2. Risk aversion, prudence and temperance

We start with a technical characterization of risk aversion, prudence and temperance using the framework in Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006). Consider a decision maker whose preferences admit the classical expected utility representation, with u being the von Neumann-Morgenstern utility function (Von Neumann and Morgenstern, Reference Neumann and Morgenstern1953). Assume further that u is an increasing and continuously differentiable function. Let w be the initial wealth of this decision maker. Let $k, \delta \gt 0$ be two constants and $\{\overline{\epsilon}_i\}$ be a list of mutually independent and zero-mean random variables. The weak preference relation over lottery pairs is represented by $\succeq$, and we use $[o_1, o_2]$ represent a lottery with two possible and equally likely outcomes o ₁ and o ₂. The decision maker is risk averse if $w \succeq w+\overline{\epsilon}_i$, $\forall w$ and $\forall \overline{\epsilon}_i$. This is equivalent to the second derivative of u being negative. Risk aversion also implies that the individual prefers to face two certain losses at different states of the world, that is, $[w-k,w-\delta]\succeq[w,w-k-\delta]$, $\forall k, \delta \gt 0$.Footnote ² The decision maker is said to be prudent if she prefers to disaggregate a sure loss and a zero-mean lottery, that is she prefers to face them in different states of the world. Formally, the preferences of a prudent decision maker imply $[w-k,w+\overline{\epsilon}_i]\succeq[w,w-k+\overline{\epsilon}_i]$, $\forall w$, k, and $\forall \overline{\epsilon}_i$, as demonstrated in Figure 1. Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006) show that this is equivalent to third derivative of u being non-negative, i.e. having a convex marginal utility function. In their words, adding $\overline{\epsilon}_i$ to a higher wealth is less painful for a prudent individual.

Fig. 1 Prudence according to Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006)

On the other hand, the decision is maker is said to be temperate if she prefers to face two statistically independent zero-mean lotteries, $\overline{\epsilon}_i$ and $\overline{\epsilon}_j$, at different states of the world. Formally, the preferences of a temperate decision-maker imply $[w+\overline{\epsilon}_i,w+\overline{\epsilon}_j]\succeq[w,w+\overline{\epsilon}_i+\overline{\epsilon}_j]$, $\forall w$, $\forall \overline{\epsilon}_i$, and $\forall \overline{\epsilon}_j$, as demonstrated in Figure 2. Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006) show that this equivalent to fourth derivative of u being less than or equal to zero.

Fig. 2 Temperance according to Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006)

3. Experimental tasks from previous literature

We begin by outlining the design aspects of key experimental studies that examine higher-order risk preferences, followed by a summary of their behavioral outcomes.

Deck and Schlesinger Reference Deck and Schlesinger(2010) use a text-based design supplemented with figures representing binary lotteries. Let $[+\epsilon,-\epsilon]$ be the zero mean lottery and k > 0 be a constant. In their prudence tasks, participants start with an initial amount (w > 0) and are presented with two equally likely states, Heads or Tails, determined by a coin flip. They are instructed to (i) choose the state under which the outcome of the zero mean lottery would be added to their initial amount, and (ii) select the state in which they would prefer an additional fixed amount (k) to be added to their earnings. For temperance tasks, a different zero mean lottery is used instead of k, and the choice procedure remains similar. The outcome of binary lotteries is determined by a spinner with a half-green (high payoff) and half-red (low payoff) configuration. Figure 3 provides an example task where participants circle their choices in the underlined sections of the italicized text. By circling “Head” (or “Tails”) along with “Same” in this task, participants exhibit prudence as they are willing to face the zero mean lottery in the favorable state. On the other hand, subjects circling “Different” instead of “Same” exhibit imprudence in this task.

Fig. 3 A compound lottery in Deck and Schlesinger Reference Deck and Schlesinger(2010)

In Deck and Schlesinger Reference Deck and Schlesinger(2014), compound lotteries are depicted using a representation similar to Figure 4 below. The divided portions of a pie shape correspond to different states, while a smaller pie within a larger one represents an additional lottery accompanying the realization of a specific state. The outcomes of these lotteries are determined using a spinner, similar to the approach in Deck and Schlesinger Reference Deck and Schlesinger(2010). This design allows for the combination of lotteries by drawing smaller pie shapes inside larger ones, enabling the construction of complex gambles. In their study, participants are presented with a choice between two compound lotteries for each task, resulting in a total of 38 tasks.

Fig. 4 A compound lottery in Deck and Schlesinger Reference Deck and Schlesinger(2014)

In Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014), compound lotteries are represented as shown in Figure 5 below. In this particular compound lottery, subjects first engage in a binary lottery with two equally likely outcomes. The first outcome (€90) occurs if the dice throw is equal to 1, 2, or 3, while the second outcome (also €90 in this example) occurs if the dice throw is equal to 4, 5, or 6. In the latter case, subjects face two additional binary lotteries, each with two equally likely outcomes determined by separate dice throws. This example illustrates a scenario where two zero mean lotteries are encountered in the same state, representing an intemperate choice. The alternative option involves facing these gambles in different states, with the second zero mean lottery, with respective outcomes €50 and -€50, received in tandem with the realization of the first outcome of the initial lottery (when the first dice throw is equal to 1, 2, or 3). This combination would indicate a temperate choice.

Fig. 5 A compound lottery in Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014)

A simplified representation of the compound lotteries used in Ebert and Wiesen Reference Ebert and Wiesen(2011) is shown in Figure 6 : the initial 50/50 gamble is depicted as a ballot box containing two balls labeled “Up” and “Down.” If the “Up” ball is drawn, the subject incurs a loss of 2, and a second zero-mean risk lottery follows. The second lottery is represented by another ballot box containing multiple balls, with four balls shown in this simplified version. Yellow balls indicate a loss, while white balls indicate a gain. If the “Down” ball is drawn from the first ballot box, no loss occurs, and no second lottery follows. A preference for this particular compound lottery would indicate an imprudent choice.

Fig. 6 A compound lottery in Ebert and Wiesen Reference Ebert and Wiesen(2011)

In a recent experimental study, Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) investigate whether higher-order risk preferences exhibit differences between gain and loss domains, and examine the existence of a reflection effect similar to the one observed for risk aversion. Figure 7 presents a simplified representation of the options used in their experiment. In this representation, lotteries with binary outcomes are depicted as the act of drawing a random ball from an urn. In the given example, the subject is confronted with a binary lottery offering a payout of €5 or €10. If the latter outcome is realized, the subject then faces an additional binary lottery where there is an equal probability of winning or losing €3. This combination would indicate a prudent choice since the subject faces the second lottery in tandem with the good outcome of the first lottery.

Fig. 7 A compound lottery in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022)

In addition to the lottery pairs approach described in the previous examples, alternative methods have been employed for the experimental measurement of higher-order risk preferences. In Ebert and Wiesen Reference Ebert and Wiesen(2014), a multiple price list approach is utilized, where uncertainties are resolved through draws from ballot boxes. The graphical representations of these lotteries are similar to Figure 6, but each time a subject compares two such lotteries, varying amounts of compensation are added to one of the two lotteries, forming a price list. Another approach is presented in Schneider et al. Reference Schneider, Ibanez and Riener(2022), where certainty equivalents are initially elicited and used to derive utility points. These utility points are then employed in a spline regression to construct a non-parametric utility function. From this function, n^th order derivatives can be obtained, enabling the classification of higher-order risk preferences. This methodology is also applied by Schneider and Sutter Reference Schneider and Sutter(2020).

The experimental findings from Deck and Schlesinger Reference Deck and Schlesinger(2010) indicate evidence of prudence among subjects, although to a limited extent. However, intemperate behavior is observed more frequently than temperance. In their subsequent study, Deck and Schlesinger Reference Deck and Schlesinger(2014) find that subjects display a strong inclination towards risk aversion, prudence, and a moderate preference for temperance. They also identify a significant correlation between risk-averse and temperate behavior. Haering et al. Reference Haering, Heinrich and Mayrhofer(2020) replicate this setup among subjects from the USA, China, and Germany and they observe a behavior similar to the findings of Deck and Schlesinger Reference Deck and Schlesinger(2014). They also explore the reduced-form equivalents of compound lotteries for a subset of participants, highlighting certain behavioral patterns, such as reduced prevalence of prudent and temperate choices, that are influenced by this framing.

Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) reports similar findings in their sample, which includes both individuals from the general population and a group of students. They observe a high degree of prudence among both groups, while temperance is less prevalent. Ebert and Wiesen Reference Ebert and Wiesen(2011) observe an aggregate level of prudence comparable to that reported in Deck and Schlesinger Reference Deck and Schlesinger(2010). They also find that most prudent individuals exhibit a skewness seeking behavior, while the reverse may not generaly hold. Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) examine higher-order risk attitudes under three treatments: losses, 50-50 gains, and small probability gains. The 50-50 gains treatment aligns with previous studies utilizing binary lotteries. In this treatment, subjects exhibit significant risk aversion, and moderate prudence, but also a tendency toward intemperate behavior.

In summary, the general results from these studies suggest that prudence is commonly observed, while the presence of temperance is less consistent. Several studies demonstrate a higher frequency of intemperate behavior. A comprehensive review of the literature on higher-order risk preferences can be found in Trautmann and van de Kuilen Reference Trautmann and van de Kuilen(2018).

4. Fork Game: Design

In our experimental design, lotteries are represented by pipes, which were typically forked as in Figure 8, with some variations on their horizontal size. In this figure, a lottery with two equally likely outcomes that has the payments of $[2,-2]$ is represented.Footnote ³

Fig. 8 A simple lottery, represented with a pipe

In all the games corresponding to each of the concepts, the resolution of uncertainty is represented by a falling ball which goes left or right with probability 50% whenever it encounters a fork on its path.

The task in each round of the Risk Aversion treatment is straightforward and the subject is asked to choose one forking pipe amongst two, differing in the payoffs shown in their tags. The main advantage of our design does not show itself in this game, but it serves as a nice introduction for the next two games, which will use the same image files and similar designs.

In a given round of the Prudence treatment, the subject is expected to place a small forking pipe after one of the two ends of a big forking pipe. The first and bigger pipe represents the first lottery with each side assigned a payoff as usual. The slots to place the smaller pipe are aligned directly under the exits of the bigger pipe, so the ball can continue its path, if it happens to reach the second pipe.

In Figure 9, a sample round for Prudence treatment is shown. The subject can construct one of two distinct lotteries by placing the smaller pipe to the slot at the left end or the right end of the larger pipe: [(3 + [2, −2]) ,9] and [3, (9 + [2, −2])]. In this example, we see that the subject went with the former. From the figure, we (and in the experiment the participant) also see the result of this specific round, since the ball reached to the right hand side, gaining the participant payoff of 9 for this specific round.

Fig. 9 End result for a prudence round

Finally, for our Temperance treatment we introduce another type of pipe, as shown in Figure 10. The main point of introducing this pipe is to allow sublotteries to be placed after one another, independent from their result. While the payoff in this case is same with the case with Figure 8, here there is additional utility that we can chain one lottery after another.

Fig. 10 Pipes for the temperance treatment

In the Temperance treatment, the participant has to choose the placement for two forking pipes given, and there are four slots to choose from. However not all subsets of these four slots are available for selection, since the below two slots are only available if the slot directly above is already filled.

In Figure 11, the end of an example temperance round is shown. Here, the participant is presented two lotteries:[(5 + [2, −2]), (5 + [2, −2])] and [5, (5 + [2, −2] + [2, −2])] and from what we observe they went with the latter one. Note that, the payoffs are symmetric here, so it does not make a difference if the lotteries are placed one after another in the right side or the left side. The random selection chose the right side, resulting in a total payoff of 5. If left side of this contraption was chosen randomly instead, they would face with two additional lotteries of [2, −2].

Fig. 11 End result for a temperance round

This setup is easily extendable to represent risk apportionment of any degree so as to elicit even higher order risk preferences. In the Online Appendix, we describe both theoretically and graphically how the Fork Game can be used to elicit edginess attitudes, which is equivalent to 5^th derivative of the respective von Neumann-Morgenstern utility function, $u(.)$, being positive.

5. Experimental procedures

The experiments were conducted in the Economics Laboratory at Boğaziçi University in two separate phases. The first phase took place between December 2022 and March 2023, while the second phase occurred between November 2024 and December 2024. The experimental sessions in the first phase involved the Fork Game, whereas those in the second phase included either the Fork Game, or a replication of the experiment in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022), or a replication of the experiment in Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). In replicating Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014), we closely followed their original design aspects. However, in our implementation, uncertainties were resolved immediately, and subjects observed their round-specific earnings at the end of each round. This design choice ensured consistency across all experimental formats in our study. For example, during the replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022), the balls selected from the urns were highlighted to indicate the outcome.

Participants were recruited via the online recruiting system ORSEE (Greiner, Reference Greiner2015). A total of 402 subjects participated in the experiments: 134 in the first phase and 268 in the second phase, over 67 sessions.Footnote ⁴ All participants were students at Boğaziçi University, with an average age of 21.01. Among them, 47.3% identified as female and 52.7% as male.

In all experiments, participants first encountered the Risk Aversion treatment, followed by the Prudence and Temperance treatments. The order of these latter two treatments was randomized for each participant, as in Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). Lotteries within each treatment were also presented in a random order. Additionally, the graphical placement of options and the placement of potential rewards within these options were randomized. For instance, in the Fork Game, rewards were randomly assigned to the forking pipes and then for each lottery pair, the values were assigned to the left or right side of a pipe with equal probabilities.

Each treatment consisted of 12 rounds. The payoff structure was based on the 50–50 gains treatments described in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022), with the full set of parameters detailed in the Online Appendix. Payoffs were presented as “points” in all experiments. After completing the three treatments, participants answered demographic questions and completed the full set of the Global Preferences Survey. All treatments and survey questions were presented in Turkish.

The experimental software was developed as a web application in JavaScript using the Vue.js framework. It was deployed on GitHub and accessed via standard Chrome browsers in full-screen mode.Footnote ⁵ The software also included the main introduction and separate instructions for each of the three treatments.

Participant compensation included earnings from one randomly selected round plus a participation fee. To account for Turkey’s significant inflation, we adjusted the point-to-currency conversion: in the first phase, each point equaled 1.5 Turkish Lira (TL) with a 20 TL participation fee, while the second phase used 3 TL per point with a 40 TL participation fee. Average earnings including participation fee were 52.5 TL for the first phase, and 105.6 TL for the second phase. The Global Preferences Survey’s monetary values, originally from the 2012 Gallup World Poll, were also inflation-adjusted: multiplied by 5 for the first phase and by 10 for the second phase.Footnote ⁶

7. Comparison with the experimental tasks in the literature

To provide a broader comparison of methodologies, we carried out experiments that applied the designs of both Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014), using the same lottery settings. In this section, as well as in the Online Appendix, the terms Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) refer to the sessions we executed using these designs and the resulting data, rather than their original datasets.

The comparison with our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and our replication of Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) focuses on three aspects: the prevalence of risk-averse, prudent, and temperate choices; the distributional patterns of these choices; and operational aspects of the experiments, including task difficulty and completion times.

The results presented in Table 6 show broadly consistent patterns of risk preferences across all three experimental designs. In the Fork Game, subjects make risk-averse choices in 7.90 out of 12 decisions, comparable to 7.98 in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and 7.66 in Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). As shown in Table 7, these differences are not statistically significant ( $p-value = 0.71$ and $p-value = 0.56$, respectively). A notable difference emerges in prudence choices: subjects in the Fork Game and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) exhibit higher levels of prudence (9.22 and 9.25 respectively) compared to Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) (6.90). For temperance, all three designs yield similar results, with subjects making temperate choices in approximately 5-6 out of 12 decisions, with no statistically significant differences between the designs.

Table 6 Choices: Risk Aversion, Prudence and Temperance Across Experimental Designs

Notes: We report risk-averse, prudent, and temperate choices for each experiment: Fork Game, our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and our replication of Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014).

* and *** indicate the average number of choices significantly different from random choice (6 in our experiment), at 0.1, and 0.01 respectively.(Wilcoxon test)

The distribution of choices, illustrated in Figure 14, reveals several interesting patterns across the three designs. Panel A shows that the Fork Game exhibits a right-skewed distribution for risk-averse choices, with a peak at 7 choices (15.3% of observations) and about 71% of subjects making more than six risk-averse choices. This pattern aligns with Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014), though with slightly different concentrations around the peak. For prudent choices (Panel B), the Fork Game and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) show similar distributions with pronounced peaks at higher values, with approximately 84% of subjects making more than six prudent choices. The distribution of temperate choices (Panel C) shows consistency across all three designs, with modes typically falling between 4–5 choices.

Table 7 Comparison of choices between designs

Notes: This table reports the pairwise comparisons of risk-averse, prudent, and temperate choices between the three experimental designs, Fork Game and our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). The first numbers in each cell represent the mean number of choices for each design, and p-values from Wilcoxon rank-sum tests are reported in parentheses.

While all three experiments use the same underlying lottery structures derived from Eeckhoudt and Schlesinger Reference Eeckhoudt and Schlesinger(2006), the visual representation and framing differ. The Fork Game presents choices through the physical placement of forked pipes, Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) use colored balls and urns, and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) employ a decision-tree representation. The similarity between prudence levels in the Fork Game and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014), despite their different visual approaches, suggests that certain presentation methods might facilitate the understanding of prudence concepts differently.

Fig. 14 Histogram of choices across different experiments. Panel A: Fork Game (a) Risk aversion. (b) Prudence. (c) Temperance. Panel B: Our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022): (d) Risk aversion. (e) Prudence. (f) Temperance. Panel C: Our replication of Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014): (g) Risk aversion. (h) Prudence. (i) Temperance.

Our findings can be compared with those reported in Haering et al. Reference Haering, Heinrich and Mayrhofer(2020), who study the consistency of higher-order risk preferences across different presentation formats. For risk aversion, we observe similar levels, with approximately 66% risk-averse choices in our Fork Game compared to their 65–70%. However, we find somewhat stronger evidence of prudence, with about 77% prudent choices compared to their 70–75%. This pattern reverses for temperance, where we observe lower levels (47% temperate choices) compared to their findings of about 55%. Differences in prudence and temperance levels between studies may reflect the impact of experimental design on preference elicitation, a point also emphasized by Haering et al. Reference Haering, Heinrich and Mayrhofer(2020) in their comparison of compound versus reduced-form lottery presentations. In particular, our finding that the Fork Game and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) exhibit higher levels of prudence than Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) aligns with their observation that the presentation format can significantly influence the elicitation of higher-order risk preferences.

Regarding operational aspects, as reported in Table 8, Table 9 subjects rate the perceived difficulty of the Fork Game at 3.70 (on a scale of 0–10) compared to 4.20 in Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) and 4.38 in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022). Pairwise comparisons using Wilcoxon tests reveal that the Fork Game’s lower difficulty rating is statistically significant when compared to Bleichrodt and Bruggen (2022) ( $p-value = 0.01$) and marginally significant compared to Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) ( $p-value=0.09$). The difference between Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) and Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) is not statistically significant ( $p-value = 0.50$).

Table 8 Operational aspects of designs

Notes: This table reports operational measures for each experimental design, Fork Game and our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). Panel A shows the self-reported difficulty ratings on a scale of 0–10, where higher numbers indicate greater perceived difficulty. Panel B shows the average time taken to complete a round in seconds, including the entire decision-making process and outcome resolution.

The analysis of completion times reveals varying patterns across treatments as shown in Table 9 and illustrated in Figure 15. For risk aversion tasks, average response times are relatively similar across all three designs (5.50, 5.15, and 4.68 seconds for Fork Game, Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022), and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) respectively). The average round time in the Fork Game (6.58 seconds) is lower than both Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) (6.89 seconds) and Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) (7.96 seconds). While the difference between the Fork Game and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) is not statistically significant ( $p-value=0.32$), both designs show significantly faster completion times compared to Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) ( $p-value \lt 0.001$ for both comparisons).

Fig. 15 Average round time for each treatment and game

To examine these differences more rigorously, pairwise comparisons of completion times for each task (1–12 in Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022)) were performed between designs.Footnote ⁷ The results indicate that differences in completion times are more pronounced in prudence treatment, where they are statistically significant for most tasks ( $p-value \lt 0.01$). The differences are less consistent in the temperance treatment, though the Fork Game generally maintains faster completion times.

Table 9 Average round times

Notes: This table reports detailed round completion times for each treatment across experimental designs, Fork Game and our replication of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022) and Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014). All times are reported in seconds and include both the decision-making process and outcome resolution. The summary statistics include all rounds for all participants in each treatment. For task-specific comparisons, see Tables 8–10 in the Online Appendix.

Further evidence on the robustness of our findings is provided in the Online Appendix, where we present detailed task-by-task comparisons and regression analyses. The parameter-matched comparisons reveal consistent patterns in elicited preferences across different presentation formats. Task-level analysis shows that the higher prudence levels in the Fork Game and our replication of Noussair et al. Reference Noussair, Trautmann and Van de Kuilen(2014) persist across different parameter combinations, suggesting this finding reflects a genuine difference in how prudence is understood across different presentation formats rather than an artifact of specific parameter choices. Regression analyses demonstrate consistent relationships between individual characteristics and risk preferences across all three experimental designs, particularly in the relationship between general risk attitudes and both risk-averse and temperate choices.

These comparisons suggest that the Fork Game maintains the ability to elicit higher-order risk preferences consistent with existing experimental designs. The strong correlation between risk-averse choices and similar patterns in temperance across all three designs provides evidence for the robustness of these elicitation methods. Differences in prudence levels and completion times between designs warrant further investigation to better understand how various presentation methods might affect the elicitation of higher-order risk preferences.

The consistency of the results in risk aversion and temperance between designs suggests that the divergence in prudence levels is not due to systematic differences in the subject groups or general risk attitudes. Rather, these differences might be attributed to specific aspects of how prudence choices are presented in each design. Further research could explore which elements of different experimental designs most effectively facilitate the elicitation of higher-order risk preferences.

8. Summary and discussion of results

In this study, we introduce the “Fork Game,” a novel graphical device designed to elicit higher-order risk preferences. Our method utilizes forking pipes to represent the randomization aspect of the experiments where the resulting outcome is represented by a falling ball within those pipes, taking a randomly chosen path at the forks. In the treatments, participants are tasked with placing the pipes to construct the desired lottery, which is equivalent to the lottery pairs used in the previous literature. By visualizing the randomization aspect and the chosen lottery clearly, and designing all games as direct variations of the same underlying concepts, we can extend the games in a straightforward manner to analyze even more complex risk attitudes without significantly increasing the procedural complexity.

Tasks aimed at eliciting higher-order risk preferences often involve complex gambles, which can lead to distinct decision-making patterns compared to simplified versions of the tasks. The comprehension of the overall consequences of complex gambles poses a challenge for subjects. Additionally, task difficulty is a relevant factor to consider. In our study, we believe that our task is relatively simple, as the average reported difficulty is 3.7 out of 10 and no participant has reported a rating above 8 in our experiments.

Our laboratory experiments reveal consistent results with previous findings regarding higher-order risk preferences. Subjects in the “Fork Game” exhibit a high prevalence of risk aversion and prudence, while temperance is less common. Intemperate choices are observed among a significant fraction of subjects. Our results align qualitatively and quantitatively with those of Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022), indicating a positive and statistically significant correlation between our experiment and their study. Although we do not find a significant correlation for temperance, our experiment’s overall level of temperate choices is similar to that observed by Bleichrodt and Bruggen Reference Bleichrodt and van Bruggen(2022).

Furthermore, we relate binary choices in risk aversion, prudence, and temperance treatments to the demographic characteristics and responses to the Global Preference Survey (GPS). We find that the GPS risk variable is negatively associated with risk-averse and temperate choice, however, we do not observe a significant effect on prudence which further validates the observed relation between risk aversion and temperance. This observation can be attributed to the presence of mixed risk aversion and mixed risk-loving preferences, which are commonly found in the population. Mixed risk-averse individuals tend to be prudent and temperate and mixed risk-loving preferences are associated with prudence and intemperance. Despite their different risk preferences, both groups show similar attitudes towards prudence, leading to a prevalence of prudence in the population, and a positive association between risk aversion, temperance, and the overall willingness to take risks. These findings are consistent with our experimental results.