# Prospect theory in multiple price list experiments: further insights on behaviour in the loss domain

Géraldine Bocquého, Julien Jacob, Marielle Brunette

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Géraldine Bocquého, Julien Jacob, Marielle Brunette. Prospect theory in multiple price list experiments: further insights on behaviour in the loss domain. Theory and Decision, 2023, 94 (4), pp.593-636. 10.1007/s11238-022-09902-y. hal-03768070

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# Prospect theory in multiple price list experiments: further insights on behaviour in the loss domain

Géraldine Bocquého,* Julien Jacob† and Marielle Brunette‡

April 30, 2022

**Abstract**

In the theoretical description of prospect theory, distinct sets of parameters can control the curvature of the value function and the shape of the probability weighting function. There is one for the gain domain and one for the loss domain. However, in most estimations, behaviour over losses is assumed to perfectly reflect behaviour over gains, through a unique set of parameters. We examine the consequences of relaxing this simplifying assumption in the context of Tanaka et al.'s (American Economic Review 100(1):557-571, 2010) risk-elicitation procedure based on multiple price lists. We show that subjects' behaviour for gains is mostly reflected for losses at the aggregate and individual levels, and is consistent with the distinctive prospect theory fourfold pattern. Reflection is only partial as the mean curvature of the value function is slightly less convex for losses than it is concave for gains. These results are robust to a high-stake context. However, we demonstrate that assuming reflection can have huge consequences on loss-aversion measures. Incidentally, we also highlight the existence of a strong, negative and persistent pure loss-frame effect on elicited loss aversion. We recommend that future practitioners and modellers are particularly cautious about the loss-aversion values they obtain or use because these are especially sensitive to parametric assumptions and framing.

**Keywords:** risk preferences, Tanaka-Camerer-Nguyen method, behavioural economics, reflection effect, reflected behaviour, interval elicitation

**JEL Codes:** C91; D81

**Acknowledgments:** The authors thank Kene Boun My for help with the implementation of the experiment and Marie-Claire Villeval for useful advice for a previous version of the article.

**Declarations:**

**Funding:** This research was funded by the Contrat de plan État-région (CPER) Lorraine Ariane under the project RIM (Risques Industriels et Multiples). The UMR BETA is supported by a

<footnote>*Corresponding author: Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRAE, BETA, 14 rue Girardet, 54000, Nancy, France. E-mail: geraldine.bocqueho@inrae.fr, ORCID ID: 0000-0002-6256-5109.</footnote>

<footnote>†Université de Strasbourg, Université de Lorraine, CNRS, BETA, 67000, Strasbourg, France. E-mail: julienjacob@unistra.fr, ORCID ID: 0000-0002-2258-4591.</footnote>

<footnote>‡Université de Lorraine, Université de Strasbourg, AgroParisTech, CNRS, INRAE, BETA, 54000, Nancy, France. E-mail: marielle.brunette@inrae.fr, ORCID ID: 0000-0001-8192-4819. Associated researcher to the Climate Economics Chair (CEC), Paris, France.</footnote>
grant from the French National Research Agency (ANR) as part of the "Investissements d'Avenir"
programme, Lab of Excellence ARBRE (Grant No ANR-11-LABX-0002-01).

**Conflicts of interest/Competing interests:** The authors have no relevant financial or non-financial interests to disclose.

**Availability of data and material:** The experimental instructions used in this study are included in this article as supplementary material. The datasets generated and analysed during the current study are not publicly available due the fact that they constitute an excerpt of research in progress but are available from the corresponding author on reasonable request.

**Code availability:** The code used to analyse the datasets is available upon request.

**Author contributions:** All the authors contributed to the design of the experiment and preparation of the experimental material. Data collection was performed by Julien Jacob and Marielle Brunette. Data analysis was carried out by Géraldine Bocquého. The first draft of the manuscript was written by Géraldine Bocquého and all the authors commented on later versions of the manuscript. All the authors read and approved the final manuscript.
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# 1 Introduction

Kahneman and Tversky (1979) started developing prospect theory (PT) as an alternative to standard expected utility (EU) to describe individual choices in a stochastic environment, while taking important cognitive biases into account. Since then, the cumulative version of PT (Tversky and Kahneman (1992)) has proved to be one of the most promising decision theories (Harrison and Rutström, 2009; Barberis, 2013) and has progressively been disseminated into empirical applications, paving the way for a positive approach to behaviour under risk. Examples include for instance Babcock (2015) in the field of agricultural economics, Barberis et al. (2001) and Grinblatt and Han (2005) in finance, and Mercer (2005) in political science. Applying PT requires manipulating several parameters that code for several aspects of individual risk preferences. Some parameters control the curvature of the value function while others control the shape of the probability weighting function, and one captures loss aversion. Measuring these parameters in different contexts has been a major preoccupation, both to validate the relevancy of the theoretical model in describing real behaviour and to provide values for empirical applications. Due to the multidimensional nature of PT, experiments are the preferred approach, and values are now available for a variety of populations. Examples include Nguyen and Leung (2009) for Vietnamese fishermen, Booij et al. (2010) for Dutch households, Liu (2013) for Chinese cotton farmers, Bocquého et al. (2014) for French farmers, Liebenhem and Waibel (2014) for African cattle farmers, Bocquého et al. (2018) for refugees from the Middle East, L’Haridon and Vieider (2019) for students from 30 countries, Zhao and Yue (2020) for U.S. farmers. The large majority of these studies rely on lottery choices organized in multiple price lists (MPLs), using Tanaka et al.’s (2010) design (henceforth TCN).<fnref n="1" /> The success of MPLs can be explained by their simplicity of use, transparency in terms of incentives, and the possibility of directly inferring a risk attitude interval from the pattern of responses for each respondent, without requiring any estimation procedure (Harrison and Rutström, 2008).

However, while parametric methods for preference elicitation are easy to interpret, they are susceptible to a contamination effect.<fnref n="2" /> A misspecification of either the value function or probability

<footnote n="1">¹Booij et al. (2010) and L’Haridon and Vieider (2019) are exceptions in this respect as they rely on certainty and/or probability equivalents, using a matching and a choice procedure respectively. Choices made on random lottery pairs are another usual experimental design but they are more common for laboratory measures (e.g., Harrison and Rutström, 2009).</footnote>

<footnote n="2">²Non-parametric methods include Wakker and Deneffe's (1996) trade-off method, taken further by Abdellaoui (2000), Abdellaoui et al. (2007) and Abdellaoui et al. (2016), and based on measurements of indifferences. They are typically more difficult to administer and may suffer from error propagation because of their chained nature (L'Haridon and Vieider, 2019). They are also generally less efficient because more questions are needed (Abdellaoui et al., 2008). Last, they were</footnote>
weighting function will indeed bias the other (Booij et al., 2010). One common variation in these
functional forms is the use of either similar or distinct parameters for the gain and loss domains. In
Tversky and Kahneman's (1992) cumulative PT, two different parameters control the curvature of
the value function in each of the two domains but in many empirical applications these are merged.
A similar simplifying assumption is often made about the parameters of the probability weighting
function, in line with Kahneman and Tversky's (1979) original PT. However, there is mixed evidence
as to whether these two assumptions are empirically grounded and to what extent they affect the
preferences under elicitation.

In this context, our objective is to formally investigate the impact of relaxing such simplifying
assumptions in the context of the TCN MPLs. Preference elicitation is sensitive to the
elicitation procedure (Csermely and Rabas, 2016; Bauermeister et al., 2018) which means it is
not straightforward to extend the insights obtained in the PT framework but from certainty- or
probability-equivalent designs (Tversky and Kahneman, 1992; Abdellaoui et al., 2008; Booij et al.,
2010; L'Haridon and Vieider, 2019) and random lottery pairs (Harrison and Rutström, 2009). We test
the robustness of our findings on the size of the lottery stakes because the fact that risk preferences
may change over outcome levels (Markowitz, 1952) is not currently integrated into PT (Bouchouicha
and Vieider, 2017).

To this end, we carry out a series of within-subject risk experiments on a sample of students. We
test the original TCN design, and two alternative frames, a reflected *loss* frame and a *high-stake* frame,
using monetary incentives. In a first step, the reflected frame allows us to elicit individual values
for the curvature of the value and probability weighting functions specifically on loss outcomes.
In a second step, we assess the consequences for the loss aversion measures, in particular whether
loss aversion remains consistent between the *gain* and the *loss* frames, as predicted by the theory.
We develop our analysis at both the aggregate and individual levels. Bougherara et al. (2017) is
the only TCN MPL study we are aware of where distinct PT parameters are allowed in the gain
and loss domains. Unlike us, they use the original TCN design, which restricts the range of the
loss outcomes. Furthermore, they rely on a specific sample of French farmers and provide no
individual-level analysis.

This paper is organised as follows. In Section 2, we outline the relevant literature. In Section 3, we

<footnote>made incentive compatible only recently (Johnson et al., 2021). See Bauermeister et al. (2018) for a laboratory comparison
of the TCN method with Wakker and Deneffe's (1996) under the PT framework.</footnote>
describe the sample and experimental protocol, and in Section 4, we present the parametric methods
of measurement. In Section 5, we report non-parametric results on behaviour under the baseline
TCN design, *loss* frame and *high-stake* frame, as well as measures PT parameters. In Section 6, we
fine-tune the measures of loss aversion by accounting for the asymmetry in the curvature of the value
function. We discuss the results at the end of Sections 5 and 6 in specific subsections. Section 7
features the conclusion.

## 2 Literature review

The idea that individuals behave differently regarding gambles expressed as losses or gains is not
new. For instance, Slovic and Lichtenstein (1968) find that individuals give much less importance to
probabilities of losses than to probabilities of gains. This idea is at the basis of the PT Kahneman
and Tversky (1979) propose as an alternative to EU. In EU, choices depend on final wealth which
integrates all outcomes in the same way but in PT value is assigned to gains and losses, defined
with respect to a variable reference point. Consequently, PT is able to account for the *reflection
effect*: subjects who are risk averse in the domain of gains become risk seeking in the domain of
losses, and *vice versa*. The reflection effect is a well-known empirical finding backed up by numerous
experimental studies (e.g., Kühberger et al., 1999; Bosch-Domènech and Silvestre, 2006; Laury and
Holt, 2008), although a few studies do not corroborate such a result (Hershey et al., 1982). In addition,
in EU, probabilities are objective whereas, in PT, they are distorted into decision weights. This
additional feature allows PT to accommodate a more general *fourfold pattern* of risk attitudes, which
depends on outcome sign and probability level (Tversky and Kahneman, 1992):

(i) risk aversion over medium to high-probability gains

(ii) risk seeking over medium to high-probability losses

(iii) risk seeking over low-probability gains

(iv) risk aversion over low-probability losses.

This pattern has been widely investigated in experimental economics (Cohen et al., 1987; Wehrung,
1989; Harbaugh et al., 2010).

In terms of structural parameters, one may associate the reflection effect with the shape of the
value function, and the fourfold pattern with the shape of the probability weighting function (Lau
et al., 2019). Although in PT distinct parameters can control the shape of these two functions in gains
and losses, many experimental studies make the *a priori* assumption that unique parameters operate
in both domains. This is mostly the case for those based on the TCN risk experiment (Nguyen and
Leung, 2009; Tanaka et al., 2010; Liu, 2013; Bocquého et al., 2014, 2018; Zhao and Yue, 2020),
but also with some studies using other experimental designs, like Harrison and Rutström (2008,
pp. 92-95), and Harrison and Rutström (2009) as far as the weighting parameters are concerned.
These constraints on parameter values take for granted the reflected behavioural patterns described
above, while there is little evidence on whether such constraints are innocuous (Lau et al., 2019). In
particular, it may directly affect the loss aversion measure (Booij et al., 2010).

Reasons for such assumptions beyond mere simplicity hardly appear. In the case of the value
function, Wakker (2010, pp. 267–271) suggests it avoids the analytical problems the widespread
power specification brings for analyzing loss aversion. First, he demonstrates that loss aversion
cannot be defined independently from the unit of money unless the powers for gains and losses
agree.<fnref n="3" /> Second, he explains that a power value function cannot accommodate all plausible empirical
observations whenever the powers for gains and for losses differ.<fnref n="4" /> Another possible reason is Tversky
and Kahneman’s (1992) reporting of coincidental evidence for perfect reflection: the median exponent
of the power value function is exactly the same for gains and losses. In the case of the probability
weighting function, it is noteworthy that Kahneman and Tversky’s (1979) original description of PT
does assume identical parameters for the gain and the loss domains. With the cumulative version,
Tversky and Kahneman (1992) find that the median values of the weighting function parameters in
the gain and loss domains are very close, and imply a slightly less pronounced curvature for losses.

The subsequent experimental studies assuming distinct PT functions in the gain and loss domains
often report close parameter values, even when these values are statistically different. When reflection
is partial, most of the time, behaviour for losses deviates less from linearity, and thus from expected
value maximization (Wakker, 2010). The bulk of evidence thus advocates a more concave value
function for gains than convex for losses (Wakker et al., 2007, pp. 224). This is what Booij et al.’s
(2010) summary table of PT empirical estimates illustrates. The 16 studies under review use a power
value function and show that losses are always evaluated equally or more linearly than gains. Only

<footnote n="3">³The implicit scaling convention for the value function is $v(y = 1) = 1$ and $v(y = -1) = -1$. When $\sigma^+$ and $\sigma^-$ are not the same, the power specification implies $\lambda$ is defined relative to the unit of money used for the $y$ outcomes (Wakker, 2010, pp. 267–268).</footnote>

<footnote n="4">⁴Let us assume the following value functions defined separately over the gain and loss domains: for $y > 0$, $v(y) = y^a$ and, for $y < 0$, $v(y) = -\lambda(-y)^b$, where $\lambda > 0$ is the coefficient of loss aversion and $a \neq b$. Wakker (2010) recalls it implies there is always a part of the gain domain where $v(y) > -v(-y)$. However, it is empirically plausible that $v(y) \le -v(-y)$ for all $y > 0$.</footnote>
the values reported for Fennema and van Assen (1998) and Abdellaoui et al. (2008) show a large
difference in the parameters of the value function. However, more detailed reading of the articles
reviewed gives a more nuanced picture.<fnref n="5" /> For instance, in Abdellaoui et al. (2007), several correlation
coefficients show there is less support for reflection at the individual level than at the aggregate level.
Harrison and Rutström (2009) report that, when they use a structural mixture model allowing for
EU and PT to coexist in the sample instead of a simple PT model, they find different power value
functions for gains and losses. In addition, curvature is much more pronounced for losses than for
gains.

A few other authors obtain more convexity for losses than concavity for gains, as highlighted
by Wakker et al. (2007, pp. 224). Booij and van de Kuilen (2009) report analogous evidence at
the sample level but not for the underlying population. A more original pattern, illustrated in Booij
et al.’s (2010) table by Abdellaoui et al. (2008) and Fehr-Duda et al. (2006), is a slightly concave
value function for losses. In the first case, the value function is also concave for gains while, in the
second case, it is linear. Abdellaoui et al.’s (2008) concave-concave pattern applies to both individual
and pooled median data. In this respect, we also note the work of Attema et al. (2013, 2016) who
find concavity in both domains in experiments related to health outcomes, suggesting differences in
PT valuations when involving monetary or health issues. Attema (2012) explains that health issues
imply concavity, at least because of the time dimension associated: time discounting can be seen as
concavity of life duration.

As regards the probability weighting parameters, similarity applies to all of the 19 weighting
functions reviewed by Booij et al. (2010). The predominant shape for the weighting function is the
standard ‘inverse S’ in both the gain and the loss domains. In the 2-parameter forms, though, the
elevation parameter is often higher for losses, while the degree of curvature is very similar between
gains and losses. This is confirmed at a large scale by Booij et al.’s (2010) own dataset, collected from
a representative panel of Dutch households. More recently, L’Haridon and Vieider (2019) used a large
sample of students from 30 countries and found that the two parameters describing the probability
weighting function are significantly different between gains and losses. However, they reaffirmed
that the two functions can be seen as economically very similar.

Last, Lau et al. (2019) revisit data from five experimental studies in order to examine specifically
the reflection effect and the fourfold pattern of risk attitudes. They account for three different

<footnote n="5">⁵See Wakker et al. (2007, pp. 224) for additional studies with unclear or balanced findings.</footnote>
criteria at the aggregate level (population means of relevant parameters) and individual level (within-individual correlation in parameters between gain and loss domains, and population share of individuals who display the reflection effect or fourfold pattern). They find that only two of the five datasets under scrutiny meet at least two of the three criteria.

# 3 Experimental design

## 3.1 Participants and incentives

The experiment was conducted at the Laboratory of Experimental Economics of Strasbourg (LEES) in October 2017 using computers (EconPlay platform). We recruited 191 students from different study courses at the University of Strasbourg, France. Each subject participated in one session only. Before running the experiment, we collected a few socio-demographic characteristics. Descriptive statistics for the corresponding variables are given in Table 1. The students were on average 22 years old and half were female. About 25% were engaged in master studies and half were studying economics, management, mathematics, or political sciences.

We ran 8 sessions of a lottery choice experiment based on TCN, with lottery payoffs in a virtual currency called ECU (Experimental Currency Unit). Each session lasted approximately 1 hour. Before starting, we informed subjects that two lottery choices would be randomly selected for payment at the end of the experiment. We also informed subjects that the currency exchange rate was ECU 200 for € 1. Subjects received € 5 as a show-up fee to compensate for travel expenses and € 17 as an initial endowment to ensure final earnings would always be positive. Final payment ranged from € 21.05 to € 24.98, with an average of € 22.09.

<table><caption>Table 1: Descriptive statistics of socio-demographics</caption><thead><tr><th></th><th>Mean</th><th>Std. Dev.</th></tr></thead><tbody><tr><td>age (in years)</td><td>21.77</td><td>3.44</td></tr><tr><td>dummy if female</td><td>0.50</td><td>0.50</td></tr><tr><td>dummy if education level is at least master</td><td>0.25</td><td>0.43</td></tr><tr><td>dummy if education background is Econ., Mgmt, Maths or Pol. Sc.</td><td>0.50</td><td>0.50</td></tr><tr><td>Nb. of obs.</td><td>191</td><td></td></tr></tbody></table>

Econ: Economics, Mgmt: Management, Pol. Sc.: Political Sciences
## 3.2 Treatments

We defined and used four treatments by manipulating two factors of two levels each, in a full factorial design: the main stake domain (G = *gain* and L = *loss*) and the stake level (LLo = *low* and Hi = *high*). Our baseline treatment, called GLo, is a replication of TCN experiment in which we substituted ECUs for thousands of Vietnamese dongs. In this treatment, subjects are confronted successively with three series of choices between two binary lotteries A and B, as displayed in Table 2. The first and second series consist each of fourteen choices between all-gain lotteries. The third series consists of seven choices only, between lotteries that mix positive with negative payoffs. The probabilities of winning are always 50% in this last series. An important distinction with usual MPLs like Holt and Laury's (2002) is the enforcement of monotonic switching by asking subjects about their switching point in each series, i.e., the row at which they start to prefer lottery B rather than lottery A. As a consequence, they cannot switch back and forth.

The GHi treatment is identical to the baseline GLo treatment except that all stakes are multiplied by 2 in all three series. In the LLo treatment, all payoffs from the baseline are multiplied by -1 and the two columns in which lotteries are displayed for subjects to choose one are reversed (see Appendix A). Last, the LHi treatment combines the *loss* frame with the *high-stake* frame. In each session, subjects successively faced all treatments. To counterbalance potential order effects, we varied the order of treatments across sessions using a balanced Latin square design. We then randomly assigned participants to sessions. Table 3 provides details about the order of treatments and number of subjects in each of the 8 sessions.

The two lottery choices used for the final payment are selected such as, in a first step, the *high* or *low* frame is randomly selected. In a second step, one choice from the corresponding *gain* treatment and one from the corresponding *loss* treatment is randomly drawn and played for real. We provide the original French version of the experimental instructions, an English version of this and a few illustrative screen shots in the Online Resource.
<table><caption>Table 2: Lottery options corresponding to the baseline GLo treatment</caption><thead><tr><th rowspan="2"></th><th colspan="4">Payoffs</th><th rowspan="2">(Expected value A-B)</th></tr><tr><th colspan="2">Option A</th><th colspan="2">Option B</th></tr></thead><tbody><tr><td colspan="6"><strong>Series 1</strong></td></tr><tr><td>Probabilities</td><td>0.30</td><td>0.70</td><td>0.10</td><td>0.90</td><td></td></tr><tr><td>row 1</td><td>40</td><td>10</td><td>68</td><td>5</td><td>(7.7)</td></tr><tr><td>row 2</td><td>40</td><td>10</td><td>75</td><td>5</td><td>(7.0)</td></tr><tr><td>row 3</td><td>40</td><td>10</td><td>83</td><td>5</td><td>(6.2)</td></tr><tr><td>row 4</td><td>40</td><td>10</td><td>93</td><td>5</td><td>(5.2)</td></tr><tr><td>row 5</td><td>40</td><td>10</td><td>106</td><td>5</td><td>(3.9)</td></tr><tr><td>row 6</td><td>40</td><td>10</td><td>125</td><td>5</td><td>(2.0)</td></tr><tr><td>row 7</td><td>40</td><td>10</td><td>150</td><td>5</td><td>(-0.5)</td></tr><tr><td>row 8</td><td>40</td><td>10</td><td>185</td><td>5</td><td>(-4.0)</td></tr><tr><td>row 9</td><td>40</td><td>10</td><td>220</td><td>5</td><td>(-7.5)</td></tr><tr><td>row 10</td><td>40</td><td>10</td><td>300</td><td>5</td><td>(-15.5)</td></tr><tr><td>row 11</td><td>40</td><td>10</td><td>400</td><td>5</td><td>(-25.5)</td></tr><tr><td>row 12</td><td>40</td><td>10</td><td>600</td><td>5</td><td>(-45.5)</td></tr><tr><td>row 13</td><td>40</td><td>10</td><td>1,000</td><td>5</td><td>(-85.5)</td></tr><tr><td>row 14</td><td>40</td><td>10</td><td>1,700</td><td>5</td><td>(-155.5)</td></tr><tr><td colspan="6"><strong>Series 2</strong></td></tr><tr><td>Probabilities</td><td>0.90</td><td>0.10</td><td>0.70</td><td>0.30</td><td></td></tr><tr><td>row 1</td><td>40</td><td>30</td><td>54</td><td>5</td><td>(-0.3)</td></tr><tr><td>row 2</td><td>40</td><td>30</td><td>56</td><td>5</td><td>(-1.7)</td></tr><tr><td>row 3</td><td>40</td><td>30</td><td>58</td><td>5</td><td>(-3.1)</td></tr><tr><td>row 4</td><td>40</td><td>30</td><td>60</td><td>5</td><td>(-4.5)</td></tr><tr><td>row 5</td><td>40</td><td>30</td><td>62</td><td>5</td><td>(-5.9)</td></tr><tr><td>row 6</td><td>40</td><td>30</td><td>65</td><td>5</td><td>(-8.0)</td></tr><tr><td>row 7</td><td>40</td><td>30</td><td>68</td><td>5</td><td>(-10.1)</td></tr><tr><td>row 8</td><td>40</td><td>30</td><td>72</td><td>5</td><td>(-12.9)</td></tr><tr><td>row 9</td><td>40</td><td>30</td><td>77</td><td>5</td><td>(-16.4)</td></tr><tr><td>row 10</td><td>40</td><td>30</td><td>83</td><td>5</td><td>(-20.6)</td></tr><tr><td>row 11</td><td>40</td><td>30</td><td>90</td><td>5</td><td>(-25.5)</td></tr><tr><td>row 12</td><td>40</td><td>30</td><td>100</td><td>5</td><td>(-32.5)</td></tr><tr><td>row 13</td><td>40</td><td>30</td><td>110</td><td>5</td><td>(-39.5)</td></tr><tr><td>row 14</td><td>40</td><td>30</td><td>130</td><td>5</td><td>(-53.5)</td></tr><tr><td colspan="6"><strong>Series 3</strong></td></tr><tr><td>Probabilities</td><td>0.50</td><td>0.50</td><td>0.50</td><td>0.50</td><td></td></tr><tr><td>row 1</td><td>25</td><td>-4</td><td>30</td><td>-21</td><td>(6.0)</td></tr><tr><td>row 2</td><td>4</td><td>-4</td><td>30</td><td>-21</td><td>(-4.5)</td></tr><tr><td>row 3</td><td>1</td><td>-4</td><td>30</td><td>-21</td><td>(-6.0)</td></tr><tr><td>row 4</td><td>1</td><td>-4</td><td>30</td><td>-16</td><td>(-8.5)</td></tr><tr><td>row 5</td><td>1</td><td>-8</td><td>30</td><td>-16</td><td>(-10.5)</td></tr><tr><td>row 6</td><td>1</td><td>-8</td><td>30</td><td>-14</td><td>(-11.5)</td></tr><tr><td>row 7</td><td>1</td><td>-8</td><td>30</td><td>-11</td><td>(-13.0)</td></tr></tbody></table>

Design adapted from Tanaka et al. (2010). Lottery payoffs are in ECUs. Information on expected value is not displayed to respondents.
<table><caption>Table 3: Characteristics of the experimental sessions</caption><thead><tr><th>Order of treatments</th><th>Number of subjects</th></tr></thead><tbody><tr><td>1:GLo-GHi-LHi-LLo</td><td>25</td></tr><tr><td>2:GHi-LLo-GLo-GHi</td><td>25</td></tr><tr><td>3:LLo-LHi-GHi-GLo</td><td>22</td></tr><tr><td>4:LHi-GLo-LLo-GHi</td><td>27</td></tr><tr><td>5:GLo-GHi-LHi-LLo</td><td>21</td></tr><tr><td>6:GHi-LLo-GLo-LHi</td><td>22</td></tr><tr><td>7:LLo-LHi-GHi-GLo</td><td>24</td></tr><tr><td>8:LHi-GLo-LLo-GHi</td><td>25</td></tr><tr><td>Total:</td><td>191</td></tr></tbody></table>

# 4 Parametric measurement methods

## 4.1 Choice model

We measure individual risk preference parameters using the PT framework. Like most experimental studies, including Tversky and Kahneman's (1992), we assumed that subjects' reference point corresponds to the status quo, or, equivalently, their asset position before making the choices of interest. In the context of our experiment, subjects' assets include their personal assets, the show-up fee and the initial endowment. It means the reference point which distinguishes gains from losses is zero, and subjects perceive positive lottery payoffs as gains and negative ones as losses. Let us also assume that subjects value lottery payoffs $y$ through a two-part power value function of the form (Tversky and Kahneman, 1992):

$$v(y) = \begin{cases} y^{\sigma^{+}} & \text{if } y > 0 \\ 0 & \text{if } y = 0. \\ -\lambda(-y)^{\sigma^{-}} & \text{if } y < 0 \end{cases} \qquad (1)$$

In this specification, parameter $\sigma^+$ ($> 0$) determines the shape of the value function in the gain domain and acts as an *anti*-index of concavity. In the loss domain, parameter $\sigma^-$ ($> 0$) controls curvature as an index of concavity, while parameter $\lambda$ ($> 0$) modifies the slope. The power function belongs to the parametric family of constant relative risk aversion (CRRA). It means that preferences between two prospects are not affected if all payoffs are multiplied by the same positive number. Parameter $\lambda$ is the decision maker's coefficient of loss aversion. The value function is concave (resp. convex) in the
gain domain when $\sigma^+ < 1$ (resp. $\sigma^+ > 1$). In the loss domain, $\sigma^- < 1$ means the value function is convex. The decision maker is more (resp. less) sensitive to losses than to gains when $\lambda > 1$ (resp. $\lambda < 1$). The usual empirical finding is $\lambda > 1$ which means loss aversion, along with $\sigma^+ < 1$ and $\sigma^- < 1$, which characterises a concave value function for gains, but convex for losses.

We shall now define decision weights over cumulative probabilities. The prospect value of any binary lottery $(y_1, p; y_2)$ is:

$$PV(y_1, p; y_2) = \begin{cases} \omega^d(p) \cdot v^d(y_1) + [1 - \omega^d(p)] \cdot v^d(y_2) & \text{if } y_1 \ge y_2 \ge 0 \text{ or } y_1 \le y_2 \le 0 \\ \omega^d(p) \cdot v^d(y_1) + \omega^d(1-p) \cdot v^d(y_2) & \text{if } y_1 < 0 < y_2 \end{cases} \quad (2)$$

where the probability weighting function $\omega^d(\cdot)$ is continuous, strictly increasing from the unit interval into itself, and satisfies $\omega^d(0) = 0$ and $\omega^d(1) = 1$; and the superscript $d$ indicates the payoff domain and can take the values + for gains and – for losses. In PT, risk behaviour results from the interplay of the curvature of the value function, loss aversion and probability weighting. For the weighting function, in line with TCN, we adopt the one-parameter form of Prelec's (1998) specification:

$$\omega^d(p) = \exp[-(-\ln p)^{\gamma^d}] \quad (3)$$

where $d$ again indicates gains and losses; and $\gamma^d (> 0)$ controls the curvature of the probability weighting function. This parameter can be interpreted as an index of likelihood sensitivity: when $\gamma^d < 1$ the distinction between different levels of probability is blurred, whereas $\gamma^d > 1$ means the probability gap is exacerbated. Extreme behaviours are thus described by values close to nullity where subjects tend to perceive probabilities as all being equal (extreme likelihood insensitivity) and by very high values where subjects perceive probabilities as either 0 or 1 (extreme likelihood sensitivity). The usual empirical observation is $\gamma^d < 1$, giving the weighting function an ‘inverse S-shape’. In the case of a binary prospect, it characterizes an overweighting of the low-probability outcome and an underweighting of the high-probability outcome. If $\gamma^d > 1$, the function takes the less conventional ‘S-shape’.
## 4.2 Parameter elicitation

For each subject and each treatment, we calculate the three individual PT parameters using the TCN analytical interval approach. In their experiment, the choices are carefully designed so bounds for $\sigma^+$ and $\gamma^+$ can be jointly inferred by crossing responses to Series 1 and Series 2. In a second step, conditionally to the $\sigma^+$ value previously elicited, bounds for $\lambda$ are inferred from the switching point in Series 3. Parameter values are approximated by taking the midpoint of intervals. When there is no switch, the value at the boundary is used.

For the choices expressed in the *gain* frame (treatments GLo and GHi), we use the correspondence TCN gives between the combination of switching points in the three series and the parameter values. In the *loss* frame (reflected treatments LLo and LHi), the correspondence between the couple of switching points in Series 1 and Series 2 and the couple of $\sigma^-$ and $\gamma^-$ values is the same. However, we need to recalculate $\lambda$ values from Series 3 choices. We provide the calculation details in Appendix B.

We define the $\lambda$ value for each treatment (conditional to $\sigma^+$ or $\sigma^-$) according to the choices subjects make for the three series of the same treatment. We also define two alternative values. We calculate a $\lambda_{oppos}$ for each treatment from the choices in Series 3 of the same treatment, but conditional to $\sigma^d$ from the corresponding opposite *gain* or *loss* treatment. The opposite pairs are treatments GLo and LLo on the one hand, and GHi and LHi on the other hand. We also calculate a $\lambda_{gen}$ for each treatment from the choices in Series 3 of the same treatment, but conditional to both $\sigma^+$ and $\sigma^-$. In other words, the two value function parameters are applied to Series 3 payoffs depending whether they are a gain or a loss. More precisely, we calculate $\lambda_{gen}$ in treatments GLo and LLo using $\sigma^+$ from GLo and $\sigma^-$ from LLo, and $\lambda_{gen}$ in treatments GHi and LHi using $\sigma^+$ from GHi and $\sigma^-$ from LHi. It means that, for $\lambda_{gen}$, we relax the initial assumption that a single parameter controls the curvature of the value function in the gain and loss domains.

# 5 Results about frame effects

We shall start by presenting the pattern of behaviour each treatment reveals using the total number of risky or safe choices and the parameter values. We shall then investigate the effect of the *loss* and *high-stake* frames on these two measures.
## 5.1 Pattern of behaviour

Table 4 shows the proportion of left-hand side (LHS) choices by row in the four treatments.<fnref n="6" /> In the *gain* treatments GLo and GHi, the LHS lottery is the *safe* lottery, while the right-hand side (RHS) lottery is the *risky* one.<fnref n="7" /> The contrary is true for the reflected *loss* treatments LLo and LHi. In Series 1 of the *gain* treatments, those who still choose the LHS lottery (i.e., option A in Table 2) as of row 7 can be qualified as risk averse as the expected value of the RHS lottery (i.e., option B) starts to exceed the expected value of the LHS lottery. Conversely, those choosing the RHS lottery at earlier rows can be qualified as risk lovers. In Series 2 this occurs as soon as row 1, and in Series 3 by row 2.

In Series 1 of the baseline GLo treatment, 46% of the subjects choose the LHS lottery at row 7 while 40% choose the RHS lottery at row 6. It means that at least 46% of the subjects are risk averse and at least 40% are risk lovers. The behaviour of the remaining 14% is close to neutrality. The low proportion of risk-averse subjects compared to usual observations under classical EU can be explained by subjects being risk seeking over low-probability gains, typically the best outcomes of the RHS lotteries (see lotteries B of Series 1 in Table 2). This is consistent with item (iii) of the distinctive fourfold pattern of behaviour described in Section 2. In Series 2 of the baseline treatment, 79% of the subjects choose the LHS lottery (i.e., lottery A) at row 1, meaning at least 79% of the subjects are risk averse. This time, there is no best outcome with a low probability, and subjects are predominantly risk averse. This is consistent with item (i) of the fourfold pattern of behaviour. We should recall at this point that Series 3 features 50-50 lotteries mixing gains and losses. In the baseline treatment, 68% of the subjects choose the LHS lottery (i.e., lottery A) at row 2, meaning at least 68% of the subjects are risk averse. We cannot directly comment on this result with respect to the fourfold pattern as lotteries are mixed. Nevertheless, it should be noted that it is consistent with the usual empirical finding of loss aversion as the RHS lottery features high losses.

We shall now discuss the alternative treatments. Choosing the LHS lottery in the bold italic rows of Table 4 is an indicator of risk aversion in the *gain* treatment but conversely it is an indicator of risk seeking in the *loss* treatments. In Series 1 of the *loss* treatment LLo, 36% of the subjects choose the LHS lottery at row 7 while 52% choose the RHS lottery at row 6. It means that at least 36% of

<footnote n="6">⁶Appendix C gives the distribution of subjects’ switching points in the baseline GLo treatment. Similarly to other TCN risk experiments carried out in rural areas (e.g., Tanaka et al., 2010; Bocquého et al., 2014), we find that students massively choose extreme switching points in all three series. The *never* switch option in Series 1 is an exception, though.</footnote>

<footnote n="7">⁷As Bosch-Domènech and Silvestre (2013) put it "*the terms safe and risky, used by Holt and Laury (2002), must be understood in a loose sense and relative to each other: in a given pair, [the risky] lottery (...) gives a larger good payoff, but a lower bad payoff, than [the safe lottery]."</footnote>
<table><caption>Table 4: Proportion of left-hand side choices by row and treatment</caption><thead><tr><th></th><th>Treatment GLo</th><th>Treatment GHi</th><th>Treatment LLo</th><th>Treatment LHi</th></tr></thead><tbody><tr><td colspan="5"><strong>Series 1</strong></td></tr><tr><td>row 1</td><td>0.83</td><td>0.80</td><td>0.77</td><td>0.67</td></tr><tr><td>row 2</td><td>0.83</td><td>0.80</td><td>0.75</td><td>0.64</td></tr><tr><td>row 3</td><td>0.81</td><td>0.77</td><td>0.70</td><td>0.60</td></tr><tr><td>row 4</td><td>0.77</td><td>0.71</td><td>0.63</td><td>0.55</td></tr><tr><td>row 5</td><td>0.69</td><td>0.64</td><td>0.54</td><td>0.49</td></tr><tr><td>row 6</td><td>0.60</td><td>0.60</td><td>0.48</td><td>0.43</td></tr><tr><td>row 7</td><td>0.46</td><td>0.46</td><td>0.36</td><td>0.29</td></tr><tr><td>row 8</td><td>0.38</td><td>0.34</td><td>0.28</td><td>0.20</td></tr><tr><td>row 9</td><td>0.27</td><td>0.28</td><td>0.21</td><td>0.16</td></tr><tr><td>row 10</td><td>0.18</td><td>0.19</td><td>0.16</td><td>0.10</td></tr><tr><td>row 11</td><td>0.12</td><td>0.13</td><td>0.12</td><td>0.07</td></tr><tr><td>row 12</td><td>0.07</td><td>0.05</td><td>0.08</td><td>0.06</td></tr><tr><td>row 13</td><td>0.04</td><td>0.03</td><td>0.07</td><td>0.05</td></tr><tr><td>row 14</td><td>0.02</td><td>0.02</td><td>0.06</td><td>0.05</td></tr><tr><td colspan="5"><strong>Series 2</strong></td></tr><tr><td>row 1</td><td>0.79</td><td>0.75</td><td>0.81</td><td>0.75</td></tr><tr><td>row 2</td><td>0.77</td><td>0.73</td><td>0.79</td><td>0.74</td></tr><tr><td>row 3</td><td>0.76</td><td>0.70</td><td>0.73</td><td>0.69</td></tr><tr><td>row 4</td><td>0.74</td><td>0.68</td><td>0.68</td><td>0.62</td></tr><tr><td>row 5</td><td>0.69</td><td>0.62</td><td>0.58</td><td>0.53</td></tr><tr><td>row 6</td><td>0.68</td><td>0.59</td><td>0.50</td><td>0.48</td></tr><tr><td>row 7</td><td>0.61</td><td>0.53</td><td>0.41</td><td>0.38</td></tr><tr><td>row 8</td><td>0.52</td><td>0.49</td><td>0.33</td><td>0.31</td></tr><tr><td>row 9</td><td>0.48</td><td>0.45</td><td>0.27</td><td>0.23</td></tr><tr><td>row 10</td><td>0.39</td><td>0.39</td><td>0.20</td><td>0.17</td></tr><tr><td>row 11</td><td>0.30</td><td>0.31</td><td>0.17</td><td>0.14</td></tr><tr><td>row 12</td><td>0.19</td><td>0.21</td><td>0.12</td><td>0.10</td></tr><tr><td>row 13</td><td>0.17</td><td>0.17</td><td>0.10</td><td>0.09</td></tr><tr><td>row 14</td><td>0.13</td><td>0.15</td><td>0.10</td><td>0.09</td></tr><tr><td colspan="5"><strong>Series 3</strong></td></tr><tr><td>row 1</td><td>0.92</td><td>0.94</td><td>0.94</td><td>0.96</td></tr><tr><td>row 2</td><td>0.68</td><td>0.73</td><td>0.64</td><td>0.69</td></tr><tr><td>row 3</td><td>0.46</td><td>0.58</td><td>0.43</td><td>0.56</td></tr><tr><td>row 4</td><td>0.31</td><td>0.42</td><td>0.21</td><td>0.32</td></tr><tr><td>row 5</td><td>0.15</td><td>0.15</td><td>0.12</td><td>0.12</td></tr><tr><td>row 6</td><td>0.08</td><td>0.11</td><td>0.08</td><td>0.09</td></tr><tr><td>row 7</td><td>0.07</td><td>0.07</td><td>0.07</td><td>0.09</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td><td>191</td><td>191</td></tr></tbody></table>

For treatments GLo and GHi, the bold italic rows are those where left-hand side choices denote risk aversion. For the other two treatments, the bold italic rows are those where left-hand side choices denote risk seeking.
the subjects are risk lovers and at least 52% are risk averse. The behaviour of the remaining 12% is close to risk neutrality. The low proportion of risk-seeking subjects can be explained by subjects being very risk averse over low-probability losses which are typically the worst outcomes of the LHS lotteries. This is again in accordance with the fourfold pattern, item (iv). In Series 2 of LLo, 81% of the subjects choose the LHS lottery at row 1, meaning at least 81% of the subjects are risk seeking. In the absence of a worst outcome with a low probability, subjects are predominantly risk seeking as predicted by item (ii) of the fourfold pattern. Finally, in Series 3 of LLo, 64% of the subjects choose the LHS lottery at row 2, meaning at least 64% of the subjects are risk lovers. This time, this result cannot be explained by loss aversion as it is the LHS lottery which features the worst losses.

In the *high-stake* treatments (GHi and LHi), the number of LHS choices follows a similar pattern as in the corresponding *low-stake* treatments. This leads to the same conclusion about subjects mostly following the PT fourfold pattern of behaviour.

The aggregate parameter values we calculate based on the methodology described in Section 4 confirm the non-parametric analysis. The first column of Table 5 displays the means of $\sigma^d$, $\gamma^d$ and $\lambda$ for the underlying population, and the corresponding standard errors in the baseline GLo treatment. We find respective values of about 0.65, 0.64 and 2.46. This means that, on average, subjects have a concave value function in the gain domain (and convex in the loss domain), tend to overweight low-probability outcomes, and value losses more than twice as much as gains. This is again consistent with the fourfold pattern of behaviour. The other three columns of Table 5 show that parameter values in the alternative treatments vary, in particular in the case of loss aversion. However, they determine roughly the same aggregate behaviour, i.e., $\sigma^d < 1$, $\gamma^d < 1$ and $\lambda > 1$.

At the individual level, the picture is less clear-cut. Table 6 classifies subjects into four categories according to the shape of their value function in the gain and loss domains. The majority of our sample displays a standard ‘S-shape’ value function, concave in gains and convex in losses (i.e., $\sigma^+ < 1$ and $\sigma^- < 1$). This amounts to 72% of the subjects in the low-stake conditions and 67% in the high-stake conditions. Another category is quite well represented: 16 to 18% of the subjects exhibit a concave value function in the two gain and loss domains ($\sigma^+ < 1$ and $\sigma^- > 1$). The two other categories of Table 6, corresponding to a convex value function for gains, are anecdotal. Subjects with an everywhere convex value function represent 8 to 10% of the sample, and only 4 to 5% of subjects were found to behave oppositely to the standard.
<table><caption>Table 5: Population mean parameter values by treatment</caption><thead><tr><th></th><th>Treatment GLo<br>Mean/(S.E.)</th><th>Treatment GHi<br>Mean/(S.E.)</th><th>Treatment LLo<br>Mean/(S.E.)</th><th>Treatment LHi<br>Mean/(S.E.)</th></tr></thead><tbody><tr><td>σ<sup>d</sup></td><td>0.65<br>(0.032)</td><td>0.69<br>(0.033)</td><td>0.76<br>(0.029)</td><td>0.84<br>(0.028)</td></tr><tr><td>γ<sup>d</sup></td><td>0.64<br>(0.012)</td><td>0.64<br>(0.026)</td><td>0.66<br>(0.018)</td><td>0.64<br>(0.023)</td></tr><tr><td>λ</td><td>2.46<br>(0.165)</td><td>2.76<br>(0.197)</td><td>1.21<br>(0.092)</td><td>1.11<br>(0.124)</td></tr><tr><td>Nb. of obs.</td><td>191</td><td>191</td><td>191</td><td>191</td></tr><tr><td>Wald test σ<sup>d</sup> =1</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.001</td></tr><tr><td>Wald test γ<sup>d</sup> =1</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td></tr><tr><td>Wald test λ =1</td><td>0.000</td><td>0.000</td><td>0.060</td><td>0.423</td></tr></tbody></table>

Standard errors are clustered at the session level.

For Wald tests, the number displayed is the p-value.

S.E. stands for standard error.

<table><caption>Table 6: Proportion of subjects according to the curvature of the value function in the gain and loss domains</caption><thead><tr><th></th><th>low stakes</th><th>high stakes</th></tr></thead><tbody><tr><td>standard</td><td>0.72</td><td>0.67</td></tr><tr><td>everywhere concave</td><td>0.16</td><td>0.18</td></tr><tr><td>everywhere convex</td><td>0.08</td><td>0.10</td></tr><tr><td>anti-standard</td><td>0.04</td><td>0.05</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td></tr></tbody></table>
We can summarize the above results as follows:

**Result 1** According to both the number of LHS choices and the parameter values, at the aggregate level, subjects' behaviour is consistent with the PT fourfold pattern, whatever the treatment. Individual behaviour was found to be more heterogeneous but a large majority of our sample exhibits a standard value function, concave for gains and convex for losses.

## 5.2 Frame effects

### 5.2.1 Loss frame

We compare the *gain* treatments to the *loss* treatments to test formally whether behaviour for gains is reflected for losses. If reflection is perfect at the sample level and the fourfold pattern of behaviour is fully symmetrical then subjects should make symmetrical risk choices in the two frames. As the LHS and the RHS are reversed when lottery stakes are multiplied by −1, the proportion of LHS choices at each row should be very close between the *gain* and *loss* frames, at least in Series 1 and 2. Indeed, loss aversion does not come into play when choosing between all-loss lotteries. For instance, a similar proportion of subjects should switch at row 12 in Series 1, in the *gain* frame (very risk-averse subjects) and in the *loss* frame (very risk-seeking subjects).

This is not what Table 4 shows for Series 1 and 2 as the LHS choices are significantly lower in treatment LLo compared to baseline GLo. Paired t-tests on the means of the total number of LHS choices over all rows give p-values of 0.018 and 0.000 for Series 1 and 2 respectively. The same applies to the *high-stake* treatments (p-values of 0.000 and 0.001 for Series 1 and 2 respectively).<fnref n="8" /> In Table 7, we regress the total number of LHS choices on frame characteristics, by series, and control for the interacted frame effects and individual socio-demographics. We confirm a highly significant negative effect of the *loss* frame in Series 1 and 2. The effect is rather small (coefficients are -0.853 and -1.450 respectively, meaning the number of LHS choices decreases only by about 1) and does not depend on the stake level. It implies that, when faced with losses, subjects tend to switch comparatively earlier on average, i.e., they find the LHS lottery less attractive compared to the RHS lottery.

<footnote n="8">⁸Wilcoxon signed-rank tests for matched pairs, which are the equivalent non-parametric tests, lead to the same conclusions with the same levels of significance (p-values of 0.032 and 0.000 for Series 1 and 2 respectively). It means the distributions of the total number of LHS choices among subjects are statistically different from each other between the *gain* and *loss* frames.</footnote>
<table><caption>Table 7: Regression (OLS) of total number of left-hand side choices on frame characteristics and socio-demographics by series</caption><thead><tr><th rowspan="2">Covariate</th><th colspan="2">Series 1</th><th colspan="2">Series 2</th><th colspan="2">Series 3</th></tr><tr><th>Coef.</th><th>Std. Err.</th><th>Coef.</th><th>Std. Err.</th><th>Coef.</th><th>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>7.271***</td><td>(1.650)</td><td>6.074***</td><td>(1.606)</td><td>1.531***</td><td>(0.572)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Loss</td><td>-0.853**</td><td>(0.360)</td><td>-1.450***</td><td>(0.403)</td><td>-0.162</td><td>(0.174)</td></tr><tr><td>High</td><td>-0.267</td><td>(0.290)</td><td>-0.461</td><td>(0.365)</td><td>0.330**</td><td>(0.149)</td></tr><tr><td>Loss × High</td><td>-0.586</td><td>(0.407)</td><td>-0.010</td><td>(0.502)</td><td>0.000</td><td>(0.208)</td></tr><tr><td><em>Individual characteristics:</em></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Age</td><td>-0.064</td><td>(0.076)</td><td>0.052</td><td>(0.067)</td><td>0.042</td><td>(0.026)</td></tr><tr><td>Female</td><td>0.221</td><td>(0.396)</td><td>0.736</td><td>(0.467)</td><td>0.527***</td><td>(0.154)</td></tr><tr><td>Master</td><td>-0.244</td><td>(0.552)</td><td>-1.701***</td><td>(0.620)</td><td>-0.194</td><td>(0.209)</td></tr><tr><td>Economics</td><td>0.302</td><td>(0.411)</td><td>0.134</td><td>(0.467)</td><td>0.007</td><td>(0.154)</td></tr><tr><td>Model R-squared</td><td colspan="2">0.035</td><td colspan="2">0.049</td><td colspan="2">0.034</td></tr><tr><td>Nb. of obs. / clusters</td><td colspan="2">764/191</td><td colspan="2">764/191</td><td colspan="2">764/191</td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

OLS stands for Ordinary Least Squares.

There could be two explanations for this. As determined in Section 5.1, our sample is mostly risk averse in the gain domain and risk seeking in the loss domain, and thus over-values the LHS lottery in both domains. Thus, switching earlier when facing losses means that the gap between the prospect values of the two LHS and RHS lotteries is thinner in the loss domain than in the gain domain. This is an argument in favour of a more linear value function in the loss domain. Probability weighting may provide another explanation. Indeed, in the *loss* treatments the LHS lotteries feature low-probability losses. Yet, our sample mostly overweights low-probability events (Section 5.1), and thus under-values the LHS lottery in the *loss* treatments. Thus, switching earlier when facing losses may mean that probability weighting is stronger in the loss domain than in the gain domain. This is consistent with a more curved probability weighting function in the loss domain.

To choose between these two options, we assess the effect of the *loss* frame directly on the mean PT parameters of Table 5. In Tables 8, 9 and 10, we estimate the regression coefficients of $\sigma^d$, $\gamma^d$ and $\lambda$ respectively on frame characteristics (1) and individual socio-demographics (2). The *loss* frame has a significant positive impact on curvature $\sigma^d$ (+0.118, significant at the 1% level), but not on $\gamma^d$. We thus conclude in favour of a partial reflection of the value function at the aggregate level towards slightly more linearity in the loss domain, for both low stakes and high stakes.

As for loss aversion $\lambda$, Table 10 shows a strong and significant negative impact of the *loss* frame (-1.249, significant at the 1% level), with an additional small negative impact in the high-stake
<table><caption>Table 8: Regression (OLS) of $\sigma^d$ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>(2)<br>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>0.647***</td><td>(0.026)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td></tr><tr><td>Loss</td><td>0.118***</td><td>(0.032)</td></tr><tr><td>High</td><td>0.040</td><td>(0.026)</td></tr><tr><td>Loss×High</td><td>0.036</td><td>(0.037)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td></tr><tr><td>Age</td><td></td><td>0.002</td></tr><tr><td>Female</td><td></td><td>-0.050</td></tr><tr><td>Master</td><td></td><td>0.107**</td></tr><tr><td>Economics</td><td></td><td>-0.023</td></tr><tr><td>Model R-squared</td><td>0.041</td><td>0.065</td></tr><tr><td>Adjusted R-squared</td><td>0.038</td><td>0.056</td></tr><tr><td>Nb. of obs. /clusters</td><td>764/191</td><td>764/191</td></tr><tr><td>AIC</td><td>609.045</td><td>598.302</td></tr><tr><td>BIC</td><td>627.599</td><td>635.410</td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \* and \*\* stand for significance at the 10, 5 and 1% level respectively.

OLS stands for Ordinary Least Squares.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.

<table><caption>Table 9: Regression (OLS) of $\gamma^d$ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>(2)<br>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>0.636***</td><td>(0.020)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td></tr><tr><td>Loss</td><td>0.029</td><td>(0.025)</td></tr><tr><td>High</td><td>0.009</td><td>(0.022)</td></tr><tr><td>Loss×High</td><td>-0.034</td><td>(0.031)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td></tr><tr><td>Age</td><td></td><td>-0.006</td></tr><tr><td>Female</td><td></td><td>-0.026</td></tr><tr><td>Master</td><td></td><td>0.068*</td></tr><tr><td>Economics</td><td></td><td>0.008</td></tr><tr><td>Model R-squared</td><td>0.002</td><td>0.011</td></tr><tr><td>Adjusted R-squared</td><td>-0.002</td><td>0.002</td></tr><tr><td>Nb. of obs. /clusters</td><td>764/191</td><td>764/191</td></tr><tr><td>AIC</td><td>258.555</td><td>259.300</td></tr><tr><td>BIC</td><td>277.109</td><td>296.409</td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \* and \*\* stand for significance at the 10, 5 and 1% level respectively.

OLS stands for Ordinary Least Squares.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.
<table><caption>Table 10: Regression (OLS) of λ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>Std. Err.</th><th>(2)<br>Coef.</th><th>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>2.456***</td><td>(0.179)</td><td>2.175***</td><td>(0.665)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Loss</td><td>-1.249***</td><td>(0.201)</td><td>-1.249***</td><td>(0.202)</td></tr><tr><td>High</td><td>0.303</td><td>(0.215)</td><td>0.303</td><td>(0.216)</td></tr><tr><td>Loss × High</td><td>-0.404*</td><td>(0.239)</td><td>-0.404*</td><td>(0.240)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Age</td><td></td><td></td><td>0.004</td><td>(0.029)</td></tr><tr><td>Female</td><td></td><td></td><td>0.262*</td><td>(0.147)</td></tr><tr><td>Master</td><td></td><td></td><td>0.306</td><td>(0.249)</td></tr><tr><td>Economics</td><td></td><td></td><td>-0.049</td><td>(0.151)</td></tr><tr><td>Model R-squared</td><td>0.128</td><td></td><td>0.138</td><td></td></tr><tr><td>Adjusted R-squared</td><td>0.124</td><td></td><td>0.130</td><td></td></tr><tr><td>Nb. of obs. / clusters</td><td>764/191</td><td></td><td>764/191</td><td></td></tr><tr><td>AIC</td><td>3173.122</td><td></td><td>3172.127</td><td></td></tr><tr><td>BIC</td><td>3191.676</td><td></td><td>3209.236</td><td></td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

OLS stands for Ordinary Least Squares.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.

conditions (-0.404, significant at the 10% level). Surprisingly, this result seems to reveal that subjects are less loss averse when gain outcomes are converted into losses, and even less so when outcomes are high. It leads to λ mean values cut by more than half and ending up close to the neutrality threshold (1.21 under the low-stake conditions, and 1.11 under the high-stake conditions, Table 5). The Wald tests at the bottom of Table 5 demonstrate that mean λ is still statistically different from 1 in the low-stake case, but at the 10% level only. In the high-stake case, loss aversion is no longer significant.

The results of the above analysis may be summarized as follows:

**Result 2** At the aggregate level, subjects switch significantly earlier when Series 1 and 2 are framed as loss lotteries, whatever the stake size. It is due to a slightly more linear value function for losses than for gains. Surprisingly, subjects are far less loss averse in the loss domain on average, even leading to loss neutrality in the high-stake conditions.

### 5.2.2 High-stake frame

Although this paper focuses on the *loss* frame, the effect of the *high-stake* frame itself on behaviour merits comment. Table 4 shows that in the *high-stake* treatments (GHi and LHi), the number of LHS choices at each row seems close to that in the corresponding *low-stake* treatments. Paired t-tests on
the total number of LHS choices give mixed results. In Series 2, the means between the *high-stake* and *low-stake* treatments are statistically equivalent (p-values are 0.206 in the *gain* frame and 0.179 in the *loss* frame). In Series 1, the means are equivalent in the *gain* frame only (p-values is 0.356 in the *gain* frame, while it is 0.007 in the *loss* frame). In Series 3, the means are different from each other at the 5% level in both the *gain* and *loss* frames (p-values are 0.027 and 0.038 respectively).<fnref n="9" /> This is broadly in accordance with the regressions of Table 7 as there is no significant effect of the *high-stake* frame on the total number of LHS choices for Series 1 and 2 and a significant positive effect for Series 3. The effect is very small (+0.330), suggesting that when faced with high stakes, subjects switch comparatively later on average, i.e., they find the LHS lottery more attractive than the RHS lottery. However, the regression results in Tables 8, 9 and 10 show no significant impact of the *high-stake* frame on any of the three PT parameters.

Loss aversion can provide a sufficient explanation, but in the case of the *gain* frame only. In treatment GHi, the RHS (or risky) lottery features the highest losses. Since higher stakes mean a higher penalty for the highest losses compared to other outcomes, loss-averse subjects tend to reduce the prospect value of the RHS lottery proportionally more than the LHS lottery. This results in a higher proportion of LHS choices and of risk-averse subjects (73% for GHi vs. 68% for GLo, row 2 of Table 4). However this reasoning does not hold true in the *loss* treatment LHi as, this time, the highest losses belong to the LHS (or risky) lottery. Thus, we should observe a decrease in the proportion of LHS choices and of risk-seeking subjects. On the contrary, at least 64% of the subjects are risk seeking in treatment LLo while this amounts to 69% in treatment LHi.

We can summarize our results as follows:

**Result 3** *At the aggregate level, there is no stake effect on the choices in Series 1 and 2. In Series 3, subjects switch a bit later when stakes are higher. Mere loss aversion can explain this result, but in the gain context only. Stake size has no impact on any of the three PT parameters.*

**5.3 Discussion**

Our study shows that subjects exhibit the PT fourfold pattern of behaviour (Result 1). The orders of magnitude of the population mean values for the three preference parameters are similar to other studies using the same experimental design, parametric specifications and measurement method than

<footnote n="9">⁹Wilcoxon signed-rank tests for matched pairs give similar conclusions about the distributions of the total number of LHS choices.</footnote>
TCN (e.g., Liu (2013), Campos-Vazquez and Cruilty (2014), Bocquého et al. (2014), Bocquého et al. (2018), Bauermeister et al. (2018)). We provide a graphical comparison in Appendix D. Individual values confirm that this pattern applies to the majority of our sample. This is in line with Tversky and Kahneman (1992) who observe the entire fourfold pattern for 88% of the 25 subjects they tested. The standard concave-convex S-shape value function is also the most common in Abdellaoui et al.'s (2007) work. However, with a similar power specification, but focusing on losses and allowing a linear category, Etchart-Vincent (2004) finds that roughly only half of her sample exhibited convexity when facing small (54%) or large (46%) losses. Booij and van de Kuilen's (2009) classification of a large sample based on non-parametric individual value functions reveals that, in total, concavity in gains applied to 42% of the subjects and convexity in losses to 47%. However, only 25% of the subjects combine both into the S-shape. Nevertheless, concave value functions for losses are not uncommon (about 20% in our sample, Table 12). Abdellaoui et al. (2008) report the everywhere concave category was the most common in their study, even if it does not correspond to a significantly different proportion of subjects than the standard category. Etchart-Vincent (2004) counts 34% and 45% of subjects with a concave value function for small and large losses respectively. In Booij and van de Kuilen (2009), cited earlier, the linear category for losses is well represented (30%), and dominates the concave category (22%). The linear-concave shape is even as prevalent as the standard concave-convex S-shape.

Our study also shows that the value function is more linear in the loss domain, for both low and high stakes (Result 2). This is reported in many other experimental studies, as highlighted in Section 2 (e.g., Kahneman and Tversky, 1979; Fennema and van Assen, 1998; Abdellaoui, 2000; Laury and Holt, 2008). Let us focus on Bougherara et al. (2017), the only study documenting distinct loss and gain parameters using the TCN design. The authors rely on a structural estimation procedure instead of eliciting intervals, but like us they report close but significantly different power parameters for the curvature of the value function when they ignore loss aversion. They find as expected more linearity for losses than for gains (parameter are 0.66 and 0.60 respectively). However, they detect a small significant difference as well in the probability weighting function, even when loss aversion is accounted for. Again, there is more linearity for losses (parameters are 0.84 and 0.79 respectively). This is in line with Tversky and Kahneman (1992) and Abdellaoui (2000) who use the same one-parameter form.
Another highlight of Result 2 is the strong negative effect of the *loss* frame on loss aversion. This unexpected result is too strong to be caused by response error only. We suggest two possible explanations for this. Firstly, we suspect a pure loss-frame effect. As highlighted by many authors, loss aversion elicitation is indeed very sensitive to framing (Abdellaoui et al., 2007; Wakker, 2010). Another possible explanation is the sequential structure of the TCN procedure, and the fact that $\lambda$ is calculated in a second step conditionally on a unique $\sigma^d$ value from the first step. We will further investigate this point in Section 6.

Our last result so far is the clear absence of stake effects on Series 1 and 2 choices, but also on PT parameters (Result 3). As a consequence, there is no evidence of Markowitz's (1952) 'forgotten' fourfold pattern (Scholten and Read, 2014) at the aggregate level. This pattern depends on outcome sign and level:

* risk aversion over medium to high gains

* risk seeking over medium to high losses

* risk seeking over small gains

* risk aversion over small losses.

Manipulating payoff ranges in MPLs has been common and, like us, several studies fail to detect an effect of stake size on the curvature of the value function (Booij and van de Kuilen, 2009; Booij et al., 2010). Using a sample of nearly 2,000 students and scaling up lottery gain and loss outcomes by a factor 10, Booij et al. (2010) also report no difference between the high and low treatments when they rely on non-parametric estimates. However, the authors stress that the power coefficient for low amounts of gains is .071 lower than for high amounts, suggesting that utility is more curved for low outcomes. This is not a common finding in the literature, which mostly delivers evidence in favour of risk aversion increasing with payoff level (increasing relative risk aversion).<fnref n="10" /> For example, using different probability levels and a multiplying factor of 6 to 10, Bouchouicha and Vieider (2017) show that the mean risk premia move up across gain stakes for every probability considered. Stake effects for losses are especially inconclusive in the literature. For instance, Etchart-Vincent (2004) mentions that behaviour towards risk in general appears not to be sensitive to the magnitude of negative payoffs. In the same vein, Bouchouicha and Vieider (2017) cannot replicate for losses the pattern they describe

<footnote n="10">¹⁰In the EU context, Reynaud and Couture (2012) replicate Holt and Laury’s (2002) as well as Eckel and Grossman’s (2008) baseline lotteries, and also multiply stakes by 20. They find that, for both methods, the mean constant relative risk aversion coefficients are statistically different, subjects being more risk averse for high payoffs than for low payoffs. However, in the case of Holt and Laury’s (2002) experiment, they report that distribution of the coefficients is not modified by the payoff level.</footnote>
for gains, whatever the probability level.

Evidence regarding the probability weighting parameter is mixed. Tversky and Kahneman (1992) already suspected that decision weights might be sensitive to framing, including the level of outcomes. Booij et al. (2010) cannot find any significant effect of stake size while Etchart-Vincent (2004) reports the opposite result based on loss lotteries, but only when using a 2-parameter specification. Fehr-Duda et al. (2010) and Bouchouicha and Vieider (2017) find a significant effect of stake size in the gain domain but not in the loss domain. As regards loss aversion, Schmidt and Traub (2002), Booij and van de Kuilen (2009) and Booij et al. (2010) detect no effect of stake size on the controlling parameter, which is a similar finding to our own. However, in Abdellaoui et al. (2007), the mean value of the loss aversion coefficient decreases with the size of the gains and losses involved. Bleichrodt and Pinto (2002) observe a similar phenomenon in the health domain.

As far as the choice of the econometric model for regressing PT parameters is concerned, it could be argued that the ordinary least squares method is not suitable although it is widely used in the experimental literature based on MPLs. Actually, risk parameter values are derived from intervals, advocating for the use of interval regressions. This is what we report in Appendix E, Tables 18 to 20. <fnref n="11" /> It should be noted that regressing each parameter independently is another simplification as the TCN procedure elicits $\sigma^d$ and $\gamma^d$ intervals jointly from Series 1 and 2. Furthermore, the distribution of $\lambda$ is far from normal (see Appendix F), and thus a log-normal interval regression may even seem more adequate for this parameter. This is what the Akaike and Schwarz's Bayesian information criteria show for the log-interval regression in Appendix E, Table 21). These alternative specifications show that the results from the ordinary least square regressions are robust. The only difference is in Table 21 where the interaction term between the *loss* and *high-stake* frames becomes significant at the 5% level (instead of 1%), and a weekly significant positive effect of stake size appears.

# 6 Consequences for elicited loss aversion

In this section, we shall fine-tune the measures of the loss aversion parameter. We first investigate further the surprising negative effect of the *loss* frame on loss aversion. Secondly, we explore the consequences of relaxing the hypothesis of identical value functions in the gain and loss domains

<footnote n="11">¹¹In Tables 18 and 19, the intervals for $\sigma^d$ and $\gamma^d$ are artificially rebuilt based on the matrix of point estimates in TCN and the information the authors give about the 0.5 approximation.</footnote>
for the subsequent measures of loss aversion. We analyse the results at the aggregate and individual
levels.

6.1 A pure loss-frame effect

We aim to untangle the two components of the strong negative effect of the *loss* frame on $\lambda$ (Result 2) by identifying which part is due to a pure framing effect, the mixed lotteries of Series 3 being mirrored into the opposite domain, and which part is due to the sequential calculation procedure, $\lambda$ being calculated conditionally on $\sigma^{-}$ instead of $\sigma^{+}$ in the *loss* treatments. To this end, we use the alternative $\lambda_{oppos}$ parameter described in Section 4. In the *gain* treatments, $\lambda_{oppos}$ values are conditional to $\sigma^{-}$ elicited over losses, and in the *loss* treatments, they are conditional to $\sigma^{+}$ elicited over gains. The population mean and median of $\lambda_{oppos}$ by treatment are displayed in Table 11.

Comparing $\lambda$ of baseline GLo to $\lambda_{oppos}$ of treatment LLo allows an assessment of the pure framing
effect in the low-stake conditions because both measures use $\sigma^{+}$. The mean values still drop by about
half, from 2.46 to 1.34 (instead of 1.21). The drop in median values is slightly more pronounced,
from 1.60 to 0.65. In the high-stake conditions, the drop is even more striking. Under these last
conditions, we still cannot exclude neutrality towards loss as the Wald test on the mean in the bottom
panel shows (p-value is 0.145 instead of 0.423). It should be noted that median loss aversion is even
significantly lower than 1, indicating gain seeking. The first two lines of Table 12 give the proportions
of loss-averse respondents according to individual $\lambda$ and $\lambda_{oppos}$ values. A subject is classified as loss
averse when the parameter under scrutiny is higher than 1.<fnref n="12" /> We find that the pure framing effect
decreases the proportion of loss-averse subjects from 69% to 36% in the low-stake conditions, and
from 76% to 31% in the high-stake conditions.

<footnote n="12">¹²Corresponding distributions are represented in Appendix G, Figures 6 and 7.</footnote>
<table><caption>Table 11: Population $\lambda$, $\lambda_{oppos}$ and $\lambda_{gen}$ values by treatment</caption><thead><tr><th rowspan="2"></th><th colspan="2">Treatment GLo</th><th colspan="2">Treatment GHi</th><th colspan="2">Treatment LLo</th><th colspan="2">Treatment LHi</th></tr><tr><th>Mean (S.E.)</th><th>Median (S.E.)</th><th>Mean (S.E.)</th><th>Median (S.E.)</th><th>Mean (S.E.)</th><th>Median (S.E.)</th><th>Mean (S.E.)</th><th>Median (S.E.)</th></tr></thead><tbody><tr><td>λ</td><td>2.46 (0.16)</td><td>1.60 (0.11)</td><td>2.76 (0.20)</td><td>2.01 (0.31)</td><td>1.21 (0.09)</td><td>0.62 (0.04)</td><td>1.11 (0.12)</td><td>0.51 (0.05)</td></tr><tr><td>λ<sub>oppos</sub></td><td>2.47 (0.16)</td><td>1.65 (0.15)</td><td>2.76 (0.21)</td><td>2.02 (0.27)</td><td>1.34 (0.07)</td><td>0.65 (0.04)</td><td>1.21 (0.13)</td><td>0.51 (0.04)</td></tr><tr><td>λ<sub>gen</sub></td><td>8.43 (2.18)</td><td>1.08 (0.12)</td><td>7.47 (1.98)</td><td>1.03 (0.14)</td><td>3.37 (0.92)</td><td>0.41 (0.07)</td><td>7.66 (4.84)</td><td>0.43 (0.07)</td></tr><tr><td>Nb. of obs.</td><td>191</td><td>191</td><td>191</td><td>191</td><td>191</td><td>191</td><td>191</td><td>191</td></tr><tr><td>Wald test λ =1</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.001</td><td>0.060</td><td>0.000</td><td>0.423</td><td>0.000</td></tr><tr><td>Wald test λ<sub>oppos</sub> =1</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.000</td><td>0.002</td><td>0.000</td><td>0.145</td><td>0.000</td></tr><tr><td>Wald test λ<sub>gen</sub> =1</td><td>0.011</td><td>0.500</td><td>0.014</td><td>0.852</td><td>0.036</td><td>0.000</td><td>0.211</td><td>0.000</td></tr></tbody></table>

Standard errors are clustered at the session level.

For Wald tests, the number displayed is the p-value.

S.E. stands for standard error.

<table><caption>Table 12: Proportion of subjects exhibiting loss aversion according to various parameter specifications</caption><thead><tr><th></th><th>low stakes</th><th>high stakes</th></tr></thead><tbody><tr><td>λ in <i>gain</i> frame</td><td>0.69</td><td>0.76</td></tr><tr><td>λ<sub>oppos</sub> in <i>loss</i> frame</td><td>0.36</td><td>0.31</td></tr><tr><td>λ<sub>oppos</sub> in <i>gain</i> frame</td><td>0.68</td><td>0.76</td></tr><tr><td>λ<sub>gen</sub> in <i>gain</i> frame</td><td>0.52</td><td>0.50</td></tr><tr><td>λ<sub>gen</sub> in <i>loss</i> frame</td><td>0.33</td><td>0.31</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td></tr></tbody></table>
Comparing $\lambda$ of baseline GLo to $\lambda_{oppos}$ of the same treatment allows an assessment of the sequentiality effect in the low-stake conditions because the two measures do not use the same ($\sigma^d$). Sequentiality slightly increases the population mean (from 2.46 to 2.47) and median loss aversion (from 1.60 to 1.65, Table 11). In the high-stake conditions (GHi treatment), no salient difference is found. This can be explained by our previous finding that the curvature of the value function has significantly different but close values in the gain and loss domains, whatever the size of stakes (Result 2). The first and third lines of Table 12 demonstrate that the sequentiality effect is not prominent at the individual level either. The proportions of loss-averse subjects are unchanged: 68% vs. 69% when low stakes, and 76% vs. 76% when high stakes.

We summarize our conclusions as follows:

**Result 4** *The very low means and medians for loss aversion in the loss frame are for the most part due to a pure framing effect rather than to a sequentiality effect, and it holds for both the low-stake and high-stake conditions.*

## 6.2 A more general loss aversion parameter

We fully relax the assumption about equal curvature in the gain and loss domains and assess the consequences for elicited loss aversion. We expect that accounting jointly for $\sigma^+$ and $\sigma^-$ when measuring loss aversion will provide slightly more accurate values. Moreover, it might moderate the pure framing effect we have just isolated. We use the more general $\lambda_{gen}$ parameter defined in Section 4, which is calculated conditionally on both $\sigma^+$ and $\sigma^-$. We compare, $\lambda$ with $\lambda_{gen}$ in the *gain* treatments, and also $\lambda_{oppos}$ with $\lambda_{gen}$ in the *loss* treatments. This allows us to assess the effect of applying $\sigma^-$ to Series 3 negative payoffs instead of $\sigma^+$ when calculating loss aversion.

Table 11 shows it dramatically increases mean loss aversion in the *gain* treatments: from 2.46 to 8.43 in the low-stake conditions, and from 2.76 to 7.47 in the high-stake conditions. This is partly due to extreme outliers with very high loss aversion values, including 2- and 3-digit numbers. Extreme values in the vicinity of 0 do also appear. It leads to high standard errors, and despite a high mean value, $\lambda_{gen}$ becomes significantly different from 1 at the 5% level instead of the 1% level (p-value is 0.011 in the low-stake context and 0.014 in the high-stake context). The extreme $\lambda_{gen}$ values belong to subjects whose $\sigma^+$ and $\sigma^-$ are on either side of the 1 threshold, meaning an everywhere-concave or everywhere-convex value function. Adding up the second and third lines of Table 6 shows these
non-standard shapes represent up to 28% of our sample. Given the highly skewed distribution of loss aversion and large influence of outliers on the means, we also look at the medians and individual values, which give opposite results. Indeed, median loss aversion decreases from 1.60 to 1.08 in the low-stake conditions, and from 2.01 to 1.03 in the high-stake conditions (Table 11). Loss neutrality cannot be excluded anymore (respective p-values of 0.500 and 0.852 by Wald tests). The first and fourth lines of Table 12 about individual measures reveal that the proportion of loss-averse subjects falls from 69% to 52% for low stakes, and even more from 76% down to 50% for high stakes. As a consequence, loss-neutral and gain-seeking behaviour appear much more widespread than usually considered.

In the *loss* treatments, applying $\sigma^{-}$ to loss outcomes instead of $\sigma^{+}$ entails similar effects although much less pronounced. In the low-stake conditions, mean loss aversion jumps from 1.34 to 3.37 (and from 1.21 to 7.66 in the high-stake conditions), but median loss aversion decreases from 0.65 to 0.41 (and from 0.51 to 0.43 in the high-stake conditions). The proportion of loss-averse subjects only decreases slightly in the low-stake conditions from 36% to 33% and it remains unchanged in the high-stake conditions (31%).

These analyses give the following result:

**Result 5** Assuming a unique curvature for the value function leads to strongly underestimating mean loss aversion in the TCN method. On the contrary, it overestimates median loss aversion. At the individual level, it skews distribution of loss aversion to the right, decreases heterogeneity, and artificially inflates the proportion of loss-averse subjects.

To assess specifically whether using two distinct power parameters in the value function mitigates the pure loss-frame effect, we compare $\lambda_{gen}$ in the *gain* treatments to $\lambda_{gen}$ in the *loss* treatments. In the low-stake conditions we find that the mean and median loss aversion values now drop by more than half: from 8.43 to 3.37 and from 1.08 to 0.41 respectively (Table 11). The proportion of loss-averse subjects also strongly decreases from 52% to 33% (Table 12). In the high-stake conditions, the results are almost similar: mean loss aversion remains the same (from 7.47 to 7.66), median loss aversion drops (from 1.03 to 0.43), and the proportion of loss-averse subjects sharply declines (from 50% to 31%). These findings may be summarized as follows:

**Result 6** The pure loss-frame effect towards a sharp decrease in loss aversion is substantial, both at the aggregate and individual levels, even when relaxing the constraint on the identity of the curvature
## 6.3 Discussion

Our findings about a strong and negative pure loss-frame effect on loss aversion (Results 4 and 6) give credit to the volatility of loss aversion and its sensitivity to small details of framing as Wakker (2010) put it. In particular, Wakker (2010) explains violations of asset integration entailed by reference dependence are highly irrational, and can explain part of this volatility. Andersen et al. (2006) and Harrison and Rutström (2008) also discuss the crucial role of the reference point as a source of volatility, especially in dynamic tasks and in the field. As a remedy, based on several studies, Wakker (2010) recommends proper learning and incentives. More broadly, Tversky and Kahneman (1992) highlight heuristics of choices, and thus PT parameters, might be sensitive to the formulation of the problem, the method of elicitation and the context of choice. Technical reasons may also explain the wide variation in loss-aversion measures between studies, such as different parametric assumptions about the other PT components, the use of either mean or median values for loss aversion indices, and different definitions of loss aversion itself, including the money unit assumed for scaling the power utility (Abdellaoui et al., 2007).

This study also shows that allowing distinct curvatures in gains and losses for the value function in the TCN method leads to much higher means for loss aversion, but much lower medians, as well as much fewer loss-averse subjects (Result 5). Examples of high mean loss aversion values are quite limited in the literature, even when reflection is not imposed. Abdellaoui et al. (2007) and Harrison and Rutström (2009) are exceptions in this respect. Using a non-parametric elicitation procedure and a certainty-equivalent design, Abdellaoui et al. (2007) report a value of 8.27 but only when they use Köbberling and Wakker's (2005) definition of loss aversion; the four other definitions lead to values equal or lower than 2. Harrison and Rutström (2009) find a mean of 5.78, but only in the case of their mixture EU-PT model.

Other experimental studies with no gain-loss parameter constraint report more reasonable means for loss aversion. They most often rely on the structural estimation procedure based on pooled observations. Using the TCN design, Bougherara et al. (2017) estimate a mean value of 1.37 for French farmers' loss aversion, when there is no constraint on the probability weighting function. With the same power value function but using other lottery designs, Andersen et al. (2006), Abdellaoui et al.
(2008), Harrison and Rutström (2009), and Booij et al. (2010) find mean loss aversion values of 1.07, 1.60, 1.38, and 1.58 respectively. The second study assumes no constraint on the weighting function only, the first and the third studies assume no constraint on the value function only. In the last study both the value and weighting functions are free from gain-loss constraints. Schmidt and Traub (2002) estimate an average pooled estimate of loss aversion of 1.43, with no constraint on the weighting function and a piecewise linear specification which *de facto* imposes reflection on the value function.

As far as medians are concerned, the value of 2.25 Tversky and Kahneman (1992) report is usually considered as a benchmark and probably explains the conventional wisdom that losses count twice as gains. It is based on individual estimates, assuming a power value function, where subjects are free to behave differently over gains and losses. It thus shares several characteristics with the 1.08 and 0.41 median values we obtain when there are low stakes, in the *gain* and *loss* treatments respectively (1.03 and 0.43 when there are high stakes, Table 11).

Let us come back to Abdellaoui et al. (2007) who, similarly to us, specifically evaluate how bad it is to measure loss aversion when imposing reflection on the power value function. At the aggregate level they do not find any major distortion in the medians (medians are 0.43 to 2.54 over the five loss aversion coefficients they test, 1.69 under Kahneman and Tversky's (1979) definition as in our study), but considerable distortions arise at the individual level. Individual differences can be rather large, particularly for Köbberling and Wakker's (2005) measure of loss aversion. In addition, the classification of a prominent proportion of subjects with respect to their attitude towards losses changes (by 8% to 37.5%, 33% under Kahneman and Tversky's (1979) definition).

Schmidt and Traub (2002), cited earlier, also provide results at the individual level, finding mixed evidence for loss aversion as we do. Their tests indeed classify only 30% of their student sample as strictly loss averse, and 25% as gain seeking, while about 45% cannot be classified. Andersen et al. (2006) estimate three variants of a structural model including PT preference parameters, and similarly to us report strikingly large range of individual values, with important loss-neutral and gain-seeking modes in the distributions.

# 7 Conclusion

In this article, we examine the consequences of relaxing the simplifying assumption of similar PT parameter values over gains and losses, in the context of the TCN MPLs. We implement the original
risk experiment and additional treatments involving a *loss* frame and a *high-stake* frame. We show that subjects' behaviour for gains is mostly reflected for losses at the aggregate and individual levels, and is consistent with the PT fourfold pattern. Reflection is partial only, as the mean curvature of the value function is slightly less convex for losses than it is concave for gains. On the contrary, the weighting function is statistically the same in both domains. These results are robust to a high-stake context, and add to the mixed evidence drawn from other lottery designs. However, we demonstrate that assuming reflection when measuring loss aversion is not innocuous either at the aggregate or individual level. It leads to underestimating mean loss aversion while overestimating median loss aversion. It also skews distribution to the right, decreases heterogeneity and inflates the proportion of loss-averse subjects. Incidentally, we also highlight the existence of a strong, negative and persistent loss-frame effect on loss aversion, regardless of whether reflection is assumed or not.

Our results argue in favour of a more refined analysis of loss aversion, because loss aversion may not be the dominant behaviour in all circumstances. In particular, the loss-neutral and gain-seeking behaviours appear to be more common than usually presumed. The TCN experimental design is popular to elicit risk preferences in the field because it combines the simplicity of the MPL format with the sophistication of PT. We recommend that future practitioners are particularly cautious about the loss-aversion values they obtain because they are especially sensitive to assumptions on parameters and to framing. Testing for various parametric specifications in a more systematic way and acknowledging explicitly for the possible effects of frame variations would help increase the internal and external validity of the risk preference measures of specific populations. More generally, we also call for deeper analyses that account for subjects' heterogeneity, e.g., by reporting percentile values at the aggregate level and by carrying out individual-level investigations. Beyond the experimental sphere, it would help real-life choice modellers who utilize experimental risk preference values to explain and represent behavioural heterogeneity.

The MPL format allows the original TCN procedure to elicit parameter intervals directly from observed responses. It is one element explaining its simplicity of use for non-specialists. The drawbacks are that individual response error is not addressed and the elicitation structure is sequential. This is the main limitation of our study. The extent to which structural econometric estimation, which is able to accommodate these concerns, may dampen the consequences of the reflection assumption and of framing effects is an opportunity for further research. It would probably lessen the huge
preference heterogeneity we report. Another limitation of our study is the small multiplying factor between the *low-stake* and *high-stake* treatments. Using factors of 10, 20, 50, or even 90, instead of 2, may invalidate our result about the absence of stake effects. In other experimental contexts, such factors produced a large increase in risk aversion.

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# Appendices

## A Lotteries of the LLo treatment
<table><caption>Table 13: Lottery options corresponding to treatment LLo</caption><thead><tr><th rowspan="2"></th><th colspan="4">Payoffs</th><th rowspan="2">(Expected value C-D)</th></tr><tr><th>Option C</th><th></th><th>Option D</th><th></th></tr></thead><tbody><tr><td colspan="6"><strong>Series 1</strong></td></tr><tr><td>Probabilities</td><td>0.10</td><td>0.90</td><td>0.30</td><td>0.70</td><td></td></tr><tr><td>row 1</td><td>-68</td><td>-5</td><td>-40</td><td>-10</td><td>(7.7)</td></tr><tr><td>row 2</td><td>-75</td><td>-5</td><td>-40</td><td>-10</td><td>(7.0)</td></tr><tr><td>row 3</td><td>-83</td><td>-5</td><td>-40</td><td>-10</td><td>(6.2)</td></tr><tr><td>row 4</td><td>-93</td><td>-5</td><td>-40</td><td>-10</td><td>(5.2)</td></tr><tr><td>row 5</td><td>-106</td><td>-5</td><td>-40</td><td>-10</td><td>(3.9)</td></tr><tr><td>row 6</td><td>-125</td><td>-5</td><td>-40</td><td>-10</td><td>(2.0)</td></tr><tr><td>row 7</td><td>-150</td><td>-5</td><td>-40</td><td>-10</td><td>(-0.5)</td></tr><tr><td>row 8</td><td>-185</td><td>-5</td><td>-40</td><td>-10</td><td>(-4.0)</td></tr><tr><td>row 9</td><td>-220</td><td>-5</td><td>-40</td><td>-10</td><td>(-7.5)</td></tr><tr><td>row 10</td><td>-300</td><td>-5</td><td>-40</td><td>-10</td><td>(-15.5)</td></tr><tr><td>row 11</td><td>-400</td><td>-5</td><td>-40</td><td>-10</td><td>(-25.5)</td></tr><tr><td>row 12</td><td>-600</td><td>-5</td><td>-40</td><td>-10</td><td>(-45.5)</td></tr><tr><td>row 13</td><td>-1,000</td><td>-5</td><td>-40</td><td>-10</td><td>(-85.5)</td></tr><tr><td>row 14</td><td>-1,700</td><td>-5</td><td>-40</td><td>-10</td><td>(-155.5)</td></tr><tr><td colspan="6"><strong>Series 2</strong></td></tr><tr><td>Probabilities</td><td>0.70</td><td>0.30</td><td>0.90</td><td>0.10</td><td></td></tr><tr><td>row 1</td><td>-54</td><td>-5</td><td>-40</td><td>-30</td><td>(-0.3)</td></tr><tr><td>row 2</td><td>-56</td><td>-5</td><td>-40</td><td>-30</td><td>(-1.7)</td></tr><tr><td>row 3</td><td>-58</td><td>-5</td><td>-40</td><td>-30</td><td>(-3.1)</td></tr><tr><td>row 4</td><td>-60</td><td>-5</td><td>-40</td><td>-30</td><td>(-4.5)</td></tr><tr><td>row 5</td><td>-62</td><td>-5</td><td>-40</td><td>-30</td><td>(-5.9)</td></tr><tr><td>row 6</td><td>-65</td><td>-5</td><td>-40</td><td>-30</td><td>(-8.0)</td></tr><tr><td>row 7</td><td>-68</td><td>-5</td><td>-40</td><td>-30</td><td>(-10.1)</td></tr><tr><td>row 8</td><td>-72</td><td>-5</td><td>-40</td><td>-30</td><td>(-12.9)</td></tr><tr><td>row 9</td><td>-77</td><td>-5</td><td>-40</td><td>-30</td><td>(-16.4)</td></tr><tr><td>row 10</td><td>-83</td><td>-5</td><td>-40</td><td>-30</td><td>(-20.6)</td></tr><tr><td>row 11</td><td>-90</td><td>-5</td><td>-40</td><td>-30</td><td>(-25.5)</td></tr><tr><td>row 12</td><td>-100</td><td>-5</td><td>-40</td><td>-30</td><td>(-32.5)</td></tr><tr><td>row 13</td><td>-110</td><td>-5</td><td>-40</td><td>-30</td><td>(-39.5)</td></tr><tr><td>row 14</td><td>-130</td><td>-5</td><td>-40</td><td>-30</td><td>(-53.5)</td></tr><tr><td colspan="6"><strong>Series 3</strong></td></tr><tr><td>Probabilities</td><td>0.50</td><td>0.50</td><td>0.50</td><td>0.50</td><td></td></tr><tr><td>row 1</td><td>-30</td><td>21</td><td>-25</td><td>4</td><td>(6.0)</td></tr><tr><td>row 2</td><td>-30</td><td>21</td><td>-4</td><td>4</td><td>(-4.5)</td></tr><tr><td>row 3</td><td>-30</td><td>21</td><td>-1</td><td>4</td><td>(-6.0)</td></tr><tr><td>row 4</td><td>-30</td><td>16</td><td>-1</td><td>4</td><td>(-8.5)</td></tr><tr><td>row 5</td><td>-30</td><td>16</td><td>-1</td><td>8</td><td>(-10.5)</td></tr><tr><td>row 6</td><td>-30</td><td>14</td><td>-1</td><td>8</td><td>(-11.5)</td></tr><tr><td>row 7</td><td>-30</td><td>11</td><td>-1</td><td>8</td><td>(-13.0)</td></tr></tbody></table>

Lottery payoffs are in ECUs. Information on expected value is not displayed to respondents.
# B Calculation of the individual PT parameters according to frame

We distinguish parameters and functions between the *gain* and *loss* frames through index *d* (*d* is + in the *gain* frame, *d* is – in the *loss* frame).

## B.1 Calculation of $\sigma^d$ and $\gamma^d$

The following calculations are valid for Series 1 and 2.

### B.1.1 Gain frame

Let $A(x_A, p_A; y_A)$ be the LHS lottery and $B^l(x_{B_l}, p_B; y_B)$ the RHS lottery of row $l$ ($x_A, y_A, x_{B_l}$ and $y_B$ are strictly positive $\forall l$, and $0 < p_A < 1$ and $0 < p_B < 1$). The lottery structure is such that only $x_{B_l}$ varies over rows $l$, while the other lottery attributes remain similar. For a given subject, switching at row $s$ means the following inequalities in terms of prospect value:

$$ \left\{ \begin{array}{l} PV(A) > PV(B_{s-1}) \\ PV(A) < PV(B_s). \end{array} \right. \qquad (4) $$

From Equation (2) we know that the prospect value of any lottery $(x, p; y)$ is $v^d(y) + \omega^d(p)$. $(v^d(x) - v^d(y))$ when $xy \ge 0$ and $|x| > |y|$. Equations (1) and (3) give the functional forms for the value and probability weighting functions respectively. Thus, we obtain:

$$ \begin{array}{l} (4) \Leftrightarrow \left\{ \begin{array}{l} y_A^{\sigma^+} + \exp[-(-\ln p_A)^{\gamma^+}](x_A^{\sigma^+} - y_A^{\sigma^+}) > y_B^{\sigma^+} + \exp[-(-\ln p_B)^{\gamma^+}](x_{B_s-1}^{\sigma^+} - y_B^{\sigma^+}) \\ y_A^{\sigma^+} + \exp[-(-\ln p_A)^{\gamma^+}](x_A^{\sigma^+} - y_A^{\sigma^+}) < y_B^{\sigma^+} + \exp[-(-\ln p_B)^{\gamma^+}](x_{B_s}^{\sigma^+} - y_B^{\sigma^+}) \end{array} \right. \\ \\ \Leftrightarrow y_B^{\sigma^+} + \exp[-(-\ln p_B)^{\gamma^+}](x_{B_s-1}^{\sigma^+} - y_B^{\sigma^+}) < y_A^{\sigma^+} + \exp[-(-\ln p_A)^{\gamma^+}](x_A^{\sigma^+} - y_A^{\sigma^+}) \\ < y_B^{\sigma^+} + \exp[-(-\ln p_B)^{\gamma^+}](x_{B_s}^{\sigma^+} - y_B^{\sigma^+}). \end{array} \qquad (5) $$
### B.1.2 Loss frame

Let $C^l(x_{C_l}, p_C; y_C)$ be the LHS lottery of row $l$ and $D(x_D, p_D; y_D)$ the RHS lottery $(x_{C_l}, y_C, x_D$, and $y_D$ are strictly negative $\forall l$, and $0 < p_C < 1$ and $0 < p_D < 1$). The lottery structure is such that only $x_{C_l}$ varies over rows $l$. This time, switching at row $s$ means:

$$ \left\{ \begin{array}{l} PV(C) > PV(D_{s-1}) \\ PV(C) < PV(D_s). \end{array} \right. \qquad (6) $$

As $xy \ge 0$ still, any lottery $(x, p; y)$ has the same prospect value than in the previous section, i.e., $v^d(y) + \omega^d(p) \cdot (v^d(x) - v^d(y))$ when $|x| > |y|$. Equation (1) gives the specific value function for the loss domain $v^d(x) = -\lambda(-x)^{\sigma^d} \forall x < 0$. Thus,

$$ (6) \Leftrightarrow \left\{ \begin{array}{l} -\lambda(-y_C)^{\sigma^{-}} + \exp[-(-\ln p_C)^{\gamma^{-}}](-\lambda)((-x_{C_{s-1}})^{\sigma^{-}} - (-y_C)^{\sigma^{-}}) \\ > -\lambda(-y_D)^{\sigma^{-}} + \exp[-(-\ln p_D)^{\gamma^{-}}](-\lambda)((-x_D)^{\sigma^{-}} - (-y_D)^{\sigma^{-}}) \\ -\lambda(-y_C)^{\sigma^{-}} + \exp[-(-\ln p_C)^{\gamma^{-}}](-\lambda)((-x_{C_s})^{\sigma^{-}}(-y_C)^{\sigma^{-}}) \\ < -\lambda(-y_D)^{\sigma^{-}} + \exp[-(-\ln p_D)^{\gamma^{-}}](-\lambda)((-x_D)^{\sigma^{-}} - (-y_D)^{\sigma^{-}}). \end{array} \right. $$

Simplifying by $-\lambda$, we obtain:

$$ (6) \Leftrightarrow \left\{ \begin{array}{l} (-y_C)^{\sigma^{-}} + \exp[-(-\ln p_C)^{\gamma^{-}}]((-x_{C_{s-1}})^{\sigma^{-}} - (-y_C)^{\sigma^{-}}) \\ < (-y_D)^{\sigma^{-}} + \exp[-(-\ln p_D)^{\gamma^{-}}]((-x_D)^{\sigma^{-}} - (-y_D)^{\sigma^{-}}) \\ (-y_C)^{\sigma^{-}} + \exp[-(-\ln p_C)^{\gamma^{-}}]((-x_{C_s})^{\sigma^{-}}(-y_C)^{\sigma^{-}}) \\ > (-y_D)^{\sigma^{-}} + \exp[-(-\ln p_D)^{\gamma^{-}}]((-x_D)^{\sigma^{-}} - (-y_D)^{\sigma^{-}}). \end{array} \right. $$

As $x_{C_l} = -x_{B_l}$, $y_C = -y_B$, $x_D = -x_A$, $y_D = -y_A$, and $p_D = p_A$, $p_C = p_B$, we further obtain:

$$ (6) \Leftrightarrow \left\{ \begin{array}{l} y_B^{\sigma^{-}} + \exp[-(-\ln p_B)^{\gamma^{-}}](x_{B_{s-1}}^{\sigma^{-}} - y_B^{\sigma^{-}}) < y_A^{\sigma^{-}} + \exp[-(-\ln p_A)^{\gamma^{-}}](x_A^{\sigma^{-}} - y_A^{\sigma^{-}}) \\ y_B^{\sigma^{-}} + \exp[-(-\ln p_B)^{\gamma^{-}}](x_{B_s}^{\sigma^{-}} - y_B^{\sigma^{-}}) > y_A^{\sigma^{-}} + \exp[-(-\ln p_A)^{\gamma^{-}}](x_A^{\sigma^{-}} - y_A^{\sigma^{-}}) \end{array} \right. $$

$$ \begin{aligned} y_B^{\sigma^{-}} + \exp[-(-\ln p_B)^{\gamma^{-}}](x_{B_{s-1}}^{\sigma^{-}} - y_B^{\sigma^{-}}) &< y_A^{\sigma^{-}} + \exp[-(-\ln p_A)^{\gamma^{-}}](x_A^{\sigma^{-}} - y_A^{\sigma^{-}}) \\ &< y_B^{\sigma^{-}} + \exp[-(-\ln p_B)^{\gamma^{-}}](x_{B_s}^{\sigma^{-}} - y_B^{\sigma^{-}}). \end{aligned} \qquad (7) $$
We can see that (7) is equivalent to (5), meaning that a similar couple of switching points in Series 1 and 2 ($s_1, s_2$) in the gain domain and in the loss domain leads to a similar couple of parameters($\sigma^d, \gamma^d$).

## B.2 Calculation of $\lambda$, $\lambda_{opos}$, and $\lambda_{gen}$ conditionally to $\sigma^d$

The following calculations are valid for Series 3 only, where lotteries mix positive and negative payoffs. We distinguish $\sigma^{d1}$ and $\sigma^{d2}$, depending on which type of payoff the parameter applies to: $\sigma^{d1}$ applies to gains while $\sigma^{d2}$ applies to losses.

### B.2.1 Gain frame

Let $A^l(x_{A_l}, p; y_{A_l})$ be the LHS lottery and $B^l(x_B, p; y_{B_l})$ the RHS lottery of row $l$ ($x_{A_l}$ and $x_B$ are strictly positive $\forall l$, while $y_{A_l}$ and $y_{B_l}$ are strictly negative $\forall l$, and $p = \frac{1}{2}$). The lottery structure is such that only $x_B$ does not vary over rows. A subject switching at row $s$ means:

$$ \left\{ \begin{array}{l} \text{PV}(A_{s-1}) > \text{PV}(B_{s-1}) \\ \text{PV}(A_s) < \text{PV}(B_s). \end{array} \right. \qquad (8) $$

From Equation (2), we know that the prospect value of any binary lottery $(x, p; y)$ is $\omega^d(p) \cdot v^d(x) + \omega^d(p)(1-p) \cdot v^d(y)$ when $xy \le 0$. As $p = \frac{1}{2}$ it simplifies to $\omega^d(\frac{1}{2}) \cdot (v^d(x) + v^d(y))$. We designate as $\lambda_{12}$ the loss aversion parameter, which is equivalent to $\lambda$, $\lambda_{opos}$ or $\lambda_{gen}$ depending on the hypotheses relative to $\sigma^{d1}$ and $\sigma^{d2}$. We obtain:

$$ \begin{align*} (8) \Leftrightarrow & \begin{cases} \exp[-(-\ln \frac{1}{2})^{\gamma^d}] [(x_{A_{s-1}})^{\sigma^{d1}} + (-\lambda_{12})(-y_{A_{s-1}})^{\sigma^{d2}}] \\ > \exp[-(-\ln \frac{1}{2})^{\gamma^d}] [(x_{B_{s-1}})^{\sigma^{d1}} + (-\lambda_{12})(-y_{B_{s-1}})^{\sigma^{d2}}] \\ \exp[-(-\ln \frac{1}{2})^{\gamma^d}] [(x_{A_s})^{\sigma^{d1}} + (-\lambda_{12})(-y_{A_s})^{\sigma^{d2}}] \\ < \exp[-(-\ln \frac{1}{2})^{\gamma^d}] [(x_{B_s})^{\sigma^{d1}} + (-\lambda_{12})(-y_{B_s})^{\sigma^{d2}}] \end{cases} \\ \Leftrightarrow & \begin{cases} (x_{A_{s-1}})^{\sigma^{d1}} - \lambda_{12}(-y_{A_{s-1}})^{\sigma^{d2}} > (x_{B_{s-1}})^{\sigma^{d1}} - \lambda_{12}(-y_{B_{s-1}})^{\sigma^{d2}} \\ (x_{A_s})^{\sigma^{d1}} - \lambda_{12}(-y_{A_s})^{\sigma^{d2}} < (x_{B_s})^{\sigma^{d1}} - \lambda_{12}(-y_{B_s})^{\sigma^{d2}} \end{cases} \\ \Leftrightarrow & \begin{cases} \lambda_{12} [(-y_{B_{s-1}})^{\sigma^{d2}} - (-y_{A_{s-1}})^{\sigma^{d2}}] > (x_{B_{s-1}})^{\sigma^{d1}} - (x_{A_{s-1}})^{\sigma^{d1}} \\ \lambda_{12} [(-y_{B_s})^{\sigma^{d2}} - (-y_{A_s})^{\sigma^{d2}}] < (x_{B_s})^{\sigma^{d1}} - (x_{A_s})^{\sigma^{d1}}. \end{cases} \end{align*} $$
Last, as $y_{A_l}$ and $y_{B_l}$ are negative payoffs such as $|y_{A_l}| < |y_{B_l}|$, $\sigma^{d1} > 0$, and $\sigma^{d2} > 0$, we can write:

$$ (8) \Leftrightarrow \left\{ \begin{array}{l} \lambda_{12} > \frac{(x_{B_{s-1}})^{\sigma^{d1}} - (x_{A_{s-1}})^{\sigma^{d1}}}{(-y_{B_{s-1}})^{\sigma^{d2}} - (-y_{A_{s-1}})^{\sigma^{d2}}} \\ \lambda_{12} < \frac{(x_{B_s})^{\sigma^{d1}} - (x_{A_s})^{\sigma^{d1}}}{(-y_{B_s})^{\sigma^{d2}} - (-y_{A_s})^{\sigma^{d2}}}. \end{array} \right. \qquad (9) $$

Inequations (9) define the bound values of $\lambda$ when $\sigma^{d1} = \sigma^{d2} = \sigma^+$, $\lambda_{oppos}$ when $\sigma^{d1} = \sigma^{d2} = \sigma^-$, and $\lambda_{gen}$ when $\sigma^{d1} = \sigma^+$ and $\sigma^{d2} = \sigma^-$.

### B.2.2 Loss frame

Let $C^l(x_C, p; y_{C_l})$ be the LHS lottery and $D^l(x_{D_l}, p; y_{D_l})$ the RHS lottery of row $l$ ($x_C$ and $x_{D_l}$ are strictly negative $\forall l$, while $y_{C_l}$ and $y_{D_l}$ are strictly positive $\forall l$, and $p = \frac{1}{2}$). The lottery structure is such that only $x_C$ does not vary over rows. A subject switching at row $s$ means:

$$ \left\{ \begin{array}{l} \mathrm{PV}(C_{s-1}) > \mathrm{PV}(D_{s-1}) \\ \mathrm{PV}(C_s) < \mathrm{PV}(D_s). \end{array} \right. \qquad (10) $$

As $xy \le 0$ still, any lottery $(x, p; y)$ has the same prospect value than in the previous section, i.e., $\omega^d(\frac{1}{2}) \cdot (v^d(x) + v^d(y))$. Equation (1) gives the specific value function for the loss domain $v^d(x) = -\lambda_{12}(-x)^{\sigma^d} \forall x < 0$. We obtain:

$$ \begin{align*} (10) \Leftrightarrow & \left\{ \begin{array}{l} \exp [-(\ln \frac{1}{2})\gamma] [(-\lambda)(-x_{C_{s-1}})^{\sigma^{d2}} + (y_{C_{s-1}})^{\sigma^{d1}}] \\ > \exp [-(\ln \frac{1}{2})\gamma] [(-\lambda_{12})(-x_{D_{s-1}})^{\sigma^{d2}} + (y_{D_{s-1}})^{\sigma^{d1}}] \\ \exp [-(\ln \frac{1}{2})\gamma] [(-\lambda)(-x_{C_s})^{\sigma^{d2}} + (y_{C_s})^{\sigma^{d1}}] \\ < \exp [-(\ln \frac{1}{2})\gamma] [(-\lambda_{12})(-x_{D_s})^{\sigma^{d2}} + (y_{D_s})^{\sigma^{d1}}] \end{array} \right\} \\ \Leftrightarrow & \left\{ \begin{array}{l} (y_{C_{s-1}})^{\sigma^{d1}} - \lambda_{12}(-x_{C_{s-1}})^{\sigma^{d2}} > (y_{D_{s-1}})^{\sigma^{d1}} - \lambda_{12}(-x_{D_{s-1}})^{\sigma^{d2}} \\ (y_{C_s})^{\sigma^{d1}} - \lambda_{12}(-x_{C_s})^{\sigma^{d2}} < (y_{D_s})^{\sigma^{d1}} - \lambda_{12}(-x_{D_s})^{\sigma^{d2}} \end{array} \right\} \\ \Leftrightarrow & \left\{ \begin{array}{l} \lambda [(-x_{D_{s-1}})^{\sigma^{d2}} - (-x_{C_{s-1}})^{\sigma^{d2}}] > (y_{D_{s-1}})^{\sigma^{d1}} - (y_{C_{s-1}})^{\sigma^{d1}} \\ \lambda [(-x_{D_s})^{\sigma^{d2}} - (-x_{C_s})^{\sigma^{d2}}] < (y_{D_s})^{\sigma^{d1}} - (y_{C_s})^{\sigma^{d1}}. \end{array} \right. \end{align*} $$
Last, as $x_C$ and $x_{D_t}$ are negative payoffs such as $|x_C| > |x_{D_t}|$, $\sigma^{d1} > 0$, and $\sigma^{d2} > 0$, we can write:

$$ (10) \Leftrightarrow \left\{ \begin{array}{l} \lambda_{12} < \frac{(y_{D_{s-1}})^{\sigma^{d1}} - (y_{C_{s-1}})^{\sigma^{d1}}}{(-x_{D_{s-1}})^{\sigma^{d2}} - (-x_{C_{s-1}})^{\sigma^{d2}}} \\ \lambda_{12} > \frac{(y_{D_s})^{\sigma^{d1}} - (y_{C_s})^{\sigma^{d1}}}{(-x_{D_s})^{\sigma^{d2}} - (-x_{C_s})^{\sigma^{d2}}}. \end{array} \right. \qquad (11) $$

As $x_C = -x_B$, $y_C = -y_B$, $x_D = -x_A$ and $y_D = -y_A$, then Equation (11) can be written as:

$$ \left\{ \begin{array}{l} \lambda_{12} < \frac{(-y_{A_{s-1}})^{\sigma^{d1}} - (-y_{B_{s-1}})^{\sigma^{d1}}}{(x_{A_{s-1}})^{\sigma^{d2}} - (x_{B_{s-1}})^{\sigma^{d2}}} \\ \lambda_{12} > \frac{(-y_{A_s})^{\sigma^{d1}} - (-y_{B_s})^{\sigma^{d1}}}{(x_{A_s})^{\sigma^{d2}} - (x_{B_s})^{\sigma^{d2}}} \end{array} \right. $$

$$ \Leftrightarrow \left\{ \begin{array}{l} \lambda_{12} < \frac{(-y_{B_{s-1}})^{\sigma^{d1}} - (-y_{A_{s-1}})^{\sigma^{d1}}}{(x_{B_{s-1}})^{\sigma^{d2}} - (x_{A_{s-1}})^{\sigma^{d2}}} \\ \lambda_{12} > \frac{(-y_{B_s})^{\sigma^{d1}} - (-y_{A_s})^{\sigma^{d1}}}{(x_{B_s})^{\sigma^{d2}} - (x_{A_s})^{\sigma^{d2}}}. \end{array} \right. \qquad (12) $$

Inequations (12) define the bound values of $\lambda$ when $\sigma^{d1} = \sigma^{d2} = \sigma^{-}$, $\lambda_{oppos}$ when $\sigma^{d1} = \sigma^{d2} = \sigma^{+}$, and $\lambda_{gen}$ when $\sigma^{d1} = \sigma^{+}$ and $\sigma^{d2} = \sigma^{-}$.

As a last remark, in the special case when $\sigma^{d1} = \sigma^{d2}$, note that the bound values of the loss aversion parameter in the *gain* and in the *loss* frames are linked. Recall that, in the *gain* frame, the bounds of the loss aversion parameter are (Equation (9)):

$$ \left\{ \begin{array}{l} \lambda_{12,min}^{G} = \frac{(x_{B_{s-1}})^{\sigma^{d1}} - (x_{A_{s-1}})^{\sigma^{d1}}}{(-y_{B_{s-1}})^{\sigma^{d2}} - (-y_{A_{s-1}})^{\sigma^{d2}}} \\ \lambda_{12,max}^{G} = \frac{(x_{B_s})^{\sigma^{d1}} - (x_{A_s})^{\sigma^{d1}}}{(-y_{B_s})^{\sigma^{d2}} - (-y_{A_s})^{\sigma^{d2}}}. \end{array} \right. $$

Comparing with Equation (12) provides the following equivalencies for $\lambda$ bounds, which also apply to $\lambda_{oppos}$:

$$ \left\{ \begin{array}{l} \lambda_{max}^{L} = 1/\lambda_{min}^{G} \\ \lambda_{min}^{L} = 1/\lambda_{max}^{G}. \end{array} \right. $$
C Raw results

<table><caption>Table 14: Distribution of switching points for treatment GLo</caption><thead><tr><th rowspan="2">Row</th><th colspan="3">Percentage of respondents</th></tr><tr><th>Series 1</th><th>Series 2</th><th>Series 3</th></tr></thead><tbody><tr><td>1</td><td>16.8</td><td>20.9</td><td>8.4</td></tr><tr><td>2</td><td>0.0</td><td>1.6</td><td>23.6</td></tr><tr><td>3</td><td>2.6</td><td>1.6</td><td>22.0</td></tr><tr><td>4</td><td>3.1</td><td>1.6</td><td>15.2</td></tr><tr><td>5</td><td>8.9</td><td>5.2</td><td>16.2</td></tr><tr><td>6</td><td>8.4</td><td>1.6</td><td>6.8</td></tr><tr><td>7</td><td>14.1</td><td>6.3</td><td>0.5</td></tr><tr><td>8</td><td>8.4</td><td>9.4</td><td></td></tr><tr><td>9</td><td>10.5</td><td>3.7</td><td></td></tr><tr><td>10</td><td>8.9</td><td>8.9</td><td></td></tr><tr><td>11</td><td>6.8</td><td>9.4</td><td></td></tr><tr><td>12</td><td>4.2</td><td>10.5</td><td></td></tr><tr><td>13</td><td>3.1</td><td>2.1</td><td></td></tr><tr><td>14</td><td>2.6</td><td>4.7</td><td></td></tr><tr><td>never</td><td>1.6</td><td>12.6</td><td>7.3</td></tr></tbody><tfoot><tr><td>Total</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td><td>191</td></tr></tfoot></table>
<table><caption>Table 15: Distribution of switching points for treatment GHi</caption><thead><tr><th rowspan="2">Row</th><th colspan="3">Percentage of respondents</th></tr><tr><th>Series 1</th><th>Series 2</th><th>Series 3</th></tr></thead><tbody><tr><td>1</td><td>20.4</td><td>25.1</td><td>6.3</td></tr><tr><td>2</td><td>0.0</td><td>2.1</td><td>20.9</td></tr><tr><td>3</td><td>2.6</td><td>2.6</td><td>14.7</td></tr><tr><td>4</td><td>5.8</td><td>2.6</td><td>16.2</td></tr><tr><td>5</td><td>6.8</td><td>5.2</td><td>26.7</td></tr><tr><td>6</td><td>4.7</td><td>3.7</td><td>4.2</td></tr><tr><td>7</td><td>14.1</td><td>5.2</td><td>4.2</td></tr><tr><td>8</td><td>11.5</td><td>4.2</td><td></td></tr><tr><td>9</td><td>5.8</td><td>4.2</td><td></td></tr><tr><td>10</td><td>9.4</td><td>6.3</td><td></td></tr><tr><td>11</td><td>5.8</td><td>7.9</td><td></td></tr><tr><td>12</td><td>7.9</td><td>9.9</td><td></td></tr><tr><td>13</td><td>2.6</td><td>3.7</td><td></td></tr><tr><td>14</td><td>1.0</td><td>2.1</td><td></td></tr><tr><td>never</td><td>1.6</td><td>15.2</td><td>6.8</td></tr></tbody><tfoot><tr><td>Total</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td><td>191</td></tr></tfoot></table>

<table><caption>Table 16: Distribution of switching points for treatment LLo</caption><thead><tr><th rowspan="2">Row</th><th colspan="3">Percentage of respondents</th></tr><tr><th>Series 1</th><th>Series 2</th><th>Series 3</th></tr></thead><tbody><tr><td>1</td><td>23.0</td><td>18.8</td><td>5.8</td></tr><tr><td>2</td><td>1.6</td><td>2.1</td><td>29.8</td></tr><tr><td>3</td><td>5.2</td><td>6.3</td><td>21.5</td></tr><tr><td>4</td><td>7.3</td><td>5.2</td><td>21.5</td></tr><tr><td>5</td><td>8.9</td><td>9.9</td><td>9.9</td></tr><tr><td>6</td><td>5.8</td><td>7.3</td><td>3.1</td></tr><tr><td>7</td><td>12.0</td><td>9.4</td><td>1.0</td></tr><tr><td>8</td><td>8.4</td><td>7.9</td><td></td></tr><tr><td>9</td><td>6.8</td><td>5.8</td><td></td></tr><tr><td>10</td><td>4.7</td><td>7.3</td><td></td></tr><tr><td>11</td><td>4.2</td><td>3.1</td><td></td></tr><tr><td>12</td><td>3.7</td><td>5.2</td><td></td></tr><tr><td>13</td><td>1.0</td><td>1.0</td><td></td></tr><tr><td>14</td><td>1.6</td><td>0.5</td><td></td></tr><tr><td>never</td><td>5.8</td><td>9.9</td><td>7.3</td></tr></tbody><tfoot><tr><td>Total</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td><td>191</td></tr></tfoot></table>
<table><caption>Table 17: Distribution of switching points for treatment LHi</caption><thead><tr><th rowspan="2">Row</th><th colspan="3">Percentage of respondents</th></tr><tr><th>Series 1</th><th>Series 2</th><th>Series 3</th></tr></thead><tbody><tr><td>1</td><td>33.0</td><td>25.1</td><td>4.2</td></tr><tr><td>2</td><td>2.6</td><td>1.0</td><td>26.7</td></tr><tr><td>3</td><td>4.7</td><td>4.7</td><td>13.1</td></tr><tr><td>4</td><td>4.7</td><td>7.3</td><td>24.1</td></tr><tr><td>5</td><td>6.3</td><td>8.4</td><td>19.9</td></tr><tr><td>6</td><td>5.2</td><td>5.2</td><td>2.6</td></tr><tr><td>7</td><td>14.7</td><td>10.5</td><td>0.5</td></tr><tr><td>8</td><td>8.4</td><td>6.3</td><td></td></tr><tr><td>9</td><td>4.2</td><td>8.9</td><td></td></tr><tr><td>10</td><td>6.3</td><td>5.8</td><td></td></tr><tr><td>11</td><td>2.6</td><td>3.1</td><td></td></tr><tr><td>12</td><td>1.6</td><td>3.7</td><td></td></tr><tr><td>13</td><td>0.5</td><td>1.0</td><td></td></tr><tr><td>14</td><td>0.5</td><td>0.0</td><td></td></tr><tr><td>never</td><td>4.7</td><td>8.9</td><td>8.9</td></tr><tr><td>Total</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Number of observations</td><td>191</td><td>191</td><td>191</td></tr></tbody></table>
D Comparison of PT parameter values from various studies

<figure><img src="image_4.png" /><figcaption>Figure 1: Mean parameter values from studies using the TCN methodology.</figcaption></figure>
E Alternative regressions

<table><caption>Table 18: Interval regression of $\sigma^d$ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>Std. Err.</th><th>(2)<br>Coef.</th><th>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>0.649***</td><td>(0.027)</td><td>0.626***</td><td>(0.146)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Loss</td><td>0.120***</td><td>(0.034)</td><td>0.120***</td><td>(0.034)</td></tr><tr><td>High</td><td>0.041</td><td>(0.027)</td><td>0.041</td><td>(0.027)</td></tr><tr><td>Loss × High</td><td>0.040</td><td>(0.039)</td><td>0.040</td><td>(0.039)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Age</td><td></td><td></td><td>0.002</td><td>(0.006)</td></tr><tr><td>Female</td><td></td><td></td><td>-0.054</td><td>(0.037)</td></tr><tr><td>Master</td><td></td><td></td><td>0.110**</td><td>(0.049)</td></tr><tr><td>Economics</td><td></td><td></td><td>-0.027</td><td>(0.037)</td></tr><tr><td>Constant</td><td>-0.973***</td><td>(0.038)</td><td>-0.986***</td><td>(0.039)</td></tr><tr><td>Nb. of obs. / clusters</td><td>764/191</td><td></td><td>764/191</td><td></td></tr><tr><td>Estimated standard error of the model</td><td>0.378</td><td></td><td>0.373</td><td></td></tr><tr><td>AIC</td><td>4035.529</td><td></td><td>4025.424</td><td></td></tr><tr><td>BIC</td><td>4058.722</td><td></td><td>4067.171</td><td></td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

The estimated standard error of the model is comparable to the root mean squared error that would be obtained in an OLS regression.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.
<table><caption>Table 19: Interval regression of $\gamma^d$ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>(2)<br>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>0.634***</td><td>(0.021)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td></tr><tr><td>Loss</td><td>0.030</td><td>(0.025)</td></tr><tr><td>High</td><td>0.008</td><td>(0.023)</td></tr><tr><td>Loss×High</td><td>-0.033</td><td>(0.032)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td></tr><tr><td>Age</td><td></td><td>-0.006</td></tr><tr><td>Female</td><td></td><td>-0.026</td></tr><tr><td>Master</td><td></td><td>0.068*</td></tr><tr><td>Economics</td><td></td><td>0.010</td></tr><tr><td>Constant</td><td>-1.241***</td><td>(0.032)</td></tr><tr><td>Nb. of obs. /clusters</td><td>764/191</td><td>764/191</td></tr><tr><td>Estimated standard error of the model</td><td>0.289</td><td>0.288</td></tr><tr><td>AIC</td><td>3752.779</td><td>3753.742</td></tr><tr><td>BIC</td><td>3775.971</td><td>3795.489</td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

The estimated standard error of the model is comparable to the root mean squared error that would be obtained in an OLS regression.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.

<table><caption>Table 20: Interval regression of $\lambda$ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>(2)<br>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>2.396***</td><td>(0.197)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td></tr><tr><td>Loss</td><td>-1.323***</td><td>(0.228)</td></tr><tr><td>High</td><td>0.289</td><td>(0.241)</td></tr><tr><td>Loss×High</td><td>-0.445</td><td>(0.280)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td></tr><tr><td>Age</td><td></td><td>-0.003</td></tr><tr><td>Female</td><td></td><td>0.292*</td></tr><tr><td>Master</td><td></td><td>0.398</td></tr><tr><td>Economics</td><td></td><td>-0.049</td></tr><tr><td>Constant</td><td>0.757***</td><td>(0.077)</td></tr><tr><td>Nb. of obs. /clusters</td><td>764/191</td><td>764/191</td></tr><tr><td>Estimated standard error of the model</td><td>2.132</td><td>2.116</td></tr><tr><td>AIC</td><td>3969.289</td><td>3968.305</td></tr><tr><td>BIC</td><td>3992.481</td><td>4010.052</td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

The estimated standard error of the model is comparable to the root mean squared error that would be obtained in an OLS regression.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.
<table><caption>Table 21: Log-interval regression of λ on frame characteristics and socio-demographics</caption><thead><tr><th>Covariate</th><th>(1)<br>Coef.</th><th>Std. Err.</th><th>(2)<br>Coef.</th><th>Std. Err.</th></tr></thead><tbody><tr><td>Constant</td><td>0.459***</td><td>(0.084)</td><td>0.580</td><td>(0.383)</td></tr><tr><td><em>Frames:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Loss</td><td>-0.900***</td><td>(0.122)</td><td>-0.900***</td><td>(0.122)</td></tr><tr><td>High</td><td>0.170*</td><td>(0.095)</td><td>0.169*</td><td>(0.095)</td></tr><tr><td>Loss×High</td><td>-0.329**</td><td>(0.141)</td><td>-0.329**</td><td>(0.141)</td></tr><tr><td><em>Socio-demographics:</em></td><td></td><td></td><td></td><td></td></tr><tr><td>Age</td><td></td><td></td><td>-0.011</td><td>(0.017)</td></tr><tr><td>Female</td><td></td><td></td><td>0.092</td><td>(0.081)</td></tr><tr><td>Master</td><td></td><td></td><td>0.191</td><td>(0.127)</td></tr><tr><td>Economics</td><td></td><td></td><td>0.069</td><td>(0.085)</td></tr><tr><td>Constant</td><td>0.039</td><td>(0.062)</td><td>0.036</td><td>(0.061)</td></tr><tr><td>Nb. of obs. /clusters</td><td>764/191</td><td></td><td>764/191</td><td></td></tr><tr><td>Estimated standard error of the model</td><td>1.040</td><td></td><td>1.036</td><td></td></tr><tr><td>AIC</td><td>3204.499</td><td></td><td>3207.211</td><td></td></tr><tr><td>BIC</td><td>3227.692</td><td></td><td>3248.958</td><td></td></tr></tbody></table>

Standard errors are clustered at the individual level.

\*, \*\* and \*\*\* stand for significance at the 10, 5 and 1% level respectively.

The estimated standard error of the model is comparable to the root mean squared error that would be obtained in an OLS regression.

AIC and BIC stand for Akaike and Schwarz's Bayesian information criteria respectively.
F Distribution of individual PT parameter values

<figure><img src="image_5.png" /><figcaption>Figure 2: Distribution of individual PT parameters for baseline treatment GLo.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 3: Distribution of individual $\sigma^d$ parameters over treatments.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 4: Distribution of individual $\gamma^d$ parameters over treatments.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5: Distribution of individual λ parameters over treatments.</figcaption></figure>
G Distribution of selected individual loss-aversion values

<figure><img src="image_9.png" /><figcaption>Figure 6: Distribution of individual $\lambda$ values in baseline GLo and $\lambda_{oppos}$ values in treatment LLo.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7: Distribution of individual λ values in treatment GHi and λ<sub>oppos</sub> values in treatment LHi.</figcaption></figure>
<figure><img src="image_11.png" /></figure>

# Supplementary material

## A Experimental instructions - French version

L'expérience à laquelle vous allez participer est destinée à l'étude de la prise de décision individuelle en situation risquée. Vous allez être confronté à 12 séries de choix entre deux options A et B. Les gains/pertes associés à ces options seront exprimés en une monnaie virtuelle appelée ECU (Experimental Currency Unit).

A la fin de l'expérience, vous seront posées des questions concernant vos caractéristiques personnelles (âge, sexe, études, etc).

Prenons deux exemples afin de vous familiariser avec les choix à effectuer. Voici deux séries ressemblant à celles auxquelles vous allez être confronté.

### SERIE 1

<table><thead><tr><th rowspan="3">Choix</th><th colspan="2">option A</th><th colspan="2">option B</th></tr><tr><th colspan="2">composition de l'urne</th><th colspan="2">composition de l'urne</th></tr><tr><th>3 boules jaunes</th><th>7 boules bleues</th><th>1 boule jaune</th><th>9 boules bleues</th></tr><tr><th>si jaune</th><th>si bleu</th><th>si jaune</th><th>si bleu</th></tr></thead><tbody><tr><td>1</td><td>20</td><td>5</td><td>34</td><td>2,5</td></tr><tr><td>2</td><td>20</td><td>5</td><td>37,5</td><td>2,5</td></tr><tr><td>3</td><td>20</td><td>5</td><td>41,5</td><td>2,5</td></tr><tr><td>4</td><td>20</td><td>5</td><td>46,5</td><td>2,5</td></tr><tr><td>5</td><td>20</td><td>5</td><td>53</td><td>2,5</td></tr><tr><td>6</td><td>20</td><td>5</td><td>62,5</td><td>2,5</td></tr><tr><td>7</td><td>20</td><td>5</td><td>75</td><td>2,5</td></tr><tr><td>8</td><td>20</td><td>5</td><td>92,5</td><td>2,5</td></tr><tr><td>9</td><td>20</td><td>5</td><td>110</td><td>2,5</td></tr><tr><td>10</td><td>20</td><td>5</td><td>150</td><td>2,5</td></tr><tr><td>11</td><td>20</td><td>5</td><td>200</td><td>2,5</td></tr><tr><td>12</td><td>20</td><td>5</td><td>300</td><td>2,5</td></tr><tr><td>13</td><td>20</td><td>5</td><td>500</td><td>2,5</td></tr><tr><td>14</td><td>20</td><td>5</td><td>850</td><td>2,5</td></tr></tbody></table>

Cette Série 1 contient 14 choix à effectuer entre deux options, l'option A et l'option B.

### Détaillons le choix 1:

**option A:** Considérons une urne composée de 10 boules dont 3 sont jaunes et 7 sont bleues. Si une boule jaune était tirée au sort alors vous obtiendriez 20 ECUS et si la boule tirée au sort était bleue alors vous obtiendriez 5 ECUS.
**option B:** Considérons une urne composée de 10 boules dont 1 est jaune et 9 sont bleues. Si une boule jaune était tirée au sort alors vous obtiendriez 34 ECUS et si la boule tirée au sort était bleue alors vous obtiendriez 2,5 ECUS.

La question qui vous sera posée concernant cette série est la suivante:

- Je choisis l'option A pour les choix 1 à ___.

- Je choisis l'option B pour les choix ___ à 14.

Cela revient à choisir entre l'option A et l'option B pour chacun des 14 choix. Supposons que vous choisissiez l'option A pour les choix 1 à 3, et l'option B pour les choix 4 à 14, devrait alors apparaître respectivement dans les cases ci-dessus les chiffres 3 et 4. Il est également possible de choisir l'option A pour tous les choix. Il faut dans ce cas saisir dans la première case le chiffre 14 et X dans la deuxième case. Il est également possible de choisir l'option B pour tous les choix. Il faut dans ce cas saisir dans la première case la lettre X et 1 dans la deuxième case.

## SERIE 2

<table><thead><tr><th rowspan="3">Choix</th><th colspan="2">option A</th><th colspan="2">option B</th></tr><tr><th colspan="2">composition de l'urne</th><th colspan="2">composition de l'urne</th></tr><tr><th>5 boules jaunes</th><th>5 boules bleues</th><th>5 boules jaunes</th><th>5 boules bleues</th></tr><tr><th></th><th>si jaune</th><th>si bleu</th><th>si jaune</th><th>si bleu</th></tr></thead><tbody><tr><td>1</td><td>12,5</td><td>-2</td><td>15</td><td>-10,5</td></tr><tr><td>2</td><td>2</td><td>-2</td><td>15</td><td>-10,5</td></tr><tr><td>3</td><td>0,5</td><td>-2</td><td>15</td><td>-10,5</td></tr><tr><td>4</td><td>0,5</td><td>-2</td><td>15</td><td>-8</td></tr><tr><td>5</td><td>0,5</td><td>-4</td><td>15</td><td>-8</td></tr><tr><td>6</td><td>0,5</td><td>-4</td><td>15</td><td>-7</td></tr><tr><td>7</td><td>0,5</td><td>-4</td><td>15</td><td>-5,5</td></tr></tbody></table>

Cette Série 2 contient 7 choix à effectuer entre deux options, l'option A et l'option B.

### Détaillons le choix 1:

**option A:** Considérons une urne composée de 10 boules dont 5 sont jaunes et 5 sont bleues. Si une boule jaune était tirée au sort alors vous obtiendriez 12,5 ECUS et si la boule tirée au sort était bleue alors vous perdriez 2 ECUS.

**option B:** Considérons une urne composée de 10 boules dont 5 sont jaunes et 5 sont bleues. Si une boule jaune était tirée au sort alors vous obtiendriez 15 ECUS et si la boule tirée au sort était bleue
alors vous perdriez 10,5 ECUS.

La question qui vous sera posée concernant cette série est la suivante:

- Je choisis l'option A pour les choix 1 à __.

- Je choisis l'option B pour les choix __ à 7.

Cela revient à choisir entre l'option A et l'option B pour chacun des choix. Supposons que vous choisissiez l'option A pour les choix 1 à 3, et l'option B pour les choix 4 à 7, devrait alors apparaître respectivement dans les cases ci-dessus les chiffres 3 et 4. Il est également possible de choisir l'option A pour tous les choix. Il faut dans ce cas saisir dans la première case le chiffre 7 et X dans la deuxième case. Il est également possible de choisir l'option B pour tous les choix. Il faut dans ce cas saisir dans la première case la lettre X et 1 dans la deuxième case.

Deux séries de choix viennent de vous être présentées en exemple. Au total, vous serez confrontés à 12 séries différentes.

Vous devez prendre le temps nécessaire pour choisir les réponses qui correspondent réellement à vos préférences. Il n'y a pas de bonne ou de mauvaise réponse, juste des comportements différents à observer. Vous n'avez pas non plus de contrainte horaire, vous disposez de tout le temps dont vous avez besoin. A la fin de l'expérience, deux de vos choix seront tirés au hasard par l'ordinateur afin de déterminer votre rémunération pour l'expérience. Cette rémunération correspondra à une somme d'argent. Les ECUS (Experimental Currency Units) seront convertis au taux de 200 ECUS = 1 €. Comme certains choix peuvent se traduire par des pertes, nous vous donnons dès à présent une somme fixe de 17 €. A cette somme se rajoutera: i) le gain ou la perte que vous réaliserez durant l'expérience; ii) un forfait de 5 € pour compenser vos frais de déplacement et le temps passé. Nous vous demanderons de signer un reçu pour la somme finale obtenue. Le protocole de paiement a été construit de telle manière que vous obteniez un montant final positif quels que soient vos choix.

Pour les besoins de l'expérience, vous devez impérativement répondre à toutes les questions. Vos réponses seront enregistrées par le réseau informatique et traitées de façon anonyme. La confidentialité des informations contenues dans ce questionnaire est assurée par l'anonymat du répondant. Vos réponses resteront donc tout à fait confidentielles. Les résultats seront présentés
sous forme synthétique dans des publications scientifiques en respectant scrupuleusement l'anonymat des réponses. L'absence de communication entre les participants est un gage de réussite. Nous vous demandons donc de ne pas discuter avec les autres participants.

Au cours de l'expérience n'hésitez pas à poser des questions aux organisateurs. Ils sont à votre disposition.

## B Experimental instructions - English version

The experiment you are going to participate is focused on individual decision making in risky situations. You will be given 12 series of choices between two options A and B. The gains/losses associated with these options will be expressed in a virtual currency called ECU (Experimental Currency Unit).

At the end of the experiment you will be asked questions about your personal characteristics (age, gender, education, etc.).

Let's take an example to become familiar with the choices you are about to make. Here are two series which look like the ones you will be facing.

### SERIES 1

<table><thead><tr><th rowspan="3">Choice</th><th colspan="2">option A</th><th colspan="2">option B</th></tr><tr><th colspan="2">urn content</th><th colspan="2">urn content</th></tr><tr><th>3 yellow balls</th><th>7 blue balls</th><th>1 yellow ball</th><th>9 blue balls</th></tr><tr><th></th><th>if yellow</th><th>if blue</th><th>if yellow</th><th>if blue</th></tr></thead><tbody><tr><td>1</td><td>20</td><td>5</td><td>34</td><td>2.5</td></tr><tr><td>2</td><td>20</td><td>5</td><td>37.5</td><td>2.5</td></tr><tr><td>3</td><td>20</td><td>5</td><td>41.5</td><td>2.5</td></tr><tr><td>4</td><td>20</td><td>5</td><td>46.5</td><td>2.5</td></tr><tr><td>5</td><td>20</td><td>5</td><td>53</td><td>2.5</td></tr><tr><td>6</td><td>20</td><td>5</td><td>62.5</td><td>2.5</td></tr><tr><td>7</td><td>20</td><td>5</td><td>75</td><td>2.5</td></tr><tr><td>8</td><td>20</td><td>5</td><td>92.5</td><td>2.5</td></tr><tr><td>9</td><td>20</td><td>5</td><td>110</td><td>2.5</td></tr><tr><td>10</td><td>20</td><td>5</td><td>150</td><td>2.5</td></tr><tr><td>11</td><td>20</td><td>5</td><td>200</td><td>2.5</td></tr><tr><td>12</td><td>20</td><td>5</td><td>300</td><td>2.5</td></tr><tr><td>13</td><td>20</td><td>5</td><td>500</td><td>2.5</td></tr><tr><td>14</td><td>20</td><td>5</td><td>850</td><td>2.5</td></tr></tbody></table>

Series 1 features 14 choices between two options, option A and option B.
Let's discuss Choice 1:

**option A:** Consider an urn that contains 10 balls, of which 3 are yellow and 7 are blue. If a yellow ball was drawn, then you would get ECU 20, and if it was a blue ball, then you would get ECU 5.

**option B:** Consider an urn that contains 10 balls, of which 1 is yellow and 9 are blue. If a yellow ball was drawn, then you would get ECU 34, and if it was a blue ball, then you would get ECU 2.5.

The question you will be asked about this series is:

- I choose option A for choices 1 to __.

- I choose option B for choices __ to 14.

It is equivalent to choosing between option A and option B for each of the 14 choices. Suppose you choose option A for rows 1 to 3, and option B for rows 4 to 14, then you must fill in the blanks above numbers 3 and 4 respectively. It is also possible to choose option A for all rows. In this case, fill the first blank with number 14 and the second one with letter X. It is also possible to choose option B for all rows. In this case, fill the first blank with letter X and the second with number 1.

SERIES 2

<table><thead><tr><th rowspan="3">Choice</th><th colspan="2">option A</th><th colspan="2">option B</th></tr><tr><th colspan="2">urn content</th><th colspan="2">urn content</th></tr><tr><th>5 yellow balls</th><th>5 blue balls</th><th>5 yellow balls</th><th>5 blue balls</th></tr><tr><th></th><th>if yellow</th><th>if blue</th><th>if yellow</th><th>if blue</th></tr></thead><tbody><tr><td>1</td><td>12.5</td><td>-2</td><td>15</td><td>-10.5</td></tr><tr><td>2</td><td>2</td><td>-2</td><td>15</td><td>-10.5</td></tr><tr><td>3</td><td>0.5</td><td>-2</td><td>15</td><td>-10.5</td></tr><tr><td>4</td><td>0.5</td><td>-2</td><td>15</td><td>-8</td></tr><tr><td>5</td><td>0.5</td><td>-4</td><td>15</td><td>-8</td></tr><tr><td>6</td><td>0.5</td><td>-4</td><td>15</td><td>-7</td></tr><tr><td>7</td><td>0.5</td><td>-4</td><td>15</td><td>-5.5</td></tr></tbody></table>

Series 2 features 7 choices between two options, option A and option B.

Let's discuss Choice 1:

**option A:** Consider an urn that contains 10 balls, of which 5 are yellow and 5 are blue. If a yellow ball was drawn, then you would get ECU 12.5, and if it was a blue ball, then you would lose ECU 2.

**option B:** Consider an urn that contains 10 balls, of which 5 are yellow and 5 are blue. If a yellow ball was drawn, then you would get ECU 15, and if it was a blue ball, then you would lose ECU 10.5.
The question you will be asked about this series is:

- I choose option A for choices 1 to ___.

- I choose option B for choices ___ to 7.

This is equivalent to choosing between option A and option B for each of the 7 choices. Suppose you choose option A for rows 1 to 3, and option B for rows 4 to 7, then you must fill in the blanks above numbers 3 and 4 respectively. It is also possible to choose option A for all rows. In this case, fill the first blank with number 7 and the second one with letter X. It is also possible to choose option B for all rows. In this case, fill the first blank with letter X and the second with number 1.

We have just presented two series of choices as examples. In total, you will be confronted with 12 different series.

Take the time to choose the answers that really fit your preferences. There is no right or wrong answer, just different forms of behaviour that we will observe. There is no time constraint, you have all the time you need. At the end of the experiment, two of your choices will be randomly drawn by the computer to determine your payment for participating in the experiment. This remuneration will correspond to a sum of money. The ECUs (Experimental Currency Units) will be converted at the rate of ECU 200 = €1. As some choices may result in losses, we are now giving you an initial endowment of €17. We will add to this amount: i) the gain or loss that you will realize during the experiment; ii) a show-up fee of €5 to compensate for travel expenses and time spent. We will ask you to sign a receipt for the final amount obtained. We built the payment protocol in such a way that you get a positive final amount regardless of your choices.

To meet the objectives of the experiment, you must answer all the questions. Your answers will be recorded by the computer network and processed anonymously. The confidentiality of the information collected is ensured by the anonymity of the respondent. Your answers will therefore remain completely confidential. The results will be presented in a synthetic format in scientific publications and will respect the anonymity of the respondents. The absence of communication between participants is a guarantee of success. We ask you not to talk with other participants.

During the experience do not hesitate to ask questions to the organizers. They are at your disposal.
C A few screen shots

Série 1

<table><thead><tr><th rowspan="3">Choix</th><th colspan="2">Option A</th><th colspan="2">Option B</th></tr><tr><th colspan="2">Composition de l'urne :<br>3 boules ● et 7 boules ●</th><th colspan="2">Composition de l'urne :<br>1 boule ● et 9 boules ●</th></tr><tr><th>Si ●</th><th>Si ●</th><th>Si ●</th><th>Si ●</th></tr></thead><tbody><tr><td>1</td><td>40</td><td>10</td><td>68</td><td>5</td></tr><tr><td>2</td><td>40</td><td>10</td><td>75</td><td>5</td></tr><tr><td>3</td><td>40</td><td>10</td><td>83</td><td>5</td></tr><tr><td>4</td><td>40</td><td>10</td><td>93</td><td>5</td></tr><tr><td>5</td><td>40</td><td>10</td><td>106</td><td>5</td></tr><tr><td>6</td><td>40</td><td>10</td><td>125</td><td>5</td></tr><tr><td>7</td><td>40</td><td>10</td><td>150</td><td>5</td></tr><tr><td>8</td><td>40</td><td>10</td><td>185</td><td>5</td></tr><tr><td>9</td><td>40</td><td>10</td><td>220</td><td>5</td></tr><tr><td>10</td><td>40</td><td>10</td><td>300</td><td>5</td></tr><tr><td>11</td><td>40</td><td>10</td><td>400</td><td>5</td></tr><tr><td>12</td><td>40</td><td>10</td><td>600</td><td>5</td></tr><tr><td>13</td><td>40</td><td>10</td><td>1000</td><td>5</td></tr><tr><td>14</td><td>40</td><td>10</td><td>1700</td><td>5</td></tr></tbody></table>

- Je choisis l'option A pour les choix 1 à

- Je choisis l'option B pour les choix à 14.

Valider
Vous voulez valider vos choix pour cette séquence.

Veuillez confirmer votre décision.

Confirmer

Annuler

Série 2

<table><thead><tr><th rowspan="3">Choix</th><th colspan="2">Option A</th><th colspan="2">Option B</th></tr><tr><th colspan="2">Composition de l'urne :<br>9 boules ● et 1 boule ●</th><th colspan="2">Composition de l'urne :<br>7 boules ● et 3 boules ●</th></tr><tr><th>SI ●</th><th>SI ●</th><th>SI ●</th><th>SI ●</th></tr></thead><tbody><tr><td>15</td><td>40</td><td>30</td><td>54</td><td>5</td></tr><tr><td>16</td><td>40</td><td>30</td><td>56</td><td>5</td></tr><tr><td>17</td><td>40</td><td>30</td><td>58</td><td>5</td></tr><tr><td>18</td><td>40</td><td>30</td><td>60</td><td>5</td></tr><tr><td>19</td><td>40</td><td>30</td><td>62</td><td>5</td></tr><tr><td>20</td><td>40</td><td>30</td><td>65</td><td>5</td></tr><tr><td>21</td><td>40</td><td>30</td><td>68</td><td>5</td></tr><tr><td>22</td><td>40</td><td>30</td><td>72</td><td>5</td></tr><tr><td>23</td><td>40</td><td>30</td><td>77</td><td>5</td></tr><tr><td>24</td><td>40</td><td>30</td><td>83</td><td>5</td></tr><tr><td>25</td><td>40</td><td>30</td><td>90</td><td>5</td></tr><tr><td>26</td><td>40</td><td>30</td><td>100</td><td>5</td></tr><tr><td>27</td><td>40</td><td>30</td><td>110</td><td>5</td></tr><tr><td>28</td><td>40</td><td>30</td><td>130</td><td>5</td></tr></tbody></table>

- Je choisis l'option A pour les choix 15 à

16

- Je choisis l'option B pour les choix

17

à 28.

Valider
Série 3

<table><thead><tr><th rowspan="3">Choix</th><th colspan="2">Option A</th><th colspan="2">Option B</th></tr><tr><th colspan="2">Composition de l'urne :<br>5 boules &bull; et 5 boules &bull;</th><th colspan="2">Composition de l'urne :<br>5 boules &bull; et 5 boules &bull;</th></tr><tr><th>Si &bull;</th><th>Si &bull;</th><th>Si &bull;</th><th>Si &bull;</th></tr></thead><tbody><tr><td>29</td><td>25</td><td>-4</td><td>30</td><td>-21</td></tr><tr><td>30</td><td>4</td><td>-4</td><td>30</td><td>-21</td></tr><tr><td>31</td><td>1</td><td>-4</td><td>30</td><td>-21</td></tr><tr><td>32</td><td>1</td><td>-4</td><td>30</td><td>-16</td></tr><tr><td>33</td><td>1</td><td>-8</td><td>30</td><td>-16</td></tr><tr><td>34</td><td>1</td><td>-8</td><td>30</td><td>-14</td></tr><tr><td>35</td><td>1</td><td>-8</td><td>30</td><td>-11</td></tr></tbody></table>

- Je choisis l'option A pour les choix 29 à

- Je choisis l'option B pour les choix à 35.

Valider