Accommodating Stake Effects under Prospect Theory - University of Reading · 2016. 4. 18. · †Corresponding author: Department of Economics, University of Reading, UK; email: [email protected];Tel

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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

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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

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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.

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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.

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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.

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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).

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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.

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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,

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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

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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.

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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.

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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

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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

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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


Task 3


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


3

27

Task 6


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


6

30

Task 9


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


9

33

Task 12


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


12

36

Task 15


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


15

39

Task 18


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


18

42

Task 21


19

43

Documents

Accommodating Stake Effects under Prospect Theory - University of Reading · 2016. 4. 18. · †Corresponding author: Department of Economics, University of Reading, UK; email: [email protected];Tel