Regional differences in cognitive dissonance in evacuation behavior at the time of the 2011 Japan earthquake and tsunami

This paper constructs an evacuation decision-making model that takes cognitive dissonance into consideration. The purpose of this construction is to clarify the psychological mechanism for the evacuation behavior of residents during an emergency, based on Akerlof and Dickens (Akerlof and Dickens Am Econ Rev 72:307–319, 1982) "The economic consequences of cognitive dissonance". Specifically, we empirically explore people’s psychological mechanism (e.g., cognitive dissonance) for evacuation behavior when a tsunami disaster occurs. As a result, we show that the level of anxiety depends on the area where residents live and that the average anxiety of residents is mostly correlated with the level of damage of past disasters, and that it is affected also by the ages of residents. Since the level of anxiety largely affects an individual’s evacuation behavior, this result can indicate for what kinds of people intervention and assistance are required based on the level of anxiety. A high level of anxiety basically promotes evacuation. Since our results show that anxiety is increased by the experience of tsunamis, education having people virtually experience tsunamis may increase evacuation rates efficiently.


Introduction
Some natural disasters take many human lives instantly. A tsunami is one such disaster. The Great East Japan Earthquake on March 11, 2011, which recorded a magnitude of 9.0, produced a tsunami that struck a wide area about 32 min after the earthquake, and over 18,000 residents lost their lives. The Indian Ocean Earthquake off the coast of Sumatra on The list of authors is in alphabetical order. College of Sport and Health Science, Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577, Japan December 26, 2004, which recorded a magnitude of 9.1, produced a tsunami that struck many Asian countries, killing approximately 220,000 people. The island of Sumatra, Indonesia was struck by the first wave approximately 15 min after the earthquake. From records of past such disasters, it is well known that achieving early evacuation is effective as a method of minimizing the damage caused by a tsunami, according to the Japanese Central Disaster Management Council (2003). On the other hand, there are many reports that residents often hesitate to evacuate even after the emergency public announcement of the forecast of a big tsunami. So, in order to achieve early evacuation, we need to elucidate the psychological mechanism and then implement disaster prevention policies in accordance with this mechanism.
As will be shown in Sect. 2, none have modeled the evacuees' psychological mechanism for people's evacuation behavior. Although many evacuation simulations have been developed, in the simulations, the start point of an individual's evacuation was given by the temporal distribution of starting evacuation based on the past records or the results of surveys conducted after the evacuation. To see the effects of various disaster prevention policies, we have to clarify the psychological mechanism for starting their evacuation.
Various disaster prevention policies can be implemented to achieve the early evacuation of residents (e.g., establishing evacuation routes and raising awareness of evacuation through disaster prevention education). In order to ascertain what policies improve their early evacuation rate, it is essential to understand the decision-making mechanism that expresses how each resident decides whether or not to evacuate.
A logical decision-making model can be considered for evacuation by minimizing costs or maximizing utility in the evacuation. However, people do not necessarily behave logically, and it has been noted that, particularly during a disaster, people hesitate to evacuate due to various irrational factors, one of which is cognitive dissonance (Hirose 2004). In addition, it is pointed out that they have a normalization bias in which the subjective probability for risk is irrationally low. Cognitive dissonance is the occurrence of dissonance when a person holds multiple cognitions that are inconsistent with each other. When cognitive dissonance occurs, it is thought that it produces the motivation to reduce the cognitive dissonance that causes psychological discomfort.
In the context of tsunami evacuation behavior, after a big earthquake, people have two cognitions: (1) a big tsunami might occur and take their lives, and (2) evacuation might be an overreaction to this earthquake. In this situation, people avoid this cognitive dissonance by supposing that the subjective probability of occurrence of a big tsunami is very low or zero. This conforms with the normalization bias.
The current paper, based on a microfounded model basically following Akerlof and Dickens (1982), estimates the psychological parameters of the model, using questionnaires on the actual evacuation behaviors at the time of Japan's 2011 earthquake and tsunami. This approach leads to empirical elucidation of the psychological decision-making mechanisms for evacuation. A quantitative analysis of the mechanisms is indispensable for evaluating evacuation policies. To perform quantitative analyses, we need a theoretical model. Actually, a mathematical model representing the effect of cognitive dissonance on people's behavior has been constructed by Akerlof and Dickens (1982). To our knowledge, our analysis is the first quantitative analysis of the Akerlof and Dickens model.
The remainder of the paper is organized as follows. In Sect. 2, we review past studies to contrast the purpose of our paper with the past studies. In Sect. 3, we construct an evacuation behavior model that modifies the model of Akerlof and Dickens (1982). In Sect. 4, we introduce an evacuation behavior questionnaire and arrange the data used for the empirical analysis. Section 5 sets the exogenous cost parameters and estimate the model's parameters.
In Sect. 7, we discuss our results in the context of the past studies and design evacuation policies based on our estimated anxiety costs. In Sect. 7, we provide a conclusion.
There are several approaches to explain non-evacuation behavior. Applying the prospect theory of Kahneman and Tversky (1979) to this behavior is valid. In their prospect theory, the value function is normally convex when people experience losses. That is why people take risk-seeking actions when experiencing losses. As a result, they choose not to evacuate because staying at home has a higher level of uncertainty. However, this does not explain why people suppose an irrationally low probability of occurrence of tsunami.
The theory of cognitive dissonance proposed by Festinger (1957) is frequently discussed in sociology and psychology, while in economics it has been mathematically modeled by Akerlof and Dickens (1982) and Rabin (1994). There have also been studies such as Dickens (1986), which modified and developed the model in his previous study and modeled the relationship between the mechanism causing crime to occur and cognitive dissonance, and Balestrino and Ciardi (2008), who modeled the relationship between the timing of marriage and cognitive dissonance. However, these studies, including Akerlof and Dickens (1982), were limited to a qualitative analysis and did not have a quantitative analysis. Other studies (e.g., Wood and Miller (2020)) includes cognitive dissonance when discussing evacuation, but they have not constructed a mathematical model.
To our knowledge, in the field of economics, only Ida et al. (2015) quantitatively analyze the effect of cognitive dissonance although many economic papers cite the economic theoretical modeling of cognitive dissonance by Akerlof and Dickens (1982). They examined a choice-induced preference change, which is captured by the same questionnaire method as in some previous psychological papers (e.g., Brehm 1956;Steele et al. 1993;Chen and Risen 2010;Izuma and Murayama 2013). Ida et al. asked people first to rate several choices according to their preference, and second to choose between two of the items that had similar preference ratings in the first stage. Finally, people are asked to rate their preference for the chosen items one more time. If the second rating is different from the first rating, that represents cognitive dissonance because it implies that their choice behaviors affect their preference (i.e., "I like it because I chose it"), as Ida et al. explain. Ida et al. succeeded in showing the existence of cognitive dissonance in Japanese people's attitude about the trade-off between nuclear power generation and avoiding an increase in electricity rates.
However, while they show the difference in people's preference before and after their choice behavior, Ida et al. (2015) do not clarify the microfounded generation mechanisms of the cognitive dissonance.
1 3 2 Methodology: the evacuation behavior selection model Akerlof and Dickens (1982) mathematically explain the phenomenon that workers working in a hazardous environment, in which their lives could be in danger, may underestimate the risk due to cognitive dissonance. The time period in the model consists of two periods: In the first period, the workers cannot avoid the risk of accidents, but in contrast, in the second period, they can avoid this risk by purchasing safety equipment. The workers who correctly estimate the risk of accidents purchase safety equipment, but the workers who underestimate the risk due to cognitive dissonance do not purchase this equipment.

The Akerlof and Dickens (1982) model and its modifications
The purchasing behavior is expressed by dividing it into three stages. In the first stage, the worker's threshold value of his subjective probability for whether or not to purchase safety equipment is derived. In the second stage, the worker uses the derived threshold value of subjective probability and selects the subjective probability that minimizes costs when purchasing or that when not purchasing the equipment. Finally, in the third stage, the worker compares the costs and decides whether or not to actually purchase the equipment. Figure 1 shows the behavior decision process in Akerlof and Dickens (1982). The variables underlined in Fig. 1 are the selections at each of the stages and the equation number under the line denotes the corresponding equation in the current paper. In the current paper, in the third stage, "wait" or "evacuate" is chosen instead of "No purchase" or "Purchase" in Akerlof and Dickens (1982).
We will formulate the costs in the events of evacuation and non-evacuation in a situation in which there are fears that a tsunami will strike. Following the sensations of the tremors from the earthquake and the issuance of a tsunami warning, for the residents in coastal areas where it is thought that a tsunami will strike, residents select 1) the subjective probability of death from the tsunami and 2) whether or not to evacuate. This setting is essentially the same as that in Akerlof and Dickens (1982), in which workers decide 1) the subjective probability that an accident will occur and 2) whether or not to purchase safety equipment.
On applying Akerlof and Dickens (1982) to evacuation behavior, we made the following three modifications.
Modification 1: In Akerlof and Dickens (1982), as the results of the modeling, in the event that the worker selects to not purchase safety equipment, the subjective probability = 0. This implies that cognitive dissonance results in a complete elimination of the subjective risk. However, this result is an extreme case. So, we introduce a sense of resistance Fig. 1 Behavior decision process to the divergence between the objective probability and the subjective probability of death from the tsunami to our model. As a result, residents will not completely eliminate subjective risks.
Modification 2: In Akerlof and Dickens (1982), before purchasing safety equipment, in the first period, which corresponds to before the start of the tsunami evacuation in this study, they assume that workers have never encountered risks associated with an accident. However, in a tsunami disaster in reality, it can be considered that people will have been killed by the tsunami before the start of the evacuation, so we modified the model so that it takes into consideration the possibility of people having died before the start of the evacuation.
Modification 3: In Akerlof and Dickens (1982), while the anxiety cost term is set to be proportional to the subjective probability, the expected cost of death is set to be proportional to only the objective probability, not the subjective probability. This inconsistent treatment seems unrealistic and unnatural. Therefore, in this study, we have modified the model so that both anxiety and the expected cost of death are proportional to the subjective probability.
All parts except for the three modifications follow Akerlof and Dickens (1982). Table 1 shows the comparison of our model with the Akerlof-Dickens model with respect to several factors. The following expresses our model mathematically. The symbols used for the formulation of costs are as follows: p : subjective probability of death, q : objective probability of death, C d : cost of death, C f : cost of anxiety, C m : cost of moving, h : the minimum time required to prepare for evacuation, h : the time from the start of the evacuation until the warning is issued (where h + h = 1 without loss of generality) , and : the sense-ofresistance parameter. Figure 2 shows the summary of the model. Residents pay the cost of anxiety and the cost of death until evacuation is completed, also, the cost of moving when evacuating. Meanwhile, residents pay the cost of expected anxiety, the cost of death, and the cost of resistance when not evacuating.
The time span of our evacuation model is totally different from the Akerlof and Dickens model. In our case, cognitive dissonance arises in a short time after an earthquake. People in danger from a tsunami do not hurry to evacuate because the first tsunami will come between about 20 min and 20 h after the earthquake. When the tsunami comes after the earthquake, it depends on the geographical position of the seismic center and the magnitude of the earthquake. In addition to the interval between the quake and the tsunami, the possibility of a tsunami is very low, and the precision of the forecasted scale is not fine. So, even if they immediately decide to evacuate, they will prepare for evacuation. For example, they take out some clothes from their closets for several days of evacuation and may move some things upstairs for them to prevent them from being soaked by a tsunami. We set this fixed time for preparation for evacuation as h . This is important when applying the Akerlof and Dickens model to evacuation behavior.

Details of each stage
(a) The first stage: determining the threshold value of subjective probability p Residents compare the total cost of evacuation with that of staying in order to decide whether to evacuate. The costs are composed of the following items: C m , the cost of moving incurred from the evacuation after the start of the evacuation behavior; phC d , the cost of expected death incurred when there is no evacuation; phC f , the cost of expected anxiety; and h(q∕p) , the cost of resistance. The cost of resistance becomes larger as the subjective probability of death becomes smaller than the objective probability of death, and it expresses the cost of resistance due to fear of misjudgment of the probability of death caused by the tsunami. We derive the threshold value of subjective probability. The inequality condition (1) expresses a judgment when a resident selects to evacuate or not. This judgment depends on what level the resident sets his subjective probability, p , at. The threshold p is expressed by inequalities (2) and (3), which are derived from the judgment formula (1).
(b) The second stage: determining the subjective probability. In the second stage, from the range of the subjective probability selections obtained in the first stage, the subjective probability that minimizes the total cost is selected separately when the resident decides to evacuate and when she/he decides to not evacuate. As a result, the subjective probability p 1 shown in Formula (6) is selected when evacuating and p 2 in Formula (7) when not evacuating. In Akerlof and Dickens (1982), people chose a subjective probability, corresponding to p 2 , of zero. In our model, since we introduce a sense of resistance to the divergence between the objective probability and the subjective probability of death, residents will not completely eliminate subjective risks. In the third stage, the selected subjective probability in the second stage is used, both costs across the first and second periods are compared, and whether or not to evacuate is selected. As a result, the cost when evacuating can be expressed by Formula (8), and the cost when not evacuating by Formula (9), and each agent selects the smaller of these costs.

Comparative statics with respect to parameters
First, in order to analyze the effects each of the parameters in this model have on evacuation behavior, the evacuation behavior judgment formula f is defined by subtracting the cost of not evacuating Formula (9), from the cost of evacuating, Formula (8), and this becomes Formula (10). By differentiating Formula (10) with respect to each parameter, we can obtain the changes in the evacuation trends that occur with changes in each parameter.
If Formula (10) is negative, "evacuate" is selected, and if Formula (10) is positive, "do not evacuate" is selected. Therefore, by differentiating Formula (10) with each parameter and finding out its functional form, we can obtain the tendency toward evacuating or not evacuating. Each parameter shall be within the following range from what is realistically valid.
Consequently, Formulas (12) to (15) are obtained. We understand that evacuation becomes difficult as the cost of moving C m increases, and that evacuation becomes more likely as the cost of death C d , the cost of anxiety C f , and the objective probability p increase, confirming that an evacuation tendency conforming to actual evacuation behavior was appropriately expressed.
The disaster case and the evacuation behavior questionnaire

Introduction
The 2011 Japan Earthquake which recorded a magnitude of 9.0 occurred at 14:46 (Japan time) on March 11 at the east and southeast off the coast of the Oshika Peninsula, Miyagi Prefecture. The evacuation alert information was given as follows. First, at 14:49 on March 11, the Meteorological Agency issued a tsunami warning (forecasted height, 3 m) in Iwate Prefecture, Miyagi Prefecture, and Fukushima Prefecture. After that, the level of the forecasted tsunami was increased at 15:14 on March 11. The arrival of the tsunami was observed from 15:20 this day. In order to estimate the parameters, we employ the evacuation behavior questionnaires performed by the City Bureau of the Ministry of Land, Infrastructure and Transport, Japan. The questionnaire, "Digital Archive of the Great East Japan Earthquake and Tsunami Disaster Recovery Support Survey of Damaged Cities" was carried out for 49 municipalities in 6 prefectures. In this survey, personal attribute data of 10,240 samples and a total of 24 items such as evacuation place, route, and flooded area are collected.

Samples used for analysis
In this research, since it is necessary to secure a certain number of samples for analysis for each municipality, we target 22 municipalities that have more than 100 samples (total number of samples: 8369). Table 2 shows the municipalities used for analysis, the number of samples and the evacuation rate. The municipalities are listed downwards in terms of evacuation rates. The highest evacuation rate was about 92% in Oduchi, and the lowest evacuation rate was 48% in Asahi.
Considering that the characteristics of people's evacuation behavior are likely to be affected by their age, we set three population categories: age group 1 (49 years old or younger), age group 2 (50 to 69 years old), and age group 3 (over 70 years old). We estimate the parameters of evacuation decision by age group.
The number of samples and the evacuation rate of these three categories are shown in Figs. 3 and 4. In these classifications, the evacuation rates are in the range between 70 and 78%, depending on the gender and the age. The minimum rate is about 71% in the case of males in age group 1, while the maximum rate is about 78% in the case of females in age group 2.

Introduction
We will show the policy for the setting of the parameters. We formulized the evacuation decision-making model shown by Formula (10) using the objective probability of death from the tsunami q , cost of death C d , cost of moving C m , cost of anxiety C f , time ratio from the start of the evacuation action to the end of the evacuation action h , the shortest preparation time required for the evacuation behavior from the occurrence of the disaster h , and the sense-of-resistance parameter . The exogenous cost parameters other than the cost of anxiety can be calculated from the previous studies. Based on these exogenous parameters, we estimate the cost-of-anxiety parameter. We divide the setting of the parameters in the evacuation decision-making model into two stages: (1) setting the exogenous costs and the various parameters, and (2) using the non-aggregated questionnaire data to estimate the distribution of the cost-of-anxiety parameter.

Setting the exogenous cost parameters
First, in order to set the exogenous costs and various parameters, we referred to previous research involving cost-benefit calculations, and studies of the statistical values of life based on the willingness to pay for the objective probability of death from a tsunami p , the cost of death C d , the cost of moving C m , evacuation preparation time, the estimated evacuation time, and the sense-of-resistance parameter. We will explain the parameters one by one as follows (a) The objective probability of death q The objective probability of death of an individual from the tsunami, q, is calculated by dividing the total deceased and missing people in the 2011 Japan Earthquake and Tsunami by the daytime population municipality by municipality. The number of the daytime population, the total number of deceased and missing people, and the objective probability are shown for each municipality in Table 3.

(b) The cost of death C d
We calculated the cost of death C d . In this study, the cost of death C d is the value of the lives of the residents themselves that are lost when they die in the tsunami. In this study, this cost is set as the willingness to pay for avoiding the risk of death. We set this with reference to the survey-research report on an economic analysis of damages and loss from traffic accidents (2012) by the Director General for Economic and Fiscal Management, the Cabinet Office, who statistically estimates the value of life (and who is responsible for the policies for a cohesive society) (Itaoka et al. (2005)). In the survey-research report on an economic analysis of damages and loss from traffic accidents (2012), the statistical value of life was calculated based on the willingness to pay for the risk of dying in a traffic accident. According to this report, the per capita loss from death is 213 million yen. Also, in the study of Itaoka et al. (2005), they estimated the willingness-to-pay amount for four age categories (40 to 49 years, 50 to 59 years, 60 to 69 years, and 70 years and over), calculated the statistical value of life in each category, and showed that the statistical value of life for people aged 70 years and above tends to be smaller than that of people aged under 70 years. For the cost of death used in this study, we set the cost for people under 70 years old as 213 million yen, and the cost for people aged 70 years and over as 77 million yen, which was calculated in the study of Itaoka et al. (2005). Table 4 shows the cost of death.   working hours used for the calculations, we used the values from the salary census of 2010. C i time is defined by Formula (17).
C i physical+mental signifies the mental and physical costs incurred from the evacuation behavior. This cost is set based on previous studies. Although there has been no research that attempted to measure C i physical+mental as the mental and physical costs resulting from the evacuation behavior, Sato et al. (2002) converted into monetary values the mental and physical costs of transfer (between trains) behavior in the process of quantifying resistance to transfer behavior at stations as a generalized transfer cost. In this study, C i physical+mental , the mental and physical costs that are incurred during evacuation behavior, is set to be the energy value and the mental burden in the generalized transfer cost of Sato et al. Sato et al. calculated the total cost of energy value and the mental burden for the elderly aged 70 years and over to be 74.01 [yen∕min] , and the value for other pedestrians to be 26.26 [yen∕min] . We use these values. The cost of moving per unit of time C unit m calculated from the above-described sequence is shown in Table 5. The final cost of moving is obtained from Formula (18), from the moving time of each resident until arriving at the evacuation site, using the cost of moving per unit of time C unit m .
In this study, based on the results of a survey of actual conditions of natural walking conducted by Akutsu (1975), the walking speed was set by gender and age category. We use his result as the walking speeds to calculate the cost of moving. Table 6 shows walking speeds for each citizen category. Additionally, moving speed of cars and bicycles were set based on the walking speed as shown in Table 7.
Average annual salary in category i Actual working hours per month × 12(months) × 60(minutes) We use the evacuation preparation time T prepare and evacuation time T evacuate to calculate h , which is the time ratio from the start of the evacuation behavior until the time when the danger of a tsunami has passed (i.e., residents stops feeling anxiety); h , which is the time ratio of the preparation time required for the evacuation behavior; and the cost of moving C m . Evacuation preparation time T prepare is the shortest time required for evacuating. In this study, we suppose T prepare = 35minute , referring to the evacuation starting times of residents who responded to the questionnaire. Regarding the evacuation time T evacuate , we suppose T prepare = 120minute . These settings are summarized in Table 8. (e) Sense-of-resistance parameter The sense-of-resistance parameter determines the difference between the subjective probability perceived by each resident and the objective probability. This parameter should be set so that the cost-of-anxiety threshold value is nonnegative for all the samples analyzed (number of samples, n = 8,369). We assume that = 0.1.

Estimating the distribution of the cost of anxiety
By rearranging the evacuation behavior judgment formula f = 0 shown in Formula (10) with regard to individual i 's cost of anxiety C i f , the cost-of-anxiety threshold value C i fc shown in Formula (19) is obtained as Individual i evacuates when his or her anxiety C i f is greater than the anxiety threshold value C i fc C i f > C i fc . In other words, the probability that individual i will evacuate is expressed by the cumulative probability to the right of C i fc in the probability density function, as shown in Fig. 5. Therefore, if we assume that the cost of anxiety follows a certain probabilistic distribution, we can estimate the distribution, using the non-aggregated data from the results of the evacuation behavior questionnaire and the cost-of-anxiety threshold value C i fc of individual i. We set a logistic distribution as the probabilistic distribution in the following manner. The average and variance of residents' anxiety are thought to differ according to age, gender, and whether or not they heard a tsunami alert, checked the hazard map, and participated in a local evacuation drill. Therefore in this study, we take these factors into account as "categories." Anxiety C f of residents in category k, as in Formula (20), is assumed to be the sum of the average value of the residents' anxiety in each category Ĉk f and the variable term. The variable term is expressed as the product of the scale parameter k that represents the variance of the probability and the probability term i expressing the logistic distribution of parameters ( , ) = (0,1) . In other words, among residents who belong to a certain category, there will be variations in the cost-of-anxiety parameter between them as individuals, but as a whole, the value of C f will follow a logistic distribution. The probability density function of the logistic distribution and the probability function can be expressed by Formula (21) using the average parameter and the variance parameter . One problem with using a logistic distribution is that the area of the distribution on the left side of the y-axis has negative values of cost of anxiety (see Fig. 5). Negative cost of anxiety is difficult to interpret. However, if we assume that these negative values are all zero cost of anxiety, there will be no problem, noting that C i f is positive. In other words, the negative areas can be used only for conveniently calculating the probability.
As individual i evacuates when his or her anxiety C i f is greater than the anxiety threshold value C i fc C i f > C i fc , the probability of evacuation can be expressed by Formula (22). As Eq. (22) shows, function is composed of two terms: the term mainly determining the variance of the cost of anxiety, 1∕ k C i fc , and the term including the average value Ĉk f , Ĉk f ∕ k . Since the average and variance of residents' anxiety are thought to differ according to age, gender, and whether or not they heard a tsunami alert, checked the hazard map, and participated in a local evacuation drill, both terms can be expressed as the function of these attributes. So, we set Formulas (23) and (24) using dummy variables.
The dummy variables within Formulas (23) and (24) are shown as x sex : sex dummy ( x = 1 when individual i is male; x = 0, female), x age1 : young aged dummy ( x = 1 w hen individual i is age group 1; x = 0 , others), x age2 : middle aged dummy ( x = 1 w hen individual i is in age group 2; x = 0 , others), x alert : alert dummy ( x = 1 when individual i heard a tsunami alert or call for evacuation;x = 0, others ), x sign : sign dummy ( x = 1 when individual i had seen a sign marking previous tsunamis; x = 0 , others), x map : map dummy ( x = 1 w hen individual i checked the tsunami hazard map; x = 0 , others), and x drill : drill dummy ( x = 1 when individual i participated in a local evacuation drill; x = 0, others ), x municipaln : municipal dummy ( x = 1 : When individual i lives in municipality n; x = 0 , others). (21) Based on this setting, the likelihood function is set as in Formula (25) and estimated using the maximum likelihood method.
We will explain the results of the method of regression method within Formulas (23) and (24) and the examination of validity. We performed this regression analysis using statistical analysis software, namely the "multiple logistic regression analysis" tool in Excel Statistics 2010. To investigate the goodness of fit of the model, we used McFadden's pseudo-decision coefficients R 2 and AIC, and the hit rate as the indicators. We judged the validity of the parameters using the significance of the regression coefficients and the validity of the signs.
We estimated all the combinations of dummy variables, and we adopted the best combination that has the highest AIC, likelihood ratio, and hit rate. Formula (27) shows the regression equation with only significant parameters. Table 7 shows the regression coefficients of Formula (27).

Analysis of the estimated anxiety costs and their effects on evacuation rates
There are two kinds of coefficients: coefficients related to average and coefficients related to variance . Looking at Table 9, we can see that the coefficients related to variance do not satisfy the significance level for the p-value and we cannot grasp the difference by attribute. However, the coefficients related to average satisfy the significance level for the p -value and we can grasp the difference by attribute. Therefore, we examine the difference in average anxiety.
(24) 1 k = 0 + 1 x sex + 2 x age1 + 3 x age2 + 4 x alert + 5 x sign + 6 x map + 7 x drill + n x municipal n (n = 1 ∼ 22) First, we explore the difference in average by area. Figure 6 shows the average anxiety level by area. Average anxiety level shows average anxiety in each area with the anxiety of Ishinomaki as numeraire = 1. Figure 7 shows the relationship between the maximum  Fig. 6 Average anxiety level by area run-up height of the Sanriku earthquake tsunami or the Chile earthquake tsunami and the average anxiety level estimated in the current paper.
The top 8 regions with shaded bars in Fig. 6 are the area which suffered flood damage by either the 1960 Chile earthquake tsunami or the Sanriku earthquake tsunami. As shown in Figs. 6 and 7, the average anxiety of a resident who lives in the areas which suffered flood damage by a large-scale past disaster is large. Moreover, average anxiety of residents is roughly correlated with the damage of past disasters.
Next, we explore the difference in average anxiety level by sex, age and whether they heard an alert or not. In order to see what level of evacuation rate corresponds to the difference in anxiety, we consider three comparisons.
Comparison 1: A comparison of the evacuation rates between the cases when all residents have a "female" level of anxiety that the female has and the case when all residents have a "male" level of anxiety.
Comparison 2: A comparison of the evacuation rates between the case when all residents have the anxiety level of age group 3 and the case when all residents have the anxiety level of age group 1.  Comparison 3: A comparison of the evacuation rates between the case when all residents have the anxiety level of people who heard an alert and the case when all residents have the anxiety level of people who did not hear an alert. Table 10 summarizes the three comparisons in terms of average evacuation rates. As shown in Table 10, the anxiety felt by females is larger than that by males and the difference in anxiety by gender is equivalent to the difference in the evacuation rate 2.546% (comparison 1). The difference by gender category may be caused by a difference in the role in the family during evacuation. The anxiety of age group 3 is larger than that of age group 1, and the difference in anxiety is equivalent to the difference in evacuation rate 3.278% (comparison 2). The difference by age category may be caused by their past disaster experience. Anxiety of residents who heard an alert is larger than that of residents who did not hear an alert. This difference in anxiety is equivalent to the difference in the evacuation rate 2.926% (comparison 3). This result shows that it is effective to issue an alert for achieving early evacuation.

Discussion with policy design
As the literature in evacuation practices shows, cognitive dissonance is a major irrational factor in evacuation. People reduce the subjective probability of tsunami occurrence with irrational reasons. But, as shown in our literature review section, previous evacuation models have not considered the mechanisms of cognitive dissonance. Section 3 applies the Akerlof-Dickens model to evacuation behavior with minor modifications, and Sect. 5 quantitatively estimate the anxiety costs, which plays an important role in determining evacuation rates with the cognitive dissonance in the mechanism.
The estimated anxiety costs show that residents living in flooding areas have higher average anxiety costs than residents living in non-flooding areas, and that both the average anxiety cost and the variance among residents become higher as the age category becomes older. These results are novel because there are no papers exploring quantitative effects of cognitive dissonance on any evacuation.
Based on our quantitative results, we can consider what policies can manage cognitive dissonance in evacuation. A high level of anxiety basically promotes evacuation. Since our results shows that anxiety costs are increased by experience of tsunamis, education having people virtually experience tsunamis can increase evacuation rates efficiently. In addition, both the average anxiety cost and the variance among residents become higher as the age category becomes older. So, it is important to give disaster prevention education to the elderly. These policies are designed based on our results.
However, our research only targets the areas affected by the 2011 Japan Earthquake and Tsunami. Future research should target other evacuation events and obtain the distributions of the costs of anxiety in other municipalities and regions, and compare their costs of anxieties among regions. Because the level of disaster prevention education differs among regions, we can clarify the effect of the education on the anxiety costs. This can propose disaster prevention policies suitable for each region.
In addition to the limitation of the target area, our model does not explain all the effects of cognitive dissonance on evacuation. For example, people choosing not to evacuate due to cognitive dissonance might affect other people's evacuation. People are uncertain about whether large tsunamis come or not. So, someone who has always thought tsunamis were not that dangerous might be able to convince some other people of tsunamis' not coming.
For another example, people might develop their mind in a rather long term. If someone live in tsunami-threatened areas, he constructs his belief that tsunamis do not come after settling in that place. This affects the evacuation behavior. In the current model, we do not take account of this long-term effect, but only the effect of cognitive dissonance which arises after the earthquake.

Conclusion
In the current study, we modeled tsunami evacuation behavior while taking cognitive dissonance into consideration. Applying the model to the data of the evacuation behavior at the time of the 2011 Japan Earthquake and Tsunami, we quantitatively estimated the anxiety of 22 municipalities from evacuation behavior by the cross-attributes of gender, age, and whether or not they heard a tsunami alert, and whether or not they have checked a hazard map, and whether or not they participated in local evacuation drill as "categories." As a result, we mainly obtained the following four results.
(1) We show that anxiety varies largely across residential locations. (2) The average anxiety of residents is roughly correlated with the damage in the past disasters. (3) Anxiety of females is larger than that of males, and the anxiety of age group 3 is larger than that of age group 1. (4) Tsunami alert affects the anxiety and increases the evacuation rate. Additionally, the provision of a hazard map and participation in evacuation drills might affect the improvement of the anxiety and the evacuation rate. However, these effects do not satisfy significance levels.
Future research can combine this evacuation model with a dynamic tsunami evacuation simulation and explore some policies related to evacuation routes.