allow for the participating stakeholders to state positive or negative preferences regarding an action’s performance relative to ado nothing action. In order to measure conflict, the statements need to be represented in a manner allowing for such a measure to be meaningful. Thus, the attitudes must be represented using a measurable value function where the questionnaire is used as a tool for eliciting preferences. Conventional preference elicitation methods for approximating such value functions are typically considered to be cognitively demanding [15]. The nature of the actions being so loosely defined renders the use of more elaborate preference elicitation techniques, since they rely on well-defined alternatives. To resolve this situation, the questionnaire we propose is inspired by the attitude surveys often employing different versions of the Likert scale.An alternative approach to reducing the cognitive burden of the decision-maker is to use preference disaggregation techniques. The techniques do this by utilizing global preferences, e.g., a ranking of a subset of alternatives, or by a pairwise comparison of alternatives, to infer value functions; see, e.g., [16–19] for details.The questionnaire, which was previously reported in [13, Ch. 7], enables the capturing of negative and positive attitudes of the actions relative to ado nothing action. Then, methods for cardinal ranking are used to interpret the responses in terms of surrogate values and attribute weights. Lastly, the questionnaire contains a section where the respondent enters demographic information. This information can then be used to analyze the preferences of the citizens, e.g., to find stakeholder groups with conflicting interests.


## 1.1. Case Setting

In this paper, we describe a case conducted in Upplands Väsby municipality slightly north of Stockholm City. In this case, we analyzed the citizens preferences regarding a set of actions that could be implemented in the future. A previous reporting of the case can be found in Chapter 7 of Ekenberg et al. [13], in which a simplified conflict measure was utilized. In the reported method, the stakeholders were divided into two groups, the con- and the pro-group. The conflict index was then measured as the difference between the arithmetic means of the groups part-worth values. The methods presented in this paper are rather based on Ward’s method [11].Typically, the actions are described quite briefly, for example, in short sentences such as “Build residential area near the lake” or “Build apartments in the town centre”. In the paper, we use the term “action” to distinguish from the more traditional term “alternatives” employed within the field of decision analysis. In our setting, an action is a tentative project proposal without an associated cost but rather a line of direction for a future project. This is somewhat similar to the tentative proposals defined as “topics” by [14].In order to understand attitudes, the questionnaire must allow for the participating stakeholders to state positive or negative preferences regarding an action’s performance relative to ado nothing action. In order to measure conflict, the statements need to be represented in a manner allowing for such a measure to be meaningful. Thus, the attitudes must be represented using a measurable value function where the questionnaire is used as a tool for eliciting preferences. Conventional preference elicitation methods for approximating such value functions are typically considered to be cognitively demanding [15]. The nature of the actions being so loosely defined renders the use of more elaborate preference elicitation techniques, since they rely on well-defined alternatives. To resolve this situation, the questionnaire we propose is inspired by the attitude surveys often employing different versions of the Likert scale.An alternative approach to reducing the cognitive burden of the decision-maker is to use preference disaggregation techniques. The techniques do this by utilizing global preferences, e.g., a ranking of a subset of alternatives, or by a pairwise comparison of alternatives, to infer value functions; see, e.g., [16–19] for details.The questionnaire, which was previously reported in [13, Ch. 7], enables the capturing of negative and positive attitudes of the actions relative to ado nothing action. Then, methods for cardinal ranking are used to interpret the responses in terms of surrogate values and attribute weights. Lastly, the questionnaire contains a section where the respondent enters demographic information. This information can then be used to analyze the preferences of the citizens, e.g., to find stakeholder groups with conflicting interests.


## 2. Modeling of Preferences

In the decision analysis field, we distinguish between two types of preference representation functions, the value function and the utility function. A value function represents preferences over certain outcomes, as opposed to the utility function which represents preferences over lotteries with uncertain outcomes, and the two representations are based upon different axiom systems. The focus in this paper is on preference representations under certainty, and in the absence of uncertain consequences (or risk), preferences over objectsx are typically represented by means of a value function v(x)↦[0,1], such that the object x is preferred to object y (commonly denoted by x≻y) by the decision-maker if and only if v(x)>v(y). If v(x)=v(y) then the decision-maker is indifferent between the two objects. The value function is measurable and is defined over an interval scale if v(w)-v(x)>v(y)-v(z) and v(w)>v(x),v(y)>v(z) entails that exchanging x for w is preferred compared to exchanging z for y; see [20] for a comprehensive treatment.In the case of multiple decision-makers or stakeholdersS1,S2,…,Sn having different individual measurable value functions v1(),v2(),…,vn(), in group decision analysis, it is of concern to aggregate the set of individual value functions using some proper preference aggregation rule providing a group value function sharing the properties of the individual value functions. For this purpose, given that all individual value functions of stakeholder k are of the form vk and that domain and range are shared among the stakeholders, it has been shown that an additive aggregation of individual measurable value functions provides a measurable group value function W(Aj) for an alternative Aj, such that W(Aj)=∑λkvk(Aj), where vk(Aj) is the value of alternative Aj for stakeholder Sk, and λk are stakeholder scaling constants subject to λi≥0 and ∑λi=1 whenever exchange independence is satisfied (if all stakeholders are indifferent between two exchanges then the whole group must also be indifferent) [21].In the multiattribute setting, considering a set of evaluation attributesG={G1,G2,…,Gm}, the value of an alternative Aj is obtained by aggregating the corresponding set of value functions v1(x1),v2(x2),…,vm(xm). The most common way of aggregation is the additive approach such that the value V(Aj) of Aj is given by V(Aj)=∑iwivij, where vij is the value of alternative Aj under attribute Gi, and wi is the weight of attribute Gi under the condition that 0≤wi≤1 and ∑iwi=1. This additive aggregation provides a measurable multiattribute value function given that the conditions of mutually preferential independence and difference independence hold; see, e.g., [22]. The weights wi then act as attribute scaling constants, scaling the value contribution to an alternative from an attribute. The scaled value qij=wi·vij is called the “part-worth” value of attribute Gi to alternative Aj. In a group setting, the group’s value of alternative Aj is then given by(1)WAj=∑kλk∑iqijkwhereqijk is the part-worth value of attribute Gi to alternative Aj for stakeholder Sk; i.e., qijk=wik·vijk. Henceforth in this paper, qijk notation for a stakeholders part-worth value will be used, and we will denote the stakeholder Sk’s value of alternative Aj under attribute Gi with vijk. In order to obtain these values, different approaches have been suggested to elicit them from decision-makers. For an overview of elicitation methods, see, e.g., [22, 23].


## 3. Rank-Based Elicitation

The procedure in rank-ordering methods is to elicit the preferences as ranks, which are then converted into cardinal surrogate weights. Two such methods, rank-sum (RS) and rank-reciprocal (RR), are described in [24], and a third method the Rank Order Centroid (ROC) is described in [25, 26].Rank-ordering methods have desirable advantages over more precise elicitation techniques, such as (i) being easier to elicit the vague preferences (less cognitively demanding) and (ii) having increased likelihood for a group to come to agreement [27, 28]. But rank ordered weight elicitation may be problematic. For example, Jia et al. [29] point out that, in a real-life setting, uncertainty may exist regarding both the magnitudes and ordering of weights, and even though information regarding the difference in importance may exist, the information is not considered in the transformation from rank orders into weights.A method taking the difference in importance into consideration is the cardinal ranking (CAR) method [10, 30]. In CAR, a prerequisite is an ordinal ranking, to which cardinal information is added; see Figure 1 for a visualization of the difference between ordinal and cardinal ranking. The cardinal information is used to denote the strength of preference between pairs of elements in the ranking. This strength of preference is interpreted as the number of steps between each pair of elements on an underlying importance scale. The notation ≻i is used for describing this, where i is the number of steps. The cardinality can be described by semantic expressions, e.g., obtained from a linguistic analysis,~0 equally important, 0 steps≻1 slightly more important, 1 step≻2 more important, 2 steps≻3 much more important, 3 stepsFigure 1
The left picture visualizes an ordinal ranking of six elements, while the right picture visualizes a cardinal ranking of the elements.This enables the decision-maker to make statements such as the following:(i)
Attribute A is equally important (~0) as attribute B(ii)
Attribute B is slightly more important (≻1) than attribute C(iii)
Attribute C is more important (≻2) than attribute D(iv)
Attribute D is much more important (≻3) than attribute ENote that similar linguistic translations are commonly used in other MCDA methods such as AHP [31] and MACBETH [32, 33].The CAR method has been demonstrated using both linear inequalities to represent cardinal ranking statements [34] and closed formulas for obtaining surrogate weights [10, 30]. More recently, rank-based methods have been suggested for probability elicitation as well, with a particular aim for use in time scarce environments [35].


## 4. Cardinal Ranking for Conflict Evaluations

In this section we introduce an application of CAR which captures negative or positive preferences with regard to an alternative’s performance relative to ado nothing alternative over a set of attributes. The method enables respondents to express the strength of preference between ordered pairs of alternatives using steps of preference intensities and at the same time express whether the alternatives are negative or positive relative to ado nothing alternative.Assume that we have a set of alternativesA={A1,A2,…,An} which are evaluated against a set of attributes G={G1,G2,…,Gm}. To capture the negative or positive preferences we need to introduce ado nothing alternative Aα to the set of alternatives. Thedo nothing alternative represents the current state; i.e., it should be considered whether the actions under consideration are better/worse than this alternative. See, e.g., Lahdelma et al. [36] for arguments supporting this technique.We conform to the CAR method’s procedure for eliciting the alternatives’ values [10, 30] but extend it with a third step, the step where thedo nothing alternative is inserted in the ranking:(1)
An ordinal number is assigned to each position on the underlying measurement scale.(2)
The underlying scale consists ofQ positions in decreasing order of importance. Alternative Ai has a position p(i) on the scale, such that 1≤p(i)≤Q, where Q≥m. The strength, or cardinality, of preference si between two adjacent alternatives Ai≻siAj, is then si=pi-pj.(3)
A do nothing alternativeAα is inserted into the ranking by providing it with a position p.(4)
The cardinal ranking is normalized to a proportional[0,1]-value scale according to the following equation: (2)viCAR=Q-piQ-1


### 4.1. Example of CAR for Conflict Evaluations

Assume that a decision-maker evaluates the performance of four alternatives,{A1,…,A4}, with regard to attribute C1.(1)
The decision-maker orders the alternatives asA1≻A2≻A3≻A4.(2)
He/she adds cardinal information to the ordinal ranking by introducing cardinality (strength of preference) steps between pairs of alternatives:
(i)
A1 is much better (≻3) than A2,(ii)
A2 is better (≻2) than A3(iii)
A3 is much better (≻3) than A4,which gives the following cardinal rankA1≻3A2≻2A3≻3A4; see Figure 2.(3)
He/she states that alternativesA1 and A2 are considered to be positive and A3 and A4 negative relative to the do nothing alternative. The do nothing alternative Aα is therefore inserted at position 0, between A2 and A3, giving the following cardinal ranking, A1≻3A2≻1Aα≻1A3≻3A4; see Figure 2.(4)
The cardinal ranking results in the following positions on the underlying scale:
(i)
A1 at position p(4),(ii)
A2 at position p(1),(iii)
Aα at position p(0),(iv)
A3 at position p(-1),(v)
A4 at position p(-4).These positions are then mapped onto a proportional[0,1]-value scale, giving the alternatives the following values, where vij denotes alternative Aj:s value under attribute Gi. As [37] points out, the do nothing alternative is not always assigned a score of 0. Note that in our approach the do nothing alternative can be assigned a value between 0 and 1 depending on its position in the ordinal ranking. In this example, Q=9 and p(4)=1, p(1)=4, p(0)=5, p(-1)=6 and p(-4)=9.
(i)
v4=1.000,(ii)
v3=0.625,(iii)
vα=0.500,(iv)
v2=0.375,(v)
v1=0.000.Figure 2
The left picture visualizes a cardinal ranking of the four alternatives. The right picture visualizes a cardinal ranking of the alternatives, with thedo nothing alternative inserted between alternatives A2 and A3.


### 4.2. Conflicts

In a group decision analysis setting, it may be of interest to investigate whether a certain alternative is conflict-prone with respect to one or more attributes. The preferences elicited in the application of CAR for conflict evaluations can be used for assessing the level of conflict within or between stakeholder groups. In the following section, we will propose two such indices, the within-group conflict index and the between-group conflict index. Based upon these indices we can define consensus properties that textually describe the level of conflict associated with an attribute with respect to a specific alternative. A similar approach was presented by Bana e Costa [12].Before presenting the indices, we stipulate the following conditions for the concept of conflict.Definition 1 (conflict).
Given a set of stakeholdersS and a set of alternatives A, conflict exists in S if there are two or more stakeholders in S with positive scaling constants and these have differing preferences towards at least one alternative Ai∈A. Formally, there must exist Sk,Sl∈S with λk,λl>0 such that vjk<vαk and vjl≥vαl.Definition1 explicitly says that if all stakeholders share preferences towards an alternative, there is no conflict. In this sense it can be argued that this definition is a very strong definition of conflict. In the case of multiple attributes, Definition 1 can be extended to attribute conflict. Needless to say, given an attribute Gi with a positive weight wi>0 we have that vijk<viαk implies qijk<qiαk and vijk≥viαk implies qijk≥qiαk since qijk=wivijk. We can now define the meaning behind attribute conflict and measurable conflict.Definition 2 (attribute conflict).
Given a set of stakeholdersS, a set of alternatives A, and a set of evaluation attributes G, conflict exists for Gi∈G if there exist Sk,Sl∈S with λk,λl>0 such that qijk<qiαk and qijl≥qiαl.


#### 4.2.1. Measurable Conflict

The definitions above do not consider different opinions on how “good” (or “bad”) two alternatives are given that they are both considered as productive (or counter-productive) to count for conflict. In other words, the definitions do not account for value differences, which lead us to measurable conflict.In a two-stakeholder setting, given that attribute conflict exists for attributeGi such that qijk<qiαk and qijl≥qiαl, we will argue that it is reasonable to base a measure of conflict upon the value differences:(3)dijk=qiαk-qijkThe intuition behind (3) is that stakeholder Sk considers Aj to be qiαk-qijk worse than thedo nothing alternative while another stakeholder Sl considers Aj to be qiαl-qijl better than thedo nothing alternative. Given two stakeholders Sk,Sl and two alternatives A1,A2 such that(4)qi1k≤qi2k≤qiαk(5)qi1l≥qi2l≥qiαl,then conflict cannot be smaller forA2 compared to A1. Further, conflict is larger for A1 compared to A2 if Sk would rather exchange A1 for Aα than exchange A2 for Aα, or if Sl would rather exchange Aα for A1 than exchange Aα for A2.However, a generalized measure of conflict should consider more than two stakeholders. In addition to measuring how far from thedo nothing alternative two stakeholders in conflict are, we need to consider disparities between sets of stakeholders. For this reason we adopt a cluster distance approach to conflict measurement by partitioning the stakeholders into two sets based upon their preferences towards the alternative.LetS={S1,S2,…,Sn} be a set of stakeholders. For each attribute Gi and alternative Aj, we partition the stakeholder set S into two partitions, the con-group Sij- and the pro-group Sij+. The members of the Sij- evaluated vijk to be less than the value of thedo nothing alternative viαk, and the stakeholders in Sij+ evaluated vijk to be greater than or equal to viαk; see the following equations:(6)Sij-=Sk∈S:vijk<viαkk=1n(7)Sij+=Sk∈S:vijk≥viαkk=1nHence, thedo nothing alternative Aα is used to separate the stakeholders into either Sij- or Sij+. The intuition behind the conflict index is to measure the distance between these two groups such that if two stakeholders Sk∈Sij-,Sl∈Sij+ in general disagree to a large extent, i.e., they both have a large differences qiαk-qijk and qiαl-qijl, then the conflict is greater than if they have smaller differences. Further, it can be argued that if the power balance between Sij- and Sij+ is more equal, then the conflict is stronger since it has been demonstrated that less powerful stakeholders are more willing to accept an alternative even though they deem it counterproductive [38]. In our case, this is represented as follows: if the difference (8)∑Sk∈Sij-λk-∑Sl∈Sij+λl is small, then the power balance is more equal entailing larger conflict.To measure these properties reflecting the distance between the groups, we utilize three sums of squares similarly to how Ward’s clustering method measures distances between clusters [11, p. 466]. We obtain the sum of squared differences between each dijk and the group’s mean distance, i.e., the differences (9)dijk-∑Sk∈SijdijkSij for the con-group (12), the pro-group (13), and the combined group (14), respectively.Definition 3 (within-group conflict index).
A within-group conflict indexdijS for stakeholder set S with two or more stakeholders, under attribute Gi and alternative Aj, is given by(10)dijS=βTijS-CijS+PijSwhere (11)β=1∑Sk∈Sλk2(12)CijS=∑Sk∈Sij-λk2dijk-∑Sk∈Sij-dijkSij-2(13)PijS=∑Sk∈Sij+λk2dijk-∑Sk∈Sij+dijkSij+2(14)TijS=∑Sk∈Sijλk2dijk-∑Sk∈SijdijkSij2In (10), β is used to normalize the value onto [0,1]. It can be noted that 0≤dijS≤1, where dijS=1 occurs when all stakeholders in the con-group rank such that qiα=1 and qij=0, and all stakeholders in the pro-group rank such that qiα=0 and qij=1 and the power balance is equal. Further, dijS=0 occurs if either the con-group or the pro-group is empty.


### 4.3. Consensus Properties

The semantic meaning of the conflict indices can then be explained by dividing the conflict range[0,1] into a number of subintervals each describing a certain level of consensus. We define the points a and b as the interval’s lower and upper bound and stipulate n interval points {x1,x2,…,xn-1,xn}. These points are to divide the scale into the n intervals such as a,x1,[x1,x2],…,[xn-1,b]. Each interval is then associated with a consensus property label describing the level of consensus, e.g.,consensus aligned attribute,potentially controversial attribute, andcontroversial attribute, with regard to a specific action.


### 4.4. Example

Assume that ten stakeholdersS={S1,…,S10} evaluate the performance of five actions {A1,…,A5}, with regard to attribute G1 using CAR for conflict evaluations. The actions’ positions on the underlying measurement are presented in Table 1 and the normalized values in Table 2. Attribute G1 is for illustrative purposes given the weight of 1.Table 1
The positions on the underlying measurement scale produced by the cardinal rankings.
Stakeholder Position S k p ( A 1 ) p ( A 2 ) p ( A 3 ) p ( A 4 ) p ( A 5 ) p ( A α ) S 1 8 6 8 1 1 8 S 2 8 6 8 1 1 8 S 3 8 7 8 2 1 8 S 4 8 7 8 3 1 8 S 5 8 7 8 3 1 8 S 6 1 2 1 5 8 1 S 7 1 2 1 5 8 1 S 8 1 2 1 6 8 1 S 9 1 3 1 7 8 1 S 10 1 3 2 7 8 1Table 2
The stakeholders’ cardinal rankings normalized to a proportional[0,1]-value scale.
Stakeholder Value Weight S k v 1,1 k v 1,2 k v 1,3 k v 1,4 k v 1,5 k v 1 , α k w 1 S 1 0.00 0.29 0.00 1.00 1.00 0.00 1.00 S 2 0.00 0.29 0.00 1.00 1.00 0.00 1.00 S 3 0.00 0.14 0.00 0.86 1.00 0.00 1.00 S 4 0.00 0.14 0.00 0.71 1.00 0.00 1.00 S 5 0.00 0.14 0.00 0.71 1.00 0.00 1.00 S 6 1.00 0.86 1.00 0.43 0.00 1.00 1.00 S 7 1.00 0.86 1.00 0.43 0.00 1.00 1.00 S 8 1.00 0.86 1.00 0.29 0.00 1.00 1.00 S 9 1.00 0.71 1.00 0.14 0.00 1.00 1.00 S 10 1.00 0.71 0.86 0.14 0.00 1.00 1.00Properties ofG1 for each action can now be defined. First, we stipulate classes of controversy based upon the magnitude of conflict. Assume four such classes which range from “consensus aligned”, i.e., low conflict, to “very controversial” where the conflict is high; see Table 3. Note that the subranges and the semantical labels are defined by the authors for illustrative purposes. In a real setting the ranges and labels should be defined by the decision-maker.Table 3
The table shows four conflict index intervals, each associated with a consensus property label which semantically describes the level of consensus.
Conflict index interval Consensus property label [ 0.00 ,  0.10 ) Consensus aligned [ 0.10 ,  0.20 ) Potentially controversial [ 0.20 ,  0.50 ) Controversial [ 0.50 ,  1.00 ] Very controversialTo analyze conflict within the stakeholder group, for each action we form a con-group (6) and a pro-group (7) and subsequentially calculate the squared distance within the con-group (12) and the pro-group (13), the squared distance between the con- and pro-group (14), and the within-group conflict index dijS (10).The results are shown in Table4. Action A1 has d1,1S=0; i.e., there exists no conflict between the stakeholders since all stakeholders are members of the pro-group and no stakeholders are members of the con-group. As seen in Table 2, all stakeholders placed A1 on the same position as Aα in the cardinal ranking. However, stakeholders S1,…,S5 stated that all other actions were better than A1, and stakeholders S6,…,S10 stated that all other actions were worse than action A1. Thus, when the ranking was normalized to proportional [0,1]-value scale, the value of action A1 for stakeholders S1,…,S5 was v1,1k=0, and for stakeholders S6,…,S10 the value was v1,1k=1. Action A2 has d1,2S=0.2, since both the con-group and the pro-group estimated the value of action to lie close to Aα. Action A3 has an even lower conflict index, d1,3S=0.0429, since all stakeholders except one are members of the pro-group. The pro-group members estimated the value of A3 to be equal to the value of Aα, and the member of the con-group estimated the value of A3 to be slightly less than the value of Aα. Action A4 has d1,4S=0.7857 since, as seen in Table 2, the stakeholders of both groups estimated it to lie closer towards the most preferred or least preferred alternative for both groups, action A5. Lastly, action A5 has d1,5S=1, since all members of the con- and pro-group estimate it to be the worst and, respectively, best action. See the classification of the actions in Table 5.Table 4
The distance within the con- and pro-group, the distance between the con- and pro-group, and the conflict index.
A 1 A 2 A 3 A 4 A 5 C 1 j S 0.0000 0.0002 0.0000 0.0008 0.0000 P 1 j S 0.0000 0.0002 0.0000 0.0008 0.0000 T 1 j S 0.0000 0.0045 0.0002 0.0634 0.1000 d 1 j S 0.0000 0.2000 0.0429 0.7857 1.0000Table 5
The classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned ×  × Potentially controversial  × Controversial Very controversial    × ×Note that the conflict index measures the disagreement between the con-group and the pro-group. If either the con-group or the pro-group is empty,CjiS (or PijS) cancels TijS out.


### 4.5. Between-Group Conflict Index

Of interest for the application case was first to measure the conflict within one group of stakeholders, i.e., the within-group conflict index. Second, it was of interest to measure the conflict between two stakeholder groups, e.g., people living in different parts of the town or belonging to different generations.The between-group conflict index measures the conflict between two stakeholder groups. The intuition behind the between-group conflict index is that if the two groups disagree to the same extent on the same matters, there is low conflict between the groups. The between-group conflict should thus reveal to what extent two groups disagree differently from each other. LetD and E be two subsets of S. For each alternative Aj and attribute Gi, we partition D and E into two partitions: the con-group SijD- and the pro-group SijD+ for D, and the con-group SijE- and the pro-group SijE+ for E. The members of SijD- and SijE- estimated the part-worth value qijk to be less than the part-worth value of thedo nothing alternative qiαk, and the members of SijD+ and SijE+ estimated it to be greater than or equal to qiαj; see the following equation:(15)SijD-=Sk∈D:qijk<qiαkj=1nSijD+=Sk∈D:qijk≥qiαkj=1nSijE-=Sk∈E:qijk<qiαkj=1nSijE+=Sk∈E:qijk≥qiαkj=1nThe squared distances within the con-groupCijD,CijE,CijD,E (12), and the pro-group PijD,PijE,PijD,E (13), and the squared distance between the con- and pro-group TijD,TijE,TijD,E (14) for groups D and E are then obtained according to the given equations. The conflict with respect to alternative Aj under attribute Gi between the two groups D and E can now be represented by the between-group conflict index dijD,E.Definition 4 (between-group conflict index).
(16) d i j D , E = β T i j D , E - T i j D + T i j E - ( C D , E - C i j D + C i j E + P i j D , E - P i j D + P i j E  w h e r e   β = 1 ∑ S k ∈ S λ k 2The intuition behind the between-group conflict index is that conflict increases when two stakeholder groups’ con- and pro-groups have greater value differences between them. This can be noted by observing that0≤dijD,E≤1, where dijD,E=1 when the two stakeholders groups have opposing preferences; for example, CijD=0, CijE=0, CijD,E=0, PijD=0, PijE=0, PijD,E=0, TijD=0, TijE=0, TijD,E=1; i.e., (1-(0+0))-((0-(0+0))+(0-(0+0)))=1.


### 4.6. Example

Using the preference data in Table2, we divide the group of stakeholders into two sets, D={S1,S2,S3,S4,S5} and E={S6,S7,S8,S9,S10}, based on some demographic variable, e.g., age or sex. These sets are further divided into the con-group sets SijD- and SijE- and the pro-group sets SijD+ and SikE+. The conflict between the stakeholder groups is then given by the conflict index dijD,E. The results of the calculations are presented in Table 6.Table 6
The con- and pro-indices for both groups and the conflict between the groups.
A 1 A 2 A 3 A 4 A 5 C i 1 D 0.0000 0.0000 0.0000 0.0000 0.0000 P i 1 D 0.0000 0.0002 0.0000 0.0008 0.0000 T i 1 D 0.0000 0.0002 0.0000 0.0008 0.0000 C i 1 E 0.0000 0.0002 0.0000 0.0008 0.0000 P i 1 E 0.0000 0.0000 0.0000 0.0000 0.0000 T i 1 E 0.0000 0.0002 0.0002 0.0008 0.0000 C i 1 D , E 0.0000 0.0002 0.0000 0.0008 0.0000 P i 1 D , E 0.0000 0.0002 0.0000 0.0008 0.0000 T i 1 D , E 0.0000 0.0045 0.0002 0.0634 0.1000 d i 1 D , E 0.0000 0.2000 0.0143 0.7857 1.0000As in the within-group conflict index, the between-group conflict index range is[0,1]. In this example we use the consensus properties defined in the within-group example in Section 4.4; see the conflict index intervals and the associated consensus properties in Table 3.As seen in Table6, the result for action A1 is d1,1D,E=0, since there is no conflict between the groups. Table 2 shows that all stakeholders of both subsets D (S1,…,S5) and E (S6,…,S10) are members of the pro-group and that they all placed A1 on the same position as Aα in the cardinal ranking; i.e., no con-groups exist. The stakeholders in D expressed that all other actions were better than A1, and stakeholders in E expressed that all other actions were worse than action A1. Thus, when the ranking was normalized to proportional [0,1]-value scale, the value for A1 was v1,1k=0 for S1,…,S5, and v1,1k=1 for S6,…,S10. Action A2 has d1,2D,E=0.2, since the two stakeholder groups have similar preferences. Both groups estimated the value of the action to lie close to Aα. Action A3 has d1,3D,E=0.0143, note that this is because only stakeholder S10 estimates that the action is negative, and all other stakeholders estimate it to be equal to Aα. Action A4 has d1,4D,E=0.7857, since the two stakeholder groups have opposing preferences, as seen in Table 2. Action A5 has d1,5D,E=1, since all members of D estimate that the action is the best of all actions, and all members of E estimate that the action is the worst of all actions. The results of the classification of the actions are presented in Table 7.Table 7
The classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned ×  × Potentially controversial  × Controversial Very Controversial    × ×


## 4.1. Example of CAR for Conflict Evaluations

Assume that a decision-maker evaluates the performance of four alternatives,{A1,…,A4}, with regard to attribute C1.(1)
The decision-maker orders the alternatives asA1≻A2≻A3≻A4.(2)
He/she adds cardinal information to the ordinal ranking by introducing cardinality (strength of preference) steps between pairs of alternatives:
(i)
A1 is much better (≻3) than A2,(ii)
A2 is better (≻2) than A3(iii)
A3 is much better (≻3) than A4,which gives the following cardinal rankA1≻3A2≻2A3≻3A4; see Figure 2.(3)
He/she states that alternativesA1 and A2 are considered to be positive and A3 and A4 negative relative to the do nothing alternative. The do nothing alternative Aα is therefore inserted at position 0, between A2 and A3, giving the following cardinal ranking, A1≻3A2≻1Aα≻1A3≻3A4; see Figure 2.(4)
The cardinal ranking results in the following positions on the underlying scale:
(i)
A1 at position p(4),(ii)
A2 at position p(1),(iii)
Aα at position p(0),(iv)
A3 at position p(-1),(v)
A4 at position p(-4).These positions are then mapped onto a proportional[0,1]-value scale, giving the alternatives the following values, where vij denotes alternative Aj:s value under attribute Gi. As [37] points out, the do nothing alternative is not always assigned a score of 0. Note that in our approach the do nothing alternative can be assigned a value between 0 and 1 depending on its position in the ordinal ranking. In this example, Q=9 and p(4)=1, p(1)=4, p(0)=5, p(-1)=6 and p(-4)=9.
(i)
v4=1.000,(ii)
v3=0.625,(iii)
vα=0.500,(iv)
v2=0.375,(v)
v1=0.000.Figure 2
The left picture visualizes a cardinal ranking of the four alternatives. The right picture visualizes a cardinal ranking of the alternatives, with thedo nothing alternative inserted between alternatives A2 and A3.


## 4.2. Conflicts

In a group decision analysis setting, it may be of interest to investigate whether a certain alternative is conflict-prone with respect to one or more attributes. The preferences elicited in the application of CAR for conflict evaluations can be used for assessing the level of conflict within or between stakeholder groups. In the following section, we will propose two such indices, the within-group conflict index and the between-group conflict index. Based upon these indices we can define consensus properties that textually describe the level of conflict associated with an attribute with respect to a specific alternative. A similar approach was presented by Bana e Costa [12].Before presenting the indices, we stipulate the following conditions for the concept of conflict.Definition 1 (conflict).
Given a set of stakeholdersS and a set of alternatives A, conflict exists in S if there are two or more stakeholders in S with positive scaling constants and these have differing preferences towards at least one alternative Ai∈A. Formally, there must exist Sk,Sl∈S with λk,λl>0 such that vjk<vαk and vjl≥vαl.Definition1 explicitly says that if all stakeholders share preferences towards an alternative, there is no conflict. In this sense it can be argued that this definition is a very strong definition of conflict. In the case of multiple attributes, Definition 1 can be extended to attribute conflict. Needless to say, given an attribute Gi with a positive weight wi>0 we have that vijk<viαk implies qijk<qiαk and vijk≥viαk implies qijk≥qiαk since qijk=wivijk. We can now define the meaning behind attribute conflict and measurable conflict.Definition 2 (attribute conflict).
Given a set of stakeholdersS, a set of alternatives A, and a set of evaluation attributes G, conflict exists for Gi∈G if there exist Sk,Sl∈S with λk,λl>0 such that qijk<qiαk and qijl≥qiαl.


### 4.2.1. Measurable Conflict

The definitions above do not consider different opinions on how “good” (or “bad”) two alternatives are given that they are both considered as productive (or counter-productive) to count for conflict. In other words, the definitions do not account for value differences, which lead us to measurable conflict.In a two-stakeholder setting, given that attribute conflict exists for attributeGi such that qijk<qiαk and qijl≥qiαl, we will argue that it is reasonable to base a measure of conflict upon the value differences:(3)dijk=qiαk-qijkThe intuition behind (3) is that stakeholder Sk considers Aj to be qiαk-qijk worse than thedo nothing alternative while another stakeholder Sl considers Aj to be qiαl-qijl better than thedo nothing alternative. Given two stakeholders Sk,Sl and two alternatives A1,A2 such that(4)qi1k≤qi2k≤qiαk(5)qi1l≥qi2l≥qiαl,then conflict cannot be smaller forA2 compared to A1. Further, conflict is larger for A1 compared to A2 if Sk would rather exchange A1 for Aα than exchange A2 for Aα, or if Sl would rather exchange Aα for A1 than exchange Aα for A2.However, a generalized measure of conflict should consider more than two stakeholders. In addition to measuring how far from thedo nothing alternative two stakeholders in conflict are, we need to consider disparities between sets of stakeholders. For this reason we adopt a cluster distance approach to conflict measurement by partitioning the stakeholders into two sets based upon their preferences towards the alternative.LetS={S1,S2,…,Sn} be a set of stakeholders. For each attribute Gi and alternative Aj, we partition the stakeholder set S into two partitions, the con-group Sij- and the pro-group Sij+. The members of the Sij- evaluated vijk to be less than the value of thedo nothing alternative viαk, and the stakeholders in Sij+ evaluated vijk to be greater than or equal to viαk; see the following equations:(6)Sij-=Sk∈S:vijk<viαkk=1n(7)Sij+=Sk∈S:vijk≥viαkk=1nHence, thedo nothing alternative Aα is used to separate the stakeholders into either Sij- or Sij+. The intuition behind the conflict index is to measure the distance between these two groups such that if two stakeholders Sk∈Sij-,Sl∈Sij+ in general disagree to a large extent, i.e., they both have a large differences qiαk-qijk and qiαl-qijl, then the conflict is greater than if they have smaller differences. Further, it can be argued that if the power balance between Sij- and Sij+ is more equal, then the conflict is stronger since it has been demonstrated that less powerful stakeholders are more willing to accept an alternative even though they deem it counterproductive [38]. In our case, this is represented as follows: if the difference (8)∑Sk∈Sij-λk-∑Sl∈Sij+λl is small, then the power balance is more equal entailing larger conflict.To measure these properties reflecting the distance between the groups, we utilize three sums of squares similarly to how Ward’s clustering method measures distances between clusters [11, p. 466]. We obtain the sum of squared differences between each dijk and the group’s mean distance, i.e., the differences (9)dijk-∑Sk∈SijdijkSij for the con-group (12), the pro-group (13), and the combined group (14), respectively.Definition 3 (within-group conflict index).
A within-group conflict indexdijS for stakeholder set S with two or more stakeholders, under attribute Gi and alternative Aj, is given by(10)dijS=βTijS-CijS+PijSwhere (11)β=1∑Sk∈Sλk2(12)CijS=∑Sk∈Sij-λk2dijk-∑Sk∈Sij-dijkSij-2(13)PijS=∑Sk∈Sij+λk2dijk-∑Sk∈Sij+dijkSij+2(14)TijS=∑Sk∈Sijλk2dijk-∑Sk∈SijdijkSij2In (10), β is used to normalize the value onto [0,1]. It can be noted that 0≤dijS≤1, where dijS=1 occurs when all stakeholders in the con-group rank such that qiα=1 and qij=0, and all stakeholders in the pro-group rank such that qiα=0 and qij=1 and the power balance is equal. Further, dijS=0 occurs if either the con-group or the pro-group is empty.


## 4.2.1. Measurable Conflict

The definitions above do not consider different opinions on how “good” (or “bad”) two alternatives are given that they are both considered as productive (or counter-productive) to count for conflict. In other words, the definitions do not account for value differences, which lead us to measurable conflict.In a two-stakeholder setting, given that attribute conflict exists for attributeGi such that qijk<qiαk and qijl≥qiαl, we will argue that it is reasonable to base a measure of conflict upon the value differences:(3)dijk=qiαk-qijkThe intuition behind (3) is that stakeholder Sk considers Aj to be qiαk-qijk worse than thedo nothing alternative while another stakeholder Sl considers Aj to be qiαl-qijl better than thedo nothing alternative. Given two stakeholders Sk,Sl and two alternatives A1,A2 such that(4)qi1k≤qi2k≤qiαk(5)qi1l≥qi2l≥qiαl,then conflict cannot be smaller forA2 compared to A1. Further, conflict is larger for A1 compared to A2 if Sk would rather exchange A1 for Aα than exchange A2 for Aα, or if Sl would rather exchange Aα for A1 than exchange Aα for A2.However, a generalized measure of conflict should consider more than two stakeholders. In addition to measuring how far from thedo nothing alternative two stakeholders in conflict are, we need to consider disparities between sets of stakeholders. For this reason we adopt a cluster distance approach to conflict measurement by partitioning the stakeholders into two sets based upon their preferences towards the alternative.LetS={S1,S2,…,Sn} be a set of stakeholders. For each attribute Gi and alternative Aj, we partition the stakeholder set S into two partitions, the con-group Sij- and the pro-group Sij+. The members of the Sij- evaluated vijk to be less than the value of thedo nothing alternative viαk, and the stakeholders in Sij+ evaluated vijk to be greater than or equal to viαk; see the following equations:(6)Sij-=Sk∈S:vijk<viαkk=1n(7)Sij+=Sk∈S:vijk≥viαkk=1nHence, thedo nothing alternative Aα is used to separate the stakeholders into either Sij- or Sij+. The intuition behind the conflict index is to measure the distance between these two groups such that if two stakeholders Sk∈Sij-,Sl∈Sij+ in general disagree to a large extent, i.e., they both have a large differences qiαk-qijk and qiαl-qijl, then the conflict is greater than if they have smaller differences. Further, it can be argued that if the power balance between Sij- and Sij+ is more equal, then the conflict is stronger since it has been demonstrated that less powerful stakeholders are more willing to accept an alternative even though they deem it counterproductive [38]. In our case, this is represented as follows: if the difference (8)∑Sk∈Sij-λk-∑Sl∈Sij+λl is small, then the power balance is more equal entailing larger conflict.To measure these properties reflecting the distance between the groups, we utilize three sums of squares similarly to how Ward’s clustering method measures distances between clusters [11, p. 466]. We obtain the sum of squared differences between each dijk and the group’s mean distance, i.e., the differences (9)dijk-∑Sk∈SijdijkSij for the con-group (12), the pro-group (13), and the combined group (14), respectively.Definition 3 (within-group conflict index).
A within-group conflict indexdijS for stakeholder set S with two or more stakeholders, under attribute Gi and alternative Aj, is given by(10)dijS=βTijS-CijS+PijSwhere (11)β=1∑Sk∈Sλk2(12)CijS=∑Sk∈Sij-λk2dijk-∑Sk∈Sij-dijkSij-2(13)PijS=∑Sk∈Sij+λk2dijk-∑Sk∈Sij+dijkSij+2(14)TijS=∑Sk∈Sijλk2dijk-∑Sk∈SijdijkSij2In (10), β is used to normalize the value onto [0,1]. It can be noted that 0≤dijS≤1, where dijS=1 occurs when all stakeholders in the con-group rank such that qiα=1 and qij=0, and all stakeholders in the pro-group rank such that qiα=0 and qij=1 and the power balance is equal. Further, dijS=0 occurs if either the con-group or the pro-group is empty.


## 4.3. Consensus Properties

The semantic meaning of the conflict indices can then be explained by dividing the conflict range[0,1] into a number of subintervals each describing a certain level of consensus. We define the points a and b as the interval’s lower and upper bound and stipulate n interval points {x1,x2,…,xn-1,xn}. These points are to divide the scale into the n intervals such as a,x1,[x1,x2],…,[xn-1,b]. Each interval is then associated with a consensus property label describing the level of consensus, e.g.,consensus aligned attribute,potentially controversial attribute, andcontroversial attribute, with regard to a specific action.


## 4.4. Example

Assume that ten stakeholdersS={S1,…,S10} evaluate the performance of five actions {A1,…,A5}, with regard to attribute G1 using CAR for conflict evaluations. The actions’ positions on the underlying measurement are presented in Table 1 and the normalized values in Table 2. Attribute G1 is for illustrative purposes given the weight of 1.Table 1
The positions on the underlying measurement scale produced by the cardinal rankings.
Stakeholder Position S k p ( A 1 ) p ( A 2 ) p ( A 3 ) p ( A 4 ) p ( A 5 ) p ( A α ) S 1 8 6 8 1 1 8 S 2 8 6 8 1 1 8 S 3 8 7 8 2 1 8 S 4 8 7 8 3 1 8 S 5 8 7 8 3 1 8 S 6 1 2 1 5 8 1 S 7 1 2 1 5 8 1 S 8 1 2 1 6 8 1 S 9 1 3 1 7 8 1 S 10 1 3 2 7 8 1Table 2
The stakeholders’ cardinal rankings normalized to a proportional[0,1]-value scale.
Stakeholder Value Weight S k v 1,1 k v 1,2 k v 1,3 k v 1,4 k v 1,5 k v 1 , α k w 1 S 1 0.00 0.29 0.00 1.00 1.00 0.00 1.00 S 2 0.00 0.29 0.00 1.00 1.00 0.00 1.00 S 3 0.00 0.14 0.00 0.86 1.00 0.00 1.00 S 4 0.00 0.14 0.00 0.71 1.00 0.00 1.00 S 5 0.00 0.14 0.00 0.71 1.00 0.00 1.00 S 6 1.00 0.86 1.00 0.43 0.00 1.00 1.00 S 7 1.00 0.86 1.00 0.43 0.00 1.00 1.00 S 8 1.00 0.86 1.00 0.29 0.00 1.00 1.00 S 9 1.00 0.71 1.00 0.14 0.00 1.00 1.00 S 10 1.00 0.71 0.86 0.14 0.00 1.00 1.00Properties ofG1 for each action can now be defined. First, we stipulate classes of controversy based upon the magnitude of conflict. Assume four such classes which range from “consensus aligned”, i.e., low conflict, to “very controversial” where the conflict is high; see Table 3. Note that the subranges and the semantical labels are defined by the authors for illustrative purposes. In a real setting the ranges and labels should be defined by the decision-maker.Table 3
The table shows four conflict index intervals, each associated with a consensus property label which semantically describes the level of consensus.
Conflict index interval Consensus property label [ 0.00 ,  0.10 ) Consensus aligned [ 0.10 ,  0.20 ) Potentially controversial [ 0.20 ,  0.50 ) Controversial [ 0.50 ,  1.00 ] Very controversialTo analyze conflict within the stakeholder group, for each action we form a con-group (6) and a pro-group (7) and subsequentially calculate the squared distance within the con-group (12) and the pro-group (13), the squared distance between the con- and pro-group (14), and the within-group conflict index dijS (10).The results are shown in Table4. Action A1 has d1,1S=0; i.e., there exists no conflict between the stakeholders since all stakeholders are members of the pro-group and no stakeholders are members of the con-group. As seen in Table 2, all stakeholders placed A1 on the same position as Aα in the cardinal ranking. However, stakeholders S1,…,S5 stated that all other actions were better than A1, and stakeholders S6,…,S10 stated that all other actions were worse than action A1. Thus, when the ranking was normalized to proportional [0,1]-value scale, the value of action A1 for stakeholders S1,…,S5 was v1,1k=0, and for stakeholders S6,…,S10 the value was v1,1k=1. Action A2 has d1,2S=0.2, since both the con-group and the pro-group estimated the value of action to lie close to Aα. Action A3 has an even lower conflict index, d1,3S=0.0429, since all stakeholders except one are members of the pro-group. The pro-group members estimated the value of A3 to be equal to the value of Aα, and the member of the con-group estimated the value of A3 to be slightly less than the value of Aα. Action A4 has d1,4S=0.7857 since, as seen in Table 2, the stakeholders of both groups estimated it to lie closer towards the most preferred or least preferred alternative for both groups, action A5. Lastly, action A5 has d1,5S=1, since all members of the con- and pro-group estimate it to be the worst and, respectively, best action. See the classification of the actions in Table 5.Table 4
The distance within the con- and pro-group, the distance between the con- and pro-group, and the conflict index.
A 1 A 2 A 3 A 4 A 5 C 1 j S 0.0000 0.0002 0.0000 0.0008 0.0000 P 1 j S 0.0000 0.0002 0.0000 0.0008 0.0000 T 1 j S 0.0000 0.0045 0.0002 0.0634 0.1000 d 1 j S 0.0000 0.2000 0.0429 0.7857 1.0000Table 5
The classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned ×  × Potentially controversial  × Controversial Very controversial    × ×Note that the conflict index measures the disagreement between the con-group and the pro-group. If either the con-group or the pro-group is empty,CjiS (or PijS) cancels TijS out.


## 4.5. Between-Group Conflict Index

Of interest for the application case was first to measure the conflict within one group of stakeholders, i.e., the within-group conflict index. Second, it was of interest to measure the conflict between two stakeholder groups, e.g., people living in different parts of the town or belonging to different generations.The between-group conflict index measures the conflict between two stakeholder groups. The intuition behind the between-group conflict index is that if the two groups disagree to the same extent on the same matters, there is low conflict between the groups. The between-group conflict should thus reveal to what extent two groups disagree differently from each other. LetD and E be two subsets of S. For each alternative Aj and attribute Gi, we partition D and E into two partitions: the con-group SijD- and the pro-group SijD+ for D, and the con-group SijE- and the pro-group SijE+ for E. The members of SijD- and SijE- estimated the part-worth value qijk to be less than the part-worth value of thedo nothing alternative qiαk, and the members of SijD+ and SijE+ estimated it to be greater than or equal to qiαj; see the following equation:(15)SijD-=Sk∈D:qijk<qiαkj=1nSijD+=Sk∈D:qijk≥qiαkj=1nSijE-=Sk∈E:qijk<qiαkj=1nSijE+=Sk∈E:qijk≥qiαkj=1nThe squared distances within the con-groupCijD,CijE,CijD,E (12), and the pro-group PijD,PijE,PijD,E (13), and the squared distance between the con- and pro-group TijD,TijE,TijD,E (14) for groups D and E are then obtained according to the given equations. The conflict with respect to alternative Aj under attribute Gi between the two groups D and E can now be represented by the between-group conflict index dijD,E.Definition 4 (between-group conflict index).
(16) d i j D , E = β T i j D , E - T i j D + T i j E - ( C D , E - C i j D + C i j E + P i j D , E - P i j D + P i j E  w h e r e   β = 1 ∑ S k ∈ S λ k 2The intuition behind the between-group conflict index is that conflict increases when two stakeholder groups’ con- and pro-groups have greater value differences between them. This can be noted by observing that0≤dijD,E≤1, where dijD,E=1 when the two stakeholders groups have opposing preferences; for example, CijD=0, CijE=0, CijD,E=0, PijD=0, PijE=0, PijD,E=0, TijD=0, TijE=0, TijD,E=1; i.e., (1-(0+0))-((0-(0+0))+(0-(0+0)))=1.


## 4.6. Example

Using the preference data in Table2, we divide the group of stakeholders into two sets, D={S1,S2,S3,S4,S5} and E={S6,S7,S8,S9,S10}, based on some demographic variable, e.g., age or sex. These sets are further divided into the con-group sets SijD- and SijE- and the pro-group sets SijD+ and SikE+. The conflict between the stakeholder groups is then given by the conflict index dijD,E. The results of the calculations are presented in Table 6.Table 6
The con- and pro-indices for both groups and the conflict between the groups.
A 1 A 2 A 3 A 4 A 5 C i 1 D 0.0000 0.0000 0.0000 0.0000 0.0000 P i 1 D 0.0000 0.0002 0.0000 0.0008 0.0000 T i 1 D 0.0000 0.0002 0.0000 0.0008 0.0000 C i 1 E 0.0000 0.0002 0.0000 0.0008 0.0000 P i 1 E 0.0000 0.0000 0.0000 0.0000 0.0000 T i 1 E 0.0000 0.0002 0.0002 0.0008 0.0000 C i 1 D , E 0.0000 0.0002 0.0000 0.0008 0.0000 P i 1 D , E 0.0000 0.0002 0.0000 0.0008 0.0000 T i 1 D , E 0.0000 0.0045 0.0002 0.0634 0.1000 d i 1 D , E 0.0000 0.2000 0.0143 0.7857 1.0000As in the within-group conflict index, the between-group conflict index range is[0,1]. In this example we use the consensus properties defined in the within-group example in Section 4.4; see the conflict index intervals and the associated consensus properties in Table 3.As seen in Table6, the result for action A1 is d1,1D,E=0, since there is no conflict between the groups. Table 2 shows that all stakeholders of both subsets D (S1,…,S5) and E (S6,…,S10) are members of the pro-group and that they all placed A1 on the same position as Aα in the cardinal ranking; i.e., no con-groups exist. The stakeholders in D expressed that all other actions were better than A1, and stakeholders in E expressed that all other actions were worse than action A1. Thus, when the ranking was normalized to proportional [0,1]-value scale, the value for A1 was v1,1k=0 for S1,…,S5, and v1,1k=1 for S6,…,S10. Action A2 has d1,2D,E=0.2, since the two stakeholder groups have similar preferences. Both groups estimated the value of the action to lie close to Aα. Action A3 has d1,3D,E=0.0143, note that this is because only stakeholder S10 estimates that the action is negative, and all other stakeholders estimate it to be equal to Aα. Action A4 has d1,4D,E=0.7857, since the two stakeholder groups have opposing preferences, as seen in Table 2. Action A5 has d1,5D,E=1, since all members of D estimate that the action is the best of all actions, and all members of E estimate that the action is the worst of all actions. The results of the classification of the actions are presented in Table 7.Table 7
The classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned ×  × Potentially controversial  × Controversial Very Controversial    × ×


## 5. Application of CAR for Conflict Evaluations at Upplands Väsby Municipality

In a current research project, researchers from Stockholm University cooperate with the Royal Institute of Technology in developing models for public planning and decision processes in Swedish municipalities. We investigate how to increase expressiveness of the media used for communication between the parties in these processes. The focus here lies on developing tools that enrich this kind of communication. As a part of the project we conduct three case studies, of which one is in cooperation with civil servants and politicians at Upplands Väsby (UV), a municipality in the Stockholm region.


### 5.1. A Case Study

As reported in [13, Ch. 7], the politicians and civil servants wanted to get in-depth information regarding the citizens preferences regarding actions, which possibly could be implemented in the future. It was especially of interest to investigate if there were any actions that potentially could lead to citizen conflicts. To facilitate this, it was decided to use a web-based questionnaire (the Appendix) as a front-end for preference elicitation. The content of the questionnaire was developed together with civil servants at UV. The questionnaire consisted of four parts:Part I consisted of ten questions regarding different “focus areas” (or criteria in an MCDA setting). Under each focus area, the respondents used an implementation of CAR for conflict evaluations to rank five actions; see Figure 3.Figure 3
Question 1 from Part 1 of the questionnaire translated to English. A handle on the slider represents one action. Actions to the left of “Neither good nor bad” are regarded to be worse, and actions to the right are considered to be better than thedo nothing action. The preference intensity is represented by a colored gradient on the slider (red to the left of Neither good nor bad, and green to the right). A textual description of the strength of preference between pairs of actions is found underneath the slider. The questionnaire text was originally in Swedish.Part II consisted of one question where the ten focus areas are given weights using an implementation of CAR; see Figure 4.Figure 4
The weighting of the focus areas translated to English. A handle on the slider represents one focus area. The importance of a focus area is represented by the blue colored gradient on the slider, and the rightmost focus area is the most important one. A textual description of the strength of preference between pairs of actions is found underneath the slider. The questionnaire text was originally in Swedish.Part III consisted of three questions where the respondents stated their preferences regarding two contradicting actions.Part IV consisted of questions regarding demographic information.An invitation letter containing a URL to the questionnaire was sent by mail to 10,000 citizens. The sample was chosen by conducting simple random sampling on a sampling frame consisting of 31,408 citizens. In the section below we present an analysis of the results from the eighth focus area,School. In total we received 939 answers; of these 465 were male, 456 female, 15 did not want to disclose their gender, and 3 selected other/agender. The analysis consists of two parts, the analysis of (i) the difference in preferences in the total population and (ii) the difference in preference between females and males.


#### 5.1.1. Results

Five actionsA1,…,A5 are evaluated under the focus area/attribute G8School:(i)
A1 reduces preschool child groups.(ii)
A2 raises the quality of teaching.(iii)
A3 increases professional development for schools and teachers.(iv)
A4 increases modern information technology (IT) in education.(v)
A5 involves caretakers more in school.The respondents assess the affect associated with each action with regard to ado nothing alternative (current state). Note that this assessment may not be related to the cost of implementing the action, rather the feeling/affect related to the implementation. In the questionnaire implementation of the CAR for conflict evaluations (Figure 3) thedo nothing alternative is represented by the tick mark located in the middle of the scale.Analysis I: Conflict in the Population. In the first analysis we investigate the difference in preference in the total population. We used four consensus property labels to semantically describe the level of conflict. The within-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the subranges and the consensus properties are defined by the authors for illustrative purposes.The results of this analysis are presented in Table8. The results show that all actions have a very low conflict index, indicating that the respondents have similar preferences. Actions A1 (dS8,1=0.0444), A4 (dS8,1=0.0440), and A5 (dS8,5=0.0412) have the largest conflict indices. Actions A3 and A2 have the lowest conflict indices of dS8,3=0.0244 and dS8,2=0.0197, respectively. The attribute properties are presented in Table 9, showing that all actions are consensus aligned.Table 8
Analysis I: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict in the total population.
A 1 A 2 A 3 A 4 A 5 | S 8 j - | 88 11 24 133 120 | S 8 j + | 851 928 915 806 819 C 8 j S 1.04 ⋅ 10 - 7 2.74 ⋅ 10 - 8 7.32 ⋅ 10 - 8 1.64 ⋅ 10 - 7 1.52 ⋅ 10 - 7 P 8 j S 4.91 ⋅ 10 - 6 5.57 ⋅ 10 - 6 5.54 ⋅ 10 - 6 4.87 ⋅ 10 - 6 4.68 ⋅ 10 - 6 T 8 j S 7.12 ⋅ 10 - 6 6.01 ⋅ 10 - 6 6.25 ⋅ 10 - 6 7.09 ⋅ 10 - 6 6.64 ⋅ 10 - 6 d 8 j S 0.0444 0.0197 0.0244 0.0440 0.0412Table 9
Analysis I: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very ControversialAnalysis II: The Difference in Preference between Females and Males. In the second analysis we investigate the difference in preference between females and males. We use the same consensus property labels as in the first analysis to semantically describe the level of conflict. The between-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the consensus properties are defined by the authors for illustrative purposes.The results of the calculations are presented in Table10. All actions have very low conflict indices, which also is reflected in the attribute properties (Table 11), where all actions areconsensus aligned. The action with the highest conflict index is A1 (dS8,1=0.0107), followed by A4 (dS8,4=0.0052), A3 (dS8,3=0.0027), A5 (dS8,5=0.0019), and A2 (dS8,2=0.0005). The difference in value between the actions’ conflict indices is small since both groups have very similar preferences.Table 10
Analysis II: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict between females and males.
A 1 A 2 A 3 A 4 A 5 | S 8 j D - | 25 6 10 60 57 | S 8 j D + | 431 450 446 396 399 | S 8 j E - | 60 5 13 69 61 | S 8 j E + | 405 460 452 396 404 C 8 j D 3.60 ⋅ 1 0 - 8 1.29 ⋅ 1 0 - 8 6.00 ⋅ 1 0 - 8 6.45 ⋅ 1 0 - 8 6.87 ⋅ 1 0 - 8 P 8 j D 2.39 ⋅ 1 0 - 6 2.70 ⋅ 1 0 - 6 2.68 ⋅ 1 0 - 6 2.07 ⋅ 1 0 - 6 2.31 ⋅ 1 0 - 6 T 8 j D 3.20 ⋅ 1 0 - 6 2.94 ⋅ 1 0 - 6 3.07 ⋅ 1 0 - 6 2.95 ⋅ 1 0 - 6 3.20 ⋅ 1 0 - 6 C 8 j E 6.82 ⋅ 1 0 - 8 1.56 ⋅ 1 0 - 8 6.20 ⋅ 1 0 - 9 9.61 ⋅ 1 0 - 8 8.70 ⋅ 1 0 - 8 P 8 j E 2.57 ⋅ 1 0 - 6 3.02 ⋅ 1 0 - 6 3.03 ⋅ 1 0 - 6 2.85 ⋅ 1 0 - 6 2.52 ⋅ 1 0 - 6 T 8 j E 3.86 ⋅ 1 0 - 6 3.24 ⋅ 1 0 - 6 3.33 ⋅ 1 0 - 6 4.23 ⋅ 1 0 - 6 3.62 ⋅ 1 0 - 6 C 8 j D , E 1.05 ⋅ 1 0 - 7 2.85 ⋅ 1 0 - 8 7.39 ⋅ 1 0 - 8 1.61 ⋅ 1 0 - 7 1.57 ⋅ 1 0 - 7 P 8 j D , E 5.05 ⋅ 1 0 - 6 5.72 ⋅ 1 0 - 6 5.71 ⋅ 1 0 - 6 4.99 ⋅ 1 0 - 6 4.83 ⋅ 1 0 - 6 T 8 j D , E 7.27 ⋅ 1 0 - 6 6.18 ⋅ 1 0 - 6 6.41 ⋅ 1 0 - 6 7.22 ⋅ 1 0 - 6 6.82 ⋅ 1 0 - 6 d 8 j D , E 0.0107 0.0005 0.0027 0.0052 0.0019Table 11
Analysis II: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very Controversial


## 5.1. A Case Study

As reported in [13, Ch. 7], the politicians and civil servants wanted to get in-depth information regarding the citizens preferences regarding actions, which possibly could be implemented in the future. It was especially of interest to investigate if there were any actions that potentially could lead to citizen conflicts. To facilitate this, it was decided to use a web-based questionnaire (the Appendix) as a front-end for preference elicitation. The content of the questionnaire was developed together with civil servants at UV. The questionnaire consisted of four parts:Part I consisted of ten questions regarding different “focus areas” (or criteria in an MCDA setting). Under each focus area, the respondents used an implementation of CAR for conflict evaluations to rank five actions; see Figure 3.Figure 3
Question 1 from Part 1 of the questionnaire translated to English. A handle on the slider represents one action. Actions to the left of “Neither good nor bad” are regarded to be worse, and actions to the right are considered to be better than thedo nothing action. The preference intensity is represented by a colored gradient on the slider (red to the left of Neither good nor bad, and green to the right). A textual description of the strength of preference between pairs of actions is found underneath the slider. The questionnaire text was originally in Swedish.Part II consisted of one question where the ten focus areas are given weights using an implementation of CAR; see Figure 4.Figure 4
The weighting of the focus areas translated to English. A handle on the slider represents one focus area. The importance of a focus area is represented by the blue colored gradient on the slider, and the rightmost focus area is the most important one. A textual description of the strength of preference between pairs of actions is found underneath the slider. The questionnaire text was originally in Swedish.Part III consisted of three questions where the respondents stated their preferences regarding two contradicting actions.Part IV consisted of questions regarding demographic information.An invitation letter containing a URL to the questionnaire was sent by mail to 10,000 citizens. The sample was chosen by conducting simple random sampling on a sampling frame consisting of 31,408 citizens. In the section below we present an analysis of the results from the eighth focus area,School. In total we received 939 answers; of these 465 were male, 456 female, 15 did not want to disclose their gender, and 3 selected other/agender. The analysis consists of two parts, the analysis of (i) the difference in preferences in the total population and (ii) the difference in preference between females and males.


### 5.1.1. Results

Five actionsA1,…,A5 are evaluated under the focus area/attribute G8School:(i)
A1 reduces preschool child groups.(ii)
A2 raises the quality of teaching.(iii)
A3 increases professional development for schools and teachers.(iv)
A4 increases modern information technology (IT) in education.(v)
A5 involves caretakers more in school.The respondents assess the affect associated with each action with regard to ado nothing alternative (current state). Note that this assessment may not be related to the cost of implementing the action, rather the feeling/affect related to the implementation. In the questionnaire implementation of the CAR for conflict evaluations (Figure 3) thedo nothing alternative is represented by the tick mark located in the middle of the scale.Analysis I: Conflict in the Population. In the first analysis we investigate the difference in preference in the total population. We used four consensus property labels to semantically describe the level of conflict. The within-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the subranges and the consensus properties are defined by the authors for illustrative purposes.The results of this analysis are presented in Table8. The results show that all actions have a very low conflict index, indicating that the respondents have similar preferences. Actions A1 (dS8,1=0.0444), A4 (dS8,1=0.0440), and A5 (dS8,5=0.0412) have the largest conflict indices. Actions A3 and A2 have the lowest conflict indices of dS8,3=0.0244 and dS8,2=0.0197, respectively. The attribute properties are presented in Table 9, showing that all actions are consensus aligned.Table 8
Analysis I: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict in the total population.
A 1 A 2 A 3 A 4 A 5 | S 8 j - | 88 11 24 133 120 | S 8 j + | 851 928 915 806 819 C 8 j S 1.04 ⋅ 10 - 7 2.74 ⋅ 10 - 8 7.32 ⋅ 10 - 8 1.64 ⋅ 10 - 7 1.52 ⋅ 10 - 7 P 8 j S 4.91 ⋅ 10 - 6 5.57 ⋅ 10 - 6 5.54 ⋅ 10 - 6 4.87 ⋅ 10 - 6 4.68 ⋅ 10 - 6 T 8 j S 7.12 ⋅ 10 - 6 6.01 ⋅ 10 - 6 6.25 ⋅ 10 - 6 7.09 ⋅ 10 - 6 6.64 ⋅ 10 - 6 d 8 j S 0.0444 0.0197 0.0244 0.0440 0.0412Table 9
Analysis I: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very ControversialAnalysis II: The Difference in Preference between Females and Males. In the second analysis we investigate the difference in preference between females and males. We use the same consensus property labels as in the first analysis to semantically describe the level of conflict. The between-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the consensus properties are defined by the authors for illustrative purposes.The results of the calculations are presented in Table10. All actions have very low conflict indices, which also is reflected in the attribute properties (Table 11), where all actions areconsensus aligned. The action with the highest conflict index is A1 (dS8,1=0.0107), followed by A4 (dS8,4=0.0052), A3 (dS8,3=0.0027), A5 (dS8,5=0.0019), and A2 (dS8,2=0.0005). The difference in value between the actions’ conflict indices is small since both groups have very similar preferences.Table 10
Analysis II: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict between females and males.
A 1 A 2 A 3 A 4 A 5 | S 8 j D - | 25 6 10 60 57 | S 8 j D + | 431 450 446 396 399 | S 8 j E - | 60 5 13 69 61 | S 8 j E + | 405 460 452 396 404 C 8 j D 3.60 ⋅ 1 0 - 8 1.29 ⋅ 1 0 - 8 6.00 ⋅ 1 0 - 8 6.45 ⋅ 1 0 - 8 6.87 ⋅ 1 0 - 8 P 8 j D 2.39 ⋅ 1 0 - 6 2.70 ⋅ 1 0 - 6 2.68 ⋅ 1 0 - 6 2.07 ⋅ 1 0 - 6 2.31 ⋅ 1 0 - 6 T 8 j D 3.20 ⋅ 1 0 - 6 2.94 ⋅ 1 0 - 6 3.07 ⋅ 1 0 - 6 2.95 ⋅ 1 0 - 6 3.20 ⋅ 1 0 - 6 C 8 j E 6.82 ⋅ 1 0 - 8 1.56 ⋅ 1 0 - 8 6.20 ⋅ 1 0 - 9 9.61 ⋅ 1 0 - 8 8.70 ⋅ 1 0 - 8 P 8 j E 2.57 ⋅ 1 0 - 6 3.02 ⋅ 1 0 - 6 3.03 ⋅ 1 0 - 6 2.85 ⋅ 1 0 - 6 2.52 ⋅ 1 0 - 6 T 8 j E 3.86 ⋅ 1 0 - 6 3.24 ⋅ 1 0 - 6 3.33 ⋅ 1 0 - 6 4.23 ⋅ 1 0 - 6 3.62 ⋅ 1 0 - 6 C 8 j D , E 1.05 ⋅ 1 0 - 7 2.85 ⋅ 1 0 - 8 7.39 ⋅ 1 0 - 8 1.61 ⋅ 1 0 - 7 1.57 ⋅ 1 0 - 7 P 8 j D , E 5.05 ⋅ 1 0 - 6 5.72 ⋅ 1 0 - 6 5.71 ⋅ 1 0 - 6 4.99 ⋅ 1 0 - 6 4.83 ⋅ 1 0 - 6 T 8 j D , E 7.27 ⋅ 1 0 - 6 6.18 ⋅ 1 0 - 6 6.41 ⋅ 1 0 - 6 7.22 ⋅ 1 0 - 6 6.82 ⋅ 1 0 - 6 d 8 j D , E 0.0107 0.0005 0.0027 0.0052 0.0019Table 11
Analysis II: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very Controversial


## 5.1.1. Results

Five actionsA1,…,A5 are evaluated under the focus area/attribute G8School:(i)
A1 reduces preschool child groups.(ii)
A2 raises the quality of teaching.(iii)
A3 increases professional development for schools and teachers.(iv)
A4 increases modern information technology (IT) in education.(v)
A5 involves caretakers more in school.The respondents assess the affect associated with each action with regard to ado nothing alternative (current state). Note that this assessment may not be related to the cost of implementing the action, rather the feeling/affect related to the implementation. In the questionnaire implementation of the CAR for conflict evaluations (Figure 3) thedo nothing alternative is represented by the tick mark located in the middle of the scale.Analysis I: Conflict in the Population. In the first analysis we investigate the difference in preference in the total population. We used four consensus property labels to semantically describe the level of conflict. The within-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the subranges and the consensus properties are defined by the authors for illustrative purposes.The results of this analysis are presented in Table8. The results show that all actions have a very low conflict index, indicating that the respondents have similar preferences. Actions A1 (dS8,1=0.0444), A4 (dS8,1=0.0440), and A5 (dS8,5=0.0412) have the largest conflict indices. Actions A3 and A2 have the lowest conflict indices of dS8,3=0.0244 and dS8,2=0.0197, respectively. The attribute properties are presented in Table 9, showing that all actions are consensus aligned.Table 8
Analysis I: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict in the total population.
A 1 A 2 A 3 A 4 A 5 | S 8 j - | 88 11 24 133 120 | S 8 j + | 851 928 915 806 819 C 8 j S 1.04 ⋅ 10 - 7 2.74 ⋅ 10 - 8 7.32 ⋅ 10 - 8 1.64 ⋅ 10 - 7 1.52 ⋅ 10 - 7 P 8 j S 4.91 ⋅ 10 - 6 5.57 ⋅ 10 - 6 5.54 ⋅ 10 - 6 4.87 ⋅ 10 - 6 4.68 ⋅ 10 - 6 T 8 j S 7.12 ⋅ 10 - 6 6.01 ⋅ 10 - 6 6.25 ⋅ 10 - 6 7.09 ⋅ 10 - 6 6.64 ⋅ 10 - 6 d 8 j S 0.0444 0.0197 0.0244 0.0440 0.0412Table 9
Analysis I: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very ControversialAnalysis II: The Difference in Preference between Females and Males. In the second analysis we investigate the difference in preference between females and males. We use the same consensus property labels as in the first analysis to semantically describe the level of conflict. The between-group conflict index range [0,1] was divided into four subranges each representing one level of consensus; see Table 3. Note that the consensus properties are defined by the authors for illustrative purposes.The results of the calculations are presented in Table10. All actions have very low conflict indices, which also is reflected in the attribute properties (Table 11), where all actions areconsensus aligned. The action with the highest conflict index is A1 (dS8,1=0.0107), followed by A4 (dS8,4=0.0052), A3 (dS8,3=0.0027), A5 (dS8,5=0.0019), and A2 (dS8,2=0.0005). The difference in value between the actions’ conflict indices is small since both groups have very similar preferences.Table 10
Analysis II: the number of members of the con- and pro-groups, the con- and pro-indices, and the conflict between females and males.
A 1 A 2 A 3 A 4 A 5 | S 8 j D - | 25 6 10 60 57 | S 8 j D + | 431 450 446 396 399 | S 8 j E - | 60 5 13 69 61 | S 8 j E + | 405 460 452 396 404 C 8 j D 3.60 ⋅ 1 0 - 8 1.29 ⋅ 1 0 - 8 6.00 ⋅ 1 0 - 8 6.45 ⋅ 1 0 - 8 6.87 ⋅ 1 0 - 8 P 8 j D 2.39 ⋅ 1 0 - 6 2.70 ⋅ 1 0 - 6 2.68 ⋅ 1 0 - 6 2.07 ⋅ 1 0 - 6 2.31 ⋅ 1 0 - 6 T 8 j D 3.20 ⋅ 1 0 - 6 2.94 ⋅ 1 0 - 6 3.07 ⋅ 1 0 - 6 2.95 ⋅ 1 0 - 6 3.20 ⋅ 1 0 - 6 C 8 j E 6.82 ⋅ 1 0 - 8 1.56 ⋅ 1 0 - 8 6.20 ⋅ 1 0 - 9 9.61 ⋅ 1 0 - 8 8.70 ⋅ 1 0 - 8 P 8 j E 2.57 ⋅ 1 0 - 6 3.02 ⋅ 1 0 - 6 3.03 ⋅ 1 0 - 6 2.85 ⋅ 1 0 - 6 2.52 ⋅ 1 0 - 6 T 8 j E 3.86 ⋅ 1 0 - 6 3.24 ⋅ 1 0 - 6 3.33 ⋅ 1 0 - 6 4.23 ⋅ 1 0 - 6 3.62 ⋅ 1 0 - 6 C 8 j D , E 1.05 ⋅ 1 0 - 7 2.85 ⋅ 1 0 - 8 7.39 ⋅ 1 0 - 8 1.61 ⋅ 1 0 - 7 1.57 ⋅ 1 0 - 7 P 8 j D , E 5.05 ⋅ 1 0 - 6 5.72 ⋅ 1 0 - 6 5.71 ⋅ 1 0 - 6 4.99 ⋅ 1 0 - 6 4.83 ⋅ 1 0 - 6 T 8 j D , E 7.27 ⋅ 1 0 - 6 6.18 ⋅ 1 0 - 6 6.41 ⋅ 1 0 - 6 7.22 ⋅ 1 0 - 6 6.82 ⋅ 1 0 - 6 d 8 j D , E 0.0107 0.0005 0.0027 0.0052 0.0019Table 11
Analysis II: the classification of the actions consensus properties.
Consensus Property A 1 A 2 A 3 A 4 A 5 Consensus aligned × × × × × Potentially controversial Controversial Very Controversial


## 6. Concluding Remarks

Public decision problems can be complex. They involve multiple actions and multiple stakeholders which may have conflicting preferences with regard to the actions. A problem in such situations is that these opposing preferences might lead to conflicts between stakeholders, which may lead to delays in the decision process. An approach that can be used to increase the understanding of the stakeholders’ opinions is to allow them to state their opinions, e.g., by using a web-based questionnaire.In this paper, we showed how the CAR method can be applied, enabling respondents to state negative or positive preferences with regard to an alternative’s performance relative to ado nothing alternative. We applied CAR for conflict evaluations in a case study conducted in cooperation with Upplands Väsby municipality. The citizens’ preferences were elicited by a web-based questionnaire that used an implementation of the method. We showed how the method can be used to highlight conflict between and within different stakeholders groups and how the conflict can be conceptualized into semantic attribute properties. The results of such an analysis can aid decision-makers in the process of making well-informed decisions by clarifying the actions that can be conflict-prone.



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*Source: 8290434-2018-09-27.xml*