1 Introduction

It is becoming increasingly important to empirically elucidate the mechanisms of behavior under ambiguity and predict people’s behavior. Ambiguity refers to a situation in which the probability distribution of outcomes is unknown (e.g., Trautmann & van de Kuilen, 2015). For example, understanding the principles of people’s disaster mitigation behaviors is critical for designing natural disaster countermeasures that encourage such behavior. Similarly, predicting vaccination decisions is essential for implementing effective infection control. As the probability that a natural disaster will occur or that people will become infected is uncertain, such behaviors take place in ambiguity.

Based on Ellsberg-type experiments, previous studies have shown that people are ambiguity-averse. However, recent research has shown that ambiguity attitudes vary depending on the source of ambiguity, domain of outcome, and likelihood of uncertain events (e.g., Kocher et al., 2018; l’Haridon et al., 2018; Li et al., 2018; Trautmann & van de Kuilen, 2015). For example, ambiguity attitudes toward artificial sources, such as the Ellsberg urns, differ from those toward natural sources (e.g., Li et al., 2018); ambiguity-seeking (or ambiguity-averse) attitudes have been observed for low-likelihood events in the gain (or loss) domain, and ambiguity-averse (or ambiguity-seeking) attitudes have been observed for high-likelihood events in the gain (or loss) domain (Trautmann & van de Kuilen, 2015).

Additionally, recent studies have shown that not only ambiguity aversion, but also ambiguity-generated insensitivity (hereafter, a-insensitivity) is essential to describe behaviors under ambiguity (Anantanasuwong et al., 2020; Dimmock et al., 2016b; Li, 2017). Ambiguity aversion is a motivational component of ambiguity attitudes, reflecting either liking or disliking ambiguity. However, a-insensitivity is a cognitive component of ambiguity attitudes that captures the perceived level of ambiguity, indicating a general lack of understanding of likelihood (Baillon et al., 2012; Baillon et al., 2018b; Li, 2017; Li et al., 2018). These rich ambiguity attitudes may explain real-world behaviors more precisely and drive practical policy implications. For example, as a-insensitivity, unlike ambiguity aversion, can be improved by learning (Baillon et al., 2018a; Baillon et al., 2018b), providing information that improves people’s perception of the likelihood of uncertain events may improve a-insensitivity, and facilitate disaster mitigation measures and infectious disease control.

Only a few studies have investigated the characteristics and heterogeneity of ambiguity attitudes (ambiguity aversion and a-insensitivity) toward natural sources across people (e.g., Baillon et al., 2018b; Li, 2017), or examined the relationship between ambiguity attitudes toward natural sources and real-world behaviors (Anantanasuwong et al., 2020; Gaudecker et al., 2022). However, these studies focus only on the gain domain. Owing to the findings of previous studies that ambiguity attitudes toward artificial sources differ between the gain and loss domains, it is necessary to examine the characteristics of ambiguity attitudes and the relationship between ambiguity attitudes and real-world behaviors in the gain and loss domains of natural sources. To the best of our knowledge, no study has previously examined the relationship between attitudes toward ambiguity and real-world behaviors across the gain and loss domains of natural and artificial sources; this study’s aim is to address this gap.

We also examine how cognitive ability affects ambiguity attitudes, and investigate the relationship between ambiguity attitudes and real-world behaviors by cognitive abilities. Frederick (2005) has shown that cognitive reflection ability affects risk attitudes and time preferences. Additionally, as a-insensitivity captures the extent of having an understanding of likelihood, cognitive reflection ability is expected to be related to a-insensitivity. Thus, examining the relationship between ambiguity attitudes and cognitive reflection abilities may assist in understanding interindividual heterogeneity in ambiguity attitudes and the relationship between ambiguity attitudes and behavior.

We elicit ambiguity attitudes—ambiguity aversion and a-insensitivity—toward natural and artificial sources in the gain and loss domains from participants in various age groups, income levels, and educational backgrounds, rather than just students. We consider precipitation during the rainy season as a natural source that ordinary people experience regularly, and the Ellsberg urn as an artificial source. We examine the relationship between ambiguity attitudes and real-world behaviors using the elicited ambiguity attitude indexes. We take flood preparedness behaviors as examples of real-world behaviors related to the source of precipitation. To do this, we conduct web experiments applying Baillon et al.’s (2018b) method to measure ambiguity attitudes toward the natural source, and Dimmock et al. (2016b) method to measure ambiguity attitudes toward the artificial source; both experiments are conducted in the gain and loss domains. As both methods measure the ambiguity aversion and a-insensitivity indexes, ambiguity attitudes elicited through both methods are comparable.

By comparing ambiguity attitudes toward natural and artificial sources in the gain and loss domains and investigating the relationship between ambiguity attitudes and real-world behaviors, this study contributes to the literature by providing the following new findings. 1) People are more a-insensitive toward natural sources than artificial sources, even though the level and distribution of outcomes remain the same. 2) People are ambiguity averse in the gain domain and more ambiguity averse toward natural sources than artificial sources; by contrast, people are ambiguity seeking in the loss domain, with similar levels of ambiguity seeking between the natural and artificial sources. 3) People with low cognitive reflection ability tend to be more a-insensitive than those with high cognitive reflection ability. 4) In the group with high cognitive reflection ability, people with higher a-insensitivity are less likely to adopt risk mitigation behaviors (i.e., flood preparedness behaviors) in the gain domain of natural sources. However, a statistically significant correlation does not exist between ambiguity attitudes and flood preparedness behaviors in the group with low cognitive reflection ability.

2 Elicitation of ambiguity attitude indexes

This section explains the methods of eliciting ambiguity attitudes for our application. We apply Baillon et al.’s (2018b) method for the natural source and Dimmock et al. (2016b) method for the artificial source. Sources of ambiguity refer to groups of events generated by the same uncertainty mechanism (Abdellaoui et al., 2011, p. 969). Subsection 2.1 describes, first, the conventional method of eliciting ambiguity attitudes toward artificial sources, followed by the method of eliciting ambiguity attitudes toward natural sources.

2.1 Elicitation of ambiguity attitudes toward artificial sources

We use a box containing 100 balls of 10 different colors: red, blue, yellow, purple, orange, green, black, white, gray, and brown. The study participants do not know how many balls of each color are present in the box. This box is considered to be the artificial source of ambiguity, to distinguish it from the natural source.

Participants are instructed to consider a gamble, \({1000}_{{E}_{j}}0\), where the participant gains 1,000 (JPY) if \({E}_{j}\) occurs and 0 otherwise (1,000 JPY is approximately 7.7 USD). In our experiment, we consider six events, including event \({E}_{j}\) of j winning colors with \(j=\) 1, 3, 4, 6, 7, 9. For example, on event \({E}_{3}\), three of the ten colors are winning colors.

Under this setting, we define matching probability \(m\) as follows: \({1000}_{{E}_{j}}0 \sim {1000}_{m\left(\frac{j}{10}\right)}0\). In other words, matching probability is the probability that the subject is indifferent to gaining 1,000 (JPY) with objective probability \(\frac{j}{10}\) and 1,000 (JPY) under event \({E}_{j}\). Individuals with ambiguity-neutral attitudes assume \(m\left(\frac{j}{10}\right)\) is equivalent to \(\frac{j}{10}\); this is called ambiguity-neutral probability. Thus, we can control ambiguity-neutral probability in artificial source experiments.

In the loss domain, participants are asked to consider a gamble, \({1000}_{{E}_{j}}0\), where the participant loses 1,000 (JPY) if \({E}_{j}\) occurs and 0 otherwise, and the matching probability is defined as follows: \({-1000}_{{E}_{j}}0\sim {-1000}_{m\left(\frac{j}{10}\right)}0\).

Using the matching probabilities for ambiguity-neutral probabilities, we parametrically measure the indexes of (global) ambiguity attitudes (Dimmock et al., 2016b). The indexes of a-insensitivity and ambiguity aversion are defined as follows:

$$\text{A-insensitivity index: } a=1-s,$$
$$\text{Ambiguity aversion index in the gain domain: } b=1-s-2c,$$
$$\text{Ambiguity aversion index in the loss domain: } b=s+2c-1.$$

s and c are the parameters of the following linear model with an error term e: \(m\left(p\right)=c+sp+e\). We estimate these parameters individually by using the ordinary least squares method, as these matching probabilities and ambiguity-neutral probabilities are observed for each individual.

\(b>0\) indicates ambiguity aversion, \(b=0\) ambiguity neutrality, and \(b<0\) ambiguity seeking. Note that we define the ambiguity aversion index in the gain domain as \(b=1-s-2c\) (Dimmock et al., 2016b), and the ambiguity aversion index in the loss domain as \(b=s+2c-1\) (Dimmock et al., 2015), where a greater \(b\) implies a more pessimistic attitude toward gain and loss outcomes. If ambiguity-neutral probability \(p\) increases, matching probability \(m(p)\) should also increase; thus, monotonicity requires \(s\ge 0\), which is equivalent to \(a\le 1\). Therefore, the case of \(a>1\) indicates a violation of monotonicity (Li et al., 2018).

2.2 Elicitation of ambiguity attitudes toward natural sources

In the natural source experiments, we cannot measure matching probability for ambiguity-neutral probabilities because we cannot know the ambiguity-neutral probabilities for events of a natural source. Baillon et al. (2018b) proposed an innovative method to elicit the two indexes of ambiguity attitudes toward natural sources in the case of three events. Let \({m}_{1}\), \({m}_{2}\), and \({m}_{3}\) represent the matching probabilities of three single events, and \({m}_{23}\), \({m}_{31}\), and \({m}_{12}\) denote the matching probabilities of the complements of these single events, respectively. For example, \({m}_{1}=m({E}_{1})\) and \({m}_{23}=m({E}_{2}\cup {E}_{3})\) for \(\Omega ={E}_{1}\cup {E}_{2}\cup {E}_{3}\). They define the indexes of ambiguity aversion \(b\) and a-insensitivity \(a\) as follows:

$$\text{A-insensitivity index: } a=3\times \left(\frac{1}{3}-\left({\overline{m} }_{c}-{\overline{m} }_{s}\right)\right),$$
$$\text{Ambiguity aversion index in the gain domain: } b=1-{\overline{m} }_{c}-{\overline{m} }_{s},$$
$$\text{Ambiguity aversion index in the loss domain: } b={\overline{m} }_{c}+{\overline{m} }_{s}-1,$$

where \({\overline{m} }_{s}=\frac{{m}_{1}+{m}_{2}+{m}_{3}}{3}\), \({\overline{m} }_{c}=\frac{{m}_{12}+{m}_{23}+{m}_{31}}{3}\). \(a>0\) denotes a-insensitivity, \(a<0\) a-oversensitivity, and \(a=0\) ambiguity neutrality. \(b>0\) indicates ambiguity aversion, \(b<0\) ambiguity seeking, and \(b=0\) ambiguity neutrality. In the gain domain, a large value of \(b\) connotes that the probability of a gain outcome is underestimated, indicating a more pessimistic attitude. In the loss domain, a large value of \(b\) implies that the probability of a loss outcome is overestimated, indicating a more pessimistic attitude.

In the natural source, \(a>1\) also indicates violation of monotonicity because the average matching probability for a single event should be less than the average matching probability of the complements, \({\overline{m} }_{c}\ge {\overline{m} }_{s}\) (Anantanasuwong et al., 2020, p. 7). Empirical studies have shown a certain percentage of violation of monotonicity (e.g., 20.3%–26.1% in Anantanasuwong et al., 2020, p. 30).

3 Experimental designs and data summary

3.1 Web survey

We conducted a web survey using a panel owned by Cross Marketing Inc., one of Japan’s largest Internet research companies with 5.41 million registered members (as of January 2022). The survey period was from July 14, 2022 to July 24, 2022. We limited our survey to registered members living in Osaka city to control for regional attributes. Osaka is one of Japan’s largest cities with a population of approximately 2.7 million. This area was chosen because we examine the relationship between ambiguity attitudes toward precipitation of natural sources and flood preparedness behaviors, and Osaka Bay is one of the areas in Japan most exposed to heavy coastal inundation, with expected annual loss due to coastal inundation in 2050 ranking 5th among 120 cities worldwide (Abadie et al., 2017). The survey company randomly requested participation from the registered members and accepted their responses on a first-come, first-served basis. The respondents were informed that it would take 20–30 min to complete the survey, and they were free to quit at any time. We have obtained Institutional Review Board approval and have pre-registered this experiment on the randomized controlled trials registry (see Declarations for details).

The web survey was conducted in three parts. The first part was a practice session for answering matching probability. The second part comprised experiments for eliciting matching probabilities. The third part collected information on the participants’ demographic variables, cognitive reflection ability, risk preferences, time preferences, flood preparedness behaviors, and health-related behaviors.

In the gain domain experiments, we used a monetary incentive design in which the survey respondents could earn 1,000 points if they won the lottery in the choice experiments. The value of 1 point is 1 JPY (1 USD is approximately 130 JPY), and these points could be exchanged for cash, electronic money, or merchandise. However, as in Dimmock et al. (2015), the respondents did not receive the same monetary incentives in the loss domain experiments. Instead, they were offered 500 points for participating, and the value was set equal to the anticipated average reward in the gain domain (we assumed a 50% chance of getting 1,000 points because the survey respondents did not know the exact answer). According to Etchart-Vincent and l’Haridon (2011), real and hypothetical choices do not differ significantly in the loss domain (see also Baillon & Bleichrodt, 2015).

We randomly divided the participants into four groups: the gain and loss domains of the natural source and the gain and loss domains of the artificial source. In the next section, we explain these four groups in detail.

3.2 Elicitation of matching probabilities

3.2.1 Natural source case

The respondents were presented with Option A (precipitation lottery) and Option B (probability lottery) and asked to choose between the two, as shown in Fig. 1. We focus our explanation on the gain domain case (for details about the loss domain case, refer to the online Appendix A). While making their choices, we asked the respondents to compare Option A and Option B from the top row onward and select their preference for each percentage.

Fig. 1
figure 1

An example of choice experiments in the natural source

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The payment under Option A was set at 1,000 JPY for the single event of maximum one-hour precipitation being “greater than 50 mm,” “greater than 30 mm and less than 50 mm,” and “less than 30 mm” in Osaka City in August 2022, and their complements (“less than 50 mm,” “less than 30 mm or greater than 50 mm,” and “greater than 30 mm”). Thus, respondents answered this choice question six times, varying the event for which they were rewarded. Additionally, they were asked to answer an additional (seventh) choice to check the consistency of their preference. The seventh choice was presented randomly from among the first through fifth choices. Note that the immediately preceding sixth choice was not used for this consistency check as it would have resulted in a series of identical questions.

3.2.2 Artificial source case

The survey respondents were presented with Lottery A and Lottery B (Fig. 2) and asked to choose between the two. Lottery A was the ambiguous lottery and Lottery B was the risky lottery. We asked the respondents to compare the lotteries in order from the top row onward and select their preference for each percentage. The payment under Lottery A was set at 1,000 JPY for the single events of “one ball,” “three balls,” and “four balls,” and their complements (“nine balls,” “seven balls,” and “six balls”). Thus, respondents answered this choice question six times, varying the event for which they were rewarded. The winning color was determined as the one the respondents preferred in the previous question of these choice experiments to prevent them from speculating that the investigators were intentionally selecting the winning color.

Fig. 2
figure 2

An example of choice experiments in the artificial source

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3.3 Sample data summary

Table 1 summarizes the sample data. The demographic variables are income (1,000,000 JPY), age, female (1 if a respondent is female), university (1 if a respondent is a university graduate), number of family members, and number of children. The averages of all these variables are not statistically different across the four groups—the gain and loss domains of natural and artificial sources—at the 0.05 significance level.

Table 1 Summary Statistics
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Low CRT score is a dummy variable taking one if a respondent belongs to a group with low cognitive reflection ability based on the cognitive reflection test (Frederick, 2005), which was conducted during the web survey. The cognitive reflection test (CRT) measures cognitive reflection ability, the ability or disposition to reflect on questions and resist reporting a response that first comes to mind (Frederick, 2005). The CRT score has frequently been used to examine the relationship between cognitive reflection ability and risk attitudes or time preferences (Bradford et al., 2017; Charness et al., 2023; Ioannou & Sadeh, 2016; Meissner et al., 2023). We use CRT scores to examine the relationship between cognitive reflection ability and ambiguity attitudes. The cognitive reflection test in this survey comprised the following three questions: (1) A bat and a ball cost 11,000 JPY in total. The bat costs 10,000 JPY more than the ball. How much does the ball cost? (2) If it takes 5 machines 5 min to make 5 widgets, how long would it take 100 machines to make 100 widgets? (3) In a lake, there is a patch of lotus leaves. Every day, the lotus leaves double in size. If it takes 48 days for the lotus leaves to cover the entire lake, how many days would it take for the lotus leaves to cover half the lake?

Following previous studies (Albaity et al., 2014; Oechssler et al., 2009), we define respondents who answered at least two out of three questions correctly on the cognitive reflection test as the High CRT score group and respondents who answered less than two questions correctly as the Low CRT score group. The rates of the Low CRT score group ranged from 59%–66% across the gain and loss domains of natural and artificial sources; the rate is not statistically different across the four patterns (the gain and loss domains of natural and artificial sources) at the 0.05 significance level.

Risk aversion, likelihood insensitivity, and time preference represent the respondent’s risk aversion, likelihood insensitivity, and time preference, which were elicited from the choice experiments during the web survey. Details of these experiments and the measurement of variables are presented in the online Appendix B.

We include variables on housing as control variables in the regression of flood preparedness behaviors on ambiguity attitudes; rental house or apartment (1 if the residence is a rental house or apartment) and owned apartment (1 if the residence is an owned apartment), where the baseline housing style is owned house.

4 Matching probability and indexes of ambiguity attitudes

4.1 Matching probability in the case of natural source

Table 2 shows the summary statistics of matching probability elicited from natural source experiments. We observe that the mean, median, and standard deviation of the matching probability of each event are almost identical, indicating that respondents perceive the likelihood of these events as being almost the same. This may imply that respondents have difficulty identifying the likelihood of these events owing to the ambiguity. Additionally, the standard deviations of the matching probability of each event are large, indicating that the matching probabilities are highly heterogeneous across respondents.

Table 2 Matching probability elicited from natural source experiments
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4.2 Matching probability in the case of artificial source

Table 3 presents the summary statistics of the matching probabilities elicited from artificial source experiments, and Fig. 3 presents the plot of matching probability for each ambiguity-neutral probability. We observe that m(0.1) and m(0.3) are larger than their corresponding ambiguity-neutral probabilities of 0.1 and 0.3 in both the gain and loss domains. In contrast, m(0.6), m(0.7), and m(0.9) are smaller than their ambiguity-neutral probabilities of 0.6, 0.7, and 0.9 in both the gain and loss domains. Thus, on average, people have ambiguity-seeking attitudes in the gain domain and ambiguity-averse attitudes in the loss domain on low ambiguity-neutral probabilities. Conversely, on average, people have ambiguity-averse attitudes in the gain domain and ambiguity-seeking attitudes in the loss domain on the high ambiguity-neutral probabilities. This fourfold pattern of ambiguity attitudes is consistent with the findings of previous studies: ambiguity seeking and ambiguity aversion for low likelihood in the gain and loss domains, respectively, and ambiguity aversion and ambiguity seeking for high-likelihood in the gain and loss domains, respectively (Trautmann & van de Kuilen, 2015). Figure 3 shows that the graph of matching probability is flat, implying that respondents are a-insensitive on average.

Table 3 Matching probability elicited from artificial source experiments
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Fig. 3
figure 3

Matching probabilities of the artificial source

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4.3 Consistency of responses regarding matching probabilities

To check the consistency of responses regarding matching probabilities, we elicited matching probabilities twice by randomly selecting only one of the six events (see Subsection 3.2.1). The mean differences between the first and second matching probabilities in the gain and loss domains of the natural source are 0.0037 (95% confidence interval (CI) is [-0.035, 0.042]) and -0.062 (95% CI is [-0.062, 0.028]), respectively, and -0.020 (95% CI is [-0.053, 0.014]) and -0.012 (95% CI is [-0.049, 0.026]), respectively for the artificial source. Overall, this indicates that the first and second matching probabilities are not different, and that the participants’ responses regarding matching probabilities are consistent.

4.4 Indexes of ambiguity attitudes

Table 4 presents the summary statistics of the indexes of ambiguity attitudes. A-insensitivity indexes toward the natural source are larger than those toward the artificial source, both in the gain and loss domains; the mean difference is 0.216 (95% CI is [0.116, 0.316]) and 0.220 (95% CI is [0.119, 0.321]) in the gain and loss domains, respectively. This indicates that the respondents have difficulty identifying the likelihood of the uncertain events in the natural source. Moreover, a-insensitivity indexes are slightly larger in the loss domain than in the gain domain; the mean difference is 0.0946 (95% CI is [-0.00415, 0.193] and 0.0907 (95% CI is [-0.0117, 0.193]) for the natural and artificial sources, respectively. However, the difference is not statistically significant at the significance level of 0.05.

Table 4 Summary statistics of the indexes of ambiguity attitudes
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A-insensitivity indexes in the gain domain of this study are 0.859 and 0.643 toward the natural and artificial sources, respectively, making them close to those measured in previous studies. For example, the a-insensitivity index toward various financial natural sources ranges from 0.69 to 0.84 in the gain domain in Anantanasuwong et al. (2020, Table B1, p. 38); that toward the artificial source is 0.41 in Dimmock et al. (2016b, Table 3, p. 1370) and 0.81 in Li et al. (2018, the basic Ellsberg urn of Table 3, p. 3235), both measured in the gain domain.

Ambiguity aversion indexes, on average, are positive in the gain domain of both natural and artificial sources. They are larger for the natural source than for the artificial source, with the mean difference being 0.14 (95% CI is [0.042, 0.25]). By contrast, they are negative in the loss domain of both natural and artificial sources and are similar between them, with the mean difference being 0.009 (95% CI is [-0.093, 0.11]). These values indicate that respondents are, on average, ambiguity averse in gain domains and ambiguity seeking in loss domains, which is consistent with previous studies (Trautmann & van de Kuilen, 2015).

The ambiguity aversion indexes in the gain domain of this study are 0.204 and 0.060 for the natural and artificial sources, respectively. This makes them close to or slightly lower than values in previous studies. For example, the ambiguity aversion index toward various financial natural sources in the gain domain ranges from 0.16 to 0.21 in Anantanasuwong et al., (2020, Table 1, p. 28); that toward artificial sources is 0.12 in Dimmock et al. (2016b, Table 3, p. 1370) and 0.15 in Li et al. (2018, the basic Ellsberg urn of Table 3, p. 3235).

4.5 Violation of monotonicity

The survey participants’ rates for violation of monotonicity (a > 1) are 28.5% and 32.9% in the gain and loss domains of the natural source, respectively, compared to 19.6% and 28.3% in the gain and loss domains of the artificial source, respectively. These rates are similar to those in previous studies that have measured a-insensitivity index in the gain domain; these range from 20.3%–26.1% for natural sources in Anantanasuwong et al. (2020) and from 14%–28% for artificial sources in Li et al. (2018).

5 Relationship between ambiguity attitudes and cognitive reflection ability

We consider the relationship between ambiguity attitudes and cognitive reflection ability (see Subsection 3.3 for the cognitive reflection ability). Figure 4 shows the distribution of ambiguity attitudes by groups with High and Low CRT score. The a-insensitivity index is distributed around 1 in the Low CRT score group, and is distributed at a larger value in the Low CRT score group than in the High CRT score group, indicating that the Low CRT score group is more a-insensitive. However, this tendency is weaker for the natural source cases, indicating that even the High CRT score group finds it more difficult to identify the likelihood of the natural source events. By contrast, we see no notable difference in the ambiguity aversion index between the two groups.

Fig. 4
figure 4

Distribution of ambiguity attitudes by groups with High and Low CRT score

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We use regression analysis to examine if this tendency is sustained after controlling individual characteristic variables. The dependent variables are the indexes of a-insensitivity and ambiguity aversion, and the independent variable is the dummy variable Low CRT score taking 1 if the respondent belongs to the Low CRT score group. We include the control variables of income, female, age, university, number of family members, number of children, risk aversion, likelihood insensitivity, and time preference (see Subsection 3.3 for variable definitions).

Table 5 presents the estimation results (the full results are presented in the online Appendix C). The Low CRT score group tends to have a higher a-insensitivity index. The difference of a-insensitivity index between the two groups is statistically significant for the artificial source at the 0.05 significance level. However, its difference is not statistically significant for the natural source, probably because even the High CRT score group finds it difficult to identify likelihood of the natural source events. As for the ambiguity-averse attitude, there is no statistically significant difference between the High and Low CRT score groups.

Table 5 Estimation results of regressing ambiguity attitudes on CRT score
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6 Relationship between ambiguity attitudes and flood preparedness behaviors

6.1 Flood preparedness behaviors

We use flood behavior as the index for flood preparedness behaviors, with a maximum value of 4 and a minimum value of 0 (the summary statistics are shown in Table 6). A high value of flood behavior means that people engage more in the following flood preparedness behaviors which the Japanese government recommends:

  • Stockpiling drinking water, food, and daily necessities to last for about three days in case of a landslide or flood due to heavy rain.

  • Preparing an emergency bag (drinking water, emergency food, first-aid supplies, masks, gloves, flashlight, and portable radio) in case of evacuation owing to a landslide or flood caused by heavy rain.

  • Checking evacuation sites and routes to evacuate without panicking in the event of a landslide or flood disaster caused by heavy rain.

  • Acquiring insurance to prepare for landslide and flood disasters caused by heavy rain.

Table 6 Summary statistics of flood preparedness behaviors index
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6.2 Ordered probit model

We apply the ordered probit model with flood behavior as the dependent variable and ambiguity attitudes (b: ambiguity aversion index and a: a-insensitivity index) as the independent variables.

$$Flood\; behavio{r}^{*}={\beta }_{0}+{\beta }_{1}b+{\beta }_{2}a+\gamma z+u,\;u|b, a, z \sim N\left(0, 1\right).$$

Flood behavior* is a latent variable for flood behavior, z is a vector of control variables, \(\beta\) s are coefficient parameters, \(\gamma\) is a vector of coefficient parameters, and \(u\) is an error term. We use the following control variables: Low CRT score, income, age, female, number of family members, number of children, risk aversion, likelihood insensitivity, time preference, rental house or apartment, and owned apartment.

Figure 5 presents the estimation results (the full results are shown in Table D1 of the online Appendix D). The points are estimated coefficients of ambiguity aversion b and a-insensitivity a, and the error bars indicate 95% confidence intervals.

Fig. 5
figure 5

Estimated effects of ambiguity attitudes on flood preparedness behaviors

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We find a negative correlation between a-insensitivity and flood preparedness behaviors in the gain domain of the natural source, which is statistically significant at the 0.05 level. This indicates that people who are less likely to identify the likelihood (or more likely to perceive the likelihood as the same) are unlikely to adopt flood preparedness behaviors, which is consistent with our insight (see Subsection 7.3 for a detailed discussion). However, we do not find any statistically significant correlations between a-insensitivity and behaviors in the loss domain of the natural source and the gain and loss domains of the artificial source. Additionally, we do not find any statistically significant correlations between ambiguity aversion and behaviors in the gain and loss domains of the natural and artificial sources.

6.3 Considering cognitive reflection ability

We include the cross term between Low CRT score and ambiguity attitudes (a and b) to identify the correlation between ambiguity attitudes and flood preparedness behaviors by cognitive reflection ability. The control variables included are the same as in Subsection 6.2.

$$Flood\; behavio{r}^{*}={\beta }_{0}+{\beta }_{1}b+{\beta }_{2}a+{\beta }_{3}b\times low\; CRT\; score+{\beta }_{4}a\times low\; CRT\; score+\gamma z+u.$$

Figure 6 presents the estimation results (the full results are shown in Table D2 of the online Appendix D). In the High CRT score group, as in Subsection 6.2, we find a negative correlation between a-insensitivity and flood preparedness behaviors in the gain domain of the natural source, which is statistically significant at the 0.05 level. Moreover, the effect amplifies compared to that in Subsection 6.2. However, in the Low CRT score group, we do not find statistically significant correlations between a-insensitivity and behaviors. These results indicate that a relationship exists between a-insensitivity and flood preparedness behaviors in the High CRT score group, but not in the Low CRT score group. In other words, the relationship found in Subsection 6.2 reflects the relationship found in the High CRT score group. Additionally, as in Subsection 6.2, we do not find any statistically significant correlations between ambiguity aversion and flood preparedness behaviors.

Fig. 6
figure 6

Estimated effects of ambiguity attitudes on flood preparedness behaviors by CRT score

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We estimate the same models applying the ordinary least squares method for the robustness check with heteroscedasticity-robust standard errors. The results are identical to the ordered probit models. The estimation results of the ordinary least squares method are shown in the online Appendix E.

7 Discussion

7.1 Comparison of ambiguity attitudes toward natural and artificial sources in the gain and loss domains

This study compared the indexes of a-insensitivity and ambiguity aversion across natural and artificial sources in the gain and loss domains. To the best of our knowledge, no previous study has simultaneously compared ambiguity attitudes for these four patterns. This comparison has revealed a number of novel findings which we discuss in detail below, contributing to and enhancing the literature.

First, we found that a-insensitivity toward the natural source was higher than that toward the artificial source, although the level and distribution of outcomes were identical (Table 4). The results indicate that people find it more challenging to identify probabilities for the natural source than the artificial source. This is expected as the a-insensitivity index captures the perception around the level of ambiguity, reflecting that people are more intensely aware of ambiguity with regard to the natural source than the Ellsberg-type artificial source.

Second, we observed that, in both natural and artificial sources, respondents were ambiguity averse in the gain domain and ambiguity seeking in the loss domain (Table 4), which is consistent with the results of previous studies (e.g., Dimmock et al., 2015). We also found that respondents were more ambiguity averse toward the natural source than the artificial source in the gain domain. This means that the subjective probability for the natural source tends to be further from ambiguity-neutral probability than the subjective probability for the artificial source because people dislike the ambiguity of the natural source more than that of the artificial source.

7.2 Relationship between a-insensitivity and cognitive reflection ability

Additionally, this study examined the relationship between cognitive reflection ability and ambiguity attitudes. We found that respondents with Low CRT score were more a-insensitive (Table 5). Thus, people with low cognitive reflection ability have more difficulty in identifying the likelihood of ambiguous events and tend to view the likelihood of all uncertain events to be equal. Conversely, we did not find any correlation between ambiguity aversion and cognitive reflection ability. These results are consistent with our expectations because a-insensitivity and ambiguity aversion indexes measure the perception of ambiguity and the level of liking or disliking toward ambiguity, respectively (Baillon et al., 2021; Li et al., 2018). Our results are also consistent with various related studies. For example, Choi et al. (2022) found that cognitive ability is negatively related to likelihood insensitivity, and Baillon et al. (2018b) showed that the time pressure for answering is detrimental to cognitive understanding and increases a-insensitivity, although time pressure does not affect ambiguity aversion.

7.3 Relationship between ambiguity attitudes and real-world behaviors

7.3.1 Natural source of ambiguity

In the High CRT score group, we found that respondents with higher a-insensitivity were less likely to adopt flood preparedness behaviors in the gain domain of the natural source (Fig. 6). This may be attributable to the fact that people with high a-insensitivity cannot discriminate the different likelihood levels and perceive the likelihood of any event to be alike. In other words, people with high a-insensitivity cannot discriminate the likelihood of flood damage and overestimate the likelihood before and after flood preparedness. Therefore, since people with high a-insensitivity perceive the likelihood of flood damage when they take flood preparedness as being as high as the likelihood of flood damage when they do not take flood preparedness, they have low incentives to take flood preparedness. In contrast, people with low a-insensitivity can discriminate the different likelihoods of different events; therefore, they perceive the likelihood of flood damage in a situation where they take flood preparedness as low and are likely to take flood preparedness.

A negative correlation between a-insensitivity and behaviors under ambiguity, such as stock market participation, has been observed in previous studies (e.g., Dimmock et al., 2016b). However, the interpretation of the negative correlation differs between this and previous studies. As stated above, people with high a-insensitivity tend not to take flood preparedness measures because of the overestimation of flood damage, both before and after behaviors. In contrast, in the case of stock market participation, people are exposed to ambiguity only after stock market participation. In this case, because people with high a-insensitivity tend to underestimate or overestimate extreme probabilities under ambiguity after stock market participation (e.g., they underestimate the likelihood of stock gain or overestimate the likelihood of stock loss), they do not tend to enter the stock market.

However, we did not find statistically significant correlations between a-insensitivity and flood preparedness behaviors in the Low CRT score group. A possible reason is that the distribution of a-insensitivity clusters around 1 (Fig. 4) in this group, and the variation in a-insensitivity is small. As a result, the relationship between a-insensitivity and flood preparedness behaviors is not statistically significant. The same reason explains why no correlation is found between a-insensitivity and flood preparedness behaviors in the loss domain, where the distribution of a-insensitivity is around 1, irrespective of cognitive reflection ability.

Finally, no correlation was found between ambiguity aversion and flood preparedness behaviors (Fig. 6), irrespective of cognitive reflection ability. This is consistent with the results of previous studies that demonstrated no correlation between ambiguity aversion and real-world behaviors after controlling a-insensitivity (e.g., Dimmock et al., 2016b).

7.3.2 Artificial source of ambiguity

With respect to the artificial source, we did not see any correlations between ambiguity attitudes (a-insensitivity and ambiguity aversion) and flood preparedness behaviors. A possible reason is that the artificial source of ambiguity—the Ellsberg-type box—did not relate to flood preparedness behaviors. This result may suggest that ambiguity attitudes toward natural sources related to behaviors (in this study, the precipitation as the natural source related to flood preparedness behaviors) can explain real-world behaviors that ambiguity attitudes toward artificial sources, which previous studies have frequently measured, cannot explain. Thus, the results suggest that eliciting ambiguity attitudes toward natural sources is worth considering, specifically when explaining or predicting real-world behaviors based on ambiguity attitudes.

Note that this result does not imply that ambiguity attitudes toward natural sources can necessarily explain real-world behaviors better than artificial sources. The relationship between source and behavior is fundamental for the explanatory power of ambiguity attitudes toward real-world behavior. For example, although we investigate the relationship between ambiguity attitudes toward the natural source regarding precipitation and health-related behaviors, we do not observe the correlations between them (see the online Appendix F).

7.3.3 Policy implications

Most previous studies that examined the relationship between ambiguity attitudes and real-world behaviors have focused on ambiguity aversion and not a-insensitivity (e.g., Dimmocket al., 2016a; Sutter et al., 2013). For example, Dimmock et al. (2016a) observe that ambiguity aversion negatively correlates with stock market participation; Sutter et al. (2013) find that people with higher ambiguity aversion are less likely to smoke, but do not observe any relationship between ambiguity aversion and drinking and savings. However, focusing on a-insensitivity is important in examining the relationship between real-world behaviors and ambiguity attitudes because recent studies have shown that a-insensitivity describes behaviors appropriately under ambiguity (e.g., Dimmock et al., 2016b). The same was observed in this study regarding the relationship between flood preparedness behaviors and a-insensitivity, although only among people with high cognitive reflection ability.

A-insensitivity captures the level of understanding of likelihood, and people with higher a-insensitivity may adopt suboptimal behaviors (Li, 2017). However, a-insensitivity can be improved through learning (Baillon et al., 2018a). Thus, providing information regarding the likelihood of events, or taking measures to improve the perception of likelihood may improve a-insensitivity and encourage people to make better choices. For example, providing citizens with information about flood probability could promote flood preparedness even if the probability is uncertain. Attempts to increase cognitive reflection ability may also be effective for improving a-sensitivity because cognitive reflection ability negatively correlates with a-insensitivity.

8 Conclusion

We compared ambiguity attitudes across natural and artificial sources in both gain and loss domains and examined the relationships between ambiguity attitudes and real-world behaviors. We found higher a-insensitivity toward the natural source than the artificial source. Furthermore, we observed ambiguity-averse attitudes in the gain domain, with a greater ambiguity aversion toward the natural source than the artificial source. In contrast, we observed ambiguity-seeking attitudes in the loss domain, with a similarity in degree between both sources. Additionally, in the High CRT score group, people with higher a-insensitivity were less likely to adopt flood preparedness behaviors in the gain domain of the natural source.

Our findings show that the relationship between ambiguity attitudes and real-world behaviors depends on the source of ambiguity, indicating that applying ambiguity attitudes toward natural sources is worth considering, specifically when explaining real-world behaviors based on ambiguity attitudes. Future research examining the relationship between ambiguity attitudes toward various natural and artificial sources and real-world behaviors will further illuminate and clarify the relationship between ambiguity attitudes and real-world behaviors.