Abstract
Representing ambiguity in the laboratory using a Bingo Blower (which is transparent and not manipulable) and asking the subjects a series of allocation questions, we obtain data from which we can estimate by maximum likelihood methods (with explicit assumptions about the errors made by the subjects) a significant subset of particular parameterisations of the empirically relevant models of behaviour under ambiguity, and compare their relative explanatory and predictive abilities. Our results suggest that not all recent models of behaviour represent a major improvement in explanatory and predictive power, particularly the more theoretically sophisticated ones.
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Notes
 1.
Though we should admit that the issue of the ‘best’ way to elicit preferences (whether by pairwise choices, HoltLaury prices lists, the BeckerDeGrootMarschak mechanism, by allocation questions or by some other method) is an open one.
 2.
Which reads “In practice, the probability of one of the ‘ambiguous’ states was drawn from the uniform distribution over [0,2/3]. This distribution was not announced to the subjects.”
 3.
One way of thinking about this distinction is through indifference curves in outcome space. With the smooth models indifference curves are smoothly continuous, in contrast with kinked models, where there is a kink (a change in the slope) in the indifference curves along the certainty line.
 4.
(Our note) MEU, CEU and αMEU are respectively MaxMin Expected Utility, Choquet Expected Utility, and Alpha Expected Utility (all of which we consider specifically later).
 5.
If they had estimated the weighting function at all points, rather than estimating the parameters of the particular functional form, it would have been precisely Choquet.
 6.
Such as, for example, MaxMin (in which the decisionmaker looks at the worst that can happen and makes that as good as possible) and MaxMax (in which the decisionmaker looks at the best that can happen and makes that as good as possible). See Hey et al. (2010) for the empirical evidence against such theories.
 7.
In the context of our experiment, where there are three outcomes and hence six capacities, then the relationship between the two theories is given by the following, where p _{1},p _{2},p _{3} are the objective probabilities and w(.) is the weighting function, and the capacities for CEU are as denoted above:
$$\begin{array}{@{}rcl@{}} &&{}w_{E_{i}}=w(p_{i})\,\, \text{for} \,\, i=1,2,3\,\, \text{and}\\ &&{}w_{E_{j}\cup E_{k}}=w(p_{j}+p_{k})\,\ \text{for}\,\, j\neq k\in 1,2,3 \end{array} $$  8.
These assumptions were made after private communication with Marciano Siniscalchi, though we do not imply that our modelling of the VEU model has his approval.
 9.
We do not deny that some subjects could suspect that different coloured balls had different weights, but that could have been checked after the experiment. No subject asked for such a check.
 10.
One criticism concerning the implementation of ambiguity in the lab using the Bingo Blower comes from Morone and Ozdemir (2012). The criticism consists of the observation that the ability of getting the right probabilities is subject specific; that is, subjects have different counting skills, or might have problems in the perception of colours. This criticism may be true but it is not clear how this could affect the validity of the Bingo Blower in generating ambiguity in the lab. Moreover since we analyse the data subjectbysubject, it is unimportant if different subjects have different perceptions of the amount of ambiguity.
 11.
There are also problems, though of a different nature, involved with using our Random Lottery Incentive mechanism. But see http://people.few.eur.nl/wakker/miscella/debates/randomlinc.htm
 12.
In the HoltLaury price lists subjects are presented with a set of pairwise choices arranged in a list. In each pair subjects are asked to choose between some ambiguous lottery and some certain amount of money. As one goes down the list, the certain amount increases. The subject’s certainty equivalent is revealed by the point at which the subject switches from choosing the lottery to choosing the certain amount. See Holt and Laury (2002).
 13.
See Andersen et al. 2006.
 14.
Chambers et al. (2010) also investigate a generic Multiple Priors model.
 15.
It should be noted that the authors admit that the assumptions were quite strong and that they discuss the serious identification problems with twostageprobability models.
 16.
Because the subjects received the 76 questions in different orders (and with the colours on the left and the colours on the right randomly selected) this means that the position of the 60 estimation questions (and hence the 16 prediction questions) varied from subject to subject, but for each subject they were randomly positioned.
 17.
Note that with our combination since the variables to be explained are the proportions of the endowment allocated the various colours, in order to make the loglikelihoods comparable with those from other specifications, we need to subtract from the maximised loglikelihoods the sum of the natural logarithms of the amounts to be allocated in the relevant problems.
 18.
This is given by \(k\ln (n)2LL\), where k is the number of estimated parameters, n the number of observations and LL the maximised loglikelihood.
 19.
With the other preference functionals we note the following, as far as the mean parameters are concerned:

(1)
with CEU the estimated mean capacities are almost additive, but get slightly less so in Treatment 2;

(2)
with AEU, the mean lower bounds on the probabilities are close to the SEU subjective probabilities, and get closer to equality in Treatment 2;

(3)
with VEU the mean δ parameter is close to 0 and similar in the two treatments;

(4)
with COM the mean lower bounds on the probabilities are close to the SEU probabilities and the λ is close to 0.5;

(5)
we note that the α parameter in the AEU model is on average lower than the λ parameter in the COM model. This suggests that the Steiner point is a less important consideration to the subjects than the maximum expected utility.

(1)
 20.
We run 100 replications using a coefficient of risk aversion r equal to 0.8 and a coefficient of precision equal to 12. For each preference functional we set the following parameters’ values: SEU: p _{1}=0.2, p _{2}=0.3, p _{3}=0.5; CEU: w _{ E(1)}=0.10, w _{ E(2)}=0.20, w _{ E(3)}=0.30, \(w_{E(2)\cup E(3)}=0.85\), \(w_{E(3)\cup E(1)}=0.75\), \(w{E(2)\cup E(2)}=0.65\); AEU: \(\underline {p}_{1}=0.10\), \(\underline {p}_{2}=0.15\), \( \underline {p}_{3}=0.25\), α=0.5; VEU: p _{1}=0.2, p _{2}=0.3, p _{3}=0.5, δ=0.10; COM: \(\underline {p}_{1}=0.10\), \(\underline {p} _{2}=0.20\), \(\underline {p}_{3}=0.30\), λ=0.75.
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Acknowledgments
The authors would like to thank the Editor of this journal and a referee for very helpful comments which led to significant improvements in both the analysis of our results and their presentation.
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Hey, J.D., Pace, N. The explanatory and predictive power of non twostageprobability theories of decision making under ambiguity. J Risk Uncertain 49, 1–29 (2014). https://doi.org/10.1007/s1116601491988
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Keywords
 Alpha model
 Ambiguity
 Bingo blower
 Choquet expected utility
 Contraction model
 Rank dependent expected utility
 Subjective expected utility
 Vector expected utility
JEL Classifications
 D81
 C91