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# Paying none, some or all? Between-subject random incentives and preferences towards risk and time

Noémi Berlin, Emmanuel Kemel, Vincent Lenglin, Antoine Nebout

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Noémi Berlin, Emmanuel Kemel, Vincent Lenglin, Antoine Nebout. Paying none, some or all? Between-subject random incentives and preferences towards risk and time. Journal of Economic Psychology, 2025, 112, pp.102870. 10.1016/j.joep.2025.102870. hal-05403681

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# Paying none, some or all? Between-subject random incentives and preferences towards risk and time

Noémi Berlin<sup>a</sup>

Emmanuel Kemel<sup>b *</sup>

Vincent Lenglin<sup>c</sup>

Antoine Nebout<sup>d</sup>

<sup>a</sup> CNRS, Economix, University of Paris Nanterre. 200 Avenue de la République, 92000 Nanterre, France.
noemi.berlin@parisnanterre.fr

<sup>b</sup> CNRS & HEC Paris. 1 Rue de la Libération, 78350 Jouy-en-Josas, France. kemel@hec.fr

<sup>c</sup> ETHICS EA 7446, Catholic University of Lille. 60 Boulevard Vauban, 59800 Lille, France.
vincent.lenglin@univ-catholille.fr

<sup>d</sup> University Paris-Saclay, INRAE, AgroParisTech, Paris-Saclay Applied Economics. 22 Place de l'Agronomie, 91120 Palaiseau, France. antoine.nebout@inrae.fr

## Abstract

The within-subject random incentive system (Within-RIS) is the standard real-incentive mechanism in economic experiments: each subject has one choice randomly selected for payment. In a variation, Within-RIS is applied to a fraction of subjects only: each subject has a probability $\pi$ of being selected and has one choice randomly selected for payment. This paper assesses the validity of this mechanism called Hybrid random incentive system (Hybrid-RIS), by investigating if/how the selection probability affects elicited risk and time preferences. We report an experiment conducted with 335 subjects across four different treatments, varying the selection probability: 0, 0.1, 0.5, and 1. Using model-free statistics and structural estimations, we find that elicited time preferences remain stable, while risk aversion is lower when the selection probability is null. However, there are no significant differences among the non-null probabilities. These findings suggest that Hybrid-RIS is not only cost-effective, but also a valid mechanism to avoid hypothetical bias.

JEL classification: C11, C91, D81

Keywords: Incentives, risk and time preferences, experiment

The replication material is on https://doi.org/10.3886/E209071V1

*Corresponding author: kemel@hec.fr
# 1 Introduction

While incentives are essential in experimental economics to ensure that subjects reveal their true preferences, the way they are administered is equally crucial, as it can significantly affect the preferences elicited. In this regard, the *Within Random Incentive System* (Within-RIS) is the standard mechanism for implementing real incentives in experiments that elicit risk and time preferences. With this mechanism, each subject makes multiple choices, and only one is randomly selected and actually paid.<fnref n="1" /> However, as this approach can be costly given researchers’ budget constraints, an alternative is the Between-RIS, where each subject makes only one choice, and a selection probability $\pi$ determines the fraction of subjects whose decision is implemented and paid. This method mitigates experimental costs by a factor of $\pi < 1$, allowing for larger payoffs or increased sample sizes<fnref n="2" /> but also limits the amount of data collected per subject. Although Within- and Between-RIS have received theoretical (Azrieli et al., 2018) and empirical support (e.g., Starmer and Sugden, 1991; Cubitt et al., 1998; Baltussen et al., 2012), their respective limitations have led to the development of a third incentive mechanism: the Hybrid-RIS. With this mechanism, each subject makes multiple choices, and has a probability $\pi$ of being selected and having Within-RIS applied to her choices. Hybrid-RIS combines the advantages of Within-RIS and Between-RIS by preserving the ability to assess individual consistency across multiple choices while reducing experimental costs through the selection probability $\pi$. This study evaluates the validity of Hybrid-RIS by analyzing whether / how the selection probability $\pi$ affects elicited risk and time preferences.

Managing experimental costs is both a budgetary and scientific challenge. When real payments are involved, budget constraints limit the outcome range that can be studied. This is problematic since economic decisions often involve large stakes (e.g., retirement savings, stock market investments), and magnitude effects influence both risk preferences (Fehr-Duda et al., 2010; Kachelmeier and Shehata, 1992) and time preferences (Frederick et al., 2002). Behaviors observed with small experimental amounts may thus differ from real-world

<footnote n="1">¹One might wonder why paying for all choices would not be preferable. As noted by Charness et al. (2016), paying for every decision could create interactions between choices, which would manifest through portfolio effects (hedging) or wealth effects. Such interactions could distort individual risk-taking behavior, making it harder to isolate pure preference measurements. Within-RIS has been designed with the objective of avoiding such effects (e.g., Azrieli et al., 2018).</footnote>

<footnote n="2">²Subject compensation must still meet minimum payment requirements for legal compliance and subject-pool maintenance, which may require adjusting the show-up fee accordingly.</footnote>
decisions. Large payoffs are also needed for studying time preferences over long horizons (Kemel and Paraschiv, 2023) or capturing utility curvature, which is expected to be linear for small stakes (Rabin, 2000). Hybrid-RIS is also useful when the incentive procedure is cumbersome, such as when researchers investigate non-monetary outcomes. For example, Abdellaoui et al. (2018) designed an experiment where subjects could be offered a real job contract involving a fixed salary, but whose number of working hours varied depending on the outcome of the experiment. Such a complex procedure can only be implemented for a subset of participants; hence, the use of Hybrid-RIS can be particularly useful in such contexts.

The main motivation for our investigation is therefore very concrete and addresses a question often raised in studies using a Hybrid-RIS design: “is $\pi$ high enough to motivate subjects?”. However, beyond this pragmatic concern, varying $\pi$ also provides an opportunity to investigate additional empirical questions. Specifically, the selection probability introduces a lottery that is compounded with the Within-RIS, raising the question of how subjects perceive this compound lottery. Do they treat each choice in isolation, ignoring the selection probability, or do they compound it with the selection probability? These two possibilities generate different behavioral implications. If subjects compound by $\pi$, two empirical patterns may be observed. First, lower values of $\pi$ may mitigate their motivation, leading to increased choice variability and less consistent decisions. We call this, the *dilution effect*. Second, compounding risk or time choices by $\pi$ may generate a *common ratio effect*. We elaborate on these hypotheses in Section 2.5, where we also describe our empirical strategy for testing them.

Three recent studies have investigated the impact of Hybrid-RIS on elicited risk or time preferences.<fnref n="3" /> Anderson et al. (2023) propose an ingenious within-subjects design that varies the selection probability ($\pi$ = 1 or $\pi$ = 1/8) and the magnitude of consequences. Using Within-RIS as a benchmark, the authors find that subjects exhibit lower risk aversion when $\pi$ = 1/8 compared to $\pi$ = 1, and an even larger reduction in risk aversion when payoffs are divided by 8 (and $\pi$ = 1). This suggests that changing the selection probability $\pi$ has less impact than changing the face value of outcomes. Aydogan et al. (2024) observe similar evaluations of five prospects involving risk or ambiguity in a between-subjects comparison

<footnote n="3">³See also Umer (2023) who reports a meta-analysis supporting the validity of Within-RIS and Hybrid-RIS for dictator games.</footnote>
of Within-RIS and Hybrid-RIS with $\pi = 1/10$. Brañas-Garza et al. (2023) compare different incentive mechanisms for elicited time preferences: hypothetical choices, Within-RIS, and Hybrid-RIS with $\pi = 1/10$. The authors do not observe any differences between these mechanisms and call for further investigation involving different levels of non-extreme payment probabilities: “However, further studies should analyze [Hybrid-RIS in more detail, for example, considering different payment probabilities (we only test for 1/10 probability)” (p. 415).

This study extends previous contributions by being the first to systematically test the impact of multiple, non-extreme values of $\pi$ on elicited risk and time preferences. We report an experiment involving 335 subjects, designed to elicit risk and time preferences across four treatments that manipulate the Hybrid-RIS selection probability $\pi$ using a between-subjects design: $\pi = 0, \pi = 0.1, \pi = 0.5$, and $\pi = 1$. When $\pi = 1$, the mechanism corresponds to Within-RIS, whereas when $\pi = 0$, all choices are hypothetical and not incentivized. For non-extreme values of $\pi$ (i.e., $0 < \pi < 1$), the treatments correspond to Hybrid-RIS. Preferences are elicited using a series of simple binary choices. For risk, we measure the certainty equivalents (the certain amount of money an individual considers equivalent to a risky prospect) of four binary lotteries; for time, we measure the sooner equivalents (which are sooner amounts that an individual would consider equivalent to a delayed payment) of four delayed payments. Both types of equivalents are elicited through an incentive-compatible bisection procedure (see Section 2). We analyze the data using both model-free and structural estimations of decision models. We consider Discounted Expected Utility (DEU), a model of rational decisions in risk and time, and a behavioral extension that accounts for nonlinear probability weighting and non-exponential discounting (Abdellaoui et al., 2019). A Bayesian approach is employed to draw conclusions from possible non-rejection of the null hypothesis.

We report two key patterns. First, incentives matter in the context of risk: subjects are less risk-averse in the hypothetical treatment ($\pi = 0$) compared to the real-incentive treatments ($\pi > 0$). However, the magnitude of the selection probability does not affect elicited risk preferences. Second, we find that elicited time preferences are unaffected by any level of $\pi$. In particular, no significant difference is observed between $\pi = 0$ (hypothetical choices) and $\pi = 1$ (Within-RIS). Overall, we do not provide evidence of either a dilution effect or a
common ratio effect induced by the use of Hybrid-RIS.

The remainder of the paper is organized as follows. Section 2 presents the experimental design and hypotheses. Section 3 describes the statistical analysis. Section 4 reports the results regarding the impact of $\pi$ on elicited risk and time preferences data (raw-data analysis) and preference parameters (structural-model estimations). Finally, Section 5 discusses and concludes on our findings.

# 2 Experimental design

We implement a between-subjects design in which subjects are randomly assigned to one of four treatments varying the selection probability $\pi \in \{0, 0.1, 0.5, 1\}$. Specifically, all subjects complete the same set of risk and time decision tasks, only the selection probability $\pi$ varies between subjects. In this section, we first introduce the notation used to describe risk and time prospects. We then present the decision tasks and the procedure used to elicit our experimental data: certainty and sooner equivalents. Finally, we detail the experimental implementation, including sample characteristics, session procedures, and how the incentive mechanism was communicated across treatments.

## 2.1 Notation

In the risk domain, $x_{py}$ denotes a risk prospect that gives an outcome $x$ with probability $p$ and $y$ with probability $1-p$. The expected gain of the risk prospect $x_{py}$ is $px + (1-p)y$. We consider binary choices between a prospect $x_{py}$ and a sure outcome $c$. The certainty equivalent of the risk prospect $x_{py}$ is the outcome $c^*$ such that $c^* \sim x_{py}$, where $\sim$ denotes indifference. By definition, there is risk aversion (resp. risk seeking) when $c^* < px + (1-p)y$ (resp. $c^* > px + (1-p)y$).

In the time domain, $z_t$ denotes an outcome received at time $t$ with certainty. When the outcome is received immediately (i.e., at $t=0$), we simplify the notation to $z$. We consider binary choices of type $z_t$ vs. $x_{t+\tau}$, where $t \ge 0$ and $\tau > 0$. The sooner equivalent in $t$ of a delayed outcome $x_{t+\tau}$ is the outcome $s_t^*$ such that $s_t^* \sim x_{t+\tau}$.
## 2.2 Choice tasks and equivalents

We adopt the elicitation method proposed by Abdellaoui et al. (2013), and the stimuli used by Nebout et al. (2023), which involve eliciting certainty equivalents of four binary lotteries $x_{py}$ and sooner equivalents of four delayed outcomes $x_{t+\tau}$. The parameter values of $x$, $y$, $p$, $t$, and $\tau$ for these prospects are summarized in Table 1. The maximum outcome in the experiment is €100, and most prospects involve an outcome of €80, which corresponds approximately to ten times the net hourly minimum wage in France, where the experiment was conducted.

<table><caption>Table 1: Parameter values of risk and time prospects used to elicit certainty and sooner equivalents. In the risk domain, $p$ represents the probability of receiving $x$. In the time domain, $\tau$ indicates the delay (in months) before receiving $x$, while $t$ represents the delay (in months) before receiving the sooner outcome.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Risk</th><th colspan="3">Time</th></tr><tr><th>x</th><th>y</th><th>p</th><th>x</th><th>t</th><th>τ</th></tr></thead><tbody><tr><td>R1</td><td>80</td><td>0</td><td>0.25</td><td>T1</td><td>80</td><td>0</td><td>3</td></tr><tr><td>R2</td><td>80</td><td>0</td><td>0.50</td><td>T2</td><td>80</td><td>0</td><td>6</td></tr><tr><td>R3</td><td>80</td><td>0</td><td>0.75</td><td>T3</td><td>80</td><td>0</td><td>12</td></tr><tr><td>R4</td><td>100</td><td>20</td><td>0.50</td><td>T4</td><td>80</td><td>6</td><td>6</td></tr></tbody></table>

For the risk tasks, the lotteries are designed to cover the main patterns of risk preferences. Since risk preferences typically vary with probability levels, lotteries R1 to R3 only differ in their associated probability levels. Lottery R4 shares the same probability as R2, but its outcomes are increased by €20. Including a lottery with two strictly positive outcomes, such as R4, is essential for separately identifying utility and probability weighting functions in structural estimations.

The time tasks involve choices over delayed monetary outcomes, where the later payoff remains fixed while the timing of the sooner outcome is varied. Prospects T1 to T3 are designed to detect decreasing impatience by varying the time gap between options. Prospect T4 maintains the same temporal gap as T2 (a six-month delay between the two options) but introduces a common front-end delay of six months. This allows us to isolate present bias, in the same way that lottery R4 helps identify probability weighting in the risk tasks.

To elicit a subject's equivalent for a given risk (resp. time) prospect, we ask the subject
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Figure 1: Displays for risky (left) and inter-temporal (right) choices, translated from French.</figcaption></figure>

to indicate her preference between the given prospect and a series of 15 certain (resp. sooner) outcome. Figure 1 illustrates the displays used for these binary risk and inter-temporal choices. In order to speed up the completion of the 15 choices for each prospect, a bisection procedure is implemented. Subjects answer four successive binary choices following a bisection algorithm that halves the interval containing their equivalent at each step, as illustrated in Figure 2. The subjects are informed that the bisection procedure is used only as a way to ease the measurement of their equivalent, but in the case of real incentives, any of the 15 choices is equally likely to be selected for real payment.

This adaptive design, implemented in Kemel and Paraschiv (2021), is simple, fast, and incentive-compatible, allowing for the collection of data equivalent in informational content to that of a 15-choice list through straightforward binary choices, while mitigating the framing effects typically associated with choice lists (see, e.g., Ert and Erev 2013).

For risk (resp. time) prospects, monotonicity (and non-negative discounting) implies that the certainty equivalent (resp. sooner equivalent) of a prospect $x_{py}$ (resp. $x_t$) lies within the interval $(x, y)$ (resp. $(0, x]$). Initially, indifference lies within an interval of length 80. After four successive binary choices using the bisection method, the subject's indifference interval is reduced to a length of 5. This procedure is equivalent to completing a choice list of 15 choices between the prospect ($x_{py}$) (resp. $x_t$) and varying certain (resp. sooner) amounts $v \in \{y+5, \dots, x-5\}$, ($\text{resp. } \{5, \dots, 75\}$), in which only one switching point is
allowed and indifference is not permitted.

<figure><img src="image_5.png" /><figcaption>Figure 2: Bisection algorithm</figcaption></figure>

Consider, for example, the risky prospect $R_1 = (80_{0.5}0)$, where the subject has a 50% chance of obtaining €80. The first question asks if she prefers a certain outcome of €40 over the risky prospect $R_1$. Her response indicates whether her certainty equivalent for $R_1$ is above or below €40. Accepting (resp. declining) implies that her certainty equivalent lies within (0, 40) (resp. (40, 80)). The second question repeats this procedure, replacing the certain amount with the midpoint of the previously identified interval. As shown in Figure 2, repeating this process four times yields intervals of length 5. The midpoint of the final interval provides an equivalent estimate accurate to approximately €2.5.

The expected values of the risky prospects range between €20 and €60, while all inter-temporal prospects involve a delayed outcome of €80. Therefore, the average (non-discounted) expected gain across the eight prospects is €62.5. This value is relatively high, as it corresponds to approximately 7.5 hours of net minimum wage labor, whereas completing the eight elicitation tasks takes about 10 minutes. Accounting for the fact that the Hybrid-RIS procedure assigns a probability $(1 - \pi)$ of receiving zero and therefore scales down the expected gain by a factor of $\pi$, the expected experimental earnings for a risk-neutral and infinitely patient subject are €6.25 in the treatment $\pi = 0.1$ and €31.25 in the treatment $\pi = 0.5$. Overall, our treatments allow us to explore a wide range of expected rewards, spanning from €0 (for $\pi = 0$) to €62.5 (for $\pi = 1$).
## 2.3 Procedure

The experiment was conducted through online group sessions comprising 10 to 23 subjects each.<fnref n="4" /> In total, 335 subjects were randomly allocated to four treatment groups: 84 subjects in the $\pi = 0$ treatment, 81 in $\pi = 0.1$, 88 in $\pi = 0.5$, and 82 in $\pi = 1$.<fnref n="5" />

Subjects were recruited from the INSEAD Behavioral Lab subject pool in Paris and invited to participate in a one-hour experiment conducted via Zoom at scheduled times. At the start of each Zoom session, subjects received a 10-minute presentation delivered by the experimenter (detailed instructions are available in Subsection 5.2 of the Online Appendix). Subjects could communicate privately with the experimenter through Zoom chat or publicly using their microphones to clarify instructions at any point during the session. Communication among subjects was prohibited. All subjects were required to keep their cameras on throughout the session, and those without functional cameras were excluded from participation. Each session corresponded exclusively to one treatment, ensuring that all subjects within a given session were assigned to the same treatment group and received identical instructions.

Subjects in the $\pi = 0$ treatment were informed that all of their choices were hypothetical, but that they should respond according to their true preferences to ensure the scientific quality of the experiment. Subjects in the $\pi \in \{0.1, 0.5\}$ treatments were informed that, at the end of the experiment, they would play a lottery with probability $\pi$ of having one of their choices randomly selected and paid for real. This part was skipped for subjects in treatment $\pi = 1$. Then, for the three treatments with $\pi > 0$, the standard Within-RIS procedure was described sequentially: first, a lottery would determine whether the choice to be paid for real would come from a risk or a time task; second, a lottery would select one of the four corresponding prospects ($R_i$ or $T_i$); and finally, one of the 15 choices within the associated choice list would be randomly drawn for payment. Overall, each binary choice in the 8 choice lists had a probability of being selected equal to $q = \frac{1}{2} \times \frac{1}{4} \times \frac{1}{15} = \frac{1}{120}$.

<footnote n="4">⁴A detailed schedule of these sessions is provided in Table 8 in the Online Appendix.</footnote>

<footnote n="5">⁵Risk and time elicitation tasks were part of a broader survey, which included additional, unrelated questions that were not incentivized. The survey consisted of four distinct parts: Part A involved a food consumption survey asking subjects to report their average consumption over the past year (approximately 20 minutes duration); Part B contained the risk elicitation tasks; Part C included the time elicitation tasks; and Part D comprised general sociodemographic and lifestyle questions. The order of Parts A, B, and C was randomized for each subject, whereas Part D was always presented last. Within the risk and time sections, the order of the prospects was individually randomized.</footnote>
This Within-RIS payment scheme was explained with particular emphasis on its incentive compatibility, as well as on the fact that the bisection procedure provided a convenient and straightforward way to complete each list, while being non-manipulable.

## 2.4 Sample size

The sample size was chosen to ensure sufficient statistical power for our between-subjects treatment comparisons. Power calculations require prior information about the standard deviation of the equivalents. Our analysis is based on data collected by Nebout et al. (2023) using the same stimuli in a hypothetical context. Regarding certainty equivalents, we expected a standard deviation of 20. We wanted to capture a significant absolute change in €4, which corresponds to a 20% change in the expected value of $80_{0.25}0$, 10% in the expected value of $80_{0.5}0$, and 7% of the expected value of $80_{0.75}0$. For time, a difference of €4 corresponds to 10% of the mean sooner equivalents measured by Nebout et al. (2023); a standard deviation of 20 also seems to be a reasonable assumption.

This leads to an effect size of 0.25, which is generally considered as a “small” effect size (Cohen, 2013). With a type 1 errors risk of $\alpha = 0.05$ and type II error risk of $1 - \beta = 0.2$, a sample size of 70 was recommended for each group to run an ANOVA on equivalents comparing each of the four treatments in a between-subjects design (pwr.anova.test function from the pwr R package). For the ANOVA comparing the three treatments with $\pi > 0$, a sample size of 81 was recommended for each group. Therefore, all treatment groups included at least 81 subjects.

Power analysis requires informed estimates of the anticipated variance in the data. In the absence of prior data on effects related to the values of $\pi$ considered in this study, we could not perform power calculations for pairwise treatment comparisons. Therefore, we based our power analysis solely on the between-treatment ANOVAs.

## 2.5 Hypotheses

In our experimental design, subjects make a series of decisions and are enrolled in a lottery with a probability $\pi$, where one of their decisions is randomly selected and played for real. Factually, this procedure introduces a multi-stage lottery that raises a concern: how do people edit and value such lotteries? The editing stage relates to the framing of decisions,
whereas the valuation stage relates to the preferences towards objects once edited under a
particular frame (Tversky and Kahneman 1981; Kahneman and Tversky 1979). When stages
are presented sequentially, like in our treatments, subjects can either isolate each choice (i.e.,
ignore the other stages), or compound them (i.e. reconstitute the multi-stage into a large
compound lottery). The isolation hypothesis solves the concern of the multi-stages, because
each choice is assumed to be perceived as a single choice. Under the compounding hypoth-
esis, additional assumptions regarding preferences are needed in order to derive preferences
from choices. Under the model of rational choice, i.e. Expected Utility, multi-stages do not
impact preferences towards each individual choices. This convenient feature does not always
hold under behavioral models.

We now present the hypotheses related to the editing and treatment of multi-stage lotteries
in our design. We distinguish between the hypotheses we assume and those we test.

Assumed hypotheses

We focus here on the case of Within-RIS where $\pi = 1$. In this case, subjects are guaranteed to have one of their choices selected for real payment. In our design, the selection of the choice selected for payment follows several stages: (i) the selection of one of the two question types (risk vs. time), (ii) the selection of one of the four prospects (within the question type) and (iii) the selection of one decision out of 15 within the choice list of the selected prospect. Overall, these stages constitute a compound lottery, which, when reduced, results in a 1/120 probability of each choice being played and paid. Previous empirical studies support the isolation hypothesis in this situation (Starmer and Sugden, 1991; Cubitt et al., 1998): the subjects treat each choice in isolation and ignore the other stages. This hypothesis cannot be tested within our design. We therefore assume that subjects isolate, which allows us to derive preferences from single choices.

Tested hypotheses

Hybrid-RIS introduces an additional stage in the incentivization process, with a selection
probability $\pi$. Do subjects account for this stage when considering each choice, or do
they ignore it? Our design addresses this question by varying the values of $\pi$ between
treatments. If subjects isolate, they will ignore the selection stage and $\pi$ should not impact
choices. If subjects do not isolate and compound by $\pi$, each choice will be edited as a two-
stage lottery. Risky choices are turned into two-stage lotteries, and sure time prospects are
turned into risky-time prospects. How does this impact choices? If subjects comply with
the axioms of rational choice, compounding each choice by $\pi$ should not impact observed
behavior. Otherwise, theoretical predictions depend on the specific rules of rationality that
are violated (e.g. reduction of compound lotteries, independence axiom) and it is difficult
to make general theoretical predictions. Instead, we formulate empirical predictions, based
on stylized facts related to compound lotteries and risk-and-time prospects.

The first type of stylized fact is the *dilution effect*. For both risk and time, compounding by $\pi$ reduces the expected value of the prospects. This dilution of incentives may increase decision noise (Smith and Walker, 1993). When the expected value of decision consequences decreases, subjects have less motivation to invest cognitive effort, resulting in greater variability and less consistent choices. According to this dilution hypothesis, decision errors increase as $\pi$ decreases. Dilution may also lead to larger deviations from rational decision models. We formally introduce how these deviations are measured in Section 3.2.

The second type of stylized fact is the *common ratio effects*. For risk, compounding by $\pi$ could trigger the well-documented common ratio effect (Kahneman and Tversky, 1979; Allais, 1953). For small values of $\pi$, it could lead subjects to favor riskier alternatives compared to scenarios involving certain outcomes (Nebout and Dubois, 2014). We refer to this phenomenon as the *risk common ratio effect*. In the context of time, compounding choice options by $\pi$ introduces risk into intertemporal choices. This has been shown to influence discount rates, typically enhancing patience and reducing present bias (Epper and Fehr-Duda, 2023; Weber and Chapman, 2005). We refer to this phenomenon as the *time common ratio effect*.

## Summary of the tested hypotheses

Overall, we build on the literature and assume that subjects do not compound the multiple (Within-RIS) choices: they treat each choice in isolation. Our design focuses on the impact of the selection probability $\pi$: do people isolate (from $\pi$) or do they compound (by $\pi$)? If subjects isolate choices from $\pi$, no treatment effect should be observed. Thus, isolation serves as our null hypothesis. However, if subjects do not isolate, two effects can be observed. The dilution effect predicts an increase in decision errors for lower values of $\pi$ in both risk
and time domains. The common ratio effect predicts changes in risk attitudes and time preferences when $\pi < 1$. These hypotheses are summarized in Table 2.

<table><caption>Table 2: Hypotheses about the impact of $\pi$ on elicited risk and time preferences.</caption><thead><tr><th></th><th>For risk</th><th>For time</th></tr></thead><tbody><tr><th>Isolation</th><td>H0: no difference</td><td>H0: no difference</td></tr><tr><th>Dilution</th><td>DR: more inverse S and larger errors when $\pi$ decreases</td><td>DT: more present bias and larger errors when $\pi$ decreases</td></tr><tr><th>Common ratio</th><td>CR: more risk taking for $0 < \pi < 1$ than for $\pi = 1$</td><td>CT: more patience and less present bias for $0 < \pi < 1$ than for $\pi = 1$</td></tr></tbody></table>

# 3 Statistical analysis

We now present the analysis plan for the experimental data, which is examined at two levels: preference data and preference parameters. Preference data are directly derived from choices, while preference parameters are obtained through model estimations.

## 3.1 Preference data

The raw experimental data consist in certainty equivalents and sooner equivalents, which respectively measure risk preferences and time preferences. The lower the certainty equivalent (resp. sooner equivalent), the higher the risk aversion (resp. impatience). We can thus assess the impact of $\pi$ on measured risk and time preferences by comparing the equivalents between treatments. We use a series of t-tests to compare means across treatments pairwise. A series of ANOVA tests, examining the effects of treatments in a between-subjects design, completes the analysis.

In the framework of classical statistics, failure to reject the null does not provide evidence. In contrast, the Bayesian approach allows support for the null by calculating the Bayesian factors. This factor is the ratio of the likelihood of the data computed using two different models: one that enforces the null, and one that does not. When the model in the denominator enforces the null, the Bayesian factor captures the weight of the evidence against (when it is greater than one) or for (when it is smaller than one) the null. We complement our analysis of preference data with a series of Bayesian ANOVAs (Kass and Raftery, 1995).
to assess the weight of evidence supporting the null hypothesis that $\pi$ has no impact on measurements.

## 3.2 Preference parameters

We also examine preferences using behavioral choice models estimated from the preference data. These models allow us to characterize several dimensions of preferences and compare the estimated parameters between treatments to determine which dimension of preferences is impacted by $\pi$.

### 3.2.1 Behavioral models

We consider three models for risk and time preferences: DEU, HDRDU and HDRDU2.

DEU is the benchmark of rational decision models and involves two components: an increasing utility function $u(x) = x^{\theta}$ whose concavity parameter $\theta$ captures risk attitudes, and a decreasing exponential discount function $D(t) = e^{-\delta t}$ whose rate $\delta$ captures impatience.

The second model is hyperbolic discounted rank dependent utility (HDRDU) which accounts for two major limitations of DEU: probability weighting and non-exponential discounting (Abdellaoui et al., 2019). We model probability weighting by a Prelec specification $w(p) = e^{-(\log(p))^{\gamma}}$ where $\gamma$ captures deviations from linear probability weighting $^6$. Non-exponential discounting is modeled by a quasi-hyperbolic specification $D = \beta e^{-\delta t}$ when $t > 0$, where $\beta$ captures present bias. Note that DEU corresponds to a specific case of HDRDU where $\gamma = 1$ and $\beta = 1$. When $\gamma < 1$ subjects exhibit inverse-S-shaped probability weighting; when $\gamma > 1$, which is less common in the literature, subjects exhibit S-shaped probability weighting. When $\beta < 1$ subjects exhibit present bias. Parameters $\gamma$ and $\beta$ can thus be used to measure deviations from rational decision making.

The third model, HDRDU2, involves the same parameters as HDRDU, but assumes linear utility for time. This allows for a disentangled estimation of parameters for risk and time prospects and aligns with empirical stylized facts, where utility for risk is generally concave, while utility for time is closer to linear (e.g. Abdellaoui et al., 2013, 2018).

<footnote>$^6$Two-parameter specifications that separate elevation and sensitivity may offer a better goodness of fit, but their elevation parameter contributes to risk attitudes. This complicates the interpretation of utility which is no longer the only driver of risk attitudes. The one-parameter Prelec imposes $w(1/e) = 1/e$, so utility can be interpreted as the EU utility when $p = 1/e$. Another one-parameter specification, the Goldstein-Einhorn that imposes $w(1/2) = 1/2$ was tested and gave qualitatively identical results.</footnote>
The structural behavioral equations mapping the preference data to the preference parameters, for each model, are provided in Table 3.

<table><thead><tr><th colspan="3">Models</th></tr><tr><th></th><th>CE: c ~ x<sub>py</sub></th><th>SE: s<sub>t</sub> ~ x<sub>t+τ</sub></th></tr></thead><tbody><tr><td>DEU</td><td>c = u<sup>-1</sup>[p(u(x) - u(y)) + u(y)]</td><td>s = u<sup>-1</sup>[e<sup>-rτ</sup>u(x)]</td></tr><tr><td>HDRDU</td><td>c = u<sup>-1</sup>[w(p)(u(x) - u(y)) + u(y)]</td><td>s = u<sup>-1</sup>[D(t + τ)/D(t)u(x)]</td></tr><tr><td>HDRDU2</td><td>c = u<sup>-1</sup>[w(p)(u(x) - u(y)) + u(y)]</td><td>s = D(t + τ)/D(t)x</td></tr></tbody></table>

<table><caption>Table 3: Decision models (top) and their parametric specifications (bottom).</caption><thead><tr><th colspan="2">Parametric Specification</th></tr></thead><tbody><tr><td>Power utility</td><td>u(x) = x<sup>θ</sup></td></tr><tr><td>Prelec probability weighting</td><td>w(p) = exp[(-log p)<sup>γ</sup>]</td></tr><tr><td>QH discounting</td><td>D(t) = βe<sup>-δt</sup>, D(0) = 1</td></tr></tbody></table>

### 3.2.2 Econometric estimation

The preference data consist of indifference values (certainty equivalents or sooner equivalents) derived from choice lists completed by bisection (see Section 2.2). A decision error made during any choices of the bisection would result in an erroneous observation. To account for such errors, for individuals i and prospects k, we distinguish observed indifference values $c_{i,k}$ and $s_{i,k}$ from the theoretical indifference $\hat{c}_{i,k}$ and $\hat{s}_{i,k}$ which are predicted by the structural equations of our models (cf. Table 3). Errors are assumed to be normally distributed across individuals and prospects, with the following structure:

$$c_{i,k} = \hat{c}_{i,k} + \nu_i + \epsilon_{i,k}^r \quad (1)$$

$$s_{i,k} = \hat{s}_{i,k} + \nu_i + \epsilon_{i,k}^t \quad (2)$$

where $\nu_i \sim N(0, \rho_r)$ are random coefficients that account for the clustered nature of our data (we have several observations per individual). Coefficients $\epsilon_i^r \sim N(0, \sigma_r)$ and $\epsilon_i^t \sim N(0, \sigma_t)$ account for idiosyncratic errors.
For each treatment, the objective is to estimate the preference parameters $\theta, \delta$ under the rational model DEU, and $\theta, \delta, \gamma, \beta$ under the behavioral models HDRDU and HDRDU2. The noise parameters $\sigma_r$ and $\sigma_t$ are informative as they capture decision errors. We estimate these parameters at the aggregated level, and assume that they capture the preferences of a representative agent. Our design involves 8 observations per subject, which does not allow to precisely capture parameter heterogeneity.

More specifically, we are particularly interested in the differences in parameters between treatments. Using a set of dummy variables for each treatment, we estimate treatment effects in the form of parameter differences across treatments. For example, $\theta_{0.5} - \theta_{0.1}$ represents the difference in the utility parameter $\theta$ between treatment $\pi = 0.5$ and treatment $\pi = 0.1$, while $\sigma_1 - \sigma_{0.5}$ represents the difference in the error parameter $\sigma$ between treatment $\pi = 1$ and treatment $\pi = 0.5$. For DEU, we estimate 4 parameters ($\theta, \delta, \sigma_r$ and $\sigma_t$) for $\pi = 0$, and $4 \times 3$ parameter differences between treatments. For HDRDU and HDRDU2, we estimate 6 parameters ($\theta, \delta, \gamma, \beta, \sigma_r$ and $\sigma_t$) for $\pi = 0$ and $6 \times 3$ parameter differences between treatments.

The parameters are estimated with Bayesian MCMC procedure implemented in Stan (details about the likelihood and specifications are provided in Section 2 of the Online Appendix). Regarding the priors on the model parameters ($\theta, \delta, \gamma, \beta, \sigma_r$ and $\sigma_t$), we consider uniform priors over the domain of plausible values. More importantly, we also apply uniform priors for the treatment effects. The prior differences are expressed on the log scale to capture prior relative differences. These priors range from -2 to 2, which correspond to a factor between 0.135 and 7.4. Estimations with smaller prior effects, ranging from -0.5 to 0.5 (corresponding to a factor between 0.6 and 1.65), yield qualitatively identical results.

# 4 Results

## 4.1 Impact of $\pi$ on preference data

This section presents our main results on preference data, addressing the hypotheses related to the selection probability $\pi$.

Figure 3 reports the mean and its 95% confidence interval for the four (within-subjects) measurements under the four (between-subjects) treatments, for risk (top) and time (bottom).
<figure><img src="image_6.png" /><figcaption>Figure 3: Mean certainty equivalents (top) and sooner equivalents (bottom) and their 95% confidence intervals.</figcaption></figure>

Note: brackets report the results of test (t-test for 2 by 2 comparisons, Anova otherwise): *, p < 0.05, ***, p < 0.001.
tom)$^7$. Regarding risk, for the four prospects, the mean certainty equivalents are larger for treatment $\pi = 0$ than for other treatments. The average certainty equivalents are similar across the different types of treatments with real incentives ($\pi > 0$). This suggests that the average subject exhibits more risk seeking in the absence of incentives but that the type of incentive mechanism does not impact risk preferences.

A series of tests confirms this pattern (significance of the tests is reported on Figure 3). First, we run ANOVAs with treatment as a four-level (between-subjects) factor for each of the four prospects, and observe a significant impact of treatment for each prospect ($p < 0.05$) except for $80_{0.25}0$. Additional tests refine the analysis using t-tests to compare treatments two-by-two. The difference between treatment $\pi = 0$ and $\pi = 0.1$ is statistically significant for two prospects ($p < 0.05$). Other two-by-two comparisons between treatments do not capture significant differences for any prospect. Finally, significant differences are observed between hypothetical ($\pi = 0$) and non-hypothetical treatments ($\pi > 0$). This latter point is supported by a series of ANOVA focusing on the three treatments with $\pi > 0$ that fails to capture differences across treatments ($p > 0.05$ for the four prospects). In addition to the mean patterns, comparing the variance of certainty equivalents across treatments can reveal the impact of treatment on the noisiness of the responses. Indeed, the dilution effect predicts that, because of random choices, the variance of responses increases when the expected value of the incentive decreases. We compare the variances for each prospect across treatments two-by-two. None of the p-values from the 12 F-tests reaches the 5% significance level.

We then apply the same analysis to the time prospects. As shown in bottom Figure 3 (time), the mean sooner equivalents are similar across all treatments for the four prospects. The ANOVAs and t-tests reveal no significant effects. The analysis does not capture any impact of the treatments on time measurements. We also run a series of variance tests comparing the treatment levels two-by-two for each prospect. None of the p-values from these 12 F-tests reach the 5% significance level.

Overall, the analysis of preference data offers a simple and clear picture. Our subjects exhibit more risk seeking with hypothetical choices ($\pi = 0$) than with incentivized choices ($\pi > 0$). Among incentivized treatments, the level of the selection probability $\pi$ does not affect choices. In addition, the variance in the measurements is stable across all treatments.

<footnote>$^7$Tables 1 to 4 in the Online Appendix present these descriptive statistics in detail.</footnote>
The lower risk aversion observed when $\pi = 0$ rejects the isolation hypothesis. However, the lack of differences in variance between treatments or means between incentivized treatments does not support the dilution nor the common-ratio effects. Regarding time, no treatment difference is observed in terms of means nor variance. Therefore, we cannot reject the assumption of isolation for time choices. Importantly, running tests for each of the four prospects separately creates a risk of family wise errors. Despite this liberal bias, our analysis fails to capture any significant differences.

We now report the results under a Bayesian approach. For risk, we focus on the treatments with $\pi > 0$ and run several Bayesian ANOVAs, to assess differences between treatments. The Bayesian factors are below 0.1 for prospects 1 and 4, providing “strong evidence” in favor of the null hypothesis (H0); for Prospects 2 and 3, they lie between 0.1 and 1/3, indicating “moderate evidence” for the null. Overall, the Bayesian analysis clearly supports the null hypothesis – that is, the incentive mechanism does not impact the risk data. For time, the Bayesian factors lie between 1/10 and 1/30, again providing strong evidence in favor of the null. Therefore, our analysis offers solid evidence for the conclusion that incentives do not influence time measurements.

Overall, these results allow us to test some of the hypotheses presented in Table 2. The absence of significant differences between treatments $\pi = 1$ and $\pi < 1$ invalidates the common ratio predictions (CR and CT) and supports the isolation hypothesis (H0).

The Online Appendix (Section 3) reports a series of additional analyses based on preference data. Indices of deviations from rational decision-making are introduced, based on violations of monotonicity, probability dependence, and non-stationarity. None of these indices are affected by the treatments.

## 4.2 Impact of $\pi$ on parameters of structural models

This section reports the results of the structural estimations of the decision models that provide preference parameters. The econometric analysis also estimates the error parameters of the decision models. We focus on the impact of $\pi$ on preference and error parameters, in order to identify which aspects of choices are affected by $\pi$: preferences (utility, discount function, weighting function) or errors.
<table><caption>Table 4: Estimations with quasi-hyperbolic discounting
(Mean and 95% credible interval)</caption><thead><tr><th>Model</th><th>DEU</th><th>HDRDU</th><th>HDRDU2</th></tr></thead><tbody><tr><td>Parameter</td><td>Mean [95% CI]</td><td>Mean [95% CI]</td><td>Mean [95% CI]</td></tr><tr><td>θ<sub>0</sub></td><td>0.837 [0.755, 0.927]</td><td>0.893 [0.821, 0.968]</td><td>0.891 [0.821, 0.967]</td></tr><tr><td>δ<sub>0</sub></td><td>0.054 [0.048, 0.061]</td><td>0.046 [0.038, 0.054]</td><td>0.051 [0.042, 0.061]</td></tr><tr><td>β<sub>0</sub></td><td></td><td>0.859 [0.807, 0.913]</td><td>0.844 [0.787, 0.904]</td></tr><tr><td>γ<sub>0</sub></td><td></td><td>0.666 [0.586, 0.746]</td><td>0.666 [0.588, 0.746]</td></tr><tr><td>σ<sup>r</sup><sub>0</sub></td><td>11.941 [10.758, 13.277]</td><td>10.709 [9.640, 11.941]</td><td>10.719 [9.651, 11.948]</td></tr><tr><td>σ<sup>t</sup><sub>0</sub></td><td>18.680 [16.989, 20.566]</td><td>18.473 [16.740, 20.346]</td><td>18.450 [16.728, 20.213]</td></tr><tr><td>Effects</td><td></td><td></td><td></td></tr><tr><td rowspan="3">θ</td><td>θ<sub>0.1</sub> - θ<sub>0</sub></td><td>-0.309 [-0.432, -0.188]</td><td>-0.160 [-0.269, -0.054]</td><td>-0.159 [-0.264, -0.054]</td></tr><tr><td>θ<sub>0.5</sub> - θ<sub>0.1</sub></td><td>0.009 [-0.106, 0.119]</td><td>-0.005 [-0.106, 0.094]</td><td>-0.005 [-0.104, 0.098]</td></tr><tr><td>θ<sub>1</sub> - θ<sub>0.5</sub></td><td>0.096 [-0.023, 0.212]</td><td>0.041 [-0.064, 0.146]</td><td>0.043 [-0.064, 0.148]</td></tr><tr><td rowspan="3">δ</td><td>δ<sub>0.1</sub> - δ<sub>0</sub></td><td>-0.017 [-0.025, -0.009]</td><td>-0.012 [-0.024, -0.001]</td><td>-0.005 [-0.020, 0.009]</td></tr><tr><td>δ<sub>0.5</sub> - δ<sub>0.1</sub></td><td>0.002 [-0.005, 0.009]</td><td>0.004 [-0.006, 0.014]</td><td>0.006 [-0.010, 0.021]</td></tr><tr><td>δ<sub>1</sub> - δ<sub>0.5</sub></td><td>0.004 [-0.003, 0.010]</td><td>0.002 [-0.007, 0.012]</td><td>0.000 [-0.013, 0.014]</td></tr><tr><td rowspan="3">β</td><td>β<sub>0.1</sub> - β<sub>0</sub></td><td></td><td>0.002 [-0.069, 0.074]</td><td>-0.028 [-0.114, 0.056]</td></tr><tr><td>β<sub>0.5</sub> - β<sub>0.1</sub></td><td></td><td>0.014 [-0.051, 0.076]</td><td>0.017 [-0.070, 0.102]</td></tr><tr><td>β<sub>1</sub> - β<sub>0.5</sub></td><td></td><td>0.012 [-0.049, 0.072]</td><td>0.024 [-0.056, 0.100]</td></tr><tr><td rowspan="3">γ</td><td>γ<sub>0.1</sub> - γ<sub>0</sub></td><td></td><td>-0.157 [-0.273, -0.039]</td><td>-0.157 [-0.273, -0.041]</td></tr><tr><td>γ<sub>0.5</sub> - γ<sub>0.1</sub></td><td></td><td>0.001 [-0.121, 0.123]</td><td>0.001 [-0.120, 0.118]</td></tr><tr><td>γ<sub>1</sub> - γ<sub>0.5</sub></td><td></td><td>0.070 [-0.056, 0.194]</td><td>0.071 [-0.059, 0.200]</td></tr><tr><td rowspan="3">σ<sup>r</sup><sub>0</sub></td><td>σ<sup>r</sup><sub>0.1</sub> - σ<sup>r</sup><sub>0</sub></td><td>2.831 [0.878, 4.827]</td><td>1.199 [-0.510, 3.009]</td><td>1.177 [-0.575, 2.986]</td></tr><tr><td>σ<sup>r</sup><sub>0.5</sub> - σ<sup>r</sup><sub>0.1</sub></td><td>0.186 [-1.950, 2.389]</td><td>0.310 [-1.648, 2.261]</td><td>0.338 [-1.611, 2.257]</td></tr><tr><td>σ<sup>r</sup><sub>1</sub> - σ<sup>r</sup><sub>0.5</sub></td><td>0.032 [-2.102, 2.152]</td><td>1.027 [-0.951, 3.056]</td><td>1.014 [-0.950, 3.035]</td></tr><tr><td rowspan="3">σ<sup>t</sup><sub>0</sub></td><td>σ<sup>t</sup><sub>0.1</sub> - σ<sup>t</sup><sub>0</sub></td><td>1.450 [-1.205, 4.187]</td><td>1.762 [-0.941, 4.449]</td><td>1.817 [-0.906, 4.563]</td></tr><tr><td>σ<sup>t</sup><sub>0.5</sub> - σ<sup>t</sup><sub>0.1</sub></td><td>-2.218 [-4.979, 0.485]</td><td>-2.196 [-5.088, 0.558]</td><td>-2.227 [-5.090, 0.542]</td></tr><tr><td>σ<sup>t</sup><sub>1</sub> - σ<sup>t</sup><sub>0.5</sub></td><td>-0.677 [-3.217, 1.827]</td><td>-0.930 [-3.441, 1.705]</td><td>-0.937 [-3.427, 1.631]</td></tr><tr><td>LL</td><td></td><td>-6946.279</td><td>-6730.169</td><td>-6730.576</td></tr></tbody></table>

The first six rows of Table 4 report the estimated preference parameters for the hypothetical treatment ($\pi = 0$). The remaining rows present the estimated differences in parameters across treatments. Under DEU, a concave power utility function is estimated according to the estimate of 0.84, and its credible interval does not include the value 1.

The monthly discount rate is estimated at 5% and is significantly greater than zero. In the hypothetical treatment, subjects exhibit mild risk aversion and impatience. The standard
deviation of the choice error is higher for time choices (€18.7) than for risk choices (€11.9).

Regarding the impact of treatments, we observe a significant difference for the utility pa-
rameter between treatments π = 0.1 and π = 0. The utility parameter is 0.31 lower when
π = 0.1. This means that the utility function is more concave and, thus, subjects exhibit
greater risk aversion in this treatment than in the treatment without incentives. Under
DEU, we also find a significant difference in the discount rate, which is 1.7 points lower
when π = 0.1 than when π = 0. This suggests that, under DEU, subjects are more patient
in the presence of real incentives. The magnitude of the standard deviation of errors is also
impacted. For risk, it is €2.8 euros larger when π = 0.1, suggesting larger deviations from
DEU predictions. The differences of model parameters between other levels of selection
probability π (0.5 versus 0.1 and 1 versus 0.5) are not significant, according to their cred-
ible intervals. Therefore, the analysis under DEU concludes that subjects exhibit greater
risk aversion and patience with real incentives (π > 0). However, it does not capture any
differences between the different non-zero values of the selection probability π.

We now assess the robustness of these findings under the behavioral models, starting with HDRDU when π = 0. The coefficient of the probability weighting function is estimated at 0.86 and is significantly lower than 1. This indicates that our average subject exhibits an inverse-S probability weighting. Since probability weighting accounts for part of risk aversion, the utility coefficient is estimated at 0.89; this corresponds to a slight but statistically significant concavity. The change in the estimate of the curvature of the utility (as compared to DEU) entails a change in the estimate of the discount rate of 4.6% in the HDRDU model. Under HDRDU2, the estimate is 5.1% because the model assumes a linear inter-temporal utility. The estimate of the present bias parameter is qualitatively identical in the two behavioral models: around 0.85 and significantly lower than 1. Therefore, the analysis under the behavioral models suggests that our average subject exhibits deviations from DEU that are typically observed: inverse S probability weighting and present bias. Accounting for these patterns improves their log-likelihood beyond that of DEU.

We now describe how these preference parameters vary across treatments. Starting with the $\gamma$ coefficient that drives probability weighting, we observe that its estimate is lower for treatment $\pi = 0.1$ than for treatment $\pi = 0$. This implies that the average subject exhibits a more pronounced probability weighting in the presence of real incentives. No
treatment effect is captured on the present bias parameter $\beta$. Under the behavioral models, the error parameter for risk does not change significantly between treatments $\pi = 0.1$ and $\pi = 0$, unlike under DEU. A possible explanation for this discrepancy lies in the change in probability weighting. Since DEU does not directly capture this dimension, the increase in probability weighting instead manifests as a change in the error parameter. In contrast, under the behavioral models, this change is explicitly captured by a shift in the $\gamma$ parameter, leaving the error parameter unaffected.

Overall, the behavioral analysis confirms that the presence of incentives increases the con-cavity of the utility function. It provides a slightly different picture on impatience as no treatment difference is captured on discount rates. Regarding behavioral parameters, prob-ability weighting is found to be more pronounced in the presence of real incentives, and no effect is observed for present bias. Overall, both DEU and behavioral models capture differences only between treatments $\pi = 0.1$ and $\pi = 0$, only. No differences are observed between the other treatments.

In Subsection 4.1 of the Online Appendix, we perform two complementary analyses exploring alternative specifications of the effects of $\pi$ on the model parameters. First, we report estimations where the levels $\pi = 0.1$ and $\pi = 0.5$ are grouped and taken as the reference category. This category corresponds to Hybrid-RIS and allows for a direct comparison of these types of incentive mechanisms with hypothetical choices ($\pi = 0$) or Within-RIS ($\pi = 1$). These results are consistent with those obtained without grouping the Hybrid-RIS categories.

Another complementary analysis focuses on the three treatments where $\pi > 0$ and tests for a linear effect of $\pi$ (see Section 4.2 of the Online Appendix). In other words, the treatment variable is treated as a continuous variable, and the linear specification allows us to save degrees of freedom and increase statistical power, enabling us to capture the effect of $\pi$ on model parameters. However, we observe no effect. For each model parameter, no linear effect of $\pi$ is captured.

Overall, the analysis of preference parameters estimated from structural-equation models is consistent with the analysis of preference data (Section 4.1). Our results do not support the hypotheses based on dilution (DR and DT) nor common ratio effects (CR and CT). Therefore, the hypothesis based on isolation holds both for risk and time risk among treatments
with $\pi > 0$. Nevertheless, differences are observed between treatments $\pi = 0$ and $\pi > 0$. This does not align with our hypotheses.

# 5 Discussion and Conclusion

The study reports the first analysis of the impact of the selection probability $\pi$ in the Hybrid-RIS, on elicited risk and time preferences. Considering four different values of $\pi$, including extreme ones, we also address the cases of no incentives ($\pi = 0$) and within-subject random incentive ($\pi = 1$). The face values used in our experiment are large enough to stimulate the attention of subjects even when $\pi = 0.1$. Therefore, our experiment responds to the call made by Baltussen et al. (2012) for "a [...] design with either larger face values or larger selection probabilities", as our design indeed considers large face values and systematically varies the selection probability. Our experiment is the first to assess the impact of incentives for both risk and time preferences, as previous studies considered these two dimensions separately. Most studies on RIS have focused on risk, probably because the mechanism can be modeled as a lottery. Fewer studies have investigated the impact of RIS on time preferences. Considering both risk and time allows us to test several hypotheses regarding the impact of $\pi$ predicted by the dilution and compounding effects. Our results are based on model-free statistical analyses, completed by structural estimations of decision models.

We summarize our key results. For risk, real incentives matter, but the magnitude of the Hybrid-RIS selection probability $\pi$ does not. For time, we observe no effect of the selection probability, even between hypothetical choices ($\pi = 0$) and choices incentivized by Within-RIS ($\pi = 1$). Strictly speaking, these findings are non-results as they are based on the non-rejection of the null (due to the absence of statistically significant differences). However, the Bayesian approach, used both on preference data and preference parameters, allows us to consider that our data provide evidence to support the null. These results have direct implications for experimental practice and support a "convenient truth": (i) experimentalists do not need to implement real incentives for observing time preferences; and (ii) for risk, experimentalists can avoid the hypothetical bias, while minimizing the experimental costs, using a Hybrid-RIS with a low selection probability.

Our analysis also allows us to investigate how subjects integrate the selection probability. According to the literature on risk attitudes, the selection probability can affect risk choices
based on two possible effects: dilution and compounding. Dilution predicts more errors and
deviations from the rational model, DEU. We find no evidence of such an effect. Specifically,
we do not observe significant differences in the variances of preference data, suggesting that
the magnitude of errors is similar across treatments. Furthermore, our structural estimations
capture more pronounced probability weighting, a deviation from DEU, when $\pi > 0$ than
when $\pi = 0$. This either contradicts the dilution effect or suggests that probability weighting
is not an error due to a lack of motivation. Regarding the common ratio effect, we do not
observe differences in risk preferences between treatments where $0 < \pi < 1$ and $\pi = 1$.
This absence of an effect is consistent with the assumption that subjects did not compound
the risky choices with the selection probability. Overall, our data support the assumption
that our subjects isolate risky choices from the lottery introduced by Hybrid-RIS. The
prior literature supports that subjects isolate choices in the Within-RIS. The present paper
extends this result to Hybrid-RIS, suggesting that subjects also isolate choices from the
selection probability.

A puzzling result is the absence of monotonic effect of $\pi$ on risky choices: we indeed capture a
difference between $\pi = 0$ and $\pi = 0.1$ but find no differences between cases where $\pi > 0$. This
may be because a threshold in the expected value of the incentives must be reached in order
to interest subjects. Under this explanation, the responses of interested and non-interested
subjects would differ; but the responses of interested subjects would not be sensitive to the
magnitude of the incentive. In our experiment, this threshold may have been reached in
the treatment where $\pi = 0.1$. Indeed, in our experiment, the expected values of the risk
prospects compounded with $\pi = 0.1$ are €6 for 80$_{0.5}$$0, €4 for 80$_{0.5}$$0, €2 for 80$_{0.25}$$0, and
€6 for 100$_{0.5}$$20. These values are in the range of those used to elicit risk attitudes (Holt
and Laury 2002) and are considered sufficiently large to motivate subjects (Crosetto and
Filippin 2016).

The absence of a monotonic effect of $\pi$ on risky choices can also be discussed in light of the
recently developed concept of aversion to complexity (e.g., Oberholzer et al., 2024). One
may speculate that complexity increases with the number of stages (or lotteries) that must
be resolved to determine the final payment. In our experiment this number is equal to 4 in
treatments $\pi = 0.1$ and $\pi = 0.5$, and is equal to 3 when $\pi = 1$. The absence of difference
between treatments $0 < \pi < 1$ and treatment $\pi = 1$ suggests that the increase in complexity
generated by the Hybrid-RIS does not impact elicited preferences.

The treatments with real incentives ($\pi > 0$) are also more complex than the treatment with hypothetical choices ($\pi = 0$). This increased complexity may explain our observed differences between these treatments. Indeed, when $\pi > 0$, the payment mechanism involves more stages, and its explanation to the subjects (description and incentive compatibility) requires lengthier and more complex instructions. The higher complexity generated by the real incentive procedure may reduce the appeal of lottery options (perceived as more complex) for individuals who are averse to complexity. Aversion to complexity may thus account for the greater risk aversion observed under real incentives compared to the hypothetical condition but fails to explain the absence of differences between the Hybrid-RIS and Within-RIS conditions.

Besides the impact of incentives, our experiment also informs about how usual decision models capture risk and time preferences. Under DEU, the same utility function applies to both risk and time. Under this model, risk attitudes are directly connected to inter-temporal substitution. Therefore, a change in risk attitudes (measured by certainty equivalents) should necessarily come with a change in time preferences (measured by sooner equivalents). In our experiment, we observe that introducing a non-zero selection probability impacts certainty equivalents but does not impact sooner equivalents. This constitutes a new piece of evidence showing that risk preferences are not time preferences (e.g. Abdellaoui et al., 2013). As a consequence, the technique that consists of measuring the inter-temporal utility from risky choices and using it to correct the estimated discount rate can produce misleading results. Implemented on our data, this technique suggests that the discount rate is impacted by the treatments. However, according to the model-free analysis, the treatments do not impact time preference data. The observed impact of incentives on the discount rate is therefore likely to be a spurious effect, due to the impact of the treatments on the risky utility function.
## Acknowledgments

We thank the INSEAD-Sorbonne University Behavioural Lab for technical support. We thank the subjects of the iRisk Seminar (IESEG Lille) and 2024 ASFEE conference for their feedback on this paper. We are also grateful to the Editor and the two anonymous referees for their detailed reviews and constructive comments.

## Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this manuscript, the author(s) used ChatGPT-4.0 to assist with language editing of several sentences. All content generated with the help of this tool was subsequently reviewed and edited by the author(s), who take full responsibility for the final version of the manuscript.

## Funding sources

This work (under the name of the "PsychAlim" project) was supported by the CNRS/INRAE MITI program.

## Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Online Appendix

"Paying none, some or all? Between-subject random
incentives and preferences towards risk and time"

Noémi Berlin¹ Emmanuel Kemel² * Vincent Lenglin³ Antoine Nebout⁴

¹CNRS, Economix, University of Paris Nanterre

²CNRS & HEC Paris

³Lille Catholic University

⁴University Paris-Saclay, INRAE

Abstract

This document presents additional results, as well as the instructions used for the
experiment reported in the paper.

<footnote>*Corresponding author: emmanuel.kemel@greg-hec.com</footnote>
# 1 Descriptive Statistics

<table><caption>Table 1: Risk domain: Mean (std) for measured certainty equivalents (CEs)</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">Measurement (CE)</th></tr><tr><th>80<sub>0.25</sub>0</th><th>80<sub>0.5</sub>0</th><th>80<sub>0.75</sub>0</th><th>100<sub>0.5</sub>20</th></tr></thead><tbody><tr><td rowspan="4">Treatment</td><td>π = 0</td><td>19.23 (11.04)</td><td>35.54 (15.01)</td><td>49.29 (15.67)</td><td>52.68 (13.87)</td></tr><tr><td>π = 0.1</td><td>16.14 (13.23)</td><td>26.08 (15.86)</td><td>39.78 (18.99)</td><td>48.30 (14.69)</td></tr><tr><td>π = 0.5</td><td>14.60 (11.39)</td><td>27.16 (16.57)</td><td>38.69 (18.77)</td><td>47.10 (14.88)</td></tr><tr><td>π = 1</td><td>15.37 (11.92)</td><td>30.00 (18.13)</td><td>42.93 (17.68)</td><td>46.95 (15.63)</td></tr></tbody></table>

<table><caption>Table 2: Risk domain: The p-values of tests comparing CEs across treatments</caption><thead><tr><th></th><th>80<sub>0.25</sub>0</th><th>80<sub>0.5</sub>0</th><th>80<sub>0.75</sub>0</th><th>100<sub>0.5</sub>20</th></tr></thead><tbody><tr><td>0 vs 0.1</td><td>0.107</td><td>&lt;0.001</td><td>&lt;0.001</td><td>0.051</td></tr><tr><td>0.1 vs 0.5</td><td>0.421</td><td>0.666</td><td>0.708</td><td>0.599</td></tr><tr><td>0.5 vs 1</td><td>0.670</td><td>0.289</td><td>0.132</td><td>0.949</td></tr><tr><td>Anova (ρ ∈ {0, 0.1, 0.5, 1})</td><td>0.063</td><td>0.001</td><td>0.001</td><td>0.042</td></tr><tr><td>Anova (ρ ∈ {0.1, 0.5, 1})</td><td>0.714</td><td>0.309</td><td>0.307</td><td>0.821</td></tr><tr><td>BF (ρ ∈ {0.1, 0.5, 1})</td><td>0.057</td><td>0.121</td><td>0.122</td><td>0.050</td></tr></tbody></table>

<table><caption>Table 3: Time domain: Mean (std) for measured sooner equivalents (SEs)</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">Measurement (SE)</th></tr><tr><th>v<sub>0</sub> ~ 80<sub>3</sub></th><th>v<sub>0</sub> ~ 80<sub>6</sub></th><th>v<sub>0</sub> ~ 80<sub>12</sub></th><th>v<sub>6</sub> ~ 80<sub>12</sub></th></tr></thead><tbody><tr><td rowspan="4">Treatment</td><td>π = 0</td><td>56.37 (18.73)</td><td>49.76 (21.60)</td><td>39.82 (23.71)</td><td>56.13 (19.14)</td></tr><tr><td>π = 0.1</td><td>54.97 (20.80)</td><td>47.93 (23.23)</td><td>43.43 (23.43)</td><td>56.27 (20.41)</td></tr><tr><td>π = 0.5</td><td>54.66 (20.40)</td><td>47.61 (21.27)</td><td>40.85 (24.08)</td><td>54.15 (19.19)</td></tr><tr><td>π = 1</td><td>57.20 (18.28)</td><td>49.21 (22.16)</td><td>39.76 (23.74)</td><td>55.67 (18.91)</td></tr></tbody></table>

<table><caption>Table 4: Time domain: The p-values of tests comparing SEs across treatments</caption><thead><tr><th></th><th>v<sub>0</sub> ~ 80<sub>3</sub></th><th>v<sub>0</sub> ~ 80<sub>6</sub></th><th>v<sub>0</sub> ~ 80<sub>12</sub></th><th>v<sub>6</sub> ~ 80<sub>12</sub></th></tr></thead><tbody><tr><td>0 vs 0.1</td><td>0.651</td><td>0.601</td><td>0.327</td><td>0.965</td></tr><tr><td>0.1 vs 0.5</td><td>0.922</td><td>0.926</td><td>0.482</td><td>0.489</td></tr><tr><td>0.5 vs 1</td><td>0.394</td><td>0.633</td><td>0.765</td><td>0.603</td></tr><tr><td>Anova (ρ ∈ {0, 0.1, 0.5, 1})</td><td>0.818</td><td>0.885</td><td>0.733</td><td>0.909</td></tr><tr><td>Anova (ρ ∈ {0.1, 0.5, 1})</td><td>0.668</td><td>0.766</td><td>0.599</td><td>0.886</td></tr><tr><td>BF (ρ ∈ {0, 0.1, 0.5, 1})</td><td>0.061</td><td>0.047</td><td>0.067</td><td>0.054</td></tr></tbody></table>
## 2 Bayesian estimations method

We describe the estimations procedure of the preference parameters. For each CE $c_{i,k}$ and SE $s_{i,k}$, the structural equations give predictions $\hat{c}_{i,k} = CE(k, \Theta)$ and $\hat{s}_{i,k} = SE(k, \Theta)$, where CE() and SE() are functions given by the structural equations of the models and that take the characteristics of prospect k and vector of parameters $\Theta$ as arguments.

Our error specification maps observed preferences data and theoretical ones as follows:

$$c_{i,k} = \hat{c}_{i,k} + \nu_i + \epsilon_{i,k}^r \quad (1)$$

$$s_{i,k} = \hat{s}_{i,k} + \nu_i + \epsilon_{i,k}^t \quad (2)$$

with $\nu_i \sim N(0, \rho^2)$ are random coefficients capturing individual effect and $\epsilon_i^r \sim N(0, \sigma_r^2)$ and $\epsilon_i^t \sim N(0, \sigma_t^2)$ are idiosyncratic errors. The presence of individual effects introduces a hierarchy on the coefficients. For given values of individual parameters $\nu_i$ the likelihood density of an observed CE is

$$p(c_{i,k}) = \frac{1}{\sigma^r} \phi \left( \frac{c_{i,k} - CE(k, \Theta) - \nu_i}{\sigma^r} \right) \quad (3)$$

and the likelihood density of an observed SE

$$p(s_{i,k}) = \frac{1}{\sigma^t} \phi \left( \frac{s_{i,k} - SE(k, \Theta) - \nu_i}{\sigma^s} \right) \quad (4)$$

Our choice procedure does not provide exact measurements of CE and SE values. Instead, for each value $v$, the choice list, completed by bisection, provides bounds $v^-$ and $v^+$ such that $v \in [v^-, v^+]$ and $v^+ - v^- = 5$ euros. Therefore, the likelihood $l_k^{CE}$ of an observed CE $c_{i,k}$ follows:

$$l_{i,k}^{CE} = \Phi \left( \frac{c_{i,k} - CE(k, \Theta) - \nu_i + 2.5}{\sigma_r} \right) - \Phi \left( \frac{c_{i,k} - CE(k, \Theta) - \nu_i - 2.5}{\sigma_r} \right) \quad (5)$$

and the likelihood $l_{i,k}^{SE}$ of an observed SE follows

$$l_{i,k}^{SE} = \Phi \left( \frac{s_{i,k} - V_t(k, \Theta) - \nu_i + 2.5}{\sigma_t} \right) - \Phi \left( \frac{s_{i,k} - V_t(k, \Theta) - \nu_i - 2.5}{\sigma_t} \right) \quad (6)$$

where $\Phi$ is the cumulative normal distribution.
The parameters were estimated using a Bayesian MCMC with the Hamiltonian algorithm implemented in Stan. They were sampled on the log scale, for numerical tractability, and uniform priors where considered over a plausible domain:

$$
\begin{align*}
\log \theta, \log \gamma &\sim \mathcal{U}(\log(0.1), \log(2)), \\
\log \delta &\sim \mathcal{U}(\log(0.001), \log(1)), \\
\log \beta &\sim \mathcal{U}(\log(0.1), \log(1)), \text{ and} \\
\log \frac{\sigma_r}{100}, \log \frac{\sigma_t}{100} &\sim \mathcal{U}(\log(0.01), \log(1)),
\end{align*}
$$

where $\mathcal{U}$ stands for the uniform distribution. These priors constitute weakly informative priors and essentially aim at excluding aberrant values from the parameter space.

Treatment effects were defined additively on the log scale using a set of dummy variables for each level of $\pi$. These effects were given uniform priors $\mathcal{U}(-2, 2)$, which imply multiplicative shifts in the original parameter scale ranging from approximately 1/7.4 to 7.4. Additional estimations were run with a smaller range of uniform priors for the treatment effects (-0.5, 0.5) as a robustness check and provided virtually identical results.

The estimations were based on 1000 warm ups and 1000 draws, which was found to be enough to satisfy usual indicators of good convergence. For each simulation, the parameters were exponentiated in order to be expressed on their natural-scale. The marginal treatment effects for each parameter were also computed, as differences of parameters between treatments. Estimates are taken as the mean of the posterior distributions, and inference is based on the 95% credible interval. Given that the model includes individual random effects, these credible intervals account for individual clustering of observations.

# 3 Additional results on preference data

## 3.1 Monotonicity violation index

We examined the consistency of preference data. The dilution hypotheses, DR and DT, predict that individuals will make more errors as $\pi$ decreases. To test these predictions, we introduce an index error based on violations of monotonicity in either the risk or time domain.

With our risk stimuli, monotonicity with respect to outcomes and probabilities implies that $CE(100_{0.5}20) \ge CE(80_{0.5}0)$ and $CE(80_{0.75}0) \ge CE(80_{0.5}0) \ge CE(80_{0.25}0)$, where
$CE(P)$ denotes the certainty equivalent of a risk prospect $P$. We create a dummy variable for each subject, indicating a violation of monotonicity whenever either condition is not met.

For time, monotonicity with respect to time delays requires that $SE(80_3) \ge SE(80_6) \ge SE(80_{12})$, where $SE(P)$ denotes the sooner equivalent of a time prospect $P$. We create a dummy variable for each subject, indicating a violation of monotonicity when this condition is not met. Differences between treatments in these variables will be tested using $\chi^2$ tests.⁴

Table 5 provides the statistics on these indexes measuring violations of monotonicity. Overall, monotonicity is violated by 12.5% of subjects for risk and 16.4% for time. No pattern appears regarding the value of $\pi$ and $\chi^2$ tests do not capture differences across treatments ($p = 0.507$ for risk, and $p = 0.584$ for time), even when a treatment with $\pi > 0$ are grouped ($p = 0.712$ for risk, and $p = 0.436$ for time). These results are at odds with the dilution hypothesis.

<table><caption>Table 5: Percentages of monotonicity violations for CEs and SEs across treatments</caption><thead><tr><th></th><th>π = 0</th><th>π = 0.1</th><th>π = 0.5</th><th>π = 1</th><th>Overall</th></tr></thead><tbody><tr><th>Risk</th><td>14.3</td><td>13.6</td><td>8.0</td><td>14.6</td><td>12.5</td></tr><tr><th>Time</th><td>13.1</td><td>21.0</td><td>15.9</td><td>15.8</td><td>16.4</td></tr></tbody></table>

## 3.2 Probability-dependence index

We introduce indices of deviation from the DEU and test for changes in the distribution of these indices across treatments. We start with an index of ‘probability dependence’ of elicited risk attitudes.

For probability dependence of risk attitudes, we consider $c_1$ the certainty equivalent of $80_{0.250}$ and $c_3$ the certainty equivalent of $80_{0.750}$ and consider $\frac{60-c_3}{60} - \frac{20-c_1}{20}$ as an index of probability dependence of the relative risk premium (RRP). This index equals zero if the RRP does not depend on the probabilities. It takes positive (negative) values when RRP is larger (smaller) for $p = 0.25$ than for $p = 0.75$.

The mean value of our index is 0.105 and is significantly different from zero ($p < 0.001$). This suggests that our subjects exhibit a larger relative risk premium for a probability of 0.75 than for a probability of 0.25. An ANOVA with treatment as a four level between-subject factor did not capture any difference on this index ($p = 0.608$), even when the treatments
with real incentives ($\pi > 0$) are grouped ($p = 0.474$). This suggests that deviations from DEU in terms of probability dependence are stage across treatments, which does not accord with the dilution hypothesis.

### 3.3 Non-constant impatience index

For time, we introduce two indexes of non-stationarity, that also constitutes a deviation from DEU.

We consider $s_3$, the sooner equivalent of $80_3$ and $s_{12}$ the sooner equivalent of $80_{12}$. We consider the difference of the raw monthly discount rate implied by these two sooner equivalents as an index of non constant impatience: $log(c_{3}/80)/3 - log(c_{12}/80)/12$. This index equals zero when the raw monthly discount rate remains constant and positive (negative) when it is decreasing (increasing). We consider a similar index for present bias from $s_6$, the sooner equivalent of $80_6$, and $s_6'$, the sooner equivalent such that $s_6' \sim 80_{12}$. We compare the raw monthly discount rates for the same delay of 6 months with or without the front-end delay: $log(s_6/80)/6 - log(s_6'/80)/6$. This index equals zero in the absence of a “common difference effect” and positive (negative) values when impatience is decreasing (increasing).

The mean difference in the monthly discount rate is 0.18 and statistically different from 0 according to a t-test ($p < 0.001$). An ANOVA with treatment as a four-level between-subject factor did not capture any difference on this index ($p = 0.850$), even when the treatments with real incentives ($\pi > 0$) are grouped ($p = 0.506$). This suggests that deviations from DEU in terms non-exponential discounting are stable across treatments. This result does not align with the dilution hypothesis, nor with the common-ratio effect hypothesis.
# 4 Additional econometric results on preference parameters

## 4.1 Grouping HRIS levels

The following tables report the results of estimations using different contrasts between treatments. The treatments with $0 < \pi < 1$, corresponding to Hybrid-RIS are grouped. Therefore, comparisons are made between $\pi = 0$ (hypothetical choices) and Hybrid-RIS; and between Hybrid-RIS and $\pi = 1$ (Within-RIS). These results are consistent with those obtained without grouping the Hybrid-RIS categories.

<table><caption>Table 6: Estimations with quasi-hyperbolic discounting (Mean and 95% credible interval)</caption><thead><tr><th>Model</th><th>DEU</th><th>HDRDU</th><th>HDRDU2</th></tr></thead><tbody><tr><td>Parameter</td><td>Mean [95% CI]</td><td>Mean [95% CI]</td><td>Mean [95% CI]</td></tr><tr><td>θ<sub>0</sub></td><td>0.838 [0.754, 0.932]</td><td>0.892 [0.818, 0.970]</td><td>0.893 [0.822, 0.968]</td></tr><tr><td>δ<sub>0</sub></td><td>0.054 [0.049, 0.060]</td><td>0.046 [0.038, 0.054]</td><td>0.052 [0.042, 0.061]</td></tr><tr><td>β<sub>0</sub></td><td></td><td>0.859 [0.807, 0.910]</td><td>0.845 [0.788, 0.905]</td></tr><tr><td>γ<sub>0</sub></td><td></td><td>0.667 [0.587, 0.750]</td><td>0.666 [0.589, 0.749]</td></tr><tr><td>σ<sup>r</sup><sub>0</sub></td><td>11.935 [10.796, 13.228]</td><td>10.736 [9.661, 11.931]</td><td>10.711 [9.665, 11.908]</td></tr><tr><td>σ<sup>t</sup><sub>0</sub></td><td>18.687 [16.888, 20.592]</td><td>18.442 [16.775, 20.339]</td><td>18.476 [16.724, 20.342]</td></tr><tr><td>Effects</td><td></td><td></td><td></td></tr><tr><td rowspan="2">θ</td><td>θ<sub>0</sub><sub>&lt;π&lt;1</sub> - θ<sub>0</sub></td><td>-0.306 [-0.419, -0.201]</td><td>-0.164 [-0.263, -0.069]</td><td>-0.164 [-0.259, -0.075]</td></tr><tr><td>θ<sub>1</sub> - θ<sub>0</sub><sub>&lt;π&lt;1</sub></td><td>0.102 [0.006, 0.206]</td><td>0.043 [-0.051, 0.136]</td><td>0.041 [-0.049, 0.132]</td></tr><tr><td rowspan="2">δ</td><td>δ<sub>0</sub><sub>&lt;π&lt;1</sub> - δ<sub>0</sub></td><td>-0.016 [-0.023, -0.010]</td><td>-0.010 [-0.020, -0.001]</td><td>-0.002 [-0.014, 0.009]</td></tr><tr><td>δ<sub>1</sub> - δ<sub>0</sub><sub>&lt;π&lt;1</sub></td><td>0.005 [-0.001, 0.010]</td><td>0.004 [-0.004, 0.013]</td><td>0.003 [-0.009, 0.014]</td></tr><tr><td rowspan="2">β</td><td>β<sub>0</sub><sub>&lt;π&lt;1</sub> - β<sub>0</sub></td><td></td><td>0.011 [-0.048, 0.073]</td><td>-0.019 [-0.090, 0.053]</td></tr><tr><td>β<sub>1</sub> - β<sub>0</sub><sub>&lt;π&lt;1</sub></td><td></td><td>0.018 [-0.037, 0.073]</td><td>0.032 [-0.040, 0.101]</td></tr><tr><td rowspan="2">γ</td><td>γ<sub>0</sub><sub>&lt;π&lt;1</sub> - γ<sub>0</sub></td><td></td><td>-0.156 [-0.259, -0.051]</td><td>-0.156 [-0.258, -0.054]</td></tr><tr><td>γ<sub>1</sub> - γ<sub>0</sub><sub>&lt;π&lt;1</sub></td><td></td><td>0.067 [-0.048, 0.181]</td><td>0.070 [-0.044, 0.183]</td></tr><tr><td rowspan="2">σ<sup>r</sup><sub>0</sub></td><td>σ<sup>r</sup><sub>0</sub><sub>&lt;π&lt;1</sub> - σ<sup>r</sup><sub>0</sub></td><td>2.880 [1.213, 4.509]</td><td>1.286 [-0.223, 2.743]</td><td>1.314 [-0.177, 2.732]</td></tr><tr><td>σ<sup>r</sup><sub>1</sub> - σ<sup>r</sup><sub>0</sub><sub>&lt;π&lt;1</sub></td><td>0.171 [-1.608, 1.976]</td><td>1.215 [-0.470, 2.984]</td><td>1.209 [-0.426, 3.072]</td></tr><tr><td rowspan="2">σ<sup>t</sup><sub>0</sub></td><td>σ<sup>t</sup><sub>0</sub><sub>&lt;π&lt;1</sub> - σ<sup>t</sup><sub>0</sub></td><td>0.303 [-2.001, 2.562]</td><td>0.668 [-1.672, 2.831]</td><td>0.611 [-1.630, 2.844]</td></tr><tr><td>σ<sup>t</sup><sub>1</sub> - σ<sup>t</sup><sub>0</sub><sub>&lt;π&lt;1</sub></td><td>-1.719 [-3.937, 0.692]</td><td>-2.010 [-4.299, 0.241]</td><td>-1.938 [-4.169, 0.374]</td></tr><tr><td>LL</td><td></td><td>-6947.318</td><td>-6730.106</td><td>-6730.243</td></tr></tbody></table>
## 4.2 Linear effect of $\pi$ for treatments with $\pi > 0$

The following table reports the results of estimations focusing on the treatments with $0 < \pi$ and testing a linear effect of $\pi$ on the preference parameters. In other words, the treatment variable $\pi$ is considered as a continuous variable. For each parameter $\rho$, a log-linear specification is considered: $\rho = \rho_0 e^{(k \times \pi)}$. Testing for a linear effect, captured by $k$ saves degrees of freedom and increases the statistical power.

The results reported in Table 7 capture no effect of $\pi$ on the log parameters. In the table, the estimated values $\rho_0$ correspond to the intercepts of the log linear model on each parameter. They correspond to the predicted parameter values at $\pi = 0$ when predictions come from estimations based on the treatments with $\pi > 0$. The values differ from the values estimated in other tables because of the difference between treatment $\pi = 0$ and treatments $\pi > 0$.

<table><caption>Table 7: Estimations with quasi-hyperbolic discounting, with a linear effect of $\pi$ (Mean and 95% credible interval)</caption><thead><tr><th>Model</th><th>DEU</th><th>HDRDU</th><th>HDRDU2</th></tr><tr><th>Parameter</th><th>Mean [95% CI]</th><th>Mean [95% CI]</th><th>Mean [95% CI]</th></tr></thead><tbody><tr><td>θ<sub>0</sub></td><td>0.511 [0.445, 0.582]</td><td>0.716 [0.650, 0.783]</td><td>0.715 [0.649, 0.780]</td></tr><tr><td>δ<sub>0</sub></td><td>0.036 [0.032, 0.041]</td><td>0.034 [0.027, 0.041]</td><td>0.047 [0.038, 0.057]</td></tr><tr><td>β<sub>0</sub></td><td></td><td>0.861 [0.814, 0.908]</td><td>0.811 [0.750, 0.872]</td></tr><tr><td>γ<sub>0</sub></td><td></td><td>0.493 [0.412, 0.580]</td><td>0.491 [0.408, 0.575]</td></tr><tr><td>σ<sub>0</sub><sup>r</sup></td><td>0.148 [0.133, 0.164]</td><td>0.116 [0.104, 0.130]</td><td>0.116 [0.104, 0.130]</td></tr><tr><td>σ<sub>0</sub><sup>t</sup></td><td>0.200 [0.180, 0.222]</td><td>0.203 [0.183, 0.225]</td><td>0.204 [0.184, 0.226]</td></tr><tr><th colspan="4">Effects k</th></tr><tr><td>θ</td><td>θ = θ<sub>0</sub>e<sup>(k × π)</sup></td><td>0.218 [-0.010, 0.447]</td><td>0.060 [-0.096, 0.218]</td><td>0.063 [-0.092, 0.220]</td></tr><tr><td>δ</td><td>δ = δ<sub>0</sub>e<sup>(k × π)</sup></td><td>0.165 [-0.035, 0.361]</td><td>0.178 [-0.118, 0.477]</td><td>0.112 [-0.189, 0.422]</td></tr><tr><td>β</td><td>β = β<sub>0</sub>e<sup>(k × π)</sup></td><td></td><td>0.032 [-0.049, 0.114]</td><td>0.055 [-0.051, 0.167]</td></tr><tr><td>γ</td><td>γ = γ<sub>0</sub>e<sup>(k × π)</sup></td><td></td><td>0.147 [-0.116, 0.408]</td><td>0.150 [-0.113, 0.421]</td></tr><tr><td>σ<sup>r</sup></td><td>σ<sup>r</sup> = σ<sub>0</sub><sup>r</sup>e<sup>(k × π)</sup></td><td>0.015 [-0.148, 0.174]</td><td>0.126 [-0.052, 0.293]</td><td>0.129 [-0.045, 0.297]</td></tr><tr><td>σ<sup>t</sup></td><td>σ<sup>t</sup> = σ<sub>0</sub><sup>t</sup>e<sup>(k × π)</sup></td><td>-0.168 [-0.324, -0.000]</td><td>-0.188 [-0.356, -0.023]</td><td>-0.192 [-0.350, -0.033]</td></tr><tr><td>LL</td><td></td><td>-5251.572</td><td>-5076.501</td><td>-5076.278</td></tr></tbody></table>
# 5 Experimental procedure

## 5.1 List of the sessions

The following table reports the dates of the sessions, the treatment implemented and the number of subjects.

<table><caption>Table 8: Sessions' schedule</caption><thead><tr><td>Session</td><td>date</td><td>treatment</td><td>number of subjects</td></tr></thead><tbody><tr><td>1</td><td>March 4</td><td>π = 0.1</td><td>20</td></tr><tr><td>2</td><td>March 23</td><td>π = 0.1</td><td>18</td></tr><tr><td>3</td><td>March 23</td><td>π = 0.5</td><td>17</td></tr><tr><td>4</td><td>March 25</td><td>π = 0</td><td>19</td></tr><tr><td>5</td><td>March 25</td><td>π = 1</td><td>20</td></tr><tr><td>6</td><td>March 25</td><td>π = 0.1</td><td>17</td></tr><tr><td>7</td><td>March 25</td><td>π = 0.5</td><td>12</td></tr><tr><td>8</td><td>March 30</td><td>π = 1</td><td>20</td></tr><tr><td>9</td><td>March 30</td><td>π = 0.1</td><td>13</td></tr><tr><td>10</td><td>March 30</td><td>π = 1</td><td>19</td></tr><tr><td>11</td><td>March 30</td><td>π = 0</td><td>17</td></tr><tr><td>12</td><td>April 7</td><td>π = 0</td><td>13</td></tr><tr><td>13</td><td>April 7</td><td>π = 0.5</td><td>22</td></tr><tr><td>14</td><td>April 8</td><td>π = 0</td><td>21</td></tr><tr><td>15</td><td>April 8</td><td>π = 0.5</td><td>23</td></tr><tr><td>16</td><td>April 8</td><td>π = 1</td><td>10</td></tr><tr><td>17</td><td>April 8</td><td>π = 0.1</td><td>13</td></tr><tr><td>18</td><td>April 27</td><td>π = 1</td><td>14</td></tr><tr><td>19</td><td>April 27</td><td>π = 0</td><td>15</td></tr><tr><td>20</td><td>April 27</td><td>π = 0.5</td><td>14</td></tr></tbody></table>
## 5.2 Instructions

This section provides the slides used to present the instructions to the subjects. The differences between treatments are emphasized in yellow.
Those are the instructions displayed at the start of the experiment and read by the experimentalist to the session's participants.

Specific instructions for the different treatment groups are highlighted in yellow.

Translated from French.

Hello and welcome!

Thank you for participating in this online experiment with INSEAD!
## Reminders

* Do not leave your browser for the entire duration of the experiment.

* If you have any questions or encounter technical difficulties, you may contact us by using the chat.

* You are not allowed to communicate with the other participants.

* The data collected in this experiment will be analysed and published for scientific purposes only. The data will be completely anonymized from the start of the experiment.

* To thank you for your participation, you will receive 10 euros through a PayPal transfer if you finish the experiment.

* To receive the transfer, you will need to enter a password that will collectively be communicated at the end of the experiment. Don't quit the experiment before that!

* $\pi=1$ + one choice actually played

* $\pi=0.5$ + 1 in 2 chances (50%) to see a choice actually played

* $\pi=0.1$ + 1 in 10 chances (10%) to see a choice actually played

* $\pi>1$ Present or future additional payoff, between 0 and 100 euros.

## This experiment is made of 4 independent parts.

* Part “Nutrition”

* Questions about your eating habits over the past 12 months.

* Part "Risk"

* Choices between uncertain payoffs.

* Part “Time”

* Choice between two amounts of money to be received at two different dates.

* Part “Lifestyle”

* Various questions about you

The order of those parts might be different.

Specific instructions will now be given to you for each part.
Part “Nutrition”

* This survey asks you to enter your average annual consumption of various food categories.

• It lasts about 20 minutes.

• More precise instructions will be given at the beginning of the survey.

Part “Risk”

• This part is made of 4 **sequences of 4 choices** between an **option A** which gives you an uncertain payoff and an **option B** which gives you a sure payoff.

* **For π>0:** Your answers to these questions will be used to determine your payment should the sequence in question be selected for the actual payment.

• We will illustrate this procedure in the following example.

• For π=0: These choices are hypothetical
Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €20 (100% chances)

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

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €20 (100% chances)

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

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

This means that for all option B values greater than €20, you prefer option B to option A.
Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €20 (100% chances)

<figure><img src="image_9.png" /><figcaption>OPTION B</figcaption></figure>

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

This means that for all option B values greater than €20, you prefer option B to option A.

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

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

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

Warning ! Once you have clicked on "NEXT" you cannot come back and modify your answer.

7

Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €10 (100% chances)

<figure><img src="image_13.png" /><figcaption>OPTION B</figcaption></figure>

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

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

8
Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €10 (100% chances)

<figure><img src="image_17.png" /></figure>

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €10 (100% chances)

<figure><img src="image_18.png" /></figure>

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

This means that for all option B values smaller than €10, you prefer option A to option B.
Here is an example of the questions that will follow

You must choose between option A or option B.
Click on your preferred option than validate

Option A gives you a 50% chance of winning €40 and a 50% chance of winning 0€.

Option B is certain : it gives you €15 (100% chances)

<figure><img src="image_19.png" /></figure>

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

For each option A, you will have a sequence of 4 choices with a different option B.

Be very careful. After 4 choices with the same option A, it changes !

Part “Time”

* It is made of 4 **sequences of 4 choices** between an **option A** which gives you a sure payoff in the distant future (3 months, 6 months, 1 year) and an **option B** which gives you a sure payoff sooner.

• For π>0: Your answers to these questions will be used to determine your payment should the sequence in question be selected for the actual payment.

• We will illustrate this procedure in the following example.

* For π=0: These choices are hypothetical.
You must choose between option A or option B.
You are asked to choose between:

Option A that gives you €40 in 6 months.
Option B that gives you €20 tomorrow.

OPTION A

<table><thead><tr><th></th><th></th><th></th><th>€40</th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></tbody></table>

O A

OPTION B

<table><thead><tr><th></th><th>€20</th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></tbody></table>

O B

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

You must choose between option A or option B.
You are asked to choose between:

Option A that gives you €40 in 6 months.
Option B that gives you €20 tomorrow.

OPTION A

<table><thead><tr><th></th><th></th><th></th><th>€40</th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></tbody></table>

O A

OPTION B

<table><thead><tr><th></th><th>€20</th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></tbody></table>

O B

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

This means that for all values of option B smaller than €20, you prefer option A over option B.

Warning! Once you have clicked on "NEXT", you will not be able to come back and change your answer.
You must choose between option A or option B.
You are asked to choose between:

Option A that gives you €40 in 6 months.
Option B that gives you €30 tomorrow.

OPTION A

<table><thead><tr><th></th><th></th><th></th><th>€40</th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td><td></td></tr></tbody></table>

O A

OPTION B

<table><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>€30</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td><td></td></tr></tbody></table>

O B

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

You must choose between option A or option B.
You are asked to choose between:

Option A that gives you €40 in 6 months.
Option B that gives you €30 tomorrow.

OPTION A

<table><thead><tr><th></th><th></th><th></th><th>€40</th><th></th><th></th></tr></thead><tbody><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td><td></td></tr></tbody></table>

O A

OPTION B

<table><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>30€</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td><td></td></tr></tbody></table>

O B

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

This means that for all values
of option B greater than €30,
you prefer option B.
You must choose between option A or option B.

You are asked to choose between:

Option A that gives you €40 in 6 months.

Option B that gives you €25 tomorrow.

OPTION A

<table><thead><tr><th></th><th></th><th></th><th>€40</th><th></th><th></th></tr></thead><tbody><tr><td></td><td>Tomorrow</td><td>3 months</td><td style="background-color: #d9e2f3;">6 months</td><td>9 months</td><td>12 months</td></tr></tbody></table>

O A

OPTION B

<table><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td></td><td>25€</td><td></td><td></td><td></td><td></td></tr><tr><td style="background-color: #d9e2f3;">Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td><td></td></tr></tbody></table>

O B

NEXT

The descriptive text above the options corresponds to the two tables. To validate the option you prefer, simply click on the A or B button that corresponds to your choice and confirm your answer by clicking on the "NEXT" button. Once you have validated, you will be done with the example and you will be able to start answering the questions according to your preferences.

For every option A, you will have a sequence of 4 choices with a different option B.

Be very careful. After 4 choices with the same option A, it will change !

## Part “Lifestyle”

* The last survey is made of various questions about your lifestyle

* Some of these questions are personal. Feel free not to answer if you do not want to.

* It lasts about 10 minutes.

Warning, once you are done with this last part, stay online to proceed to the payment !
Payment

* If you finish the whole experiment, you will receive 10 euros for compensating you your time.

* The transfer will be made tomorrow to your PayPal account. For π=0: Instructions are stopped here.
For π>1.

In addition to this compensation :

• **For π=1:** Each of you will have a question from the “risk” section or the “time” section actually paid.

• For π=0.5: Each of you has a 50% chance of having a question from the “risk” section or the “time” section actually paid.

• **For π=0.1:** Each of you has a 10% chance of having a question from the “risk” section or the “time” section actually paid.

* You will learn at the end of the experiment if you are selected to see a question really played. In that case :

• The computer will randomly draw if the question relates to the “risk” or “time” section.

• Within this section, a question will be randomly selected. The option corresponding to the preferences you indicated during the experiment will be paid.

• If it is an “risk” option, the risk will be played by the computer and the corresponding PayPal transfer will be initiated tomorrow.

• If it is a sure payoff, the PayPal transfer will be initiated either tomorrow or at the specified time.

If the “risk” part is randomly drawn

• One of the 4 options A will be randomly drawn

• For this option, a value between the maximum and minimum payoff will be randomly drawn.

• For this value, the option you prefer between option A and this value will be played.

* Therefore, it is in your best interest to provide your real preference for each choice, as to not find yourself in an undesirable situation when it comes to determining your payoffs.
## Example of payoff in the “risk” section

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

## Example of earnings in the “risk” part

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

A value will be randomly drawn among:

5, 10, 15, 20, 25, 30, 35

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

Sure payoff

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>
## If the “Time” part is drawn

* One of the 4 options A will be randomly drawn.

* For this option, a value between the maximum and the minimum short-term payoff will be randomly drawn.

* For this value, the option you prefer between the maximum payoff with a distant delay and this value with a closer delay will be played.

* Therefore, it is in your best interest to give your real preference for each choice, as to not find yourself in an undesirable situation when it comes to determining your payoffs.

## Example of earnings in the “time” part

OPTION A

<table><tr><td></td><td></td><td>€40</td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_35.png" /></figure>

OPTION A

<table><tr><td></td><td></td><td>€40</td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_36.png" /></figure>

OPTION A

<table><tr><td></td><td></td><td>€40</td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_37.png" /></figure>

OPTION B

<table><tr><td>€20</td><td></td><td></td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_38.png" /></figure>

OPTION B

<table><tr><td>25 €</td><td></td><td></td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_39.png" /></figure>

OPTION B

<table><tr><td>30 €</td><td></td><td></td><td></td><td></td></tr><tr><td>Tomorrow</td><td>3 months</td><td>6 months</td><td>9 months</td><td>12 months</td></tr></table>

<figure><img src="image_40.png" /></figure>

A value will be randomly drawn among :

5, 10, 15, 20, 25, 30, 35

<figure><img src="image_41.png" /></figure>

Option A: €40 in 6 months

Option B: €35 tomorrow
## Real incentives interface

We will now determine if you have been selected to get one of your answers on risk or time actually paid

Click on the "draw" button to see if you have been selected

DRAW

Have you been selected ?

yes

Now click on the "TIME / RISK" button to see which game was selected

TIME / RISK

The selected game is :

risk

What type of choice ? (risk or time)

## Real incentives interface

Click on the "random draw" button to see how much you've earned

RANDOM DRAW

* Situation number 1 was selected.
With option A you can get €40 with a 50% chance or 0€ with a 50% chance.

* RESULT FROM THE RANDOM DRAW : €10
The minimum amount for which you chose option B is €20.

* The minimum amount for which you chose B is above the randomly drawn number. You therefore get option A.

* Since you get option A, we will play the lottery to determine how much you have won. Click on the "activate the lottery" button

ACTIVATE THE LOTTERY

You have earned :
€40

Random draw of
a choice
Options A +
value for B

Chosen option
from your
answers.

If an uncertain
situation is
chosen, it is
played, and its
result determines
your payoff
# Summary of earnings

• For $\pi=0.5$ or $\pi=0.1$: You may be selected to have a choice played for real, with a maximum payout of 100 euros.

• All choices have the same probability to be selected.

• You do not know which choice will be drawn. If that choice is drawn, we will follow the preferences you will have indicated during the survey.

• It is in your best interest to give your real preferences for all questions, as to not be disappointed if one of the choices is played for real.

• The 10 euros payoff will be transferred to participants who reach the end of the survey, and the random draw of a question for actual payoff will take place at the end of the experiment.

27

For all treatment groups

# End of instructions

• We will provide you with a link to be copied in your browser.

• If possible, please use Chrome or Explorer browsers.

• To start the experiment, you must enter your ID number carefully. If you make a mistake, we will not be able to send you your payoff.

• Do not hesitate to raise your hand or ask questions in the private chat.

• To finish the experiment and know the amount of your payoff, you must enter a password that will be communicated in an hour.

• Therefore, there is no need to hurry. Take your time to make sure that you answer according to your preferences. It is not a race!

28
Understanding of the general
instructions

• May I ask questions during the experiment if I need help ?

YES: using the chat

• How many parts is this experiment made of ?

FOUR

• What is the monetary compensation for your participation ?

10 euros if you answer all parts.

• Can we win more or less than these 10 euros ?

For $π>0$: Yes. If you are selected to receive a paid choice, you may receive up to 100 additional euros through PayPal, issued tomorrow or later.

For $π=0$:No

• What are my chances to get a paid question?

For π=1: 100% chances : it is certain that one question will be paid.

For π=0.1: I have a 1 in 10 chances to get an paid question.

For π=0.5 : I have a 1 in 2 chances to actually get a paid question.