Lottery- and survey-based risk attitudes linked through a multichoice elicitation task

Abstract

We analyze the results from three different risk attitude elicitation methods. First, the broadly used test by Holt and Laury (2002), HL, second, the lottery-panel task by Sabater-Grande and Georgantzis (2002), SG, and third, responses to a survey question on self-assessment of general attitude towards risk (Dohmen et al. 2011). The first and the second task are implemented with real monetary incentives, while the third concerns all domains in life in general. Like in previous studies, the correlation of decisions across tasks is low and usually statistically non-significant. However, when we consider only subjects whose behavior across the panels of the SG task is compatible with constant relative risk aversion (CRRA), the correlation between HL and self-assessed risk attitude becomes significant. Furthermore, the correlation between HL and SG also increases for CRRA-compatible subjects, although it remains statistically non-significant.

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Fig. 1

Notes

  1. 1.

    Weber et al. (2002) developed a psychometric scale that assesses risk taking in five different domains: financial decisions (separately for investing vs. gambling) and health/safety, recreational, ethical, and social decisions. Nevertheless, their elicitation method is unidimensional and is the same across domains, except that the framing of the question used to elicit self-assessed risk attitude is different for each specific domain.

  2. 2.

    For a recent example of five elicitation methods and reference to such results, see Crosetto and Filippin (2016). As in previous experimental studies on the topic, they also find that the estimated risk aversion parameters vary greatly across tasks.

  3. 3.

    For example, regression to the mean has been found to affect repeated choices in the same task by García-Gallego et al. (2011). Moreover, Lévy-Garboua et al. (2012) found a significantly higher elicited risk aversion in sequential than in simultaneous treatment, in decreasing and random than in increasing treatment, in high than in low-payoff condition. Their findings suggest that subjects use available information that has no value for normative theories. Cox et al. (2014) have rationalized some of these findings by showing the role of the payment mechanism in these distortions. Indeed, they find that random-lottery incentive mechanisms—as those usually employed in risk-elicitation tasks—may decrease the proportion of risky choices in the population, compared to a one-task design. This could explain why significantly more risk aversion emerges under multiple-task than under one-task elicitation methods.

  4. 4.

    See Butler and Loomes (2007) and Cubitt et al. (2015).

  5. 5.

    See Harrison et al. (2005) and Holt and Laury (2005): the former demonstrated and the latter confirmed the possibility of order effects in HL’s original design. Indeed, the order effect (participating in a low-payment choice before making a high-payment choice) magnifies the scale effect (scaling up real payments by 10 or 20 times).

  6. 6.

    A number of studies have shown that this assumption is questionable [see Wakker (2010) for a review], although other models (e.g., prospect theory) do not seem to have significantly higher explanatory power than the expected utility (Harrison and Rutström 2009).

  7. 7.

    Notice that this problem would emerge also in the absence of enforced monotonicity. In fact, in Holt and Laury (2002) a subject who switches from one lottery to the other more than once, as the probability of the best outcome increases, is still considered as if he/she has made just one switch. This is done by assigning as switch line from one lottery to the other the one corresponding to the number of safe choices the subject has made.

  8. 8.

    See Sect. 5.3 of Attanasi et al. (2014) for a discussion on the participants’ higher trust in physical rather than computerized instruments when facing random processes in laboratory experiments. Indeed, when implementing random processes in our experiment, we mainly relied on physical rather than computerized instruments. However, the subject’s relevant choice for the final payment was selected among all those made in the task (i.e., 1 over the 19 lines of the battery of lotteries) through a computerized random draw. This was made to limit the length of the experiment and to allow sufficient heterogeneity in subjects’ final payment, especially to avoid “collective winner effects” due to emotions documented in Zeelenberg and Pieters (2004). The same motivations hold for the computerized random draw of the relevant panel of lotteries in task 2 (SG).

  9. 9.

    In Sabater-Grande and Georgantzis (2002), the payoffs are expressed in pesetas, since the experimental sessions were run in Spain before the introduction of the euro as official currency in the European Union. In the cited follow-up studies, still run in Spain, the payoffs are equivalently expressed in euros.

  10. 10.

    In contrast to the claims of Diecidue and Ven (2008), while prior works have shown that typical reference points are individuals’ own current wealth or endowment (Kahneman and Tversky 1979), recent research has included aspiration levels in the literature on risky choice as reference points (Brown et al. 2012; Hoffmann et al. 2013).

  11. 11.

    Diecidue et al. (2015) report results from two experiments without finding support for an aspiration level at zero outcome, neither for simple lotteries nor for complex lotteries.

  12. 12.

    A numerical example is available upon request.

  13. 13.

    Principal component analysis uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. This transformation is defined in such a way that the first principal component has the largest possible variance (i.e., it accounts for as much of the variability in the data as possible) and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors are an uncorrelated orthogonal basis set.

  14. 14.

    To account for the ordered nature of our dependent variable, we have also estimated an ordered logit model: the results are qualitatively unchanged and available upon request.

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Authors

Corresponding author

Correspondence to Giuseppe Attanasi.

Additional information

G. Attanasi gratefully acknowledges financial support by the European Research Council (ERC) [Starting Grant DU 283953] and by the project “Creative, Sustainable Economies and Societies” (CSES) coordinated by Robin Cowan, funded through the University of Strasbourg IDEX Unistra. The authors wish to thank Omotuyole Ambali, Luca Corazzini, Astrid Gamba, Glenn W. Harrison, Lucio Picci, Matteo Rizzolli, and Luca Stanca for their useful comments and inputs.

Appendix

Appendix

Appendix A: figures and tables

See Figures 2, 3 and Tables 9, 10, 11.

Fig. 2
figure2

Task 1, variant of Holt and Holt and Laury (2002)

Fig. 3
figure3

Task 2, Sabater-Grande and Georgantzis (2002)

Table 9 Elicited r and distribution of choices for HL
Table 10 Elicited r for SG
Table 11 Distribution of choices for SG

Appendix B: CRRA-compatible patterns in SG

Recall from Sect. 3.2.1 that the monetary gains x(p) of the SG task are designed to offer, for each winning probability p, a linearly increasing expected value in the losing probability \((1-p)\), according to

$$\begin{aligned} {p\cdot x(p)=c+(1-p)t }, \end{aligned}$$
(1)

where t is a panel-specific constant representing the incentive to accept riskier options and c is a global constant (in our design, 1€) from which all lottery panels begin at the sure \((p=1)\) payoff.

Thus, for a CRRA utility \(u(x)=\frac{x^{1-r}}{1-r}\) with \(r<1\) (\(r=0\) implies risk neutrality and \(r<0\) risk-loving behavior), using (1) above re-written as \(x(p)=\frac{c+(1-p)t}{p}\), the functional to maximize for a subject whose behavior is compatible with the expected utility maximization becomes

$$\begin{aligned} {v(x,p)=p\cdot u(x)=p\cdot \frac{{\left( \frac{c+(1-p)t}{p}\right) }^{1-r}}{1-r}}. \end{aligned}$$

The second derivative with respect to p is

$$\begin{aligned} {\frac{{\partial }^2v}{\partial p^2}= -\frac{r (c+t)^2 {\left( \frac{c+t-pt}{p}\right) }^{-1-r}}{p^3}}, \end{aligned}$$

indicating, for \(r \in (0,1)\), an interior maximum at a probability satisfying the first-order condition \(\frac{{\partial }v}{\partial p}=0\), given by

$$\begin{aligned} {p^{*} = \frac{r(c+t)}{t}}, \end{aligned}$$
(2)

which, if expressed in terms of r, gives us the risk-aversion parameter as an increasing function of the probability chosen

$$\begin{aligned} r=\frac{pt}{c+t}. \end{aligned}$$

Notice that the derivative of the optimal p in (2) with respect to t (given by \(-\frac{cr}{t^2}\)) is negative, and thus the optimal probability for a risk-averse subject (\(r \in (0,1)\)) is a decreasing function of t, implying the predicted pattern of choosing riskier lotteries as one moves from panel 1 to panel 4. For a risk-neutral (\(r=0\)) and a risk-loving (\(r<0\)) subject the optimal p in (2) is, respectively, equal to and lower than 0. Given the constraints on the discrete set of available probabilities for each panel (\(p \in \{0.1, 0.2,\ldots , 1\}\)), a risk-neutral or risk-loving subject should choose the lottery at the far right extreme of each panel in Fig. 3 (\(p=0.1\)).

Appendix C: Experimental Instructions

Welcome and thank you for participating in this experimental session. By following the instructions you will earn an amount in euros that will be paid in cash at the end of the session.

Your earnings will be based entirely on your decisions: decisions of other participants will not affect your earnings.

Decisions and earnings of each participant will remain anonymous throughout the session.

Please turn off your cell phones and do not talk or in any way communicate with other participants.

If you have a question or problem at any point in this experiment, please raise your hand and one of the assistants will answer you.

The following rules are the same for all participants.

General rules

In this session, you will participate in two different tasks.

Only one of the two tasks will be used to determine your final earnings.

More specifically, at the end of the experiment we will randomly select the task to pay to all participants by flipping a coin.

Now, we give you the instructions for Task 1. You will receive the instructions for Task 2 at the end of Task 1.

Instruction for Task 1

The following figure reports the computer screen for Task 1.

It shows 19 pairs of lotteries, numbered from line L1 to line L19. Each pair is composed of lottery A and lottery B, respectively.

All lotteries have the same structure. Each lottery consists of 20 numbered tickets and two prizes and involves randomly drawing a single ticket. The 20 tickets are in an envelop (Envelop 1) that you can check before the random draw.

For each line L1–L19, lottery A always gives the same two prizes, namely 12.00 euros and 10.00 euros, and lottery B always gives the same two prizes, namely 22.00 euros and 0.50 euros.

For each lottery, the computer screen shows how many tickets have been assigned to each prize. Within each pair, the number of tickets assigned to the highest prize of the lottery is the same for lottery A and lottery B, and corresponds to the line number, e.g., 1 ticket in L1 and 19 tickets in L19.

Please indicate the line starting from which you prefer playing lottery B rather than lottery A.

This means that: for all pairs of lotteries from L1 until the line before the indicated one, you would play lottery A; for all pairs of lotteries from the indicated line until L19, you would play lottery B.

In particular, if you indicate L1, it means that you would play lottery B for every possible line; if you indicate L20 (last empty line), it means that you would play lottery A for every possible line.

figurea

At the end of the experimental session, if this task will be selected for payment, your earnings will be determined as follows:

  • The computer will select randomly and with equal probability one of the 19 lines.

  • Given the line selected by the computer, your choice will be used to determine the lottery in which you will participate.

  • One of the assistants will draw randomly and with equal probability one of the 20 tickets from Envelop 1. The drawn ticket will determine the prize you will win in the lottery in which you have chosen to participate.

Instructions for Task 2

The following figure reports the computer screen for Task 2.

It shows four panels of ten lotteries.

In each panel, each column indicates a lottery.

All lotteries have the same structure. Each lottery consists of ten numbered tickets and two prizes and involves randomly drawing a single ticket. The ten tickets are in an envelop (Envelop 2) that you can check before the random draw.

The lowest prize is 0 euros for each lottery, while the highest prize is a positive amount of euros, this amount being different for each lottery.

For each lottery, the computer screen shows the probability of winning and the positive amount of euros you can win.

The probability of winning indicates the percentage of tickets assigned to the highest prize. For example:

  • 100% means that all the ten tickets are assigned to the highest prize; thus, whatever the drawn ticket, you win the corresponding positive amount of money;

  • 50% means that if the drawn ticket is from 1 to 5 (5 included), you win the correspondent positive amount of money; if it is from 6 to 10, you win nothing.

  • 10% means that if the drawn ticket is no. 1, you win the corresponding positive amount of money; if it is from 2 to 10, you win nothing.

For each of the four panels of lotteries below, please indicate the lottery you would like to play.

figureb

At the end of the experimental session, if this task will be selected for payment, your earnings will be determined as follows:

  • The computer will select randomly and with equal probability one of the four panels of lotteries.

  • Given the panel selected by the computer, your choice will be used to determine the lottery in which you will participate.

  • One of the assistants will draw randomly and with equal probability one of the ten tickets from Envelop 2. The drawn ticket will determine the prize you will win in the lottery in which you have chosen to participate.

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Attanasi, G., Georgantzís, N., Rotondi, V. et al. Lottery- and survey-based risk attitudes linked through a multichoice elicitation task. Theory Decis 84, 341–372 (2018). https://doi.org/10.1007/s11238-017-9613-0

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Keywords

  • Risk aversion
  • Elicitation methods
  • Lottery choices

JEL Classification

  • D81
  • C91