A Direct Test of Ambiguity Hedging

There are 5 treatments in the experiment, where participants answer a two-choice Ellsberg Urn task. In this task, participants choose between a jar with a known number of blue and orange marbles and an jar with an unknown color of blue and orange marbles. They make two choices. The first is which of the two jars they would bet on a blue marble being drawn from and the second is which of the two jars they would bet on an orange marble being drawn from. Two of the treatments elicit the beliefs about the probabilities of winning a bonus payment in a two-choice Ellsberg task. These two treatments are designed to test the incentive compatibility of the RIS, as well as determine whether the hedging opportunity in the task is understood and reported. The first treatment, Ranges: Single, pays for only one of the two questions that the participants answer. The participants know which is paid before answering the two questions. In the second treatment, Ranges: Before, a coin is flipped to determine which of the two questions is paid and participants do not know until after they answer the questions. Both of these treatments have a belief elicitation stage to determine whether participants observe and report the hedging opportunity. The third and fourth treatment are quite similar to the first two treatment, but now the hedging opportunity has been removed. This is done through the two questions being about different sets of jars. In 4 Jars: Single, only one of the questions asked is paid and participants know which it is before answering. In 4 Jars: Before, a coin is flipped to determine which of the two questions is paid and participants do not know until after they answer the questions. There is no belief elicitation stage in these treatments. The last treatment measures whether correlation between the two unknown jars in the 4 Jar treatments is perceived.

2024-08-14

2024-12-01

From the Ranges: Single and Ranges: Before treatment, we are interested in the differences between the reported beliefs between the two treatments. Specifically, we are interested in the question asking about participants' beliefs over probability of winning a bonus based on their choices in the Ellsberg task. When comparing 4 Jars: Before and Ranges: Before, we are interested in comparing the number of participants choosing to bet on the unknown jars for both questions.

We are also interested in the difference in individuals who perceive positive correlation versus negative correlation in the 4 Jar: Correlation treatment. We will compare the proportion of individuals choosing the guesses worth $2.10 across the two groups. Additionally, we are interested in whether answering the other comprehension questions correctly affects the perception of ambiguity hedging in our Ranges: Before treatment. We will use a logistic regression to determine this.

For Ranges: Single, there are two jars displayed to participants, Jar A and Jar B. Jar A has 20 marbles total. 5 marbles are orange and 15 marbles are blue in Jar A. Participants know the contents are Jar A and are shown an image of the jar. Jar B begins with 10 marbles only, which are blue. Ten more marbles will be placed into Jar B, for a total of 20 marbles. The additional 10 marbles placed in Jar B are randomly chosen from a Large Jar. This Large Jar has 100 marbles, where the number of blue and orange marbles is randomly chosen with uniform probability over all possible combinations of blue and orange. Participants do not know the color composition of the Large Jar. 10 of the marbles from the Large Jar are chosen randomly and placed into Jar B, for a total of 20 marbles in Jar B. Participants now know that there are 20 total marbles in Jar B, but only know that 10 are blue and that the other ten can be any number of blue and orange marbles.

There are two stages, where participants make choices based on Jar A and Jar B. In the first stage, participants answer the Blue Choice Problem and the Orange Choice Problem. In these problems, participants must choose whether they believe a blue(orange) marble is more likely to be chosen out of Jar A or Jar B. Before participants answer these questions, one is randomly chosen to be used for payment. Participants are told which of the Orange and Blue Choice problems if chosen for payment before they answer the questions. They still must answer both Choice Problems.

After the Choice Problems are answered, participants move on to the second stage. In Stage 2, participants report their beliefs in four different questions. One of the questions is our question of interest to determine whether hedging is identifiable. This question is "I believe my overall probability of winning a bonus is between ___ and ____." If the hedging opportunity is recognized, participants should enter 50% and 50% in these blanks. The three other questions are used as comprehension checks. For each of these questions, participants can enter a range of beliefs that are incentivized using two Multiple Price Lists (MPLs).

For the Multiple Price List incentivization, two MPLs are used, which correspond to the two probabilities that can be entered. The first MPL, which corresponds to the first number entered, is the MPL that incentivizes the event in which the question itself asks about. The number entered in the first blank corresponds to the row in the MPL that participants will switch from preferring a bet on the event to a lottery. The second MPL, which corresponds to the second number entered, is the MPL that incentivizes the complement of the event that the question asks about. 100 minus the number entered in the second blank is the row in the MPL that participants will switch from preferring a bet on the event's complement to a lottery. A screen that explains these incentives and shows how the MPLs change as numbers are entered is available for each participant to use.

After beliefs are reported for all 4 questions, the experiment is complete. Participants receive a $2.00 show-up fee and then one of the two stages is chosen for payment. If the first stage is chosen for payment, a marble is chosen from the Jar the participant chosen in whichever of the two Choice Problems was used for payment. If the color is the right color, subjects receive a bonus payment, otherwise they do not. If the second stage is chosen for payment, one of the four questions is chosen randomly. For that question, one of the two MPLs is randomly chosen and a row in that MPL is chosen and participants are paid based on their choice in that row.

The second treatment, Ranges: Before, is almost identical to Ranges: Single. There are two major differences. The first difference is that both Choice Problems may be used for payment and the participant does not know which is chosen until after they answer both. Now, before participants answer the Choice Problems, a fair coin is flipped. If the coin lands heads, the Orange Choice Problem is used for payment. If the coin lands tails, the Blue Choice Problem is used for payment. Participants are not told the outcome of the coin flip before they answer both Choice Problems.

The second difference is that there are now five belief questions in the second stage. An additional question is added to ask about the probability that participants believe they will win a bonus if one of the two choice problems is in fact chosen for a bonus. This is an additional comprehension question. Participants receive a $2.00 show-up fee and the way that bonuses are paid is the same as Ranges: Single. One of the two stages is chosen to determine the bonus. If the first stage is chosen, the coin flip is then revealed to determine which of the two Choice Problems will be used for payment.

The third treatment is 4 Jars: Single. In this treatment, participants are shown 4 jars. Two jars are round and two jars are cylinders. There is a Round Jar A and a Cylinder Jar A. In both of these jars there are 10 blue and 10 orange marbles. Participants are shown these and know the contents of both Jar As. The third and fourth jar are Round Jar B and Cylinder Jar B. These both are empty at the beginning of the experiment and are filled with 20 marbles from two different large jars.

The Round Jar B is filled with 20 marbles from a Large Round Jar. This Large Round Jar has 100 marbles in it. The composition of the Large Round Jar is randomly chosen. Participants do not know the color composition of marbles in the Large Round Jar. The Large Cylinder Jar may not be the same as the Large Round Jar, with the composition also randomly chosen. Participants are told that these Large Jars may not have the same color composition and are not told what the compositions are. 20 marbles are randomly chosen from the Large Round Jar and placed into the Round Jar B. 20 marbles are randomly chosen from the Large Cylinder Jar and placed into the Cylinder Jar B. There are now 20 marbles in each of the Jar Bs, and their color compositions do not have to be the same. Participants are told there are 20 marbles in each Jar B, but do not know the exact number of blue and orange marbles in either Jar B. Participants answer two Choice Problems. The Orange Choice problem asks whether an orange marble is more likely to be chosen from Cylinder Jar A or Cylinder Jar B. The Blue Choice problem asks whether a blue marble is more likely to be chosen from Round Jar A or Round Jar B. Before participants answer the Choice Problems, one of the two is randomly chosen to be used for payment and which is chosen is told to the participant.

After participants are told which problem counts for payment, but before they answer both Choice Problems, they answer two unincentivized questions to measure the perception of correlation between Round Jar B and Cylinder Jar B. Participants receive a $2.00 show-up fee and then a bonus that depends on their answer to the Choice Problems. A marble is chosen from whichever Jar the participant chose in

4 Jars: Before is almost identical to 4 Jars: Single. The only difference is that both Choice Problems may be used for payment and the participant does not know which is chosen until after they answer both. Now, before participants answer the Choice Problems, a fair coin is flipped. If the coin lands heads, the Orange Choice Problem is used for payment. If the coin lands tails, the Blue Choice Problem is used for payment. Participants are not told the outcome of the coin flip before they answer both Choice Problems. After the Choice Problems are answered, the outcome of the coin flip is revealed and participants are paid based on the marble drawn from the jar chosen in the Choice Problem indicated by the coin flip.

The last treatment is called 4 Jars: Correlation. In this treatment, participants are shown two empty jars. One of the jars is a Cylinder Jar and one is a Round Jar. These both are empty at the beginning of the experiment and are filled with 20 marbles from two different large jars. The Round Jar is filled with 20 marbles from a Large Round Jar. This Large Round Jar has 100 marbles in it. The composition of the Large Round Jar is randomly chosen. Participants do not know the color composition of marbles in the Large Round Jar. The Large Cylinder Jar may not be the same as the Large Round Jar, with the composition also randomly chosen. Participants are told that these Large Jars may not be the same color composition and are not told what the compositions are. 20 marbles are randomly chosen from the Large Round Jar and placed into the Round Jar. 20 marbles are randomly chosen from the Large Cylinder Jar and placed into the Cylinder Jar. There are now 20 marbles in each of the Jars, and their color compositions do not have to be the same.

Participants only choice is to decide whether a blue or orange marble will be chosen from the two jars. For half of the participants, they will receive $2.10 is they guess the marbles from both jars are the same color and are correct. If these participants guess the marbles are different colors and are correct, they will earn $2.00. The other half of the participants will receive $2.00 is they guess the marbles from both jars are the same color and are correct. If these participants guess the marbles are different colors and are correct, they will earn $2.10. After they make their guess, a marble is randomly chosen from each of the two jars and participants are paid if their guess was correct. Participants also receive a $1.00 show-up fee.

Not available

Randomization is done in office by a computer.

Randomization happens at the individual level.

No

1000 adults

1000 adults

150 adults for Ranges: Single treatment, 150 adults for 4 Jars: Single treatment, 300 adults for Ranges: Before treatment, 300 adults for 4 Jars: Before treatment, and 100 adults for 4 Jars: Correlation treatment.

For each of our intended statistical tests, we used a power of 90% and a significance level of alpha = 0.05 to determine the sample size. For the first test, we estimate a sample size of 335 that is necessary for a 10% difference in proportion between the two treatments. For the second test, we estimate necessary sample sizes of 221 for Ranges: Before and 111 for Ranges: Single for a 15% difference in the proportions of participants stating 50-50% between the two treatments. For our last question, we must assume a difference in the number of individuals choosing Jar A for both questions and Jar B for both questions. We will estimate about 35% of participants choosing Jar A for both questions and about 20% of participants choosing Jar B for both questions. For this, an estimation of an effect size of 20% difference in these proportions, we need 77 Jar B choosers and 135 Jar A choosers. For our correlation test, we estimate a necessary sample size of 52 for an effect size of 30% between those perceiving positive and negative correlation between the two jars. Using these sample sizes, we will collected 150 participants for Ranges: Single and 4 Jars: Single, 300 participants for Ranges: Before and 4 Jars: Before, and 100 participants for 4 Jars: Correlation.