5.2 A perfect mixing model
5.2.2 Simulation results
The simulation experiments are conducted using Repast Simphony. The agent model runs over a series of 20,000 periods with 1,000 public agents, a single producer agent, and a single media agent. Table 5.1 lists the input parameters and values used in the simulation. Some of these are social constants – parameters that would be expected to characterize an agent society. Ideally the values of these could be verified empirically, although in practice it seems very unlikely empirical information is actually available. Some of the parameters, however, define specific situations – for example, the high and low contamination levels. The value of these parameters is defined by the specific situation that the modeller wants to simulate. So the basis of these values is the modeller’s view of a typical, representative or simply interesting situation.
Table 5.1 Input parameters and values used in the simulation of a perfect mixing model
Input parameter Value Description
Maximum initial condition I 10-4 Defines initial risk and recreancy belief
Number of neighbours K 4 Number of neighbours in a perfectly mixed population
Low contamination level Clow
t 10-4 Level before and after crisisHigh contamination level Chigh
t 0.2 Level during the crisisContamination start period Tstart 2000 Time when the crisis starts
Contamination end period Tend 5999 Time when the crisis ends
Risk perception threshold B 0.15 Defines when a recall increases recreancy Recreancy increment D 0.25 Amount by which a recall increases
Recall voluntariness v t
0,1 Whether recall is voluntary or involuntary Recreancy variation E 0.35 Amount by which recall voluntarinesschanges recreancy
Weight of ‘event discovery’ 0.85 Weight given to ‘event discovery’ in the partial model with recreancy
Weight of ‘event discovery’ 0.65 Weight given to ‘event discovery’ in the full model
Weight of ‘recreancy’ 0.05 Weight given to ‘recreancy’ in the full model
Figures 5.3 through 5.18 present the traces of agent risk perception in a single run that correspond to the stages by which the perfect mixing model is developed. Outcome variables are mean public risk perception over agents and over time and mean risk amplification (the gap between the objective risk and public risk perception) over agents and over time. Simulation result based on equation (5.1) is not presented as the basic decision rule simply produces convergence on the mean of the initial beliefs.
First stage: public response to experienced contamination in a perfectly
mixed population
Figure 5.3 illustrates the evolution of individual risk perceptions in a single run with a contamination incident introduced (equation (5.3)). Before contamination occurs, public risk perception stays at a very low level. And during contamination it surges to a relatively high level and produces exogenous peaks that reflect the changes in risk magnitude. This assumes that direct experience is an important foundation of perception and plays a critical role in shaping subjective risk estimates of public actors, and that the risk perceived by consumers varies with the information available to them. In addition, the results evidently demonstrate that the exogenous peaks occur around the time period when the contamination incident is coming to an end – in this case because the growth of risk perception has not stopped by the time the contamination ceases, there is a turning point when the ceasing occurs. The turning point is quite sharp, so the growth in risk perception is still positive just before the turning point, but the growth clearly declines as the incident progresses. It is also essentially monotonic: there are no transient reversals in the growth until the main turning point is reached. Hence, public risk response to external influence appears somewhat predictable, given knowledge of the model parameters.
Figure 5.3 shows risk attenuation rather than risk amplification – by the end of the growth phase, the risk perception (in terms of subjective probability of contamination) is still less than the objective risk (the actual contamination probability). But it is hard to say whether this is
Figure 5.4 and Figure 5.5. Risk amplification occurs if the contamination level is very low (Figure 5.4), and risk also becomes attenuated if the contamination level is very high (Figure 5.5). This indicates that risk amplification only occurs if the contamination level is below a certain threshold. The fluctuation of standard deviation in Figure 5.4 suggests that low contamination raises the disagreement among individuals’ responses to the contamination crisis. Furthermore, it has been observed that a former model with longer period of contamination generates risk amplification with lower amplification threshold. In Chapter 7 both the contamination level and contamination duration are parameterised to inspect the sensitivity of the model to these parameters.
Figure 5.3 Trace of public risk perception in a single run of a perfect mixing model with contamination
0.2high
Figure 5.4 Trace of public risk perception in a single run of a perfect mixing model with contamination
0.001high
C t
Second stage: addition of a product recall event in a perfectly mixed
population
Figure 5.6 demonstrates the trace of agent risk perception in a single run when product recall is also considered as an exogenous effect (equation (5.4)). It shows that publicity of product recall triggers a very strong growth of risk perception – this is as expected from the model as recall is assumed to associate with a significant effect on risk perception. More importantly, there is a considerable discrepancy between mean public risk perception and the contamination level – there is evidently risk amplification. In the model the consumers’ decision rule gives a value to the recall element of 0 or 1, not the objective contamination probability. Therefore, the public estimate of the risk can become amplified above the objective level.
Figure 5.6 Trace of public risk perception in a single run of a perfect mixing model with recall (a t
2003
1)Figure 5.7 presents the comparison between model with contamination (Figure 5.3 on the left) and model with product recall (Figure 5.6 on the right). After contamination is removed, the decline rate is much faster in Figure 5.6 than in Figure 5.3. This is because the absence of product recall facilitates the relaxation of risk perception from a fairly high level to a very low
level.
Figure 5.7 Comparison between model with contamination (Figure 5.3 on the left) and model with recall (Figure 5.6 on the right)
Third stage: addition of recreancy in a perfectly mixed population
Figure 5.8 shows the trace of public risk perception and recreancy belief in a single run with a voluntary recall (equation (5.8) with v t
1). Comparison between model with recall (Figure 5.6 on the left) and model with voluntary recall (Figure 5.8 on the right) is presented in Figure 5.9. A t-test indicates that there is a significant difference between Figure 5.6 and Figure 5.8 in peak mean risk perception (t df
950.297
428.738 and p0.001) and residual mean risk perception ( t df
611.801
4,668.792 and p0.001 ) across 500 runs with a significance level of 0.05. Compared with Figure 5.6, Figure 5.8 exhibits a relatively slow growth followed by a relatively slow decay, and there is a lower degree of risk amplification. This is because voluntary recall decreases agents’ recreancy belief in the producer and thus positively affects their risk beliefs. In other words, the indirect effect (i.e. recreancy) of a voluntary recall can somewhat diminish its direct effect (i.e. product recall information) and thus lessens consumers’ perceptions of risk.In particular, in Figure 5.8 recreancy belief surges to a high level after the recall is completed and then becomes level. In the model when there is no recall in force, an agent will increase its recreancy belief by some amount if the risk it perceives is above some threshold (which is set as 0.15) and keep its recreancy belief unchanged otherwise. Therefore, recreancy belief continues to increase as risk perception stays above the threshold and becomes stabilised when risk perception falls below the threshold. And the constant high level of recreancy is the reason why risk perception stabilises at a higher level in Figure 5.8 than in
Figure 5.8 Trace of public risk perception and recreancy belief in a single run of a perfect mixing model with recall timing and voluntary recall (a t
2005
1,v t
1)Figure 5.9 Comparison between model with recall (Figure 5.6 on the left) and model with voluntary recall (Figure 5.8 on the right)
Figure 5.10 shows the trace of mean risk belief and recreancy belief over time with an involuntary recall (equation (5.8) with v t
0). Comparison between model with recall (Figure 5.6) and model with involuntary recall (Figure 5.10) is given in Figure 5.11. Figure 5.6 and Figure 5.10 are statistically different in peak risk perception ( t df
965.768
132.019 and p0.001 ) and residual risk perception(t df
522.098
5, 276.382 and p0.001) over 500 runs with a significance level of 0.05. Figure 5.11 demonstrates that involuntary recall generates a lower level of risk amplification and a much higher residue of concern. This is what the modeller expects based on the decision rule. The explanation for the lower amplification is that, involuntary recall increases recreancy belief, but risk perceived from recreancy is much lower than that from recall information, so involuntariness to some extent reduces the amplification effect of recall information and leads to a lower degree of amplification. There is no recall after the crisis, so recreancy belief continues to increase until risk perception drops to a certain threshold, resulting in a higher residual risk perception after the crisis.Figure 5.10 Trace of public risk perception and recreancy belief in a single run of a perfect mixing model with recall timing and involuntary recall (a t
2001
1,v t
0)Figure 5.11 Comparison between model with recall (Figure 5.6 on the left) and model with involuntary recall (Figure 5.10 on the right)
Figure 5.12 shows the comparison between model with voluntary recall (Figure 5.8 on the left) and model with involuntary recall (Figure 5.10 on the right). There is a statistically significant difference between them in terms of peak mean risk perception ( t df
996.147
326.958 and p0.001 ), peak mean recreancy belief ( t df
684.217
2,894.413 and p0.001 ), and post-crisis risk perception (t df
692.890
2,846.114 and p0.001) over 500 runs with a significance level of 0.05. This is in line with the decision rules in the model. During the crisis an agent increases its recreancy belief when the producer implements an involuntary recall and decreases its recreancy belief when a recall is made voluntarily. The magnitude of risk amplification is therefore higher in Figure 5.10 than in Figure 5.8. Also, Figure 5.10 displays a higher stabilised risk perception and recreancy belief after the crisis. This is mainly due to the reason that in an involuntary recall event it takes a longer time to reduce agent risk beliefs to the threshold that defines when a recall increases recreancy, leading to a higher recreancy belief and residual risk perception. In reality, when an involuntary recall comes into force, consumers tend to feel that the company is not socially responsible in dealing with the crisis, to have a more negative impression of the company, and to perceive the product as more dangerous.Figure 5.12 Comparison between model with voluntary recall (Figure 5.8 on the left) and model with involuntary recall (Figure 5.10 on the right)
Fourth stage: addition of media in a perfectly mixed population
The effects of roles that the media assumes on public perceived risk are examined in the light of voluntary recall and involuntary recall. Figure 5.13 demonstrates the trace of individual agent beliefs when the producer makes a recall voluntarily and the media acts as an objective leader, a mixed leader-follower, and a public follower, respectively (equation (5.10) with
1v t ). Risk amplification – that is, a collective perception that exceeds the objective risk
level – occurs regardless of the role that media assumes. The objectivity of media coverage appears to be inversely related to risk amplification: a media that simply follows public opinion is associated more strongly with exaggerated risk perceptions than an objective one. Another insight is that for all of the three roles of media, risk amplification seems to decrease with the contamination level. Particularly, risk amplification will increase significantly if the contamination level is very low (Figure 5.14) and decease considerably if the contamination level is very high (Figure 5.15). Sensitivity analysis presented in Chapter 7 explores how contamination level affects peak risk amplification.
It is significant that when the crisis finishes, and, the contamination has fallen to its original level, when the media is a public follower the public risk perception remains very high – it is not corrected by the reduction in objective risk. So risk amplification and the role of media are important not just at the start of a crisis but at the end. It will be hard for crisis managers to end a crisis if the media is a strong follower of public opinion.
Figure 5.13 Trace of public risk perception in a single run of a full perfect mixing model with voluntary recall (a t
2008
1,v t
1) and contamination Chigh
t 0.2Figure 5.14 Trace of public risk perception in a single run of a full perfect mixing model with voluntary recall (a t
2008
1,v t
1) and contamination Chigh
t 0.05Figure 5.15 Trace of public risk perception in a single run of a full perfect mixing model with voluntary recall (a t
2008
1,v t
1) and contamination Chigh
t 0.95Figure 5.16 shows that if the producer issues a recall involuntarily, risks will be intensified regardless of the role of media (equation (5.10) with v t
0). Both risk amplification and residual risk perception are higher than in the case where a recall is made voluntarily. There is an inverse relationship between the objectivity of media coverage and risk amplification. Figure 5.16, Figure 5.17, and Figure 5.18 show that social risk amplification decreases with the level of contamination.The model does not explore possibilities in which the media leads public opinion, but leads it with a belief that is different from the objective level of risk. A theory that claims the media will communicate exaggerated stories in order to sell more newspapers or TV viewing might lead with a very high risk belief.
Figure 5.16 Trace of public risk perception in a single run of a full perfect mixing model with involuntary recall (a t
2003
1,v t
0) and contamination Chigh
t 0.2Figure 5.17 Trace of public risk perception in a single run of a full perfect mixing model with involuntary recall (a t
2003
1,v t
0) and contamination Chigh
t 0.05Figure 5.18 Trace of public risk perception in a single run of a full perfect mixing model with involuntary recall (a t