## 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 (Am Econ Rev 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.

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## 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.

## Notes

Booij 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 & Rutström, 2009).

Non-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 & Vieider, 2019). They are also generally less efficient because more questions are needed (Abdellaoui et al., 2008). Last, they were 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.

The 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).Let 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 \(a \ne b\). 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 \(y>0\).

See Wakker et al. (2007, p. 224) for additional studies with unclear or balanced findings.

Appendix 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.Wilcoxon 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.Wilcoxon signed-rank tests for matched pairs give similar conclusions about the distributions of the total number of LHS choices.

In 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.

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## Acknowledgements

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.

## 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 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).

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All the authors contributed to the design of the experiment and preparation of the experimental material. Data collection was performed by JJ and MB. Data analysis was carried out by GB. The first draft of the manuscript was written by GB and all the authors commented on later versions of the manuscript. All the authors read and approved the final manuscript.

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## Appendices

### Appendix A: Lotteries of the LLo treatment

See Table 13.

### Appendix 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).

### 1.1 B.1 Calculation of \(\sigma ^{d}\) and \(\gamma ^{d}\)

The following calculations are valid for Series 1 and 2.

#### 1.1.1 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:

From Eq. (2), we know that the prospect value of any lottery (*x*, *p*; *y*) is \(v^d(y)+ \omega ^d(p) \cdot (v^d(x)-v^d(y))\) 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

#### 1.1.2 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

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,

Simplifying by \(-\lambda\), we obtain

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

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\((\sigma ^d,\gamma ^d)\).

### 1.2 B.2 Calculation of \(\lambda\), \(\lambda _\mathrm{oppos}\), and \(\lambda _\mathrm{gen}\) 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.

#### 1.2.1 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

From Eq. (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 \(\lambda\), \(\lambda _\mathrm{oppos}\) or \(\lambda _\mathrm{gen}\) depending on the hypotheses relative to \(\sigma ^{d1}\) and \(\sigma ^{d2}\). We obtain

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

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

#### 1.2.2 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

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

Last, as \(x_{C}\) and \(x_{D_{l}}\) are negative payoffs such as \(|x_{C}|>|x_{D_{l}}|\), \(\sigma ^{d1} >0\), and \(\sigma ^{d2} >0\), we can write:

As \(x_C = -x_B\) , \(y_C = -y_B\), \(x_D = -x_A\) and \(y_D = -y_A\), then Eq. (11) can be written as

Inequations (12) define the bound values of \(\lambda\) when \(\sigma ^{d1}=\sigma ^{d2}=\sigma ^{-}\), \(\lambda _\mathrm{oppos}\) when \(\sigma ^{d1}=\sigma ^{d2}=\sigma ^{+}\), and \(\lambda _\mathrm{gen}\) 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 (Eq. (9)):

Comparing with Eq. (12) provides the following equivalencies for \(\lambda\) bounds, which also apply to \(\lambda _\mathrm{oppos}\):

### Appendix C: Raw results

### Appendix D: Comparison of PT parameter values from various studies

See Fig. 1.

### Appendix E: Alternative regressions

### Appendix F: Distribution of individual PT parameter values

### Appendix G: Distribution of selected individual loss-aversion values

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Bocquého, G., Jacob, J. & Brunette, M. Prospect theory in multiple price list experiments: further insights on behaviour in the loss domain.
*Theory Decis* **94**, 593–636 (2023). https://doi.org/10.1007/s11238-022-09902-y

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DOI: https://doi.org/10.1007/s11238-022-09902-y