An evolution model of risk preference influenced by extremists in large group emergency consensus process

Abstract

This study uses opinion dynamics to explore the influence of extremists in the consensus process of large group decision-making. When moderates are exposed to extremists, their risk preference will be affected. By using the opinion leader theory for reference, the influence model of extremists is constructed. To better study the influence of extremists, the similarity of risk preference between extremists and moderates is modeled to measure their similarity degree. From this model, for every moderate, the extremists are divided into two groups: homogeneous group and heterogeneous group. Finally, the risk preference evolution model is structured by considering that moderates change their risk preference dynamically according to their initial preference, their attitude towards the homogeneous groups, and the heterogeneous groups. Finding from data analysis shows that moderates with high acceptance toward the influence of extremists are more likely to reach group consensus. It is also found that the preference trend of moderates with a certain degree of acceptance toward heterogeneous groups fluctuates with a ‘W’ shape. This study bridges the gap between opinion dynamics and group decision making. Meanwhile, the model inspires new explanations and new perspectives for the group consensus process.

Keywords

Extremists opinion dynamics group emergency decision-making group consensus risk preference evolution

1 Introduction

As the growth of the economic blossoms in globalization, this also causes an increase in the number of large-scale projects which are however at the expense of the accidents causing many people injury, harm, and even death unavoidably, such as Rio de Janeiro Building collapse, Colombia Motorway Bridge collapses, Miami Pedestrian Bridge collapses and so on. The rate of injury in large-scale project accidents is higher than any other industry. Risk assessment effectively and rapidly is vital to control the accident which has focused attention on the need for emergency decision making. Emergency decisions on evaluating risks have far-reaching consequences, yet the decision-makers often have to be under severe time pressure. Meanwhile, due to the different experiences, specialties and statues, the decision-making group composes of the construction party, the owner party and the related decision-makers in the large-scale project have different risk preference values. Therefore, it’s very difficult to form a highly consistent decision plan in a short time.

The emergency decision-making group consists of the interaction of multiple decision-makers to evaluate risks from high to low under the time pressure. To obtain the final evaluation result, the consensus process is a necessary part to know how the decision-makers interact with each other and how the maximum consensus degree is reached. Decision-makers are the key elements in the consensus process and their decision-making behavior may vary depending on their risk preferences. Decision-maker’s risk preference can be expressed by his opinions on certain risks. When their opinions towards risks interact with each other, their risk preferences may be changed accordingly. Through the ongoing opinion exchange, the decision maker’s risk preference will be updated on the influence of other decision-makers which makes the group consistency more efficient.

By extension of opinion dynamics, this study explores the risk preference evolution of moderates with a certain acceptance to the influence of the extremists and persistence to their initial risk preference. Extremists are divided into homogeneous groups and heterogeneous groups depending on similarities of their risk preference to every single moderate. The similarity model is put forward to recognize the different groups in extremists. When the moderates interact with various extremist groups, the acceptance is changed according to their personality. Meanwhile, the influence model is built to measure the influence of extremists. To better study the influence of extremists to moderates, this paper assumes that moderates have no influence on extremists and there is also no influence between moderates. After the interaction with extremists, the risk preference of moderates would evolve until the group consensus reached. This study bridges the gap between opinion dynamics and group decision making and provides a new perspective for group consensus process.

2 Problem statement and research structure

Opinion dynamics theory exploits how individuals change their initial opinions through interaction in the complex groups. The bounded confidence model is the most popular one in opinion dynamics. Deffuant, Hegselmann and Krause explore the bounded confidence model [1 –3] and their research focuses on how the agent’s opinion will be affected by the neighboring agents whose opinions to a certain confidence level are similar to his own opinion. The difference between these two models is that the agents in DK model are pair interaction while the agent in HK model will interact with all the agents under a certain confidence level [4]. On this basis, the influence of the specific agents, such as radical groups, charismatic leaders to the normal agents or ordinary agents are further studied [5 –7].

A large group can be divided into ‘extremist’ agents and ‘moderate’ agents [8]. The extremists influence moderate agents with their ‘very radical’ and ‘very strong messages’. In the evolution process, the opinions of moderate agents are updated by their interaction with the extremists’ strong information [6]. By the application of the bounded confidence model, uncertainties of the moderates would be decreased when interacting with extremists. When the proportion of extremists and uncertainty are fixed, the opinion density distribution remains stable [8]. Based on the BC model with fixed uncertainties, the resilience of the moderate agents’ opinion is researched. When the arrival of a new group called tipping energy is added to the initial group and the resilience of the moderate mean opinion would be changed [9].

Regarding the connection between opinion dynamics and group decision making, the opinion dynamics on groups of stakeholders are linked in social networks. By using an agent-based model, the group decision making process in a network of interacting stakeholders is analyzed [10]. Taking group consistency degree and consistency time as performance indexes, the influence of the uncertainty of decision-makers’ preference on the group consistency is studied with opinion dynamics [11]. The connection between opinion dynamics and group decision making is analyzed and the group decision making in the complex network should be put into the key interdisciplinary research [12]. Opinion dynamics is an opinion fusion process. Dong, et al. review the fusion process in opinion dynamics is reviewed and opinion dynamics could be used as a potential tool to develop consensus reaching models in group decision making is pointed [13].

According to the literature review, it is found that the existing research focuses on the fusion process of opinions under a certain confidence degree, fixed uncertainties, special groups and verities noises. Meanwhile, the opinion dynamics began to be applied to group decision-making areas. However, in the actual consensus process of emergency decision making, to obtain information under the time pressure, moderate’s risk preference maybe not only affected by the decision-makers whose risk preference is similar to himself, but also by the ones who have the different risk preference. To the limited knowledge, fewer researches focus on the heterogeneous opinion interaction between moderates and extremists in the emergency decision-making process.

Therefore, this paper is focused on the influence of extremists on the risk preference of the moderates and is structured as follows. First, there is a short description of the influence mechanism of extremists on the risk preference of moderates. Then follows the evolution process and decision-making mechanism in the emergency events. After that, the evolution model of risk preference is constructed based on the structuring influence model of extremists, the similarity model, and the identification mechanism. Next, a case study is described and discussed with data simulation. Then the comparison and discussion between the existing works and our research are conducted. The paper ends with a discussion about the main results and the needs for further research.

3 The characteristics of extremists and moderates in emergency decision-making

In this paper, it is believed that the decision-making group is composed of two kinds of decision-making members, namely, extremists and moderates. In the emergency events, the extremists with the strong message are more likely to evaluate the risks radically to play an authoritative role to influence the moderates. In this way, the consensus will be reached in a shorter time, which is an effective indicator to reduce life property loss, under the influence of extremes. In Table 1, the characteristics of extremists and moderates in decision-making experience, decision time, risk preference value, decision role, risk preference certainty, and personal influence are concluded.

Table 1
The characteristics of extremists and moderates in large decision groups

Decision Members\Characteristics Extremists Moderates

Experience Extensive knowledge or similar experience Lack of experience or related knowledge

Decision Time Rapid Slow

Risk preference values Fully positive or negative Ambiguous

Decision Role Experts or leaders Common ones

Deterministic degree to risk preference Extremely high determination Uncertainty

Personal influence A certain influence degree Less or even no influence

Decision Members\Characteristics	Extremists	Moderates
Experience	Extensive knowledge or similar experience	Lack of experience or related knowledge
Decision Time	Rapid	Slow
Risk preference values	Fully positive or negative	Ambiguous
Decision Role	Experts or leaders	Common ones
Deterministic degree to risk preference	Extremely high determination	Uncertainty
Personal influence	A certain influence degree	Less or even no influence

To make a wiser decision and obtain more comprehensive information, decision-makers will interact with each other to adjust their risk preferences. In the interaction process, based on the characteristics of extremists and moderates in Table 1, the moderates in the emergency group would like to accept strong messages from extremists. Their risk preference is often influenced by extremists during the group discussion process. In the actual decision-making practice, moderates may have different acceptance to the influence of extremists. Some moderates are more willing to accept opinions similar to their preferences, just like the bounded confidence model. On the contrary, some moderates believe that it’s meaningless to refer to opinions with a similar preference and they are more willing to accept those different or even opposite opinions.

4 Evolution model of risk preference in large group emergency decision making

4.1 Basic definitions

Definition 1 Large group of mixed preferences in decision-making. The risk factor set of the emergency event is X ={ x₁, x₂, ⋯ , x_n } and n≥2. A large group of mixed preferences in decision-making is $Ω = {E^{*}, E} = {(e_{1}^{*}, e_{2}^{*}, \dots, e_{p}^{*}), (e_{1}, e_{2}, \dots, e_{q})}$ and p + q = m,p≥1,q≥19, m≥20.

The decision-making group Ω consists of two subsets E* and E. E* is the set of extremists who evaluate risks with the maximum or the minimum risk value in the emergency event. And E is the set of moderates who will not make an absolute high or low-risk choice when the risk factors are evaluated in the emergency decision-making.

In general, decision-makers have different risk preferences under different risk factors, and their risk preferences can be a relatively comprehensive representation of decision-makers’ opinions from different dimensions. Therefore, to comprehensively measure the risk preference of decision-makers, the risk preference value under n risk factors is considered and the risk preference vectors under the n-dimensional risk factors are established. The definitions of extremists’ risk preference vector and moderates’ risk preference vector will be given below.

Definition 2 Extremist risk preference vector. The preference value of the ith extremist in E* under the jth risk factor in the emergency event is $v_{ij}^{*} = 1$ or 0, in which the value 1 is the maximum risk preference value that decision-makers may choose in evaluating this risk factor and the value 0 is the minimum risk preference value, and i = 1,2, ... .,p, j = 1,2, ... ... ,n. The risk preference vector $V_{i}^{*} = (v_{i 1}^{*}, v_{i 2}^{*}, \dots, v_{ip}^{*})$ is called the extremist risk preference vector of the ith decision member in E*.

Definition 3 Moderate risk preference vector. The preference value of the lth moderate in E under the jth risk factor in the emergency event isv_lj, and l = 1,2, ... .,q, j = 1,2, ... ... ,n, 0 < v_lj < 1, that is, the risk preference value of the moderate is chosen between the maximum and minimum of risk preference. The risk-preference vector V_l = (v_l1, v_l2, …, v_lq) is called the moderate risk preference vector of the lth decision member in E.

4.2 Extremist risk preference influence model

It is assumed that the risk preference value of $e_{i}^{*}$ in set E* has an impact at time t on each moderate in set E. And αⁱ (t) represents the impact degree.

When time t = 0, the initial influence value of extremists is αⁱ (0). And because it is assumed that moderates do not influence other decision-makers, initial influence value αⁱ (0) of $e_{i}^{*}$ does not take into account the opinion of the decision-makers in set E, but only considers the number of the extremists with opposite views in set E* and the number of decision members m in decision-making groups Ω.

The mathematical operation exclusive OR is used to distinguish the number of extremists with opposite views against $e_{i}^{*}$ in set E*. Assuming that a and b are real numbers, if a = b, then a ⊕ b = 0; if a ¬= b, then a ⊕ b = 1. According to the above definition, the formula of the initial influence αⁱ (0)of $e_{i}^{*}$ in E* is given in Formula (1). $α^{i} (0) = 1 - \frac{\sum_{j = 1}^{n} \sum_{k = 1, k \neq i}^{p} v_{ij}^{*} \oplus v_{kj}^{*}}{p \cdot n}$ (1)

In the actual emergency event, the influence of extremists declines as time goes on, that is, the influence of extremists is the most influential at the first moment of the emergency event, and then the dominance of extremists will weaken over time. The power-law attenuation function $y^{'} (t) = N_{0} t^{- γ} e^{- \frac{t}{τ}}$ expressed by an exponential truncation is proposed in the literature [15] studying on opinion leader’s influence.

In this function, the opinion leader’s influence is affected by four factors, that is, initial influence N₀, time of transit t, the speed of influence decay γ and extinction time τ of message propagation from the opinion leader. This function is similar to the influence of the extremists studied in this paper. Therefore, the function is used for reference to derive the influence model of extremists without considering the speed of influence decay γ (that is, γ = 0). The influence model is shown in Formula (2). $α^{i} (t + 1) = α^{i} (t) \cdot e^{- \frac{t}{τ}}$ (2)

τ is a given constant that represents the influence duration of the extremist. When the decision time t is far less than τ, the influence of the extremist does not change. And the extremist will have no influence when the decision time t is increased to exceed the influence duration τ.

4.3 Similarity model of risk preference and group consistency model of large groups with extremist risk preferences

The preference similarity is to evaluate the similarity degree between two preference vectors. The smaller the value, the lower the similarity between the two preference vectors is and the greater the difference is. The study of preference similarity includes the inner product method, cosine method, DICE coefficient, and Jaccard coefficient at home and abroad. The distance measurement is mainly to measure the space distance between two preference vectors. The longer the distance, the greater the difference is. The main methods of distance measurement are Ming Distance, the Manhattan Distance, Euclidean distance.

The preference angle cosine measure method and the preference distance measure method have different focuses. The former considers the spatial angle of the two preference vectors to measure the direction consistency, which reflects the gap in distance between the preference vectors. And the latter considers the absolute distance between the two preference vector nodes. Therefore, this paper combines the characteristics of the two methods and constructs a risk preference similarity model which considers the preference vector direction and the preference vector distance based on previous studies.

$e_{i}^{*}$ and e_l are selected respectively from set E* and set E, and the similarity of risk preference between their risk preference vectors $V_{i}^{*}$ and V_l is shown in Formula (3).

$\begin{matrix} r_{il} (V_{i}^{*}, V_{l}) \\ = {\frac{(| V_{i}^{*} - {\bar{V}}_{i}^{*} |) \cdot (| V_{l} - {\bar{V}}_{l} |)^{T}}{{∥ V_{i}^{*} - {\bar{V}}_{i}^{*} ∥}_{2} \cdot {∥ V_{l} - {\bar{V}}_{l} ∥}_{2}} \times 1 - \sqrt{\frac{\sum_{j = 1}^{n} v_{ij}^{*} - v_{lj}^{2}}{n}}}^{1 / 2} \end{matrix}$ (3) where $v_{ij}^{*}, v_{ij} \in [0, 1]$ , $0 ⩽ \frac{(| V_{i}^{*} - {\bar{V}}_{i}^{*} |) \cdot (| V_{l} - {\bar{V}}_{l} |)^{T}}{{∥ V_{i}^{*} - {\bar{V}}_{i}^{*} ∥}_{2} \cdot {∥ V_{l} - {\bar{V}}_{l} ∥}_{2}} ⩽ 1$ [16], therefore, it can be drawn $0 ⩽ r_{il} (V_{i}^{*}, V_{l}) ⩽ 1$ clearly. The model shows that the smaller the angle and the closer the distance between the two vectors, the higher the similarity of risk preference vectors between the two decision-makers is.

The large group of mixed risk preferences is formed by extremists and moderates. The preference consistency of the mixed preference group is determined by the similarities of the decision-making members’ risk preferences within the large group. The higher the preference similarity, the higher the preference consistency of the large group is and the easier the consensus is to be reached in the large group. The process of consistency calculation in the large group is followed as the literature [16].

4.4 Identification mechanism of the homogeneous group and the heterogeneous group

The risk preference similarities r_1l, r_2l, …, r_pl between one single moderate in set E and all extremists are calculated according to the risk preference similarity model. Based on this, the set E* is clustered. In the process of clustering, the threshold value γ is used to determine the similarity of risk preference vectors between the moderate e _l and any one of the extremists in the set E*: when the similarity between two decision-makers is larger than the threshold γ, then the corresponding extremist is placed in e _lsquo s risk preference homogeneous group $E_{s}^{l}$ , otherwise, he will be placed in e _lsquo s risk preference heterogeneous group $E_{d}^{l}$ . When each moderate in E has his corresponding risk preference homogeneous group and risk preference heterogeneous group, the algorithm stops.

4.5 Risk preference evolution model of the moderates

At time t, the risk preference vector of moderate e_l in set E is V_l (t) (l = 1, ... ,q) and his corresponding homogeneous group and heterogeneous group in E* are $E_{s}^{l}$ and $E_{d}^{l}$ respectively. The risk preference vector of decision member i₁ in set $E_{s}^{l}$ is $V_{i_{1}}^{*} (t)$ (i₁ = 1, ... ,p₁) and the risk preference vector of decision member i₂ in set $E_{d}^{l}$ is $V_{i_{2}}^{*} (t)$ (i₂ = 1, ... ,p₂). p₁ and p₂ are the number of extremists in homogeneous and heterogeneous groups respectively, and 0≤p1, p2≤p,p₁ + p₂ = p. Because extremists are not affected by other decision-makers, their risk-preference vectors are the same at all stages as the initial stage, that is, $V_{i_{1}}^{*} (t) = V_{i_{1}}^{*} (0)$ , $V_{i_{2}}^{*} (t) = V_{i_{2}}^{*} (0)$ .

At the next stage t + 1, the moderate will update his risk preference vector according to his acceptance degree to extremists that affect him at stage t.

Before the updated model is established, the influence degree of homogeneous and heterogeneous groups in extremist group is normalized to make the value of updated vectors between 0 and 1, the reasonable value set in this article. Formula (4) represents the normalized calculation. ${\hat{α}}^{i_{1}} (t) = \frac{α^{i_{1}} (t)}{\sum_{i_{1} = 1}^{p_{1} (j, t)} α^{i_{1}} (t)}, {\hat{α}}^{i_{2}} (t) = \frac{α^{i_{2}} (t)}{\sum_{i_{2} = 1}^{p_{2} (j, t)} α^{i_{2}} (t)}$ (4)

The risk preference evolution model of e_l at stage t + 1 is shown in Formula (5):

$\begin{matrix} V_{l} (t + 1) = 1 - λ_{s} - λ_{d} \cdot V_{l} (t) \\ + λ_{s} \cdot \sum_{i_{1} = 1}^{p_{1} (j, t)} {\hat{α}}^{i_{1}} (t) \cdot V_{i_{1}}^{*} (0) + λ_{d} \cdot \sum_{i_{2} = 1}^{p_{2} (j, t)} {\hat{α}}^{i_{2}} (t) \cdot V_{i_{2}}^{*} (0) \end{matrix}$ (5)

The risk preference vector of the jth risk factor in V_l (t + 1) is v_lj (t + 1). Because of 0 ⩽ λ_s, λ_d ⩽ 1, 0 < v_lj (t) <1, $0 ⩽ {\hat{α}}^{i_{1}} (t), {\hat{α}}^{i_{2}} (t) ⩽ 1$ , $\sum_{i_{1} = 1}^{p_{1} (j, t)} {\hat{α}}^{i_{1}} (t) = \sum_{i_{2} = 1}^{p_{2} (j, t)} {\hat{α}}^{i_{2}} (t) = 1$ , thus 0 ⩽ v_lj (t + 1) ⩽1.

5 Parametric sensitivity analysis and simulation of risk preference evolution in large group emergency decision-making considering the influence of extremists

5.1 Case background

Beach-Hill Gold Mine project of Da Chai-Dan Mining Company in Qinghai province is composed of 4 adjacent mines with the area of 342 square kilometers. The mine was discovered by the first Geological Brigade of Qinghai Province in the 1990’s, and was established in 2006 by Da Chai-Dan Mining Company.

In the implementation phase of Beach-Hill Gold Mine project, disagreeing with the original design plan, the local power company urgently proposed to redesign the scheme suggesting that the 35Kv high voltage collocated indoors be changed outdoors’ one transformer be changed for two and the design project of substation be redesigned according to the checkup opinion of the power company. After the project department received the emergency of design change application, project department managers of the general contractor of Construction Management, heads of the owners and personnel of the project-related party, a total of 70 persons were convened to assess the possible risks caused by the design change application.

According to the case data, the eight major risks arising from the emergency of the redesign are collected, that is, the risks of environmental deterioration, increased CAPEX, delayed project progress, poor quality, operational difficulties, poor maintainability, health and safety hazards and expanded scope of work.

The value of risk preference is [0,1]. If one’s risk preference towards a factor value = 1, that means the decision-maker considers that the risk will cause the entire project to fail when the emergency occurs. On the contrary, if value = 0, that means the decision-maker thinks the risk has no impact on the project plan. The greater the decision maker’s risk preference value, the greater the decision-maker believes the impact of the risk on the project and vice versa.

In the emergency decision-making meeting, 70 decision-making members assess risk according to their own experience and the current situation. At the end of the meeting, the risk assessment results of 70 decision-making members are extracted from the conference data. According to the definition of 4.1, the group of 70 mixed preference members consists of 20 extremists and 50 moderates, as shown in Fig. 1.

Fig. 1

Risk preference values of 20 extremists and 50 moderates.

Then programming and simulation are conducted in MATLAB as the simulation process in Fig. 2.

Fig. 2

Data simulation process.

5.2 Parametric sensitivity analysis

In this paper, the consistency ρ (t) of moderates at t + 1 stage depends on the risk preference values of the moderates. According to the above model construction structure, the risk preference value of the moderates is affected by four uncertain parameters λ_s,λ_d, τ and γ.

To study the sensitivity of the four uncertain parameters to the risk preference value, four parameters will be analyzed sensitively to get the sensitivity coefficient. And the greater the parameter sensitivity coefficient, the greater the influence of the parameter to the result. Formula (6) represents the parameter sensitivity, that is, $E = \frac{Δ v}{Δ F}$ (6)

ΔF indicates the change rate of the uncertainty parameter F. In this paper, the uncertainty parameter F are λ_s, λ_d, τ and γ. Δv stands for the change rate of the final result-risk preference Value V when the uncertainty parameter F is changed.

Next, the risk preference value of the moderate x₁ in t = 1 is taken as an example to analyze the sensitivity coefficients E of these four uncertainties. Because of the relatively independent parameters in this article, to get a more comprehensive sensitivity analysis of the four parameters, it is necessary to conduct dynamic sensitivity analysis. To be specific, one of the parameters will be analyzed with a certain regularity of change, and the other three parameters to the fixed interval value of the synchronized dynamic changes.

The variations range of parameters λ_s and λ_d are 0 ⩽ λ_s, λ_d ⩽ 1,0 ⩽ λ_s + λ_d ⩽ 1. The changes of the risk preference value V (t + 1) of the moderate x₁ in T = 1 are analyzed. According to the definition of the preceding article, when t > τ, the decision will be terminated. To get the analytical results, the variations range of the parameter τ is 2 ⩽ τ ⩽ 8. The variations range of the parameters γ is 0.4 ⩽ γ ⩽ 0.8 (calculated from Formula (3)).

(1) Sensitivity analysis of parameter λ_s and λ_d

When the value varies of λ_s is between 0 and 1 and the interval value is 0.1, that is Δλ_s= 0.1. In order to study the sensitivity of parameter λ_s, the values of λ_s, γ, and τ are simulated according to the parameter adjustment formula shown in Formula (7), (8), and (9). $λ_{d} = 0 + 0.1 * θ$ (7) $γ = 0.4 + 0.04 * θ$ (8) $τ = 2 + 0.6 * θ$ (9) where θ ∈ [0, 10], θ are the integers coefficient of an interval between 0 and 10.

Make θ = 0, the V-value v₀ and the sensitivity coefficients E₀ will be calculated when the parameter λ_s is changed. After the completion of the calculation, set θ = θ + 1, then a new set of three parameters are obtained which are input to get a new set of V value v_θ and the sensitivity coefficient E_θ of the parameter λ_s. And then input gradually, until θ = 10. Dynamic sensitivity coefficients of the parameter λ_s are shown in Table 2 and Fig. 3.

Table 2

Dynamic sensitivity coefficients of the parameter λ_s

λ_d = 0 +0.1 * θ, γ = 0.4 + 0.04 * θ, τ = 2 +0.6 * θ
θ	λ _s	v_θ	E_θ	θ	λ _s	v_θ	E_θ	θ	λ _s	v_θ	E_θ
0	0	0.613	∖∖∖	2	0	0.304	∖∖∖	5	0	0.178	∖∖∖
	0.1	0.582	0.498		0.1	0.289	0.474		0.1	0.260	4.597
	0.2	0.558	0.420		0.2	0.297	0.261		0.2	0.245	0.581
	0.3	0.539	0.341		0.3	0.313	0.523		0.3	0.248	0.148
	0.4	0.524	0.265		0.4	0.337	0.775		0.4	0.249	0.035
	0.5	0.514	0.195		0.5	0.358	0.630		0.5	0.250	0.029
	0.6	0.507	0.134		0.6	0.375	0.466	6	0	0.261	∖∖∖
	0.7	0.503	0.082		0.7	0.389	0.381		0.1	0.291	1.146
	0.8	0.501	0.043		0.8	0.400	0.285		0.2	0.273	0.620
	0.9	0.500	0.016	3	0	0.321	∖∖∖		0.3	0.281	0.267
	1	0.500	0.002		0.1	0.292	0.881		0.4	0.289	0.312
1	0	0.447	∖∖∖	0.2	0.243	1.697	7	0	0.450	∖∖∖
	0.1	0.436	0.245		0.3	0.259	0.678		0.1	0.405	17.004
	0.2	0.429	0.150		0.4	0.282	0.860		0.2	0.389	0.387
	0.3	0.426	0.065		0.5	0.301	0.697		0.3	0.350	1.000
	0.4	0.427	0.006		0.6	0.318	0.549	8	0	0.500	∖∖∖
	0.5	0.429	0.062		0.7	0.331	0.426		0.1	0.444	1.111
	0.6	0.434	0.101	4	0	0.354	∖∖∖		0.2	0.400	1.000
	0.7	0.439	0.124		0.1	0.285	1.930	9	0	0.500	∖∖∖
	0.8	0.445	0.130		0.2	0.248	1.296		0.1	0.450	1.000
	0.9	0.450	0.121		0.3	0.229	0.800	10	0	0.500	∖∖∖
				0.4	0.250	0.941
				0.5	0.262	0.456
				0.6	0.270743	0.352

Fig. 3

Trend chart of dynamic sensitivity coefficients for the parameter λ_s.

It can be seen from Table 5 that when the parameter λ_s changes according to certain rules, the moderates’ preference values v_θ also change, and the sensitivity coefficients of the parameter λ_s also show the changing trend. When θ = 5λ_s = 0.1, that is, λ_d = 0.4, γ = 0.56, τ = 4.4, the sensitivity coefficients of the parameter λ_s is up to the highest value E₅ = 4.597 in all dynamic fixed parameters. And when the parameter λ_s changes one unit, the risk preference value of the moderate changes of 4.597 units.

It can be seen from the trend chart of dynamic sensitivity coefficients for the parameterλ_s that the sensitivity coefficients are also changed under different fixed parameters. When θ = 0 and θ = 1, the sensitivity coefficients of the parameters are under 0.5. And at the time θ = 7, 8, 9, most of the sensitivity coefficients of the parameterλ_s are more than 1. The overall sensitivity coefficients trend of the parameterλ_s fluctuates between 0.006 and 4.597.

Similarly, the sensitivity analysis of the parameterλ_d is conducted. The trend of dynamic sensitivity coefficients of the parameter λ_d is shown in Fig. 4.

Fig. 4

Trend chart of dynamic sensitivity coefficients for the parameter λ_d.

According to the overall fluctuation trend in Fig. 4, the parameterλ_d shows different sensitivity coefficients under different fixed parameters, but their values are all above 1, which shows that the influence degree of the parameter λ_d on the results is high, and the fluctuation range of sensitivity coefficients fluctuates between 0.01 and 52.507.

(2) Sensitivity analysis of parameterγ

Make the parameterγchanged at 0.04 intervals between 0.4 and 0.8 to observe the effect of the parameter γ on the result. And the values of λ_s, λ_d, and τ are simulated according to the parameter adjustment formula shown in Formula (10), (11), and (12). $λ_{s} = 0 + 0.1 * θ$ (10) $λ_{d} = 0 + 0.1 * θ$ (11) $τ = 2 + 0.6 * θ$ (12) where θ ∈ [0, 10], θ are the integers coefficient of an interval between 0 and 10.

Because of 0 ⩽ λ_d + λ_s ⩽ 1, the fixed value θ is taken an integer value between 0 and 5.

From the result, it is found that the value v_θ of the moderate’s preference is different from the fixed parameters under different values of θ, but in the fixed parameters under the same values of θ, the sensitivity coefficients of the parameter γ is zero, that is, the parameter γ has no effect on the result.

(3) Sensitivity analysis of parameterτ

In the same way, the parameter τ is changed from 2 to 8 at the interval of 0.4 to be observed the effect of the parameter τ on the result.

Make fixed parametersλ_s = 0 +0.1 * θ, λ_d = 0 +0.1 * θ, γ = 0.4 + 0.06 * θ. θ is taken an integer value between 0 and 5. The trend of dynamic sensitivity coefficients of the parameter τ is shown in Fig. 5.

Fig. 5

Trend chart of dynamic sensitivity coefficients for the parameter τ.

It can be seen from Fig. 5 that the sensitivity coefficient of the parameter τ is between 0–0.6, and most of the sensitivity coefficient is 0, which shows that this parameter has a minimal effect on the result.

Through the above analysis, it indicates that the parameter λ_d has the maximum sensitivity coefficient relatively. It indicates that the effect of the parameter λ_d on the results of the greatest degree. This result is also consistent with the actual situation [17], that is, in the actual group decision-making process, once the different decision opinions are accepted, that will have the big influence in the group which may cause the qualitative change in decision results; the parameterλ_s has certain influence to the result, but the influence degree is lower than the parameterλ_d; the sensitivities of the parameters λ_s and λ_d are extremely low which show that the two parameters have no impact or minimal impact on the results.

5.3 Simulation results

According to the parametric sensitivity analysis, parameter γ and τ have little effect on moderates’ risk preference value. Therefore, parameter γ and τ can be fixed to any value with the range of the values. In this paper, let γ = 0.55 and τ = 8, and focus on the simulation discussion on the parameter λ_d and λ_s.

From the evolution model of risk preference, it can be found that the moderates’ risk preference in the next stage depends on three parts. The first part is the moderates’ retention to their risk preference value in the previous stage; the second part is the moderates’ acceptance to the homogenous group; the third part is the moderates’ acceptance to the heterogeneous group. Therefore, parameters λ_d and λ_s will be simulated and discussed based on these three parts.

(1) Whenλ_s = 0, λ_d = 0, that is, moderates will not be affected by any extremists. In this situation, the risk preference of the moderates will not evolve. Their risk preference values at each stage are the same as the previous stage and there is no group interaction.

(2) When λ_s = 1, λ_d = 0, that is, moderates will accept the influence of homogenous groups instead of the heterogeneous group and abandon their initial risk preference values. The moderates’ risk preference evolution graph of 8 risk factors is as show in Fig. 6.

Fig. 6

The moderates’ risk preference evolution graph of 8 risk factors under the full acceptance to the influence of the homogenous group.

As can be seen from the Fig. 6, when the moderates fully accept the influence of the homogeneous group, their risk preference values to 8 risk factors reach an agreement at t = 1. The risk preference value fluctuates at t = 2 and is no longer changed until t = 3. The reasons for this trend are that the moderates give up their initial risk preference and fully accept the influence of the homogeneous group. In this case, based on the influence of the homogeneous group, moderates’ risk preference reaches the same value at t = 1. But at the same time, the homogeneous group of the moderates is re-divided according to the new risk preference values at t = 1. Therefore, the evolution trend is changed slightly at t = 2. The risk preference value does not evolve at the stage of t = 3 and the risk preference values become stationary.

(3) When λ_s = 0, λ_d = 1, that is, moderates will accept the influence of heterogeneous groups instead of homogeneous groups and abandon their initial risk preference values. The moderates’ risk preference evolution graph of 8 risk factors is as shown in Fig. 7.

Fig. 7

The moderates’ risk preference evolution graph of 8 risk factors under the full acceptance to the influence of the heterogeneous group.

When the moderates fully accept the influence of the heterogeneous group, the consensus time is the same as the situation of the full acceptance to the homogeneous group, that is, the risk preference values reach an agreement at t = 1. But the difference is that the evolution trend results in W-type fluctuations. It shows that even after the moderates reach a consistent agreement with the risk factors at the current stage, the risk preference value will evolve with the development of time. Because of the huge difference between moderates and the extremists in the heterogeneous group, the moderates’ risk preference values are changed significantly at the next stage comparing with the values at the previous stage when the moderates accept the risk preference of the heterogeneous group. Every time the risk preference values are updated, and then a new set of heterogeneous groups is generated. At the next stage, the moderates fully re-accept the influence of the new heterogeneous group. This big change in preference values results in W-type fluctuation or reversal of the risk preference. According to the analysis above, although the agreement is reached at t = 1, the risk preference will fluctuate over time. The risk preference values of the moderates are repeatedly changed which will not reach a stable consensus. In this situation, a consistent risk preference value can’t be obtained in a certain period.

(4) When 1 - λ_s - λ_d = 0, that is, the persistence degree of the moderates to their initial risk preference is 0. The different values of λ_sand λ_d in the range of λ_s, λ_d ∈ (0, 1) are discussed as shown in Fig. 8.

Fig. 8

The moderates’ risk preference evolution graph of 8 risk factors with different values of λ_s and λ_d under the absence of persistence degree.

We can see from the Fig. 8 that, when the persistence degree of the moderates is absent, the risk preference values of the moderate group to 8 risk factors also reach an agreement at t = 1. Whenλ_s > λ_d, the trend curve of risk preference fluctuates less with the increase of λ_d. Until λ_s = λ_d = 0.5, the trend curve of risk preference reach the most stable state at t = 1 when the risk preference values are no longer changed. When λ_s < λ_d, the trend curve of risk preference fluctuated more with the increase of λ_d. And a significant W-shape fluctuation appeared.

The reason for this trend is that when λ_s ≠ λ_d ≠ 0.5, the moderates used different acceptance degrees to the influence of the homogeneous group and heterogeneous group respectively to obtain new risk preference values. And the new homogeneous group and the heterogeneous group are formed continuously during the evolution of their risk preference value. Different acceptability to the homogeneous and heterogeneous groups led to fluctuations in the curve. At the time of λ_s = λ_d = 0.5, the moderates gave up their risk preference completely. According to the evolution formula of risk preference, no matter how the set of the homogeneous groups and the heterogeneous groups are changed following the change of risk preference values, the moderates adopted the same acceptance degree to all the extremists after the stage t = 1. In that way, the moderate group’s risk preference values reach a completely stable state when t = 1.

From the above discussion, it is found that the trend of risk preference for 8 risk factors is the same in a certain stage. Therefore, to be easily observed and discussed, the risk factor x₁ is taken as an example.

(5) When λ_s = 0 and λ_d ∈ (0, 1), that is, the moderates don’t accept the influence of the homogeneous group, but accept the influence of heterogeneous group and retain their risk preference values to a certain extent. The simulation process is shown in Fig. 9.

Fig. 9

The moderates’ risk preference evolution graph of x1 with different values of λ_d under λ_s = 0.

As can be seen from the Fig. 9, when the moderates do not accept the influence of the homogeneous group, the consensus time is gradually shortened with the increase of λ_d. When λ_d = 0.5, W-type fluctuation gradually appears in the trend of the risk preference.

The reason for this trend is that when λ_d in a small value (0 < λ_d < 0.5), the persistence degree of the moderates to their risk preference is in a larger value. At this time, the moderates’ risk preference values are dominated by maintaining the initial risk preference value. It would lead to a lack of consensus among the moderate group in a short period. When λ_d is in a larger value (1 > λ_d ⩾ 0.5), risk preference values of the moderates are dominated by accepting the influence of the heterogeneous groups. Their risk preference values are changed gradually and largely from their initial risk preference values. This process makes the consensus time shorten and the W-type curve amplified gradually.

(6) When λ_d = 0 and λ_s ∈ (0, 1), the moderates don’t accept the influence of the heterogeneous groups, but accept the influence of homogeneous group and retain their risk preference values to a certain extent. The risk factor x1 is taken as an example to analyze shown in Fig. 10.

Fig. 10

The moderates’ risk preference evolution graph of x1 with different values of λ_s under λ_d = 0.

From Fig. 10, it is found that when the moderates don’t accept the influence of heterogeneous groups, the consensus time is also gradually shortened with the increase of λ_s.

The reason for the above trend is that when the moderates completely don’t accept the influence of the heterogeneous group, with the increase of λ_s, the risk preference values of the moderates are changed largely according to the risk preference of the homogeneous group and less according to their initial risk preference. In this process, the consensus time is shortened gradually.

From the above simulation results, it can be seen that the parameters λ_s and λ_d affect the evolution trend of the moderates’ risk preference and the consensus time in different degrees. The bigger the acceptance of the moderates to the extremists, the shorter the consensus time is. When λ_d > 0.5, the evolution trend of moderates’ risk preference appears W-type curve which shows that the moderate group can’t obtain an accurate risk preference value for a risk factor at a certain time.

6 Comparisons and discussions

In consensus reaching process of large-scale group decision making, the existing research focus is mainly from the perspective of opinion dynamics feedback mechanism, the influence of specific decision-makers, and linguistic expressions during the interaction of the process. Next, the characteristics of the proposed approach through comparison with existing studies from the above three perspectives are discussed.

(1) Opinion dynamics feedback mechanism. In [18], Zhang et al. applied the bounded confidence opinion dynamics model to provide the feedback under acceptable confidence level for experts to modify their opinions in the consensus reaching process. In [19], Li et al. studied cognitive dissonance behaviors by applying the bounded confidence model and concluded that the consensus opinion of all the agents will become more extreme with the increase in both the bounded confidence and connection probabilities. These studies reflect the decision-makers’ acceptance of the opinions in the bounded confidences and ignore the opinions outside the bounded confidences. However, in the actual emergency decision-making context, decision-makers may have contradictory psychology to uncertain opinions and will be open to all the opinions. The process of opinion evolution by distinguishing the acceptance of similar opinions and different opinions are discussed and simulated in our research.

(2) Influence of specific decision-makers. In [18], Zhang et al. applied a network partition algorithm to detect sub-networks of experts and further identified the opinion leaders. The decision-makers with the lowest individual consensus level will refer to the leaders’ opinions to adjust his/her opinion. In [20], Dong et al. proposed an algorithm to identify the stable agents with stable opinions and the oscillation agents with fluctuating opinions in the process of forming collective opinions. these studies indicate the influence of specific decision-makers, but these studies ignore the existence and the influence of the decision-makers with extreme preference. Our proposed approach classified the decision-making group into extremists and moderates from the perspective of risk preference in emergency context and focus more on the detailed study of the influence of extremists for consideration of influence quantification and influence declination.

(3) Linguistic expressions. In [21], Yu et al. applied the fuzzy envelope method to transform multi-granular fuzzy linguistic terms sets into trapezoidal fuzzy numbers to adjust evaluation values under the conditions of the minimum adjust cost in the consensus reaching process. In [22], Li et al. proposed an average consistency-driven model to set personalized numerical scales for linguistic terms with comparative linguistic expressions. These studies only focus on how the language itself more truly reflects the expression of the decision-makers but ignores the meaning of the language itself. They are missing studies that assess the extremes of language which is compensated in our research. The extreme opinions are identified and the behavior of the decision-makers with extreme opinions is explained in this research.

Our proposed approach proposes a novel evolution model of risk preference to study the influence of extremists on moderates with different acceptance in the group consensus reaching process under the context of emergency.

7 Conclusion

To make a wiser decision and obtain more comprehensive information, decision-makers will interact with each other to adjust their risk preferences. This research focuses on the influence of extremists on moderates and explores the risk preference interaction between the extremists and moderates in the emergency decision-making process of large-scale projects by using opinion dynamic theory. The main contributions are as follows.

(1) The characteristics of extremists and moderates in group emergency decision-making are recognized from the perspective of risk preferences. On this basis, the interaction decision-making process of extremists and moderates by introducing opinion dynamic theory is described.

(2) A new similarity model of risk preference is constructed considering directionality and distance of risk preference vectors. By using this model, the extremist group is classified into the homogeneous group and the heterogeneous group depending on the similarity to the moderates.

(3) The influence model of extremists considering influence duration and influence recession is constructed according to the actual emergency events. Then the acceptance degree of moderates to the influence of extremists in the homogeneous group and heterogeneous group is introduced.

(4) The risk preference evolution model of the moderates considering the influence of extremists is constructed and a case study of the large-scale projects is carried out.

Based on the models and case data analysis, it is found that the acceptance of the moderates to the influence of the extremists have an effect on the group consensus time and the evolution trend of risk preference. This study bridges the gap between opinion dynamics and group decision making. Meanwhile, the model inspires new explanations and new perspectives for the group emergency consensus process.

This research limits in studying the influence of extremists on the risk preference of the moderates. In the actual decision-making process, the interaction between the decision members is complex because there is more than the influence of extremists on moderates. In future research, interaction influence should be considered more widely.

Footnotes

Acknowledgments

This work is supported by the projects for the National Natural Science Foundation of China (No.71671189, 72073041), the Independent Exploration and Innovation Project of Graduate Students of Central South University (2017zzts046) and China Scholarship Council (201706370105).

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