Socioeconomic Institute
Sozialökonomisches Institut
Working Paper No. 0705
Risk and Rationality: Uncovering Heterogeneity in
Probability Distortion
Adrian Bruhin, Helga Fehr-Duda, and Thomas F. Epper
July 2007, revised version
Socioeconomic Institute
University of Zurich
Working Paper No. 0705
Risk and Rationality: Uncovering Heterogeneity in Probability Distortion
July 2007, revised version
Authors’ addresses: Adrian Bruhin
E-mail: adrian.bruhin@sts.uzh.ch
Helga Fehr-Duda
E-mail: fehr@econ.gess.ethz.ch
Thomas F. Epper
E-mail: epper@econ.gess.ethz.ch
Publisher Sozialökonomisches Institut
Bibliothek (Working Paper)
Rämistrasse 71
CH-8006 Zürich
Phone: +41-44-634 21 37
Fax: +41-44-634 49 82
URL: www.soi.uzh.ch
E-mail: soilib@soi.uzh.ch
Risk and Rationality:
Uncovering Heterogeneity in
Probability Distortion
∗
Adrian Bruhin Helga Fehr-Duda Thomas F. Epper
July 18, 2007
Abstract
It has long been recognized that there is considerable heterogeneity in individual risk
taking behavior but little is known about the distribution of risk taking types. We present
a parsimonious characterization of risk taking behavior by estimating a finite mixture
regression model for three different experimental data sets, two Swiss and one Chinese,
over a large number of real gains and losses. We find two distinct types of individuals:
In all three data sets, the choices of roughly 80% of the subjects exhibit significant
deviations from rational probability weighting consistent with prospect theory. 20%
of the subjects weight probabilities linearly and behave essentially as expected value
maximizers. Moreover, the individuals are assigned to one of these two groups with
probabilities of close to one resulting in a low measure of entropy. The reliability and
robustness of our classification suggest using a mix of preference theories in applied
economic modeling.
KEYWORDS: Individual Risk Taking Behavior, Latent Heterogeneity, Finite Mixture
Regression Models
JEL CLASSIFICATION: D81, C49
∗
Authors’ affiliation: Swiss Federal Institute of Technology, Chair of Economics, Weinbergstrasse 35, CH-
8092 Zurich, Switzerland, phone: +41 44 632 4625, email of corresponding author: adrian.bruhin@sts.uzh.ch
1 Introduction
Risk is a ubiquitous feature of social and economic life. Many of our everyday choices, and
often the most important ones, such as what trade to learn and where to live, involve risky
consequences. While it has long been recognized that individuals differ in their risk taking
attitudes, surprisingly little is known about the distribution of risk preferences in the pop-
ulation (for an exception see Dohmen, Falk, Huffman, Sunde, Schupp, and Wagner (2005)).
Since preferences are one of the ultimate drivers of behavior, knowledge of the comp osition
of risk attitudes is paramount to predicting economic behavior. Economic models often allow
for heterogeneity, but this heterogeneity is usually confined to remain within the boundaries
of the standard model of preferences, expected utility theory (EUT). The empirical evidence,
however, reveals that heterogeneity in risk taking behavior is of a substantive kind, i.e. some
people evaluate risky prospects consistently with EUT, whereas other people deviate substan-
tially from expected utility maximization (Hey and Orme, 1994). Moreover, it seems to b e the
case that rational decision makers revealing EUT-preferences constitute only a minority of the
population. To improve descriptive performance a plethora of alternative theories have been
developed (for an overview see Starmer (2000)). Unfortunately, no single b est fitting model
has been identified so far (Harless and Camerer, 1994) and, depending on the individual, one
or the other model fits better. This finding poses a serious problem for applied economics.
What the mo deler needs is a parsimonious representation of risk preferences which is empir-
ically well grounded and robust, and not a host of different functionals. Providing such a
parsimonious characterization of heterogeneity in risk taking behavior is the objective of this
paper.
Our method is based on a literature on classifying individuals which has recently emerged in
the social sciences. On the basis of statistical classification procedures, such as finite mixture
regression models, investigators have tried to discover which decision rules people actually
use when playing games or dealing with complex decision situations (El-Gamal and Grether,
1995; Houser, Keane, and McCabe, 2004). The finite mixture regression approach does not
require fitting a model for each individual which is - given the usual quality of the choice data
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- frequently impossible. Instead, our approach reveals latent heterogeneity by estimating the
relative sizes of distinct behavioral groups and by endogenously assigning each individual to
a spe cific group characterized by a unique set of parameter values.
We apply such a finite mixture regression mode l to choice data from three different exper-
iments, two of which were conducted in Zurich, Switzerland. The third experiment took place
in Beijing, People’s Republic of China. We analyze 452 subjects’ decisions over real monetary
gains and losses, which comprise a total of about 18,000 choices. All three experiments were
designed in a similar manner and served to elicit certainty equivalents for binary lotteries.
Using a fle xible sign-dependent functional as basic behavioral model, we show the following
results.
First, in all three data sets, we find two distinct behavioral types of risk taking behavior.
Second, the ratios of the different types in their respective populations are practically equal
in both the Swiss and the Chinese data sets and amount to roughly 20:80. Third, without
putting any a priori restrictions on parameter values we find that one of the two types, which
comprises about 20% of the individuals in each data set, exhibits near linear probability
weighting functions and value functions. Therefore, this group can essentially be characterized
as expected value maximizers. This result is particularly interesting in the light of Rabin’s
calibration theorem (Rabin, 2000) which shows that expected utility maximizers should be
approximately risk neutral for small stakes typically encountered in laboratory experiments.
Therefore, we label subjects belonging to this group of risk neutral people as “EUT-types”.
Fourth, in each data set, the second group, which comprises about 80% of the individuals,
is characterized by significant deviations from linear probability weighting and can be con-
veniently described as prospect theory types. Fifth, almost all the experimental subjects are
unambiguously assigned to one of the two distinct type s. Measuring the quality of classifica-
tion by the average normalized entropy (El-Gamal and Grether, 1995) we obtain an extent
of ambiguous assignments of less than 5% of the maximum entropy, a value which is, to our
knowledge, unequaled in the literature. Thus, we observe almost no “ambiguous” types, i.e.
individuals that are assigned a high probability (of say 0.4) of being one type and a high
probability (of say 0.6) of being another type are practically absent. This c lean classifica-
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