The Power weighting function has parameter γ:
𝑤(𝑝) = 𝑝𝛾 (3)
where 0 ≤ p ≤ 1. For γ >1 this function is everywhere convex and subjects underweigh the probabilities; for γ < 1 the function is everywhere concave and subjects overweigh the
probabilities.64 For γ = 1 in the pure risk lottery, the subject is an EU maximizer. In Appendix D
64 In the Power weighting function γ = 0 raises issues because it would imply that the agent places a probability of 1 for an event regardless of its objective probability. We do not encounter this issue in the estimation, though one could still constrain γ > 0.
Figures BB3.1 – BB3.3 provide examples of what the weighting function would look like under
different values of γ and its implied decision weights.
The Power function has the least flexible functional form compared to the Prelec or TK
function, given that the former can only accommodate the index of pessimism, not likelihood
insensitivity. Comparing γ between the pure risk and uncertainty lotteries reveals possible
uncertainty aversion that is due to difference in pessimism.
Two hypotheses are tested with respect to the Power function:
Hypothesis I – For the uncertainty lotteries, 𝛾 differs significantly from 1, controlling for utility curvature.
𝛾𝑠𝑐𝑟𝑎𝑚𝑏𝑙𝑒𝑑 ≠ 1; 𝛾𝑏𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑 ≠ 1
Hypothesis II – Pessimism for the uncertainty lottery exceeds that for the pure risk lotteries, such
that the weighting function for the uncertainty lotteries is more convex than the one for the pure
risk lottery.
𝛾𝑝𝑢𝑟𝑒 𝑟𝑖𝑠𝑘 < 𝛾𝑠𝑐𝑟𝑎𝑚𝑏𝑙𝑒𝑑; 𝛾𝑝𝑢𝑟𝑒 𝑟𝑖𝑠𝑘 < 𝛾𝑏𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑
3.5 Empirical Analysis
The weighting functions are jointly estimated with risk attitudes assuming RDU with
CRRA utility functions. This joint estimation approach builds on previous work on structural
estimation of risk attitudes by Andersen, Harrison, Lau and Rutström (2008), Harrison and
3.5.1 Estimation Approach
Data are pooled across the three types of lotteries. The main assumption is: different
sources of uncertainty generate different source functions but not different utility functions.
Therefore, we estimate a common CRRA utility function but allow the weighting parameters to
vary with type of lotteries. The reference lottery is the pure risk lottery, and the covariates are
the scrambled and blackened lotteries. The econometric approach is illustrated using the Prelec
weighting function shown in (1). This approach can be easily extended to the TK and Power
weighting functions.
Recall the Prelec weighting function in (1):
𝑤(𝑝) = 𝑒𝑥𝑝 (−𝜂(−𝑙𝑛𝑝)𝜑)
where 𝑤(. ) is the function that transforms p into decision weight, 𝑤(𝑝), and
𝑤(. ) weighs the best option and (1 − 𝑤(. )) weighs the worst option.
The parameters, 𝜂 and 𝜑, are allowed to vary with the exogenous treatments for type of lotteries. With the pure risk lottery as the reference, both 𝜂 and 𝜑 are estimated conditional on dummy covariates Scrambled and Blackened:
𝜂 = 𝜂𝑝𝑢𝑟𝑒 𝑟𝑖𝑠𝑘 + 𝜂𝑠𝑐𝑟𝑎𝑚𝑎𝑏𝑙𝑒𝑑 × 𝑆𝑐𝑟𝑎𝑚𝑏𝑙𝑒𝑑 + 𝜂𝑏𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑 × 𝐵𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑 (4)
𝜑 = 𝜑𝑝𝑢𝑟𝑒 𝑟𝑖𝑠𝑘 + 𝜑𝑠𝑐𝑟𝑎𝑚𝑎𝑏𝑙𝑒𝑑 × 𝑆𝑐𝑟𝑎𝑚𝑏𝑙𝑒𝑑 + 𝜑𝑏𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑 × 𝐵𝑙𝑎𝑐𝑘𝑒𝑛𝑒𝑑 (5) with 𝜂 and 𝜑 each constrained to be non-negative.
𝑅𝐷𝑈𝑟𝑖𝑠𝑘𝑦 = 𝑒𝑥𝑝 (−𝜂(−𝑙𝑛𝑝)𝜑) ∗ (𝑥𝐻
𝑟𝑖𝑠𝑘𝑦(1−𝑟)
(1−𝑟) ) + (1 – 𝑒𝑥𝑝 (−𝜂(−𝑙𝑛𝑝)𝜑)) ∗ (
𝑥𝐿𝑟𝑖𝑠𝑘𝑦(1−𝑟) (1−𝑟) ) (6) where r is the coefficient of relative risk aversion, xH is the high prize, xL is the low prize, and
p is the objective probability of the high prize, xH. Similarly, the evaluation for the safe option
is: 𝑅𝐷𝑈𝑠𝑎𝑓𝑒 = 𝑒𝑥𝑝 (−𝜂(−𝑙𝑛𝑝)𝜑) ∗ (𝑥𝐻 𝑠𝑎𝑓𝑒(1−𝑟) (1−𝑟) ) + (1 – 𝑒𝑥𝑝 (−𝜂(−𝑙𝑛𝑝)𝜑)) ∗ ( 𝑥𝐿𝑠𝑎𝑓𝑒(1−𝑟) (1−𝑟) ) (7) The latent preferences for evaluating each option, or the RDU for each lottery pair, is
calculated for the candidate estimates of 𝑟, 𝜂 and 𝜑. Following (6) and (7), the index
∆𝑅𝐷𝑈 = 𝑅𝐷𝑈𝑟𝑖𝑠𝑘𝑦− 𝑅𝐷𝑈𝑠𝑎𝑓𝑒 (8)
is the difference in valuation between the risky option and the safe option. The index is then linked
to observed choices by using a “logit” likelihood function that is denoted as Λ(∆𝐸):
𝑝𝑟𝑜𝑏 (𝑐ℎ𝑜𝑜𝑠𝑒 𝑟𝑖𝑠𝑘𝑦 𝑜𝑝𝑡𝑖𝑜𝑛) = Λ(∆𝑅𝐷𝑈) (7)
The risky option is assumed to be chosen when Λ(∆𝑅𝐷𝑈) > ½.
Thus the likelihood of the observed responses, conditional on the RDU specification,
CRRA utility function and the Prelec probability weighting function specifications being true,
depends on the estimated parameters 𝑟, 𝜂 and 𝜑 given the above stochastic specification and the observed choices, y. The log-likelihood is then
ln 𝐿 ( 𝑟, 𝜂, 𝜑; 𝑦) = Σ[ lnΛ(∇𝑅𝐷𝑈) × 𝐈(𝑦 = 1) + ln(1 − Λ(𝛻𝑅𝐷𝑈)) × 𝐈(𝑦 = 0)] (9) where 𝐈(∙) is the indicator function and y = l (0) denotes the choice of the risky (safe) option.
An important extension of the core model is to apply contextual utility, due to Wilcox
(2011), and to allow for subjects to make some behavioral errors. The latent index in (8) then
becomes:
∆𝑅𝐷𝑈 = [(𝑅𝐷𝑈𝑟𝑖𝑠𝑘𝑦 − 𝑅𝐷𝑈𝑠𝑎𝑓𝑒) /ν] /𝜇 (8-) where ν is a normalizing term defined as the difference between the maximum and the minimum utility in each lottery pair. The parameter 𝜇 > 0 is a structural Fechner “noise parameter” used to allow some error when evaluating the difference in RDU between the two options. A
common Fechner error is assumed for the three types of lotteries.
We extend the likelihood specification to include the noise parameter, 𝜇, and maximize
ln 𝐿 (𝑟, 𝜂, 𝜑, 𝜇 ; 𝑦) by estimating r, 𝜂, 𝜑 and 𝜇, given observations on y. The estimation is performed using maximum likelihood.
3.5.2 Descriptive Statistics
The characteristics of the subject pool are described in Table C3. Each gender is about
evenly represented in the overall sample. Income is divided into two groups: 40% have
household income above $100,000 and the rest have household income of $100,000 or below.
Age is divided into two groups: 56% are between the ages of 18 and 40 and the rest between the
ages of 41 and 75. A majority hold a college education (80%).
The underlying objective probabilities (i.e., 0.1, …, 0.9) are randomly assigned to subjects. If subjects are evenly distributed across the 9 objective probabilities, then the
is true for the pure risk lottery tasks where each subject completes four tasks; Table C4 shows
that the proportion of subjects in each objective probability is about 11%. This is also true for
many of the probabilities in the uncertainty lottery tasks. However, since each subject only
completes one scrambled and one blackened lottery task, the sample set there is smaller, and
thus, it is harder to achieve an even representation of subjects for all probability assignments. A
few of the probability assignments have proportions that are as low as 6% or as high as 19% of
the sample size (relative to the ideal 11%).
The proportion of safe choices made by subjects is illustrated in Figure C4.1; the solid,
dash and dotted curves represent the pure risk, scrambled and blackened lotteries, respectively.
For the pure risk lotteries, as the probability of getting the high prize increases, the proportion of
safe choices decreases, as shown by the downward-sloping solid curve. This behavior is
expected since subjects know the probabilities of all possible outcomes. In contrast, the curves
for the uncertainty lotteries are flatter, which suggests that behavior is similar across the range of
underlying probabilities. The same general pattern is observed across demographic sub-samples.
Thus, there is preliminary evidence that in the uncertainty lotteries subjects tend to overweigh
low likelihood events and underweigh high likelihood events, displaying likelihood insensitivity.
Comparing across the three curves, in the low probabilities the pure risk curve is below
the uncertainty curves, whereas in the high probabilities the pure risk curve is above the
uncertainty curves. To examine if the three curves are significantly different from one another, a
proportions test is performed to examine if the proportions of safe choices are significantly
different across the three types of lotteries.65 When comparing between the uncertainty lotteries,
65 The proportions test, by the Stata command prtesti, tests whether the same proportion from two classes are significantly different.
behavior is not significantly different. When comparing between the uncertainty and the pure
risk lotteries, behavior is significantly different. This is shown in Figure C4.2, where the “black
dot” denotes if a given uncertainty lottery has proportion of safe choices that is significantly different from the pure risk lotteries. Overall there is preliminary evidence showing that subjects
behave differently in scenarios with known vs. unknown risk.