Eliciting ambiguity aversion in unknown and in compound lotteries: a smooth ambiguity model experimental study

After Becker and Brownson (1964), the idea that information which reduces ambiguity has a positive value for ambiguity-averse subjects has been clearly stated within different decision-theoretic models, e.g., Quiggin (2007), using Machina (2004) concept of almost-objective acts; Attanasi and Montesano (2012), relying on the Choquet expected utility model. Moreover, focusing on a specific adaptation of KMM, Snow (2010) has proved that the value of information that resolves ambiguity increases with greater ambiguity and with greater ambiguity aversion. Attanasi and Montesano (2012) have obtained similar results within the Choquet model.

For treatment 1, two sessions were run, respectively, with 17 and 18 students. Treatment 2 had three sessions, respectively, with 16, 10, and 9 students. Treatment 3 also had three sessions, respectively, with 12, 10, and 13 subjects.

Experimental instructions are available upon request.

Given that the subject has to set the price at which to sell a random “initial endowment”, she is assigned a lottery that she has just declared to prefer among four possible lotteries (task 4). Therefore, her “initial endowment” in task 5 (and, as will be shown, in tasks 6–9) depends on the choice made in task 4, although the subject does not know this in task 4.

More specifically, there are four different envelopes, labeled respectively with letter A, B, C and D, i.e., one for each lottery available in task 4. Each of these envelopes contains eleven different numbered tickets. The distance between two subsequent numbers on the tickets in an envelope is the same, so as to have the same number of tickets in each envelope, with the lowest numbered ticket being equal to \(\underline{x} _{4}^{j}\) and the highest being equal to \(\overline{x}_{4}^{j}\). In particular, the eleven tickets inside envelope \(A\) are \(14,14.5,\ldots ,18.5,19\) ; those inside envelope \(B\) are \(12,13,\ldots ,21,22\); those inside envelope \(C\) are \(8,10,\ldots ,26,28\); those inside envelope \(D\) are \(4,7,\ldots ,31,34\). The eleven tickets in envelope \(j\) represent the set of possible prices of lottery \(l_{4}^{j_{4}}\), with \(j=A,B,C,D\). A ticket is randomly drawn from each envelope. The ticket drawn from envelope \(j\) determines the random “buying price” for lottery \(j\). Then, without knowing this price, the subject states her smallest selling price (reservation price) for her lottery \(l_{4}^{j_{4}}\), by choosing one among the eleven possible prices for lottery \(j\). In case task 5 is selected for payment at the end of the experiment, the following happens: if, for the lottery the subject owned in task 5, the subject’s smallest selling price is lower than the respective random “buying price”, the subject sells her lottery and is paid the latter price. Otherwise, she will have to play her lottery, and her payoff (\( \overline{x}_{4}^{j_{4}}\) or \(\underline{x}_{4}^{j_{4}}\)) will depend on the ball randomly drawn from the small urn.

The term “construction urn” is borrowed from Klibanoff et al. (2012). Epstein (2010) calls this “second-order urn”.

The ten decision tasks are shown to the subject always in the same order. The reason why tasks 1–5 (which rely on the 5–5 balls small urn) are proposed always before tasks 6–10 is to elicit the subject’s risk-aversion before introducing unknown/multiple small urns. About task 5 coming before tasks 6–9, Halevy (2007) has shown that the (usually) higher reservation price for the 5–5 balls small urn (task 5 here) is not a consequence of this urn being proposed before the unknown/multiple ones (tasks 6–9 here). Finally, about the order of tasks 6–9, our theoretical results for the subject’s reservation price in tasks \(t=6,\ldots ,9\) do not suggest that this price should be always increasing or always decreasing with \(t\). Rather, the trend of the subject’s reservation price over tasks 6–9 should depend on the “sign” of her attitude toward ambiguity (e.g., see (5) and (7) below). A similar argument holds for task 4 always coming before task 10: different signs of the ambiguity attitude lead to different predictions about if and how the subject’s choice varies between the two tasks.

For tasks 1–4 and 10, performing the task means playing the chosen lottery (random draw of one ball from the 10-ball small urn). For tasks 5–9, it means playing the assigned lottery only if the subject’s selling price is not lower than the random “buying price” for that lottery.

More specifically, in treatment 1, the drawer is given the chance to check (in front of all experimental subjects in the session) that the number of white and orange balls in the transparent big urn is 50-50. Then, together with the transparent big urn, he/she is brought by the experimenter behind a screen where he/she performs the random draw of 10 balls (one after the other, with replacement) from the big urn. The screen being inside the laboratory, experimental subjects can “listen” to the random draw but they cannot see the color of the ten randomly drawn balls. After each of the ten random draws, the drawer shows the ball to the experimenter, records its color on a paper sheet, and puts the ball back in the urn. At the end of the ten random draws, the drawer puts 10 balls in an opaque small urn according to the colors recorded on the paper sheet, comes out from behind the screen and shows the opaque small urn to all experimental subjects in the session (they are informed about this procedure before it takes place). Then, he/she places the opaque small urn on a table in front of all experimental subjects and task 6 begins. At the end of the experiment, if task 6 is randomly selected (by the drawer him/herself) to determine participants’ final earnings, the drawer randomly draws one of the 10 balls from the opaque small urn. If the randomly selected task is one among tasks 7–9, the drawer will eliminate some possible compositions of the 10-ball opaque small urn (according to the rules specified above) before randomly drawing one of the 10 balls. The procedure in treatment 3 is as in treatment 1 apart from two features. First, the big urn is opaque and neither the drawer nor any experimental subject in the session may check the number of white and orange balls in the opaque big urn (although this urn is shaken by the drawer in front of everybody to show that there are many balls inside). Second, before the beginning of the experiment the opaque big urn is placed on a table in front of all experimental subjects and the drawer makes a preliminary random draw from a transparent 2-ball urn containing 1 white ball and 1 orange ball. The color of the randomly drawn ball is assigned to the highest of the two outcomes in each lottery in all the ten tasks of the experiment. Then, before the beginning of task 6 the drawer uses the opaque big urn to determine the composition of the 10-ball opaque small urn, according to the same random draw procedure of treatment 1. In treatment 2 the drawer is given the chance to check (in front of all experimental subjects in the session) the composition of each of the 11 transparent small urns inside the transparent construction urn. Then, he/she places this big “urn of all urns” on a table in front of all experimental subjects and task 6 begins. At the end of the experiment, if task 6 is randomly selected (by the drawer him/herself) to determine participants’ final earnings, the drawer will first randomly draw one of the 11 transparent small urns from the transparent big urn and then randomly draws one of the 10 balls from this small urn. If the randomly selected task is one among tasks 7–9, the drawer will take out of the transparent big urn some of the 11 transparent small urns (according to the rules specified above) before randomly drawing one of the remaining ones from the transparent big urn.

When checking if a behavioral pattern in tasks 1–4 is compatible with \(\textit{CARA}\), we allow up to only one possible deviation of at most one lottery \(l_{t}^{j_{t}}\) from each of the theoretical patterns. For example, we assign a \(\textit{CARA}\) index to pattern \((B,C,B,B)\), namely index 7, but we assign no index to \((B,D,B,B)\) or to \((C,C,B,B)\).

As for Table 4, when checking if a behavioral pattern in tasks 1–4 is compatible with \(\textit{CARA}\), we allow up to only one possible deviation of at most one lottery \(l_{t}^{j_{t}}\) from each of the theoretical patterns. For example, we assign a \(\textit{CRRA}\) index to pattern \((B,B,C,B)\), namely index 8, but we assign no index to \((B,C,C,B)\) or to \((C,B,C,B)\).

Tasks 1–4 contain lotteries whose expected payoffs are between the expected (real) payoff of lotteries 1X and 20X in Holt and Laury (2002). Although \(\textit{RRA}\) intervals are not perfectly coincident between Table 5 in this paper and their Table 3, the similarity of results is impressive: our study finds 5.26 % of subjects with \(\textit{RRA}\in (1.320,+\infty )\), and they find 1 % and 6 % of subjects with \(\textit{RRA}\in (1.370,+\infty )\), respectively, in the “1X real” and the “20X real” payoffs task (stay in bed); our study finds 5.26 % of subjects with \(\textit{RRA}\in (0.890,1.320)\), and they find 3 % and 11 % of subjects with \(\textit{RRA}\in (0.970,1.370)\), respectively, in “1X real” and “20X real” (highly-risk-averse); our study finds 63.16 % of subjects with \(\textit{RRA}\in (0.123,0.890)\), and they find 62 % and 64 % of subjects with \(\textit{RRA}\in (0.150,0.970)\), respectively, in “1X real” and “20X real” (from very-risk-averse to slightly-risk-averse); our study finds 26.32 % of subjects with \(\textit{RRA}\in (-\infty ,0.123)\), and they find 34 % and 19 % of subjects with \(\textit{RRA}\in (-\infty ,0.150)\), respectively, in “1X real” and “20X real” (from risk-neutral to highly-risk-loving). Notice also that the distribution of \(\textit{RRA}\) in Table 5 in this paper is not very different from 1X and 20X real-payoff single unordered tasks in Holt and Laury (2005) and from 1X and 10X real-payoff single unordered tasks in Harrison et al. (2005). For example, the percentage of non-risk-averse subjects (from risk-neutral to highly-risk-loving) under \(\textit{CRRA}\) in tasks 1–4 of our experiment (23.32 %) is between those found by Harrison et al. (2005) in 1X and 10X real-payoff single unordered tasks (31.71 % and 12.73 %, respectively).

In particular, in treatment 1, the objective second-order probabilities are as follows: in tasks 6 and 10, \(q_{10}=q_{0}=1/1024\simeq 0.1\,\%\), \( q_{9}=q_{1}=10/1024\simeq 1\,\%\), \(q_{8}=q_{2}=45/1024\simeq 4.4\,\%\), \( q_{7}=q_{3}=120/1024\simeq 11.7\,\%\), \(q_{6}=q_{4}=210/1024\simeq 20.5\,\%\), and \(q_{5}=252/1024\simeq 24.6\,\%\); in task 7, \(q_{7}=q_{3}=1/16=6.25\,\%\), \( q_{6}=q_{4}=4/16=25\,\%\), and \(q_{5}=6/16=37.5\,\%\); in task 8, \( q_{10}=q_{3}=1/128\simeq 0.8\,\%\), \(q_{9}=q_{4}=7/128\simeq 5.5\,\%\), \( q_{8}=q_{5}=21/128\simeq 16.4\,\%\), \(q_{7}=q_{6}=35/128\simeq 27.3\,\%\); in task 9, \(q_{7}=q_{0}=1/128\simeq 0.8\,\%\), \(q_{6}=q_{1}=7/128\simeq 5.5\,\%\), \( q_{5}=q_{2}=21/128\simeq 16.4\,\%\), \(q_{4}=q_{3}=35/128\simeq 27.3\,\%\). All other \(q_{\theta }\) are zero. In treatment 2, the objective second-order probabilities are: in tasks 6 and 10, \(q_{\theta }=1/11\simeq 9.1\,\%\) for every \(\theta =0,1,\ldots ,10\); in task 7, \(q_{\theta }=1/5\) for every \(\theta =3,4,\ldots ,7\); in task 8, \(q_{\theta }=1/8\) for every \(\theta =3,4,\ldots ,10\); in task 9, \(q_{\theta }=1/8\) for every \(\theta =0,1,\ldots ,7\). All other \( q_{\theta }\) are zero.

All raw data and statistical codes are available on request.

More precisely, \(\textit{CE}(L_{5})\) predicted by the \(\textit{CARA}\) ordering is calculated as the average between \(\textit{CE}(l_{4}^{j_{4}};\overline{ARA}_{h})\) and \(\textit{CE}(l_{4}^{j_{4}};\underline{ARA}_{h})\) in (1), for \( h=1,2,\ldots ,9 \). Similarly, \(\textit{CE}(L_{5})\) predicted by the \(\textit{CRRA}\) ordering is calculated as the average between \(\textit{CE}(l_{4}^{j_{4}};\overline{\textit{RRA}}_{k})\) and \(\textit{CE}(l_{4}^{j_{4}};\underline{\textit{RRA}}_{k})\) in (2), for \(k=1,2,\ldots ,12\).

The Kruskal–Wallis equality-of-populations rank test verifies the hypothesis that several samples are from the same population.

The Kolmogorov-Smirnov test compares two observed distributions \(f(\cdot )\) and \(g(\cdot )\). The procedure involves forming the cumulative frequency distributions \(F(\cdot )\) and \(G(\cdot )\) and finding the size of the largest difference between these. The hypothesis tested is whether the two observed distributions are equal (pairwise comparisons between treatments 1–2, treatments 1–3, and treatments 2-3).

According to the Kruskal–Wallis test the null hypothesis of equality of distributions (P value \(=0.401\) for CARA and P value \( =0.357\) for CRRA) cannot be rejected. The Kolmogorov-Smirnov test confirms this result.

EU maximizing subjects for the Binomial and Uniform treatments and subjective expected utility (henceforth, SEU) maximizing subjects for the Unknown are both value-ambiguity-neutral and choice-ambiguity-neutral, hence coherent-ambiguity-neutral. However, given that the choice set in all our experimental tasks is discrete, it cannot be excluded that weekly non-EU (and non-SEU) maximizing subjects may fall into the group of coherent-ambiguity-neutral subjects. For example, consider a non-EU maximizing subject with a strictly concave \(\phi \) function, hence being ambiguity-averse. If the concavity of her \(\phi \) function is small, then in our discrete choice set she could make the same choice as another subject with a linear \(\phi \) function, hence ending up being classified as coherent-ambiguity-neutral.

Although the number of unclassified subjects is lower in the Binomial than in the other two treatments, unclassified subjects are not statistically different from classified ones both with respect to CARA or CRRA ordering and with respect to the lottery chosen in task 4.

These results are not shown but can be made available upon request.

This intuition is reinforced by the fact that the correlation between (strong) value-ambiguity-aversion and (strong) choice-ambiguity-aversion found above in all the sample of classified subjects is higher (coeff. \(= 0.45\)) and significant (P value \(=0.007\)) only if the analysis is restricted at the Binomial treatment. In this treatment, it is plausible that only highly-ambiguity-averse subjects show at the same time \( \textit{CE}(L_{6})<\textit{CE}(L_{5})\) and \(j_{10}\prec j_{4}\).

The \(t\) test is any statistical hypothesis test (parametric) in which the test statistic follows a Student’s t distribution if the null hypothesis is supported. Here, a two-sample \(t\) test is run for a difference in mean (the null hypothesis is that the two samples have the same mean).

In order to verify the equality in the distribution of each risk-aversion index among different signs of the ambiguity attitude, two different tests have been performed: Kruskal–Wallis equality-of-populations rank test and Kolmogorov-Smirnov equality-of-distributions test. Both under CARA and under CRRA, the null hypothesis of equality in distributions can be rejected according to the Kruskal–Wallis equality-of-populations rank test (respectively for CARA and CRRA: P value \(=0.010\) , P value \(=0.010\)). By performing the Kolmogorov-Smirnov equality-of-distributions test with a pairwise comparison between different signs of the ambiguity attitude, the results are found to be consistent with the Kruskal–Wallis test.

Again, a Kruskal–Wallis test has been performed to check whether the distribution of \(l_{4}^{j_{4}}\) is different by treatment. According to this test, the null hypothesis of equality in distribution (P value \( =0.000 \)) can be rejected. A Kolmogorov-Smirnov test has also been performed with a pairwise comparison between different signs of the ambiguity attitude, and the results are consistent with the Kruskal–Wallis test.

The guess in task 6 is positively correlated (coeff. \(=0.20\), P value \(=0.045\)) with the “normalized” \(\textit{CE}(L_{6}) \). In Sect. 4.4, the specific meaning of “normalized” certainty equivalent is explained.

Recall that each unknown small urn in tasks 6–9 has 10 balls inside. Hence, the set of possible guesses in each of these tasks is \(\{0,1,\ldots ,10\}\).

Through the Kruskal–Wallis test, we can reject the null hypothesis of equality in distribution (P value \(=0.000\)). The Kolmogorov-Smirnov test confirms this result.

P value \(=0.546\) for CARA, P value \(=0.841\) for CRRA.

P value \(=0.023\) for \(\textit{CE}(L_{6})\), P value \(=0.054\) for \( \textit{CE}(L_{7})\).

P value \(=0.002\) for both \(\textit{CE}(L_{6})\) and \(\textit{CE}(L_{7})\).

If \(\textit{CE}(L_{6})\) is taken as reference, P value \(=0.202\) for CARA, P value \(=0.278\) for CRRA. If \(\textit{CE}(L_{7})\) is taken as reference, P value \(=0.216\) for CARA, P value \(=0.333\) for CRRA.

P value \(=0.838\) for CARA, P value \(=0.765\) for CRRA.

According to the Kruskal–Wallis test on the equality in distribution of guesses by treatment, the null hypothesis cannot be rejected (respectively, for task 6, 8, 9, P value: \(0.739\), \(0.375\), \(0.175\)). The Kolmogorov-Smirnov equality-of-distributions test with a pairwise comparison between treatments confirms this result.

However, \(\textit{CE}(L_{6})\) is positively correlated (coeff. \(=0.35\)) with the guess in task 6 only in the Uniform treatment (P value \(=0.037\)). Also, \(\textit{CE}(L_{9})\) is positively correlated (coeff. \(=0.36\)) with the guess in task 9 only in the Binomial treatment (P value \( =0.035 \)).

In tasks 5–9, each subject always has the possibility to choose among eleven possible selling prices. Therefore, for every \(t=5,6,\ldots ,9\) , index 1 can always be assigned to \(\textit{CE}(L_{t})=\underline{x}_{4}^{j_{4}}\), index 11 to \( \textit{CE}(L_{t})=\overline{x}_{4}^{j_{4}}\)  and internal \(\textit{CE}(L_{t})\) can be indexed accordingly. See footnote 5.

A relevant exception is again represented by the (normalized) certainty equivalent in task 7. Controlling for CARA and treatment and taking the Binomial as reference treatment, the Unknown treatment has a positive and significant effect (P value \(=0.059\)) over \(\textit{CE}(L_{7})\). Controlling for CRRA and treatment, the Uniform treatment also has a positive and significant effect (P value \(=0.041\)) over \(\textit{CE}(L_{7})\).

To be more precise, only in the regressions for \(\textit{CE}(L_{7})-\textit{CE}(L_{6})>0\) (\(\textit{CE}\) normalized), the Uniform treatment significantly increases the probability that \(\textit{CE}(L_{7})-\textit{CE}(L_{6})>0\) with respect to the Binomial treatment (P value \(=0.040\)). This result also holds when controlling for the difference in the guesses about the number of winning balls in task 7 and task 6.

Ghirardato (2004) also discusses the issue of the formal definition of ambiguity and ambiguity attitude.

Early literature is surveyed in Camerer and Weber (1992) and Camerer (1995).

See for example Hey et al. (2010) and Hey and Pace (2014): both experimental designs are aimed at testing only non-two stage probability models.

In particular, Halevy (2007) found that 15–20% of his subjects are ambiguity-neutral and able to reduce compound lotteries. Another 35 % of subjects exhibit ambiguity aversion (proneness) together with aversion (proneness) to mean-preserving spreads in the second-order distribution. Both these categories of subjects are consistent with KMM.

Indeed, both papers use Ellsberg-type urns. In particular, Conte and Hey (2013) adopted the same urns 2 and 3 as Halevy (2007) that reproduce two-stage lotteries. However, Conte and Hey (2013) use exclusively pairwise questions to reduce the number of parameters to be estimated, no meaning to estimate a utility function. Differently, Halevy (2007) asks subjects to state certainty equivalents through the BDM mechanism so as to infer a utility function from the subject’s answers.

The authors found results in favor of KMM both through individual estimates (56 % of subjects show behavior consistent with KMM) and by classifying subjects according to posterior probabilities of each of them being coherent with one out of four types of preferences (50 % for KMM).

Ahn et al. (2011) implemented an experimental design where subjects are asked to choose between different lotteries that duplicate the return of a portfolio containing a safe asset and an ambiguous asset.

Among several studies finding a positive correlation, see Lauriola and Levin (2001), Di Mauro and Maffioletti (2004), Bossaerts et al. (2010), and Ahn et al. (2011).

Among the few detecting no correlation, see Cohen et al. (1987) and Cohen et al. (2011).

For example, the set of possible lotteries \(l_{t}^{j}\) (with \(j=A,B,C,D\), and \(t=1,2,3,4\)) in Table 2 may be enriched with just one lottery in task 4, namely \(l_{4}^{E}=(40,0.5;0,0.5)\). Then, under the \(\textit{CRRA}\) specification, subjects with \(\textit{RRA}\in (-0.146,0.123)\) would pick \( (l_{1}^{j_{1}},l_{2}^{j_{2}},l_{3}^{j_{3}},l_{4}^{j_{4}})=(D,D,D,D)\) and subjects with \(\textit{RRA}\in (-\infty ,-0.146)\) would pick \( (l_{1}^{j_{1}},l_{2}^{j_{2}},l_{3}^{j_{3}},l_{4}^{j_{4}})=(D,D,D,E)\). Notice that this would lead to disentangle risk-neutral and risk-loving subjects as in Table 3 (p. 1649) of Holt and Laury (2002).

See Harrison and Rutström (2008) for a deeper analysis of these and additional issues, as well as alternative elicitation methods. They also review useful techniques to estimate behavioral errors that could arise in the calculation of reservation prices through the BDM mechanism.

Differently from what recommended by Plott and Zeiler (2005), we did not introduce any practice round before task 5. This is because we did not want subjects to have any feedback about random draws of buying prices before going through the BDM mechanism in tasks 5–9. However, before task 5, both the drawer and each subject participating in the experiment were given the chance to check that each of the four envelopes—through which the BDM was physically implemented—contained the eleven numbered tickets here indicated in footnote 5. Further, the fact that instructions of each new task were given prior to that task allowed subjects to focus on a BDM at a time.

To test for possible endowment effects, our experimental design should be extended to allow subjects to act as lottery sellers in some tasks (as in our experiment) and as lottery buyers in others.

See footnote 9.

A possible theoretical criticism of the random-lottery incentive mechanism, put forward by Holt (1986), was demonstrated not to occur empirically by Starmer and Sugden (1991).

Among the most recent, see Hey and Lee (2005) and Lee (2008).

Cox et al. (2012) have compared the performance of several payment mechanisms in individual choice tasks and found a new mechanism—with the same expected value of payoff incentives as RLIM—that is less biased than RLIM. With this new mechanism, at the end of the experiment, one state of nature is randomly drawn, and then all (comonotonic) lotteries chosen in the experiment are paid out for this state of nature and the payoff divided by the number of tasks. Moreover, Harrison and Swarthout (2012) have shown that preference estimates obtained under RLIM differ from those obtained in a one-task design and, more generally, documented concerns about the use of payment protocols over multiple choices that, as RLIM, assume the validity of the independence axiom. Similar concerns may be ascribed to the fact that the application of RLIM as payment protocol requires no violation of the reduction of compound lotteries axiom (henceforth ROCL). In this regard, Harrison et al. (2012) have found no violation of ROCL when subjects are presented with only one choice and violation of ROCL when subjects are presented with many choices and RLIM is used as a payment protocol.

Their actual finding is that the hypergeometric case is the one having the strongest relationship with ambiguity attitude.