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© Department of Economics, University of Reading 2016
Accommodating Stake Effects under Prospect Theory
by Ranoua Bouchouicha and Ferdinand M. Vieider
Department of Economics
Discussion Paper No. 2016-120 Department of Economics University of Reading Whiteknights Reading RG6 6AA United Kingdom www.reading.ac.uk
Accommodating stake effects under prospect theory
Ranoua Bouchouicha∗
Ferdinand M. Vieider†
February 6, 2016
Abstract
We investigate how to accommodate qualitative changes in risk preferences over out-comes and probabilities under prospect theory. We find a double two-fold patternof risk preferences for gains, one over probabilities and one over outcomes. Whilesuch patterns over probabilities are an integral part of prospect theory, a solutionon how to incorporate two-fold patterns over outcomes has only recently been pro-posed by Scholten and Read (2014) [‘Prospect theory and the “forgotten” fourfoldpattern of risk preferences. JRU 48(1)’]. We use this insight to address violationsof probability-outcome separability under prospect theory as stakes increase [Fehr-Duda, Bruhin, Epper, and Schubert (2010). ‘Rationality on the Rise: Why RelativeRisk Aversion Increases with Stake Size’. JRU 40(2)]. We replicate the violationsusing traditional functional forms such as power or exponential utility. Using log-arithmic utility instead makes the violations disappear, showing the importance ofaccommodating changing risk preferences across the outcome dimension.
Keywords: risk preferences; prospect theory; probability-outcome separability
JEL-classification: C51, C52, C91, D03, D90
∗Henley Business School, University of Reading, UK†Corresponding author: Department of Economics, University of Reading, UK; email:
[email protected]; Tel. +44-118-347 8208
1
1 Introduction
Almost any important real world decision involves considerable levels of risk. It thus
comes as no surprise that attitudes towards such risks and how to model them have
received considerable attention in economics. After the axiomatisation by von Neumann
and Morgenstern (1944) of Daniel Bernoulli’s 1738 expected utility theory (reprinted
in Bernoulli, 1954), it soon became clear that a setup richer than a function with one
subjective dimension defined over lifetime wealth would be needed to model real world
behaviour such as the coexistence of insurance uptake and lottery play (for an early dis-
cussion of these issues, see Vickrey, 1945). Markowitz (1952) provided such a framework
by allowing preferences to differ between gains and losses relative to a reference point
given by current wealth. Psychologists Preston and Baratta (1948) proposed a different
solution, involving the subjective transformation of probabilities instead of outcomes—a
solution that they considered to be psychologically more realistic (Lopes, 1987).
The two approaches of subjectively transforming changes in wealth into utilities and
subjectively transforming probabilities into decision weights were finally combined in
prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992). Prospect
theory is recognised by most scholars to be the leading descriptive theory of decision mak-
ing under risk today (Barberis, 2013; Starmer, 2000; Wakker, 2010). Nevertheless, its
descriptive accuracy continues to be debated. Scholten and Read (2014) recently pointed
out how prospect theory has generally neglected the type of changes in risk attitudes
taking place purely over outcomes while keeping probabilities constant, as originally pro-
posed by Markowitz (1952). Fehr-Duda, Bruhin, Epper and Schubert (2010) uncovered
issues in the separability principle underlying prospect theory—a principle according to
which changes in attitudes over outcomes ought to be reflected purely in utility, while
changes in attitude over probabilities ought to be reflected in probability weighting.
We revisit this recent debate using data from an incentivised experiment over gains
that allow us to disentangle attitudes over outcomes from attitudes over probabilities. We
find relative risk aversion to increase in stakes at all probability levels, with qualitative
reversals from risk seeking to risk aversion as stakes increase for some probability levels,
as described in Markowitz’s original thought experiment. We also replicate qualitatively
similar probability distortions across different stake levels as originally found by Preston
and Baratta (1948). We then address modelling issues arising from this double two-fold
2
pattern under prospect theory. In particular, we investigate whether the conclusions
reached by Scholten and Read (2014) generalise to probabilities larger than their p ≤ 0.1.
This further allows us to revisit the separability violations pinpointed by Fehr-Duda et
al. (2010), and thus to examine the descriptive validity of prospect theory in our setup.
We do so by examining the fit of different functional forms to our data. A two-
parameter expo-power utility function is found to fit our data significantly better than
commonly used power or exponential utility functions. We do, however, obtain an even
better fit by using the decreasingly-elastic, one-parameter logarithmic utility function
proposed by Scholten and Read (2010) for intertemporal decisions, and applied to risk
by Scholten and Read (2014). This confirms the finding that this utility function can
accommodate changes in risk preferences over the outcome space when combined with
a suitable probability weighting function. We further expand this finding to the whole
probability space, showing that a two-parameter weighting function is essential to de-
scribe decision patterns ranging over the whole probability space.
Finally, we revisit the separability issue raised by Fehr-Duda et al. (2010). Examining
risk preferences over low and high stakes, the latter found stake effects and the resulting
increase in relative risk aversion to be reflected in probability weighting rather than in
utility curvature (see also Hogarth and Einhorn, 1990). This violates the separability
precept of prospect theory, whereby changes in outcomes ought to be purely reflected in
utility curvature. We replicate their finding of high stakes shifting probability weighting
downwards using a setup similar to the original one. We then show that combining
our two-parameter weighting function with a logarithmic utility function as proposed by
Scholten and Read (2014) these stake effects on probability weighting disappear. In other
words, separability violations seem to derive from the traditional neglect of qualitative
changes in risk attitudes over outcomes, and from the inability of traditional power and
exponential utility functions to account for such patterns. A decreasingly elastic utility
function combined with a two-parameter weighting function solves this issue.
This paper proceeds as follows. Section 2 introduces our theoretical apparatus and
discusses the econometric approach. Section 3 describes the experiment. The results are
presented in section 4. Finally, section 5 discusses the results and concludes the paper.
3
2 Theory and econometrics
Theoretical setup
We start from a description of the theoretical setup. We work with experimentally
elicited indifferences cei ∼ (xi, pi; yi) throughout, whereby cei is the certainty equivalent
of a prospect giving a pi chance to obtain xi, and a complementary chance of 1 − pi at
yi < xi. We are interested in particular in three models: i) cumulative prospect theory
(CPT ); ii) Markowitz-expected utility (MEU ); and iii) Dual-expected utility (DEU ).
The most general model we want to estimate based on our data takes the following form:
v(cei) = w(pi)v(xi) + [1− w(pi)]v(yi), (1)
where v(.) represents a utility or value function with a fixed point at 0, and w(.) represents
a probability weighting function with w(0) ≡ 0 and w(1) ≡ 1. MEU obtains from this
as a special case by setting w(pi) ≡ pi. We refer to this function as MEU even though
we are focusing on gains only, since we are interested in a utility function over changes
in wealth that can take on first convex and then concave patterns.1 We can thus rewrite
the equation above as:
u(cei) = piu(xi) + (1− pi)u(yi), (2)
where the use of u(.) instead of v(.) serves to emphasise that the two utility functions esti-
mated with and without probability weighting will not generally be the same (Bleichrodt,
Abellan-Perpiñan, Pinto-Prades and Mendez-Martinez, 2007; Schmidt and Zank, 2008).
Finally, the dual function obtains from equation 1 by setting v(x) ≡ x. This will
result in the following equation to be estimated:
cei = w(pi)xi + [1− w(pi)]yi, (3)
where the tilde on top of the weighting function again indicates that this will generally
not be the same function as in the complete prospect theory formulation. This function
has been axiomatised for the dual of rank-dependent utility by Yaari (1987). Schmidt1Losses are difficult to investigate, since we could not impose large monetary losses on our student
subjects. Furthermore, both Scholten and Read (2014) and Fehr-Duda et al. (2010) established theirmain results for gains, with less consistent patterns obtaining for losses.
4
and Zank (2007) provide an axiomatisation for the dual of reference-dependent MEU.
Functional forms
We next need to determine the functional forms to be used. For MEU, we will assume a
2-parameter utility function capable of capturing changes in curvature. We will use the
two-parameter version of an expo-power functional form proposed by Saha (1993). The
function takes the following form:
u(x) = 1− exp (−µxρ), (4)
where µ is the coefficient of the exponential function, and the parameter ρ indicates
the curvature of a power function. The parameter ρ can be interpreted as an indicator
of absolute risk aversion, with ρ < 1 indicating decreasing absolute risk aversion, ρ =
1 constant absolute risk aversion, and ρ > 1 increasing absolute risk aversion. The
parameter µ indicates levels of relative risk aversion, with µ < 0 indicating decreasing
relative risk aversion, and µ > 0 increasing relative risk aversion. The exponential
function and power function obtain from this as a special case. The power function
u(x) = xρ was used by Tversky and Kahneman (1992) to fit prospect theory parameters,
and exhibits constant elasticity.2 This is likely the most popular functional form in the
estimation of prospect theory models, and it was also employed by Fehr-Duda et al.
(2010). The exponential function u(x) = 1− exp(−µx) exhibits decreasing elasticity.3
We furthermore use a logarithmic utility function, as proposed by Scholten and Read
(2014). It has the advantage of incorporating decreasing elasticity to a greater extent
than exponential utility, which makes it better suited to accommodate Markowitz-type
patterns over outcomes under prospect theory. The function takes the following form:
u(x) =1
γlog (1 + γx) , (5)
where γ > 0 indicates decreasing sensitivity or risk aversion, γ = 0 indicates risk neutral-
ity, and γ < 0 risk seeking for gains. The function has the important property of being2Elasticity is defined as εv = ∂log[v(x)]
∂log(x). See Scholten and Read (2014) for a technical discussion.
3Scholten and Read (2014) use a normalised version of this utility function proposed by Köbberlingand Wakker (2005) for the study of mixed prospects, taking the form u(x) = 1−exp(−µx)
µ. We use
the simpler form in our analysis because it is nested in the expo-power function. None of the resultspresented below change if we use the normalised version instead.
5
decreasingly elastic, implying that for small stakes it can accommodate risk seeking by
w(p)v(x) > v(px), while this pattern may reverse as all outcomes are increased so that
w(p)v(kx) < v(pkx) for some k > 1.
We use the two-parameter probability weighting function developed by Prelec (1998):
w(p) = exp(−β(−ln(p))α) (6)
where β governs mostly the elevation of the weighting function, with higher values in-
dicating a less elevated function. The parameter can thus be interpreted as capturing
probabilistic pessimism for gains. The parameter α governs the slope of the probability
weighting function and hence probabilistic sensitivity. A value of α = 1 indicates lin-
earity of the weighting function, and α < 1 represents the typical case of probabilistic
insensitivity (Abdellaoui, 2000; Tversky and Fox, 1995; Wu and Gonzalez, 1996).4
Econometric approach
The model just presented is deterministic in nature. To accommodate the possibility
that people make mistakes, we now develop an explicit stochastic structure. Given that
we aim to model preference relations cei ∼ (xi, pi; yi), we can represent the certainty
equivalent predicted by our model, cei, as follows:
cei = v−1[w(pi)v(xi) + (1− w(pi)) v(yi)] (7)
The actual certainty equivalent we observe will now be equal to the certainty equivalent
calculated from our model plus some error term, or cei = cei + εi. We assume this error
to be normally distributed with mean zero, εi ∼ N(0, σ2i ) (see Train, 2009). Follow-
ing Bruhin, Fehr-Duda and Epper (2010), we can now express the probability density
function ψ(.) for a given subject n and prospect i as follows
ψin(θ, σi) =1
σiφ
(cei − cei
σi
)(8)
where φ is the standard normal density function, and θ indicates the vector of parameters4All results presented below remain qualitatively unchanged if we use an alternative two-parameter
function developed by Goldstein and Einhorn (1987) instead. However, the two-parameter function byPrelec provides a significantly better fit to our data, a result that holds in combination with differentutility functions (p < 0.01 in all cases; Vuong, 1989, test).
6
to be estimated. The subscript i indicates that we allow the error term to depend on
the specific prospect, or rather, on the difference between the high and low outcome in
the prospect, such that σi = σ|xi − yi|.5 This allows the error term to differ for choice
lists of different lengths, since the sure amount always varies in equal steps between xi
and yi in our experimental design.
These parameters can now be estimated by standard maximum likelihood procedures.
To obtain the overall likelihood function, we now need to take the product of the density
functions above across prospects for each subject:
Ln(θ) =∏i
ψin(θ, σi) (9)
where θ is the vector of parameters to be estimated such as to maximize the likelihood
function, and the subscript n indicates the subject-specific likelihood. Taking logs and
summing over decision makers n we obtain
LL(θn) =N∑n=1
ln [ψn(θ, σi)] (10)
We estimate this log-likelihood function in Stata 13 using the Broyden-Fletcher-Goldfarb-
Shanno optimization algorithm. Errors are always clustered at the subject level.
3 The Experiment
Participants and setting : The experiment was run as a classroom experiment at the be-
ginning of a class in advanced microeconomics at the University of Reading, UK, in the
fall term 2015. A total of 47 students showed up on the first day of class and participated
in the experiment. We eliminate two students because they exhibited multiple switching
behaviour in some choice lists. Students had been taught the basics of expected utility
theory in microeconomics 2 the previous year. This discussion followed a standard ex-
position, and stressed neither initial wealth integration, nor were any violations of EUT
discussed in class.5Wilcox (2011) pointed out a potential problem when applying such a model in a discrete choice setup,
whereby the probability of choosing the riskier prospect may be increasing in risk aversion in some cases.This probelm does not apply in our setting. Also, Apesteguia and Ballester (2014) have shown that thisprobelm does not occur even in discrete choice models when a derived certainty equivalent is comparedto a sure amount, as in our setup.
7
Students were told that their answers were to be kept anonymous and would not
be traced back to them. Students were also told that there were no right and wrong
answers, and that they only needed to record their preferences. They were also told that
the lecturer would be using average responses (but never individual ones) as examples of
behaviour during class, and that they may find it interesting to look at their preferences
as the course progressed.
Stimuli : We use two-outcome prospects thoughout. The stimuli used in the experiment
are reported in table 1. We included three different probability levels, 0.1, 0.5, and 0.9.
The stakes ranged from £10 to £200. High stakes were needed in order to meaningfully
test the Markowitz patterns over outcomes with constant probability. Finally, non-zero
lower outcomes were needed to separate utility curvature from probability weighting
in parametric estimates of prospect theory. The stimuli were balanced in the sense
that all stake levels were included for all probabilities. The order of the tasks was
counterbalanced.
Table 1: Experimental stimuli
probability low stakes high stakes prospect theory
0.1: (10,0) (20,0) (60,0) (100,0) (200, 0) (200,20) (200,60)0.5: (10,0) (20,0) (60,0) (100,0) (200, 0) (200,20) (200,60)0.9: (10,0) (20,0) (60,0) (100,0) (200, 0) (200,20) (200,60)
Incentives: Participants were told that two individuals in the class would be randomly
selected to play one of their choices for real money. The selection took place based on
random numbers attached to the questionnaires, in order to guarantee the anonymity
of the selected students. Although paying for one randomly selected task may raise
theoretical concerns under non-expected utility models, it is the standard procedure in
this type of experiment. Empirically test of this issue did not find a difference between
deterministic incentives and incentivisation based on randomly chosen rounds (Bardsley,
Cubitt, Loomes, Moffatt, Starmer and Sugden, 2010; Cubitt, Starmer and Sugden, 1998).
They constitute the only solution if one wants to obtain rich measurements of preferences
for each subject. Randomisation between subjects is also a standard procedure followed
in the literature when high stakes are offered (Abdellaoui, Bleichrodt and L’Haridon,
2008; Harrison, Lau and Rutström, 2007). Some papers explicitly tested whether paying
only some randomly selected subjects made a difference, and found none (Armantier,
8
2006; Bolle, 1990; Harrison et al., 2007). Full instructions are included in the appendix.
4 Results
4.1 Nonparametric analysis
We start by showing some descriptive results. To this end we obtain a nomalised risk
premium as follows. First, we normalise the certainty equivalent to cei−yixi−yi . This is a
measure of risk tolerance, and has the advantage that it is comparable across outcome
levels. Conveniently, it corresponds to a decision weight w(pi) under DEU. To obtain
also comparability across probabilities, we now further subtract it from the probability
of winning in a given prospect, to obtain the relative risk premium πi = pi − cei−yixi−yi .
This can be interpreted as a measure of risk aversion. It is a measure of whether the
decision weight for a given prospect under linear utility is a) lower than the probability
itself (πi > 0, risk aversion); b) higher than the probability (πi < 0, risk seeking); or
c) equal to the probability (πi = 0, risk neutrality). The risk premia are now perfectly
comparable across both stakes and probabilities.
Figure 1: Mean normalised CEs with 95% confidence intervals
Figure 1 shows the mean normalised risk premia for all prospects with zero lower
outcomes. Two general patterns stand out. First, within each stake level risk aversion
increases strongly in probabilities. For the smallest probability of p = 0.1, we find
9
significant risk seeking for all stake levels. For p = 0.9, we find significant risk aversion
across all stake levels. Risk preferences for p = 0.5 are always intermediate between
those of the more extreme probabilities. Second, across stakes the mean risk premia
move up for every probability, indicating relative risk aversion increasing in stakes. For
the intermediate probability of p = 0.5 we indeed find qualitatively different patterns
across stakes as predicted by Markowitz—risk seeking for the smallest prize, and risk
aversion for the largest prize, with risk neutrality for intermediate stakes.6 The levels of
risk aversion we find may seem low. The trend of relatively low levels of risk aversion is,
however, consistent with the results from the international comparison of risk preferences
reported by Vieider, Chmura and Martinsson (2012), where Britain constituted an outlier
amongst rich countries.7 Notice also that this may be an advantage in comparing our
results to those of Fehr-Duda et al. (2010), who also find relatively low levels of risk
aversion in their Chinese data.
We can represent these preferences non-parametrically by assuming either MEU or
DEU. Since utility is only unique up to an affine transformation, we can normalise the
lowest and highest outcomes in a series of prospects arbitrarily. By choosing u(x) ≡ 1
and u(y) ≡ 0, we can plot the non-parametric functions that result from probability
variations while keeping the outcomes in the prospect constant. While not corresponding
to Markowitz’s original thought experiment (where probabilities were held constant and
outcomes varied), this representation is perfectly legitimate in terms of Markowitz’s
theory. Figure 2(a) shows such a plot of utility for two stake levels (plots for other stake
levels are similar). In both cases, we observe the expected pattern of risk seeking for
relatively small expected outcomes, and risk aversion for larger expected outcomes.
Problems become apparent when comparing the stake ranges over which we observe
these patterns. For the low stake prospect offering a chance at £20 or else 0, we observe
risk seeking up to an expected outcome of £10.8, and risk aversion from £16.8, with
preferences changing somewhere in between. If one looks at the curve drawn using
the £200 stake level, however, risk seeking ranges to well above £43. This is a clear6Markowitz used a small probability of p = 0.1 in his thought experiment. The fact that we find
uniforme risk seeking behaviour for this probability level reflects the issue that we could not offer millionsin our incentivised experiment, as Markowitz does in his hypothetical one.
7It is still somewhat unclear why that may be the case. Vieider et al. (2012) speculated that thismay be due to the large proportion of students of foreign origin participating in the study. A similarobservation may hold in our own data set. Given the size of the data set, however, it does not make senseto test for this, and knowing this we have consciously chosen not to collect any data on demographiccharacteristics from the outset.
10
contradiction with the risk aversion starting at £16.8 in the lower stake prospect. What
this goes to show is simply that Markowitz-expected utility is not good at handling
variation in probabilities.8 Subjectively transforming probabilities likely constitutes a
better way of capturing this type of variation (Preston and Baratta, 1948; Yaari, 1987).
u(£)
£0 20
1
6
0.1
10.8
0.5
16.8
0.9
43.0
0.1
83.7
0.5
152.4
0.9
200
(a) MEU
w(p)
p0 1
1
0.1
0.1
0.5
0.42
0.9
0.76
0.1
0.3
0.5
0.54
0.9
0.85
(b) DEU
Figure 2: Modelling of risk preferences in prospects (200, pi) and (20, pi)
Probability transformations combined with linear utility are shown in figure 2(b)
for the same two stake levels. For both stake levels, risk seeking for small probabilities
is now reflected in a subjective weight that is larger than the objective probability,
w(0.1) > 0.1. This is the probability attributed to winning the prize, and can thus
be interpreted as a measure of optimism. For the largest probability, this pattern is
inverted, i.e. w(0.9) < 0.9. Problems surface when looking at differences in stakes.
Passing from £20 to £200, the function shifts systematically downwards. However, this
cannot be captured in any of the subjective transformations, since the probabilities
remain identical. In other words, we observe dual violations for the dual theory—it
performs very badly at handling outcome variations.
The issues just shown constituted the rationale for the development of prospect theory
(Kahneman and Tversky, 1979; Tversky and Kahneman, 1992), which combines the
non-linear reference-dependent outcomes transformations proposed by Markowitz with8One could imagine that the curve may be initially convex, then concave, and then convex again.
Originally proposed as a solution for the contradictions found in original expected utility theory byFriedman and Savage (1948), such a fix quickly runs into contradictions itself. We can show this in ourown data recurring to other stake levels. For instance, the prospect offering £60 with a 0.9 probabilityshows clear risk aversion at £47—very close to the risk seeking for £43 found for £200 obtaining with aprobability of 0.1.
11
non-linear transformations of probabilities into decision weights. Estimating CPT based
on our data requires structural estimations to fit parametric functions (see Abdellaoui,
2000 and Bleichrodt and Pinto, 2000, for non-parametric estimations). This, however,
raises the issue of whether differences across stake levels will purely be reflected in utility
curvature, and differences across probabilities purely in probability weighting. Using
state-of-the-art structural estimation techniques, Fehr-Duda et al. (2010) indeed found
increases in stake sizes to register in probability weighting rather than in utility curvature.
This poses a fundamental challenge to prospect theory, as it violates the separability
postulate whereby patterns over outcomes ought to be captured by utility curvature,
whereas patterns over probabilities ought to be captured by probability weighting. The
answer to that question may, however, depend on the choice of functional forms in
the presence of the qualitative changes in risk preferences across both outcomes and
probabilities we observe. We now address issues arising from the fitting of functional
forms, and test the separability hypothesis under different functional assumptions.
4.2 Fitting prospect theory parameters
Prospect theory has typically not incorporated the qualitative changes from risk seeking
to risk aversion over outcomes emphasised by Markowitz (Scholten and Read, 2014). In
this section, we will start fitting functional forms to the data. We start by estimating a
classical prospect theory model with a two-parameter weighting function and commonly
used one-parameter utility functions. We then test this model against a prospect the-
ory model using a two-parameter utility function. We can also test the latter against
simplified models assuming linearity in probabilities (MEU) or outcomes (DEU). We fur-
thermore test Markowitz against the Dual directly, to determine the more costly amongst
the two simplifications in our experimental data. This will give us a better idea of the
variability across stakes versus probabilities in our experimental data.
Table 2 shows parameter estimates of our main models, and table 3 shows a matrix
of tests pitching the models against each other. Test statistics shown are based on like-
lihood ratio tests for nested models, and on Vuong tests for non-nested models (Vuong,
1989; all test results are qualitatively unaffected if we use a Clark sign-test instead).
Some interesting results emerge. For instance, CPT with a two-parameter expo-power
utility function fits the data significantly better than PT with power utility. It also
12
performs better than CPT with exponential utility. Testing CPT with exponential util-
ity directly against PT with power utility, we find the former to perform significantly
better. This goes against the conclusion reached by Stott (2006), who found PT with
power utility and a two-parameter weighting function to provide the best fit. Notice
also how the confidence intervals estimated around the point estimates are much nar-
rower with exponential utility than under power utility, which may indicate collinearity
between utility curvature and the elevation of the probability weighting function using
the power function. Indeed, PT with power utility performs no better than DEU in our
setting—notwithstanding the clear violations of DEU we showed in the nonparametric
analysis above. Directly comparing MEU and DEU yields a clear verdict in favour of
the Dual. This reflects the fact that for the stake sizes here employed, variation of risk
preferences across the probabilistic dimension is much more important than variation
across stakes (Fehr-Duda and Epper, 2012). This could change when stakes get truly
important, but is clearly outside of the range that can be investigated in experiments.
Table 2: Parameter estimations for different models
prospect theory Markowitz-EU Dual-EUexpo-power expo power log
µ 0.012 0.005 0.001 0.594[0.004,0.020] [0.004,0.007] [0.000,0.001] [0.559,0.629]
ρ 0.872 1.106 1.470[0.751,0.992] [0.832,1.380] [1.253,1.687]
γ 0.016[0.004,0.027]
α 0.641 0.634 0.593 0.635[0.571,0.711] [0.594,0.673] [0.525,0.660] [0.564,0.707]
β 0.613 0.720 0.935 0.664 0.857[0.524,0.702] [0.676,0.765] [0.735,1.135] [0.570,0.757] [0.827,0.887]
σ 0.158 0.158 0.162 0.157 0.181 0.162[0.132,0.184] [0.151,0.165] [0.137,0.188] [0.131,0.183] [0.152,0.209] [0.155,0.170]
Subjects 45 45 45 45 45 45ll -3494.95 -3497.51 -3520.28 -3492.72 -3620.67 -3521.1495% Confidence intervals in paranetheses
Finally, we test PT with logarithmic utility against other functional forms. Scholten
and Read (2014) found a CPT model with logarithmic utility to fit their data involving
probabilities smaller than or equal to 0.1 significantly better than power utility. We
confirm this results, and extend it to exponential utility, and even expo-power utility.
This result obtains because the decreasing elasticity of the utility ensures that for small
stakes the probability weights overpower the utility function, thus accommodating risk
13
seeking pattern. For larger stakes, on the other hand, utility trumps weighting. We find
this pattern to hold not only for small probabilities as Scholten and Read (2014) do,
but also for moderate probabilities of p = 0.5. This is indeed where we find the most
pronounced two-fold pattern over outcomes. This is accommodated by the combination
of a utility parameter indicating decreasing sensitivity, with a rather elevated weighting
function, as indicated by the value of β significantly smaller than 1. A two-parameter
probability weighting function is thus essential to accommodate the behavioural patterns
observed in our data. Notice also that a similar elevation of the probability weighting
function is observed in combination with the expo-power and exponential utility func-
tions. This reflects the generally high levels of risk taking we find in our data—we will
further discuss the implications of this below.
Table 3: Matrix of test results
PT expo-power PT power PT expo PT log MEU
PT power χ2(1) = 50.67∗∗∗ −−−−−−−−
PT expo χ2(1) = 5.11∗∗ z = −3.71∗∗∗ −−−−−−−−
PT log z = −1.31∗ z = −3.68∗∗∗ z = −1.74∗∗ −−−−−−−−
MEU χ2(2) = 251.44∗∗∗ z = 7.86∗∗∗ z = 12.00∗∗∗ z = 11.45∗∗∗ −−−−−−−−
DEU χ2(2) = 52.39∗∗∗ χ2(1) = 1.72 χ2(1) = 47.27∗∗∗ z = 3.65∗∗∗ z = −7.39∗∗∗∗p < 0.1, ∗∗p < 0.05, ∗∗∗p < 0.01; positive sign of test statistic indicates test in favour of model in heading
We have modelled the double two-fold pattern of risk preferences across stakes and
probabilities using a prospect theory formulation combining a logarithmic utility function
with a two-parameter probability weighting function. Nevertheless, stake effects have
been shown to result in a more fundamental challenge to this type of modelling. Fehr-
Duda et al. (2010) used low and high stake prospects of a type similar to the ones
employed here to estimate a prospect theory model. Letting the parameters of the model
depend linearly on a high stakes treatment dummy, they found that stake effects were
reflected in the probability weighting function rather than in the utility function. Their
results thus cast doubt on the separability of outcome and probability transformations
underlying models such as prospect theory and rank dependant utility.
We can now attempt to replicate this pattern in our data. Since we only have
nonzero lower outcomes for the highest stakes in our data, however, we cannot follow their
approach of making both the utility parameter and the weighting parameters dependent
on stake levels. Instead we always estimate the utility function over the whole outcome
range, without making it a function of the stake level. Probability weighting, on the
14
other hand, can be made a function of stakes, since we have variation over probabilities
at all stake levels. We hereby follow the same strategy of Fehr-Duda et al. (2010), by
letting the two parameters of the weighting function (as well as the noise parameter)
be a linear function of stakes. If utility fully picks up our outcome variation, no stake
effects on probability weighting should be found. If, on the other hand, separability is
violated, we would expect to find stake effects in our data.
Table 4: Estimation of stake effect on probability weighting
Parameters: ρ α β σ
constant 0.905*** 0.613*** 0.639*** 0.167***(0.091) (0.032) (0.054) (0.015)
high stakes -0.022 0.214*** -0.013(0.029) (0.050) (0.012)
Parameters: γ α β σ
constant 0.011 0.622*** 0.669*** 0.167***(0.008) (0.035) (0.047) (0.015)
high stakes 0.013 0.035 -0.013(0.032) (0.079) (0.012)
∗p < 0.1, ∗∗p < 0.05, ∗∗∗p < 0.01;
We estimate this model using power utility as Fehr-Duda et al. (2010), and with
our best-fitting logarithmic utility function from above. As an independent variable, we
use a dummy indicator for high stakes, which takes the value 1 if the stakes are £60 or
higher (the results are not sensitive to where the line between high and low stakes is
drawn exactly; they are also stable to inserting the monetary outcomes directly). Table
4 shows the regressions. The upper panel shows the results using power utility. We find
a strong and highly significant effect of stakes on the probability weighting function. The
effect goes in the direction of pessimism increasing in stakes. This effect replicates the
effect found by Fehr-Duda et al. (2010). The strength can be seen from figure 3—the high
stake weighting function is shifted downwards relative to the low stakes one, and crosses
the 45 degree line at a much lower point. Indeed, we do no longer find probabilistic
optimism (or risk seeking, given the linearity of utility) for p = 0.5 under high stakes.
We do not find a significant difference in probabilistic sensitivity. Similar effects obtain
with an exponential utility function (not shown for parsimony).
We next look at the regression results using log utility instead of power utility, shown
in the lower panel of table 4. We now do not find any significant effect of stakes on the
probability weighting function. This is due to the better fit of the utility function, which
is capable of accommodating the two-fold pattern of risk preferences over outcomes we
15
Figure 3: Probability weighting function for low and high stakes, power utility
found. This is due to the decreasing elasticity of the value function. This also means
that the separability property of prospect theory is restored when using this parametric
setup. Previous results showing violations of this property may thus have been due to
the particular parametric functions chosen, rather than a more fundamental theoretical
failure. From a behavioural point of view, what created the modelling issues can thus be
identified with changes in risk preferences over stakes as initially discussed by Markowitz,
and until recently not incorporated into prospect theory models.
5 Discussion and conclusion
We obtained data varying stakes and probabilities for gains in a symmetric fashion. The
data show a double two-fold pattern of risk preferences. First, they showed risk seeking
for small stakes and risk aversion for large stakes at intermediate probabilities, as well as
a general increase of relative risk aversion with stakes. Second, we found that relative risk
aversion systematically increases in probabilities, thus replicating typical patterns of risk
seeking for small probabilities and risk aversion for large probabilities at different stake
levels. At the same time, modelling approaches relying on a single subjective dimension,
such as Markowitz-expected utility and Dual-expected utility, were clearly shown to be
rejected in the nonparametric analysis.
16
We then moved on to fitting functional forms, and in particular cumulative prospect
theory models transforming both outcomes and probabilities subjectively. We found that
a two-parameter utility function combined with a two-parameter probability weighting
function fit the data significantly better than traditional combinations of two-parameter
weighting functions with exponential or power utility. More interestingly, a single param-
eter logarithmic function provided the best fit to our data. The reason for this superior
fit lies in the ability of this functional form to accommodate two-fold patterns in risk
preferences over outcomes as found in our data. We thereby extended a recent insight
by Scholten and Read (2014) to moderate and large probability levels, and showed that
a combination with a two-parameter weighting function is essential in the presence of
high levels of risk tolerance. This finding is important, inasmuch as high levels of risk
tolerance have been found to be common in developing and middle-income countries
(L’Haridon and Vieider, 2015).
Our data and the novel prospect theory parametrisation just discussed furthermore
allowed us to revisit the debate on whether separability between utility and probability
weighting may be violated in prospect theory models (Fehr-Duda et al., 2010; Hogarth
and Einhorn, 1990). Using traditional modelling frameworks based on power or expo-
nential utility, we were able to replicate findings suggesting that separability between
probabilities and outcomes may be violated when stake variations are large. However,
by providing the flexibility to accommodate changing preference patterns occurring over
outcomes, the logarithmic utility function could handle the stake variations in conjunc-
tion with a two-parameter weighting function, making any previously registered influence
of stake variations on probability weighting disappear. Prospect theory was thus found
to be able to accommodate stake variations for gains (no problems had been found for
losses to start with), thus restoring the validity of the separability assumption using com-
binations of functional forms amenable to capturing a double set of two-fold patterns over
gain—risk seeking for small probabilities followed by risk aversion for large probabilities,
and risk seeking for small outcomes followed by risk aversion for large outcomes.
Other than Scholten and Read (2014), we used a two-parameter probability weight-
ing function. In our setting such a function was indeed found to be absolutely essential,
given the low levels of risk aversion we found, and the observation of the two-fold pat-
tern over outcomes for p = 0.5. Such high levels of risk taking may not be exceptional.
L’Haridon and Vieider (2015) examined prospect theory functionals across 30 countries,
17
and found quite elevated weighting functions to be the norm in developing and middle
income countries. This may indeed also explain the more elevated weighting function
found by Fehr-Duda et al. (2010) in their Chinese data. L’Haridon and Vieider (2015)
found similarly high levels of risk seeking as found in this experiment also in their British
sample. Notice how the somewhat extreme weighting pattern estimated by Scholten and
Read (2014) for probabilities outside of their range of observations (p ≤ 0.1) may be
accounted for by a more elevated probability weighting function, which can be accom-
modated using two-parameter formulations. This is exactly what we find in our data.
The general elevation of the function may be dependent on the general level of risk
aversion observed, and hence on the probabilities for which the two-fold pattern over
outcomes is observed. In our case and given our stimuli, we observed this pattern
most strongly for p = 0.5. This requires an elevated weighting function to a point
where w(0.5) > 0.5, so as to guarantee that w(0.5)v(2x) > v(x) for low stake levels, as
described by Scholten and Read (2014). Prospect theory can easily accommodate such
an elevated weighting function under its modern form. While for instance the original
formulation of prospect theory required subcertainty to explain violations such as the
Allais paradox (see Kahneman and Tversky, 1979, p. 281-282), under the modern version
incorporating rank-dependance (Tversky and Kahneman, 1992), this principle is replaced
by a considerably milder condition that does not rely on w(0.5) < 0.5.9
The fact that such elevated weighting functions have not been observed frequently
before may result from a combination of factors. For one, most Western countries tend to
exhibit higher levels of risk aversion in the outcome range here investigated (L’Haridon
and Vieider, 2015). What is more, the prevalent use of a power utility function—often
in combination with one-parameter weighting functions—may well have hidden such
patterns, at the cost of an inferior fit to the data. Our results show that combining two-
parameter weighting functions with a decreasingly elastic value function provides the best
fit for our data. They also show that previously observed violations of separability can
be resolved in this framework. We thus conclude that prospect theory can accommodate
the two-fold patterns over outcomes and over probabilities for gains we observed in our9Following the example used by Kahneman and Tversky (1979), assume we observe the preferences
2400 � (0.33, 2500; 0.66, 2400; 0) and (2400, 0.34; 0) ≺ (2500, 0.33; 0). Subcertainty would have requiredw(0.34)+w(0.66) < 1. Under CPT, this is replaced by the condition w(0.34)+ [w(0.99)−w(0.66)] < 1.This condition is much milder, and generally compatible with w(0.5) > 0.5. It can also easily becalculated that this condition is fulfilled by the weighting function estimated on our data.
18
experiment when sufficiently flexible functional forms are adopted.
19
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A Full-lenght instructions (order 1)
24
A Classroom Experiment
InstructionsBelow you are asked to indicate your choices between a lottery and different sure amounts of money.Please indicate your choice for each single line of each single task. Your answers to these tasks will bestored anonymously and will not be traced back to you. To make sure of that, each questionnaire hasa number attached to the front, which is the same number reported on the questionnaire itself. Pleasedetach that number now and keep it in a safe place. This will make sure that you can participatein the extraction for real money according to the procedures further described below. Please alsowrite down your student number here: . This will ensure that you can find yourquestionnaire again afterwards, since you will use your own choices in classroom exercises.
In each task, your are asked to choose repeatedly between a sure amount of money and a lottery. Figure1 provides an example of such a task. Option B on the right hand side consists in a lottery offering youa 50% chance of winning £5, or else nothing. Option A consists of sure amounts of money that increaseas you move down the list. Most likely, you will prefer to play the lottery rather than obtaining zero forsure, so that you will cross the right-hand option ‘Choose B’ in the first line. In the last line, you willmost likely prefer the sure amount of £5 to the lottery, which pays £5 if you win but may pay nothingif you lose. If that is the case you will cross the left-hand option ‘Choose A’. In between, it is entirelyup to your preference whether you prefer the sure amount or the lottery. Please indicate a choice foreach line, as it will not be possible to pay you if you have not indicated a preference.
Figure 1: Example of choice task
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £5 or else nothing£1 for sure O O a 50% chance to win £5 or else nothing£2 for sure O O a 50% chance to win £5 or else nothing£3 for sure O O a 50% chance to win £5 or else nothing£4 for sure O O a 50% chance to win £5 or else nothing£5 for sure O O a 50% chance to win £5 or else nothing
In the following pages, you are presented with a total of 21 such choice tasks. The procedure is the samefor each task—please indicate your preference for each line. Pay also close attention when moving fromone task to the next, as both the outcomes and the probabilities change between tasks. Makesure you pay close attention to these details, as your winnings may depend on it if selected for real play.
Once everybody is finished filling in their answers, I will collect all experimental questionnaires. I willthen randomly select two out of the returned questionnaire numbers to be played for realmoney. If you have one of the selected numbers, your decisions will be played out for real money at theend of the class. In particular, one of the 21 choice tasks will be randomly selected to be played. Then,one of the lines in the selected choice task will be randomly selected. If you chose the sure amount forthat line, you will simply be paid that amount. You will then need to sign a receipt, after which youmay leave with the money. If you have selected the lottery, we will play out the lottery according tothe probabilities indicated. You will then be paid the amount you won (if any), and sign a receipt, afterwhich you may leave with the money.
Please do not talk during the experiment! If you have a question at any time, raise your hand andI will come and assist you. Good luck!
1
25
Experimental tasks
Task 1
Option A Choose A Choose B Option B£0 for sure O O a 10% chance to win £10 or else nothing£1 for sure O O a 10% chance to win £10 or else nothing£2 for sure O O a 10% chance to win £10 or else nothing£3 for sure O O a 10% chance to win £10 or else nothing£4 for sure O O a 10% chance to win £10 or else nothing£5 for sure O O a 10% chance to win £10 or else nothing£6 for sure O O a 10% chance to win £10 or else nothing£7 for sure O O a 10% chance to win £10 or else nothing£8 for sure O O a 10% chance to win £10 or else nothing£9 for sure O O a 10% chance to win £10 or else nothing£10 for sure O O a 10% chance to win £10 or else nothing
Task 2
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £10 or else nothing£1 for sure O O a 50% chance to win £10 or else nothing£2 for sure O O a 50% chance to win £10 or else nothing£3 for sure O O a 50% chance to win £10 or else nothing£4 for sure O O a 50% chance to win £10 or else nothing£5 for sure O O a 50% chance to win £10 or else nothing£6 for sure O O a 50% chance to win £10 or else nothing£7 for sure O O a 50% chance to win £10 or else nothing£8 for sure O O a 50% chance to win £10 or else nothing£9 for sure O O a 50% chance to win £10 or else nothing£10 for sure O O a 50% chance to win £10 or else nothing
Task 3
Option A Choose A Choose B Option B£0 for sure O O a 90% chance to win £10 or else nothing£1 for sure O O a 90% chance to win £10 or else nothing£2 for sure O O a 90% chance to win £10 or else nothing£3 for sure O O a 90% chance to win £10 or else nothing£4 for sure O O a 90% chance to win £10 or else nothing£5 for sure O O a 90% chance to win £10 or else nothing£6 for sure O O a 90% chance to win £10 or else nothing£7 for sure O O a 90% chance to win £10 or else nothing£8 for sure O O a 90% chance to win £10 or else nothing£9 for sure O O a 90% chance to win £10 or else nothing£10 for sure O O a 90% chance to win £10 or else nothing
2
26
Task 4
Option A Choose A Choose B Option B£0 for sure O O a 10% chance to win £20 or else nothing£1 for sure O O a 10% chance to win £20 or else nothing£2 for sure O O a 10% chance to win £20 or else nothing£3 for sure O O a 10% chance to win £20 or else nothing£4 for sure O O a 10% chance to win £20 or else nothing£5 for sure O O a 10% chance to win £20 or else nothing£6 for sure O O a 10% chance to win £20 or else nothing£7 for sure O O a 10% chance to win £20 or else nothing£8 for sure O O a 10% chance to win £20 or else nothing£9 for sure O O a 10% chance to win £20 or else nothing£10 for sure O O a 10% chance to win £20 or else nothing£11 for sure O O a 10% chance to win £20 or else nothing£12 for sure O O a 10% chance to win £20 or else nothing£13 for sure O O a 10% chance to win £20 or else nothing£14 for sure O O a 10% chance to win £20 or else nothing£15 for sure O O a 10% chance to win £20 or else nothing£16 for sure O O a 10% chance to win £20 or else nothing£17 for sure O O a 10% chance to win £20 or else nothing£18 for sure O O a 10% chance to win £20 or else nothing£19 for sure O O a 10% chance to win £20 or else nothing£20 for sure O O a 10% chance to win £20 or else nothing
Task 5
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £20 or else nothing£1 for sure O O a 50% chance to win £20 or else nothing£2 for sure O O a 50% chance to win £20 or else nothing£3 for sure O O a 50% chance to win £20 or else nothing£4 for sure O O a 50% chance to win £20 or else nothing£5 for sure O O a 50% chance to win £20 or else nothing£6 for sure O O a 50% chance to win £20 or else nothing£7 for sure O O a 50% chance to win £20 or else nothing£8 for sure O O a 50% chance to win £20 or else nothing£9 for sure O O a 50% chance to win £20 or else nothing£10 for sure O O a 50% chance to win £20 or else nothing£11 for sure O O a 50% chance to win £20 or else nothing£12 for sure O O a 50% chance to win £20 or else nothing£13 for sure O O a 50% chance to win £20 or else nothing£14 for sure O O a 50% chance to win £20 or else nothing£15 for sure O O a 50% chance to win £20 or else nothing£16 for sure O O a 50% chance to win £20 or else nothing£17 for sure O O a 50% chance to win £20 or else nothing£18 for sure O O a 50% chance to win £20 or else nothing£19 for sure O O a 50% chance to win £20 or else nothing£20 for sure O O a 50% chance to win £20 or else nothing
3
27
Task 6
Option A Choose A Choose B Option B£0 for sure O O a 90% chance to win £20 or else nothing£1 for sure O O a 90% chance to win £20 or else nothing£2 for sure O O a 90% chance to win £20 or else nothing£3 for sure O O a 90% chance to win £20 or else nothing£4 for sure O O a 90% chance to win £20 or else nothing£5 for sure O O a 90% chance to win £20 or else nothing£6 for sure O O a 90% chance to win £20 or else nothing£7 for sure O O a 90% chance to win £20 or else nothing£8 for sure O O a 90% chance to win £20 or else nothing£9 for sure O O a 90% chance to win £20 or else nothing£10 for sure O O a 90% chance to win £20 or else nothing£11 for sure O O a 90% chance to win £20 or else nothing£12 for sure O O a 90% chance to win £20 or else nothing£13 for sure O O a 90% chance to win £20 or else nothing£14 for sure O O a 90% chance to win £20 or else nothing£15 for sure O O a 90% chance to win £20 or else nothing£16 for sure O O a 90% chance to win £20 or else nothing£17 for sure O O a 90% chance to win £20 or else nothing£18 for sure O O a 90% chance to win £20 or else nothing£19 for sure O O a 90% chance to win £20 or else nothing£20 for sure O O a 90% chance to win £20 or else nothing
4
28
Task 7
Option A Choose A Choose B Option B£0 for sure O O a 10% chance to win £60 or else nothing£2 for sure O O a 10% chance to win £60 or else nothing£4 for sure O O a 10% chance to win £60 or else nothing£6 for sure O O a 10% chance to win £60 or else nothing£8 for sure O O a 10% chance to win £60 or else nothing£10 for sure O O a 10% chance to win £60 or else nothing£12 for sure O O a 10% chance to win £60 or else nothing£14 for sure O O a 10% chance to win £60 or else nothing£16 for sure O O a 10% chance to win £60 or else nothing£18 for sure O O a 10% chance to win £60 or else nothing£20 for sure O O a 10% chance to win £60 or else nothing£22 for sure O O a 10% chance to win £60 or else nothing£24 for sure O O a 10% chance to win £60 or else nothing£26 for sure O O a 10% chance to win £60 or else nothing£28 for sure O O a 10% chance to win £60 or else nothing£30 for sure O O a 10% chance to win £60 or else nothing£32 for sure O O a 10% chance to win £60 or else nothing£34 for sure O O a 10% chance to win £60 or else nothing£36 for sure O O a 10% chance to win £60 or else nothing£38 for sure O O a 10% chance to win £60 or else nothing£40 for sure O O a 10% chance to win £60 or else nothing£42 for sure O O a 10% chance to win £60 or else nothing£44 for sure O O a 10% chance to win £60 or else nothing£46 for sure O O a 10% chance to win £60 or else nothing£48 for sure O O a 10% chance to win £60 or else nothing£50 for sure O O a 10% chance to win £60 or else nothing£52 for sure O O a 10% chance to win £60 or else nothing£54 for sure O O a 10% chance to win £60 or else nothing£56 for sure O O a 10% chance to win £60 or else nothing£58 for sure O O a 10% chance to win £60 or else nothing£60 for sure O O a 10% chance to win £60 or else nothing
5
29
Task 8
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £60 or else nothing£2 for sure O O a 50% chance to win £60 or else nothing£4 for sure O O a 50% chance to win £60 or else nothing£6 for sure O O a 50% chance to win £60 or else nothing£8 for sure O O a 50% chance to win £60 or else nothing£10 for sure O O a 50% chance to win £60 or else nothing£12 for sure O O a 50% chance to win £60 or else nothing£14 for sure O O a 50% chance to win £60 or else nothing£16 for sure O O a 50% chance to win £60 or else nothing£18 for sure O O a 50% chance to win £60 or else nothing£20 for sure O O a 50% chance to win £60 or else nothing£22 for sure O O a 50% chance to win £60 or else nothing£24 for sure O O a 50% chance to win £60 or else nothing£26 for sure O O a 50% chance to win £60 or else nothing£28 for sure O O a 50% chance to win £60 or else nothing£30 for sure O O a 50% chance to win £60 or else nothing£32 for sure O O a 50% chance to win £60 or else nothing£34 for sure O O a 50% chance to win £60 or else nothing£36 for sure O O a 50% chance to win £60 or else nothing£38 for sure O O a 50% chance to win £60 or else nothing£40 for sure O O a 50% chance to win £60 or else nothing£42 for sure O O a 50% chance to win £60 or else nothing£44 for sure O O a 50% chance to win £60 or else nothing£46 for sure O O a 50% chance to win £60 or else nothing£48 for sure O O a 50% chance to win £60 or else nothing£50 for sure O O a 50% chance to win £60 or else nothing£52 for sure O O a 50% chance to win £60 or else nothing£54 for sure O O a 50% chance to win £60 or else nothing£56 for sure O O a 50% chance to win £60 or else nothing£58 for sure O O a 50% chance to win £60 or else nothing£60 for sure O O a 50% chance to win £60 or else nothing
6
30
Task 9
Option A Choose A Choose B Option B£0 for sure O O a 90% chance to win £60 or else nothing£2 for sure O O a 90% chance to win £60 or else nothing£4 for sure O O a 90% chance to win £60 or else nothing£6 for sure O O a 90% chance to win £60 or else nothing£8 for sure O O a 90% chance to win £60 or else nothing£10 for sure O O a 90% chance to win £60 or else nothing£12 for sure O O a 90% chance to win £60 or else nothing£14 for sure O O a 90% chance to win £60 or else nothing£16 for sure O O a 90% chance to win £60 or else nothing£18 for sure O O a 90% chance to win £60 or else nothing£20 for sure O O a 90% chance to win £60 or else nothing£22 for sure O O a 90% chance to win £60 or else nothing£24 for sure O O a 90% chance to win £60 or else nothing£26 for sure O O a 90% chance to win £60 or else nothing£28 for sure O O a 90% chance to win £60 or else nothing£30 for sure O O a 90% chance to win £60 or else nothing£32 for sure O O a 90% chance to win £60 or else nothing£34 for sure O O a 90% chance to win £60 or else nothing£36 for sure O O a 90% chance to win £60 or else nothing£38 for sure O O a 90% chance to win £60 or else nothing£40 for sure O O a 90% chance to win £60 or else nothing£42 for sure O O a 90% chance to win £60 or else nothing£44 for sure O O a 90% chance to win £60 or else nothing£46 for sure O O a 90% chance to win £60 or else nothing£48 for sure O O a 90% chance to win £60 or else nothing£50 for sure O O a 90% chance to win £60 or else nothing£52 for sure O O a 90% chance to win £60 or else nothing£54 for sure O O a 90% chance to win £60 or else nothing£56 for sure O O a 90% chance to win £60 or else nothing£58 for sure O O a 90% chance to win £60 or else nothing£60 for sure O O a 90% chance to win £60 or else nothing
7
31
Task 10
Option A Choose A Choose B Option B£0 for sure O O a 10% chance to win £100 or else nothing£4 for sure O O a 10% chance to win £100 or else nothing£8 for sure O O a 10% chance to win £100 or else nothing£12 for sure O O a 10% chance to win £100 or else nothing£16 for sure O O a 10% chance to win £100 or else nothing£20 for sure O O a 10% chance to win £100 or else nothing£24 for sure O O a 10% chance to win £100 or else nothing£28 for sure O O a 10% chance to win £100 or else nothing£32 for sure O O a 10% chance to win £100 or else nothing£36 for sure O O a 10% chance to win £100 or else nothing£40 for sure O O a 10% chance to win £100 or else nothing£44 for sure O O a 10% chance to win £100 or else nothing£48 for sure O O a 10% chance to win £100 or else nothing£52 for sure O O a 10% chance to win £100 or else nothing£56 for sure O O a 10% chance to win £100 or else nothing£60 for sure O O a 10% chance to win £100 or else nothing£64 for sure O O a 10% chance to win £100 or else nothing£68 for sure O O a 10% chance to win £100 or else nothing£72 for sure O O a 10% chance to win £100 or else nothing£76 for sure O O a 10% chance to win £100 or else nothing£80 for sure O O a 10% chance to win £100 or else nothing£84 for sure O O a 10% chance to win £100 or else nothing£88 for sure O O a 10% chance to win £100 or else nothing£92 for sure O O a 10% chance to win £100 or else nothing£96 for sure O O a 10% chance to win £100 or else nothing£100 for sure O O a 10% chance to win £100 or else nothing
8
32
Task 11
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £100 or else nothing£4 for sure O O a 50% chance to win £100 or else nothing£8 for sure O O a 50% chance to win £100 or else nothing£12 for sure O O a 50% chance to win £100 or else nothing£16 for sure O O a 50% chance to win £100 or else nothing£20 for sure O O a 50% chance to win £100 or else nothing£24 for sure O O a 50% chance to win £100 or else nothing£28 for sure O O a 50% chance to win £100 or else nothing£32 for sure O O a 50% chance to win £100 or else nothing£36 for sure O O a 50% chance to win £100 or else nothing£40 for sure O O a 50% chance to win £100 or else nothing£44 for sure O O a 50% chance to win £100 or else nothing£48 for sure O O a 50% chance to win £100 or else nothing£52 for sure O O a 50% chance to win £100 or else nothing£56 for sure O O a 50% chance to win £100 or else nothing£60 for sure O O a 50% chance to win £100 or else nothing£64 for sure O O a 50% chance to win £100 or else nothing£68 for sure O O a 50% chance to win £100 or else nothing£72 for sure O O a 50% chance to win £100 or else nothing£76 for sure O O a 50% chance to win £100 or else nothing£80 for sure O O a 50% chance to win £100 or else nothing£84 for sure O O a 50% chance to win £100 or else nothing£88 for sure O O a 50% chance to win £100 or else nothing£92 for sure O O a 50% chance to win £100 or else nothing£96 for sure O O a 50% chance to win £100 or else nothing£100 for sure O O a 50% chance to win £100 or else nothing
9
33
Task 12
Option A Choose A Choose B Option B£0 for sure O O a 90% chance to win £100 or else nothing£4 for sure O O a 90% chance to win £100 or else nothing£8 for sure O O a 90% chance to win £100 or else nothing£12 for sure O O a 90% chance to win £100 or else nothing£16 for sure O O a 90% chance to win £100 or else nothing£20 for sure O O a 90% chance to win £100 or else nothing£24 for sure O O a 90% chance to win £100 or else nothing£28 for sure O O a 90% chance to win £100 or else nothing£32 for sure O O a 90% chance to win £100 or else nothing£36 for sure O O a 90% chance to win £100 or else nothing£40 for sure O O a 90% chance to win £100 or else nothing£44 for sure O O a 90% chance to win £100 or else nothing£48 for sure O O a 90% chance to win £100 or else nothing£52 for sure O O a 90% chance to win £100 or else nothing£56 for sure O O a 90% chance to win £100 or else nothing£60 for sure O O a 90% chance to win £100 or else nothing£64 for sure O O a 90% chance to win £100 or else nothing£68 for sure O O a 90% chance to win £100 or else nothing£72 for sure O O a 90% chance to win £100 or else nothing£76 for sure O O a 90% chance to win £100 or else nothing£80 for sure O O a 90% chance to win £100 or else nothing£84 for sure O O a 90% chance to win £100 or else nothing£88 for sure O O a 90% chance to win £100 or else nothing£92 for sure O O a 90% chance to win £100 or else nothing£96 for sure O O a 90% chance to win £100 or else nothing£100 for sure O O a 90% chance to win £100 or else nothing
10
34
Task 13
Option A Choose A Choose B Option B£0 for sure O O a 10% chance to win £200 or else nothing£4 for sure O O a 10% chance to win £200 or else nothing£8 for sure O O a 10% chance to win £200 or else nothing£12 for sure O O a 10% chance to win £200 or else nothing£16 for sure O O a 10% chance to win £200 or else nothing£20 for sure O O a 10% chance to win £200 or else nothing£24 for sure O O a 10% chance to win £200 or else nothing£28 for sure O O a 10% chance to win £200 or else nothing£32 for sure O O a 10% chance to win £200 or else nothing£36 for sure O O a 10% chance to win £200 or else nothing£40 for sure O O a 10% chance to win £200 or else nothing£44 for sure O O a 10% chance to win £200 or else nothing£48 for sure O O a 10% chance to win £200 or else nothing£52 for sure O O a 10% chance to win £200 or else nothing£56 for sure O O a 10% chance to win £200 or else nothing£60 for sure O O a 10% chance to win £200 or else nothing£64 for sure O O a 10% chance to win £200 or else nothing£68 for sure O O a 10% chance to win £200 or else nothing£72 for sure O O a 10% chance to win £200 or else nothing£76 for sure O O a 10% chance to win £200 or else nothing£80 for sure O O a 10% chance to win £200 or else nothing£84 for sure O O a 10% chance to win £200 or else nothing£88 for sure O O a 10% chance to win £200 or else nothing£92 for sure O O a 10% chance to win £200 or else nothing£96 for sure O O a 10% chance to win £200 or else nothing£100 for sure O O a 10% chance to win £200 or else nothing£104 for sure O O a 10% chance to win £200 or else nothing£108 for sure O O a 10% chance to win £200 or else nothing£112 for sure O O a 10% chance to win £200 or else nothing£116 for sure O O a 10% chance to win £200 or else nothing£120 for sure O O a 10% chance to win £200 or else nothing£124 for sure O O a 10% chance to win £200 or else nothing£128 for sure O O a 10% chance to win £200 or else nothing£132 for sure O O a 10% chance to win £200 or else nothing£136 for sure O O a 10% chance to win £200 or else nothing£140 for sure O O a 10% chance to win £200 or else nothing£144 for sure O O a 10% chance to win £200 or else nothing£148 for sure O O a 10% chance to win £200 or else nothing£152 for sure O O a 10% chance to win £200 or else nothing£156 for sure O O a 10% chance to win £200 or else nothing£160 for sure O O a 10% chance to win £200 or else nothing£164 for sure O O a 10% chance to win £200 or else nothing£168 for sure O O a 10% chance to win £200 or else nothing£172 for sure O O a 10% chance to win £200 or else nothing£176 for sure O O a 10% chance to win £200 or else nothing£180 for sure O O a 10% chance to win £200 or else nothing£184 for sure O O a 10% chance to win £200 or else nothing£188 for sure O O a 10% chance to win £200 or else nothing£192 for sure O O a 10% chance to win £200 or else nothing£196 for sure O O a 10% chance to win £200 or else nothing£200 for sure O O a 10% chance to win £200 or else nothing
11
35
Task 14
Option A Choose A Choose B Option B£0 for sure O O a 50% chance to win £200 or else nothing£4 for sure O O a 50% chance to win £200 or else nothing£8 for sure O O a 50% chance to win £200 or else nothing£12 for sure O O a 50% chance to win £200 or else nothing£16 for sure O O a 50% chance to win £200 or else nothing£20 for sure O O a 50% chance to win £200 or else nothing£24 for sure O O a 50% chance to win £200 or else nothing£28 for sure O O a 50% chance to win £200 or else nothing£32 for sure O O a 50% chance to win £200 or else nothing£36 for sure O O a 50% chance to win £200 or else nothing£40 for sure O O a 50% chance to win £200 or else nothing£44 for sure O O a 50% chance to win £200 or else nothing£48 for sure O O a 50% chance to win £200 or else nothing£52 for sure O O a 50% chance to win £200 or else nothing£56 for sure O O a 50% chance to win £200 or else nothing£60 for sure O O a 50% chance to win £200 or else nothing£64 for sure O O a 50% chance to win £200 or else nothing£68 for sure O O a 50% chance to win £200 or else nothing£72 for sure O O a 50% chance to win £200 or else nothing£76 for sure O O a 50% chance to win £200 or else nothing£80 for sure O O a 50% chance to win £200 or else nothing£84 for sure O O a 50% chance to win £200 or else nothing£88 for sure O O a 50% chance to win £200 or else nothing£92 for sure O O a 50% chance to win £200 or else nothing£96 for sure O O a 50% chance to win £200 or else nothing£100 for sure O O a 50% chance to win £200 or else nothing£104 for sure O O a 50% chance to win £200 or else nothing£108 for sure O O a 50% chance to win £200 or else nothing£112 for sure O O a 50% chance to win £200 or else nothing£116 for sure O O a 50% chance to win £200 or else nothing£120 for sure O O a 50% chance to win £200 or else nothing£124 for sure O O a 50% chance to win £200 or else nothing£128 for sure O O a 50% chance to win £200 or else nothing£132 for sure O O a 50% chance to win £200 or else nothing£136 for sure O O a 50% chance to win £200 or else nothing£140 for sure O O a 50% chance to win £200 or else nothing£144 for sure O O a 50% chance to win £200 or else nothing£148 for sure O O a 50% chance to win £200 or else nothing£152 for sure O O a 50% chance to win £200 or else nothing£156 for sure O O a 50% chance to win £200 or else nothing£160 for sure O O a 50% chance to win £200 or else nothing£164 for sure O O a 50% chance to win £200 or else nothing£168 for sure O O a 50% chance to win £200 or else nothing£172 for sure O O a 50% chance to win £200 or else nothing£176 for sure O O a 50% chance to win £200 or else nothing£180 for sure O O a 50% chance to win £200 or else nothing£184 for sure O O a 50% chance to win £200 or else nothing£188 for sure O O a 50% chance to win £200 or else nothing£192 for sure O O a 50% chance to win £200 or else nothing£196 for sure O O a 50% chance to win £200 or else nothing£200 for sure O O a 50% chance to win £200 or else nothing
12
36
Task 15
Option A Choose A Choose B Option B£0 for sure O O a 90% chance to win £200 or else nothing£4 for sure O O a 90% chance to win £200 or else nothing£8 for sure O O a 90% chance to win £200 or else nothing£12 for sure O O a 90% chance to win £200 or else nothing£16 for sure O O a 90% chance to win £200 or else nothing£20 for sure O O a 90% chance to win £200 or else nothing£24 for sure O O a 90% chance to win £200 or else nothing£28 for sure O O a 90% chance to win £200 or else nothing£32 for sure O O a 90% chance to win £200 or else nothing£36 for sure O O a 90% chance to win £200 or else nothing£40 for sure O O a 90% chance to win £200 or else nothing£44 for sure O O a 90% chance to win £200 or else nothing£48 for sure O O a 90% chance to win £200 or else nothing£52 for sure O O a 90% chance to win £200 or else nothing£56 for sure O O a 90% chance to win £200 or else nothing£60 for sure O O a 90% chance to win £200 or else nothing£64 for sure O O a 90% chance to win £200 or else nothing£68 for sure O O a 90% chance to win £200 or else nothing£72 for sure O O a 90% chance to win £200 or else nothing£76 for sure O O a 90% chance to win £200 or else nothing£80 for sure O O a 90% chance to win £200 or else nothing£84 for sure O O a 90% chance to win £200 or else nothing£88 for sure O O a 90% chance to win £200 or else nothing£92 for sure O O a 90% chance to win £200 or else nothing£96 for sure O O a 90% chance to win £200 or else nothing£100 for sure O O a 90% chance to win £200 or else nothing£104 for sure O O a 90% chance to win £200 or else nothing£108 for sure O O a 90% chance to win £200 or else nothing£112 for sure O O a 90% chance to win £200 or else nothing£116 for sure O O a 90% chance to win £200 or else nothing£120 for sure O O a 90% chance to win £200 or else nothing£124 for sure O O a 90% chance to win £200 or else nothing£128 for sure O O a 90% chance to win £200 or else nothing£132 for sure O O a 90% chance to win £200 or else nothing£136 for sure O O a 90% chance to win £200 or else nothing£140 for sure O O a 90% chance to win £200 or else nothing£144 for sure O O a 90% chance to win £200 or else nothing£148 for sure O O a 90% chance to win £200 or else nothing£152 for sure O O a 90% chance to win £200 or else nothing£156 for sure O O a 90% chance to win £200 or else nothing£160 for sure O O a 90% chance to win £200 or else nothing£164 for sure O O a 90% chance to win £200 or else nothing£168 for sure O O a 90% chance to win £200 or else nothing£172 for sure O O a 90% chance to win £200 or else nothing£176 for sure O O a 90% chance to win £200 or else nothing£180 for sure O O a 90% chance to win £200 or else nothing£184 for sure O O a 90% chance to win £200 or else nothing£188 for sure O O a 90% chance to win £200 or else nothing£192 for sure O O a 90% chance to win £200 or else nothing£196 for sure O O a 90% chance to win £200 or else nothing£200 for sure O O a 90% chance to win £200 or else nothing
13
37
Task 16
Option A Choose A Choose B Option B£20 for sure O O a 10% chance to win £200 or else £20£24 for sure O O a 10% chance to win £200 or else £20£28 for sure O O a 10% chance to win £200 or else £20£32 for sure O O a 10% chance to win £200 or else £20£36 for sure O O a 10% chance to win £200 or else £20£40 for sure O O a 10% chance to win £200 or else £20£44 for sure O O a 10% chance to win £200 or else £20£48 for sure O O a 10% chance to win £200 or else £20£52 for sure O O a 10% chance to win £200 or else £20£56 for sure O O a 10% chance to win £200 or else £20£60 for sure O O a 10% chance to win £200 or else £20£64 for sure O O a 10% chance to win £200 or else £20£68 for sure O O a 10% chance to win £200 or else £20£72 for sure O O a 10% chance to win £200 or else £20£76 for sure O O a 10% chance to win £200 or else £20£80 for sure O O a 10% chance to win £200 or else £20£84 for sure O O a 10% chance to win £200 or else £20£88 for sure O O a 10% chance to win £200 or else £20£92 for sure O O a 10% chance to win £200 or else £20£96 for sure O O a 10% chance to win £200 or else £20£100 for sure O O a 10% chance to win £200 or else £20£104 for sure O O a 10% chance to win £200 or else £20£108 for sure O O a 10% chance to win £200 or else £20£112 for sure O O a 10% chance to win £200 or else £20£116 for sure O O a 10% chance to win £200 or else £20£120 for sure O O a 10% chance to win £200 or else £20£124 for sure O O a 10% chance to win £200 or else £20£128 for sure O O a 10% chance to win £200 or else £20£132 for sure O O a 10% chance to win £200 or else £20£136 for sure O O a 10% chance to win £200 or else £20£140 for sure O O a 10% chance to win £200 or else £20£144 for sure O O a 10% chance to win £200 or else £20£148 for sure O O a 10% chance to win £200 or else £20£152 for sure O O a 10% chance to win £200 or else £20£156 for sure O O a 10% chance to win £200 or else £20£160 for sure O O a 10% chance to win £200 or else £20£164 for sure O O a 10% chance to win £200 or else £20£168 for sure O O a 10% chance to win £200 or else £20£172 for sure O O a 10% chance to win £200 or else £20£176 for sure O O a 10% chance to win £200 or else £20£180 for sure O O a 10% chance to win £200 or else £20£184 for sure O O a 10% chance to win £200 or else £20£188 for sure O O a 10% chance to win £200 or else £20£192 for sure O O a 10% chance to win £200 or else £20£196 for sure O O a 10% chance to win £200 or else £20£200 for sure O O a 10% chance to win £200 or else £20
14
38
Task 17
Option A Choose A Choose B Option B£20 for sure O O a 50% chance to win £200 or else £20£24 for sure O O a 50% chance to win £200 or else £20£28 for sure O O a 50% chance to win £200 or else £20£32 for sure O O a 50% chance to win £200 or else £20£36 for sure O O a 50% chance to win £200 or else £20£40 for sure O O a 50% chance to win £200 or else £20£44 for sure O O a 50% chance to win £200 or else £20£48 for sure O O a 50% chance to win £200 or else £20£52 for sure O O a 50% chance to win £200 or else £20£56 for sure O O a 50% chance to win £200 or else £20£60 for sure O O a 50% chance to win £200 or else £20£64 for sure O O a 50% chance to win £200 or else £20£68 for sure O O a 50% chance to win £200 or else £20£72 for sure O O a 50% chance to win £200 or else £20£76 for sure O O a 50% chance to win £200 or else £20£80 for sure O O a 50% chance to win £200 or else £20£84 for sure O O a 50% chance to win £200 or else £20£88 for sure O O a 50% chance to win £200 or else £20£92 for sure O O a 50% chance to win £200 or else £20£96 for sure O O a 50% chance to win £200 or else £20£100 for sure O O a 50% chance to win £200 or else £20£104 for sure O O a 50% chance to win £200 or else £20£108 for sure O O a 50% chance to win £200 or else £20£112 for sure O O a 50% chance to win £200 or else £20£116 for sure O O a 50% chance to win £200 or else £20£120 for sure O O a 50% chance to win £200 or else £20£124 for sure O O a 50% chance to win £200 or else £20£128 for sure O O a 50% chance to win £200 or else £20£132 for sure O O a 50% chance to win £200 or else £20£136 for sure O O a 50% chance to win £200 or else £20£140 for sure O O a 50% chance to win £200 or else £20£144 for sure O O a 50% chance to win £200 or else £20£148 for sure O O a 50% chance to win £200 or else £20£152 for sure O O a 50% chance to win £200 or else £20£156 for sure O O a 50% chance to win £200 or else £20£160 for sure O O a 50% chance to win £200 or else £20£164 for sure O O a 50% chance to win £200 or else £20£168 for sure O O a 50% chance to win £200 or else £20£172 for sure O O a 50% chance to win £200 or else £20£176 for sure O O a 50% chance to win £200 or else £20£180 for sure O O a 50% chance to win £200 or else £20£184 for sure O O a 50% chance to win £200 or else £20£188 for sure O O a 50% chance to win £200 or else £20£192 for sure O O a 50% chance to win £200 or else £20£196 for sure O O a 50% chance to win £200 or else £20£200 for sure O O a 50% chance to win £200 or else £20
15
39
Task 18
Option A Choose A Choose B Option B£20 for sure O O a 90% chance to win £200 or else £20£24 for sure O O a 90% chance to win £200 or else £20£28 for sure O O a 90% chance to win £200 or else £20£32 for sure O O a 90% chance to win £200 or else £20£36 for sure O O a 90% chance to win £200 or else £20£40 for sure O O a 90% chance to win £200 or else £20£44 for sure O O a 90% chance to win £200 or else £20£48 for sure O O a 90% chance to win £200 or else £20£52 for sure O O a 90% chance to win £200 or else £20£56 for sure O O a 90% chance to win £200 or else £20£60 for sure O O a 90% chance to win £200 or else £20£64 for sure O O a 90% chance to win £200 or else £20£68 for sure O O a 90% chance to win £200 or else £20£72 for sure O O a 90% chance to win £200 or else £20£76 for sure O O a 90% chance to win £200 or else £20£80 for sure O O a 90% chance to win £200 or else £20£84 for sure O O a 90% chance to win £200 or else £20£88 for sure O O a 90% chance to win £200 or else £20£92 for sure O O a 90% chance to win £200 or else £20£96 for sure O O a 90% chance to win £200 or else £20£100 for sure O O a 90% chance to win £200 or else £20£104 for sure O O a 90% chance to win £200 or else £20£108 for sure O O a 90% chance to win £200 or else £20£112 for sure O O a 90% chance to win £200 or else £20£116 for sure O O a 90% chance to win £200 or else £20£120 for sure O O a 90% chance to win £200 or else £20£124 for sure O O a 90% chance to win £200 or else £20£128 for sure O O a 90% chance to win £200 or else £20£132 for sure O O a 90% chance to win £200 or else £20£136 for sure O O a 90% chance to win £200 or else £20£140 for sure O O a 90% chance to win £200 or else £20£144 for sure O O a 90% chance to win £200 or else £20£148 for sure O O a 90% chance to win £200 or else £20£152 for sure O O a 90% chance to win £200 or else £20£156 for sure O O a 90% chance to win £200 or else £20£160 for sure O O a 90% chance to win £200 or else £20£164 for sure O O a 90% chance to win £200 or else £20£168 for sure O O a 90% chance to win £200 or else £20£172 for sure O O a 90% chance to win £200 or else £20£176 for sure O O a 90% chance to win £200 or else £20£180 for sure O O a 90% chance to win £200 or else £20£184 for sure O O a 90% chance to win £200 or else £20£188 for sure O O a 90% chance to win £200 or else £20£192 for sure O O a 90% chance to win £200 or else £20£196 for sure O O a 90% chance to win £200 or else £20£200 for sure O O a 90% chance to win £200 or else £20
16
40
Task 19
Option A Choose A Choose B Option B£60 for sure O O a 10% chance to win £200 or else £60£64 for sure O O a 10% chance to win £200 or else £60£68 for sure O O a 10% chance to win £200 or else £60£72 for sure O O a 10% chance to win £200 or else £60£76 for sure O O a 10% chance to win £200 or else £60£80 for sure O O a 10% chance to win £200 or else £60£84 for sure O O a 10% chance to win £200 or else £60£88 for sure O O a 10% chance to win £200 or else £60£92 for sure O O a 10% chance to win £200 or else £60£96 for sure O O a 10% chance to win £200 or else £60£100 for sure O O a 10% chance to win £200 or else £60£104 for sure O O a 10% chance to win £200 or else £60£108 for sure O O a 10% chance to win £200 or else £60£112 for sure O O a 10% chance to win £200 or else £60£116 for sure O O a 10% chance to win £200 or else £60£120 for sure O O a 10% chance to win £200 or else £60£124 for sure O O a 10% chance to win £200 or else £60£128 for sure O O a 10% chance to win £200 or else £60£132 for sure O O a 10% chance to win £200 or else £60£136 for sure O O a 10% chance to win £200 or else £60£140 for sure O O a 10% chance to win £200 or else £60£144 for sure O O a 10% chance to win £200 or else £60£148 for sure O O a 10% chance to win £200 or else £60£152 for sure O O a 10% chance to win £200 or else £60£156 for sure O O a 10% chance to win £200 or else £60£160 for sure O O a 10% chance to win £200 or else £60£164 for sure O O a 10% chance to win £200 or else £60£168 for sure O O a 10% chance to win £200 or else £60£172 for sure O O a 10% chance to win £200 or else £60£176 for sure O O a 10% chance to win £200 or else £60£180 for sure O O a 10% chance to win £200 or else £60£184 for sure O O a 10% chance to win £200 or else £60£188 for sure O O a 10% chance to win £200 or else £60£192 for sure O O a 10% chance to win £200 or else £60£196 for sure O O a 10% chance to win £200 or else £60£200 for sure O O a 10% chance to win £200 or else £60
17
41
Task 20
Option A Choose A Choose B Option B£60 for sure O O a 50% chance to win £200 or else £60£64 for sure O O a 50% chance to win £200 or else £60£68 for sure O O a 50% chance to win £200 or else £60£72 for sure O O a 50% chance to win £200 or else £60£76 for sure O O a 50% chance to win £200 or else £60£80 for sure O O a 50% chance to win £200 or else £60£84 for sure O O a 50% chance to win £200 or else £60£88 for sure O O a 50% chance to win £200 or else £60£92 for sure O O a 50% chance to win £200 or else £60£96 for sure O O a 50% chance to win £200 or else £60£100 for sure O O a 50% chance to win £200 or else £60£104 for sure O O a 50% chance to win £200 or else £60£108 for sure O O a 50% chance to win £200 or else £60£112 for sure O O a 50% chance to win £200 or else £60£116 for sure O O a 50% chance to win £200 or else £60£120 for sure O O a 50% chance to win £200 or else £60£124 for sure O O a 50% chance to win £200 or else £60£128 for sure O O a 50% chance to win £200 or else £60£132 for sure O O a 50% chance to win £200 or else £60£136 for sure O O a 50% chance to win £200 or else £60£140 for sure O O a 50% chance to win £200 or else £60£144 for sure O O a 50% chance to win £200 or else £60£148 for sure O O a 50% chance to win £200 or else £60£152 for sure O O a 50% chance to win £200 or else £60£156 for sure O O a 50% chance to win £200 or else £60£160 for sure O O a 50% chance to win £200 or else £60£164 for sure O O a 50% chance to win £200 or else £60£168 for sure O O a 50% chance to win £200 or else £60£172 for sure O O a 50% chance to win £200 or else £60£176 for sure O O a 50% chance to win £200 or else £60£180 for sure O O a 50% chance to win £200 or else £60£184 for sure O O a 50% chance to win £200 or else £60£188 for sure O O a 50% chance to win £200 or else £60£192 for sure O O a 50% chance to win £200 or else £60£196 for sure O O a 50% chance to win £200 or else £60£200 for sure O O a 50% chance to win £200 or else £60
18
42
Task 21
Option A Choose A Choose B Option B£60 for sure O O a 90% chance to win £200 or else £60£64 for sure O O a 90% chance to win £200 or else £60£68 for sure O O a 90% chance to win £200 or else £60£72 for sure O O a 90% chance to win £200 or else £60£76 for sure O O a 90% chance to win £200 or else £60£80 for sure O O a 90% chance to win £200 or else £60£84 for sure O O a 90% chance to win £200 or else £60£88 for sure O O a 90% chance to win £200 or else £60£92 for sure O O a 90% chance to win £200 or else £60£96 for sure O O a 90% chance to win £200 or else £60£100 for sure O O a 90% chance to win £200 or else £60£104 for sure O O a 90% chance to win £200 or else £60£108 for sure O O a 90% chance to win £200 or else £60£112 for sure O O a 90% chance to win £200 or else £60£116 for sure O O a 90% chance to win £200 or else £60£120 for sure O O a 90% chance to win £200 or else £60£124 for sure O O a 90% chance to win £200 or else £60£128 for sure O O a 90% chance to win £200 or else £60£132 for sure O O a 90% chance to win £200 or else £60£136 for sure O O a 90% chance to win £200 or else £60£140 for sure O O a 90% chance to win £200 or else £60£144 for sure O O a 90% chance to win £200 or else £60£148 for sure O O a 90% chance to win £200 or else £60£152 for sure O O a 90% chance to win £200 or else £60£156 for sure O O a 90% chance to win £200 or else £60£160 for sure O O a 90% chance to win £200 or else £60£164 for sure O O a 90% chance to win £200 or else £60£168 for sure O O a 90% chance to win £200 or else £60£172 for sure O O a 90% chance to win £200 or else £60£176 for sure O O a 90% chance to win £200 or else £60£180 for sure O O a 90% chance to win £200 or else £60£184 for sure O O a 90% chance to win £200 or else £60£188 for sure O O a 90% chance to win £200 or else £60£192 for sure O O a 90% chance to win £200 or else £60£196 for sure O O a 90% chance to win £200 or else £60£200 for sure O O a 90% chance to win £200 or else £60
19
43