Representative Agents in Lottery Choice Experiments:
One Wedding and A Decent Funeral
by
Glenn W. Harrison and E. Elisabet Rutström*
August 2005
Abstract. A methodology is proposed that allows economic behavior to be
characterized as heterogeneous in terms of the specific parameters of a given model
as well as the processes of decision-making proposed by each model. A finite
mixture model is proposed as a statistical device for estimating the parameters of the
model at the same time as one estimates the probability that each model applies to
the sample. A case study is presented in which the parameters of expected utility
theory and prospect theory are jointly estimated. The methodology provides the
wedding invitation, and the data consummates the ceremony with a decent funeral
for the representative agent model that assumes only one type of decision process
and homogenous preferences. The evidence suggests support for each theory, and
goes further to identify under what domains one can expect to see one theory
perform better than the other. The methodology is likely to be broadly applicable to
a range of debates over competing theories generated by experimental data.
*
Department of Economics, College of Business Administration, University of Central Florida,
USA. E-mail contacts: [email protected] and [email protected].
Rutström thanks the U.S. National Science Foundation for research support under grants NSF/IIS
9817518, NSF/MRI 9871019 and NSF/POWRE 9973669. We thank David Millsom and Bob
Potter for research assistance. Supporting data and instructions are stored at the ExLab Digital
Library at http://exlab.bus.ucf.edu.
One of the enduring contributions of behavioural economics is that we now have a rich set
of competing models of behaviour in many settings. Debates over the validity of these models have
often been framed as a horse race, with the winning theory being declared on the basis of some
statistical test. The problem with this approach is that it glosses the heterogeneity of the population,
by implicitly assuming that one theory fits every individual. In other words, we seem to pick the best
theory by “majority rule.” If one theory explains more of the data than another theory, we declare it
the better theory and discard the other one.
This approach is acceptable only if there is no reason to believe that decisions can differ
within the sample considered. This may be appropriate for some settings, in which we only want one
theory that best accounts for the data. But it is inappropriate when the objective is to understand the
appropriate domain of applicability of each theory. If we want to identify which people behave
according to what theory, and when, then we need to allow the data to reveal that information.1
One way that heterogeneity can be recognized is by collecting information on observable
characteristics, and allowing for unobserved individual characteristics when appropriate with panel
data. For example, a given theory might allow some individuals to be more risk averse than others as
a matter of personal preference. When we measure risk attitudes we can condition on individual
characteristics and evaluate if there is heterogeneity. But this approach only recognizes heterogeneity
within a given theory. This may be important to correct invalid inferences that theory cannot explain
data,2 but it does not allow for heterogeneous theories to co-exist in the same sample.
1
There is also a purely statistical reason to want to recognize heterogeneity in behavior. If we have
reason to believe that there are two or more population processes generating the observed sample, one can
make more appropriate inference if the data are not forced to fit a specification that assumes one population
process.
2
For example, Holt and Laury [2002] infer that one must add a “stochastic error” story to explain
observed responses in an experiment eliciting risk attitudes, when this inference is due to the extreme
assumption that every individual has the same risk attitude. As it happens, the data does support some such
stochastic error story even when one conditions on individual characteristics, but not as a logical necessity.
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One way to allow explicitly for alternative theories is to simply collect sufficient data at the
individual level, and test the different theories for that individual. One classic example of this
approach is the experimental study of expected utility theory by Hey and Orme [1994]. In one
experimental treatment they collected 100 binary choice responses from each of 80 subjects, and
were able to test alternative estimated models for each individual.3 Thus, at the beginning of their
discussion of results (p.1300), they state simply, “We assume that all subjects are different. We
therefore fit each of the 11 preference functionals discussed above to the subject’s stated preferences
for each of the 80 subjects indvidually.” Their approach would not be so remarkable if it were not for
the prevailing popularity of the “representative agent” assumption in previous tests of expected
utility theory.
We consider the more familiar case in which one cannot collect considerable data at the level
of an individual. How is one to account for heterogeneous population processes in such settings?
One solution is to use statistical mixture models, in which the “grand likelihood” of the data is
specified as a probability-weighted average of the likelihood of the data from each of K>1 models.4
In other words, with two modeling alternatives the overall likelihood of a given observation is the
probability of model A being true times the likelihood conditional on model A being true, plus the
3
They also collected an additional 100 observations from the same individuals in a later session, so
one could view these data as consisting of 200 responses per subject.
4
Mixture models have an astonishing pedigree in statistics: Pearson [1894] examined data on the ratio
of forehead to body length of 1000 crabs to illustrate “the dissection of abnormal frequency curves into
normal curves.” In modern parlance he was allowing the observed data to be generated by two distinct
Gaussian processes, and estimated the two means and two standard deviations. Modern surveys of the
evolution of mixture models are provided by Titterington, Smith and Makov [1985], Everitt [1996] and
McLachlan and Peel [2000]. Mixture models are also virtually identical to “latent class models” used in many
areas of statistics, marketing and econometrics, even though the applications often make them seem quite
different (e.g., Goodman [1974a][1974b] and Vermunt and Magidson [2003]). In experimental economics, ElGamal and Grether [1995; §6] estimates a finite mixture model of Bayesian updating behavior, and contrast it
to a related approach in which individual subject behavior is classified completely as one type of the other.
Related applications of mixture models in experiments include Bardsley and Moffatt [1995] and Harrison, List
and Tra [2005].
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probability of model B being true times the likelihood conditional on model B being true.
One challenge with mixture models of this kind is the joint estimation of the probabilities
and the parameters of the conditional likelihood functions. If these are conditional models that each
have some chance of explaining the data, then mixture models will be characterized numerically by
relatively flat likelihood functions. On the other hand, if there are indeed K distinct latent processes
generating the overall sample, allowing for this richer structure is inferentially useful. Most of the
applications of mixture modeling in economics have been concerned with their use in better
characterizing unobserved individual heterogeneity in the context of a given theory about behavior,5
although there have been some applications in environmental economics that consider individual
heterogeneity over alternative behavioral theories.6
One obvious application of this approach is to offer a statistical reconciliation of the debate
over alternative theories of choice under risk. A canonical setting for these theories has been the
choice over pairs of monetary lotteries, as reviewed by Camerer [1995], for example. We collect data
in a setting in which subjects make decisions in a gain frame, a loss frame, and a mixed frame. Such
settings allow one to test the competing models on the richest domain. All data is collected using
standard procedures in experimental economics in the laboratory: no deception, no field referents,
fully salient choices, and with information on individual characteristics of the subjects. In fact, these
experiments represent a replication and modest extension of the procedures of Hey and Orme
5
For example, see Heckman and Singer [1984], Geweke and Keane [1999] and Araña and León
[2005]. The idea here is that the disturbance term of a given specification is treated as a mixture of processes.
Such specifications have been used in many settings that may be familiar to economists under other names.
For example, stochastic frontier models rely on (unweighted) mixture specifications of a symmetric “technical
efficiency” disturbance term and an asymmetric “idiosyncratic” disturbance term; see Kumbhakar and Lovell
[2000] for an extensive review. Wang and Fischbeck [2004] provide an application to framing effects within
prospect theory, using hypothetical field survey data. Their approach is to view the frames as different data
generation processes for responses.
6
For example, see Werner [1999], who uses mixture models to characterize the “spike at zero” and
“non-spike” responses common in contingent valuation surveys.
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[1994].
The major extension is to consider lotteries in which some or all outcomes are framed as
losses, as well as the usual case in which all outcomes are frames as gains. The subject received an
initial endowment that resulted in their final outcomes being the same as those in the other frames.
Thus the final income level of the subjects is essentially equalized across all frames. There may be
some debate over whether this endowment serves as a “reference point” to define losses, but that is
one thing that is being tested with the different models.7 The experimental procedures are described
in section 1.
To keep things sharply focused, two alternative models are posited. One is a simple expected
utility theory (EUT) specification, assuming a CRRA utility function defined over the “final
monetary prize” that the subject would receive if the lottery were played out. That is, the argument
of the utility function is the prize plus the initial endowment, which are jointly always non-negative.
The other model is a popular specification of prospect theory (PT) due to Kahneman and Tversky
[1979], in which the utility function is defined over gains and losses separately, and there is a
probability weighting function that converts the underlying probabilities of the lottery into subjective
probabilities.8 The three critical features of the PT model are (i) that the arguments of the utility
function be gains and losses relative to some reference point, taken here to be the endowment; (ii)
that losses loom larger than gains in the value function; and (iii) that there be a nonlinearity in the
transformed probabilities that could account for apparently different risk attitudes for different
7
Some might argue that these would count as losses only when the subject “earns” the initial stake.
This is a fair point, but testable by simple modification of the design. There is evidence from related settings
that such changes in how subjects receive their initial endowment can significantly change behaviour: see
Rutström and Williams [2000] and Cherry, Frykblom and Shogren [2002].
8
We use the language of EUT, but prospect theorists would instead refer to the utility function as a
“value function,” and to the transformed probabilities as “decision weights.”
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lottery probabilities. We specify the exact forms of the models tested in section 2.
It is apparent that there are many variants of these two models, and many others that deserve
to be considered. Wonderful expositions of the major alternatives can be found in Camerer [1995]
and Hey and Orme [1994], among others. But this is not the place to array every feasible alternative.
Instead, the initial exploration of this approach should consider two major alternatives and the
manner in which they are characterized. Many of the variants involve nuances that would be hard to
empirically evaluate with the data available here, rich as it is for the task at hand.9 But the
methodological point illustrated here is completely general.
The “wedding” arises from the specification and estimation of a grand likelihood function
that allows each model to have different weights. The data can then identify what support each
model has. The most intuitive interpretation of these estimated probabilities is that the sample
consists of two types of decision-makers: those that make choices that are relatively well
characterized by EUT, and those that make choices that are relatively well characterized by PT. The
wedding is consummated by the maximum likelihood estimates converging on probabilities that
apportion non-trivial weight to both models. Thus the data, when allowed, indicates that one should
not think exclusively of EUT or PT as the correct model, but as models that are correct for distinct
parts of the sample.
The “funeral” arises from the ability to identify which individual characteristics are
associated with subjects being better characterized as EUT decision-makers rather than as PT
9
For example, on the EUT side one would prefer to consider specifications that did not assume
constant relative risk aversion after the empirical “wake up call” provided by Holt and Laury [2002]. But the
reliable estimation of the parameters of such specifications likely requires a much wider range of prizes than
considered here: hence the design of Holt and Laury [2002], where prizes were scaled by a factor of 20 for
most of their sample in order to estimate such a flexible specification. For an example on the PT side, there
are a plethora of specifications available for the probability weighting function, as noted by Wu and Gonzalez
[1996] and Prelec [2002], among others.
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decision-makers. Moreover, within each type, we can identify those individual characteristics
associated with EUT decision-makers that are more or less risk averse, for example. Or those PT
decision-makers that are more or less averse to losses. The “decent funeral” arises from the ability to
identify empirically domains under which each theory has some validity, once one recognizes the
dangers of statistical inference using representative agent models with homogenous preferences.10
Kirman [1992; p.119] considers the dangers of using representative agent characterizations in
macroeconomics, and argues that “... it is clear that the ‘representative’ agent deserves a decent
burial, as an approach to economics analysis that is not only primitive, but fundamentally
erroneous.”
The primary methodological contribution is to illustrate how one can move from results that
do not condition on individual characteristics, and assume one or other model as the correct
specification, to a richer characterization that allows for heterogeneity. And heterogeneity in the
decision-making process as well as the specific parameters within a given process.
Although this case study is limited to lottery choices over money, laboratory experiments,
and only two competing models, the methodology has broader implications. For example, when List
[2004] identifies domains in the field where PT appears to perform poorly, despite it apparently
performing well in the laboratory, this could be due to the characteristics that those subjects in the
field have that differentiate them from the subjects typically found in the field. It could be many
other factors, of course, as emphasized by Harrison and List [2004]. But we will see that the
approach offers insights that extend beyond the immediate application.
10
Camerer and Ho [1994] remains the most impressive work of this genre, and contains a very clear
statement of the strengths and weaknesses of the approach (p.186). Kirman [1992] reviews arguments against
the representative agent approach from a broader perspective.
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Section 1 reviews the experimental data,11 section 2 the competing models, section 3 presents
the results of estimating a mixture model for both models, and section 4 draws conclusions.
1. Experimental Design
Subjects were presented with 60 lottery pairs, each represented as a “pie” showing the
probability of each prize. The subject could choose the lottery on the left or the right, or explicitly
express indifference (in which case the experimenter would flip a coin on the subject’s behalf). After
all 60 lottery pairs were evaluated, three were selected at random for payment. The lotteries were
presented to the subjects in color on a private computer screen,12 and all choices recorded by the
computer program. This program also recorded the time taken to make each choice. An appendix
presents the instructions given to our subjects, as well as an example of the “screen shots” they saw.
In the “gain frame” experiments the prizes in each lottery were $0, $5, $10 and $15, and the
probabilities of each prize varied from choice to choice, and lottery to lottery. In the “loss frame”
experiments subjects were given an initial endowment of $15, and the corresponding prizes from the
gain frame lotteries were transformed to be -$15, -$10, -$5 and $0. Hence the final outcomes,
inclusive of the endowment, were the same in the gain frame and loss frame. In the “mixed frame”
experiments subjects were given an initial endowment of $8, and the prizes were transformed to be $8, -$3, $3 and $8, generating final outcomes inclusive of the endowment of $0, $5, $11 and $16.13
11
We are grateful to Nathaniel Wilcox for permission to use data we jointly collected.
The Behavioural Research Laboratory at the College of Business Administration of the University
of Central Florida has 24 subject stations. Each screen is “sunken” into the desk, and subjects were typically
separated by several empty stations due to the staggered recruitment procedures. No subject could see what
the other subjects were doing, let alone mimic what they were doing since each subject was started
individually at different times.
13
These final outcomes differ by $1 from the two highest outcomes for the gain frame and mixed
frame, because we did not want to offer prizes in fractions of dollars.
12
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In addition to the fixed endowment, each subject received a random endowment between $1
and $10. This endowment was generated using a uniform distribution defined over whole dollar
amounts, operationalized by a 10-sided die.
The probabilities used in each lottery ranged roughly evenly over the unit interval. Values of
0, 0.13, 0.25, 0.37, 0.5, 0.62, 0.75 and 0.87 were used.14
Subjects were recruited at the University of Central Florida, primarily from the College of
Business Administration. Each subject received a $5 fee for showing up to the experiments, and
completed an Informed Consent form. Subjects were deliberately recruited for “staggered” starting
times, so that the subject would not pace their responses by any other subject. Each subject was
presented with the instructions individually, and taken through the practice sessions at an individual
pace. Since the rolls of die were important to the implementation of the objects of choice, the
experimenters took some time to give each subject “hands-on” experience with the (10-sided, 20sided and 100-sided) die being used. Subjects were free to make their choices as quickly or as slowly
as they wanted.
The presentation of a given lottery on the left or the right was determined at random, so that
the “left” or “right” lotteries did not systematically reflect greater risk or greater prize range than the
other.
Our data consists of responses from 158 subjects making 9311 choices that do not involve
indifference. Only 1.7% of the choices involved explicit choice of indifference, and to simplify we
drop those.15 Of these 158 subjects, 63 participated in gain frame tasks, 37 participated in mixed
14
To ensure that probabilities summed to one, we also used probabilities of 0.26 instead of 0.25, 0.38
instead of 0.37, 0.49 instead of 0.50 or 0.74 instead of 0.75.
15
For the specification of likelihoods of strictly binary responses, such observations add no
information. However, one could augment the likelihood for the strict binary responses with a likelihood
defined over “fractional responses” and assume that the fraction for these indifferent responses was exactly
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frame tasks, and 58 participated in loss frame tasks.
2. Meet the Bride and Groom
A. Expected Utility Specification
We assume that utility of income is defined by U(s, x) = (s+x)r where s is the fixed
endowment provided at the beginning of the experiment, x is the lottery prize, and r is a parameter
to be estimated. With this specification we assume “perfect asset integration” between the
endowment and lottery prize.16 Probabilities for each outcome k, p(k), are those that are induced by
the experimenter, so expected utility is simply the probability weighted utility of each outcome in
each lottery. Since there were up to 4 implicit outcomes in each lottery i, EUi = 3k [ p(k) × U(k) ]
for k = 1, 2, 3, 4.
A simple stochastic specification was used to specify likelihoods conditional on the model.
The EU for each lottery pair was calculated for a candidate estimate of r, and the difference LEU =
EUR - EUL calculated, where EUL is the left lottery in the display and EUR is the right lottery. A
deterministic choice EUT could then be specified by assuming some cumulative probability
distribution function, G(@), such as the logistic. Instead we incorporate a simple “stochastic error”
specification and allow a Fechner-type noise process to cause some difference between the
preference that the subject acts on. Specifically, we assume LEU = EUR - EUL + :, and then use
this index to define the cumulative probability of the observed choice using the logistic: G(LEU) =
7(LEU) = exp(LEU)/[1+exp(LEU)]. Thus, as : 6 0, this specification collapses to the
½. Such specifications are provided by Papke and Wooldridge [1996], but add needless complexity for present
purposes given the small number of responses involved.
16
A valuable extension, inspired by Cox and Sadiraj [2005], would be to allow s and x to be
combined in some linear manner, with weights to be estimated.
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deterministic choice EUT model, but as : gets larger and larger the choice essentially becomes
random. This is one of several different types of error story that could be used.17
Thus the likelihood, conditional on the EUT model being true, depends on the estimates of
r and : given the above specification and the observed choices. The conditional log-likelihood is
ln LEUT(r, :; y,X) = 3i [ (ln 7(LEU) * yi=1) + (ln 7(LEU) * yi=0) ]
where yi =1(0) denotes the choice of the right (left) lottery in task i, and X is a vector of individual
characteristics.
B. Prospect Theory Specification
Tversky and Kahneman [1992] propose a popular specification, which is employed here.
There are two components, the utility function and the probability weighting function.
Tversky and Kahneman [1992] assume a power utility function defined separately over gains
and losses: U(x) = x" if x $ 0, and U(x) = -8(-x)$ for x < 0. So " and $ are the risk aversion
parameters, and 8 is the coefficient of loss aversion.
There are two variants of prospect theory, depending on the manner in which the probability
weighting function is combined with utilities. The original version proposed by Kahneman and
Tversky [1979] posits some weighting function which is separable in outcomes, and has been
usefully termed Separable Prospect Theory (SPT) by Camerer and Ho [1994; p. 185]. The alternative
version, proposed by Tversky and Kahneman [1992], posits a weighting function defined over the
17
See Harless and Camerer [1994], Hey and Orme [1994] and Loomes and Sugden [1995] for the first
wave of empirical studies including some formal stochastic specification in the version of EUT tested. There
are several species of “errors” in use, reviewed by Hey [1995][2002], Loomes and Sugden [1995], Ballinger
and Wilcox [1997], and Loomes, Moffatt and Sugden [2002]. Some place the error at the final choice between
one lottery or the other after the subject has decided deterministically which one has the higher expected
utility; some place the error earlier, on the comparison of preferences leading to the choice; and some place
the error even earlier, on the determination of the expected utility of each lottery.
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cumulative probability distributions. In either case, the weighting function proposed by Tversky and
Kahneman [1992] has been widely used: it assumes weights w(p) = p(/[ p( + (1-p)( ]1/(.
Assuming that SPT is the true model, prospective utility PU is defined in much the same
manner as when EUT is assumed to be the true model. The PT utility function is used instead of the
EUT utility function, and w(p) is used instead of p, but the steps are otherwise identical. The same
error process is assumed to apply when the subject forms the preference for one lottery over the
other. Thus the difference in prospective utilities is defined similarly as LPU = PUR - PUL + :.
Thus the likelihood, conditional on the SPT model being true, depends on the estimates of
", $, 8 and : given the above specification and the observed choices.18 The conditional loglikelihood is
ln LPT(", $, 8, :; y,X) = 3i l iPT = 3i [ (ln 7(LPU) * yi=1) + (ln 7(LPU) * yi=0) ].
C. The Nuptial
If we let BEUT denote the probability that the EUT model is correct, and BPT = (1-BEUT)
denote the probability that the PT model is correct, the grand likelihood can be written as the
probability weighted average of the conditional likelihoods. Thus the likelihood for the overall
model estimated is defined by
ln L(r, ", $, 8, :; y,X) = 3i ln [ (BEUT × l iEUT ) + (BPT × l iPT ) ].
This log-likelihood can be maximized to find estimates of the parameters.19
18
The fact that the SPT specification has three more parameters than the EUT specification is
meaningless. One should not pick models on some arbitrary parameter-counting basis. We refer to SPT or PT
interchangeably.
19
This approach assumes that any one observation can be generated by both models, although it
admits of extremes in which one or other model wholly generates the observation. One could alternatively
define a grand likelihood in which observations or subjects are classified as following one model or the other
on the basis of the latent probabilities BEUT and BPT. El-Gamal and Grether [1995] illustrate this approach in
the context of identifying behavioral strategies in Bayesian updating experiments.
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We allow each parameter to be a linear function of the observed individual characteristics of
the subject. This is the X vector referred to above. We consider six characteristics: binary variables
to identify Females, subjects that self-reported their ethnicity as Black, those that reported being
Hispanic, those that had a Business major, and those that reported a low cumulative GPA (below
3¼). We also included Age in years. The estimates of each parameter in the above likelihood function
actually entails estimation of the coefficients of a linear function of these characteristics. So the
estimate of r, ^r , would actually be
^r = ^r 0 + (r^FEMALE × FEMALE) + (r^BLACK × BLACK) + (r^HISPANIC × HISPANIC) +
(r^BUSINESS × BUSINESS) + (r^GPAlow × GPAlow) + (r^AGE × AGE)
where ^r 0 is the estimate of the constant. If we collapse this specification by dropping all individual
characteristics, we would simply be estimating the constant terms for each of r, ", $, 8, :. Obviously
the X vector could include treatment effects as well as individual effects, or interaction effects.
The estimates allow for the possibility of correlation between responses by the same subject,
so the standard errors on estimates are corrected for the possibility that the 60 responses are
clustered for the same subject. The use of clustering to allow for “panel effects” from unobserved
individual effects is common in the statistical survey literature.20
20
Clustering commonly arises in national field surveys from the fact that physically proximate
households are often sampled to save time and money, but it can also arise from more homely sampling
procedures. For example, Williams [2000; p.645] notes that it could arise from dental studies that “collect data
on each tooth surface for each of several teeth from a set of patients” or “repeated measurements or recurrent
events observed on the same person.” The procedures for allowing for clustering allow heteroskedasticity
between and within clusters, as well as autocorrelation within clusters. They are closely related to the
“generalized estimating equations” approach to panel estimation in epidemiology (see Liang and Zeger
[1986]), and generalize the “robust standard errors” approach popular in econometrics (see Rogers [1993]).
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3. Results
A. Heterogeneity of Process
Table 1 presents the maximum likelihood estimates of the conditional models and the
mixture model when we assume no individual covariates. The estimates for the conditonal models
therefore represent the traditional representative agent assumption, that every subject has the same
preferences and behaves in accord with one theory or the other. The estimates for the mixture model
allow each theory to have positive probability, and therefore makes one major step towards
recognizing heterogeneity of preferences.
The first major result is that the estimates for the probability of the EUT and PT specifications indicate
that each is equally likely. Each estimated probability is significantly different from zero.21 EUT wins by
a (quantum) nose, with an estimated probability of 0.549 rather than 0.450, but the whole point of
this approach is to avoid such rash declarations of victory. The data suggest that the sample is
roughly evenly split between those that are best characterized by EUT and those that are best
characterized by PT. In fact, one cannot reject the formal hypothesis that the probabilities of each
theory are identical and equal to 0.5 (p-value = 0.493).
The second major result is that the estimates for the PT specification are very weakly
consistent with the a priori predictions of that theory when the specification is assumed to fit every
subject, but strikingly consistent with the predictions of the theory when it is only assumed to fit some of the subjects.
When the conditional PT model is assumed for all subjects, the loss aversion parameter is greater
21
The likelihood function actually estimates the log odds in favor of one model or the other. If we
denote the log odds as 6, one can recover the probability for the EUT model as BEUT = 1/(1+exp(6)). This
non-linear function of 6 is calculated using the “delta method,” which also provides estimates of the standard
errors and p-values. Since BEUT = ½ when 6=0, the standard p-value on the estimate of 6 provides the
estimate for the null hypothesis H0: BEUT=BEUT listed in Table 1.
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than 1, but only equals 1.381. This is significantly different from 1 at the 5% significance level, since
the p-value is 0.088 and a one-sided test is appropriate here given the priors from PT that 8>1. But
the size of the effect is much smaller than the 2.25 estimated by Tversky and Kahneman [1992] and
almost universally employed.22 However, when the same PT specification is estimated in the mixture
model, where it is only assumed to account for the behavior of some of the subjects, the estimated
value for 8 jumps to 5.78 and is even more clearly greater than 1 (although the estimated standard
error also increases, from 0.223 to 1.609). Similarly, the ( parameter is estimated to be 0.942 with
the conditional PT model, and is not statistically different from 1 (p-value = 0.266), which is the
special EUT case of the probability weighting function where w(p)=p. But when the mixture model
is estimated, the value of ( drops to 0.681 and is significantly smaller than 1. Finally, the risk
aversion coefficients " and $ are not significantly different from each other under the conditional
model, but are different from each other under the mixture model. This difference is not critical to
PT, and is often assumed away in applied work with PT, but it is worth noting since it points to
another sense in which gains and losses are viewed differently by subjects, which is a primary tenet
of PT.23
The third major result is that stochastic errors appear to play virtually no role. This is true for both
22
For example, Benartzi and Thaler [1995; p.79] assume this value in their evaluation of the “equity
premium puzzle,” and note (p.83) that it is the assumed value of the loss aversion parameter that drives their
main result.
23
Köbberling and Wakker [2005; p. 128] argue against using this Constant Relative Risk Aversion
(CRRA) specification of the utility function, or at least constraining " = $, so that a general index of loss
aversion that they propose is well-defined. They suggest using a Constant Absolute Risk Aversion utility
function instead. The popularity of the CRRA specification justifies studying it, and it should be an easy
matter to replicate the analysis using alternative specifications. However, their concerns with the CRRA
specification derive from their assumption (p.121) that the utility function should have well-defined
directional derivatives at a zero income level, which is a sufficient condition for their index to be scaleinvariant. It is not obvious why this assumption is as “plausible” as they claim.
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conditional models, and the mixture model.24 In all cases one cannot reject the hypothesis that :=0.
It might have been expected that as one moved from the conditional models to the mixture model
that the contribution of the stochastic error would diminish, due to it having to explain
“misclassification” of EUT (PT) subjects as PT (EUT) subjects. Although there is a reduction in the
estimated size of the coefficient from 0.010 to 0.003, and the p-value for the null that :=0 increases
from 0.6 to 0.9, stochastic errors seem to simply play little role here. Of course, this does not
consider the possibility that different types of stochastic errors might be appropriate under the
different specifications, a point stressed by Loomes, Moffatt and Sugden [2002].
B. Heterogeneity of Process and Parameters
Table 2 presents estimates from the mixture model with all individual characteristics
included.25 In general we see many individual coefficients that are statistically significant, and an
overall F-test that all coefficients are jointly zero can be easily rejected with a p-value of 0.0003.
Apart from indicating marginal statistical significance, the detailed coefficient estimates of Table 2
are not particularly informative. It is more instructive to examine the distribution of coefficients
across the sample, generated by predicting each parameter for each subject by evaluating the linear
function with the characteristics of that subject.26 Such predictions are accompanied by estimated
standard errors, just as the coefficient estimates in Table 1 are, so we also take the uncertainty of the
predicted parameter value into account.
24
The estimates for : in the EUT and PT conditional models are virtually identical, but this is what
the estimates happen to be. The values listed for : in Table 1 for the mixture model estimates are identical,
since the same parameter is assumed to characterize stochastic errors for EUT and PT.
25
The likelihood function would not converge reliably with any covariates for the $ parameter, so it
is represented as a scalar.
26
For example, a black 21-year-old female that was not hispanic, did not have a business major, and
had an average GPA, would have an estimate of r equal to ^r 0 + ^r FEMALE + ^r BLACK + ^r AGE × 21. She would
have the same estimate of r as anyone else that had exactly the same set of characteristics.
-15-
Figure 1 displays the main result, a distribution of predicted probabilities for the competing
models. The two panels are, by construction, mirror images of each other since we only have two
competing models, but there is pedagogic value in displaying both. The support for the EUT
specification is extremely high for some people, but once that support wanes the support for PT
picks up. It is not the case that there are two sharp modes in this distribution, which is what one
might have expected with priors that there are two types of subjects. Instead, subjects are better
characterized as either “clearly EUT” or “probably PT.” It is as if EUT is fine for city folk, but PT
rules the hinterland. No doubt this is due to many subjects picking lotteries according to their
estimate of the expected value of the lotteries, which would make them EUT-consistent subjects
that are risk neutral (r=1). The average probability for the EUT model is 0.59, but the median
probability is 0.69, reflecting the skewness in the distributions in Figure 1.27
Figure 2 indicates the uncertainty of these predicted probabilities. Consistent with the use of
discrimination functions such as the logistic, uncertainty is smallest at the end-points and greatest at
the mid-point. Since EUT has more of it’s support from subjects that are at one extreme probability,
this implies that one can be more confident about declaring a subject as better characterized by EUT
than one can about the alternative model.
Figures 3, 4, 5 and 6 display stratifications of the probability of the EUT model being correct
in terms of individual characteristics. The results are striking. Figure 3 shows that men have a much
stronger probability of behaving in accord with the EUT model than women. Figure 4 shows that
“whites” have a dramatically higher probability of being EUT decision makers than “blacks” or
“hispanics.” Consider the implications of these two Figures for the dramatic differences that List
27
The oft-cited parameter point estimates of Tversky and Kahneman [1992; p.59] were generated
using the median of 25 estimated coefficients. They estimated a model using for each subject based on
responses to 64 tasks, and then reported the median over the 25 estimates for each parameter.
-16-
[2002][2003[2004] finds between subjects in field experiments on the floor of sportscard shows and
conventional lab experiments with college students. The vast majority of participants on those shows
are white males. No doubt there is much more going on in these field experiments than simply
demographic mix, as stressed by Harrison and List [2004], but one might be able to go a long way to
explaining the differences just on these two demographics alone.28
Figure 5 shows that business majors exhibit behavior that is much more consistent with
EUT, helping us track “homo economicus” down among, thankfully, the Econ. A similar display,
not shown, demonstrates that there is no difference between those with a low GPA as compared to
others. Figure 6 shows some modest effects of age on support for EUT. Our sample obviously has a
limited range in terms of age, with these ages also confounded with “standing” within the
undergraduate program. But there does appear to be a marked reduction in support for EUT as
subjects age or get more university education. In fact, a simple OLS regression of the predicted
probability of EUT and age estimates the effect at -0.03 of a percentage point per year (p-value <
0.001). Although this estimate is purely a descriptive statistical estimate, it does match the behavior
shown in Figure 6.
Finally, Figures 7 and 8 display the distributions of the parameters of each model. The raw
predictions for all 158 subjects include some wild estimates, but for completely sensible reasons. If
some subject has a probability of behavior in accord with PT of less than 0.0001, then one should
not expect the estimates of ", $, 8 or ( to be precise for that subject. In fact, most of the extreme
28
Haigh and List [2005] report contrary evidence, using tasks involving risky lotteries and comparing
field traders from the Chicago Board of Trade and students. However, the behavior they observe can be easily
explained under EUT using a specification of the utility function that allows non-constant relative risk
aversion, such as the expo-power specification favored by Holt and Laury [2002] in their experiments. Thus
one should not count this as contrary evidence to the claim in the text unless one constrains EUT to the
special case of CRRA.
-17-
estimates are associated with parameters that have very low support for that subject, or that have
equally high standard errors. Thus, we restrict the displays in Figures 7 and 8 to those subjects
whose probability of behaving in accord with PT or EUT, respectively, is at least ¼. The distribution
for $ is generated randomly using the estimates in Table 2, since it is only estimated as a scalar.
Figure 7 displays results that are generally consistent with those shown in Table 1. The
parameters " and $ are generally less than 1, consistent with mild risk aversion in the usual sense.
The average estimate for " is 0.59, with a median of 0.50. The loss aversion coefficient 8 is generally
much greater than 1, consistent with those subjects behaving in accord with PT having significant
loss aversion. The average estimate for 8 is 6.3, with a median of 5.4. Of course, this applies only to
the fraction of the sample deemed to have certain level of support for PT; the value of 8 for EUT
subjects is of course 1, so the weighted average for the whole sample would be much smaller (as
Table 1 indicates, it is 1.381 when estimated over all subjects). The probability weighting coefficient
( is generally less than 1, which is the value at which it collapses back to the standard EUT
assumption that w(p)=p. The average estimate is 0.79, and the median is 0.78.
Figure 8 displays results for the EUT subjects that are also broadly consistent with Table 1.
The average EUT subject has a CRRA coefficient r equal to 0.85, with a median of 0.88. There are
two modes close to the risk-neutral case, r=1, although some subjects exhibit considerable risk
aversion (r<1) and some even exhibit mild risk-loving behavior (r>1).
Although the central tendencies for the distributions in Figures 7 and 8 are close to those for
the mixture model estimates in Table 1, this is what one would expect. It is, of course, possible to
stratify these distributions by demographic characteristic. We do so only for the risk aversion
coefficients, " and r, in Figure 8, since there has been considerable controversy over whether
-18-
women are more risk averse than men (see Eckel and Grossman [2005] for a review).29 If we just
focus on the exponent of the utility function, we see that women do tend to be more risk averse
than men for those that are better characterized by either model. However, one could easily envisage
a sample such that the males were better characterized by EUT (e.g., white males in a Business
school) whereas the females were better characterized by PT (e.g., black female English majors), and
one would obtain significant evidence of greater risk aversion in women. Or one could envisage a
sample such that the males were better characterized by PT (e.g., black male engineers) and the
females by EUT (e.g., white females in a Business school), such that there would be little difference
on average. In the latter case, however, one might expect to see a difference in variance, exhibited as
heteroskedasticity. These are not intended as predictions as to precisely when one would or would
not conclude that women are more risk averse than men, so much as an illustration of the
importance of characterizing the heterogeneity with respect to parameters and models.
4. Conclusion
Characterizing behavior in lottery choice tasks as generated by potentially heterogenous
processes generates several important insights. First, it provides a framework for systematically
resolving debates over competing models. This method does not rest on exploiting structural aspects
of one type of task or another, such as have been a staple in tests of EUT. Such extreme domains
are still interesting of course, just as the experiments employed here deliberately generated loss
frames and mixed frames. But the test of the competing models should not be restricted to such
trip-wire tests, which might not characterize behavior in a broader domain. Second, it provides a
29
Of course, “risk aversion” in PT is a function of all of the parameters, and not just the exponent of
the utility function. But the tests for differential risk aversion have usually been conducted from an EUT
perspective.
-19-
more balanced metric for deciding which theory does better in a given task, rather than extreme
declarations of winners and losers. If some theory has little support, this approach will still show
that. Third, the approach can provide some insight into when one theory does better than another,
through the effects of individual characteristics and treatments on estimated probabilities of support.
This insight might be fruitfully applied to other settings in which apparently conflicting findings
have been reported. Fourth, the approach readily generalizes to include more than two models,
although likelihoods are bound to become flatter as one mixes more and more models that have any
support from the data. The obvious antidote for those problems is to generate larger samples, to
pool data across comparable experiments, and to marshall more exhaustive numerical methods.
Finally, the approach readily generalizes to other contexts, such as competing models of bidding
behavior in auctions30 or competing specifications of social preferences.31 It might therefore provide
some insight into the varying performance of the standard model of the “selfish and individually
rational” agent in different domains, such as impersonal market exchange settings versus personal
social interactions.32
30
See Harrison, List and Tra [2005] for an application.
See Engelman and Strobel [2004] for a flavor of the debates, and some tests of competing models
using representative agent models assuming homogenous preferences. The original models being tested were
quite explicit about the heterogeneity of behavior that they were assuming: see Fehr and Schmidt [1999;
p.818], Bolton and Ockenfels [2000; Assumption 4, p.172] and Charness and Rabin [2002; p.818].
32
For example, Smith [2003] and List [2005].
31
-20-
Table 1: Estimates of Parameters of Models With No Individual Covariates
Estimates from Conditional Models
Parameter
or Test
Estimate Standard Error
Estimates from Mixture Model
p-value
Lower 95%
Confidence
Interval
Upper 95%
Confidence
Interval
Estimate
Standard Error
p-value
Lower 95%
Confidence
Interval
Upper 95%
Confidence
Interval
r
:
0.837
0.010
0.025
0.019
0.000
0.602
0.788
-0.027
0.886
0.047
0.846
0.003
0.044
0.024
0.000
0.895
0.759
-0.044
0.933
0.050
"
$
8
(
:
0.730
0.733
1.381
0.942
0.010
0.033
0.062
0.223
0.052
0.019
0.000
0.000
0.000
0.000
0.613
0.665
0.611
0.943
0.841
-0.028
0.796
0.854
1.820
1.044
0.048
0.614
0.312
5.780
0.681
0.003
0.057
0.132
1.609
0.047
0.024
0.000
0.019
0.000
0.000
0.895
0.501
0.052
2.601
0.587
-0.044
0.727
0.573
8.958
0.774
0.050
0.549
0.450
0.072
0.072
0.000
0.000
0.407
0.308
0.692
0.593
BEUT
BPT
H0: BEUT=BPT
H0: "=$
H0: 8=1
H0: (=1
0.493
0.973
0.088
0.266
0.046
0.003
0.000
-21-
Table 2: Estimates of Parameters of Mixture Model With Individual Covariates
Parameter
Variable
Estimate
Standard Error
p-value
r
Constant
Female
Black
Hispanic
Age
Business
GPAlow
Constant
Female
Black
Hispanic
Age
Business
GPAlow
Constant
Constant
Female
Black
Hispanic
Age
Business
GPAlow
Constant
Female
Black
Hispanic
Age
Business
GPAlow
Constant
Female
Black
Hispanic
Age
Business
GPAlow
Constant
Female
Black
Hispanic
Age
Business
GPAlow
0.017
-0.199
-0.111
-0.280
0.049
-0.051
0.031
-1.136
0.046
0.100
0.144
0.072
-0.077
0.139
0.584
74.288
-73.734
-4.087
-3.936
0.444
-4.885
-1.761
0.657
-0.106
-0.120
-0.188
0.011
0.079
-0.090
-0.288
-0.001
-0.001
-0.027
0.014
0.032
0.003
-18.238
5.656
3.578
0.881
0.589
1.754
2.401
0.442
0.095
0.134
0.213
0.023
0.082
0.072
0.693
0.358
0.311
0.267
0.018
0.250
0.157
0.073
90.417
89.372
2.146
1.565
0.170
2.142
1.666
0.151
0.049
0.084
0.066
0.003
0.072
0.079
0.189
0.059
0.067
0.084
0.010
0.055
0.053
6.142
1.769
1.013
1.096
0.226
0.989
0.867
0.970
0.038
0.409
0.190
0.038
0.540
0.663
0.103
0.898
0.747
0.591
0.000
0.757
0.377
0.000
0.413
0.411
0.059
0.013
0.010
0.024
0.292
0.000
0.031
0.153
0.005
0.000
0.273
0.259
0.130
0.990
0.990
0.746
0.174
0.564
0.952
0.003
0.002
0.001
0.423
0.010
0.078
0.006
"
$
8
(
:
6
Lower 95%
Confidence
Interval
-0.857
-0.386
-0.377
-0.699
0.003
-0.213
-0.110
-2.504
-0.661
-0.515
-0.384
0.037
-0.571
-0.171
0.440
-104.302
-250.260
-8.326
-7.027
0.108
-9.117
-5.052
0.358
-0.201
-0.286
-0.317
0.006
-0.063
-0.246
-0.661
-0.116
-0.132
-0.193
-0.006
-0.077
-0.101
-30.369
2.161
1.577
-1.284
0.142
-0.199
0.687
Note: 6 is the log odds of the probabilities of each model, and BEUT = 1/(1+exp(6)) .
-22-
Upper 95%
Confidence
Interval
0.890
-0.011
0.154
0.140
0.094
0.112
0.173
0.232
0.753
0.716
0.672
0.108
0.416
0.449
0.729
252.878
102.793
0.151
-0.844
0.779
-0.653
1.531
0.955
-0.010
0.045
-0.058
0.016
0.221
0.067
0.086
0.115
0.131
0.139
0.035
0.140
0.107
-6.108
9.151
5.579
3.045
1.035
3.707
4.114
Figure 1: Probability of Competing Models
EUT Model
40
40
30
Percent
Percent
30
PT Model
20
10
20
10
0
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9
Probability
1
0 .1 .2 .3 .4 .5 .6 .7 .8 .9
Probability
1
Figure 2: Uncertainty In Predicted Probability of Models
Predicted probabilities and standard errors from estimates in Table 2
Quadratic regression line added
.3
Predicted
Standard
Error of
Probability
.2
.1
0
0
.2
.4
.6
.8
Predicted Probability of EUT Model
-23-
1
Figure 3: Sex, EUT and Prospect Theory
Males
80
60
Percent
60
Percent
Females
80
40
20
40
20
0
0
0
.1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
Figure 4: Race, EUT and Prospect Theory
60
Blacks
60
60
40
Percent
Percent
20
20
20
0
0
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
Others
40
Percent
40
Hispanics
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
-24-
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
Figure 5: Business Majors, EUT and Prospect Theory
Business Majors
Non-Business Majors
50
40
40
30
30
Percent
Percent
50
20
20
10
10
0
0
0
.1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
Figure 6: Age, EUT and Prospect Theory
60
17 or 18
60
60
40
Percent
Percent
20
20
0
20
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
20 and Over
40
Percent
40
19
0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
-25-
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
Probability of EUT Model
Figure 7: Parameters of the PT Model
beta
Percent
Percent
alpha
.2
.4
.6
.8
1
.2
.4
.6
.8
gamma
Percent
Percent
lam bda
1
0
5
10
15
.4
.6
.8
1
1.2
Figure 8: CRRA Parameter of the EUT Model
25
Percent
20
15
10
5
0
.4
.6
.8
-26-
1
1.2
Figure 8: Sex and Risk Aversion
r Males
Percent
Percent
alpha Males
.2
.4
.6
.8
1
1.2
.2
.6
.8
1
1.2
.6
.8
1
1.2
r Females
Per cent
Percent
alpha Females
.4
.2
.4
.6
.8
1
1.2
-27-
.2
.4
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Mixture-Model Approach,” Journal of Risk & Uncertainty, 29(2), 2004, 181-197.
Watt, Richard, “Defending Expected Utility Theory,” Journal of Economic Perspectives, 16(2), Spring
2002, 227-228.
Williams, Rick L., “A Note on Robust Variance Estimation for Cluster-Correlated Data,” Biometrics,
56, June 2000, 645-646.
Wu, George, and Gonzalez, R., “Curvature of the Probability Weighting Function,” Management
Science, 42, 1996, 1676-1690.
-31-
Appendix A: Experimental Instructions (NOT FOR PUBLICATION)
A. Instructions for Control Experiment
This is an experiment about choosing between lotteries with varying prizes and chances of
winning. You will be presented with a series of lotteries where you will make choices between pairs
of them. There are 60 pairs in the series. For each pair of lotteries, you should indicate which of the
two lotteries you prefer to play. You will actually get the chance to play three of the lotteries you
choose, and will be paid according to the outcome of those lotteries, so you should think carefully
about which lotteries you prefer.
Here is an example of what the computer display of such a pair of lotteries will look like.
-32-
The display on your screen will be bigger and easier to read.
The outcome of the lotteries will be determined by the draw of a random number between 1
and 100. Each number between (and including) 1 and 100 is equally likely to occur. In fact, you will
be able to draw the number yourself using a special die that has all the numbers 1 through 100 on it.
In the above example the left lottery pays five dollars ($5) if the number on the ball drawn is
between 1 and 40, and pays fifteen dollars ($15) if the number is between 41 and 100. The yellow
color in the pie chart corresponds to 40% of the area and illustrates the chances that the ball drawn
will be between 1 and 40 and your prize will be $5. The black area in the pie chart corresponds to
60% of the area and illustrates the chances that the ball drawn will be between 41 and 100 and your
prize will be $15.
We have selected colors for the pie charts such that a darker color indicates a higher prize.
White will be used when the prize is zero dollars ($0).
Now look at the pie in the chart on the right. It pays five dollars ($5) if the number drawn is
between 1 and 50, ten dollars ($10) if the number is between 51 and 90, and fifteen dollars ($15) if
the number is between 91 and 100. As with the lottery on the left, the pie slices represent the
fraction of the possible numbers which yield each payoff. For example, the size of the $15 pie slice
is 10% of the total pie.
Each pair of lotteries is shown on a separate screen on the computer. On each screen, you
should indicate which of the lotteries you prefer to play by clicking on one of the three boxes
beneath the lotteries. You should click the LEFT box if you prefer the lottery on the left, the
RIGHT box if you prefer the lottery on the right, and the DON’T CARE box if you do not prefer
one or the other.
You should approach each pair of lotteries as if it is one of the three that you will play out. If
-33-
you chose DON’T CARE in the lottery pair that we play out, you will pick one using a standard six
sided die, where the numbers 1-3 correspond to the left lottery and the numbers 4-6 to the right
lottery.
After you have worked through all of the pairs of lotteries, raise your hand and an
experimenter will come over. You will then roll a 20 sided die three times to determine which pairs
of lotteries that will be played out. One lottery pair from the first 20 pairs will be selected, one from
the next 20 pairs, and finally one from the last 20 pairs. If you picked DON’T CARE for one of
those pairs, you will use the six sided die to decide which one you will play. Finally, you will roll the
100 sided die to determine the outcome of the lottery you chose.
For instance, suppose you picked the lottery on the left in the above example. If the random
number was 37, you would win $5; if it was 93, you would get $15. If you picked the lottery on the
right and drew the number 37, you would get $5; if it was 93, you would get $15.
Therefore, your payoff is determined by three things:
•
by which three lottery pairs that are chosen to be played out in the series of 60 such pairs
using the 20 sided die;
•
by which lottery you selected, the left or the right, for each of these three pairs; and
•
by the outcome of that lottery when you roll the 100 sided die.
This is not a test of whether you can pick the best lottery in each pair, because none of the
lotteries are necessarily better than the others. Which lotteries you prefer is a matter of personal
taste. The people next to you will have difference lotteries, and may have different tastes, so their
responses should not matter to you. Please work silently, and make your choices by thinking
carefully about each lottery.
All payoffs are in cash, and are in addition to the $5 show-up fee that you receive just for
-34-
being here. In addition to these earnings you will also receive a randomly determined amount of
money before we start. This can be any amount between $1 and $10 and will be determined by
rolling the 10 sided die.
When you have finished reading these instructions, please raise your hand. We will then
come to you, answer any questions you may have, let you roll the 10 sided die for the additional
money, and start you on a short series of practice choices. The practice series will not be for
payment, and consists of only 6 lottery pairs. After you complete the practice we will start you on
the actual series for which you will be paid.
As soon as you have finished the actual series, and after you have rolled the necessary dice,
you will be paid in cash and are free to leave.
Practice Session
In this practice session you will be asked to make 6 choices, just like the ones that you will be
asked to make after the practice. You may take as long as you like to make your choice on each
lottery.
You will not be paid your earnings from the practice session, so feel free to use this practice
to become familiar with the task. When you have finished the practice you should raise your hand so
that we can start you on the tasks for which you will be paid.
B. Instructions for Experiment With Information and Minimum Time Constraints
This is an experiment about choosing between lotteries with varying prizes and chances of
winning. You will be presented with a series of lotteries where you will make choices between pairs
of them. There are 60 pairs in the series. For each pair of lotteries, you should indicate which of the
-35-
two lotteries you prefer to play. You will actually get the chance to play three of the lotteries you
choose, and will be paid according to the outcome of those lotteries, so you should think carefully
about which lotteries you prefer.
Here is an example of what the computer display of such a pair of lotteries will look like.
The display on your screen will be bigger and easier to read.
The outcome of the lotteries will be determined by the draw of a random number between 1
and 100. Each number between (and including) 1 and 100 is equally likely to occur. In fact, you will
-36-
be able to draw the number yourself using a special die that has all the numbers 1 through 100 on it.
In the above example the left lottery pays five dollars ($5) if the number on the ball drawn is
between 1 and 40, and pays fifteen dollars ($15) if the number is between 41 and 100. The yellow
color in the pie chart corresponds to 40% of the area and illustrates the chances that the ball drawn
will be between 1 and 40 and your prize will be $5. The black area in the pie chart corresponds to
60% of the area and illustrates the chances that the ball drawn will be between 41 and 100 and your
prize will be $15.
We have selected colors for the pie charts such that a darker color indicates a higher prize.
White will be used when the prize is zero dollars ($0).
Now look at the pie in the chart on the right. It pays five dollars ($5) if the number drawn is
between 1 and 50, ten dollars ($10) if the number is between 51 and 90, and fifteen dollars ($15) if
the number is between 91 and 100. As with the lottery on the left, the pie slices represent the
fraction of the possible numbers which yield each payoff. For example, the size of the $15 pie slice
is 10% of the total pie.
Above each lottery you will be told what the average payoff would be if this lottery were
played 1000 times. For the particular lotteries shown in our example here, the average payoff would
be $11 for the lottery on the left and $8 for the lottery on the right. The average payoffs will typically
vary from choice to choice. You will only play the lottery one time, not 1000 times, if it is selected.
You will also be told the difference between the largest prize and the smallest prize in each
lottery. Since the prizes for the left lottery are $5 and $15, this difference is $10. The prizes for the
right lottery are $5, $10 and $15, so the difference is $10 in that case.
Each pair of lotteries is shown on a separate screen on the computer. On each screen, you
should indicate which of the lotteries you prefer to play by clicking on one of the three boxes
-37-
beneath the lotteries. You should click the LEFT box if you prefer the lottery on the left, the
RIGHT box if you prefer the lottery on the right, and the DON’T CARE box if you do not prefer
one or the other.
You should approach each pair of lotteries as if it is one of the three that you will play out. If
you chose DON’T CARE in the lottery pair that we play out, you will pick one using a standard six
sided die, where the numbers 1-3 correspond to the left lottery and the numbers 4-6 to the right
lottery.
There is a small timer displayed in the lower right corner of the screen. This shows an
amount of time before which you cannot click the CONFIRM CHOICE box, and it will count
down to zero. This amount of time can vary across lottery pairs, but will always be kept the same for
five pairs in a row. You should therefore pay attention to this timer as you first see each screen.
After you have worked through all of the pairs of lotteries, raise your hand and an
experimenter will come over. You will then roll a 20 sided die three times to determine which pairs
of lotteries that will be played out. One lottery pair from the first 20 pairs will be selected, one from
the next 20 pairs, and finally one from the last 20 pairs. If you picked DON’T CARE for one of
those pairs, you will use the six sided die to decide which one you will play. Finally, you will roll the
100 sided die to determine the outcome of the lottery you chose.
For instance, suppose you picked the lottery on the left in the above example. If the random
number was 37, you would win $5; if it was 93, you would get $15. If you picked the lottery on the
right and drew the number 37, you would get $5; if it was 93, you would get $15.
Therefore, your payoff is determined by three things:
•
by which three lottery pairs that are chosen to be played out in the series of 60 such pairs
using the 20 sided die;
-38-
•
by which lottery you selected, the left or the right, for each of these three pairs; and
•
by the outcome of that lottery when you roll the 100 sided die.
This is not a test of whether you can pick the best lottery in each pair, because none of the
lotteries are necessarily better than the others. Which lotteries you prefer is a matter of personal
taste. The people next to you will have difference lotteries, and may have different tastes, so their
responses should not matter to you. Please work silently, and make your choices by thinking
carefully about each lottery.
All payoffs are in cash, and are in addition to the $5 show-up fee that you receive just for
being here. In addition to these earnings you will also receive a randomly determined amount of
money before we start. This can be any amount between $1 and $10 and will be determined by
rolling the 10 sided die.
When you have finished reading these instructions, please raise your hand. We will then
come to you, answer any questions you may have, let you roll the 10 sided die for the additional
money, and start you on a short series of practice choices. The practice series will not be for
payment, and consists of only 6 lottery pairs. After you complete the practice we will start you on
the actual series for which you will be paid.
As soon as you have finished the actual series, and after you have rolled the necessary dice,
you will be paid in cash and are free to leave.
Practice Session
In this practice session you will be asked to make 6 choices, just like the ones that you will be
asked to make after the practice. In the practice session you will see two numbers in the bottom
right corner of the page. You can ignore the second number, the one to the right.
-39-
The first number shows the minimum number of seconds that must pass before you can
move on to the next problem. This minimum will always be 10 seconds or more. We will keep the
same number of minimum seconds for the first 3 choices, and then it will change for the next 3
choices.
In the real sessions to follow the practice, the minimum time will change every 5 choices.
You may take as long as you like to make your choice on each lottery.
You will see that the minimum seconds count down. You can make your choice while it is
counting down, but you cannot CONFIRM you choice until after the number has counted down to
0. However, you may take as long as you like to make your choice on each lottery. That is, you do
not need to be ready to CONFIRM by the time the clock has counted down to 0.
You will not be paid your earnings from the practice session, so feel free to use this practice
to become familiar with the task. When you have finished the practice you should raise your hand so
that we can start you on the tasks for which you will be paid.
C. Individual Characteristics Questionnaire33
Stage 2: Some Questions About You
In this survey most of the questions asked are descriptive. We will not be grading your
answers and your responses are completely confidential. Please think carefully about each question
and give your best answers.
1.
What is your AGE? ____________ years
2.
What is your sex? (Circle one number.)
33
In the original experiments question #35 was inadvertently a repeat of question #31. We show
here the correct question #35, but the responses for these experiments for question #35 are invalid. This has
no effect on any of the results in the text, since these questions were not employed in the statistical analysis
reported here.
-40-
01
3.
Male
02
Female
Which of the following categories best describes you? (Circle one number.)
01
02
03
04
05
White
African-American
African
Asian-American
Asian
06
07
08
09
Hispanic-American
Hispanic
Mixed Race
Other
4. What is your major? (Circle one number.)
01
02
03
04
05
06
07
08
09
10
11
12
13
14
5.
What is your class standing? (Circle one number.)
01
02
03
6.
04
05
06
Senior
Masters
Doctoral
Bachelor’s degree
Master’s degree
Doctoral degree
First professional degree
What was the highest level of education that your father (or male guardian) completed?
(Circle one number)
01
02
03
04
05
8.
Freshman
Sophomore
Junior
What is the highest level of education you expect to complete? (Circle one number)
01
02
03
04
7.
Accounting
Economics
Finance
Business Administration, other than Accounting, Economics, or Finance
Education
Engineering
Health Professions
Public Affairs or Social Services
Biological Sciences
Math, Computer Sciences, or Physical Sciences
Social Sciences or History
Humanities
Psychology
Other Fields
Less than high school
GED or High School Equivalency
High school
Vocational or trade school
College or university
What was the highest level of education that your mother (or female guardian) completed?
(Circle one number)
01
02
03
04
05
Less than high school
GED or High School Equivalency
High School
Vocational or trade school
College or university
-41-
9.
What is your citizenship status in the United States?
01
02
03
04
10.
Are you a foreign student on a Student Visa?
01
02
11.
Single and never married?
Married?
Separated, divorced or widowed?
On a 4-point scale, what is your current GPA if you are doing a Bachelor’s degree, or what
was it when you did a Bachelor’s degree? This GPA should refer to all of your coursework,
not just the current year. Please pick one:
01
02
03
04
05
06
07
08
13.
Yes
No
Are you currently...
01
02
03
12.
U.S. Citizen
Resident Alien
Non-Resident Alien
Other Status
Between 3.75 and 4.0 GPA (mostly A’s)
Between 3.25 and 3.74 GPA (about half A’s and half B’s)
Between 2.75 and 3.24 GPA (mostly B’s)
Between 2.25 and 2.74 GPA (about half B’s and half C’s)
Between 1.75 and 2.24 GPA (mostly C’s)
Between 1.25 and 1.74 GPA (about half C’s and half D’s)
Less than 1.25 (mostly D’s or below)
Have not taken courses for which grades are given.
How many people live in your household? Include yourself, your spouse and any
dependents. Do not include your parents or roommates unless you claim them as
dependents.
___________________
14.
Please circle the category below that describes the total amount of INCOME earned in 2002
by the people in your household (as “household” is defined in question 13).
[Consider all forms of income, including salaries, tips, interest and dividend payments,
scholarship support, student loans, parental support, social security, alimony, and child
support, and others.]
01
02
03
04
05
06
07
08
$15,000 or under
$15,001 - $25,000
$25,001 - $35,000
$35,001 - $50,000
$50,001 - $65,000
$65,001 - $80,000
$80,001 - $100,000
over $100,000
-42-
15.
Please circle the category below that describes the total amount of INCOME earned in 2002
by your parents. [Consider all forms of income, including salaries, tips, interest and dividend
payments, social security, alimony, and child support, and others.]
01
02
03
04
05
06
07
08
09
16.
Do you work part-time, full-time, or neither? (Circle one number.)
01
02
03
17.
Part-time
Full-time
Neither
Before taxes, what do you get paid? (fill in only one)
01
02
03
04
18.
$15,000 or under
$15,001 - $25,000
$25,001 - $35,000
$35,001 - $50,000
$50,001 - $65,000
$65,001 - $80,000
$80,001 - $100,000
over $100,000
Don’t Know
_____ per hour before taxes
_____ per week before taxes
_____ per month before taxes
_____ per year before taxes
Do you currently smoke cigarettes? (Circle one number.)
00
01
No
Yes
If yes, approximately how much do you smoke in one day? _______ packs
For the remaining questions, please select the one response that best matches your reaction
to the statement.
19.
“I believe that fate will mostly control what happens to me in the years ahead.”
Strongly
Disagree
20.
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“I am usually able to protect my personal interests.”
Strongly
Disagree
21.
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“When I get what I want, it's usually because I'm lucky.”
Strongly
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
-43-
Strongly
Agree
22.
“In order to have my plans work, I make sure that they fit in with the desires of people who
have power over me.”
Strongly
Disagree
23.
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“Whether or not I get into a car accident depends mostly on how good a driver I am.”
Strongly
Disagree
30.
Strongly
“Getting what I want requires pleasing those people above me.”
Strongly
Disagree
29.
Slightly Slightly Agree
Disagree
Agree
“When I make plans, I am almost certain to make them work.”
Strongly
Disagree
28.
Disagree
“I feel like other people will mostly determine what happens to me in the future.”
Strongly
Disagree
27.
Agree
“Chance occurrences determined most of the important events in my past.”
Strongly
Disagree
26.
Strongly
“Whether or not I get into a car accident depends mostly on the other drivers.”
Strongly
Disagree
25.
Slightly Slightly Agree
Disagree
Agree
“I have mostly determined what has happened to me in my life so far.”
Strongly
Disagree
24.
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“Often there is no chance of protecting my personal interests from bad luck.”
Strongly
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
-44-
Strongly
Agree
31.
“When I get what I want, it's usually because I worked hard for it.”
Strongly
Disagree
32.
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“I think that I will mostly control what happens to me in future years.”
Strongly
Disagree
35.
Strongly
“Whether or not I get into a car accident is mostly a matter of luck.”
Strongly
Disagree
34.
Slightly Slightly Agree
Disagree
Agree
“Most of my personal history was controlled by other people who had power over me.”
Strongly
Disagree
33.
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“People like myself have very little chance of protecting our personal interests when they
conflict with those of strong pressure groups.”
Strongly
Disagree
36.
Disagree
Slightly Slightly Agree
Disagree
Agree
Strongly
Agree
“It's not always wise for me to plan too far ahead because many things turn out to be a
matter of good or bad fortune.”
Strongly
Disagree
Disagree
Slightly Slightly Agree
Disagree
Agree
-45-
Strongly
Agree