# Unraveling Ambiguity Aversion

Ilke Aydogan, Loïc Berger, Valentina Bosetti

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Ilke Aydogan, Loïc Berger, Valentina Bosetti. Unraveling Ambiguity Aversion. Review of Economics and Statistics, 2023, pp.1-32. 10.1162/rest_a_01358. hal-04071242

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iRisk WORKING PAPER SERIES

2023-iRisk-01

Unraveling Ambiguity Aversion

Ilke Aydogan

IESEG School of Management, Univ. Lille, CNRS, UMR 9221 - LEM - Lille Economie' Management, F-59000 Lille, France; and iRisk Research Center on Risk and Uncertainty (i.aydogan@ieseg.fr)

Loïc Berger

CNRS, Univ. Lille, IESEG School of Management, UMR 9221 - LEM - Lille Economie' Management, F-59000 Lille, France; iRisk Research Center on Risk and Uncertainty; RFF-CMCC European Institute on Economics and the Environment (EIEE), and Centro Euro-Mediterraneo sui Cambiamenti Climatici, Italy (l.berger@ieseg.fr)

Valentina Bosetti

Department of Economics and IGIER, Bocconi University, and RFF-CMCC European Institute on Economics and the Environment (EIEE), Centro Euro-Mediterraneo sui Cambiamenti Climatici, Italy (valentina.bosetti@unibocconi.it)

IÉSEG School of Management Lille Catholic University 3, rue de la Digue F-59000 Lille Tel: 33(0)3 20 54 58 92
www.ieseg.fr

Staff Working Papers describe research in progress by the author(s) and are published to elicit comments and to further debate. Any views expressed are solely those of the author(s) and so cannot be taken to represent those of IÉSEG School of Management or its partner institutions.

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# UNRAVELING AMBIGUITY AVERSION*

Ilke Aydogan†

Loïc Berger‡

Valentina Bosetti§

## Abstract

We report the results of two experiments designed to better understand the mechanisms driving decision-making under ambiguity. We elicit individual preferences over different sources of uncertainty (risk, compound risk, model ambiguity, and Ellsberg ambiguity), which entail different degrees of complexity, from subjects with different sophistication levels. We show that (1) ambiguity aversion is robust to sophistication, but the strong relationship that has been previously reported between attitudes toward ambiguity and compound risk is not. (2) Ellsberg ambiguity attitude can be partly explained by attitudes toward complexity for less sophisticated subjects, but not for more sophisticated ones. Overall, and regardless of the subject's sophistication level, the main driver of Ellsberg ambiguity attitude is a specific treatment of unknown probabilities. These results leave room for using ambiguity models in applications with prescriptive purposes.

**Keywords:** Ambiguity aversion, reduction of compound risk, model uncertainty, complexity

**JEL Classification:** C91-C93-D81

<footnote>*We thank Mohammed Abdellaoui, Aurélien Baillon, Laure Cabantous, Thomas Epper, Chen Li, Massimo Marinacci, Sujoy Mukerji, Vincent Théroude, Uyanga Turmunkh, and Peter Wakker for helpful comments and suggestions. We also thank seminar and conference participants at D-TEA, FUR, MUSEES, Bocconi University, Ghent University, and LEM for insightful comments and discussions. We are grateful to Hans-Joachim Zwiesler for the opportunity to run the experiment at ICA 2018 and thank the organizing committee for their help with the experiment. We also thank Diana Valerio for her help with the laboratory experiment. This research is supported by a grant from the French Agence Nationale de la Recherche (ANR-17-CE03-0008-01 INDUCED), the Region Hauts-de-France (2021.00865 CLAM), and the I-SITE UNLE (project IBEBACC). Logistic supports from the Bocconi Experimental Laboratory in the Social Sciences (BELSS) and from the Anthropo-Lab are kindly acknowledged.</footnote>

<footnote>†IESEG School of Management, Univ. Lille, CNRS, UMR 9221 - LEM - Lille Économie Management, F-59000 Lille, France; and iRisk Research Center on Risk and Uncertainty (i.aydogan@ieseg.fr). Declarations of interest: none.</footnote>

<footnote>‡Corresponding author. CNRS, Univ. Lille, IESEG School of Management, UMR 9221 - LEM - Lille Économie Management, F-59000 Lille, France; iRisk Research Center on Risk and Uncertainty; RFF-CMCC European Institute on Economics and the Environment (EIEE), and Centro Euro-Mediterraneo sui Cambiamenti Climatici, Italy (l.berger@ieseg.fr). Declarations of interest: none.</footnote>

<footnote>§Department of Economics and IGIER, Bocconi University, and RFF-CMCC European Institute on Economics and the Environment (EIEE), Centro Euro-Mediterraneo sui Cambiamenti Climatici, Italy (valentina.bosetti@unibocconi.it). Declarations of interest: none.</footnote>
# 1 Introduction

For several decades, the standard way to make rational decisions under uncertainty has been to follow Savage's (1954) subjective expected utility (SEU) theory. In 1961, Ellsberg proposed several experiments challenging canonical axioms of SEU. These experiments have given rise to a vast literature studying the phenomenon of ambiguity aversion (i.e., the preference for known probabilities, or risk, over unknown probabilities, or ambiguity) at both theoretical and empirical levels. However, whether this deviation from SEU constitutes an irrational response to uncertainty or not remains an open question (Gilboa et al., 2009, 2010, 2012; Gilboa and Marinacci, 2013). As ambiguity is present and plays an important role in most real-life decision problems,<fnref n="1" /> such a question is critical for normative interpretations of ambiguity attitudes and for the use of ambiguity models in applications with prescriptive purposes. Hence, it has profound implications for policymaking (Berger et al., 2021). Our goal, in this paper, is to clarify the extent to which ambiguity aversion is tied to an arguable mistake, such as the failure to reduce compound lotteries, and study its relationship with the decision-makers' potential limitations or the complexity of a situation.

We explore experimentally decision-making under uncertainty along three dimensions. (1) The first dimension concerns the sources of uncertainty.<fnref n="2" /> We investigate attitudes toward different sources of risk (presented in simple or compound forms) and ambiguity (presented in the form of *model ambiguity* à la Marinacci (2015)<fnref n="3" /> or *ambiguity* à la Ellsberg (1961)). Under SEU, the distinction between these sources is irrelevant: all ambiguous sources are treated as risks through the assignment of subjective probabilities, whereas compound risks are reduced to simple risks in accordance with the reduction of compound lotteries axiom. (2) The second dimension concerns the subjects' level

<footnote n="1">¹For example, in financial economics, Mukerji and Tallon (2001) show how ambiguity aversion may lead to incompleteness of financial markets, while Easley and O'Hara (2009) show how it can explain low participation in the stock market despite the potentially high benefits. In the health domain, Berger et al. (2013) show that ambiguity aversion affects treatment decisions. In climate change economics, Drouet et al. (2015) and Berger et al. (2017) show how ambiguity aversion affects optimal emission policies.</footnote>

<footnote n="2">²Sources of uncertainty are defined as “groups of events that are generated by the same mechanism of uncertainty, which implies that they have similar characteristics” (Abdellaoui et al., 2011, p. 696).</footnote>

<footnote n="3">³Model ambiguity arises when the decision-maker is not able to identify a single probability distribution (among a set of probability models) corresponding to the phenomenon of interest (Hansen, 2014; Marinacci, 2015).</footnote>
of sophistication. We investigate the preferences of a unique pool of risk professionals
(working in insurance related jobs and possessing a high level of education in the fields of
mathematics, statistics, or actuarial science) and compare them to those of a convenience
sample of university students. Given their background and their training in dealing
with computationally complex problems requiring proficiency in probabilistic reasoning,
risk professionals can be considered as being more quantitatively “sophisticated” than
students. (3) Finally, the third dimension relates to the complexity of the problem. By
proposing tasks with varying degrees of complexity within the same source, we are able
to isolate the role of complexity in decision-making under risk and ambiguity.

We elicit individual preferences using a two-color Ellsberg-type setting. The large
body of existing literature using such a setting has so far highlighted two stylized facts:

**SF1: Individuals are ambiguity averse.**

**SF2: There exists a strong relationship between attitudes towards ambiguity and compound risk.**

SF1 results from the many experiments that have formally tested Ellsberg's (1961) idea,
typically using student subjects (L'Haridon et al., 2018; see also the reviews of Machina
and Siniscalchi, 2014; Trautmann and van de Kuilen, 2015).<fnref n="4" /> It also typically generalizes
to alternative subject pools, including risk professionals (Cabantous, 2007; Cabantous
et al., 2011; Hogarth and Kunreuther, 1989).<fnref n="5" /> While ambiguity and compound risks have
distinct properties, SF2 has been put forward by Halevy (2007), who documented strong
similarities in attitudes towards Ellsberg ambiguity and compound risks among student
subjects.<fnref n="6" /> Such findings have been replicated on other student samples (e.g., Chew
et al., 2017; Dean and Ortoleva, 2019; Gillen et al., 2019), and on a representative sample

<footnote n="4">⁴We note that ambiguity seeking is also common for ambiguous events with low likelihoods. This local ambiguity seeking attitude is shown to be due to an ambiguity-generated likelihood insensitivity (Dimmock et al., 2015). Our study focuses on probabilities around 50% and does not consider this other component of ambiguity attitudes.</footnote>

<footnote n="5">⁵Note that these studies use non-incentivized surveys with decision tasks in specific insurance contexts. The typical result is that insurers charge higher insurance premia when faced with ambiguity than when the probability of a loss is known.</footnote>

<footnote n="6">⁶Halevy (2007) suggested that non-reduction of compound risk is necessary for a non-neutral attitude toward ambiguity, noting “Subjects who reduced compound lotteries were almost always ambiguity neutral, and most subjects who were ambiguity neutral reduced compound lotteries appropriately.” (Halevy, 2007, p. 531) “The results suggest that failure to reduce compound (objective) lotteries is the underlying factor of the Ellsberg paradox.” (Halevy, 2007, p. 532). See also the discussion in Abdellaoui et al. (2015, p. 1307).</footnote>
of the U.S. population (Chapman et al., 2018). Whether explicitly or implicitly, SF2 has been invoked to challenge the normative status of ambiguity aversion. Specifically, if –as is often the case– non-reduction of compound (or, more generally, of complex) risks is considered as a mistake (possibly related to computational difficulties), and if the subjects making this mistake are mainly those who are ambiguity non-neutral, there would be little room for using ambiguity models for normative purposes.

Although results in line with SF1 and SF2 have consistently emerged from the literature, their relationship with the subjects’ level of sophistication has received little attention so far. Exceptions are the studies of Chew et al. (2018), who investigate the role of subjects’ level of comprehension for SF1; and Abdellaoui et al. (2015), and Berger and Bosetti (2020), who report somewhat weaker relationships between ambiguity and compound risk attitudes among engineering students and climate policymakers respectively. Moreover, the role of complexity as a factor contributing to explain SF1 and SF2 remains largely understudied, with the exceptions of Armantier and Treich (2016), and Kovářík et al. (2016). Our paper attempts to fill these gaps by examining two research questions:

**RQ1:** Are SF1 & SF2 robust to sophistication?

**RQ2:** What are the main drivers of ambiguity attitude?

To answer these questions, we specifically targeted a selected sample of risk professionals, who possess a high level of sophistication in probabilistic reasoning. To our knowledge, we are the first to study the preferences of such a unique pool of subjects in an incentivized experiment with a simple, context-free, design allowing us to make direct comparisons with other subject pools. Although focusing on such a unique sample necessarily sacrifices representativeness, it enables us to answer our first research question and to bring novel insights on the role of sophistication. Our second research question aims at disentangling the driving mechanisms of the Ellsberg paradox. Different explanations have been proposed in the literature: Following theories that equate ambiguity aversion to compound risk aversion (e.g., Segal, 1987; Seo, 2009), the driving factor of the Ellsberg paradox is the failure to reduce compound risks.<fnref n="7" /> Along similar lines, some recent studies have

<footnote n="7">⁷Segal (1987, p. 179) wrote: "In other words, risk aversion and ambiguity aversion are two sides of the same</footnote>
suggested that ambiguity aversion can be related to an aversion towards complex risks (e.g., Armantier and Treich, 2016; Kovářík et al., 2016). Alternatively, according to a variety of theoretical models with normative underpinnings (see Gilboa and Marinacci, 2013, for a review), Ellsberg-type behaviors are primarily driven by a specific attitude towards unknown probabilities.<fnref n="8" /> In what follows, we analyze the effect of these factors to understand their respective roles in explaining SF1 and SF2, and relate them to the subjects' sophistication level.

# 2 Experiments

We report the results of two experiments. The data were collected in the context of a broad project investigating the layers of uncertainty with additional treatments (see Aydogan et al., 2022). Addressing different research questions, the current study presents distinct analyses and results.

## 2.1 Samples

The experiments were conducted with two different samples. The main experiment is an artefactual field experiment run on a unique pool of risk professionals (actuaries). The control experiment is a standard laboratory experiment with university students.

**Actuaries at ICA** We collected data from 84 risk professionals during the 31st International Congress of Actuaries (ICA).<fnref n="9" /> The average age was around 40 and 44% of the subjects were female. The subjects were highly educated: 58 subjects (69%) reported a master's degree as the highest level of education completed and 18 subjects (21%) reported a PhD degree. 46 subjects reported that their highest degree was obtained in a

<footnote>coin, and the rejection of the Ellsberg urn does not require a new concept of ambiguity aversion, or a new concept of risk aversion.”</footnote>

<footnote n="8">⁸For example, the theories of Klibanoff et al. (2005), Nau (2006), and Ergin and Gul (2009) represent ambiguity as two-stage uncertainty and account for ambiguity attitudes by distinguishing between objective and subjective probabilities present at each stage without relating it to compound risk non-reduction. Note also that a recent study of Esponda and Vespa (2021) suggests that failures in contingent reasoning may also explain Ellsberg Paradox.</footnote>

<footnote n="9">⁹ICA is a conference organized by the International Actuarial Association every four years. It gathers more than 2,500 actuaries, academics, and high-ranking representatives from the international insurance and financial industry. The 31st congress was held from June 4 to 8, 2018, in Berlin, Germany.</footnote>
field related to mathematics and statistics, while 17 subjects reported it related to actuarial sciences. The remaining subjects reported diplomas in physics (2), engineering (1), finance (1), economics and management (3), or did not report anything (14). Finally, the subjects had an average of 13 years of relevant work experience.

**University students** We collected data from 125 social science students at Bocconi University, Italy. At the time of the experiment, 80 of them (64%) were in a bachelor's program while 34 (27%) were in a master's program, and the rest (9%) were in a PhD program. 42% of the subjects were female, and the average age was 20.5.

In what follows, we characterize sophistication by the background of the subjects. In that sense, actuaries are considered as more “sophisticated” than students. Such a distinction is justified on the ground that actuaries are experts in decision-making under uncertainty and experienced risk evaluators, who make decisions in situations of ambiguity in both their private and professional roles. They also possess a high training in statistics, probability and decision theory. It should, however, be clear that different dimensions may arguably contribute to making the pool of actuaries different than that of students. In particular, the two samples differ in the following dimensions: (1) *Curriculum*: 79% of the actuaries possess a training in STEM (science, technology, engineering, and mathematics), while Bocconi is a school offering programs in humanities; (2) *Level of education*: 90% of the actuaries reported to hold at least a master's degree, while this is the case for only 9% of the students; (3) *Experience*: actuaries reported an average 13 years of relevant work experience, whereas most students had no work experience at all. In addition, the age difference between the two groups could be seen as a confounder to what precedes. As our objective in this paper is to investigate the robustness of SF1 and SF2 for a non-convenience sample, rather than to identify the precise factors driving sophistication, we report the results of a within-sample heterogeneity analysis in Online Appendix A.
## 2.2 Design

**Sources of uncertainty** We use a within subject design to study individual choices under risk and ambiguity. The experiment entails betting on the color of a card drawn from a deck in different situations. We consider the following four distinct sources of uncertainty that are constructed in a two-color Ellsberg-type setting (see also Figure 1 (a)).

1. (R) *Risk* entails a deck that contains equal proportions of black and red cards.

2. (CR) *Compound Risk* entails, with equal probabilities, either a deck that contains *p%* red and (1 − *p*)% black cards, or a deck that contains *p%* black and (1 − *p*)% red cards.

3. (MA) *Model Ambiguity* entails, with unknown probabilities, either a deck that contains *p%* red and (1−*p*)% black cards, or a deck that contains *p%* black and (1−*p*)% red cards.

4. (E) *Ellsberg ambiguity* entails a deck of 100 cards that contains an unknown proportion of black and red cards.

The sources *R* and *CR* are two sources of risk (known probabilities), whereas the sources *MA* and *E* are ambiguous (unknown probabilities). The sources *CR* and *MA* are compounded as they explicitly entail two stages, with two potential deck compositions. They differ from each other in the type of uncertainty they entail in the first stage. Specifically, the two possible deck compositions are unambiguously assigned an objective 50% probability under *CR*, whereas these probabilities are unknown in the case of *MA*. On the basis of a symmetry argument, a 50% probability could be assigned to the two possible deck compositions under *MA*, but these probabilities would then necessarily be subjectively determined.<fnref n="10" /> *E* is the standard ambiguous source originally proposed by Ellsberg (1961).

<footnote n="10">¹⁰In our experiment, symmetry in the prior distribution stems from the indifference between betting on a red or black card. The symmetry condition can also be justified on the grounds of a general symmetry of information argument: given the information available, there is a priori no reason to believe that one model deserves more weight than another.</footnote>
**Complexity** In the spirit of Kovářík et al. (2016), we consider a notion of complexity related to the *number of stages of uncertainty* a situation features. Accordingly, for each source *CR* and *MA*, we propose two distinct cases that are characterized by different levels of complexity. In the first case, we consider *p* = 0 so that the deck features a degenerate distribution: it contains either only red or only black cards. We denote the corresponding situations *CR0* and *MA0*.11 This case is of minimal complexity: although the situation is presented in two stages, it entails only one stage of uncertainty (as all the uncertainty stems from the first stage). The second case considers *p* = 25, so that the deck contains either 25% red (and 75% black) cards or 25% black (and 75% red) cards. We denote the corresponding situations *CR25* and *MA25*.12

**Procedure** Our experiments measure individual preferences over the situations *R*, *CR0*, *CR25*, *MA0*, *MA25*, and *E*. For each of them, the subjects faced a bet on the color of a card randomly drawn from a deck. For every bet, the winning color was determined by the subjects themselves.13 We elicited the certainty equivalents (CEs) of the bets using a choice-list design. We use the midpoint of an indifference interval implied by a switching point as a proxy for the CE of a bet. In view of the stark income gap between risk professionals and students (see Online Appendix A.5), we adjusted the stakes offered to the two groups by a factor of 10. Specifically, bets yielded either €200 or €0 in the experiment with actuaries and either €20 or €0 in the experiment with students. We used a standard prior within-subject random incentive mechanism in the lab (i.e., all students were paid based on one of their choices in the experiment) but adopted a between-subject random incentive system in the field (i.e., one-in-ten actuaries was paid) due to budgetary and logistical constraints.14 The details of the experimental procedures

<footnote n="11">¹¹In the literature, *CR0* has also been used to study the hypothesis of *time neutrality* (Segal, 1987, 1990; Dillenberger, 2010; Nielsen, 2020), i.e., the indifference between early and late resolution of risk. It should be clear that the time dimension is not considered in our experiment.</footnote>

<footnote n="12">¹²Note that our characterization of complexity can also be seen as referring to the number of *branches* of a lottery. In that sense, it is consistent with Chew et al. (2017, see footnote 20) in the case of compound risk, and with the notion of complexity under simple risk, which is typically assessed by the number of different outcomes of the lottery (Sonsino et al., 2002; Moffatt et al., 2015; Puri, 2018).</footnote>

<footnote n="13">¹³This feature eliminates potential suspicion effect. Furthermore, as pointed out by a referee, if anything, it may bias the results towards ambiguity neutrality if subjects self-randomize over the colors as a hedge against ambiguity. In that sense, the significant ambiguity aversion observed in our data should be seen as conservative.</footnote>

<footnote n="14">¹⁴Previous literature has reported no systematic difference between paying all subjects or paying one-of-N (Beaud and Willinger 2015; Clot et al. 2018; Berlin et al. 2022). Note also that to further encourage risk professionals to reveal their preferences *conditional* on being selected for payment, we carried out the between-</footnote>
are provided in Online Appendix B.

## 3 A relative premium measure

To examine attitudes toward the different sources of uncertainty, we introduce the following premia measure relative to risk (R).

**Definition 1.** The relative premium $\Pi_{R,j}$ is the difference between the CE of the bet on $R$ ($CE_R$) and the CE of the bet on $j$ ($CE_j$), expressed in % of the CE of the bet on the most preferred situation:

$$ \Pi_{R,j} = \frac{CE_R - CE_j}{\max\{CE_R, CE_j\}} \quad \forall j \in \{CR0, CR25, MA0, MA25, E\}. \quad (1) $$

Intuitively, two cases can be distinguished. If the individual is relatively more averse to the uncertainty present in situation $j$, the preferred bet is the one on $R$, and the relative premium represents the percentage of extra money that an individual would be ready to sacrifice to avoid betting on $j$, relative to the value of the bet on $R$. Symmetrically, if the preferred situation is $j$, the relative premium $\Pi_{R,j}$ represents the extra money that would be sacrificed to avoid betting on $R$, relative to the value of the bet on $j$. This index possesses some desirable properties. First, $\Pi_{R,j}$ is symmetric around zero across relatively more or less averse preferences. Second, $\Pi_{R,j}$ belongs to the interval $[-1; 1]$, which also makes it easy to interpret in terms of percentages. Lastly, the normalization with respect to the maximum CE allows more robust comparisons among subject pools by controlling for differences in payoffs and subjects’ overall level of uncertainty attitudes.<fnref n="15" />

In the literature, $\Pi_{R,E}$ has been commonly referred to as the *Ellsberg-ambiguity premium* (see, e.g., Maccheroni et al., 2013), and $\Pi_{R,CR}$ as the *compound risk premium* (see, e.g., Abdellaoui et al., 2015). In the same vein, $\Pi_{R,MA}$ represents the *model ambiguity*

<footnote>subject randomization prior to the experiment, thus enhancing the isolation assumption of Kahneman and Tversky (1979) (see Johnson et al., 2021 for more discussion on the prior random incentive mechanisms). Under the isolation assumption, the higher stakes offered to actuaries compensate for the large income gap between risk professionals and students (see Online Appendix A).</footnote>

<footnote n="15">¹⁵In Online Appendix A, we discuss some alternative measures that have been proposed in the literature, such as $\Pi_{R,j} = \frac{CE_R - CE_j}{CE_R}$ or $\Pi_{R,j} = \frac{CE_R - CE_j}{CE_R + CE_j}$ (Sutter et al., 2013; Trautmann and van de Kuilen, 2015). Our conclusions do not differ when using these alternative definitions.</footnote>
*premium*. These premia are represented by the arrows 1.a-c in Figure 1(b). The relative premium can furthermore be used to measure the effects of complexity and of a specific attitude toward unknown probabilities (in the first stage), as illustrated by arrows 2 and 3 in Figure 1(c). Specifically, as the sources *CR* and *MA* both present a relatively less and more complex case (i.e., one stage of uncertainty when *p* = 0 vs. two stages of uncertainty when *p* = 25), the effect of complexity, within each source, can be examined by ($\Pi_{R,CR25} - \Pi_{R,CR0}$) and ($\Pi_{R,MA25} - \Pi_{R,MA0}$). These differences indicate whether the compound risk or model ambiguity premia are larger in more complex cases than in less complex ones. Similarly, ($\Pi_{R,MA0} - \Pi_{R,CR0}$) and ($\Pi_{R,MA25} - \Pi_{R,CR25}$) capture the effect of the distinct treatment of known and unknown probabilities in the first stage within situations entailing the same degree of complexity.

# 4 Results

Our data consist of six choice lists per subject. Observations with multiple-switching, reverse-switching, or no-switching patterns are not included in the analysis as they do not provide clear measurements of the CEs.<fnref n="16" /> We do not detect any order effect on treatments (see Online Appendix A).

## 4.1 General attitudes toward different sources of uncertainty

Table 1 presents the mean relative premia. We observe that both groups of subjects are comparable in terms of ambiguity premia, exhibiting aversion toward the sources *MA* and *E* (*t*-tests, *p*<0.001).<fnref n="17" /> This suggests that ambiguity aversion (SF1) is robust to the subjects’ level of sophistication. Regarding the source *CR*, the average relative premium for *CR25* is positive for students (*p*<0.001), indicating aversion toward compound risk, but we cannot reject the null hypothesis that $\Pi_{R,CR25} = 0$ for actuaries (*t*-test, *p*=0.03

<footnote n="16">¹⁶The proportions of subjects affected by such inconsistencies in at least one of the choice lists are 11.9% for actuaries (10 out of 84) and 10.4% for students (13 out of 125) and do not differ across the two samples (two-sample Z-test of proportions, *p*=0.73). Discarding four actuaries who show inconsistent patterns in *all* lists, suggesting a lack of attention to the experiment, inconsistencies were present in 16 out of 480 lists (3.3%) for actuaries and in 25 out of 750 lists (3.3%) for students. These proportions are notably lower than what is typically observed in the literature (Yu et al., 2021).</footnote>

<footnote n="17">¹⁷Testing multiple hypotheses (e.g., testing $H_0: \Pi_{R,j} = 0$ for all $j \in \{CR0, CR25, MA0, MA25, E\}$) may require Bonferroni corrections. To allow for direct comparisons with previous literature, we report the original *p*-values, together with Bonferroni corrections when these affect the results.</footnote>
but *p*=0.17 after Bonferroni correction). The difference between actuaries and students is particularly marked for this premium (*t*-test, *p*<0.001). In contrast, the average relative premium for the less complex case *CR*0 does not differ from zero for both groups (*t*-test, *p*=0.59 for actuaries, and *p*=0.55 for students).

## 4.2 The relationship between ambiguity and compound risk attitudes

Following the existing literature, we investigate the relationship between attitudes towards ambiguity and compound risk within our two subject pools and test the robustness of SF2 to sophistication. Table 2 reports the Pearson correlation coefficients between compound risk premia and ambiguity premia.<fnref n="18" /> In line with SF2, we observe a significant correlation between attitudes toward ambiguity and compound risk for students (except between $\Pi_{R,CR0}$ and $\Pi_{R,E}$, *p*=0.475). However, such a relationship is absent in the case of actuaries.<fnref n="19" />

Next, we examine the links between ambiguity neutrality and reduction of compound risk. We adopted a comprehensive definition of ambiguity neutrality according to which a subject is considered as ambiguity neutral if $\Pi_{R,E} = \Pi_{R,MA0} = \Pi_{R,MA25} = 0$.<fnref n="20" /> Similarly, a subject is said to be reducing compound risk if $\Pi_{R,CR0} = \Pi_{R,CR25} = 0$.

The proportion of ambiguity non-neutrality among subjects who do not reduce compound risk is 95% (=20/21) for actuaries and 94% (=77/82) for students. These proportions, which are in line with the literature, suggest that non-reduction of *CR* is sufficient for ambiguity non-neutrality, irrespective of the subjects' sophistication level. Turning to necessity, we find that 80% (=77/96) of ambiguity non-neutral students are also not reducing *CR*. However, this proportion is 57% (=20/35) for actuaries, which is significantly less than for students (two-sample test of proportions, *p*=0.008). This result indicates

<footnote n="18">¹⁸The same conclusions are obtained with Spearman rank correlations, which measure monotonic –rather than linear– relationships between premia (see Online Appendix A).</footnote>

<footnote n="19">¹⁹We also analyzed correlations based on the method of *obviously related instrumental variables* (ORIV) developed by Gillen et al. (2019) to correct for measurement errors. ORIV uses an instrumental variable approach to compute correlations when there are multiple measurements of behavioral variables. We used ORIV in our data by using multiple elicitations of preferences under compound risk and model ambiguity (i.e., with *P* = 0 and *P* = 25). We observe that, although the correlations between compound risk and ambiguity using ORIV are consistently high for students: $corr(\Pi_{R,CR}, \Pi_{R,MA})=0.988$ and $corr(\Pi_{R,CR}, \Pi_{R,E})=0.916$, they remain remarkably low for actuaries: $corr(\Pi_{R,CR}, \Pi_{R,MA})=0.369$ and $corr(\Pi_{R,CR}, \Pi_{R,E})=0.057$.</footnote>

<footnote n="20">²⁰We also considered alternative definitions of ambiguity neutrality under *MA* and *E* separately (i.e., *MA* neutrality if $\Pi_{R,MA0} = \Pi_{R,MA25} = 0$, and *E*-ambiguity neutrality if $\Pi_{R,E} = 0$). Our conclusions are robust to the use of these alternative definitions (see Online Appendix A).</footnote>
that, although compound risk non-reduction appears to be also necessary for ambiguity
non-neutrality when less sophisticated subjects are considered, this is not the case for
more sophisticated ones. Overall, this first set of results enables us to answer RQ1.

**Result 1** (a) *Ambiguity aversion is robust to the subjects' sophistication level, but (b) the strong relationship between attitudes toward ambiguity and compound risk is not.*

**4.3 Complexity and ambiguity**

We now focus on compound sources to examine the effects of complexity and ambiguity
in different subject pools. For this, we run a regression analysis with random effects at
individual level, where the relative premia $\Pi_{R,j}$ for $j \in \{CR0, CR25, MA0, MA25\}$ are
regressed on a dummy for complexity (taking value 1 if $j \in \{CR25, MA25\}$), a dummy
for the presence of ambiguity (taking value 1 if $j \in \{MA0, MA25\}$), and their interaction.
The baseline is the behavior in a compound risk situation with minimal complexity. To
test the effect of sophistication, we run a regression by pooling data from the two samples
and using a dummy for actuaries.

Table 3 reports the results. We observe positive coefficients for complexity and model
ambiguity, indicating that the relative premia are higher when the situation is more
complex or does not entail objective probabilities, in comparison to the less complex
situation with objective probabilities (i.e., CR0). Therefore, both students and actuaries
can be said to be averse to complexity and unknown probabilities in the first-stage.
However, the effect of complexity is significantly lower among actuaries than among
students, although there is no difference between the two groups regarding the effect of
unknown probabilities. We also observe a negative interaction between the variables,
suggesting that the effect of complexity is less pronounced in the presence of model
ambiguity. This interaction is significant for students but not for actuaries.

**4.4 Explaining Ellsberg ambiguity**

Based on what precedes, we now investigate the roles of attitudes toward complex-
ity and unknown probabilities, together with the failure of the reduction principle in
explaining Ellsberg-ambiguity attitude. We use the following OLS regression:
$$E-AMB_i = \beta_0 + \beta_1 COMPX_i + \beta_2 UNKNOWNN_i + \beta_3 RED_i + \varepsilon_i, \quad (2)$$

where Ellsberg ambiguity attitude (E-AMB) for subject i is computed by $\Pi_{R,E}$. Attitudes toward complexity and unknown probabilities (COMPX and UNKNOWNN, respectively) are both measured with respect to $\Pi_{R,CR0}$ to isolate their pure effects and avoid interactions between them. Specifically, complexity attitude is captured by $(\Pi_{R,CR25} - \Pi_{R,CR0})$, which computes the difference between the compound risk premia under different degrees of complexity.<fnref n="21" /> Attitude toward unknown probabilities is measured by $(\Pi_{R,MA0} - \Pi_{R,CR0})$, which captures the difference in relative premia between two compound situations presenting the same degree of complexity, but different type of probabilities in their first stage.<fnref n="22" /> Finally, the measure of reduction (RED) is based on $\Pi_{R,CR0}$: As CR0 is arguably the most easily reducible compound risk situation, its non-reduction shows a clear failure of the reduction principle (rather than a failure to deal with complexity). The dummy for reduction takes 1 if $\Pi_{R,CR0} \neq 0$ and 0 otherwise.

Table 4 reports the results of the regressions. We find that attitude toward unknown probabilities has a positive and significant impact for both actuaries and students. The magnitudes of the coefficients indicate that one percentage point increase in $(\Pi_{R,MA0} - \Pi_{R,CR0})$ leads to 0.74 percentage points increase in $\Pi_{R,E}$ for actuaries, and to 0.43 percentage points increase for students. The difference in this coefficient between the two groups is significant ($p = 0.03$), indicating a stronger effect of attitude toward unknown probabilities for actuaries. In contrast, complexity attitude has a positive and significant impact for students only. The difference in the magnitude of the coefficients between the groups suggests that the effect of complexity is also more pronounced for students ($p=0.04$). Finally, the positive coefficients of the reduction variable suggest that failure of the reduction principle increases the ambiguity premium, although the coefficients are neither significant nor different in the two groups. Overall, this second set of results enables us to answer RQ2.

<footnote n="21">²¹Note that the alternative, which is to use $(\Pi_{R,MA25} - \Pi_{R,MA0})$, could be confounded by ambiguity attitudes because MA0 may also be seen as being *more ambiguous* than MA25 due to a larger spread of first stage probabilities (see Jewitt and Mukerji, 2017; Berger, 2021).</footnote>

<footnote n="22">²²The alternative, which is to use $(\Pi_{R,MA25} - \Pi_{R,CR25})$ could be confounded by risk attitudes because of the presence of risk in the second stage (see the discussion in Berger and Bosetti, 2020).</footnote>
**Result 2:** (a) For both actuaries and students, the main driver of Ellsberg-ambiguity attitude is a specific treatment of unknown probabilities. (b) A specific attitude toward complexity is found to play a significant role in explaining Ellsberg-ambiguity attitude for students only.

# 5 Concluding remarks

Decisions made by unusual subject pools, such as climate policymakers (Berger and Bosetti, 2020), professional traders (Fox et al., 1996; Haigh and List, 2005), professional chess players (Levitt et al., 2011), or golf players (Pope and Schweitzer, 2011) have been the focus of studies trying to explain important behavioral phenomena. Following this line of research, we focus on a unique pool of risk professionals to re-examine two stylized facts about ambiguity attitudes, which have emerged in the literature. Because these professionals routinely price risk and uncertainty at work, their occupational practice makes them of special interest for studying decision-making under uncertainty.

Our results show that this selected group of subjects is as much affected by ambiguity as a standard pool of university students. However, attitudes towards ambiguity and compound risk are less closely related for risk professionals than for students. In particular, compound risk non-reduction is found sufficient but not necessary for ambiguity non-neutrality for these more sophisticated subjects. We argue that attitudes toward complexity may be driving these findings. Indeed, if ambiguity is viewed as a compound source of uncertainty, or presented as such (as in model ambiguity situations), non-reduction of compound risk can be sufficient for ambiguity non-neutrality. On the other hand, if complexity makes compound risk situations being perceived as ambiguous by some subjects, those who are ambiguity non-neutral will also exhibit compound risk non-reduction (and hence the necessity). Interestingly, this effect is significantly weaker for more sophisticated subjects, who are less affected by the complexity of a situation. Consistent with this interpretation, we observe that a non-negligible proportion of ambiguity non-neutral actuaries *do* actually reduce compound risk.

The paper closest to ours is Abdellaoui et al. (2015), who compared two student samples differing in their training (engineering vs. non-engineering fields). While they
also report a somewhat weaker link between compound risk and ambiguity for more
quantitatively sophisticated students, the differences they find between their two student
samples are not as stark as those between students and actuaries. By studying a pool of
risk professionals, whose contrast with students is more extreme, our study may be seen
as more revealing for the role of sophistication in decision-making. Yet we also note that
differences in sophistication might exist within the populations studied. An additional
analysis of our data indeed indicates some heterogeneity among students but not among
risk professionals (for the details, see Online Appendix). For example, undergraduate
students are found to be more affected by complexity than graduate ones, whereas work
experience (or age) is not found to play any role among actuaries.

We argue that our findings may have important implications for different ambiguity
models. Overall, by suggesting that ambiguity aversion is mainly driven by a genuine
preference for known probabilities over unknown ones, but not necessarily by an inabil-
ity/aversion to deal with the compoundness or complexity of a situation, the results
we report in this paper are more consistent with the predictions of ambiguity theories
with normative underpinnings. Thus, they leave room for using ambiguity models in
applications with prescriptive purposes.
<figure><img src="image_2.png" /><figcaption>(a) Illustration of the four sources of uncertainty (here $p = 25$ in CR and MA)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 1: Sources of uncertainty and their characteristics</figcaption></figure>
<table><caption>Table 1: AVERAGE PREMIA RELATIVE TO RISK</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Two-sample tests<br>(p-value)</th></tr></thead><tbody><tr><td>Π<sub>R,CR0</sub></td><td>-0.007 (0.0127)</td><td>0.010 (0.0171)</td><td>0.472</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.023* (0.0108)</td><td><strong>0.136</strong>*** (0.0216)</td><td><strong>&lt;0.001</strong></td></tr><tr><td>Π<sub>R,MA0</sub></td><td><strong>0.121</strong>*** (0.0324)</td><td><strong>0.130</strong>*** (0.0219)</td><td>0.814</td></tr><tr><td>Π<sub>R,MA25</sub></td><td><strong>0.106</strong>*** (0.0227)</td><td><strong>0.181</strong>*** (0.0227)</td><td>0.027</td></tr><tr><td>Π<sub>R,E</sub></td><td><strong>0.190</strong>*** (0.0316)</td><td><strong>0.191</strong>*** (0.0249)</td><td>0.982</td></tr></tbody></table>

Notes: Standard errors in parentheses. The tests are based on two-sided *t*-tests. *** significant at 0.001, ** significant at 0.01, * significant at 0.05. Significance stars are based on *p*-values before Bonferroni correction. Values that are significant at 0.05 after Bonferroni correction are bolded.
<table><caption>Table 2: PEARSON CORRELATIONS BETWEEN COMPOUND RISK AND AMBIGUITY PREMIA</caption><thead><tr><th></th><th colspan="3">Actuaries</th><th colspan="3">Students</th></tr><tr><th></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th></tr></thead><tbody><tr><th>Π<sub>R,CR0</sub></th><td>0.169</td><td>-0.033</td><td>-0.078</td><td>Π<sub>R,CR0</sub></td><td><strong>0.315</strong>***</td><td><strong>0.344</strong>***</td><td>0.067</td></tr><tr><th>Π<sub>R,CR25</sub></th><td>0.135</td><td>0.107</td><td>0.109</td><td>Π<sub>R,CR25</sub></td><td><strong>0.407</strong>***</td><td><strong>0.652</strong>***</td><td><strong>0.475</strong>***</td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05. Significance stars are based on *p*-values before Bonferroni correction. Values that are significant at 0.05 after Bonferroni correction are bolded.
<table><caption>Table 3: RANDOM EFFECTS REGRESSIONS OF RELATIVE PREMIA</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Effect of sophistication<br>(pooled data)</th></tr></thead><tbody><tr><td>Complexity</td><td>0.031*<br>(0.015)</td><td>0.123***<br>(0.023)</td><td>-0.092***<br>(0.027)</td></tr><tr><td>Model ambiguity</td><td>0.128***<br>(0.033)</td><td>0.120***<br>(0.023)</td><td>0.008<br>(0.040)</td></tr><tr><td>Complexity × Model ambiguity</td><td>-0.046<br>(0.031)</td><td>-0.075**<br>(0.026)</td><td>0.028<br>(0.040)</td></tr><tr><td>Constant</td><td>-0.006<br>(0.013)</td><td>0.010<br>(0.017)</td><td>-0.016<br>(0.021)</td></tr><tr><td>Observations</td><td>299</td><td>471</td><td>770</td></tr></tbody></table>

Notes: Cluster-robust standard errors in parentheses. *** significant at 0.001, ** significant at 0.01, * significant at 0.05. Similiar results are obtained when controlling for age, gender, income, and education (see Online Appendix A).
<table><caption>Table 4: OLS REGRESSIONS OF ELLSBERG AMBIGUITY PREMIUM</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Difference between<br>groups (pooled data)</th></tr></thead><tbody><tr><td><i>COMPX</i></td><td>-0.170<br>(0.167)</td><td>0.253*<br>(0.116)</td><td>-0.423*<br>(0.203)</td></tr><tr><td><i>UNKNOWN</i></td><td>0.735***<br>(0.091)</td><td>0.434***<br>(0.106)</td><td>0.302*<br>(0.140)</td></tr><tr><td><i>RED</i></td><td>0.137<br>(0.074)</td><td>0.036<br>(0.048)</td><td>0.101<br>(0.088)</td></tr><tr><td>Constant</td><td>0.079**<br>(0.026)</td><td>0.091***<br>(0.024)</td><td>-0.012<br>(0.036)</td></tr><tr><td>Observations</td><td>74</td><td>114</td><td>188</td></tr></tbody></table>

*Notes:* Robust standard errors in parentheses, *** significant at 0.001, ** significant at 0.01, * significant at 0.05. Similar results are obtained when controlling for age, gender, income, and education.
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# Online Appendix

## A Further results

### A.1 Alternative measures of premia

The literature contains several alternative definitions of ambiguity/compound risk premia. In this section, we compare our measure (1) with some measures that have recently been proposed and show that our results are robust to these alternative definitions.

#### A.1.1 Premia relative to risk

(Trautmann and van de Kuilen, 2015) use two different measures of the ambiguity premium. The first, closest to our measure (1), is defined in relation to $CE_R$ only. In our notations, it reads as follows:

$$ \Pi_{R,j} \equiv \frac{(CE_R - CE_j)}{CE_R} \quad \forall j \in \{CR0, CR25, MA0, MA25, E\}. \quad (A.1) $$

One inconvenience of this definition is that it does not satisfy the symmetry across aversion and seeking attitudes and does not necessarily belong to [-1; 1]. In particular, while the aversion indices are bounded between 0 and 1, seeking indices can take values less than -1, which may introduce a bias in this direction.

Table A.1 replicates the results shown in Table 1 using this alternative definition of premia. As can be observed, the results are comparable to the one obtained using Definition 1.

<table><caption>Table A.1: AVERAGE PREMIA RELATIVE TO RISK</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Two-sample tests<br>(p-value)</th></tr></thead><tbody><tr><td>Π<sub>R,CR0</sub></td><td>-0.023</td><td>-0.008</td><td>0.634</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.022</td><td><strong>0.129</strong>***</td><td><strong>&lt;0.001</strong></td></tr><tr><td>Π<sub>R,MA0</sub></td><td><strong>0.110</strong>**</td><td><strong>0.116</strong>***</td><td>0.875</td></tr><tr><td>Π<sub>R,MA25</sub></td><td><strong>0.104</strong>***</td><td><strong>0.174</strong>***</td><td>0.049</td></tr><tr><td>Π<sub>R,E</sub></td><td><strong>0.190</strong>***</td><td><strong>0.181</strong>***</td><td>0.841</td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05. The tests are based on t-tests. Values that are significant at 5% after Bonferroni correction are bolded.

#### A.1.2 Premia relative to expected value

The second definition of ambiguity premium used by (Trautmann and van de Kuilen, 2015) is expressed relative to the expected value (EV) of the bet:

$$ \Pi_{R,j} \equiv \frac{(CE_R - CE_j)}{EV} \quad \forall j \in \{CR0, CR25, MA0, MA25, E\}. \quad (A.2) $$
While this measure satisfies symmetry in attitudes, it does not belong to [-1; 1].<sup>23</sup> Its main drawback is that it is unable to take into account the differences in subjects' risk attitudes. In particular, the magnitude of ambiguity/compound risk attitudes is bounded above by the degree of risk aversion. To illustrate, suppose that EV = 100 and consider an individual who is risk averse, such that $CE_R^1 = 40$. If this individual is also ambiguity averse, her maximal theoretical Ellsberg ambiguity premium is $\Pi_E^1 = 0.4$ (i.e. if $CE_E^1 = 0$). Now, consider a second individual who is risk neutral: $CE_R^2 = 100$. In her case, the maximal theoretical Ellsberg ambiguity premium is 1. If this individual is not extremely ambiguity averse, but makes choices such that $CE_E^2 = 60$, her ambiguity premium is $\Pi_E^2 = 0.4$. As a consequence, even when the first individual is extremely ambiguity averse with $\Pi_E^1 = 0.4$ (whereas the second is not), the definition (A.2) yields the same ambiguity measures for the two individuals. In the context of our experiment, using such a measure would not affect our results, as shown in Table A.2.

<table><caption>Table A.2: AVERAGE PREMIA RELATIVE TO RISK</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Two-sample tests<br>(p-value)</th></tr></thead><tbody><tr><td>Π<sub>R,CR0</sub></td><td>-0.011</td><td>0.010</td><td>0.420</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.029*</td><td><strong>0.140</strong>***</td><td><strong>&lt;0.001</strong></td></tr><tr><td>Π<sub>R,MA0</sub></td><td><strong>0.122</strong>***</td><td><strong>0.117</strong>***</td><td>0.920</td></tr><tr><td>Π<sub>R,MA25</sub></td><td><strong>0.105</strong>***</td><td><strong>0.181</strong>***</td><td>0.039</td></tr><tr><td>Π<sub>R,E</sub></td><td><strong>0.190</strong>***</td><td><strong>0.186</strong>***</td><td>0.919</td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05. The tests are based on t-tests. Values that are significant at 5% after Bonferroni correction are bolded.

### A.1.3 Sutter et al.’s (2013) premia

Finally, Sutter et al. (2013) propose a measure of ambiguity attitude (or, alternatively, compound risk attitude) corresponding to

$$ \Pi_{R,j} = \frac{(CE_R - CE_j)}{(CE_R + CE_j)} \quad \forall j \in \{CR0, CR25, MA0, MA25, E\}. \qquad (A.3) $$

This measure prevents the aforementioned problems of the other definitions and respects the same properties as our measure. In particular, it controls for the absolute levels of attitudes toward risk and uncertain source *i*, and ranges from -1 (extreme seeking) to 1 (extreme aversion). As can be observed in Table A.3, it also yields the same conclusions as in Table 1, except that all premia are scaled down due to the higher denominator in expression (A.3).

<footnote n="23"><sup>23</sup>For example, if the individual is risk seeking ($CE_R = 180$) and highly ambiguity averse ($CE_E = 20$), the relative ambiguity premium may be higher than 1 (in this case, $\Pi_E = 1.6$).</footnote>
<table><caption>Table A.3: AVERAGE PREMIA RELATIVE TO RISK</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Two-sample tests<br>(p-value)</th></tr></thead><tbody><tr><td>Π<sub>R,CR0</sub></td><td>-0.005</td><td>0.008</td><td>0.370</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.014*</td><td><strong>0.090</strong>***</td><td><strong>&lt;0.001</strong></td></tr><tr><td>Π<sub>R,MA0</sub></td><td><strong>0.094</strong>**</td><td><strong>0.083</strong>***</td><td>0.710</td></tr><tr><td>Π<sub>R,MA25</sub></td><td><strong>0.069</strong>***</td><td><strong>0.119</strong>***</td><td>0.035</td></tr><tr><td>Π<sub>R,E</sub></td><td><strong>0.138</strong>***</td><td><strong>0.131</strong>***</td><td>0.820</td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05. The tests are based on *t*-tests. Values that are significant at 5% after Bonferroni correction are bolded.

## A.2 Order effects

The order of the uncertain situations presented in our experiment was randomized. In this appendix, we investigate whether the order of presentation may have affected the subjects' preferences. We look at the values of the five CEs elicited for *R*, *CR0*, *CR25*, *MA0*, and *MA25*. Each of these situations is assigned an order of presentation from 1st to 8th (as we also had three other situations with model misspecification; see footnote 8 in the main text). In Table A.4, we test whether the CEs differ across different orders by using one-way ANOVA. No order effect is detected for any of the uncertain situations. Non-parametric Kruskal Wallis tests do not change the conclusions.

<table><caption>Table A.4: VARIATION IN THE CEs AS A FUNCTION OF THE ORDER OF SCENARIOS</caption><thead><tr><th>Order</th><th>1<sup>st</sup></th><th>2<sup>nd</sup></th><th>3<sup>rd</sup></th><th>4<sup>th</sup></th><th>5<sup>th</sup></th><th>6<sup>th</sup></th><th>7<sup>th</sup></th><th>8<sup>th</sup></th><th><i>p</i>-value</th></tr></thead><tbody><tr><td><i>R</i></td><td>73.89</td><td>98.57</td><td>94.00</td><td>80.00</td><td>79.09</td><td>84.00</td><td>85.00</td><td>102.50</td><td><strong>0.44</strong></td></tr><tr><td><i>CR0</i></td><td>80.00</td><td>96.67</td><td>87.50</td><td>100.00</td><td>90.00</td><td>102.50</td><td>84.09</td><td>82.86</td><td><strong>0.83</strong></td></tr><tr><td><i>CR25</i></td><td>105.71</td><td>97.73</td><td>85.00</td><td>77.50</td><td>94.29</td><td>75.26</td><td>90.00</td><td>88.89</td><td><strong>0.50</strong></td></tr><tr><td><i>MA0</i></td><td>66.25</td><td>80.00</td><td>86.15</td><td>69.44</td><td>88.33</td><td>85.71</td><td>68.75</td><td>80.00</td><td><strong>0.86</strong></td></tr><tr><td><i>MA25</i></td><td>85.00</td><td>78.33</td><td>70.31</td><td>99.64</td><td>95.56</td><td>56.25</td><td>82.92</td><td>54.29</td><td><strong>0.11</strong></td></tr></tbody></table>

### Part I: Actuaries

<table><thead><tr><th>Order</th><th>1<sup>st</sup></th><th>2<sup>nd</sup></th><th>3<sup>rd</sup></th><th>4<sup>th</sup></th><th>5<sup>th</sup></th><th>6<sup>th</sup></th><th>7<sup>th</sup></th><th>8<sup>th</sup></th><th><i>p</i>-value</th></tr></thead><tbody><tr><td><i>R</i></td><td>8.91</td><td>10.00</td><td>9.00</td><td>8.93</td><td>10.31</td><td>10.00</td><td>11.14</td><td>8.81</td><td><strong>0.71</strong></td></tr><tr><td><i>CR0</i></td><td>9.50</td><td>9.22</td><td>8.81</td><td>10.50</td><td>9.00</td><td>9.65</td><td>10.42</td><td>7.88</td><td><strong>0.54</strong></td></tr><tr><td><i>CR25</i></td><td>7.73</td><td>9.81</td><td>8.43</td><td>8.00</td><td>8.55</td><td>7.43</td><td>7.81</td><td>7.73</td><td><strong>0.75</strong></td></tr><tr><td><i>MA0</i></td><td>7.38</td><td>9.83</td><td>8.33</td><td>9.88</td><td>8.88</td><td>6.75</td><td>8.43</td><td>6.50</td><td><strong>0.13</strong></td></tr><tr><td><i>MA25</i></td><td>8.31</td><td>6.55</td><td>8.00</td><td>6.80</td><td>6.90</td><td>10.27</td><td>6.92</td><td>8.90</td><td><strong>0.06</strong></td></tr></tbody></table>

Notes: Numbers represent the mean of the CEs of the situations shown in the corresponding order.
## A.3 Descriptive statistics

Tables A.5 and A.6 show the descriptive statistics of the data we collected in both experiments. Results are presented in terms of certainty equivalents and premia relative to risk, respectively. Figures A.1 and A.2 present the distributions of the CEs for the six uncertain situations (*R*, *CR0*, *CR25*, *MA0*, *MA25*, *E*) in the two experiments.

<table><caption>Table A.5: DESCRIPTIVE STATISTICS OF THE CEs</caption><thead><tr><th></th><th>Mean</th><th>SD</th><th>Min</th><th>Max</th><th>Obs.</th></tr></thead><tbody><tr><td colspan="6"><strong>Actuaries</strong></td></tr><tr><td><em>R</em></td><td>89.08</td><td>34.07</td><td>5</td><td>195</td><td>76</td></tr><tr><td><em>CR0</em></td><td>91.38</td><td>37.00</td><td>5</td><td>195</td><td>76</td></tr><tr><td><em>CR25</em></td><td>87.60</td><td>34.20</td><td>5</td><td>195</td><td>77</td></tr><tr><td><em>MA0</em></td><td>78.31</td><td>40.79</td><td>5</td><td>195</td><td>77</td></tr><tr><td><em>MA25</em></td><td>80.57</td><td>37.64</td><td>5</td><td>195</td><td>79</td></tr><tr><td><em>E</em></td><td>71.20</td><td>39.70</td><td>5</td><td>195</td><td>79</td></tr><tr><td colspan="6"><strong>Students</strong></td></tr><tr><td><em>R</em></td><td>9.55</td><td>3.52</td><td>0.5</td><td>19.5</td><td>120</td></tr><tr><td><em>CR0</em></td><td>9.46</td><td>3.59</td><td>0.5</td><td>19.5</td><td>122</td></tr><tr><td><em>CR25</em></td><td>8.12</td><td>3.56</td><td>0.5</td><td>19.5</td><td>119</td></tr><tr><td><em>MA0</em></td><td>8.31</td><td>3.80</td><td>0.5</td><td>19.5</td><td>121</td></tr><tr><td><em>MA25</em></td><td>7.75</td><td>3.71</td><td>0.5</td><td>19.5</td><td>120</td></tr><tr><td><em>E</em></td><td>7.75</td><td>3.83</td><td>0.5</td><td>19.5</td><td>123</td></tr></tbody></table>
<table><caption>Table A.6: DESCRIPTIVE STATISTICS OF THE RELATIVE COMPOUND RISK AND AMBIGUITY PREMIA</caption><thead><tr><th></th><th>Mean</th><th>SD</th><th>Min</th><th>Max</th><th>Obs.</th></tr></thead><tbody><tr><td colspan="6"><strong>Actuaries</strong></td></tr><tr><td>Π<sub>R,CR0</sub></td><td>-0.007</td><td>0.109</td><td>-0.6</td><td>0.286</td><td>74</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.023</td><td>0.094</td><td>-0.2</td><td>0.429</td><td>75</td></tr><tr><td>Π<sub>R,MA0</sub></td><td>0.121</td><td>0.279</td><td>-0.5</td><td>0.95</td><td>74</td></tr><tr><td>Π<sub>R,MA25</sub></td><td>0.106</td><td>0.198</td><td>-0.286</td><td>0.8</td><td>76</td></tr><tr><td>Π<sub>R,E</sub></td><td>0.190</td><td>0.273</td><td>0</td><td>0.95</td><td>75</td></tr><tr><td colspan="6"><strong>Students</strong></td></tr><tr><td>Π<sub>R,CR0</sub></td><td>0.010</td><td>0.186</td><td>-0.5</td><td>0.667</td><td>118</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.136</td><td>0.234</td><td>-0.333</td><td>0.8</td><td>117</td></tr><tr><td>Π<sub>R,MA0</sub></td><td>0.130</td><td>0.238</td><td>-0.5</td><td>0.6</td><td>119</td></tr><tr><td>Π<sub>R,MA25</sub></td><td>0.181</td><td>0.246</td><td>-0.375</td><td>0.857</td><td>117</td></tr><tr><td>Π<sub>R,E</sub></td><td>0.191</td><td>0.272</td><td>-0.429</td><td>0.95</td><td>119</td></tr></tbody></table>
Actuaries

R

<figure><img src="image_4.png" /></figure>

CR0

<figure><img src="image_5.png" /></figure>

CR25

<figure><img src="image_6.png" /></figure>

Students

R

<figure><img src="image_7.png" /></figure>

CR0

<figure><img src="image_8.png" /></figure>

CR25

<figure><img src="image_9.png" /><figcaption>Figure A.1: Distributions of CEs for *R*, *CR0*, and *CR25* for actuaries (on the left) and students (on the right)</figcaption></figure>
Actuaries

MU0

<figure><img src="image_10.png" /></figure>

MU25

<figure><img src="image_11.png" /></figure>

E

<figure><img src="image_12.png" /></figure>

Students

MU0

<figure><img src="image_13.png" /></figure>

MU25

<figure><img src="image_14.png" /></figure>

E

<figure><img src="image_15.png" /><figcaption>Figure A.2: Distributions of CEs for MA0, MA25, and E for actuaries (on the left) and students (on the right)</figcaption></figure>
## A.4 More on the relationship between compound risk and ambiguity attitudes

To complement Table 2, we here report the Spearman rank correlation coefficients between compound risk premia and ambiguity premia (Table A.7). Spearman correlations replicate the strong relationship between compound risk and ambiguity premia among students and the absence of such relationship among actuaries.

<table><caption>Table A.7: SPEARMAN CORRELATIONS BETWEEN COMPOUND RISK AND AMBIGUITY PREMIA</caption><thead><tr><th rowspan="2"></th><th colspan="3">Actuaries</th><th colspan="3">Students</th></tr><tr><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th></tr></thead><tbody><tr><th>Π<sub>R,CR0</sub></th><td>0.023</td><td>-0.083</td><td>-0.065</td><td>Π<sub>R,CR0</sub></td><td><strong>0.266**</strong></td><td><strong>0.306***</strong></td><td>0.145</td></tr><tr><th>Π<sub>R,CR25</sub></th><td>0.253*</td><td>0.188</td><td>0.146</td><td>Π<sub>R,CR25</sub></td><td><strong>0.407***</strong></td><td><strong>0.635***</strong></td><td><strong>0.449***</strong></td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05. Values that are significant at 5% after Bonferroni correction are bolded.

We then examine the links between ambiguity neutrality and compound risk reduction. Table A.8 reports the contingency results discussed in the main text.

<table><caption>Table A.8: RELATION BETWEEN COMPOUND RISK AND AMBIGUITY ATTITUDES</caption><thead><tr><th rowspan="2">Reduction of <i>CR</i></th><th colspan="3">Actuaries</th><th colspan="3">Students</th></tr><tr><th colspan="3">Ambiguity neutral</th><th colspan="3">Ambiguity neutral</th></tr><tr><th></th><th>No</th><th>Yes</th><th>Total</th><th>No</th><th>Yes</th><th>Total</th></tr></thead><tbody><tr><th>No</th><td>20</td><td>1</td><td><strong>21</strong></td><td>77</td><td>5</td><td><strong>82</strong></td></tr><tr><th>Yes</th><td>15</td><td>39</td><td><strong>54</strong></td><td>19</td><td>19</td><td><strong>38</strong></td></tr><tr><th>Total</th><td><strong>35</strong></td><td><strong>40</strong></td><td><strong>75</strong></td><td><strong>96</strong></td><td><strong>24</strong></td><td><strong>120</strong></td></tr></tbody></table>

Independence test: Fisher's exact test (2-sided): *p* < 0.001 Fisher's exact test (2-sided): *p* < 0.001

Next, we replicate the analysis for the different sources of ambiguity separately. As before, a subject is classified as reducing compound risk if $Π_{R,CR0} = Π_{R,CR25} = 0$. For ambiguity, a subject is classified as *E*-ambiguity neutral if $Π_{R,E} = 0$, and as model ambiguity neutral if $Π_{R,MA0} = Π_{R,MA25} = 0$. Table A.9 reports the contingency tables between compound risk reduction and Ellsberg and model ambiguity neutrality, respectively. In Part I, we observe that the proportion of *E*-ambiguity non-neutrality among subjects who do not reduce compound risk is 81% (=17/21) for actuaries and 74% (=60/81) for students. These proportions suggest that non-reduction of *CR* is sufficient for *E*-ambiguity non-neutrality, irrespective of the subjects' sophistication level. Turning to necessity, we find that 81% (=60/74) of *E*-ambiguity non-neutral students are also not reducing *CR*. However, this proportion is only 53% (=17/32) for actuaries, which is significantly lower than for students (two-sample test of proportions, *p*<0.001). This result replicates our main findings for *E*-ambiguity neutrality: although compound risk non-reduction can be both necessary and sufficient for *E*-ambiguity non-neutrality for less sophisticated subjects, the implication for necessity is significantly weaker for more sophisticated
subjects.

<caption>Table A.9: CONTINGENCY TABLES</caption>

## Part I: RELATION BETWEEN COMPOUND RISK AND E-AMBIGUITY ATTITUDES

<table><thead><tr><th rowspan="2">Reduction of <i>CR</i></th><th colspan="3">Actuaries</th><th colspan="3">Students</th></tr><tr><th colspan="3">E-ambiguity neutral</th><th colspan="3">E-ambiguity neutral</th></tr><tr><th></th><th>No</th><th>Yes</th><th>Total</th><th>No</th><th>Yes</th><th>Total</th></tr></thead><tbody><tr><th>No</th><td>17</td><td>4</td><td><strong>21</strong></td><td>60</td><td>21</td><td><strong>81</strong></td></tr><tr><th>Yes</th><td>15</td><td>38</td><td><strong>53</strong></td><td>14</td><td>24</td><td><strong>38</strong></td></tr><tr><th>Total</th><td><strong>32</strong></td><td><strong>42</strong></td><td><strong>74</strong></td><td><strong>74</strong></td><td><strong>45</strong></td><td><strong>119</strong></td></tr></tbody></table>

Independence test: Fisher's exact test (2-sided): *p* < 0.001 Fisher's exact test (2-sided): *p* < 0.001

## Part II: RELATION BETWEEN COMPOUND RISK AND MODEL AMBIGUITY ATTITUDES

<table><thead><tr><th rowspan="2">Reduction of <i>CR</i></th><th colspan="3">Actuaries</th><th colspan="3">Students</th></tr><tr><th colspan="3">MA neutral</th><th colspan="3">MA neutral</th></tr><tr><th></th><th>No</th><th>Yes</th><th>Total</th><th>No</th><th>Yes</th><th>Total</th></tr></thead><tbody><tr><th>No</th><td>19</td><td>2</td><td><strong>21</strong></td><td>70</td><td>12</td><td><strong>82</strong></td></tr><tr><th>Yes</th><td>12</td><td>42</td><td><strong>54</strong></td><td>17</td><td>21</td><td><strong>38</strong></td></tr><tr><th>Total</th><td><strong>31</strong></td><td><strong>44</strong></td><td><strong>75</strong></td><td><strong>87</strong></td><td><strong>33</strong></td><td><strong>120</strong></td></tr></tbody></table>

Independence test: Fisher's exact test (2-sided): *p* < 0.001 Fisher's exact test (2-sided): *p* < 0.001

Part II of Table A.9 supports the same idea: the proportion of *MA* non-neutrality among subjects who do not reduce compound risk is 90% (=19/21) for actuaries and 85% (=70/82) for students, suggesting that non-reduction of *CR* is sufficient for *MA* non-neutrality, irrespective of the sophistication level. For what concerns necessity, 80% (=70/87) of *MA* non-neutral students are also not reducing *CR*, but this proportion is only 61% (=19/31) for actuaries, which is still significantly lower than for students (two-sample test of proportions, *p*=0.033).

## A.5 Differences between Subject Pools

The two subject pools differ starkly in terms of age, income and level of education. To illustrate these differences, Figures A.3-A.5 show histogram plots for distributions of age, income, and highest level of education completed in the two samples.
<figure><img src="image_16.png" /><figcaption>Figure A.3: Distribution of Age. Median Age: 38 for actuaries, 20 for students; Average Age: 39.68 for actuaries and 20.49 for students.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure A.4: Distribution of Income. Mode Income: $35.000-$50.000 for actuaries; <$5000 for students</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure A.5: Distribution of Highest Level of Education Completed</figcaption></figure>

## A.6 Robustness analysis

Tables A.10 and A.11 reproduce Tables 3 and 4, controlling for the effects of age, gender, level of income and level of education. We use a dummy variable for income, which takes a value 1 for the subjects who reported having an annual income higher than the median level within their sample and 0 for those who reported an annual income lower than that. Similarly, we use a dummy variable for level of education, which takes a value 1 for subjects who reported having completed at least a Bachelor's degree at the time of experiment and 0 otherwise. The dummy for gender takes a value 1 for female and 0 for male subjects. Overall, the addition of the demographic controls does not alter the results of our regressions. In Table A.10, the effect of model ambiguity is significant for both actuaries and students, and not different across the two while the effect of complexity is stronger and only significant among students. In Table A.11, the role of unknown probabilities in explaining Ellsberg-ambiguity attitudes, while being significant in both groups, is stronger among actuaries, whereas the role of complexity is significant and stronger for students ($p$=0.051). Regarding the control variables, we only observe differences among actuaries, for whom gender and education have a significant effect, although the latter is driven by only three actuaries (out of 74) who do not possess a degree higher than a Bachelor's one.
<table><caption>Table A.10: RANDOM EFFECTS REGRESSIONS OF RELATIVE PREMIA</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Pooled data</th></tr></thead><tbody><tr><td>Complexity</td><td>0.030<br>(0.015)</td><td>0.122***<br>(0.023)</td><td>0.122***<br>(0.023)</td></tr><tr><td>Model ambiguity</td><td>0.128***<br>(0.033)</td><td>0.120***<br>(0.023)</td><td>0.120***<br>(0.023)</td></tr><tr><td>Complexity × Model ambiguity</td><td>-0.047<br>(0.031)</td><td>-0.075**<br>(0.026)</td><td>-0.075**<br>(0.026)</td></tr><tr><td>Sophisticated</td><td></td><td></td><td>-0.009<br>(0.033)</td></tr><tr><td>Sophisticated × Complexity</td><td></td><td></td><td>-0.092***<br>(0.027)</td></tr><tr><td>Sophisticated × Model ambiguity</td><td></td><td></td><td>0.008<br>(0.040)</td></tr><tr><td>Sophisticated × Complexity × Model ambiguity</td><td></td><td></td><td>0.027<br>(0.040)</td></tr><tr><td>Age</td><td>-0.001<br>(0.001)</td><td>0.004<br>(0.012)</td><td>-0.001<br>(0.001)</td></tr><tr><td>Gender</td><td>0.027<br>(0.029)</td><td>0.009<br>(0.033)</td><td>0.015<br>(0.023)</td></tr><tr><td>Income</td><td>-0.003<br>(0.027)</td><td>-0.015<br>(0.041)</td><td>-0.020<br>(0.023)</td></tr><tr><td>Education</td><td>0.002<br>(0.052)</td><td>-0.015<br>(0.041)</td><td>0.000<br>(0.029)</td></tr><tr><td>Constant</td><td>-0.023<br>(0.090)</td><td>-0.088<br>(0.238)</td><td>-0.005<br>(0.048)</td></tr><tr><td>Observations</td><td>299</td><td>471</td><td>770</td></tr></tbody></table>

Notes: Cluster-robust standard errors in parentheses. *** significant at 0.001, ** significant at 0.01, * significant at 0.05.
<table><caption>Table A.11: OLS REGRESSIONS OF ELLSBERG AMBIGUITY PREMIUM</caption><thead><tr><th></th><th>Actuaries</th><th>Students</th><th>Pooled data</th></tr></thead><tbody><tr><td>COMPX</td><td>-0.178<br>(0.180)</td><td>0.275*<br>(0.124)</td><td>0.264*<br>(0.123)</td></tr><tr><td>UNKNOWN</td><td>0.714***<br>(0.096)</td><td>0.421***<br>(0.108)</td><td>0.424***<br>(0.108)</td></tr><tr><td>RED</td><td>0.106<br>(0.075)</td><td>0.035<br>(0.050)</td><td>0.038<br>(0.050)</td></tr><tr><td>Sophisticated</td><td></td><td></td><td>0.009<br>(0.063)</td></tr><tr><td>Sophisticated × COMPX</td><td></td><td></td><td>-0.404<br>(0.206)</td></tr><tr><td>Sophisticated × UNKNOWN</td><td></td><td></td><td>0.301*<br>(0.145)</td></tr><tr><td>Sophisticated × RED</td><td></td><td></td><td>0.084<br>(0.088)</td></tr><tr><td>Age</td><td>-0.002<br>(0.002)</td><td>0.001<br>(0.015)</td><td>-0.002<br>(0.002)</td></tr><tr><td>Gender</td><td>0.103*<br>(0.047)</td><td>-0.008<br>(0.047)</td><td>0.036<br>(0.034)</td></tr><tr><td>Income</td><td>0.051<br>(0.048)</td><td>-0.028<br>(0.042)</td><td>-0.001<br>(0.032)</td></tr><tr><td>Education</td><td>0.179**<br>(0.065)</td><td>0.027<br>(0.050)</td><td>0.040<br>(0.040)</td></tr><tr><td>Constant</td><td>-0.178<br>(0.099)</td><td>0.081<br>(0.296)</td><td>0.066<br>(0.056)</td></tr><tr><td>Observations</td><td>74</td><td>114</td><td>188</td></tr></tbody></table>

Notes: Robust standard errors in parentheses, *** significant at 0.001, ** significant at 0.01, * significant at 0.05.

## A.7 Within sample heterogeneity

In this paper, we characterize sophistication by the background of our subject pools. As we previously argued, risk professionals can be considered as more sophisticated than students because of the differences observed in their curriculum, level of education, and work experience. Nevertheless, it may be argued that the comparison between actuaries and students may not sufficiently well isolate the role of sophistication due to the potential heterogeneity within each group. Indeed, it may, for example, be that some students, e.g., those in Master's and Ph.D. programs, possess a relatively higher level of sophistication than others, and may even considered as much sophisticated as some risk professionals. Similarly, younger actuaries, who have fewer years of work experience, may differ from older, more experienced ones. In this appendix, we test for the effect of heterogeneity within the two groups. Overall, our data confirm the presence of some heterogeneity within students but not within actuaries. Specifically, students in Master's or Ph.D. programs are less affected by complexity than their bachelor counterparts.
However, the strong relationship between compound risk and ambiguity attitudes persists for both sub-groups of students. In contrast, age and work experience have no impact on actuaries' attitudes towards complexity and ambiguity.

### A.7.1 Heterogeneity within the pool of actuaries

To test the heterogeneity among actuaries, we investigate whether there exist differences between actuaries who are younger and older than the median age in the sample (38). Note that there is a large overlap between the variables “age” and “work experience” in the industry among actuaries: The older risk professionals have on average 12.7 years more work experience than the younger ones ($p<0.001$). Tables A.12-A.14 reproduce the analyses we provide in the main body of the paper (Tables 2, 3, and 4) for younger and older actuaries separately. To take into account the effect of reduced sample sizes on significance levels, we indicate, in these tables, also the marginally significant coefficients at 10%. We observe that the correlations between compound risk and ambiguity premium are somewhat higher for older risk professionals although none of the correlations are significant, except the one between $\Pi_{R,CR0}$ and $\Pi_{R,MA0}$ ($p=0.064$). The relative premia does not differ significantly among older and younger risk professionals, and the effects of complexity and model ambiguity are comparable in the two sub-groups.

<table><caption>Table A.12: PEARSON CORRELATIONS WITHIN ACTUARIES</caption><thead><tr><th colspan="3">Older risk professionals (N = 41)</th><th colspan="4">Younger risk professionals (N = 43)</th></tr><tr><th></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th><th></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th></tr></thead><tbody><tr><th>Π<sub>R,CR0</sub></th><td>0.317<sup>ms</sup></td><td>0.092</td><td>0.022</td><th>Π<sub>R,CR0</sub></th><td>-0.0004</td><td>-0.194</td><td>-0.185</td></tr><tr><th>Π<sub>R,CR25</sub></th><td>0.142</td><td>0.105</td><td>0.034</td><th>Π<sub>R,CR25</sub></th><td>0.141</td><td>0.093</td><td>0.203</td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05, *ms* marginally significant at 0.10.

<table><caption>Table A.13: RANDOM EFFECTS REGRESSIONS OF RELATIVE PREMIA, ACTUARIES</caption><thead><tr><th></th><th>Older risk<br>professionals</th><th>Younger risk<br>professionals</th><th>Effect of age<br>(pooled data)</th></tr></thead><tbody><tr><th>Complexity</th><td>0.039<sup>ms</sup><br>(0.021)</td><td>0.022<br>(0.023)</td><td>0.017<br>(0.031)</td></tr><tr><th>Model ambiguity</th><td>0.108*<br>(0.046)</td><td>0.146**<br>(0.047)</td><td>-0.038<br>(0.066)</td></tr><tr><th>Complexity × Model ambiguity</th><td>-0.017<br>(0.033)</td><td>-0.070<br>(0.051)</td><td>0.054<br>(0.060)</td></tr><tr><th>Constant</th><td>-0.008<br>(0.021)</td><td>-0.006<br>(0.016)</td><td>-0.004<br>(0.026)</td></tr><tr><th>Observations</th><td>140</td><td>159</td><td>299</td></tr></tbody></table>

Notes: Cluster-robust standard errors in parentheses. *** significant at 0.001, ** significant at 0.01,

* significant at 0.05, *ms* marginally significant at 0.10.
<table><caption>Table A.14: OLS REGRESSIONS OF ELLSBERG AMBIGUITY PREMIUM, ACTUARIES</caption><thead><tr><th></th><th>Older risk<br>professionals</th><th>Younger risk<br>professionals</th><th>Difference between<br>groups (pooled data)</th></tr></thead><tbody><tr><td>COMPX</td><td>-0.439<sup>ms</sup><br>(0.233)</td><td>0.022<br>(0.249)</td><td>-0.461<br>(0.341)</td></tr><tr><td>UNKNOWN</td><td>0.822<sup>***</sup><br>(0.099)</td><td>0.640<sup>***</sup><br>(0.152)</td><td>0.183<br>(0.182)</td></tr><tr><td>RED</td><td>0.197<br>(0.122)</td><td>0.122<br>(0.101)</td><td>0.075<br>(0.159)</td></tr><tr><td>Constant</td><td>0.060<sup>*</sup><br>(0.028)</td><td>0.104<sup>*</sup><br>(0.047)</td><td>-0.044<br>(0.054)</td></tr><tr><td>Observations</td><td>35</td><td>39</td><td>74</td></tr></tbody></table>

Notes: Robust standard errors in parentheses, *** significant at 0.001, ** significant at 0.01, * significant at 0.05, ms marginally significant at 0.10.

## A.7.2 Heterogeneity within the pool of students

To study the heterogeneity among our student subjects and test for the effect of within-sample level of sophistication, we split this sample into two groups by separating the students who had and who had not completed their Bachelor's degree at the time of the experiment. Tables A.15-A.17 reproduce the analyses provided in Tables 2, 3, and 4 following this distinction. To take into account the effect of reduced sample sizes on significance levels, we also indicate the marginally significant values at 10%.

In Table A.15, we observe similar correlation coefficients between compound risk and ambiguity premia in the two groups (although the levels of significance differ, which is possibly due to the restrictions on the sample sizes). In Table A.16, we observe that both group of students are affected by the level of complexity but Master's and PhD students are affected significantly less by the complexity than Bachelor's students. Table A.17 also shows that complexity attitude plays a role for explaining Ellsberg ambiguity mainly for Bachelor's students although the role of attitude toward unknown probabilities is comparable in the two groups.

<table><caption>Table A.15: PEARSON CORRELATIONS WITHIN STUDENTS</caption><thead><tr><th colspan="3">Master's and PhD students (N = 45)</th><th colspan="4">Bachelor's students (N = 80)</th></tr><tr><th></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th><th></th><th>Π<sub>R,MA0</sub></th><th>Π<sub>R,MA25</sub></th><th>Π<sub>R,E</sub></th></tr></thead><tbody><tr><td>Π<sub>R,CR0</sub></td><td>0.271<sup>ms</sup></td><td>0.315<sup>*</sup></td><td>0.019</td><td>Π<sub>R,CR0</sub></td><td>0.332<sup>**</sup></td><td>0.358<sup>**</sup></td><td>0.094</td></tr><tr><td>Π<sub>R,CR25</sub></td><td>0.282<sup>ms</sup></td><td>0.625<sup>***</sup></td><td>0.311<sup>*</sup></td><td>Π<sub>R,CR25</sub></td><td>0.474<sup>***</sup></td><td>0.667<sup>***</sup></td><td>0.559<sup>***</sup></td></tr></tbody></table>

Notes: *** significant at 0.001, ** significant at 0.01, * significant at 0.05, ms marginally significant at 0.10.
<table><caption>Table A.16: RANDOM EFFECTS REGRESSIONS OF RELATIVE PREMIA, STUDENTS</caption><thead><tr><th></th><th>Master's and<br>PhD students</th><th>Bachelor's<br>students</th><th>Effect of degree<br>(pooled data)</th></tr></thead><tbody><tr><td>Complexity</td><td>0.070*<br>(0.028)</td><td>0.156***<br>(0.032)</td><td>-0.086*<br>(0.042)</td></tr><tr><td>Model ambiguity</td><td>0.124***<br>(0.036)</td><td>0.117***<br>(0.031)</td><td>0.007<br>(0.047)</td></tr><tr><td>Complexity × Model ambiguity</td><td>-0.042<br>(0.029)</td><td>-0.096*<br>(0.038)</td><td>0.054<br>(0.048)</td></tr><tr><td>Constant</td><td>0.023<br>(0.025)</td><td>0.002<br>(0.023)</td><td>0.021<br>(0.034)</td></tr><tr><td>Observations</td><td>179</td><td>292</td><td>471</td></tr></tbody></table>

*Notes*: Cluster-robust standard errors in parentheses. *** significant at 0.001, ** significant at 0.01, * significant at 0.05.

<table><caption>Table A.17: OLS REGRESSIONS OF ELLSBERG AMBIGUITY PREMIUM, STUDENTS</caption><thead><tr><th></th><th>Master's and<br>PhD students</th><th>Bachelor's<br>students</th><th>Difference between<br>groups (pooled data)</th></tr></thead><tbody><tr><td>COMPX</td><td>0.178<br>(0.184)</td><td>0.304*<br>(0.152)</td><td>-0.127<br>(0.238)</td></tr><tr><td>UNKNOWN</td><td>0.557**<br>(0.187)</td><td>0.342*<br>(0.131)</td><td>0.215<br>(0.227)</td></tr><tr><td>RED</td><td>-0.027<br>(0.094)</td><td>0.076<br>(0.060)</td><td>-0.103<br>(0.110)</td></tr><tr><td>Constant</td><td>0.114*<br>(0.049)</td><td>0.067**<br>(0.025)</td><td>0.047<br>(0.055)</td></tr><tr><td>Observations</td><td>45</td><td>69</td><td>114</td></tr></tbody></table>

*Notes*: Robust standard errors in parentheses, *** significant at 0.001, ** significant at 0.01, * significant at 0.05.
# B Experimental details

## B.1 Samples

The main experiment took place in Berlin (Germany) during the 31st International Congress of Actuaries in 2018. A specific conference room with 20 computers was made available for five days for the purpose of the study (see Figure B.1). The experiment was organized in 12 sessions, each of which lasted approximately 45 minutes, including instructions and payment.

<figure><img src="image_19.png" /><figcaption>Figure B.1: Left: The temporary laboratory room during ICA 2018. Right: example of a typical session with risk professionals (actuaries).</figcaption></figure>

The control experiment was conducted at the Bocconi Experimental Laboratory for Social Sciences (BELSS) in Bocconi University, Italy. The experiment was organized in five sessions. Each session lasted approximately one hour, including instructions and payment.

## B.2 Procedure

Both experiments were run on computers in English. Subjects were recruited on a voluntary basis and could sign up in advance for a particular time slot. Similar recruitment procedures were applied in both experiments. The experiment at ICA was advertised on the conference website and through notifications on the conference app. The participants could register online for an available time slot directly. For the experiment at the university, an internal recruitment system was used and participants could directly register online. Subjects gave their consent prior to the experiment by signing an informed consent document. The experiments were anonymized. In both cases, each subject was authorized to participate only once. The experiments were organized into different sessions taking place over several days. Each session started with the experimental instructions, examples of the stimuli, and comprehension questions. Subjects could not communicate with each other during the experiments.

**Stimuli** The implementation of *CR* and *MA* was as follows. Subjects were presented with a pile of decks and told that one deck would be picked randomly from that pile. In *CR0* (*CR*25), exactly 50% of the decks in the pile contained 0% red–100% black (25% red–75% black) cards and the remaining 50% of the decks contained 0% black–100% red (25% black–75% red) cards.
The situations under MA were similar, except that the proportions of the decks with different compositions in the pile were unknown. To implement *R* and *E*, the subjects were presented with a single deck of cards. In particular, the deck contained an equal proportion of red and black cards in *R*, and an unknown proportion of red and black cards in *E*. All the decks and piles were constructed before the experiment by one of the authors, who was not present in the room during the experimental sessions. The subjects were informed about this to avoid the effects of comparative ignorance (Fox and Tversky, 1995), i.e. an extra aversion to ambiguity induced by a comparison with a more knowledgeable someone (in this case, the experimenter).

We used a choice-list design to elicit the certainty equivalents of the different bets. Specifically, subjects were asked to make 12 binary choices between the bet and a sure monetary amount ranging from €0 to €200 (see Figure B.2). The order of the bets was randomized,

<figure><img src="image_20.png" /><figcaption>Figure B.2: Example of a typical choice list faced by subjects in the experiment</figcaption></figure>

except for *E*, which was always presented at the end, as it was the only situation in which the number of cards in the deck was explicitly mentioned. After completing the choice lists, the subjects answered a short survey with demographic questions.

**Incentives** The two experiments proposed different stakes (either €200 or €0 for actuaries, and either €20 or €0 for students) and different prior random incentive mechanisms (between-subject for actuaries,<fnref n="24" /> and within-subject for students). Overall, this enables us to equate the expected experimental costs in the two experiments while at the same time adjusting for the income difference between the two samples. Indeed, the *prior* incentive systems that we used (Johnson et al., 2021), which are aimed to enhance isolation and hence incentive compatibility

<footnote n="24">²⁴To determine the subset of actuary subjects who played for real, subjects were asked to draw one sealed envelope from a box of envelopes, among which one out of 10 contained an image of a happy face allowing them to play for real. The draws were made before the experiment started, and the subjects kept their envelopes sealed until the end of the experiment.</footnote>
(Azrieli et al., 2018), have the advantage of making our income adjustment effective by inducing the actuaries to make choices conditional on being selected for payment and thus to evaluate the prizes based on their face value.<fnref n="25" /> In practice, the choice question implemented for determining the payment was selected randomly prior to the choices and resolutions of uncertainty. Specifically, in the experiment with actuaries, each subject randomly drew, at the beginning of each experimental session, a sealed envelope that contained one of the uncertain situations printed on paper and another sealed envelope that contained one of the choice questions from the corresponding choice lists. The envelopes were kept by the subjects until the end of the experiment and opened only if the subject was selected to play his/her choice for real. A similar prior incentive system was employed in the experiment with students. In this case, the same pre-selected choice situation was implemented for determining the payment of all the subjects in the same session. In practice, at the beginning of the experiment, one volunteer in each session randomly drew an envelope containing one of the uncertain situations and another envelope containing one of the choice questions from the choice lists. The two sealed envelopes were attached to a board visible to all subjects. The contents of the envelopes were revealed only at the end of the experiment and every subject in the same session was paid based on his/her recorded decision in the choice situation corresponding to the contents of the envelopes. In both experiments, the draws from the piles and/or from the decks were made (physically) in front of the subjects according to the uncertain situation contained in the envelope. This prior incentive system ensured that subjects in both experiments made their choices knowing that every choice question could be implemented with equal chance. At the end of the experiments, actuary subjects were offered goodies and drinks for their participation, and student subjects were offered a €5 participation fee.

<footnote n="25">²⁵Incentive adjustments for groups of subjects with different income levels have been used in previous studies (e.g. Sutter et al., 2013; Vieider et al., 2015).</footnote>