In this study different types of events are presumed to constitute different sources of
uncertainty, and that the decisions made can be examined using the sourcemethod. The source
method employs a class of uncertainty models that are rank-dependent: RDU/CEU or CPT.41
These models rank the possible outcomes according to the level of attractiveness, then estimate
decision weights assuming a probability weighting function. The probability weighting function
estimated from each source of uncertainty is referred to as the source function that maps the
probabilities, p, into decision weights, w(p). A source function can be estimated from events
with known or unknown probabilities. Each source function reflects interactions between beliefs
and preferences, and by comparing two source functions attitudes toward uncertainty that reflect
a combination of beliefs and preferences are revealed. Note that a source function is presumed
to reflect a combination of beliefs and preferences; it does not separate the two. It is the
difference in two source functions that is said to reveal uncertainty aversion. Here, the term
uncertainty aversion reflects the differences of beliefs across events as well as differences of
preferences across events.42
41 ABPW (2011) perform the estimation under the name Rank Dependent Utility, whereas Kothiyal, Spinu and Wakker (2014) perform the same estimation under the name Choquet Expected Utility. Both studies use the source method.
42 In the context of the source method, the term “uncertainty aversion” reflects beliefs as well as preferences. We are aware that this way of defining uncertainty aversion may cause confusion with many uncertainty models that separate beliefs from preferences. Nevertheless, we use the source method for our analysis for the purpose of examining uncertainty aversion using decision weights.
Attitudes toward uncertainty and the degree of perceived uncertainty can be defined in a
tractable manner using two indices of uncertainty: pessimism and likelihood insensitivity. As
mentioned previously, the index of pessimism reflects the concavity or convexity of a source
function, and the difference in concavity or convexity across the graphs of the source functions is
interpreted as uncertainty aversion by ABPW. Likelihood insensitivity reflects a source function
that is inverse-S shaped, i.e., for an event that has a low probability of occurring, subjects weigh
the event higher than its underlying objective probability (i.e., w(p) > p), and for an event that
has a high probability of occurring, subjects weigh the event lower than its underlying objective
probability (i.e., w(p) < p), displaying a tendency to place equal decision weights on all possible
outcomes. The difference in insensitivity across the graphs of the source functions is interpreted
as another characteristic property of uncertainty aversion by ABPW.43 In the study these two
indices are captured using the two-parameter Prelec weighting function (Prelec (1998)).
The source method does not propose a new theoretical model of uncertainty. Instead, it
proposes a way to analyze behavior under uncertainty using a class of theoretical models that
already exist in the literature: rank-dependent models. The novelty here is to define uncertainty
preferences by means of two indices based on the decision weights.
The ABPW study is based on two experimental tasks. The first task is the classic
Ellsberg urn experiment and the second involves natural uncertainties such as the weather and a
stock index in an obscure country. In the Ellsberg task subjects are presented with two urns each
containing eight balls. The known urn K contains eight balls of different colors: red, blue,
yellow, black, green, purple, brown, and cyan. The unknown urn U contains eight balls with the
43 One should note that likelihood insensitivity, by itself, may solely reflect diffuse perceptions and may not reflect any preferences toward uncertainty at all. It is the difference in insensitivities between two source functions that is considered a characteristic property of uncertainty preferences in the source method.
same set of colors but the composition is unknown to the subjects in the sense that some colors
might appear several times and others might be absent. Using a list format, subjects are
presented with a series of choices, each between a prospect and an ascending range of a sure
payment, with the switching point taken as the certainty equivalent. One of the choices on the
list is selected for payment. The second task involves natural uncertainties, such as the French
stock index CAC40, temperature in Paris (the home city where the experiment is conducted), and
temperature in a foreign city. The elicitation method is again a list varying the amount of the
certain option. One of the choices is randomly selected for payment.44
To control for risk preferences, utilities are elicited using lotteries with known
probabilities that are presented in a list format similar to that used for the uncertainty tasks, with
the switching values taken as the certainty equivalent. The lotteries always have an objective
probability of 0.5. Utilities are estimated assuming a power utility function using nonlinear
least-squares methods.
In the uncertainty task ABPW calculate, rather than estimate, decision weights, after
which they fit these weights to a two-parameter Prelec function by minimizing quadratic
distance. The Ellsberg task compares the certainty equivalents for risk and for uncertainty. The
source functions for urns K and U significantly deviate from linearity (i.e., decision weights
deviate from the underlying objective probabilities), and display significant likelihood
insensitivity, with significantly more insensitivity in the urn U than in the urn K. In particular,
44 In the Ellsberg task, ABPW (2011) implement a random incentive payment procedure where one of the choices is randomly selected for real payment. In the task that involves natural uncertainties, ABPW implement two
treatments of payment procedures. In one treatment subjects receive a flat payment and the choices are hypothetical, thus the choices are not incentivized. In the other treatment one of the choices is randomly selected for real
payment. The money that the subjects earned is collected about three months from the date of the experiment, after the uncertainty is resolved.
for large probabilities (p > 0.5) there is more underweighting of probabilities for urn U than for
urn K; for small probabilities (p ≤ 0.5) there is no significant difference. The pessimism index is
not significantly different from zero in either urn. Pessimism in urn U, however, significantly
exceeds that in urn K. The ABPW interpret the underweighting or overweighting of the
subjective probabilities as indicative of willingness to bet,45 and report that there is more
willingness to bet for risk than is for uncertainty in the high probabilities (p > 0.5), and that the
willingness to bet is the same for both in the low probabilities (p ≤ 0.5).
The second task makes observations on the certainty equivalents for natural uncertainties.
All source functions display a common inverse-S shaped with low probabilities overweighted
and high probabilities underweighted. The insensitivity and pessimism indices are significantly
different from zero but are not significantly different across the sources of natural uncertainties.
Furthermore, the source functions for natural uncertainties are not significantly different from the
one for the uncertain urn in the Ellsberg task. These findings suggest that events with unknown
probabilities are perceived to be similar, but they are perceived differently from events with
known probabilities.