Abstract
This paper presents the Bomb Risk Elicitation Task (BRET), an intuitive procedure aimed at measuring risk attitudes. Subjects decide how many boxes to collect out of 100, one of which contains a bomb. Earnings increase linearly with the number of boxes accumulated but are zero if the bomb is also collected. The BRET requires minimal numeracy skills, avoids truncation of the data, allows the precise estimation of both risk aversion and risk seeking, and is not affected by the degree of loss aversion or by violations of the Reduction Axiom. We validate the BRET, test its robustness in a large-scale experiment, and compare it to three popular risk elicitation tasks. Choices react significantly only to increased stakes, and are sensible to wealth effects. Our experiment rationalizes the gender gap that often characterizes choices under uncertainty by means of a higher loss rather than risk aversion.
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Notes
The task is similar to, but not inspired by, the unpublished Chip Draw task proposed by Eckel et al. (2003). We were pointed to the Chip Draw task by its authors only after this paper had circulated.
This mechanism appears to have much in common with the Becker-DeGroot-Marshak (BDM) procedure (Becker et al. 1964), but this is not the case. The BDM mechanism induces subjects to truthfully reveal the reservation price for an item. It has been used as a risk elicitation task by eliciting the willingness to pay for, or the willingness to accept, a lottery ticket (see Grether and Plott (1979); Harrison (1990) for examples of its use and Karni and Safra (1987) for a critical assessment of its incentive compatibility). A random device is then used to decide whether the transaction takes place or the lottery is actually played. Instead, the BRET amounts to a choice between different lotteries and the preferred lottery is always played.
A sufficient condition ensuring an identical solution is that the expected utility function is not characterized by multiple and separate local maxima.
Even though recruitment for experiments takes place mainly on campus, the subject pool also includes some non-student workers and adults from Jena.
A z-Tree version of the BRET, though not used in the sessions, was also developed and tested to ensure the widest portability of the task. The experimental software of the BRET and its source code as well as the z-Tree version are available in the online supplementary material to be found at http://goo.gl/3eogr.
We also administered the DOSPERT risk questionnaire that weights several different domains in which risk attitudes can play a role (gambling and investment among the others), but we do not report results because it shows a correlation with the incentivized choices lower than the SOEP.
On average, the subjects who submitted dominated choices reported a higher perceived difficulty of the task compared to the rest of the sample (0.57 vs. 0.42). The difference is however not significant (Mann-Whitney, p-value 0.16).
There is instead a gender difference in the perceived complexity of the tasks, with women finding the task significantly more difficult than men. This is a result that emerges in almost all the treatments.
Indirect evidence for this interpretation can be derived from their finding that the gender gap in risk attitudes is virtually identical in a treatment supposed to be “gain only” and in a payoff equivalent treatment in which losses are explicitly considered (Eckel and Grossman 2008a).
Evidence along this line is also provided by Brooks and Zank (2005); Schmidt and Traub (2002), who find a relatively larger number of women being classified as loss averse in experimental tasks with multiple individual choices involving mixed gambles. Brooks and Zank (2005) also find a larger fraction of women being risk averse. Gaechter et al. (2010) find no evidence of a gender gap in loss aversion.
Here we omit the possibility of different curvatures in the two domains as well as issues of probability weighting, as this would go beyond the scope of this paper.
A similar procedure to induce a reference point was used, among others, by Harbaugh et al. (2010), who gave real dollar bills to the participants.
A significantly higher k∗ in the Explosion treatment could be rationalized in a Prospect Theory framework for values of the parameter such as those estimated in Tversky and Kahneman (1992). Note that the incentive structure in the Explosion treatment explicitly entails the possibility of incurring a loss in case the bomb is collected as long as subjects adjust their reference point to the amount of money earned every time they take a decision.
The alternative of showing the position of the bomb at the end of the trial period is not feasible, as we have clean evidence from a pilot experiment that it generates huge serial correlation. Subjects tend to adjust their choices as a function of the outcomes previously observed. For instance, they are prone to a form of gambler fallacy implying that they could not be lucky twice in a row thereby stopping the task earlier after having observed a relatively high b.
Note that due to rounding problems the estimated \(k^{*}_{5 \times 5}\) is actually a lower bound of the choice once reported on a 100 scale.
The Random treatment requires a predetermined sequence of deletion of the 100 cells. Subjects must therefore be made aware that, in this case, the bomb’s position cannot be automatically found in the 10 × 10 square but requires the random sequence to be disclosed for the bomb to be located.
There is no subject displaying a steadily decreasing pattern.
Many other risk elicitation mechanisms have been extensively used in the literature (for a review see Harrison and Rutström (2008)). We focus on the three above as they are fast and easy to implement and they do not result in a too high cognitive load on the subjects, and are, like the BRET, most suitable to be used as controls.Results are reported in Table 7.
Table 7 Estimates of r and risk categories for all tasks For more details and to see the different way in which the tasks map choices into r see Crosetto and Filippin (2012).
Note that the BRET itself does not induce wealth effects as long as the position of the bomb is determined at the very end of the experiment.
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Acknowledgments
We are grateful to the Max Planck Institute of Economics (Jena) for financial and logistic support and to Denise Hornberger, Nadine Marmai, Florian Sturm, and Claudia Zellmann for their assistance in the lab. We would like to thank Alexia Gaudeul and Gerhard Riener for helpful suggestions and participants at the ESA 2012 Conferences in New York and Cologne and at the Nordic Conference on Behavioral and Experimental Economics (Bergen), participants of seminars at the MPI Jena and at the University of Stirling, as well as an anonymous referee, for their comments. All remaining errors are ours.
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Appendices
Appendix A. Estimates of r for the BRET, assuming CRRA u(k) = k r
K | r | K | r | K | r |
|---|---|---|---|---|---|
1 | 0 ≤ r ≤ 0.014 | 36 | 0.551 ≤ r ≤ 0.574 | 71 | 2.39 ≤ r ≤ 2.508 |
2 | 0.015 ≤ r ≤ 0.025 | 37 | 0.575 ≤ r ≤ 0.599 | 72 | 2.509 ≤ r ≤ 2.636 |
3 | 0.026 ≤ r ≤ 0.036 | 38 | 0.6 ≤ r ≤ 0.625 | 73 | 2.637 ≤ r ≤ 2.773 |
4 | 0.037 ≤ r ≤ 0.046 | 39 | 0.626 ≤ r ≤ 0.652 | 74 | 2.774 ≤ r ≤ 2.921 |
5 | 0.047 ≤ r ≤ 0.058 | 40 | 0.653 ≤ r ≤ 0.68 | 75 | 2.922 ≤ r ≤ 3.081 |
6 | 0.059 ≤ r ≤ 0.069 | 41 | 0.681 ≤ r ≤ 0.709 | 76 | 3.082 ≤ r ≤ 3.255 |
7 | 0.07 ≤ r ≤ 0.08 | 42 | 0.71 ≤ r ≤ 0.739 | 77 | 3.256 ≤ r ≤ 3.444 |
8 | 0.081 ≤ r ≤ 0.092 | 43 | 0.74 ≤ r ≤ 0.769 | 78 | 3.445 ≤ r ≤ 3.651 |
9 | 0.093 ≤ r ≤ 0.104 | 44 | 0.77 ≤ r ≤ 0.801 | 79 | 3.652 ≤ r ≤ 3.878 |
10 | 0.105 ≤ r ≤ 0.117 | 45 | 0.802 ≤ r ≤ 0.834 | 80 | 3.879 ≤ r ≤ 4.129 |
11 | 0.118 ≤ r ≤ 0.129 | 46 | 0.835 ≤ r ≤ 0.869 | 81 | 4.13 ≤ r ≤ 4.406 |
12 | 0.13 ≤ r ≤ 0.142 | 47 | 0.87 ≤ r ≤ 0.904 | 82 | 4.407 ≤ r ≤ 4.715 |
13 | 0.143 ≤ r ≤ 0.155 | 48 | 0.905 ≤ r ≤ 0.941 | 83 | 4.716 ≤ r ≤ 5.062 |
14 | 0.156 ≤ r ≤ 0.169 | 49 | 0.942 ≤ r ≤ 0.98 | 84 | 5.063 ≤ r ≤ 5.453 |
15 | 0.17 ≤ r ≤ 0.183 | 50 | 0.981 ≤ r ≤ 1.02 | 85 | 5.454 ≤ r ≤ 5.898 |
16 | 0.184 ≤ r ≤ 0.197 | 51 | 1.021 ≤ r ≤ 1.061 | 86 | 5.899 ≤ r ≤ 6.41 |
17 | 0.198 ≤ r ≤ 0.212 | 52 | 1.062 ≤ r ≤ 1.105 | 87 | 6.411 ≤ r ≤ 7.003 |
18 | 0.213 ≤ r ≤ 0.226 | 53 | 1.106 ≤ r ≤ 1.15 | 88 | 7.004 ≤ r ≤ 7.7 |
19 | 0.227 ≤ r ≤ 0.242 | 54 | 1.151 ≤ r ≤ 1.197 | 89 | 7.701 ≤ r ≤ 8.53 |
20 | 0.243 ≤ r ≤ 0.257 | 55 | 1.198 ≤ r ≤ 1.247 | 90 | 8.531 ≤ r ≤ 9.534 |
21 | 0.258 ≤ r ≤ 0.273 | 56 | 1.248 ≤ r ≤ 1.298 | 91 | 9.535 ≤ r ≤ 10.776 |
22 | 0.274 ≤ r ≤ 0.29 | 57 | 1.299 ≤ r ≤ 1.352 | 92 | 10.777 ≤ r ≤ 12.351 |
23 | 0.291 ≤ r ≤ 0.307 | 58 | 1.353 ≤ r ≤ 1.409 | 93 | 12.352 ≤ r ≤ 14.412 |
24 | 0.308 ≤ r ≤ 0.324 | 59 | 1.41 ≤ r ≤ 1.469 | 94 | 14.413 ≤ r ≤ 17.229 |
25 | 0.325 ≤ r ≤ 0.342 | 60 | 1.47 ≤ r ≤ 1.531 | 95 | 17.23 ≤ r ≤ 21.309 |
26 | 0.343 ≤ r ≤ 0.36 | 61 | 1.532 ≤ r ≤ 1.597 | 96 | 21.31 ≤ r ≤ 27.76 |
27 | 0.361 ≤ r ≤ 0.379 | 62 | 1.598 ≤ r ≤ 1.666 | 97 | 27.761 ≤ r ≤ 39.532 |
28 | 0.38 ≤ r ≤ 0.398 | 63 | 1.667 ≤ r ≤ 1.739 | 98 | 39.533 ≤ r ≤ 68.274 |
29 | 0.399 ≤ r ≤ 0.418 | 64 | 1.74 ≤ r ≤ 1.816 | 99 | r ≤ 68.275 |
30 | 0.419 ≤ r ≤ 0.438 | 65 | 1.817 ≤ r ≤ 1.898 | ||
31 | 0.439 ≤ r ≤ 0.459 | 66 | 1.899 ≤ r ≤ 1.985 | ||
32 | 0.46 ≤ r ≤ 0.481 | 67 | 1.986 ≤ r ≤ 2.077 | ||
33 | 0.482 ≤ r ≤ 0.503 | 68 | 2.078 ≤ r ≤ 2.174 | ||
34 | 0.504 ≤ r ≤ 0.526 | 69 | 2.175 ≤ r ≤ 2.278 | ||
35 | 0.527 ≤ r ≤ 0.55 | 70 | 2.279 ≤ r ≤ 2.389 |
Appendix B. Experimental instructions
The experimental instructions were originally drafted in English, then translated into German to enable us to run the experiments in the Max Planck Institute’s lab in Jena, Germany. In what follows, we will report the original, English versions of the instructions for the Baseline versions and robustness controls. The German versions are available in the additional online material at http://goo.gl/3eogr.
1.1 B.1 Baseline BRET, static
First screen
Welcome to the Experiment. In the experiments all payoffs are expressed in euro. For your punctuality you receive 2.5 euro. The experiment consists of one short task, followed by a questionnaire. Should you have any questions or need help, please raise your hand. An experimenter will then come to your place and answer your questions in private.
Second screen
On the sheet of paper on your desk you see a field composed of 100 numbered boxes. Behind one of these boxes a time bomb is hidden; the remaining 99 boxes are empty. You do not know where the time bomb is. You only know that it can be in any place with equal probability.
Your task is to choose how many boxes to collect. Boxes will be collected in numerical order. So you will be asked to choose a number between 1 and 100.
At the end of the experiment, we will randomly determine the number of the box containing the time bomb by means of a bag containing 100 numbered tokens.
If you happen to have collected the box in which the time bomb is located – i.e., if your chosen number is greater than, or equal to, the drawn number – you will earn zero. If the time bomb is located in a box that you did not collect – i.e., if your chosen number is smaller than the drawn number – you will earn an amount in euro equivalent to the number you have chosen divided by ten.
In the next screen you will be asked to indicate how many boxes you would like to collect. You confirm your choice by hitting OK.
1.2 B.2 Baseline BRET, dynamic
The following instructions, containing no changes, were used in the Baseline dynamic, Wealth effect, and Repeated treatments. Only slight parameter and word changes were needed to adapt them to the Fast, High Stake, 5 × 5, 20 × 20, Mixed 5 × 5, No Trial and Random treatments. The slight changes are indicated within brackets in the text.
First screen:
as in Baseline static
Second screen
On the sheet of paper on your desk you see a field composed of 100 {5 × 5: 25; 20 × 20: 400} numbered boxes.
You earn 10 euro cents {5 × 5: 40 euro cents; 20 × 20: 2.5 euro cents} for every box that is collected. Every second {Fast: half a second; 20 × 20: quarter of a second; 5 × 5: four seconds} a box is collected {Mixed 5 × 5: marked for collection}, starting from the top left corner {Random: following the sequence reported on the sheet of paper}. Once collected {Mixed 5 × 5: Once 4 boxes have been marked for collection}, the box disappears {Mixed 5 × 5: 4 boxes disappear} from the screen, and your earnings are updated accordingly. At any moment you can see the amount earned up to that point.
Such earnings are only potential, however, because behind one of these boxes a time bomb is hidden that destroys everything that has been collected.
You do not know where the time bomb is. You only know that it can be in any place with equal probability. Moreover, even if you collect the bomb, you will not know it until the end of the experiment.
Your task is to choose when to stop the collecting process. You do so by hitting ‘Stop’ at any time. At the end of the experiment, we will randomly determine the number of the box containing the time bomb by means of a bag containing 100 {5 × 5: 25; 20 × 20: 400} numbered tokens. If you happen to have collected the box in which the time bomb is located, you will earn zero. If the time bomb is located in a box that you did not collect, you will earn the amount of money accumulated when hitting ‘Stop’.
We will start with a practice round. After that, the paying experiment starts. No Trial: Please note that there will be no trial period. You will take only one decision, and this will be payoff relevant. You will see now the main interface of the experiment. Take your time to observe it, before you click on ‘Start’.
1.3 B.3 Explosion
First screen:
as in baseline static
Second screen
On the sheet of paper on your desk you see a field composed of 100 numbered boxes.
You earn 10 euro cents for every box that is collected. Every second a box is collected, starting from the top left corner. Once collected, the box disappears from the screen, and your earnings are updated accordingly. At any moment you can see the amount earned up to that point.
Behind one of these boxes a bomb is hidden that destroys everything that has been collected. You do not know where the bomb is. You only know that it can be in any place with equal probability.
Your task is to choose when to stop the collecting process. You do so by hitting 1‘Stop’ at any time.
If you collect the box in which the bomb is located, the bomb will explode and you will earn zero. If you stop before collecting the bomb, you gain the points accumulated that far. The position of the bomb in the paying round has been randomly determined beforehand, and the documentation of the drawing process is available in a sealed envelope at the experimenters’ desk.
We will start with a practice round. The practice round is only meant to demonstrate how the experiment works: there will be no explosion. After that, the paying experiment starts.
1.4 B.4 Loss aversion
First screen:
Welcome to the Experiment. In the experiments all payoffs are expressed in euro. For your punctuality you receive 2.5 euro. On top of that, you have received an initial endowment of 2.5 euro, which you can see on your desk. PLEASE NOTE that this is NOT the show-up fee that will be paid at the end of the experiment, but it is given to you on top of that. Please keep this additional amount of money on your desk. The experiment consists of one short task, followed by a questionnaire. Should you have any questions or need help, please raise your hand. An experimenter will then come to your place and answer your questions in private.
Second screen
The initial endowment of 2.5 euro that lies in front of you will be at stake during the experiment, according to the following rules.
On your screen you will see a field composed of 100 boxes. Every box is worth 10 euro cents, which you receive for every box collected. Every second a box is collected, starting from the top-left corner following the sequence reported on the sheet of paper. Once collected, the box disappears from the screen.
You start losing all of the 2.5 euro. Your losses are then reduced by 10 euro cents for every box collected. If you collect enough boxes, you will not only offset the losses, but you will also earn additional money, at the same value of 10 euro cents for each box. At any moment you can see the amount of your losses or gains with respect to your initial 2.5 euro.
Such gains or losses are only potential, however, because behind one of these boxes a time bomb is hidden that destroys everything that has been collected.
You do not know where the bomb is. You only know that it can be in any place with equal probability. Moreover, even if you collect the bomb, you will not know it until the end of the experiment.
Your task is to choose when to stop the collecting process. You do so by hitting ’Stop’ at any time.
At the end of the experiment we will randomly determine the number of the box containing the bomb by means of a bag containing 100 numbered tokens. If you happen to have collected the box in which the bomb is located, you will lose all of your initial 2.5 euro. If the bomb is located in a box that you did not collect, you will earn your initial 2.5 euro plus or minus the gains or losses that you had accumulated when hitting ’Stop’.
We will start with a practice round. After that, the paying experiment starts.
1.5 B.5 HL (modeled on instructions from Holt and Laury 2002)
You will be asked to make 10 choices. Each decision is a paired choice between “Option A” and “Option B”. For each decision row you will have to choose between Option A and Option B. You may choose A for some decision rows and B for other rows, and you may change your decisions and make them in any order.
Even though you will make ten decisions, only one of these will end up affecting your earnings. You will not know in advance which decision will be used. Each decision has an equal chance of being relevant for your payoffs.
Now, please look at Decision 1 at the top. Option A pays 4 euro if the throw of the ten-sided die is 1, and it pays 3.2 euro if the throw is 2-10. Option B yields 7.7 euro if the throw of the die is 1, and it pays 0.2 euro if the throw is 2-10.
The other Decisions are similar, except that as you move down the table, the chances of the higher payoff for each option increase. In fact, for Decision 10 in the bottom row, the die will not be needed since each option pays the highest payoff for sure, so your choice here is between 4 or 7.7 euro.
To determine payoffs we will use a ten-sided die, whose faces are numbered from 1 to 10. After you have made all of your choices, we will throw this die twice, once to select one of the ten decisions to be used, and a second time to determine what your payoff is for the option you chose, A or B, for the particular decision selected.
1.6 B.6 EG (modeled on instructions from Eckel and Grossman 2008a)
You will be asked to select from among five different gambles the one gamble you would like to play. The five different gambles will appear on your screen. You must select one and only one of these gambles. Each gamble has two possible outcomes (Event A or Event B), each happening with 50% probability.
Your earnings will be determined by: 1) which of the five gambles you select; and 2) which of the two possible events occur.
At the end of the experiment, we will roll a six-sided die to determine which event will occur. If a 1, 2, or 3 is rolled, then Event A will occur. If 4, 5, or 6 are rolled, then Event B will occur.
1.7 B.7 CGP (modeled on instructions from Gneezy and Potters 1997)
You will be given 4 euros and will be asked to choose the portion of this amount (between 0 and 4 euros, in cents) that you wish to invest in a risky option. The money not invested is yours to keep.
There is a 50% chance that the investment in the risky asset will be successful. If it is successful, you receive 2.5 times the amount invested; if the investment is unsuccessful, you lose the amount invested.
The roll of a 6-sided die determines the value of the risky asset. You will be asked to choose 3 success numbers; if one of these numbers is rolled, the risky investment is successful; if not, it is not successful.
After the decisions are made the die will be rolled and then you will be paid the amount not invested plus 2.5 times the investment if it is successful and plus zero if it is not.
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Crosetto, P., Filippin, A. The “bomb” risk elicitation task. J Risk Uncertain 47, 31–65 (2013). https://doi.org/10.1007/s11166-013-9170-z
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DOI: https://doi.org/10.1007/s11166-013-9170-z