Preference for Skew in Lotteries:
Laboratory Evidence and Applications?
Thomas Astebroa , José Matab , Luı́s Santos-Pintoc,†
a
Haute École Commerciale, Paris
Universidade Nova de Lisboa, Faculdade de Economia
c
University of Lausanne, Faculty of Business and Economics
b
This version: March 25, 2009
Abstract
This paper uses a laboratory experiment to investigate how the skew of
a payoff distribution influences choices under risk. We find a statistically
significant preference for positive skew under low and high stakes. We find
that the preference for skew is driven by subjective distortion of probabilities
and the shape of the utility function. We find that 54% of subjects make
choices consistent with Expected Utility Theory (EUT) whereas 46% do not.
Among the EUT individuals 47.5% love skew, 49% are indifferent to it, and
3.5% are skew averse. Among the non-EUT individuals 58% like skew and
42% are skew averse. We discuss the economic implications of our findings for
gambling in state lotteries and inventive activity. We show that a RDU model
calibrated with our estimated parameters is able to explain why individuals
only buy from 1 to 10 lottery tickets and why some individuals become
inventors.
JEL Codes: D81; C91.
Keywords: Risk; Skew; Gambling; Inventive Activity; Lab Experiment.
?We thank Simon Parker for providing data and Xavier Gabaix, Botond Koszegi,
Glenn Harrison, and Joel Sobel for helpful comments and suggestions. We also
thank seminar participants at HEC Paris, Universidade Católica Portuguesa, and
University of Lausanne. Gordon Adomdza and Won Kno provided research assistance.
†
Corresponding author. Luı́s Santos-Pinto. University of Lausanne, Faculty of
Business and Economics, Extranef 211, CH-1015, Lausanne, Switzerland. Ph: 41216923658. Fax: 41-216923435. E-mail address: [email protected].
1
1
Introduction
People may have a preference for positive skew in risky decisions. For example, people tend to overbet on the long-shot horse with low probability
of winning large returns rather than the favorite with the greatest expected
return (Joseph Golec and Maurry Tamarkin, 1998). And when people buy
lottery tickets, Forrest, Simmons and Chesters (2000) and Garrett and Sobel
(1999) show that people are more concerned with the size of the top prize
than the expected value of the lottery.
In addition, positive skew may effect significant life choices, not just gambling. Three-quarters of all people that enter self-employment face higher
variance but lower expected return than in employment (Barton Hamilton,
2000.) Further, 97% of inventors will not break even on their investments but
face a very skew distribution of returns conditional on succeeding (Thomas
Astebro, 2003.)1
The two latter results could be explained by risk loving, but Charles Holt
and Susan Laury (2002) [HL] show that in choices over real-money lotteries
people generally are risk averse, trading off returns for less risk, and that risk
aversion increases markedly as stakes go up. Additional experiments using
their method confirm low to modest risk aversion also among entrepreneurs
(Julie Ann Elston, Glenn Harrison and Elisabet Rutstrom, 2005.)
Gambling in state lotteries may also be related to optimism, pleasure from
gambling, and/or anticipatory feelings. On the other hand, the decision to
become an entrepreneur or an inventor may be a result of preferences for
being one’s own boss or of overly confident perceptions of skill.2 However,
a preference for positive skew remains as a factor that might play a role in
these decisions.
This paper uses a laboratory experiment to test how positive skew influences risky choices. We ask subjects to make several sequences of 10 pairwise
choices between a safe and a risky lottery and we count the number of safe
1
Evidence from financial markets is consistent with investors having a preference for
positive skew. For example, securities that make the market portfolio more negatively
skewed earn positive ‘abnormal’ average returns. The estimates for the annualized premium range from 2.5%–Kraus and Litzenberger (1976)–to 3.6%–Harvey and Siddique
(2000).
2
See Hey (1984), Conlisk (1993), and Caplin and Leahy (2001). While some of these
hypotheses are empirically supported, all of them are simply qualitative in nature and
explain gambling in state lotteries without quantifying it.
2
lottery choices. The design is based on Holt and Laury (2002) with two important modifications. First, we consider lotteries that have more than two
outcomes so that we can use the third moment around the mean to manipulate skew and variance separately. Second, we use a graphical display to
represent lottery payoffs.
We consider three different skew treatments. In the first treatment the
safe and risky lotteries have symmetric prize distributions. We call this
treatment the “zero skew” condition. In the second treatment – “moderate
skew” – the safe lotteries are symmetric and the risky lotteries all have skew
equal to 2. In the third treatment – “maximum skew” – the safe lotteries are
symmetric and the risky lotteries all have skew equal to 3.2 (the maximum
possible skew in our framework). We also consider two monetary stakes
conditions: low and high stakes (20 times low stakes).
We find that the mean number of safe choice in the zero skew treatment
is 3.86 and 4.62, in the low and high stakes conditions, respectively. Introducing positive skew significantly reduces the number of safe choices. In the
moderate skew treatment, the mean number of safe choices is 3.59 and 4.11,
in the low and high stakes conditions, respectively. The mean number of
safe choices in the maximum skew treatment is 3.25 and 3.79, in the low
and high stakes conditions, respectively. Thus, the number of safe choices
decreases monotonically with an increase in skew under low and high stakes
and increasing the stakes increases the average number of safe choices for all
skew treatments. All these effects are statistically significant.
To explain these choices we resort to the structural estimation. The
results obtained in the zero skew treatment are consistent with EUT and with
findings in Holt and Laury (2002, 2005) and Harrison et al. (2005, 2007). The
only difference is that we find slightly less risk aversion than they do. The
large number of risk choices observed in the maximum skew condition could
be explained by EUT if most individuals were risk seekers. But that would
contradict the safe choices observed in for the zero skew condition. Thus,
the choices made in the zero skew and in the maximum skew conditions are
at odds with the expected utility model. Since EUT is not able to reconcile
subjects’ choices between the zero skew and the maximum skew treatments
we use consider rank-dependent utility (RDU) as an alternative model of
risky choices.3
3
It is well known that RDU models are able to accommodate probability-dependent
risk attitude; that is, the preferences are consistent with both risk-averse and risk-seeking
3
To perform the estimation we model RDU using Prelec’s (1998) probability weighting function and consider three different specifications for the
utility function: power, hyperbolic absolute risk aversion (HARA), and expopower. Power utility is the benchmark approach of most empirical work on
risk aversion. The HARA utility nests the power utility and provides a simple and flexible characterization of the trade-off between risk aversion and
preference for skew. The expo-power utility offers a more flexible representation of risk preferences across different stakes than the power or HARA
utility.
We find that RDU explains subjects’ lottery choices better than EUT
across the three utility functions. Using the HARA-RDU specification we find
an estimate of .82 for the probability weighting parameter. In the power-RDU
specification the estimate is .93 and in the expo-power RDU specification is
.97. Thus, we find evidence for an inverted s-shaped probability weighting
function across the three specifications.
We show that the HARA-RDU model explains behavior better than the
power-RDU or expo-power RDU models. These results are obtained for the
average individual. Taking advantage of our design we divide subjects into
groups. We find that 80 subjects (54%) make choices consistent with EUT
whereas 78 (46%) do not. Among the subjects whose choices are consistent
with EUT, 38 (47.5%) like skew, 39 (49%) are indifferent to it, and 3 (3.4%)
are skew averse. Among the subjects whose choices are consistent with RDU,
we find that 45 (58%) like skew and 23 (42%) are skew averse.
Our findings have implications on a number of economic settings. Consider gambling in state lotteries. Empirical evidence shows that (i) despite
a return of only $.53 on the dollar, public lotteries are extremely popular,
(ii) most players buy from 1 to 10 tickets, (iii) ticket sales are an increasing
function of the mean, a decreasing function of the variance, and an increasing
function of the skew of the prize distribution, and (iv) low-income individuals
spend a higher percentage of their income on lottery tickets than do wealthier
individuals.
It is well known that a risk neutral or a risk averse EUT individual will
never play a state lottery since such gambles offer a negative expected return.
It is also well known that buying a lottery ticket can be perceived as an
behavior depending on the probability distribution of outcomes. If individuals overweight
small probabilities and underweight medium to large probabilities in evaluating risky gambles, then they will make “risky” choices in gambles involving small probabilities of large
gains.
4
attractive gamble for a RDU individual since a lottery ticket offers a very
small probability of a very large gain and a RDU individual overestimates
small probabilities. However, the decision to buy lottery tickets has not been
studied in detail.
We formalize an individual’s decision to buy lottery tickets and calibrate it
using the estimated parameters. We find that a RDU individual with HARA
utility buys from 0 to 7 lottery tickets for probability weighting levels between
.75 and 1. Additionally, we find that, for our estimated parameters, the
HARA-RDU specification matches real world data on lottery ticket purchases
better than the power-RDU or expo-power RDU specifications. We also show
that, for most probability weighting levels, Prospect Theory (PT) predicts
that if a person wants to buy one lottery ticket, then she will want to buy at
least twenty tickets which is clearly counterfactual.
Finally, we show that a RDU model calibrate with our estimated parameters can explain the decision to become an inventor. To make this point we
use data on inventive activity taken from Astebro (2003). The data shows
that the distribution of returns to inventive activity has a negative mean,
high variance, and a positive skew. Thus, a risk neutral or a risk averse
EUT agent would not be an inventor. The data also shows that inventors
face non-negligible probabilities of making very large gains or suffering substantial losses. Since PT typically assumes that losses hurt twice as much as
equally sized gains, this makes it hard for it to explain this type of behavior.
In fact, we show that a PT agent only becomes an inventor for unreasonably
high levels of probability distortion.
The reminder of the paper is organized as follows. Section 2 explains
the experimental design. Section 3 presents the findings. Section 4 provides
a brief overview of the rank-dependent utility and shows that RDU and
HARA can explain the experimental results better than EUT and better than
RDU with power or expo-power utility. Section 5 discusses the economic
significance of the findings. The Appendix contains experimental design
details, the data and summary statistics.
5
2
2.1
Experimental Design
Holt and Laury (2002)
HL study the trade-off between risk and return using lotteries with two outcomes. Subjects are given the choice between the pairs of lotteries in Table
1.
insert Table 1 here
It is expected that subjects start by choosing the safe lottery (S) in the
top row as it has both higher expected value and lower variance. As one
proceeds down the table, the expected values of both lotteries increases,
but the expected value of the risky (R) lottery increases more. When the
probability of the high-payoff ticket in the R choice increases enough (moving
down the table), a person should cross over to choose R.
Different persons will switch at different points, the switching point being
determined by the degree of risk aversion. For example, a risk-neutral person
would choose S four times before switching to R. Even the most risk-averse
person should switch over by decision 10 in the bottom row, since R then
yields a sure payoff of $3.85. Indeed, HL find that 26% switch from S to
R on the fourth decision, another 26% switch on the fifth row, and 13%
switch on the sixth row. Using a standard constant relative risk aversion
utility function, the distribution of choices implies risk neutrality for the first
group, with slight relative risk aversion for the second and risk aversion for
the third group. HL also find that when stakes are increased 20 times or
more, the number of safe choices increases significantly. 19% delay to choose
R until the fifth row, 23% select on the sixth row and 22% on the seventh
row, and the whole distribution is shifted to the right.
2.2
Introducing Skew
To find out how positive skew influences choices under risk we make two
changes in Holt and Laury’s design. First, we consider lotteries that can
have from two to ten different prizes. Increasing the number of prizes allows
us to control the mean, variance, and skew of the sequences of lotteries.
Second, lotteries are presented in a graphical display rather than in a table
6
as in HL.
Researchers have found graphical display of data sometimes to be superior
to tabular display. The advantage seems to depend on data structure and
task complexity, where for either simple or very complex tasks or unclear
data structures there seems to be little advantage of graphical display.4
Figure 1a shows our graphical representation for the first choice between
S and R in HL lotteries. Each of the lines in HL table is depicted in one
graph of the kind in Figure 1a. People are presented with the 10 choices in
sequence and are provided with a review section where any decision may be
revisited.
Figure 1b shows one of the skew treatments. Figure 1b clearly displays
a pattern, and since the various lotteries have different patterns due to skew
we believed subjects more easily can interpret the graphical than the tabular
display.
insert Figures 1a here
insert Figure 1b here
The first comparison we will make is between the results obtained when
the experiment is conducted offering the same choice options with the tabular
display of HL and with our graphical display. (The full set of lotteries are
available in an Appendix.) We then introduce skew systematically. We
modify the lotteries to have the same mean and variance as in HL, but while
the safe option has no skew, the risky option has skew equal to 0, 2 and 3.16
in the different treatments. We focus on positive skew as this seems to be the
most interesting case in a number of circumstances, but our exercise could
be extended to lotteries with negative skew.
If the expected payoff were to be maintained the higher levels of skew
would lead to some negative payoffs. We therefore add $1 to HLs payoffs.
We test whether there is any effect of adding $1. We also introduce lotteries
with 20 times the payoffs to study the impact of skew under large stakes.
4
See, e.g. Cheri Speier, 2006; Meyer J, Shamo MK, Gopher D 1999; MIH Hwang and
BJP Wu, 1990.
7
Compared to HL, we randomize subjects’ allocation to treatments and the
order of treatments, we do not offer hypothetical gambles, and we do not run
treatments with very high stakes (50× and 90×).
2.3
Treatments and Subjects
Instructions were kept identical to those in HL, with some necessary revisions
to account for the difference in presentation formats and the fact that the
task was conducted online rather than with paper and pencil. Subjects also
took a knowledge test on their understanding of the instructions and had
to pass to move forward to the experiment. (Example instructions are in
Appendix.)
Table 2 presents the different treatments. There are three basic treatment
sequences: a) subjects perform 1 set of HL prize distribution choices (T1-T4)
followed by 3 sets of low stakes choices with 3 skews and then draw a prize
from one of the 4 sets; b) Subjects perform 3 sets of high stakes choices with
3 different skews and then draw a prize from one of the 3 sets; c) Subjects
perform 3 sets of low stakes choices with 3 different skews. Subjects are
then given the option between (A) drawing a final prize from the three low
stakes distributions and finish the experiment or (B) not drawing a prize and
moving on to the high stakes choices. If subjects choose option (B) they will
do the three sets of high stakes choices with 3 skews and draw a prize from
those. All subjects presented with this option chose B.
Using posters we recruited 148 students at the University of Waterloo
during the Spring 2008. These were randomly allocated across 10 treatments
in six pre-determined treatment sequences as outlined in Table 2. Each
subject performed three or more treatments in the sequences displayed in
Table 3. Each of the sequences in the table was performed by at least 24
subjects. At the end of the experiment one prize is chosen randomly and
subjects were awarded their prize plus a $5 participation fee.
insert Table 2 here
insert Table 3 here
8
Subjects in our sample are briefly described in Table 4. Some of the
subjects did not respond to all of the demographics questions, the number
of observations available for each item being presented along with the corresponding summary statistics.
insert Table 4 here
3
3.1
Findings
Skew and Incentive Effects
To test the hypothesis that individuals display a preference for positive skew
we considered three different treatments. In the first treatment the safe and
risky lotteries have symmetric prize distributions. We call this treatment the
“zero skew” condition. In the second treatment – “moderate skew” – the
safe lotteries are symmetric and the risky lotteries all have skew equal to 2.
In the third treatment – “maximum skew” – the safe lotteries are symmetric
and the risky lotteries all have skew equal to 3.2 (the maximum possible skew
in a lottery with ten prizes).
Table 5 shows the number of safe choices under the different conditions.
Individuals display a monotonic preference for skew in both the low stakes
and high stakes condition. We also find that increasing the stakes increases
the average number of safe choices for all skew conditions. That relative risk
aversion increases with the scale of cash payoffs is consistent with Holt and
Laury (2002).
insert Table 5 here
We performed OLS regressions with the number of safe choices as the
dependent variable and included dummies for moderate and maximum skew,
and for high stakes (see Table 6). We find that the number of safe choices
decreases by 0.34 from zero skew to moderate skew and by 0.67 from zero
skew to maximum skew. When confronted with high stakes lotteries subjects
make significantly less risky choices. The least squares regression provides
easy interpretation of estimated coefficients. The coefficient 3.904 for the
9
constant in column (i) implies that under low stakes and no skew subjects
make on average 3.9 safe choices - which is risk neutral behavior.5
insert Table 6 here
Our main results are robust to the inclusion the subjects socio-demographic
characteristics described in Table 4. The high stakes coefficient decreases
somewhat from 0.60 to 0.46, but it remains significant. By design, all subjects perform all skew conditions and skew conditions are, therefore, orthogonal to subjects. The minor differences that we observe are therefore due to
the fact that observations in the two tables are not exactly the same as we do
not observe all characteristics for all subjects. This estimation reveals that
males, non-white individuals and part-time students make riskier decisions.
All other characteristics, except those related to income, are non-significant
once controlling for everything else in the model.
Since all subjects make repeated decisions under different conditions we
can also estimate individual random and fixed effects models. These estimations deliver the same coefficients for skew conditions, but the precision is
significantly increased, because the panel-data models correct for heterogeneity in risk preferences, which appear to be large. Increased precision means
that the effect of moderate skew is now significant at the 1% level, rather
than at the 5% level. The high stakes coefficient is approximately 0.5, with
p-values below 1% (see Table 6).
Socio-demographic characteristics does not seem to be important. Only
gender and race have some effect on choices; males and non-whites make
riskier choices than females and whites, although statistical significance is
reduced when accounting for individual effects.6
5
As a robustness check, we also ran an ordered-Probit where the dependent variable is
the number of times the safe choice was made. The results from such a model are more
difficult to interpret, since the marginal impact of the coefficients depend on the level of
the covariates and the specific choice being considered. The qualitative results did not
present changes from those in Table 6.
6
A reason for including the individual characteristics explicitly is that the response of
individuals to skew might depend on demographics. To test this we included interactions
between the skew variables and each of the demographic characteristics that are captured
by a dummy (gender, decision maker in the house, race, graduate student status, major,
full time student status). In all cases we cannot reject the null that the effects are identical.
10
3.2
Robustness
Our subjects perform their choices in different sequences (recall table 3).
Moreover, some of them went through the initial round which tested framing
effects. We would therefore like to know whether choices are affected by order
and by the initial treatment. The inclusion of such controls in the regression
does not change the results regarding skew. Moreover, the controls are not
jointly significant. The set of four dummies indicating which initial treatment
(if any) the subject had performed has a p-value of 0.113, while the p-value
for the five order dummies is 0.099. Thus, we conclude that there is no
meaningful order effect.
We have noted earlier that previous research has found graphical display
of data to be superior to tabular display, in particular when the tasks are neither very simple nor very complex. Subjects making fully consistent choices
would never switch back and forth between the safe and risky options. Some
of them, however, did. We find that subjects make fewer inconsistent choices
in our graphical display as compared to the tabular display. 32% of the subjects switched back and forth when data was presented in the tabular display,
and only 6% did so in the graphical display, which seems to indicate that the
graphical display is cognitively superior to the tabular display for this task.7
There seems to be some indication in our data that the tabular display
leads subjects to make a somewhat larger number of safe choices. This may
be due to a lesser ability to fully understand the data in the tabular than
in the graphical display, as indicated by the higher proportion of switching
back and forth for the tabular display. The observed average number of safe
choices is 5.3, 3.2, 4.6 and 3.6 for treatments T1 to T4, respectively. We
reject that hypothesis that the mean number of safe choices in T1 is equal
to that in T2, but not that T3 is equal to T4. For all samples, except for
T1, we cannot reject the hypothesis of risk neutral behavior. Finally, adding
one dollar to the prizes has a small and not significant effect. However, as
we have seen, multiplying the stakes by a factor of twenty has a significant
effect.
7
We checked the robustness of our regression results by defining two additional variables. One is defined as the decision before the one in which subjects make their first
risky decision. The other is their last safe decision. The three variables are identical for
subjects that never go back and forth. Results did not show any remarkable changes, except that the effect of being in the moderate skew condition is somewhat more imprecisely
estimated, in particular when the variable is their last safe decision.
11
4
Estimation of Utility Functions
In this section we use structural estimation of utility functions to compare
alternative explanations for the experimental results. To model subjects’
choices we focus on Expected Utility Theory (EUT) and Rank-Dependent
Utility (RDU). Together with Prospect Theory (PT) these are the most influential models of decision making under risk and uncertainty. Since our
experiment does not involve losses PT reduces to RDU if there is no asset
integration.8
4.1
Rank Dependent Utility
RDU assumes that individuals subjectively distort probabilities via a probability weighting function.9 Consider a lottery (p1 , x1 ; ...; pn , xn ), yielding
outcome xj with probability pj , j = 1, .., n. The probabilities p1 , ..., pn are
nonnegative andPsum to one. According to RDU the utility of this lottery
is evaluated as nj=1 πj u(xj ),where u is the utility function and the πj s are
called decision weights. The decision weights are nonnegative and sum to
one. Expected utility is the special case of the general weighting model
where πj = pj for all j. If x1 > x2 > ... > xn , then
P
RDU (p1 , x1 ; ...; pn , xn ) = nj=1 πj u(xj ),
where, for each j,
πj = w(p1 + ... + pj ) − w(p1 + ... + pj−1 ),
with w(0) = 0 and w(1) = 1. The function w(p) is the decision weight
generated by the probability p when associated with its best outcome. Thus,
the decision weight πj of outcome j depends only on its probability pj and
its ranking position.
8
Prospect Theory has four key features. The first is reference dependence, that is,
individuals care about gains and losses relative to a status quo and not about final wealth
levels. The second is loss aversion, that is, that losses loom larger than equally sized gains.
The third is probability weighting. The fourth is that the value function is concave for
gains but convex for losses (diminishing sensitivity in the domain of gains and losses).
9
Rank-dependent models were introduced by Quiggin (1982) for decision under risk
(known probabilities) and by Schmeidler (1989) for decision under uncertainty (unknown
probabilities).
12
Many observed weighting functions are not completely convex or concave
but exhibit a mixed pattern. They are concave for small probabilities and
convex for moderate and high probabilities. This pattern is called inverted
s-shaped probability weighting. The pattern implies that subjects pay much
attention to best and worst outcomes, and little attention to intermediate
outcomes.10
To model RDU we use Prelec’s (1998) probability weighting function:
w(p) = exp(−(− ln(p))η ), for 0 < p ≤ 1 and 0 < η,
(1)
where the parameter η determines the degree of distortion of probabilities.
When η = 1 there is no distortion of probabilities and we are back to EUT.
If η ∈ (0, 1), then the function captures the inverted s-shaped pattern and
the further away η is from 1 the higher the degree of probability weighting.
If η > 1, then we have a s-shaped pattern where small probabilities are
underestimated and large probabilities are overestimated.11
4.2
The Risk-Skew Trade-off
To model the utility function we focus on hyperbolic absolute risk aversion,
power, and expo-power utilities. These three utility functions represent different types of risk preferences and risk-skew trade-offs.
Consider the utility function:
u(x) = (α + x)1−β ,
(2)
where x is wealth, β < 1, and α + x > 0.12 An individual with this utility
function is risk seeking if β < 0, risk neutral if β = 0, and risk averse if
0 < β < 1. This utility function belongs to the family of hyperbolic absolute
risk aversion utility functions, also known as HARA.13 The coefficient of
10
Inverted s-shaped probability weighting predicts that people make risky choices in
gambles that yield gains with small probabilities such as in public lotteries. It also predicts
that people make safe choices in gambles that yield losses with small probabilities, a
phenomenon relevant for insurance. See Quiggin (1982).
11
An alternative would be to use Kahneman and Tversky’s (1992) probability weighting
function. However, Ingersoll (2008) shows that Kahneman and Tversky’s function is nonmonotonic. That is not the case with Prelec’s probabilit weighting function.
12
Note that nothing prevents α from being negative.
13
The HARA family of utility functions was introduced in Merton (1971). The HARA
family has an Arrow-Pratt’s coefficient of absolute risk-aversion that is an hyperbolic
function of wealth since −u00 (x)/u0 (x) = β/(α + x).
13
relative risk aversion of the HARA is
−
u00 (x)x
β
=
.
u0 (x)
1 + α/x
(3)
From (3) we see that when α > 0 the coefficient of relative risk aversion is
increasing with wealth and is bounded above by β.
We know from Tsiang (1972) that a risk averse individual has preference
for positive skew if the third derivative of his utility of wealth is positive. As
a manifestation of such preference a risk averse individual will be prepared
to accept a lower expected return (but not a negative one) or a higher level
of overall riskiness if the distribution of payoffs is more skewed to the right.
The HARA and many other utility functions satisfy this property.14
The trade-off between risk aversion and skewness preference is given a
choice-theoretical characterization by Chiu (2005). He shows that if two
lotteries are strongly skewness comparable, then the function −u000 (x)/u00 (x)
has the interpretation of measuring the strength of an individual’s preference
for skew against his risk aversion.15 The higher the ratio −u000 (x)/u00 (x) the
more important is the preference for skew and the less important is risk
aversion.
From (2) we have that
−
1+β
u000 (x)
=
, for 0 < β < 1.
00
u (x)
α+x
(4)
The power utility function is the benchmark model of most empirical work
on risk aversion.16 A power utility with coefficient of relative risk aversion
equal to β is given by
u(x) = x1−β ,
(5)
14
Tsiang (1972) shows that decreasing absolute risk aversion implies a preference
for skew but the reverse is not true. To see this consider the coefficient of absolute
risk aversion ARA = −u00 (x)/u0 (x). The derivative of ARA with respect to wealth is
000
0
00
(x)]2
− u (x)u[u(x)−[u
. We see that for an individual to display decreasing absolute risk
0 (x)]2
aversion it must be that u000 (x) >
[u00 (x)]2
u0 (x)
> 0. In contrast, for an individual to display
00
2
increasing absolute risk aversion it must be that u000 (x) < [uu0(x)]
(x) . Thus, individuals with
decreasing ARA have a preference for skew and those with increasing ARA might prefer
skew or be averse to it.
15
This ratio also determines precautionary savings.
16
See Harrison et al. (2005, 2007) and Harrison and Rutström (2008).
14
where β < 1. We see from (5) that the HARA nests the power utility since
setting α = 0 in (2) gives us a power utility. So, the power utility, like the
HARA, also displays a preference for skew. From (5) we have that
−
1+β
u000 (x)
=
, for 0 < β < 1.
00
u (x)
x
(6)
1+β
> α+x
for any given
Comparing (4) to (6) we see that if α > 0 then 1+β
x
x. This means that, for any given wealth level, the power utility function
displays a stronger preference for skew relative to aversion to risk than the
HARA.
The expo-power was first proposed by Saha (1993) and is defined as
u(x) =
1
1 − exp −φx1−θ ,
φ
with φ > 0 and θ < 1.17 The expo-power exhibits decreasing absolute risk
aversion and increasing relative risk aversion when 0 < θ < 1 and φ > 0. It
exhibits increasing absolute and relative risk aversion when θ < 0 and φ > 0.
Thus, the expo-power is more flexible than the HARA or the power utility to
capture different types of absolute risk aversion. The main disadvantage of
the expo-power is that its function −u000 (x)/u00 (x) does not provide a simple
characterization of the risk-skew trade-off.
4.3
Estimation Results
The rank dependent utility of lottery i is
P
RDUi = nj=1 πj u(xj ),
and the expected utility of lottery i is obtained by setting πj = pj for all
prizes j. To model subjects’ choices we follow the approach of Holt and Laury
(2002) and Harrison and Rutström (2008) and use maximum likelihood to
estimate the relevant parameters. First we calculate the index
∇RDU = RDUS − RDUR ,
where RDUS is the rank dependent utility of the safe (left) lottery and RDUR
is the rank dependent utility of the risky (right) lottery. This latent index,
17
The expo-power converges to the power utility when φ converges to 0.
15
based on latent preferences, is then linked to the observed choices using a
logistic cumulative distribution function Φ(∇RDUi ). This “probit” function
takes any argument between ±∞ and transforms it into a number between
0 and 1. Thus, we have the logit link function,
Pr(Choice = S) = Λ(∇RDU ).
insert Table 7 here
The estimation results reported in Table 7 indicate that the HARA utility
with RDU is the specification that performs best. Since the HARA utility
nests the Power utility, the comparison between the two amounts to a test
on the significance of the α coefficient. This parameter is indeed significant
and we conclude that the HARA generalization is indeed useful. The ExpoPower is non-nested with the HARA but since the number of parameters
is the same in the two models the comparison between the two models are
based on the value of the likelihood function. The HARA utility performs
better than the Expo-Power, both in the EUT and RDU cases. Still, the
RDU is preferred to the EUT in all models, since the η coefficient is highly
significant in all the RDU estimations. In summary, the HARA with RDU
is the alternative that does a better job in explaining our data.
In a further extension to the regression equation in column (iv) of Table
6, we estimate subject specific responses to variation in skew. In order to
perform this exercise, we start from the observation that the effect of going
from the no skew to the high skew treatment is almost exactly twice as much
as going from no skew to medium skew. In fact, defining a variable that takes
the value 0 in the no skew case, 1 in the medium skew case, and 2 in the high
skew case, and using this variable in the regression returns an estimated skew
coefficient of 0.336 and no detectable change in the High Stakes coefficient.
We used this variable to estimate subject specific responses to skew by
estimating interaction terms between this variable and the individual fixed
effects. The estimated coefficients were then used to classify subjects into
groups according to their preference for skew. Group specific utility functions
were then estimated. Table 8 reports the results.18
18
Estimation of the parameter α was difficult in columns (iv) and (v). A grid search
over the parameter space revealed that the best value for α was close to −1 in both cases,
but also the likelihood function did not improve much over fixing the parameter in zero.
16
insert Table 8 here
The groups are formed according to whether choices are consistent with
EUT or with RDU and according to preference for skew. We classify all
subjects in columns (ii), (iii), and (iv) of Table 8 as making choices consistent
with EUT since their choices are not too extreme in the direction of skew
seeking or aversion.19 Thus, we have 80 EUT subjects and 78 RDU subjects.
Among the EUT individuals we classify the 39 subjects in column (iii)
as indifferent to skew since they either did not change their choice across
the three skew treatments or if they did they went back and forth without
displaying a particular pattern. We classify the 38 subjects in column (ii) as
being EUT skew seekers since they make systematically more risky choice as
the skew increases. We classify the 3 subjects in column (iv) as being EUT
skew averse since they make systematically less risky choices as the skew
increases.20
Among the RDU individuals we classify the 45 subjects in column (i) as
skew seekers since they display an extremely strong preference for skew which
is confirmed by an estimate for the average probability weighting parameter
of .57. We classify the 23 subjects in column (v) as skew averse since they
exhibit an very strong aversion to skew which is confirmed by an estimate
for the average probability weighting parameter of 3.686.
5
Economic Significance
In this section we show that a rank-dependent expected utility model calibrated with our estimates is able to explain several stylized facts of gambling
in public lotteries and the decision to become an inventor. We also show that
PT is not able to explain these behavior as well as RDU.
5.1
Gambling in Public Lotteries
Research on gambling in public lotteries shows that:
19
The probability weighting parameter η is not different from 1 in column (iii). In the
remaining columns, the coefficient is statistically different from unity.
20
There are only three subjects classified in the group in column (iv). The η parameter
is very imprecisely estimated and it is not statistically different from 0 or 1.
17
1. Despite a return of only $.53 on the dollar, public lotteries are extremely
popular. In a 2007 Gallup poll, 46 percent of Americans reported
participation in state lottery gambling.21
2. Most players only buy a small number of tickets. For example, a “lottery ticket” in the lottery “Euromillions” is the selection and purchase
of a combination of 5 digits (from 1 to 50) and 2 stars (from 1 to 9).
Each ticket costs e2.00. The gross expected value of buying one ticket
is approximately e.80. Most players buy from 1 to 5 tickets.22
3. Ticket sales are an increasing function of the mean of the prize distribution (better bets are more attractive ones), a decreasing function of
the variance in the prize distribution (riskier bets are less attractive),
and an increasing function of the skew of the prize distribution.23
4. Low-income individuals spend a higher percentage of their income on
lottery tickets than do wealthier individuals.24
In this section we show that RDU can explain these stylized facts but PT
cannot. To do that we start by formalizing the lottery ticket purchase problem. Let each lottery ticket win a unique prize of value G with probability p
and be sold at price c > pG > 0.25 Since buying a single lottery has negative
expected value (c > pG) a risk neutral or a risk averse EUT agent will never
buy a lottery ticket. The rank-dependent utility of buying n lottery tickets
is
RDU (n) = [1 − w(np)] u(z − nc) + w(np)u(z + G − nc),
21
http://www.gallup.com/poll/104086/One-Six-Americans-Gamble-Sports.aspx
A ticket in the US lottery “Powerball” is the selection and purchase of a combination
of five white balls out of 59 balls and one red ball out of 39 red balls. Each ticket costs
$1.00. The gross expected value of buying one ticket is approximately $.59.
23
See Walker and Young (2001).
24
See Haisley et al. (2008) and the references cited in the paper.
25
This is a simplified lottery. Real world public lotteries usally have multiple prizes
(second prize, third prize, and others).
22
18
where z is initial wealth and w(·) is the probability weighting function, given
by (1). The optimal n for a RDU agent is the solution to
max [1 − w(np)] u(z − nc) + w(np)u(z + G − nc)
n
s.t. w(p) = exp(−(− ln(p))η )
nc ≤ z
u(z) ≤ RDU (n)
To calibrate the lottery parameters we set G = $106 , p = 10−6 , and c = $2.
Thus, a person that buys a single state lottery ticket faces a lottery with mean
−$1, median −$2, standard deviation $1, 000, and skew 1000. For the power
utility we consider four values for β: 0, .25, .50, and .75.26 We calibrate the
HARA and expo-power utilities with the estimates obtained with the RDU
model: α = 60.7, β = .73, φ = .024, and θ = .502. We solve this problem for
three levels of initial wealth–$10, $100 and $1, 000–and sixteen levels of the
probability weighting parameter η.27
Table 9 summarizes the results for initial wealth of $10, Table 10 for initial
wealth of $100, and Table 11 for initial wealth $1, 000. Each cell in the tables
gives us the optimal number of lottery tickets that a RDU individual buys
for each utility function and probability weighting level.
insert Table 9 here
insert Table 10 here
26
Some recent estimates for the parameter β are: β = 0.67, β = 0.52, β = 0.48, for
private-value auctions, Cox and Oaxaca (1996), Goeree et al. (1999), and Chen and Plott
(1995), respectivaly, β = 0.44 for several asymmetric matching pennies games, Goeree et
al. (2002), and β = 0.45 for 27 one-shot matrix games, Goeree and Holt (2000). Campo
et al. (2000) estimate β = 0.56 for field data from timber auctions. Holt and Laury (2002)
find β to be between 0.3 and 0.5. Harrison et al. (2005) find β = 0.37 for low stakes and
β = 0.57 for high stakes.
27
The literature has found a variety of estimates for the parameter η in the domain of
gains. Tversky and Kahneman (1992) find η = 0.61, Camerer and Ho (1994) find η = 0.56,
Wu and Gonzales (1996) find η = 0.71, Abdellaoui et al. (2000) find η = 0.60, and
Harrison (2008) finds η = 0.92 and η = 0.95.
19
insert Table 11 here
The first column of Tables 9, 10, and 11 displays the optimal lottery
ticket purchases of a RDU individual with linear utility. Such a person buys
from 0 to 500 lottery tickets depending on his initial wealth and probability
weighting level. This happens because he overestimates the probability of
winning the lottery but suffers no disutility from facing a risky gamble so he
will buy as many tickets as his wealth permits. Clearly, the predicted lottery
ticket purchases of an individual with a linear utility are inconsistent with
real world data on lottery ticket purchases.
The second, third and fourth columns of Tables 9, 10, and 11 display
the optimal lottery ticket purchases of a RDU individual with power utility
with exponent .75, .50 and .25, respectively. We see that an increase in the
concavity of the utility function reduces lottery ticket purchases across all
probability weighting levels. An individual with power utility with exponent
.75 buys from 0 to 482 tickets while an individual with exponent .50 buys
from 0 to 167 tickets and an individual with exponent .25 only buys from 0
to 38 tickets.
The fourth column displays the optimal lottery ticket purchases of a RDU
individual with a power utility calibrated by the β estimate obtained from the
power-RDU model (the estimate is .755 which is equivalent to an exponent of
.245 ≈ .25). We see that such an individual buys public lottery tickets when
η ≤ .65 (initial wealth $10), η ≤ .775 (initial wealth $100) or η ≤ .825 (initial
wealth $1, 000). However, these thresholds for lottery ticket purchases are
far from the estimate obtained for η in the power-RDU model: .928.
The fifth column displays the optimal number of lottery ticket purchases
by a RDU individual with a HARA utility calibrated by the α and β estimates
obtained from the HARA-RDU model. Such an individual buys from 0 to 44
lottery tickets. Moreover, we see that for η between .90 and 0.75, an interval
that contains the estimate obtained for η in the HARA-RDU model–.815–,
he only buys from 0 to 7 tickets. This prediction matches closely real world
data on lottery ticket purchases.
The sixth column displays the optimal number of lottery ticket purchases
by an individual with an expo-power utility calibrated by the φ and θ estimates obtained from the expo-power RDU model. We see that such an
individual only plays a public lottery if η ≤ .60 (initial wealth $10), η ≤ .675
(initial wealth $100) or η ≤ .8 (initial wealth $1, 000). These thresholds
20
for lottery ticket purchases are far from the estimate obtained for η in the
expo-power RDU model: .974.
Thus, for our estimated parameters, the predicted behavior of a RDU
individual with a HARA utility matches real world data on lottery ticket
purchases better than the predicted behavior of a RDU individual with a
power or an expo-power utility.
Tables 9, 10, and 11 also show that increases in initial wealth lead to less
than proportional increases in lottery ticket purchases for all RDU specifications. This is consistent with the fact that low-income individuals spend a
higher percentage of their income on lottery tickets than high income individuals.
Let us now see what are the predictions of Prospect-Theory for gambling
in state lotteries. The PT value of buying n lottery tickets is
P T (n) = [1 − w(np)] v(−nc) + w(np)v(G − nc),
where v(x) is the gain-loss value function. The optimal n for a PT agent is
the solution to
max [1 − w(np)] v(−nc) + w(np)v(G − nc)
n
s.t. w(p) = exp(−(− ln(p))η )
0 ≤ P T (n)
A standard assumption in PT papers is that v(x) is a power function given
by
1−r
x , x≥0
v(x) =
,
−λ(−x)1−r , x < 0
where λ is the loss aversion coefficient.28 We set λ = 2 another standard
assumption of PT papers and consider six values for r: 0, .10, .25, .35, .50
and .75.29 Table 12 summarizes the results for PT.30
28
According to Wakker and Zank (2001) the PT model with power utility is generally
assumed in parametric tests and is currently the most used nonexpected utility form.
29
Tversky and Kahneman (1992) consider v(x) = xα for x ≥ 0 and v(x) = −λ(−x)β
for x < 0. They estimate the parameters and find that the median estimate for λ is 2.25
and the median estimate for the exponent of the value function is .88 for both gains and
losses.
30
A PT individual with an expo-power value function calibrated with the estimated
parameters does not buy lottery tickets for.60 ≤ η ≤ 1.
21
insert Table 12 here
The first column in Table 12 displays the optimal lottery ticket purchases
of a PT with a linear gain-loss value function. Such an individual buys from
0 to 5,511 lottery tickets. The remaining five columns in Table 12 display
the optimal lottery ticket purchases of a PT individual with a power value
function with exponent .90, .75, .65, .50 and .25, respectively. An increase in
the concavity of the value function reduces lottery ticket purchases across all
levels of probability weighting levels. An individual with power utility with
exponent .90 buys from 0 to 2,901 tickets, one with exponent .75 buys from
0 to 667 tickets, one with exponent .65 buys from 0 to 140 tickets, one with
exponent .50 buys from 0 to 2 tickets, and one with exponent .25 always buys
0 tickets.
Comparing the predictions of PT and RDU across power-value and powerutility functions that have the same exponent we find that: (1) a PT individual switches from not playing the lottery to playing it at higher levels
of probability distortion than a RDU individual, (2) a PT individual with a
power value function with exponent less than or equal to .50 hardly buys any
lottery tickets whereas a RDU individual with a power utility function with
exponent less than or equal to .50 buys some tickets, and (3) if probability
weighting is sufficiently high, an increment in probability distortion leads
to larger incremental ticket purchases for a PT individual than for a RDU
individual.
These results are driven by PT’s assumptions of loss aversion and convexity in the domain of losses. Switching from not playing the lottery to playing
it gives a very small probability of a large gain but a very large probability of
a small loss. Since losses are worth twice as much as equally sized gains, a PT
individual must be more “optimistic” to participate in a lottery than a RDU
individual. The interaction between loss aversion and concavity/convexity of
the value function makes it very unattractive for a PT individual to purchase
lottery tickets for sufficiently concave/convex value functions. The assumption of convexity in the domain of losses implies diminishing sensitivity to
increasing losses. Thus, for sufficiently high probability weighting levels, buying an additional lottery ticket is more attractive for a PT individual than
to a RDU individual.
In short, both RDU and PT predict no purchases of lottery tickets for low
levels of probability weighting and some purchases of lottery tickets for suf22
ficiently high levels of probability distortion. However, RDU predicts much
less purchases of lottery tickets than PT for most probability weighting level.
Thus, RDU provides a better description of gambling in state lotteries than
PT.
5.2
Inventors
Astebro (2003), using a sample of 1,091 Canadian inventors, finds that the
distribution of returns to inventions has negative mean, large standard deviation, and positive skew. Since the mean return is negative a risk neutral or
a risk averse EUT agent prefers not to become an inventor.
We will now show that RDU can explain why individuals become inventors. To perform the analysis we use Astebro’s data on the costs and returns
to invention. We take the point of view of a representative inventor that
faces a set of projects from which she will pick a single project. We assume
that the probability the representative investor selects any of these projects
is uniform.
Summary statistics for the distribution of the net present value of a typical
inventive project from this sample are: mean −$36, median −$7, 298, standard deviation $178, 978, skew 20.64, minimum −$1, 497, 456, and maximum
$4, 835, 203.31 The largest loss suffered was close to 1.5 million Canadian
dollars and the largest gain was close to 5 million Canadian dollars.
In an inventive activity the payoff from being very successful is substantially higher than the payoff from being very unsuccessful due to the positive
skew in returns. An RDU individual will be attracted to inventive activity
since he overestimates the probability of being very successful and of being
very unsuccessful (the low probability events) but underestimates the probability of being moderately successful or unsuccessful (the high probability
events).
To study the behavior of an RDU individual who faces this type of risk
we consider the same utility functions as in the previous Section. Initial
wealth is set to 1.5 million Canadian dollars so that the worst project can be
undertaken. Table 13 reports the findings. Cells are defined as in the prior
Tables.
31
The net present values are the discounted sum of development cost, opportunity cost
of labor, and revenues. Most inventions did not reach the market. In those cases the
revenue is assumed to be zero.
23
insert Table 13 here
The first row in Table 13 tells us that an RDU individual with a linear
utility chooses to invent if the probability weighting parameter is less than
.9997. We see that this condition is not at all restrictive which is not surprising since the expected value of inventing is only −$36 so it is enough to bit
just a tiny bit optimistic for a RDU individual with a linear utility to invent.
This result is not descriptive of real life since most individuals have concave
utility functions.
The second, third and fourth rows of Table 13 tell us that for a RDU
individual with a power utility, the more concave is the utility function (the
less the RDU individual likes risk for a fixed probability weighting level) the
more optimistic he has to be to become an inventor.
The fifth row in Table 13 displays the results for a RDU individual with
an HARA utility calibrated with the estimated parameters. This individual
is willing to invent as long as the probability weighting parameter is less than
.8682. The last row of Table 13 shows us that a RDU agent with a expo-power
utility calibrated with the estimated parameters does not want to invent for
all probability weighting levels. This happens because the expo-power utility
implies a large aversion to risk and a weak preference for skew.
Can Prospect Theory explain the decision to become an inventor? To
answer this question we apply the standard PT model to the decision to
become an entrepreneur. Table 14 reports the findings.
insert Table 14 here
We see from Table 14 that a PT individual with a power gain-loss value
function can become an inventor if the value function is not too concave/convex.
However, estimates of the exponent of the value function in PT papers are
usually around between 0.65 to 0.50. For this range a PT agent with a power
value function would only become an inventor for implausible levels of probability weighting. This result is driven by PT’s assumption of loss aversion.
Since an inventor can make large losses and losses are weighted twice as much
as equally sized gains, a PT needs to be extremely “optimistic” to become
an inventor.
24
6
Conclusion
This paper proposes a procedure to study the impact of skew in the distribution of payoffs on choices under risk. To procedure consists in offering
individuals three sets of ten pairwise choices between a safe and a risky lottery. In each set we keep the skew of the safe lotteries fixed at zero and
vary the skew of the risky lotteries. The procedure allows us to classify subjects into groups according to EUT and RDU theory and according to their
preference for skew.
We find that the average individual displays a strong preference for positive skew. The preference for positive skew comes from probability distortion
and the shape of the utility function. In the experiment subjects are given
objective probabilities of each lottery outcome. However, the average individual overestimates small probabilities and underestimates large probabilities.
This induces the preference for skew. We conjecture that in environments
where probabilities are subjective, probability weighting might strengthen
even further individuals’ choices towards gamble with positive skew.
There are many fields in economics and management where a better understanding of decision makers’ behavior when facing positively (and negatively) skewed payoff distributions might be important. For example, Blume
and Friend (1975) and King and Leape (1998) find that most of households
hold undiversified portfolios. We know that risk aversion implies that people
should hold diversified portfolios. Polkovnichenko (2005) shows that rankdependent utility predicts simulated portfolios quantitatively similar to those
found in the data. The preference for positive skew and probability weighting
might also be relevant for designing incentives in organizations (e.g. employee
stock options and prizes in rank-order tournaments).
25
7
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28
8
Tables
Table 1: HL’s Lotteries.
Option S
1/10 of $2.00, 9/10 of $1.60
2/10 of $2.00, 8/10 of $1.60
3/10 of $2.00, 7/10 of $1.60
4/10 of $2.00, 6/10 of $1.60
5/10 of $2.00, 5/10 of $1.60
6/10 of $2.00, 4/10 of $1.60
7/10 of $2.00, 3/10 of $1.60
8/10 of $2.00, 2/10 of $1.60
9/10 of $2.00, 1/10 of $1.60
10/10 of $2.00, 0/10 of $1.60
Option R
1/10 of $3.85, 9/10 of $0.10
2/10 of $3.85, 8/10 of $0.10
3/10 of $3.85, 7/10 of $0.10
4/10 of $3.85, 6/10 of $0.10
5/10 of $3.85, 5/10 of $0.10
6/10 of $3.85, 4/10 of $0.10
7/10 of $3.85, 3/10 of $0.10
8/10 of $3.85, 2/10 of $0.10
9/10 of $3.85, 1/10 of $0.10
10/10 of $3.85, 0/10 of $0.10
E(S)-E(R)
$1.17
$0.83
$0.50
$0.16
-$0.18
-$0.51
-$0.85
-$1.18
-$1.52
-$1.85
Table 2: Treatments.
Legend
T1
T2
T3
T4
T5L
T5H
T6L
T6H
T7L
T7H
Treatment
Tabular display of Holt and Laury’s (HL) low-payoff treatment
Graphical display of HL low-payoff treatment
Tabular display of HL low-payoff treatment plus $1.00
Graphical display of HL low-payoff treatment plus $1.00
Graphical display zero skew
Graphical display zero skew 20x (high payoff)
Graphical display moderate skew
Graphical display moderate skew 20x (high payoff)
Graphical display maximum skew
Graphical display maximum skew 20x (high payoff)
29
Table 3: Sequences of Treatments.
a i)
a ii)
a iii)
a iv)
b)
c)
T1 first, then T5L, T6L and T7L randomized in 6 different orders
T2 first, then T5L, T6L and T7L randomized in 6 different orders
T3 first, then T5L, T6L and T7L randomized in 6 different orders
T4 first, then T5L, T6L and T7L randomized in 6 different orders
T5H, T6H and T7H randomized in 6 different orders
T5L, T6L and T7L and then T5H, T6H and T7H randomized in 6
different orders
Table 4: Sample characteristics.
Age
Height
People in the house
Male
Raised in Canada
Non white
Non main decison maker in house
Non married
Non full time student
Graduate student
Major in Economics or Business
Mean
22.95
1.73
2.58
0.67
0.62
0.63
0.47
0.92
0.07
0.23
0.25
Std. Dev
3.96
0.10
1.02
Obs.
146
140
141
147
147
148
148
148
148
148
139
Table 5: Average Numbers of Safe Choices with Real Stakes: Effect of Skew.
Number of subjects
124 subjects
47 subjects
124 subjects
47 subjects
124 subjects
47 subjects
Treatment
Zero skew
Zero skew
Moderate Skew
Moderate Skew
Maximum Skew
Maximum Skew
30
Low Stakes
High Stakes
(20×)
3.86
4.62
3.59
4.11
3.25
3.79
Table 6: Regression results: Effect of Skew and Stakes.
Constant
Medium Skew
High Skew
High Stakes
OLS
(i)
3.904
(0.128)
-0.339b
(0.171)
-0.673a
(0.171)
0.603a
(0.156)
Age
Male
Raised in Canada
Major in Economics or Business
People in house
Non main decison maker in the house
Non white
Non married
Non full time student
Graduate student
R2 /R¯2
Subjects/observations
0.03/0.03
148/513
OLS
(ii)
3.633
(1.032)
-0.316b
(0.156)
-0.677a
(0.156)
0.458a
(0.153)
-0.005
(0.033)
-0.488a
(0.156)
-0.168
(0.167)
-0.072
(0.176)
0.065
(0.064)
0.651
(0.229)
-0.559a
(0.177)
0.396
(0.338)
-0.530
(0.364)
0.190
(0.263)
0.06/0.05
135/474
Random
Effects
(iii)
3.765
(1.452)
-0.316a
(0.122)
-0.677a
(0.122)
0.516a
(0.153)
-0.013
(0.047)
-0.405
(0.215)
-0.227
(0.228)
-0.021
(0.241)
0.055
(0.091)
0.570
(0.322)
-0.588b
(0.252)
0.478
(0.479)
-0.463
(0.498)
0.152
(0.370)
0.27/0.22
135/474
Fixed
Effects
(iv)
-0.339a
(0.111)
-0.673a
(0.111)
0.536a
(0.175)
0.71/0.59
148/513
Note: Standard errors in parenthesis. Superscripts a and b indicate significance at the
1% and 5% level, respectively. Estimation in columns (ii) and (iii) also includes controls
for income level, source of income and responsible for tuition.
31
Table 7: Estimation of Utility Functions
CRRA
Parameter
α
β
0.755
0.015
EUT with
HARA
ExpoPower
45.11∗
18.79
0.738
0.019
φ
CRRA
0.755
0.015
RDU with
HARA
ExpoPower
60.70
21.09
0.732
0.019
0.023
0.000
0.543
0.000
θ
η
LL
-2249.6
-2219.9
-2243.0
0.928
0.036
0.815
0.038
0.024
0.000
0.502
0.000
0.974
0.051
-2246.7
-2199.1
-2294.9
Note: Clustered standard errors in parentheses. * The coefficient indicated with
an asterisk (*) has a p-value of 0.016. All other p-values are less than 0.005.
Number of observations = 5130. Number of subjects = 148.
Table 8: Estimation of RDU with HARA by Groups
α
β
η
LL
Subj.
Obs.
Estimated skew parameter in the interval
[−2, −0.5] ] − 0.5, 0[
[0] ]0, 0.5[ [0.5, 2]
(i)
(ii)
(iii)
(iv)
(v)
26.55
86.77
50.12
(27.35)
(50.22) (44.07)
0.719
0.723
0.756
0.792
0.722
(0.034)
(0.033) (0.046) (0.041) (0.024)
0.570
0.780
0.922
3.541
3.686
(0.050)
(0.048) (0.109) (2.083) (0.971)
-535.1
-584.3
-632.8
-81.2
-284.3
45
38
39
3
23
1470
1440
1320
180
720
Note: Clustered standard errors in parentheses.
Number of subjects = 148. Number of observations = 5130.
32
Table 9: Optimal number of lottery tickets bought by a RDU agent as function of probability weighting and the utility function (initial wealth 10)
√
1−exp(−.024 x)
.27
.75
.50
.25
η\u(x)
x
x
x
x
(60.7 + x)
.024
1.00
0
0
0
0
0
0
.975
3
0
0
0
0
0
.950
5
0
0
0
0
0
.925
5
0
0
0
0
0
.900
5
0
0
0
0
0
.875
5
1
0
0
0
0
.850
5
3
0
0
0
0
.825
5
4
0
0
0
0
.800
5
5
0
0
0
0
.775
5
5
1
0
0
0
.750
5
5
1
0
1
0
.725
5
5
1
0
1
0
.700
5
5
2
0
1
0
.675
5
5
2
0
1
0
.650
5
5
3
1
2
0
.625
5
5
3
1
3
0
.600
5
5
4
1
3
1
33
Table 10: Optimal number of lottery tickets bought by a RDU agent as
function of probability weighting and the utility function (initial wealth 100)
√
1−exp(−.024 x)
.27
.75
.50
.25
η\u(x)
x
x
x
x
(60.7 + x)
.024
1.00
0
0
0
0
0
0
.975
3
0
0
0
0
0
.950
50
0
0
0
0
0
.925
50
0
0
0
0
0
.900
50
1
0
0
0
0
.875
50
6
0
0
0
0
.850
50
16
0
0
0
0
.825
50
29
1
0
0
0
.800
50
39
2
0
0
0
.775
50
45
3
0
0
0
.750
50
48
6
1
1
0
.725
50
49
9
1
1
0
.700
50
49
12
1
2
0
.675
50
50
16
2
3
1
.650
50
50
20
2
4
1
.625
50
50
23
3
6
1
.600
50
50
26
4
7
1
34
Table 11: Optimal number of lottery tickets bought by a RDU agent as function of probability weighting and the utility function (initial wealth 1,000)
√
1−exp(−.024 x)
.27
.75
0.5
.25
η\u(x)
x
x
x
x
(60.7 + x)
.024
1.00
0
0
0
0
0
0
.975
3
0
0
0
0
0
.950
220
0
0
0
0
0
.925
500
1
0
0
0
0
.900
500
10
0
0
0
0
.875
500
36
0
0
0
0
.850
500
83
3
0
0
0
.825
500
147
8
1
1
0
.800
500
219
16
2
2
1
.775
500
287
28
3
4
1
.750
500
345
44
6
7
1
.725
500
391
63
9
11
2
.700
500
425
84
14
16
3
.675
500
448
105
19
22
5
.650
500
464
126
25
29
7
.625
500
475
148
31
37
9
.600
500
482
167
38
44
11
35
Table 12: Optimal number of lottery tickets bought by a PT agent as function
of probability weighting and value function
η\v(x)
x
x.90
x.75
x.65
x.50
x.25
1.00
0
0
0
0
0
0
.975
0
0
0
0
0
0
.950
2
0
0
0
0
0
.925
36
0
0
0
0
0
.900
157
3
0
0
0
0
.875
399
26
0
0
0
0
.850
755
93
0
0
0
0
.825
1,205
224
1
0
0
0
.800
1,717
420
7
0
0
0
.775
2,264
675
26
1
0
0
.750
2,820
974
62
2
0
0
.725
3,366 1,301
120
7
0
0
.700
3,886 1,642
199
17
0
0
.675
4,370 1,984
301
35
0
0
.650
4,800 2,313
414
62
1
0
.625
5,190 2,622
539
97
1
0
.600
5,511 2,901
667
140
2
0
Table 13: The decision to become an inventor by a RDU agent as function
of the utility function and probability weighting (initial wealth 1.5 million)
u(x)\η
Invent
Not Invent
x
(0,.9997)
[.9997,1]
x.75
(0,.9744)
[.9744,1]
.50
x
(0,.9374)
[.9374,1]
.25
x
(0,.8574)
[.8574,1]
.27
(60.7 + x)
(0,.8682)
[.8682,1]
√
1−exp(−.024 x)
[0,1]
.024
36
Table 14: The decision to become an inventor by a PT agent as function of
the value function and probability weighting
v(x)\η
Invent
Not Invent
x
(0,.9051)
[.9051,1]
.90
x
(0,.8441)
[.8441,1]
x.75
(0,.7264)
[.7264,1]
.65
x
(0,.6164)
[.6164,1]
.50
x
[0,1]
x.25
[0,1]
37