hierarchical maximum likelihood parameter estimation for
TRANSCRIPT
Hierarchical maximum likelihood parameterestimation for cumulative prospect theory:Improving the reliability of individual risk
parameter estimates
Ryan O. Murphy , Robert H.W. ten Brincke
ETH Risk Center – Working Paper Series
ETH-RC-14-005
The ETH Risk Center, established at ETH Zurich (Switzerland) in 2011, aims to develop cross-disciplinary approaches to integrative risk management. The center combines competences from thenatural, engineering, social, economic and political sciences. By integrating modeling and simulationefforts with empirical and experimental methods, the Center helps societies to better manage risk.
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ETH-RC-14-005
Hierarchical maximum likelihood parameter estimation for cumu-lative prospect theory: Improving the reliability of individual riskparameter estimates
Ryan O. Murphy , Robert H.W. ten Brincke
Abstract
Individual risk preferences can be identified by using decision models with tuned parameters that max-imally fit a set of risky choices made by a decision maker. A goal of this model fitting procedure is toisolate parameters that correspond to stable risk preferences. These preferences can be modeled asan individual difference, indicating a particular decision maker’s tastes and willingness to tolerate risk.Using hierarchical statistical methods we show significant improvements in the reliability of individual riskpreference parameters over other common estimation methods. This hierarchal procedure uses popu-lation level information (in addition to an individual’s choices) to break ties (or near-ties) in the fit qualityfor sets of possible risk preference parameters. By breaking these statistical “ties” in a sensible way,researchers can avoid overfitting choice data and thus better measure individual differences in people’srisk preferences.
Keywords: Prospect theory, Risk preference, Decision making under risk, Hierarchical parameter esti-mation, Maximum likelihood
Classifications: JEL Codes: D81, D03
URL: http://web.sg.ethz.ch/ethz risk center wps/ETH-RC-14-005
Notes and Comments:
ETH Risk Center – Working Paper Series
Hierarchical maximum likelihood parameter estimation
for cumulative prospect theory: Improving the
reliability of individual risk parameter estimates
Ryan O. Murphy∗, Robert H.W. ten Brincke
Chair of Decision Theory and Behavioral Game TheorySwiss Federal Institute of Technology Zurich (ETHZ)
Clausiusstrasse 50, 8006 Zurich, Switzerland+41 44 632 02 75
Abstract
Individual risk preferences can be identified by using decision models withtuned parameters that maximally fit a set of risky choices made by a decisionmaker. A goal of this model fitting procedure is to isolate parameters thatcorrespond to stable risk preferences. These preferences can be modeled asan individual difference, indicating a particular decision maker’s tastes andwillingness to tolerate risk. Using hierarchical statistical methods we showsignificant improvements in the reliability of individual risk preference pa-rameters over other common estimation methods. This hierarchal procedureuses population level information (in addition to an individual’s choices) tobreak ties (or near-ties) in the fit quality for sets of possible risk preferenceparameters. By breaking these statistical “ties” in a sensible way, researcherscan avoid overfitting choice data and thus better measure individual differ-ences in people’s risk preferences.
Key words: Prospect theory, Risk preference, Decision-making under risk,Hierarchical parameter estimation, Maximum likelihood
∗Corresponding author.Email addresses: [email protected] (Ryan O. Murphy),
[email protected] (Robert H.W. ten Brincke)
Working paper April 16, 2014
2
1. Introduction1
People must often make choices among a number of different options3
for which the outcomes are not certain. Epistemic limitations can be pre-4
cisely quantified using probabilities, and given a well defined sets of options,5
each option with its respective probability of fruition, the elements of risky6
decision making are established (Knight 1932; Luce and Raiffa, 1957). Ex-7
pected value maximization stands as the normative solution to risky choice8
problems, but people’s choices do not always conform to this optimization9
principle. Rather decision makers (DMs) reveal different preferences for risk,10
sometimes forgoing an option with a higher expectation in lieu of an option11
with lower variance (thus indicating risk aversion; in some cases DMs reveal12
the opposite preference too thus indicating risk seeking). Behavioral theories13
of risky decision making (Kahneman and Tversky, 1979; Tversky and Kah-14
neman, 1992; Kahneman and Tversky, 2000) have been developed to isolate15
and highlight the order in these choice patterns and provide psychological in-16
sights into the underlying structure of DM’s preferences. This paper is about17
measuring those subjective preferences, and in particular finding a statistical18
estimation procedure that can increase the reliability of those measures, thus19
better capturing risk preferences and doing so at the individual level.20
1.1. Example of a risky choice21
A binary lottery (i.e., a gamble or a prospect) is a common tool for22
studying risky decision making, so common in fact that they have been called23
the fruit flies of decision research (Lopes, 1983). In these simple risky choices,24
DMs are presented with monetary outcomes xi, each associated with an25
explicit probability pi. Consider for example the lottery in Table 1, offering26
a DM the choice between option A, a payoff of $1 with probability .62 or27
$83 with .38; or option B, a payoff of $37 with probability .41 or $24 with28
probability .59. The decision maker is called upon to choose either option A29
or option B and then the lottery is played out via a random process for real30
consequences.31
1.2. EV maximization and descriptive individual level modeling32
The normative solution to the decision problem in Table 1 is straightfor-33
ward. The goal of maximizing long term monetary expectations is satisfied34
2
option A option B
$1with probability
.62
$83 .38
$37with probability
.41
$24 .59
Table 1: Here is displayed an example of a simple, one-shot risky decision. The choiceis simply whether to select option A or option B. The expected value of optionA is higher ($32.16) than that of option B ($29.33), but option A contains thepossibility of an outcome with a relatively small value ($1). As it turns out, themajority of DMs prefer option B, even though it has a smaller expected value.
by selecting the option with the highest expected value. Although this deci-35
sion policy has desirable properties, it is often not an accurate description of36
what people prefer nor what they choose when presented with risky options.37
As can be anticipated, different people have different tastes, and this hetero-38
geneity includes preferences for risk as well. For example, the majority of39
incentivized DMs select option B from the lottery shown above, indicating,40
perhaps, that these DMs have some degree of risk aversion. However the41
magnitude of this risk aversion is still unknown and it cannot be estimated42
from only one choice resting from a binary lottery. This limitation has lead43
researchers to use larger sets of binary lotteries where DMs make many in-44
dependent choices, and from the overall pattern of choices, researchers can45
draw inferences about the DM’s risk preferences.46
Risk preferences have often been modeled at the aggregate level, by fitting47
parameters to the most selected option across many individuals, as opposed48
to modeling preferences at the individual level (see Kahneman and Tversky,49
1979; Tversky and Kahneman, 1992). This aggregate approach is a use-50
ful first step in that it helps establish stylized facts about human decision51
making and further facilitates the identification of systematicity in the de-52
viations from EV maximization. However, actual risk preferences exist at53
the individual level, and thus improving the resolution of measurement and54
modeling is a natural development in this research field. Substantial work55
exists along these lines already for both the measurement of risk preferences56
as well their application (e.g. Gonzalez and Wu, 1999; Holt and Laury, 2002;57
Abdellaoui, 2000; Fehr-Duda et al., 2006). The purpose of this paper is to58
add to that literature by developing and evaluating a hierarchical estimation59
method that improves the reliability of parameter estimates of risk prefer-60
ences at the individual level. Moreover we want to do so with a measurement61
tool that is readily useable and broadly applicable as a measure of individual62
3
risk preferences.63
1.3. Three necessary components for measuring risk preferences64
There are three elementary components in measuring risk preferences65
and these parts serve as the foundation for developing behavioral models of66
risky choice. These three components are: lotteries, models, and statistical67
estimation/fitting procedures. We explain each of these components below68
in general terms, and then provide detailed examples and discussion of the69
elements in the subsequent sections of the paper.70
1.3.1. Lotteries71
Choices among options reveal DM’s preferences (Samuelson, 1938; Var-72
ian, 2006). Lotteries are used to elicit and record risky decisions and these73
resulting choice data serve as the input for the decision models and parameter74
estimation procedures. Here we focus on choices from binary lotteries (such75
as Stott, 2006; Rieskamp, 2008; Nilsson et al., 2011), but the elicitation of76
certainty equivalence values is another well-established method for quantify-77
ing risk preferences (see for example Zeisberger et al., 2010). Binary lotteries78
are arguably the simplest way for subjects to make choices and thus elicit79
risk preferences. One reason to use binary lotteries is because people appear80
to have problems with assessing a lottery’s certainty equivalent (Lichtenstein81
and Slovic, 1971), although this simplicity comes at the cost of requiring a82
DM to make many binary choices to achieve the same degree of estimation83
fidelity that other methods purport to (e.g., the three decision risk measure84
from Tanaka et al., 2010).85
1.3.2. Decision model (non-expected utility theory)86
Models may reflect the general structure (e.g., stylized facts) of DMs’ ag-87
gregate preferences by invoking a latent construct like utility. These models88
also typically have free parameters that can be tuned to improve the accu-89
racy of how they capture different choice patterns. An individual’s choices90
from the lotteries are fit to a model by tuning these parameters and thus91
identifying differences in revealed preferences and summarizing the pattern92
of choices (with as much accuracy as possible) by using particular parameter93
values. These parameters can, for example, establish not just the existence of94
risk aversion, but can quantify the degree of this particular preference. This95
is useful in characterizing an individual decision maker and isolating cog-96
nitive mechanisms that underlie behavior (e.g., correlating risk preferences97
4
with other variables, including measures of physiological processes such as98
in Figner and Murphy, 2010; Schulte-Mecklenbeck et al., 2014) as well as99
tuning a model to make predictions about what this DM will choose when100
presented with different options. Here we focus on the non-expected utility101
model of cumulative prospect theory (Kahneman and Tversky, 1979; Tver-102
sky and Kahneman, 1992; Starmer, 2000) as our decision model. Prospect103
theory is arguably the most important and influential descriptive model of104
risky choice to date and it has been used extensively in research related to105
model fitting and risky decision making.106
1.3.3. Parameter estimation procedure107
Given a set of risky choices and a particular decision model, the best108
fitting parameter(s) can be identified via a statistical estimation procedure.109
A perfectly fitting parameterized model would exactly reproduce all of a110
DM’s choices correctly (i.e., perfect correspondence). Obviously this is a lofty111
goal and it is almost never observed in practice, as there almost certainly is112
measurement error in a set of observed choices.113
A decision model, with best fitting parameters, can be evaluated in sev-114
eral different ways. One approach of evaluating the goodness of fit would be115
to determine how many correct predictions the parameterized model makes116
of a DM’s choices. For example, for a set of risky choices from binary lotter-117
ies, researchers may find that 60% of the choices are constant with expected118
value maximization (EV maximization is the normative decision policy and119
trivially is a zero parameter decision model). If researchers used a concave120
power function to transform values into utilities, this single parameter model121
may increase to 70% consistency with the selected choices. Although this122
model scoring method is straightforward and intuitive, it has some stark123
shortcomings (as we will show later in the paper). Other more sophisticated124
methods, like maximum likelihood estimation, have been developed that al-125
low for a more nuanced approach to evaluating the fit of a choice model (see126
for example Harless and Camerer, 1994; Hey and Orme, 1994). All of these127
different evaluation methods can also include both in-sample, and out-of-128
sample tests, which can be useful to diagnose and mitigate the overfitting129
choice data.130
The interrelationship between lotteries, a decision model, and an esti-131
mation procedure, is shown in Figure 1. Lotteries provide stimuli and the132
resulting choices are the input for the models; a decision model and an esti-133
mation procedure tune parameters to maximize correspondence between the134
5
Out-of-sample
(prediction)
In-sample (summarization)
Parameter
estimation
2
91
with probability
with probability .32.68
3956
with probability .99.01
-24-13
with probability .44.56
-15-62
with probability .48.52
58-5
with probability .60.40
96-40
Week 2 parametersα, λ, δ, λ
Reliability
time 1 time 2
1
.78
.223299
InputChoices
√ √
√
√
√√
Lotteries
Week 1 parametersα, λ, δ, λ
ModelProspect Theory
Figure 1: On the left we find three choices for each of the binary lotteries that serve as inputfor the model. Using an estimation method we fit the model’s risk preferenceparameters to the individual risky choices at time 1. These parameters, to someextent, summarize the choices made in time 1 by using the parameters in themodel and reproduce the choices (depending on the quality of the fit). Theout-of-sample predictive capacity of the model can be evaluated by comparingthe predictions against actual choices at time 2. The reliability of the estimatesis evaluated by comparing the time 1 parameter estimates to estimates obtainedusing the same estimation method on the time 2 data. This experimental designholds the lotteries and choices consistent, but varies the estimation procedures;moreover this is done in a test-retest design which allows the computation andcomparison of both in-sample and out-of-sample statistics.
model and the actual choices from the decision maker facing the lotteries.135
This process of eliciting choices can be repeated again using the same lot-136
teries and the same experimental subjects. This test-retest design allows for137
both the development of in-sample parameter fitting (i.e., statistical expla-138
nation or summarization), as well as out-of-sample parameter fitting (i.e.,139
prediction). Moreover, the test-retest reliability of the different parameter140
6
estimates can be computed as a correlation, as two sets of parameters exist141
for each decision maker. This is a powerful experimental design as it can142
accommodate both fitting and prediction, and further can also diagnose in-143
stances of overfitting which can undermine psychological interpretations (see144
Gigerenzer and Gaissmaier, 2011).145
1.4. Overfitting and bounding parameters146
Multi-parameter models estimation methods may be prone to overfitting147
and in doing so adjust to noise instead of real risk preferences (Roberts148
and Pashler, 2000). This can sometimes be observed when parameter values149
emerge that are highly atypical and extreme. A common solution to this150
problem is to set boundaries and limit the range of parameter values that151
are potentially estimated. Boundaries prevent extreme parameter values in152
the case of weak evidence and thus reduce overfitting, but on the downside153
they negate the possibility of reflecting extreme preferences altogether, even154
though these preferences may be real. Boundaries are also defined arbitrary155
and may create serious estimation problems due to parameter interdepen-156
dence. For example, setting a boundary at the lower range on one parameter157
may radically change the estimate of another parameter (e.g., restricting the158
range of probability distortion may unduly influence estimates of loss aver-159
sion) for one particular subject. The fact that a functional (mis)specification160
can affect other parameter estimates (see for example Nilsson et al., 2011)161
and that interaction effects between parameters have been found (Zeisberger162
et al., 2010) indicates that there are multiple ways in which prospect theory163
can account for preferences. This in turn may mean that choosing different164
parameter boundaries for one or more parameters may affect the estimates of165
other parameter estimates as well if a parameter estimate runs up against an166
imposed boundary. To circumvent the pitfalls of arbitrary parameter bound-167
aries, we use a hierarchical estimation method based on Farrell and Ludwig168
(2008) without such boundaries. At its core, this method uses estimates of169
risk preferences of the whole sample to inform estimates of risk preferences at170
the individual level. In this paper we address to what degree an estimation171
method combining group level information with individual level information172
can more reliably represent individual risk preferences, rather than using173
either individual or aggregate information exclusively.174
7
1.5. Justification for using hierarchical estimation methods175
The ultimate goal of this hierarchical estimation procedure is to obtain re-176
liable estimates of individual risk preferences that can be used to make better177
predictions about risky choice behavior contrasted to other estimation meth-178
ods. This is not modeling as a means to only maximize in-sample fit, but179
rather to maximize out-of-sample correspondence; and moreover these meth-180
ods are a way to gain insights into what people are actually motivated by181
as they cogitate about risky options and make choices when confronting ir-182
reducible uncertainty. Ideally, the parameters should not only be about just183
summarizing choices, but also about capturing some psychological mecha-184
nisms that underlie risky decision making.185
1.6. Other applications of hierarchical estimation methods186
Other researchers have applied hierarchical estimation methods to risky187
choice modeling. Nilsson et al. (2011) have applied a Bayesian hierarchical188
parameter estimation model to simple risky choice data. The hierarchical189
procedure outperformed maximum likelihood estimation in a parameter re-190
covery test. The authors however did not test out-of-sample predictions nor191
the retest reliability of their parameter estimates. Wetzels et al. (2010) ap-192
plied Bayesian hierarchical parameter estimation in a learning model and193
found it was more robust with regard to extreme estimates and misunder-194
standing of task dynamics. Scheibehenne and Pachur (2013) applied Bayesian195
hierarchical parameter estimation to risky choice data using a TAX model196
(Birnbaum and Chavez, 1997) but reported no improvement in parameter197
stability.198
1.7. Other research examining parameter stability199
There exists some research on estimated parameter stability. Glockner200
and Pachur (2012) tested the reliability of parameter estimates using a non-201
hierarchical estimation procedure. The results show that individual risk202
preferences are generally stable and that the individual parameter values203
outperform aggregate values in terms of prediction. The authors concluded204
that the reliability of parameters suffered when extra model parameters were205
added. This is contrasted against the work of Fehr-Duda and Epper (2012)206
who conclude, based on experimental data and a literature review, that those207
additional parameters (related to the probability weighting function) are nec-208
essary to reliably capture individual risk preferences. Zeisberger et al. (2010)209
also tested individual parameter reliability using a different type of risky210
8
choice (certainly equivalent vs. binary choice). They found significant differ-211
ences in parameter value estimates over time but did not use a hierarchical212
estimation procedure.213
1.8. Structure of the rest of this paper214
In this paper we focus on evaluating different statistical estimation pro-215
cedures for fitting individual choice data to risky decision models. To this216
end, we hold constant the set of lotteries DMs made choices with, and the217
functional form of the risky choice model. We then fit the same set of choice218
data using a variety of estimation methods, and then contrast the results219
to differentiate their efficiency. We also compute the test-retest reliability220
of parameter estimations that resulted from different estimation procedures.221
This broad approach allows us to evaluate different estimation methods and222
make substantiated conclusions about fit quality as well as diagnose over-223
fitting. We conclude with recommendations for estimating risk preferences224
at the individual level and briefly describe an online tool for measuring and225
estimating risk preferences that researchers are invited to use as part of their226
own research programs.227
2. Stimuli and Experimental design228
2.1. Participants229
One hundred eighty five participants from the subject pool at the Max230
Planck Institute for Human Development in Berlin volunteered to participate231
in two sessions (referred to hereafter as time 1 and time 2) that were admin-232
istered approximately two weeks apart. After both sessions were conducted,233
complete data from both sessions was available for 142 participants; these234
subjects with complete datasets are retained for subsequent analysis. The235
experiment was incentive compatible and was conducted without deception.236
The data used here are the choice only results from the Schulte-Mecklenbeck237
et al. (2014) experiment and we are indebted to these authors for their gen-238
erosity in sharing their data.239
2.2. Stimuli240
In each session of the experiment individuals made choices from a set of241
91 simple binary option lotteries. Each option has two possible outcomes242
between -100 and 100 that occur with known probabilities that sum to one.243
There were four types of lotteries for which examples are shown in Table244
9
2. The four types are: gains only; lotteries with losses only; mixed lotteries245
with both gains and losses; and mixed-zero lotteries with one gain and one246
loss and zero (status quo) as the alternative outcome. The first three types247
were included to cover the spectrum of risky decisions while the mixed-zero248
type allows for measuring loss aversion separately from risk aversion (Rabin,249
2000; Wakker, 2005). The same 91 lotteries were used in both test-retest250
sessions of this research. The set was compiled of items used by Rieskamp251
(2008), Gaechter et al. (2007) and Holt and Laury (2002); these items are252
also used by Schulte-Mecklenbeck et al. (2014). Thirty-five lotteries are gain253
only, 25 are loss only, 25 are mixed, and 6 are mixed-zero. All of the lotteries254
are listed in Appendix A.255
Lottery Option A Option B Type
132
with probability.78
99 .22
39with probability
.32
56 .68gain
2-24
with probability.99
-13 .01
-15with probability
.44
-62 .56loss
358
with probability.48
-5 .52
96with probability
.60
-40 .40mixed
4-30
with probability.50
60 .500 for sure mixed-zero
Table 2: Examples of binary lotteries from each of the four types.
2.3. Procedure256
Participants received extensive instructions regarding the experiment at257
the beginning of the first session. All participants received 10 EUR as a guar-258
anteed payment for participation in the research, and could earn more based259
on their choices. Participants then worked through several examples of risky260
choices to familiarize themselves with the interface of the MouselabWeb soft-261
ware (Willemsen and Johnson, 2011, 2014) that was used to administer the262
study. While the same 91 lotteries were used for both experimental sessions,263
the item order was randomized. Additionally the order of the outcomes (top-264
bottom) and options order (A or B) and the orientation (A above B, A left265
of B) were randomized and stored during the first session. The exact oppo-266
site spacial representation of this was used in the second session, to mitigate267
order and presentation effects.268
10
Incentive compatibility was implemented by randomly selecting one lot-269
tery at the end of each experimental session and paying the participant ac-270
cording to their choice on the selected item when it was played out with271
the stated probabilities. An exchange rate of 10:1 was used between exper-272
imental values and payments for choices. Thus, participants earned a fixed273
payment of 10 EUR plus one tenth of the outcome of one randomly selected274
lottery for each completed experimental session. As a result participants275
earned about 30 EUR (approximately 40 USD) on average for participating276
in both sessions.277
3. Model specification278
We use cumulative prospect theory (Tversky and Kahneman, 1992) to279
model risk preferences. A two-outcome lottery1 L is valued in utility u(·)280
as the sum of its components in which the monetary reward xi is weighted281
by a value function v(·) and the associated probability pi by a probability282
weighting function w(·). This is shown in Equation 1.283
u(L) = v(x1)w(p1) + v(x2)(1− w(p1)) (1)
Cumulative prospect theory has many possible mathematical specifica-284
tions. These different functional specifications have been tested and reported285
in the literature (see for example Stott, 2006) and the functional forms and286
parameterizations we use here are justifiable given the preponderance of find-287
ings.288
3.1. Value function289
In general power functions have been empirically shown to fit choice data290
better than many other functional forms at the individual level (Stott, 2006).291
Stevens (1957) also cites experimental evidence showing the merit of power292
functions in general for modeling psychological processes. Here we use a293
power value-function as displayed in Equation 2.294
1Value x1 and its associated probability p1 and is the option with the highest value forlotteries in the positive domain and x1 is the option with the lowest associated value forlotteries in the negative domain. The order is irrelevant when lotteries consist of uniformgains or losses, however in mixed lotteries it matters. This ordering is to ensure that acumulative distribution function is applied to the options systematically. See Tversky andKahneman (1992) for details.
11
−25 −15 −5 5 15 25−25
−15
−5
5
15
25
x (objective value)
v(x
) (s
ubje
ctive v
alu
e)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
p (objective probability)
w(p
) (s
ubje
ctive p
robabili
ty)
Figure 2: This figure shows typical value function (left) and probability weighting function(right) from prospect theory. The parameters used to plot the solid lines are fromTversky and Kahneman (1992) and reflect the empirical findings at the aggregatelevel. The dashed lines represent risk-neutral preferences. The solid line for thevalue function is concave over gains and convex over losses, and further exhibitsa “kink” at the reference point (in this case the origin) consistent with lossaversion. The probability weighting function overweights small probabilitiesand underweights large probabilities. It is worth noting that at the individuallevel, the plots can differ significantly from these aggregate level plots.
v(x) =
{xα x ≥ 0; α > 0−λ(−x)α x < 0; α > 0; λ > 0
(2)
The α-parameter controls the curvature of the value function. If the value295
of α is below one, there are diminishing marginal returns in the domain of296
gains. If the value of α is one, the function is linear and consistent with risk-297
neutrality (i.e., EV maximizing) in decision-making. For α values above one,298
there are increasing marginal returns for positive values of x. This pattern299
reverses in the domain of losses (values of x less than 0), which is referred to300
as the reflection effect.301
For this paper we use the same value function parameter α for both gains302
and losses, and our reasoning is as follows. First, this is parsimonious. Sec-303
ond, when using a power function with different parameters for gains and304
losses, loss aversion cannot be defined anymore as the magnitude of kink at305
the reference point (Kobberling and Wakker, 2005) but would instead need306
12
to be defined contingently over the whole curve.2 This complicates interpre-307
tation and undermines clear separability between utility curvature and loss308
aversion, and further may cause modeling problems for mixed lotteries that309
include an outcome of zero. Problems of induced correlation between the loss310
aversion parameter and the curvature of the value function in the negative311
domain have furthermore been reported when using αgain 6= αloss in a power312
utility-model (Nilsson et al., 2011).313
3.2. Probability weighting function314
To capture subjective probability, we use Prelec’s functional specification315
(Prelec, 1998; see Figure 5). Prelec’s two-parameter probability weighting316
function can accommodate both inverse S-shaped as well as S-shaped weight-317
ings and has been shown to fit individual data well (Fehr-Duda and Epper,318
2012; Gonzalez and Wu, 1999). Its specification can be found in Equation 3.319
w(p) = exp (−δ(− ln(p))γ), δ > 0; γ > 0 (3)
The γ parameter controls the curvature of the weighting function. The320
psychological interpretation of the curvature of the function is a diminish-321
ing sensitivity away from the end points: both 0 and 1 serve as necessary322
boundaries and the further from these edges, the less sensitive individuals are323
to changes in probability (Tversky and Kahneman, 1992). The δ-parameter324
controls the general elevation of the probability weighting function. It is an325
index of how attractive lotteries are considered to be (Gonzalez and Wu,326
1999) in general and it corresponds to how optimistic (or pessimistic) an327
individual decision maker is.328
The use of Prelec’s two-parameter weighting function over the original329
specification used in cumulative prospect theory (Tversky and Kahneman,330
1992) requires additional explanation. In the original specification the point331
of intersection with the diagonal changes simultaneously with the shape of332
the weighting function, whereas in Prelec’s specification, the line always in-333
tersects at the same point if δ is kept constant.3 However, changing the334
2It is possible to use αgain 6= αloss by using an exponential utility function (Kobberlingand Wakker, 2005).
3That point is 1/e ≈ 0.37 for δ = 1, which is consistent with some empirical evidence(Gonzalez and Wu, 1999; Camerer and Ho, 1994). The (aggregate) point of intersec-tion is sometimes also found to be closer to 0.5 (Fehr-Duda and Epper, 2012), which is
13
point of intersection and curvature simultaneously induces a negative corre-335
lation with the value function parameter α because both parameters capture336
similar characteristics and also does not allow for a wide variety of individ-337
ual preferences (see Fehr-Duda and Epper, 2012). Furthermore, the original338
specification of the weighting function is not monotonic for γ < 0.279 (In-339
gersoll, 2008). While that low value is generally not in the range of reported340
aggregate parameters (Camerer and Ho, 1994), this non-monotonicity may341
become relevant when estimating individual preferences, where there is con-342
siderably more heterogeneity.343
4. Estimation methods344
Individual risk preferences are captured by obtaining a set of parameters345
that best fit the observed choices implemented via a particular choice model.346
In this section we explain the workings of the simple direct maximization of347
the explained fraction (MXF) of choices, of standard maximum likelihood es-348
timation (MLE) and of hierarchical maximum likelihood estimation (HML).349
4.1. Direct maximization of explained fraction of choices (MXF)350
Arguably the most straightforward way to capture an individual’s pref-351
erences is by finding the set of parameters that maximizes the fraction of352
explained choices made over all the lotteries considered. By explained choices353
we mean the lotteries for which the utility for the chosen option exceeds that354
of the alternative option, given some decision model and parameter value(s).355
The term “explained” does not imply having established some deeper causal356
insights into the decision making process, but is used here simply to refer to357
statistical explanation (e.g., a summarization or reproducibility) of choices358
given tuned parameters and a decision model. The goal of this MXF method359
is to find the set of parametersM = {α, λ, δ, γ} that maximizes the fraction360
of explained choices from an individual DM.361
Mi = arg maxM
1
N
N∑i=1
c(yi|M) (4)
consistent with Karmarkar’s single-parameter weighting function (Karmarkar, 1979). Atwo-parameter extension of that function is provided by Goldstein and Einhorn’s function(Goldstein and Einhorn, 1987), which appears to fit about equally well as both Prelec’s,and Gonzalez and Wu’s (Gonzalez and Wu, 1999).
14
By c(·) we denote the choice itself as in Equation 5, in which u(·) is given362
by Equation 1. It returns 1 if the utility given the parameters is consistent363
with the actual choice of the subject and 0 otherwise.364
c(yi|M) =
1
if A is chosen and u(A) ≥ u(B) orif B is chosen and u(A) ≤ u(B)
0 otherwise
(5)
365
366
This is a discrete problem because we use a finite number of binary lotteries.367
Solution(s) can be found using a grid search (i.e. parameter sweep) approach,368
combined with a fine-tuning algorithm that uses the best results of the grid369
search as a starting point and then refines the estimate as needed, improving370
the precision of the parameter values. Note that this process can be compu-371
tationally intense for a high resolution estimates since the full combination372
of four parameters over their full range has to be taken into consideration.373
The MXF estimation method has some advantages. Finding the param-374
eters that maximize the explained fraction of choices is a simple metric that375
is directly related to the actual choices from a decision maker. It is easy to376
explain and it is also robust to the presence of a few aberrant choices. This377
quality prevents the estimates from yielding results that are unduly affected378
by a single decision, measurement error, or overfitting.379
An important shortcoming of MXF as an estimation method is that the380
characteristics of the explained lotteries are not taken into consideration when381
fitting the choice data. Consider the lottery at the beginning of the paper382
(Table 1), in which the expected value for option A is higher than that for B,383
but just barely. Consider a decision maker that we suspect to be generally384
risk-neutral, and therefore the EV-maximizing policy would dictate select-385
ing option A over option B. But for argument’s sake, consider if we observe386
the DM select option B; how much evidence does this provide against our387
conjecture of risk neutrality? Hardly any, one can conclude, given the small388
difference in utility between the options. Now consider the same lottery, but389
the payoff for A is $483 instead of $83. Picking B over A is now a huge390
mistake for a risk-neutral decision maker and such an observation would391
yield stronger evidence that would force one to revise a conjecture about392
the decision maker’s risk neutrality. MXF as a estimation method does not393
distinguish between these two instances as its simplistic right/wrong crite-394
15
rion lacks the nuance to integrate this additional information about options,395
choices, and the strength of evidence.396
Another limitation of MXF is that it results in numerous ties for best-397
fitting parameters. In other words, many parameter sets result in exactly398
the same fraction of choices being explained. For several subjects the same399
fraction of choices is explained using very different parameters. For one sub-400
ject just over seventy percent of choices were explained by parameters that401
included 0.6 to 1.25 for the utility curvature function, 1.5 to 6 (and possi-402
bly beyond) for loss aversion, 0.44 to 1.3 for the elevation in the weighting403
function and 0.8 to 2.5 for its curvature. We find such ties for various constel-404
lations of parameters (albeit somewhat smaller) even for subjects for which405
we can explain up to ninety percent of all choices.406
Two other estimation methods (MLE and HLM) avoid these problems407
and we turn our attention to them next.408
4.2. Maximum Likelihood Estimation (MLE)409
A strict application of cumulative prospect theory dictates that even a410
trivial difference in utility leads the DM to always choose the option with the411
highest utility. However, even in early experiments with repeated lotteries,412
Mosteller and Nogee (1951) found that this is not the case with real decision413
makers. Choices were instead partially stochastic, with the probability of414
choosing the generally more favored option increasing as the utility difference415
between the options increased. This idea of random utility maximization has416
been developed by Luce (1959) and others (e.g. Harless and Camerer, 1994;417
Hey and Orme, 1994). In this tradition, we use a logistic function that418
specifies the probability of picking one option, depending on each option’s419
utility. This formulation is displayed in Equation 6. A logistic function fits420
the data from Mosteller and Nogee (1951) well for example and has been421
shown generally to perform well with behavioral data (Stott, 2006).422
p(A � B) =1
1 + eϕ(u(B)−u(A)) , ϕ ≥ 0 (6)
The parameter ϕ is an index of the sensitivity to differences in utility. Higher423
values for ϕ diminish the importance of the difference in utility, as is illus-424
trated in Figure 3. It is important to note that this parameter operates on425
the absolute difference in utility assigned to both options. Two individuals426
with different risk attitudes but equal sensitivity, will almost certainly end up427
16
0 5 10 15 200
0.2
0.4
0.6
0.8
1
utility of A
ϕ=1
ϕ=2
pro
ba
bili
ty o
f p
ickin
g A
ove
r B
Figure 3: This figure shows two example logistic functions. Consider a lottery for whichoption B has a utility of 10 while option A’s utility is plotted on the x-axis.Given the utility of B, the probability of picking A over B is plotted on the y-axis. The option with a higher utility is more likely to be selected in general. Inthe instance when both options’ utilities are equal, the probability of choosingeach option is equivalent. The sensitivity of a DM is reflected in parameterϕ. A perfectly discriminating (and perfectly consistent) DM would have a stepfunction as ϕ → 0; less perfect discrimination on behalf of DMs is captured bylarger ϕ values.
with different ϕ parameters simply because the differences in options utilities428
are also determined by the risk attitude. While the parameter is useful to429
help fit an individual’s choice data, one cannot compare the values of ϕ across430
individuals unless both the lotteries and the individuals’ risk attitudes are431
identical. The interpretation of the sensitivity parameter is therefore often432
ambiguous and it should be considered as simply an aid to the estimation433
method.434
In maximum likelihood estimation the goal function is to maximize the435
likelihood of observing the outcome, which consists of the observed choices,436
given a set of parameters. The likelihood is expressed using the choice func-437
tion above, yielding a stochastic specification.438
We use the notation M = {α, λ, δ, γ, ϕ}, where yi denotes the choice for439
17
the i-th lottery.440
Mi = arg maxM
N∏i=1
c(yi|M) (7)
By c(·) we denote the choice itself, where p(A � B) is given by the logistic441
choice function in Equation 6.442
c(yi|M) =
{p(A � B) if A is chosen (yi is 0)1− p(A � B) if B is chosen (yi is 1)
(8)
This approach overcomes some of the major shortcomings of other methods443
discussed above. MLE takes the utility difference between the two options444
into account and therefore does not only fit the number of choices correctly445
predicted, but also measures the quality of the lottery and its fit. Ironically,446
because it is the utility difference that drives the fit (not a count of correct447
predictions), the resulting best-fitting parameters may explain fewer choices448
than the MXF method. This is because MLE tolerates several smaller pre-449
diction mistakes instead of fewer, but very large, mistakes. The fact that450
the quality of the fit drives the parameter estimates may be beneficial for451
predictions, especially when the lotteries have not been carefully chosen and452
large mistakes have large consequences. Because the resulting fit (likelihood)453
has a smoother (but still reactively lumpy) surface, finding robust solutions454
is computationally less intense than with MXF.455
On the downside, MLE is not highly robust to aberrant choices. If by456
confusion or accident the DM chooses an atypical option for a lottery that is457
greatly out of line with her other choices, the resulting parameters may be458
disproportionately affected by this single anomalous choice. While MLE does459
take into account the quality of the fit with respect to utility discrepancies,460
the fitting procedure has the simple goal of finding the parameter set that461
generates the best overall fit (even if the differences in fit among parameter462
sets are trivially divergent). This approach ignores the fact that there may be463
different parameter sets that fit choices virtually the same. Consider subject464
A in Figure 8, which is the resulting distribution of parameter estimates when465
we change one single choice and reestimate the MLE parameters. Especially466
for loss aversion and the elevation of the probability weighting function we467
find that only a single choice can make the difference between very different468
parameter values.469
The MLE method solves the tie-breaking problem of MXF, but does so470
18
by ascribing a great deal of importance to potentially trivial differences in471
fit quality. Radically different combinations of parameters may be identified472
by the procedure as nearly equivalently good, and the winning parameter set473
may emerge but only differ from the other “almost-as-good” sets by a slight-474
est of margins in fit quality. This process of selecting the winning parameter475
set from among the space of plausible combinations ignores how realistic or476
reflective the sets may be of what the DM’s actual preferences are. Moreover477
the ill behaved structure of the parameter space makes it a challenge to find478
reliable results, as different solutions can be nearly identical in fit quality479
but be located in radically different regimes of the parameter space. A minor480
change in the lotteries, or just one different choice, may bounce the resulting481
parameter estimations substantially. Furthermore, the MLE procedure does482
not consider the psychological interpretation of the parameters and therefore483
the fact that we may select a parameter that attributes preferences that are484
out of the ordinary with only very little evidence to distinguish it from more485
typical preferences.486
4.3. Hierarchical Maximum Likelihood Estimation (HML)487
The HML estimation procedure developed here is based on Farrell and488
Ludwig (2008). It is a two-step procedure that is explained below.489
Step 1. In the first step the likelihood of occurrence for each of the four490
model parameters in the population as a whole is estimated. These likeli-491
hoods of occurrence are captured using probability density distributions and492
reflect how likely (or conversely, out of the ordinary) a particular parameter493
value is in the population given everyone’s choices. These density distribu-494
tions are found by solving the integral in Equation 9. For each of the four495
cumulative prospect theory parameters we use a lognormal density distribu-496
tion, as this distribution has only positive values, is positively skewed, has497
only two distribution parameters, and is not too computationally intense.498
Because the sensitivity parameter ϕ is not independent of the other param-499
eters, and because it has no clear psychological interpretation, we do not es-500
timate a distribution of occurrence for sensitivity, but instead take the value501
we find for the aggregate data with classical maximum likelihood estimation502
for this step.4 We use the notation M = {α, λ, δ, γ, ϕ}, Pα = {µα, σα},503
4It may be that a different approach with regard to the sensitivity parameter leads tobetter performance. We cannot judge this without data of a third re-test, as it requires
19
0 1 2 30
1
2
3
4
← α (curvature)
← λ (loss aversion)
parameter value
de
nsity
Step 1
0 1 2 30
1
2
3
4
← α
← λ
parameter value
de
nsity
Step 2
0 1 2 30
1
2
3
4
← δ (elevation)
(shape) γ →
parameter value
de
nsity
0 1 2 30
1
2
3
4
← δγ →
parameter value
de
nsity
Figure 4: The two steps of the hierarchical maximum likelihood estimation procedure.In the first step we fit the data to four probability density distributions. Inthe second step we obtain the actual individual estimates. A priori and in theabsence of individual choice data the small square has the highest likelihood,which is at the modal value of each parameter distribution. The individualchoices are balanced against the likelihood of the occurrence of a parameter toobtain individual parameter estimates, depicted by big squares.
Pλ = {µλ, σλ}, Pδ = {µδ, σδ}, Pγ = {µγ, σγ}, P = {Pα,Pλ,Pδ,Pγ}, S and I504
estimation in a first session, calibration of the method using a second session, and trueout-of-sample testing of this calibration in a third session. This may be interesting but isbeyond the scope of the current paper.
20
denote the number of participants and the number of lotteries respectively.505
P = arg maxP
S∏s=1
∫∫∫∫ [N∏i=1
c(ys,i|M)
]ln(α|Pα) ln(λ|Pλ) ln(δ|Pδ) ln(γ|Pγ) dα dλ dδ dγ
(9)This first step can be explained using a simplified example which is a506
discrete version of Equation 9. For each of the four risk preference param-507
eters let us pick a set of values that covers a sufficiently large interval, for508
example {0.5, 1, ..., 5}. We then take all parameter combinations (Cartesian509
product) of these four sets, i.e. α = 0.5, λ = 0.5, δ = 0.5, γ = 0.5, then510
α = 0.5, λ = 0.5, δ = 0.5, γ = 1.0, and so forth. Each of these combi-511
nations has a certain likelihood of explaining a subject’s observed choices,512
displayed within the block parentheses in Equation 9. However, not all of the513
parameter values in these combinations are equally supported by the data.514
It could for example be that the combination showed above that contains515
γ = 0.5 is 10 times more likely to explain the observed choices than γ = 1.0.516
How likely each parameter value is in relation to another value is precisely517
what we intend to measure. The fact that one set of parameters is associ-518
ated with a higher likelihood of explaining observed choices (and therefore519
is more strongly supported by the data) can be reflected by changing the520
four log-normal density functions for each of the model parameters. Because521
we multiply how likely a set of values explains the observed choices by the522
weight put on this set of parameters through density functions, Equation 9523
achieves this. The density functions, and the product thereof, denoted by524
ln(·) within the integrals, are therefore ways to measure the relative weight of525
certain parameter values. The maximization is done over all participants si-526
multaneously under the condition and to ensure that it needs to be a density527
distribution. Applying this step to our data leads to the density distributions528
displayed in Figure 4.529
Step 2. In this step we estimate individual parameters as carried out in530
standard maximum likelihood estimation, but now weigh parameters by the531
likelihood of the occurrence of these parameter values as given by the density532
functions obtained in step 1. This likelihood of occurrence is determined by533
the distributions we obtained from step 1. Those distributions and both steps534
are illustrated in Figure 4. This second step of the procedure is described in535
Equation 10. With Mi we denote M as above for subject i. The resulting536
parameters are not only driven by individual decision data, but also by how537
21
likely it is that such a parameter combination occurs in the population.538
Mi = arg maxMi
[N∏i=1
c(yi|Mi)
]ln(α|Pα) ln(λ|Pλ) ln(δ|Pδ) ln(γ|Pγ) (10)
This HML procedure is motivated by the principle that extreme conclusions539
require extreme evidence. In the first step of the HLM procedure, one si-540
multaneously extracts density distributions of all parameter values using all541
choice data from the population of DMs. In the second step, the likelihood542
of a particular parameter value is determined by how well it fits the ob-543
served choices from an individual, and at the same time, it is weighted by544
the likelihood of observing that particular parameter value in the population545
distribution (defined by the density function obtained in step 1). The param-546
eter set with the most likely combination of both of those elements is selected.547
The weaker the evidence is, the more parameter estimates are “pulled” to548
the center of the density distributions. HML therefore counters maximum549
likelihood estimation’s tendency to fit parameters to extreme choices by re-550
quiring stronger evidence to establish extreme parameter estimates. Thus,551
if a DM makes very consistent choices, the population parameter densities552
will have very little effect on the estimates. Conversely, if a DM has greater553
inconsistency in his choices, the population parameters will have more pull554
on the individual estimates. In the extreme, if a DM has wildly inconsistent555
choices, the best parameter estimates for the DM will simply be those for556
the population average.557
This is illustrated in Figure 5 that compares the estimates for four repre-558
sentative participants by MLE (points) and HML (squares). Subject 1 is the559
same subject as in Figure 8. We find that HML estimates are different from560
MLE, but are generally still within the 95% confidence intervals. This is the561
statistical tie-breaking mechanism we referred to in the previous section. It562
is also possible that the differences in estimates are large and fall outside the563
univariate MLE confidence intervals (displayed by lines), such as for subject564
2. For the other two participants HML’s estimates are virtually identical, ir-565
respective of the confidence interval. This is largely driven by how consistent566
a DM’s choices supporting particular parameter values are.567
The estimation method has been implemented in both MATLAB and C.568
Parameter estimates in the second step were found using the simplex algo-569
rithm by Nelder and Mead (1965) and initialized with multiple starting points570
22
1
α λ δ γ
2
3
4
1 2 2 4 6 1 2 1 2
Figure 5: Parameter estimates for four participants, with each row a subject and eachcolumn a parameter. The lines are approximate univariate 95% confidence in-tervals for MLE for each of the four model parameters (sensitivity not included).The round markers indicate the best estimate from the maximum likelihood es-timation procedure. The square marker indicates its hierarchical counterpart.The first subject is subject A in Figure 8. For participants one and two, we ob-serve large differences in parameter estimates, partially explained by the largeconfidence intervals. For participants three and four we find that the results arevirtually identical.
to avoid local minima. Negative log-likelihood was used for computational571
stability when possible. For the first step of the hierarchical method one572
cannot use negative log-likelihood for the whole equation due to the presence573
of the integral, so negative log-likelihood was only used for the multiplication574
across participants. Solving the volume integral is computationally intense575
because of the number of the high dimensions and the low probability val-576
ues. Quadrature methods were found to be prohibitively slow. Because the577
integral-maximizing parameters are what is of central importance, and not578
the precise value of the integral, we used Monte Carlo integration with 2,500579
(uniform random) values for each dimension as a proxy. We repeated this580
twenty times and used the average of these estimates as the final distribution581
parameters. A possible benefit of HML is that the step 1 distribution appears582
to be fairly stable. With a sufficiently diverse population it may therefore be583
possible to simply take the distributions found in previous studies, such as584
this one. The parameter values can be found in the appendix.585
5. Results586
5.1. Aggregate results587
The aggregate data can be analyzed to establish stylized facts and general588
behavioral tendencies. In this analysis, all 12,922 choices (142 participants×589
91 items) are modeled as if they were made by a single DM and one set of risk590
23
preference parameters is estimated. This preference set is estimated using591
each of the three different estimation methods discussed before (MXF, MLE,592
HLM). For MXF, four parameters are estimated that maximize the fraction593
of explained choices; for the other two methods, five parameters (the fifth594
value is the sensitivity parameter ϕ) are estimated for each method.595
The results of the three estimation procedures, including confidence in-596
tervals, can be found in Table 3. On the aggregate, we observe behavior597
consistent with a curvature parameters implying diminishing sensitivity to598
marginal returns; α values less than 1 are estimated for all of the methods,599
namely 0.59 for MXF and 0.76 for both MLE and HLM. Choice behavior600
is also consistent with some loss aversion (λ > 1), which reflects the no-601
tion that “losses loom larger than gains” (Kahneman and Tversky, 1979).602
The highest loss-aversion value we find for any of the estimation methods603
is 1.12, which is much lower than the value of 2.25 reported in the semi-604
nal cumulative prospect theory paper (Tversky and Kahneman, 1992). Part605
of the explanation for a lower value (indicating less loss aversion) might be606
that as it is verboten to take money from participants in the laboratory and607
thus “losses” were bounded to only being reductions in participant’s show-up608
payment. The probability weighting function has the characteristic inverse609
S-shape that intersects the diagonal at around p = 0.44 for MLE and HML610
(and stays above it for MXF). Similar values for all of the estimation meth-611
ods are found in the second experimental session. The largest differences612
between the two experimental test-retest sessions are seen in the probability613
weighting parameters.614
α λ δ γ ϕ
Time 1 MXF 0.59 1.11 0.71 0.90 -
MLE 0.76 (±0.03) 1.12 (±0.04) 0.93 (±0.03) 0.63 (±0.02) 0.25 (±0.04)
HML 0.76 (±0.02) 1.12 (±0.03) 0.93 (±0.02) 0.63 (±0.02) 0.26 (±0.03)
Time 2 MXF 0.58 1.11 0.71 0.86 -
MLE 0.78 (±0.03) 1.18 (±0.04) 0.91 (±0.03) 0.65 (±0.02) 0.23 (±0.03)
HML 0.78 (±0.02) 1.18 (±0.03) 0.91 (±0.02) 0.65 (±0.02) 0.23 (±0.03)
Table 3: Median parameter values for the aggregate data found using all three estimationmethods. Approximate univariate 95% confidence intervals constructed usinglog-likelihood ratio statistics for the MLE and HML methods.
With the choice function and obtained value of ϕ we can convert the615
24
parameter estimates of the first experimental session for MLE (and HML,616
since its aggregate estimates are virtually identical) into a prediction of the617
probability of picking option B over option A for each lottery in the second618
experimental session. This is then compared to the fraction of subjects pick-619
ing option B over option A. On the aggregate, cumulative prospect theory’s620
predictions appear to perform well. The correlation between the predicted621
and observed fractions is 0.93.622
5.2. Individual risk preference estimates623
When benchmarking the explained fraction and fit of individual param-624
eters compared to their estimated aggregate median counterparts, we find625
that individual parameters provide clear benefits, especially in terms of the626
fit. After having established that individual parameters have added value, a627
more direct comparison between MXF, MLE and HML is warranted. Par-628
ticularly for HML it is important to determine to what degree its parameter629
estimates sacrifice in-sample performance, which is the performance of the630
individual parameters of the first session within that same session. Individ-631
ual risk preference parameters were estimated for each subject using all three632
estimation methods. Fits are expressed in negative log-likelihood (lower is633
better) and are directly comparable because the hierarchical component is634
excluded in the evaluation of the likelihood value. The mean and standard635
deviation of the subject’s fits are displayed in Table 4. As to be expected, the636
log-likelihood is lowest for MLE because that method attempts to find the637
set of parameters with the best possible fit. HML is not far behind, however.638
No likelihood is given for MXF, as that method does not use the sensitivity639
parameter. We also determined the explained fraction of choices for each640
individual. The means and standard deviations are also printed in Table 4.641
With 0.82 the explained fraction of choices is highest for MXF because its642
goal function is to maximize that fraction. MLE and HML follow at about643
0.76. Thus, the explained percentage of choices that cannot be explained at644
time 1 by HML estimates but can be explained by MLE estimates is small to645
naught. For some participants the HML estimates explain a larger fraction of646
choices at time 1. While this seems counterintuitive, it is the fit (likelihood)647
that eventually drives the MLE parameter estimates and not the fraction648
of explained choices. For most participants the HML estimates are differ-649
ent from the MLE estimates, but the differences are inconsequential for the650
percentage of explained choices.651
25
Fraction explained by MLE
Fra
ctio
n e
xp
lain
ed
by H
ML
0.4 0.5 0.6 0.7 0.8 0.9 10.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6: The predicted fraction of choices at time 2 when using MLE parameters fromtime 1 (x-axis) and HML parameters from time 1 (y-axis). Each point representsone DM. MLE performs better for a subject whose point appears in the lowerright region, HML performs better when it appears in the upper left region.
20 50 80 110 140 170 20020
50
80
110
140
170
200
Negative log likelihood of result with MLE
Ne
ga
tive
lo
g lik
elih
oo
d o
f re
su
lt w
ith
HM
L
Figure 7: The goodness of fit at time 2 when using MLE parameters from time 1 (x-axis)and HML parameters from time 1 (y-axis). Each point represents one DM. MLEperforms better for a DM whose point appears in the lower right region, HMLperforms better when it appears in the upper left region.
26
Explained (Time 1) Predicted (Time 2)
Fraction MXF 0.82 (0.07) 0.74 (0.10)
MLE 0.76 (0.09) 0.73 (0.09)
HML 0.76 (0.08) 0.73 (0.10)
Fit MXF - -
MLE 83.73 (21.15) 103.24 (26.66)
HML 86.37 (21.81) 98.97 (23.66)
Table 4: Mean and standard deviation (in parentheses) for the three estimation methods.Explained refers to the time 1 statistics given that time 1 estimates are used,predicted refers to time 2 statistics using parameters from time 1. The explainedfraction of choices is about the same, while the fit in time 2 which is better forHML.
So far we have established that the HML estimates do not hurt the ex-652
plained fraction of choices accounted for even if it has a somewhat lower fit653
within-sample; the next question is whether HML is worth the effort in terms654
of prediction (out-of-sample). The estimates of individual parameters of the655
first session were used to predict choices in the second session. In Table 4 we656
have listed simple statistics for the fit (again comparable by not including the657
hierarchical component) and the explained fractions of all three estimation658
methods for the second session. All three methods explain approximately the659
same fraction of choices. The large advantage MXF had over the other two660
methods has disappeared. It is interesting to compare this with the predic-661
tion that participants make the same choices in both sessions. That method662
correctly predicts .71 on average, but its application is of course limited to663
situations in which the lotteries are identical. Figure 6 shows the fraction664
of correctly predicted lotteries and Figure 7 compares the fit for MLE and665
HML parameters. Judging by the fraction of lotteries explained, it is not666
immediately obvious which method is better. A more sensitive measure to667
compare the three methods is the goodness-of-fit. When comparing HML668
to MLE, we find that HML’s out-of-sample fit is better on average and its669
negative log likelihood is lower for 100 out of 142 participants. Using a paired670
sign rank test for medians we find that the fit for HML is significantly lower671
than for MLE (p < 0.001). The results are much clearer when examining the672
fit in the figure too, which shows many points on the diagonal and a large673
number below it (a lower negative log likelihood is better). We can conclude674
that HML sacrificed part of the fit at time 1, for an improved fit at time 2.675
27
Examination of the differences in parameter estimates for those partici-676
pants for which HML outperforms MLE reveals higher MLE mean param-677
eter estimates for the individual probability weighting function parameters.678
To verify if HML did not just work because MLE returned extremely high679
parameter estimates for the weighting function, we applied the simple solu-680
tion of using (somewhat arbitrarily chosen) bounds to the MLE estimation681
method (0.2 ≤ α ≤ 2, 0.2 ≤ λ ≤ 5,0.2 ≤ δ ≤ 2,0.2 ≤ γ ≤ 1). This did not682
change the results, as HML still outperformed MLE. In many instances HML683
is also an improvement over MLE for seemingly plausible MLE estimates, so684
HML’s effect is not limited to outliers or extreme parameter values. Interest-685
ingly, some of the cases in which MLE outperforms HML are those in which686
low extreme values emerge that are consistent with choices in the second687
experimental session, such as loss seeking (less weight on losses) or weight-688
ing functions that are nearly binary step functions. This lack of sensitivity689
highlights one of the potential downsides of HML.690
5.3. Parameter reliability691
An key issue is the reliability of the risk parameter estimates. This view-692
point is consistent with using risk preferences as a trait, a stable characteristic693
of an individual decision maker, and amendable to psychological interpreta-694
tion of the parameters. Reliability can be assessed by applying all three695
estimation procedures independently on the same data from the two exper-696
imental sessions. Large changes in parameter values are a problem and an697
indication that we are mostly measuring/modeling noise instead of real indi-698
vidual risk preferences. At the individual level we can examine the test-retest699
correlations of the parameters from the three estimation methods, displayed700
in Table 5. The parameter correlations for MXF indicate some reliability701
with correlations of approximately 0.40, except for δ which approaches zero.702
MLE provides more reliable estimates for δ than MXF, but performs worse703
overall. The test-retest reliability for MLE can be improved for three out704
of four parameters by applying admittedly arbitrary bounds. The highest705
test-retest correlation is obtained using HML, which yields the most reliable706
parameter estimates for all parameters.707
Another form of parameter reliability is the change in parameter esti-708
mates resulting from small changes in choice input. One way of establishing709
this form of reliability is by temporarily changing the choice on one of a710
subject’s lotteries, after which the parameters are reestimated. This is re-711
peated for each of the 91 lotteries separately. MXF and HML’s estimates are712
28
rα rλ rδ rγMXF 0.39 0.40 0.05 0.39
MLE 0.23 0.36 0.19 0.06
MLE (bounded) 0.28 0.54 0.10 0.45
HML 0.43 0.53 0.44 0.66
Table 5: Test-retest correlation coefficients (r values) for parameter values obtained us-ing the listed estimation procedure for both sessions independently. The use ofparameter bounds (0.2 ≤ α ≤ 2, 0.2 ≤ λ ≤ 5, 0.2 ≤ δ ≤ 2, 0.2 ≤ γ ≤ 1),indicated by by bounded condition, improves the test-retest correlation for MLE.The highest test-retest correlations are obtained with HML.
robust, whereas MLE’s estimates are less so. In Figure 8 the results of rees-713
timation after perturbing one choice is plotted for two subjects using both714
MLE and HML. A single different choice can yield large changes in MLE715
parameter estimates and this contribute to unreliable parameter estimates.716
This is generally an undesirable property of a measurement method as minor717
inattention, confusion, or a mistake can lead to very different conclusions718
about that person’s risk preferences. The high robustness of HML estimates719
is a general finding among subjects in our sample. Moreover the fragility720
of the estimates quickly becomes worse as the number of aberrant choices721
increases by only a few more.722
29
α
Nu
mb
er
of
resu
lts
0 1 20
30
60
90
λ
Subject A − MLE
0 1 20
30
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δ
0 2 40
30
60
90
γ
0 1 20
30
60
90
α
Nu
mb
er
of
resu
lts
0 1 20
30
60
90
λ
Subject A − HML
0 1 20
30
60
90
δ
0 2 40
30
60
90
γ
0 1 20
30
60
90
α
Nu
mb
er
of
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lts
0 1 20
30
60
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λ
Subject B − MLE
0 1 20
30
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90
δ
0 2 40
30
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γ
0 1 20
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90
α
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of
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λ
Subject B − HML
0 1 20
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60
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0 2 40
30
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90
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0 1 20
30
60
90
Figure 8: The parameter outcomes of MLE parameter estimates for two sample subjects ifwe change a single choice for each of the 91 items. The black line corresponds tothe best MLE estimate of the original choices. The bimodal pattern we observefor loss aversion and the weighting function for subject A is not a desirablefeature. A momentary lapse of attention or a minor mistake may result in verydifferent parameter combinations. HML produces more reliable estimates interms of being less sensitive to one aberrant choice compared to MLE.
30
6. Discussion723
In this paper we illustrated the merits of a hierarchical maximum likeli-724
hood (HML) procedure in estimating individual risk parameter estimations.725
The HML estimates did reduce (albeit trivially) the in-sample performance,726
while at the same time the out-of-sample fit was significantly higher than727
MLE and its explained fraction was as high as any of the other methods.728
The reliability of the parameter estimates for test-retest was furthermore729
higher for HML than for any other method. We conclude that HML offers730
advantages over the maximization of the explained fraction and maximum731
likelihood estimation and is the preferred estimation method for this choice732
model. The use of population information in establishing individual estimates733
can improve both the out-of-sample fit and the reliability of the parameter734
estimates.735
Some caveats are in order however. The benefits of hierarchical modeling736
may, for example, disappear when more choice data is available. In cases737
where many choices have been observed, the signal can more easily be teased738
apart from the noise and hierarchical methods may become unwarranted. Its739
benefits may also depend on the set of lotteries used. For lotteries specifi-740
cally designed for the tested model, that allow a narrow range of potential741
parameter values, the choices alone may be sufficient to reliably estimate742
individual risk preference parameters and hence the MXF estimates may be743
more reliable than in our current results. As researchers however we do not744
always control the set of lottery items, especially outside of the laboratory.745
Furthermore, cumulative prospect theory and the parameterization of it we746
have chosen here is but one model. The possibility exists that other models fit747
the choice data better and that the resulting parameter estimates are highly748
robust over time; in this case there may not be added value of hierarchical749
parameter estimation.750
In this paper we have not discussed alternative hierarchical estimation751
methods. Bayesian models have gained attention in the literature recently752
(Zyphur and Oswald, 2013). The frequentist model is used here because it753
requires the addition of only one term in the likelihood function, given that754
the hierarchical distribution is known. Our data indicate that this distri-755
bution appears to be fairly stable in a population. Nevertheless a Bayesian756
hierarchical estimation method may outperform the frequentist method out-757
lined here. Additionally the hierarchical method we have applied here is of a758
relatively simple nature. It is possible to modify the extent to which the hi-759
31
erarchical method weights the individual risk preference estimates and there760
may be more sophisticated ways to do so. One way to is to fit a parameter761
that moderates the weight of the hierarchical component by maximizing the762
fit in the retest. To determine the advantages of such a method requires an763
experiment with three sessions: one to obtain parameter estimates, another764
to calibrate the parameters and a third to verify it outperforms the other765
methods. This may be of both theoretical and practical interest, but it is766
beyond the scope of this paper.767
One major take away point from the current paper is that the estima-768
tion method can greatly affect the inferences one makes about individual769
risk preferences, particularly in multi-parameter choice models. A number of770
different parameter combinations may produce virtually the same fit result.771
When the interpretation of individual parameter values is used to make pre-772
dictions about what an individual is like (e.g., psychological traits) or what773
will she do in the future (out-of-sample prediction), it is important to real-774
ize that other constellations of parameters are virtually equally plausible in775
terms of fit but may lead to vastly different conclusions about individual dif-776
ferences and predictions about behavior. Moreover, using more sophisticated777
estimation methods can help us extract more “signal” from a set of noisily778
choices and thus support better measurement of innate risk preferences.779
7. Online implementation780
Implementing the measurement and estimation methods delineated in this781
paper requires some investment of time, effort, and technical expertise; this782
may act as a barrier for the measure and method’s use and wider adoption.783
One could imagine that there are researchers interested in establishing reli-784
able measures of risk preferences at the individual level, but that these risk785
preferences may not of central interest to their research programs. Nonethe-786
less knowing these individual differences are worth some attention as they787
may covary with other variables or psychological processes under scrutiny.788
To facilitate the use of individual level risk measurement, we have devel-789
oped an online version of the risky decision task used in this research. The790
URL http://vlab.ethz.ch/prospects contains a landing page, followed by the791
91 binary lotteries (see the Appendix) presented in a randomized order to a792
decision maker to elicit their choices and preferences over well defined lot-793
teries. The DM is shown each of the binary lotteries one at a time, and the794
computer records their choice as well as their response time. After the DM795
32
has registered all of her choices, the individual’s parameters are estimated796
(using the HLM method) and the results are displayed (and emailed to the797
researcher as well). Further, researchers can have have full access to the raw798
data should they wish to run additional analyses/modeling on the choice799
data from the individual decision makers. This configuration can be used in800
a behavioral laboratory with internet connected computers, and most natu-801
rally could be used as a pre- or a post-test, in sequence with other research802
tasks.803
Acknowledgments804
The authors wish to thank Michael Schulte-Mecklenbeck, Thorsten Pachur,805
and Ralph Hertwig for the use of the experimental data analyzed in this pa-806
per. These data are a subset from a large and thorough risky choice study807
conducted at the Max Plank Institute. Thanks also to Helga Fehr-Duda and808
Thomas Epper for helpful comments and suggestions. This work was funded809
in part by the ETH Risk Center CHIRP Research Project. Any errors are810
the responsibility of the authors.811
33
References812
References813
Abdellaoui, M., 2000. Parameter-free elicitation of utility and probability814
weighting functions. Management Science 46 (11), 1497–1512.815
Birnbaum, M. H., Chavez, A., 1997. Tests of theories of decision making:816
Violations of branch independence and distribution independence. Orga-817
nizational Behavior and Human Decision Processes 71, 161–194.818
Camerer, C., Ho, T.-H., 1994. Violations of the betweenness axiom and non-819
linearity in probability. Journal of Risk and Uncertainty 8, 167–196.820
Farrell, S., Ludwig, C., 2008. Bayesian and maximum likelihood estimation821
of hierarchical response times. Psychonomic Bulletin and Review 15 (6),822
1209–1217.823
Fehr-Duda, H., de Gennaro, M., Schubert, R., 2006. Gender, financial risk,824
and probability weights. Theory and Decision 60 (2-3), 283–313.825
Fehr-Duda, H., Epper, T., 2012. Probability and risk: Foundations and eco-826
nomic implications of probability-dependent risk preferences. Annual Re-827
view of Economics 4, 567–593.828
Figner, B., Murphy, R. O., 2010. Using skin conductance in judgment and829
decision making research. New York, NY: Psychology Press, pp. 163–184.830
Gaechter, S., Johnson, E. J., Hermann, A., 2007. Individual-level loss aver-831
sion in risky and riskless choice. Working Paper, University of Nottingham.832
Gigerenzer, G., Gaissmaier, W., 2011. Heuristic decision making. Annual833
review of psychology 62, 451–482.834
Glockner, A., Pachur, T., 2012. Cognitive models of risky choice: Parameter835
stability and predictive accuracy of prospect theory. Cognition 123, 21–32.836
Goldstein, W., Einhorn, H., 1987. Expression theory and the preference re-837
versal phenomena. Psychological Review 94 (2), 236–254.838
Gonzalez, R., Wu, G., 1999. On the shape of the probability weighting func-839
tion. Cognitive Psychology 38, 129–166.840
34
Harless, D. W., Camerer, C. F., 1994. The predictive utility of generalized841
expected utility theories. Econometrica 62 (6), 1251–1290.842
Hey, J. D., Orme, C., 1994. Investigating generalizations of the expected843
utility theory using experimental data. Econometrica 62 (6), 1291–1326.844
Holt, C. A., Laury, S. K., 2002. Risk aversion and incentive effects. The845
American Economic Review 92 (5), 1644–1655.846
Ingersoll, J., 2008. Non-monotonicity of the tversky-kahneman probability-847
weighting function: A cautionary note. European Financial Management848
14 (3), 385–390.849
Kahneman, D., Tversky, A., 1979. Prospect theory: An analysis of decision850
under risk. Econometrica 47, 263–291.851
Kahneman, D., Tversky, A. (Eds.), 2000. Choices, values and frames. Cam-852
bridge University Press, Cambridge, UK, pp. p. 3 or pp. 1–16.853
Karmarkar, U., 1979. Subjectively weighted utility and the allais paradox.854
Organizational Behavior and Human Performance 24, 67–72.855
Kobberling, V., Wakker, P., 2005. An index of loss aversion. Journal of Eco-856
nomic Theory 122, 119–131.857
Lichtenstein, S., Slovic, P., 1971. Reversals of preference between bids and858
choices in gambling decisions. Journal of Experimental Psychology 89 (1),859
46–55.860
Lopes, L. L., 1983. Some thought on the psychological concept of risk. Journal861
of Experimental Psychology: Human Perception and Performance 9, 137–862
144.863
Luce, R. D., 1959. Individual choice behavior: A theoretical analysis. New864
York, NY: Wiley.865
Mosteller, F., Nogee, P., 1951. An experimental measurement of utility. The866
Journal of Political Economy 59, 371–404.867
Nelder, J. A., Mead, R., 1965. A simplex method for function minimization.868
Computer Journal 7, 308–313.869
35
Nilsson, H., Rieskamp, J., Wagenmakers, E.-J., 2011. Hierarchical bayesian870
parameter estimation for cumulative prospect theory. Journal of Mathe-871
matical Psychology 55, 84–93.872
Prelec, D., 1998. The probability weighting function. Econometrica 66, 497–873
527.874
Rabin, M., 2000. Risk aversion and expected-utility theory: A calibration875
theorem. Econometrica 68, 1281–1293.876
Rieskamp, J., 2008. The probabilistic nature of preferential choice. Journal877
of Experimental Psychology: Learning, Memory and Cognition 34 (6),878
1446–1465.879
Roberts, S., Pashler, H., 2000. How persuasive is a good fit? a comment on880
theory testing. Psychological Review 107 (2), 358–367.881
Scheibehenne, B., Pachur, T., 2013. Hierarchical bayesian modeling: Does882
it improve parameter stability? In: Knauff, M., Pauen, M., Sebanz, N.,883
Wachsmuth, I. (Eds.), Proceedings of the 35th Annual Conference of the884
Cognitive Science Society. Austin TX: Cognitive Science Society, pp. 1277–885
1282.886
Schulte-Mecklenbeck, M., Pachur, T., Murphy, R. O., Hertwig, R., 2014.887
Does prospect theory capture psychological processes. Working Paper,888
MPI Berlin.889
Starmer, C., June 2000. Development in non-expected utility theory: The890
hunt for a descriptive theory of choice under risk. Journal of Economic891
Literature 38, 332–382.892
Stevens, S. S., 1957. On the psychophysical law. The Psychological Review893
64 (3), 153–181.894
Stott, H., 2006. Cumulative prospect theory’s functional menagerie. Journal895
of Risk and Uncertainty 32, 101–130.896
Tanaka, T., Camerer, C. F., Nguyen, Q., 2010. Risk and time preferences:897
Linking experimental and household survey data from vietnam. American898
Economic Review 100 (1), 557–571.899
36
Tversky, A., Kahneman, D., 1992. Advances in prospect theory: Cumulative900
representations of uncertainty. Journal of Risk and Uncertainty 5, 297–323.901
Wakker, P., 2005. Formalizing reference dependence and initial wealth in902
rabin’s calibration theorem. Working paper, Erasmus University, available903
at http://people.few.eur.nl/wakker/pdf/calibcsocty05.pdf.904
Wetzels, R., Vandekerckhove, J., Tuerlinckx, F., Wagenmakers, E.-J., 2010.905
Bayesian parameter estimation in the expectancy valence model of the906
iowa gambling task. Journal of Mathematical Psychology 54, 14–27.907
Willemsen, M., Johnson, E., 2014.908
URL http://www.mouselabweb.org909
Willemsen, M. C., Johnson, E. J., 2011. Visiting the decision factory: Observ-910
ing cognition with mouselabweb and other information acquisition meth-911
ods. A handbook of process tracing methods for decision research: A crit-912
ical review and users guide, 21–42.913
Zeisberger, S., Vrecko, D., Langer, T., 2010. Measuring the time stability of914
prospect theory preferences. Theory and Decision 72, 359–386.915
Zyphur, M. J., Oswald, F. L., 2013. Bayesian probability and statistics in916
management research: A new horizon. Journal of Management 39 (1),917
5–13.918
37
Appendix A: Lotteries
Item p(A1) A1 p(A2) A2 p(B1) B1 p(B2) B2
1 .34 24 .66 59 .42 47 .58 642 .88 79 .12 82 .20 57 .80 943 .74 62 .26 0 .44 23 .56 314 .05 56 .95 72 .95 68 .05 955 .25 84 .75 43 .43 7 .57 976 .28 7 .72 74 .71 55 .29 637 .09 56 .91 19 .76 13 .24 908 .63 41 .37 18 .98 56 .02 89 .88 72 .12 29 .39 67 .61 6310 .61 37 .39 50 .60 6 .40 4511 .08 54 .92 31 .15 44 .85 2912 .92 63 .08 5 .63 43 .37 5313 .78 32 .22 99 .32 39 .68 5614 .16 66 .84 23 .79 15 .21 2915 .12 52 .88 73 .98 92 .02 1916 .29 88 .71 78 .29 53 .71 9117 .31 39 .69 51 .84 16 .16 9118 .17 70 .83 65 .35 100 .65 5019 .91 80 .09 19 .64 37 .36 6520 .09 83 .91 67 .48 77 .52 621 .44 14 .56 72 .21 9 .79 3122 .68 41 .32 65 .85 100 .15 223 .38 40 .62 55 .14 26 .86 9624 .62 1 .38 83 .41 37 .59 2425 .49 15 .51 50 .94 64 .06 1426 .16 -15 .84 -67 .72 -56 .28 -8327 .13 -19 .87 -56 .70 -32 .30 -3728 .29 -67 .71 -28 .05 -46 .95 -4429 .82 -40 .18 -90 .17 -46 .83 -6430 .29 -25 .71 -86 .76 -38 .24 -9931 .60 -46 .40 -21 .42 -99 .58 -3732 .48 -15 .52 -91 .28 -48 .72 -7433 .53 -93 .47 -26 .80 -52 .20 -9334 .49 -1 .51 -54 .77 -33 .23 -3035 .99 -24 .01 -13 .44 -15 .56 -6236 .79 -67 .21 -37 .46 0 .54 -9737 .56 -58 .44 -80 .86 -58 .14 -9738 .63 -96 .37 -38 .17 -12 .83 -6939 .59 -55 .41 -77 .47 -30 .53 -6140 .13 -29 .87 -76 .55 -100 .45 -2841 .84 -57 .16 -90 .25 -63 .75 -3042 .86 -29 .14 -30 .26 -17 .74 -4343 .66 -8 .34 -95 .93 -42 .07 -3044 .39 -35 .61 -72 .76 -57 .24 -2845 .51 -26 .49 -76 .77 -48 .23 -3446 .73 -73 .27 -54 .17 -42 .83 -7047 .49 -66 .51 -92 .78 -97 .22 -3448 .56 -9 .44 -56 .64 -15 .36 -8049 .96 -61 .04 -56 .34 -7 .66 -6350 .56 -4 .44 -80 .04 -46 .96 -58
(Continued on next page)
38
(Continued)Item p(A1) A1 p(A2) A2 p(B1) B1 p(B2) B2
51 .43 -91 .57 63 .27 -83 .73 2452 .06 -82 .94 54 .91 38 .09 -7353 .79 -70 .21 98 .65 -85 .35 9354 .37 -8 .63 52 .87 23 .13 -3955 .61 96 .39 -67 .50 71 .50 -2656 .43 -47 .57 63 .02 -69 .98 1457 .39 -70 .61 19 .30 8 .70 -3758 .59 -100 .41 81 .47 -73 .53 1559 .92 -73 .08 96 .11 16 .89 -4860 .89 -31 .11 27 .36 26 .64 -4861 .86 -39 .14 83 .80 8 .20 -8862 .74 77 .26 -23 .67 75 .33 -763 .91 -33 .09 28 .27 9 .73 -6764 .93 75 .07 -90 .87 96 .13 -8965 .99 67 .01 -3 .68 74 .32 -266 .48 58 .52 -5 .40 -40 .60 9667 .07 -55 .93 95 .48 -13 .52 9968 .97 -51 .03 30 .68 -89 .32 4669 .86 -26 .14 82 .60 -39 .40 3170 .88 -90 .12 88 .80 -86 .20 1471 .87 -78 .13 45 .88 -69 .12 8372 .96 17 .04 -48 .49 -60 .51 8473 .38 -49 .62 2 .22 19 .78 -1874 .28 -59 .72 96 .04 -4 .96 6375 .50 98 .50 -24 .14 -76 .86 4676 .50 -20 .50 60 .50 0 .50 077 .50 -30 .50 60 .50 0 .50 078 .50 -40 .50 60 .50 0 .50 079 .50 -50 .50 60 .50 0 .50 080 .50 -60 .50 60 .50 0 .50 081 .50 -70 .50 60 .50 0 .50 082 .10 40 .90 32 .10 77 .90 283 .20 40 .80 32 .20 77 .80 284 .30 40 .70 32 .30 77 .70 285 .40 40 .60 32 .40 77 .60 286 .50 40 .50 32 .50 77 .50 287 .60 40 .40 32 .60 77 .40 288 .70 40 .30 32 .70 77 .30 289 .80 40 .20 32 .80 77 .20 290 .90 40 .10 32 .90 77 .10 291 1 40 0 32 1.00 77 .00 2
Table 6: Lotteries used in the experiment. Each row is one lottery. The first column is theitem number. Columns 2-5 describe option A, in the form of a p(A1) probabilityof A1 and a p(A2) of A2. Columns 6-9 describe option B in the same format.
39
Appendix B: Hierarchical Distribution Parameters919
µα σα µλ σλ µδ σδ µγ σγSession 1 -0.27 0.18 0.00 0.62 -0.02 0.39 -0.31 0.59
Session 2 -0.22 0.18 0.01 0.67 -0.00 0.38 -0.34 0.67
Table 7: Estimates for the density distribution parameters of both experimental sessions.Both were obtained using Monte Carlo integration with 2,500 random values oneach dimension.
40