International Journal of Economic Issues, Vol. 4, No. 2 (July-December, 2011) : 171-189
© International Science Press
THE VALUATION OF SAFETY WITH PROBABILITY
DISTORTION IN MIND: THEORY AND METHODS
SUE CHILTON & HUGH METCALF
Newcastle University
WEI PANG
Kingston University
This paper examines the role of probability distortions in the elicitation of a value of
statistical life (VSL) by comparing the results derived using the ‘Tradeoff’ methodology
with those from the traditional Lottery Equivalent (LE) methodology. We do not find
the traditional Lottery Equivalent (LE) method under Rank Dependant Expected Utility
(RDEU) criterion is a significant improvement of the LE method under Expected Utility
criterion. The implications of the results are twofold. Firstly, it suggests that probability
distortions are not the sole causal factor of observed biases in the LE method. Secondly,
the probability weighting functions with parameter values that are used in the literature
do not improve significantly the elicitation results of the LE approach, at least within
the probability range of 0.3 to 0.8.
JEL: D03, D61, D81, J17.
Keywords: Probability Distortion, Rand-dependent Utility, Trade-Off, Lottery
Equivalence.
1. INTRODUCTION
A Value of Statistical Life (VSL) is the monetary value attributable to a reduction in
the risk of death afforded to society. It is the standard measure used in all UK
governmental cost-benefit analyses of safety projects. The VSL is generally elicited
from surveys of the public and its accuracy requires people to understand and respond
to changes in small probabilities. Unfortunately, there is a growing body of empirical
evidence — both survey and experimental – suggesting that this does not occur. Instead,
people appear susceptible to so-called Probability Distortions (PD) where, in general
they are more sensitive to changes in extremely high/low probabilities than to changes
in intermediate value of probabilities.
Consider an example. Individuals are likely to view a 2% probability of death as
twice as high as a 1% probability of death. However, they do not regard a 10%
probability of death as much different to a 9% probability of death. In addition,
individuals are likely to welcome a policy effort which reduces the probability of death
caused by one hazard from 10% to 5% but maybe indifferent to the policy which reduce
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the risk from 60% to 55%. In more general situations, a probability change for an event
from 0.01 to 0.11 will be more likely to trigger responsive action than a probability
change from 0.5 to 0.6.
Therefore, if the VSL elicitation approach is subject to such probability distortions,
the resulting VSL may be under- or over-estimated depending on the nature of the
distortion, with the consequence that any subsequent cost-benefit analysis will be
biased. Chilton and Spencer (2001) confirmed that inconsistent VSL results were
obtained from using different elicition approaches and suggested that Prospect Theory
could help explain this.
The underlining assumptions in eliciting VSL are that people evaluate probabilities
linearly and their choice behaviours are expressed as Expected utility (EU).
Unfortunately, a number of empirical observations and evidence suggest that people
violate this assumption in systematic ways (Camerer & Ho 1994, C Starmer 2000). In
response to the empirical violations of EU, alternative decision theories have been
developed. Among these so-called non-EU theories, the most well-known is Prospect
theory (PT) (Kahneman and Tversky (1979). Two important features of PT helped in
explaining successfully some major deviations from EU such as the certainty effect.
One is the nonlinear transformation of probabilities into a probability weighting
function, and the other is the loss aversion that people tend to overweight perceived
losses compared to perceived gains 1. However, the approach received criticism
because of its violation of stochastic dominance. This inspired Quiggin (1982) to
develop Rank Dependent Expected Utility approach (RDEU) which applied the
probability weighting function proposed in PT without violating stochastic
dominance. This rank-dependent approach has now become the basis of the most
popular stream within non-expected utility (Wakker (1994)). Tversky and Kahneman
(1992) adopted the rank-dependent approach for transforming probabilities and
developed a new version of PT, called Cumulative Prospect Theory (CPT) which
satisfies stochastic dominance. Fennema and Wakker (1997) showed that CPT not
only solves the theoretical problems in PT but also gives different experimental
predictions.
With this background in mind, this paper focuses on the role of probability
distortion (PD) in eliciting VSL and aims to investigate two issues. First, we examine
whether PD is the cause of inconsistent responses repeatedly found in the elicitation
of the VSL in the safety domain, particularly in methodologies that attempt to reduce
the problem. One example of such an inconsistency is scope insensitivity whereby
respondents’ elicited values are insensitive to the size of the risk reduction, i.e., the
changes of probability. Secondly, we apply the rank-dependent methodology of
probability transformation to a well established and improved VSL eliciting approach,
the Lottery Equivalent (LE) approach. The LE approach is underpinned by conventional
Expected Utility theory and would thus be subject to PD effects if the bias is present
during the elicitation process. In comparison, rank dependent probability
transformation incorporates the effects of PD into the modelling of elicited preferences.
Thus, we are able to explore whether the application of rank-dependence preference
can improve the estimates on the value of risk reduction.
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A new framework for VSL elicitation is used, and crucial to this is,an
identification and subsequent accommodation of PD in responses to the valuation
of risk reductions. This is accomplished by setting up an empirical experiment with
two treatments. The first treatment adapts the TradeOff (TO) methodology proposed
by Wakker and Deneffe (1996). The TO approach has been shown to be robust to
probability distortion and can provide a benchmark VSL to study the effects of PD.
In this treatment, our focus is on the choice between health states as opposed to risk
reductions (probability changes). This helps to reduce the impact of PD. The literature
showed that perception around probability 0.5 is least biased 2. Thus we use a
probability value of 0.5 in the experimental protocol for all choices faced by
respondents in this treatment. The second treatment comprises experimental choices
among risk reductions. We use the traditional LE approach which involves a choice
of probability from the whole unit interval of probability. Many studies reported
that the results obtained under LE approach at probabilities 0.5 and 0.75 were not
significantly different3 (see McCord and De Neufville (1986), Law, Pathak & McCord
(1998)). However, in order to confirm this conclusion, experiments, an exploration
of all regions of the probability unit interval is needed. This is important because
probabilities used in safety questions are mostly located at the extreme ends of the
probability unit interval. The full coverage of probability unit interval in this paper
therefore complements the existing literature.
To identify the PD effects, we firstly compare the results obtained from the
TO treatment and LE treatment. If there is divergence between two sets of results,
we apply rank-dependent expected utility approach to the calculations of LE results.
We then compare the RDEU-value of LE results with the results from TO treatment 4.
Because the rank-dependent probability transformation method in RDEU corrects
the probability distortion effects embedded in EU preferences, the improvement
in the discrepancies between RDEU-LE and TO results compared to LE and TO
results will verify the presence of PD effects in LE treatment. Meanwhile, it also
confirms the power of RDEU in correcting probability distortions in the traditionally
elicited VSL. However, if the application of RDEU to LE results does not show
convergence towards TO results, we shall rule out the potential role of PD in VSL
elicitation and further studies on other causes of estimation bias in VSL will be
needed.
It is notable that most recent developments in the elicitation of the VSL have been
on the empirical front, particularly in the improvement of survey methodology.
Fundamentally, we hope this paper contributes to a previously unaddressed area of
VSL, namely the extension of decision theory to accommodate peoples’ underlying,
real -as opposed to assumed- behavioural traits.
The remainder of the paper is organised as follows. In section 2, we present
probability weighting and RDEU, the theory upon which the paper is based. Section 3
describes our main study design, including the TradeOff and the Lottery Equivalent
approach and how the LE approaches can be modified with RDEU in the presence of
Probability Distortion. We present the main results and discussion in section 4. Section
5 concludes.
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2. PROBABILITY WEIGHTING AND RDEU
The classical way to model an individual’s decision making under risk is Expected
Utility Theory. Based on EU, Jones-Lee (1985, 1989 and 1995) proposed that the value
of safety improvement should be estimated through eliciting willingness-to-pay (WTP)
for a probability reduction of a safety hazard. WTP is defined as a marginal rate of
substitution (MRS) of wealth for risk of death by any particular cause. The population
mean of these individuals gives rise to the main component of the monetary values of
safety for use in public sector cost-benefit analysis, namely, WTP-based “value of
statistical life” (VSL).
Despite the theoretical soundness, empirical estimation of VSL has encountered
many behavioural biases in which people do not behave in accordance with EU.
Although there is no systematic test of empirical validity of EU in the safety-decisiondomain, critical tests of EU in the decision-making literature have suggested its
empirical invalidity. Among them, there are two most prominent, classical experiments,
Allais Paradox (Allais (1953)) and Ellsberg Paradox (Ellsberg (1961)). The former found
a “certainty effect” and the latter found more aversion to ambiguous risks (risks with
unknown probabilities).
The amounting empirical findings suggest that a utility function on its own fails
to capture well an individual’s risk attitudes completely and preferences in some
situations violate the predictions of EU. Some recent empirical research has managed
to find more systematic patterns in peoples’ violating behaviour against EU, such as
pessimism to unfavourable outcomes and optimism to favourable outcomes in certain
circumstances. The notion of risk, where probabilities are well, objectively defined,
has been extended to a broader notion of uncertainty where probabilities cannot be so
well defined.
In the past two decades, decision theorists and psychologists have all together
developed decision theories to a new stage. Systematic behavioural patterns have
entered into the modelling of the decision process so that preferences become more
generalized but still maintain the capability of solving economic problems. The common
feature of these theories is to introduce a probability weighting function, (or, nonadditive probabilities or capacities) to complete people’s risk (uncertainty) attitudes
by capturing probabilities sensitivities.
Many studies show that people overweight small probabilities and underweight
large probabilities (Abdellaoui (2000)). People are also more sensitive to changes of
probabilities from impossible to possible and from possible to certain and insufficient
sensitive to intermediate changes of probabilities (Tversky-Wakker (1995)). The most
common way to model people’s sensitivity towards probability is a probability
weighting function which allows people to weight probabilities nonlinearly. The
prevailing weighting function is an inverse-S curve shaped (Camerer & Ho (1994),
Bleichrodt & Pinto (2000), Gonzalez & Wu (1999) and Tversky& Kahneman (1992)).
As shown in Figure 1, an inverse-S shaped probability weighting function is convex
for moderate and high probabilities, and concave for small probabilities. The shapeturning point of curve is roughly located at 0.33. Many experimental studies confirmed
that people tend to overweight probabilities below approximately 0.3 and underweight
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probabilities above approximately 0.3 (Gonzalez and Wu (1999), Abdellaoui (2000),
Bleichrodt and Pinto (2000)). The slope of the probability weighting function, w2 (p),
is greater than 1 for p near 0 and near 1 and is less than 1 for intermediate p.
With this specific form of weighting function, decision makers will assign
more decision weights implying more attention to extreme outcomes, both the best
and the worst outcomes. (Wakker (2001)). An inverse-S shaped probability weighting
function also predicts the behavioural pattern that people are more sensitive
for changes from impossible to possible and from possible to certain, but less
sensitive to variations in intermediate probabilities. For example, we could have
w(1)-w(0.8)>w(0.3)-w(0.1).
Figure 1: Inverse-S Shaped Decision Weighting Function
Tversky-Kahneman (1992) suggested a parsimonious one parametric function as
the theoretical version of inverse S-shaped weighting function,
πγ
w(π) =
γ
1
γ γ
(π + (1 − π) )
If γ equals to one, we have w(π) equal to π, the preference with respect to this
probability weighting function is EU preference. For γ between 0.27 and 1, the function
results in the inverse S shape. Tversky-Kahneman (1992) found γ the value of 0.69 for
losses and 0.61 for gains.
Both CPT and RDEU model people’s preference with these choice weights. While
RDEU associates the probability weights to the usual Von Neumann – Morgenstern
utility function, CPT proposed a ‘Value’ function to incorporate loss aversion defined
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through a reference point. Since we are mainly investigating the role of PD in VSL
elicitation, we will apply the RDEU approach in this paper as an alternative to the EU
criteria used in traditional elicitation methods.
The central idea of RDEU is that the evaluation of decisions is outcome rankdependent. A probability weighting function transfers probabilities into decision
weights which are allocated to outcomes ranked in the order of their utility value.
Because the elicitation of VSL is about choice among potential losses, we define a
weighting function as follows.
Consider a decision as a lottery written in a usual way: (p1 , x1, p2, x2,...pn, xn), the
outcomes are losses and are ranked as x1 <x2‚ <...<xn. In the rank-dependent approach,
we have following definitions.
Definition: A decision weighting function is defined as a probability weighting
function w(p), satisfying the following conditions:
1. w(0) = 0; w(1) = 1; 2. w(p1)≤w (p‚ ) iff p1≤ p2;
where p is the probability associated with the maximal outcome of a lottery with
outcomes ranked in declining order.
n Definition: A lottery is evaluated by a general weighting function
w( pi )u( xi ) where w(p ) is the decision weight assigned to outcome xi and determined
∑
i =1
i
by a decision weighting function w(pi ), Σ w(pi ) =1 and w(pi ) > = 0., and u(⋅) is the utility
function.
RDEU states that an evaluation of a decision is the sum of weighted utilities of
each possible outcome. All weights are nonnegative and sum to one. However, the
weighting function assigns weights to outcomes according to their importance. For
any outcome xi of a lottery, the decision weight wi can be written as w(p1+p2+...+p
(i-1)+pi)-w(p1 +...+p (i-1)). Denote p1 +...+pi as q, which represents the ranking position
of xi, then the decision weight assigned to outcome xi is w(q)-w(q-pi). In this way, we
see that the decision weights not only depend on the probability of the outcome but
also the rank of the loss. More specific, a convex weighting function implies optimism
and losses with lower rankings receive more attention; analogously, a concave
weighting function corresponds to pessimism and gives more attention to losses with
higher ranks.
3. MAIN STUDY DESIGN
There are a few well established VSL elicition methods, the Standard Gamble method
(the certainty-equivalent (CE) method in general application); the Contingent Valuation
method (the probability-equivalent (PE) method in general application) and the RiskRisk method (the lottery equivalent (LE) method in general application). All these
methods have EU as the theoretical foundation. Thus without measurement errors,
the elicited results obtained under each method should be the same. Unfortunately,
this is hardly the case. Studies in the field have documented that the CE method is
biased by the certainty effect and the PE method suffers a probability distortion effect.
The LE method is an improvement5 but still assumes expected utility criterion and
require respondents’ familiarity with the use of probabilities.
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The Tradeoff (TO) approach was introduced by Wakker and Deneffe (1996) and is
now commonly used to measure the utility of health states. Compared to other methods
of utility measurements, Wakker and Deneffe (1996) showed that TO utilities are robust
even if probabilities are transformed. Bleichrodt et al (2001) showed that the TO
approach is also sign independent, i.e., it is independent of whether outcomes are
gains or losses. More importantly, Wakker and Deneffe (1996) showed that TO approach
is an elicitation approach that remains valid for non-expected utility models and can
elicit probability indirectly.
The literature shows that the Lottery Equivalent (LE) approach (McCord and de
Neufville (1986)) is similar to the TO approach and asks respondents to compare two
lotteries that both involve risk. While the TO approach asks respondents to obtain
their indifference between two lotteries by changing one of the outcomes, the LE
approach obtains indifference by changing one of the probabilities. Thus, while the
TO approach minimises the role of probabilities in the elicitation, the LE approach
still requires precise knowledge of probabilities. This provides the basis of the
experimental design to investigate probability distortion effects.
We now describe both methods in a hypothetical setting. Assuming a respondent
had a road accident and will receive a treatment which can lead to the following
potential states: normal health (x0), minor injuries (x1), non-permanent injuries type 1
(x2), non permanent injuries type 2 (x3),... permanent injuries type 1 (xn-2), permanent
injuries type 2 (xn-1), Death (xn).
It is natural to assume that a respondent’s Willingness-to-Pay (WTP) to avoid the
potential result is an increasing function of the damage level of the accident and the
WTP to avoid maintaining normal health is zero. Denoting the WTP to avoid the
potential state xi as m(xi), we rank the potential states according to the level of m(xi) as,
m(x0) < m(x1) < m(x2) < ... < m(xn-1) < m(xn).
3.1. The Tradeoff Treatment
We firstly choose two “reference” health states xn and xn-1, so-called ‘gauge outcomes’
in the TO methodology and a probability p for the construction of risky choice used in
the treatment6. We then ask the respondent to specify an xi from the described potential
states provided to reach their indifference point in each of the following lotteries. In
the process, we keep xn, xn-1 and the probability p constant. To reduce any potential
probability distortion effects, we choose p = 0.5.
Denote by (xi, p, xj) the two-outcome lottery that assigns probability p to outcome
xi and probability 1-p to outcome xj. Note that the gauge outcome xn (death) is worse
than xn-1 (permanent injury), and the outcome x0 (perfect health) is the best. The process
starts with asking the respondent to choose a health state, denoted as x1 to reach
indifference between the two lotteries (xn, 0.5, x0) and (xn-1, 0.5, x1). Next, by replacing
x0 with x1, the respondent will find x2, their indifference point between the next two
lotteries (xn, 0.5, x1) and (xn-1, 0.5, x2). Suppose the process goes as follows, where notation
“~” means indifference between two lotteries.
Question 1: (xn, 0.5, x0) ~ (xn-1, 0.5, x1)
Question 2: (xn, 0.5, x1) ~ (xn-1, 0.5, x2)
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Question 3: (xn, 0.5, x2) ~ (xn-1, 0.5, x3)
...
Question n: (xn, 0.5, xk-1) ~ (xn-1, 0.5, xk)
At the end of the process, we elicit the set of states which the respondents specified:
x1 , x2‚ , ... xk. Setting m(xo) = 0, i.e., the WTP to avoid perfect health is zero, based on
the principle of WTP-based VSL (Jones-Lee 1989), we have the VSL for each of these
states as,
m( xi ) =
i
m( xk ), i = 1, 2 ,..., k
k
(1)
Compared to other commonly used WTP elicitation methods, this TO approach
potentially avoids two effects: first, because the choice is specified as the possible state
of damage, we avoid the requirement of a good understanding of probabilities. Second,
since we keep probability p constant at 0.5, the effects of probability distortions is
minimised.
Moreover, it is easy to show that the equation 1 holds true no matter whether the
economic principle underlying this estimation is Expected Utility (EU) or Rankdependent Expected Utility (RDEU). This gives us a benchmark to test the consistency
in utilizing either EU principle or RDEU principle in the elicitation of VOSL.
3.2. The Lottery Equivalent Treatment and RDEU Correction
We construct LE questions for the states x1,.., xk specified through the TO treatment.
We use the lottery (xk, p, xo ) which comprises the worst outcome xk and the best outcome
xo as the benchmark. For a fixed probability p7, a respondent is asked to specify the
probability q to make the following two lotteries equivalent to her/his preference:
(xk, p, xo ) ~ (xk, q, x1 ). Since we have m(xo ) equal to zero, using EU as the estimation
principle, we have8,
pm( xk ) = qm( xk ) + (1 − q)m( x1 )
m( x1 ) =
p−q
m( xk )
1− q
(2)
We substitute x1 with x2 and repeat the question to obtain q for x2. The process is
completed when q is obtained for each xi elicited through the TO treatment.
We shall now compare the results of the equation 2 with the results of the
equation 1. We would expect to observe a disparity. Our aim is to investigate
whether RDEU will reduce this disparity. If so, it might imply that probability
distortion still has certain effects on the LE elicited results and RDEU can correct
such a distortion.
We use the one parametric probability weighting function proposed by Tversky
and Kahneman (1992) here for the elicited probability q9. Different empirical estimations
suggest slightly different value of γ. We use the value of γ equal to 0.69, as proposed by
Tversky and Kahneman (1992) for loss. Bleichrodt et al (1999) also confirmed the value
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of 0.69 for γ for health outcomes. Now applying RDEU criteria to the LE question
(xk, p, xo ) ~ (xk, q, x1 ), we have,
(1-w(1-p))⋅m(xk}) = (1-w(1-q))⋅m(xk)+w(1-q)m(x1 ),
m(x1 ) = ((w(1-q)-w(1-p))/(w(1-q))⋅m(xk)
(1 − q)0.69
where, w(1 − q) =
0.69
((1 − q)
+q
1
0.69 0.69
(3)
.
)
The comparison between the results calculated through the equation 3 and equation
1 show the probability distortion effects and RDEU’s corrective power.
4. EXPERIMENTS AND RESULTS
4.1. Implementations
We began by running extensive pre-testing and piloting on academic staff and visiting
fellows in the Economics department of Newcastle University. In the first attempt of
the TO treatment, respondents were required to choose from a set of health states 10.
The piloting revealed that one technical challenge is to create a sufficiently large set of
health states11. With the consulting from research members in Health Economic Division
and Medical School in Newcastle University, we adopt the EQ-5D Health States
Classification as the descriptions of health conditions used in our questionnaire. The
EQ-5D Health States classified 245 possible health states which are based on the data
collected from a representative survey of the UK general public. Each health states is
described through 5 dimensions: mobility, self-care, pain, normal exercise and mental
health. The health defined on each dimension can be ranked in 3 levels: normal, medium
problem and severe problems. The descriptions about each health states are similar
but with different levels of health states through 5 dimensions. The subsequent piloting
on economics staff confirmed the adequacy of these descriptions for TO questions. It
suggested that the usage of this classification not only provided a rich set of choices
but also solved a potential ranking problem of these health states12.
The participants in the experiment were 60 masters students studying economics
and finance at Newcastle University. Participants were paid £20 to participate in the
experiment which comprised three sessions. The experiment was administrated by
personal interview. In the first session, participants became familiar with the
descriptions of health states on each card. They then answered practice questions which
examined their acquaintance with probabilities and expectations, and also ensured
that they were familiar with the concept of indifference in making choices. Only
students who made consistent choices proceeded to the latter stages of the experiment.
Participants answered the TO questions in the second session and the LE questions in
the third session. Through all sessions, it was emphasised that there were no right or
wrong answers but rather that their answers should reflect their preferences.
The practice session gave an opportunity for participants to gain a degree of
familiarity with and a good understanding of the questions. In this session, the
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interviewer firstly asked the respondent to imagine she/he had an accident and that a
doctor told them that they would have to conduct an operation; otherwise the
respondent would deteriorate and may even die. There were two different types of
operation that could be undertaken, but it was only possible to have one of the
operations, and the respondent should choose between them. The interviewer then
put a template in front of the respondent which displayed the operations and the
potential outcomes, Tt, Pp, Oo and B.
The descriptions of these outcomes are as follows.
Tt
Mobility
Self-care
Usual Activities
Pain/Discomfort
Anxiety/Depression
I am confined to bed
I am unable to wash or dress myself
I am unable to perform my usual activities
I have extreme pain or discomfort
I am extremely anxious or depressed.
Pp
Mobility
Self-Care
Usual Activities
Pain/Discomfort
Anxiety/Depression
I am confined to bed
I have some problems washing or dressing myself
I have some problems with performing my usual activities
I have moderate pain or discomfort
I am extremely anxious or depressed.
Oo
Mobility
Self-Care
Usual Activities
Pain/Discomfort
Anxiety/Depression
I am confined to bed
I have some problems washing or dressing myself
I am unable to perform my usual activities
I have moderate pain or discomfort
I am moderately anxious or depressed.
B
Mobility
Self-care
Usual Activities
Pain/Discomfort
Anxiety/Depression
I have no problems in walking about
I have no problems with self-care
I have some problems with performing my usual activities
I have no pain or discomfort
I am not anxious or depressed.
The interviewer then asked:
“If you choose Operation 1, you will either end up in health state of Tt or B. Each is
equally likely to happen. In other words, there is a 50% chance of Tt and a 50% chance of B.
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If you choose Operation 2, you will either end up in health state of Pp or Oo. Each is
equally likely to happen. In other words, there is a 50% chance of Pp and a 50% chance of Oo.“
The respondents made the choice and explained the reasons for the choice. The
understanding of the descriptions and the reasoning in making such a choice was
therefore established. Then the interviewer replaced the card Oo with the card B, and
put a new card A in the place where card B initially was. The description of health
state A is as below.
A
Mobility
I have no problems in walking about
Self-care
I have no problems with self-care
Usual Activities
I have no problems with performing my usual activities
Pain/Discomfort
I have no pain or discomfort
Anxiety/Depression
I am not anxious or depressed.
The interviewer asked a similar question as in the first round and the respondent
made their choice. The interviewer also asked whether the respondent felt that it was
more difficult to choose between two operations in the second round. Thus a concept
of indifference was discussed and established.
The formal experiment began with the TO treatment. We used health states Tt
and Nn as the ‘gauge outcomes’ and health state A as the initial outcome. The
description of Nn is as below.
Nn
Mobility
I have some problems in walking about
Self-Care
I have some problems washing or dressing myself
Usual Activities
I have some problems with performing my usual activities
Pain/Discomfort
I have extreme pain or discomfort
Anxiety/Depression
I am moderately anxious or depressed.
Participants were asked to imagine that they just had a road accident and could
choose between two medical operations, each of which could have good or bad
outcomes. Each outcome would occur with probability 0.5. Table 1 illustrates the format
of the TO question in the first round.
Table 1
To Treatment Starting Round
Operation 1
Operation 2
Outcome 1 (0.5)
Tt
Nn
Outcome 2 (0.5)
A
? (Xi)
A template presenting the format of the questions was shown to the participant.
The session interviewer put down each card on which a notation xi and the description
of the health state was displayed in the place where the state was missing (? (xi)), and
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asked whether participant had a clear preference between two operations or whether
they were indifferent between the two operations. A card with health state x i was
selected when the participant reached indifference between two operations. In the
second round, the session interviewer replaced the card describing health state A with
the card describing health state x i, then repeated the process until the participant
reached their indifference point between the two operations with a health state xj filled
in the missing place. Table 2 illustrates the process.
Table 2
To Treatment Process
Outcome 1
Outcome 2
Operation 1
Operation 2
Tt (p = 0.5)
Xi
Nn (q = 0.5)
? (Xj)
Taking xo as outcome A, the TO session elicited a standard sequence x1 , ...,xk that
established indifference between (Tt, 0.5,xi-1) and (Nn, 0.5, xi), i = 1,...k.
Participants could then proceed to the third session immediately or have a short
break. In the third session, a new template was presented withan LE question, as shown
in Table 3.
Table 3
LE Treatment Process
Outcome 1
Outcome 2
Operation 1
Operation 2
Xk (p=0.8)
A
Xk (q=?)
Xi
The participant was asked to compare Operation 1 (xk, 0.8, A) and Operation 2
(xk, q,xi) and choose a value of q which gave their indifference between the two
operations. The interviewer suggested some value of q to initiate the process when a
participant found it difficult to pin down a precise number for the probability. The
process repeated for each of xi where i=1,...k-1, and a set of probabilities q1 ,...qk-1 was
obtained corresponding to x1...xk-113.
4.2. Results and Discussion
50 participants were included in the analysis. Two participants were excluded because
they failed to make any tradeoffs in the practice sessions and another 8 participants
were excluded because they gave responses to less than 3 health states in the TO
session14. We catalogued the results into 3 groups based on the number of the health
states elicited in the TO session. 30 participants elicited 5 health states, 12 participants
elicited 4 health states and 8 participants elicited 3 health states. These results are
summarised and illustrated in Table 4, 5, 6 and Figure 2, 3, 4 respectively in the
appendix.
By comparing the TO results to the LE results, we can test the EU assumption
traditionally used for VSL elicitation. If a participant holds EU preferences, the TO
results will be nearly identical to the LE results. The results in all three tables showed
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that there are significant differences between the TO results and LE results for most
elicited states. The TO results are initially higher than LE results and then become
lower than LE results. The discrepancy is firstly reduced and then increased again
along the ranked elicited health states from xi+1 to xi. Similarly, if we adopt RDEU as
the elicitation assumption, we shall have the TO results nearly identical to the RDEU
(LE) results. However, as shown in all three tables, after adjusting data under RDEU
assumption, there are discrepancies remaining between RDEU-LE results and the TO
results. Specifically, compared to the LE results, the RDEU(LE) results are adjusted
downwards by a similar amount for each elicited state. Thus, for the elicited higherranked states, x„ in Group 1, xƒ in Group 2 and x‚ in Group 3, the discrepancies between
LE results and the TO results are enlarged after RDEU adjustment. While for the lowerranked states, xƒ in Group 1, x‚ in Group 2 and x1 in Group 3, the discrepancies almost
disappeared. In general, for lower-ranked states, RDEU-LE results are closer to the
TO results, while the LE results are closer to the TO results for higher-ranked states.
The implications of our results suggested that both EU and RDEU used as elicitation
assumption are violated. While the finding about the violation of EU is not a surprise,
the violation of RDEU unexpectedly rejected the prediction that the application of
RDEU might reduce the effects of probability distortion in the LE eliciting results.
However, by looking at the results closely, it can be seen that difference between
the value of each elicited states in both LE and RDEU(LE) results are similar. While LE
values of each elicited state averagely have a sequential difference of 0.04, the RDEU(LE)
results of each elicited state averagely has a sequential difference of 0.06. In comparison,
the TO values of each elicited states has the difference of 0.2 in between each other.
Therefore RDEU (LE) results improved the slope of monetary value function for each
health state, as suggested in equation 3. Such a steeper slope implied a subtle
improvement of original LE results towards TO results15. This might imply that the
results are not as disappointing as it seems at the surface. The interviewers recorded
that most respondents had difficulties in composing a probability number through
the LE treatment session. Respondents also had difficulties mapping the changes of
probability values with the changes of health states. Thus, unavoidably, there are
probability misperceptions in the answers to LE questions, which created the basis for
the big discrepancy between the LE results and TO results. While the application of
RDEU criteria made a subtle improvement, it is not sufficient to overturn the types of
probability distortion originated in the LE session. We still need a combination of
both improvements in elicitation approach and RDEU criteria to reach more accurate
estimation of VSL.
5. CONCLUSION
We conducted the experiments to elicit relative monetary value of health states in a
safety domain. In the first part of the experiments, we elicited the health states and
their relative monetary values using TO approach. In the second part of the experiments,
we sequentially used health states elicited through TO treatment as input for the LE
treatment. We derived the monetary value of health states from the answers to the LE
questions firstly using EU criteria then using RDEU criteria. The results showed that a
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discrepancy between the TO analysis and the LE analysis will be enlarged for the
worse health states but will be reduced for the better health states under the RDEU
(LE) analysis.
The literature has argued that probability distortion is one of the reasons for the
deviations from expected utility in eliciting monetary value of the health states. To
deal with the problem, on the empirical side, it is documented that the TO approach is
immune to probability distortion in the eliciting process; on the theoretical side, rankdependent utility theory is proposed to correct the deviations from expected utility
caused by probability distortion. Although it has been suggested that the TO approach
might not be applicable in the context of evaluating health states because the method
requires continuums of outcomes, this paper successfully adopted the TO approach to
elicit health states by applying the EQ-5D Health states Scoring system. Using these
TO results as a benchmark, we tested and concluded that the RDEU analysis failed to
resolve, completely, the discrepancies between the TO eliciting results and the LE
eliciting results.
We suggest a number of the explanations to account for this uncomfortable
evidence. First, because the TO approach elicits a sequence of outcome states that
are equally spaced in a unit interval, TO monetary value of health states is linear.
Meanwhile, the elicited results under LE approach also appear to be linear. This is
arguably the bias in our results, compared to a traditionally convex shape claimed
in the literature. This bias might be partially responsible for the discrepancies
appeared in the comparisons. Second, we applied one-parameter probability
weighting function proposed by Tversky and Kahneman (1992) in our RDEU analysis
of the LE data. In our results, it appeared that RDEU(LE) results are nearly always a
downwards transformation of the EU-LE results but with slightly steeper slopes.
The range of the probabilities elicited under LE approach is mostly between 0.3 and
0.7. For this range of the probabilities, the one-parameter probability weighting
function underweight the probabilities in a nearly linear way. Given the comparison
between the TO results and the EU(LE) results, a nearly parallel down-wards shift
of the EU-LE results is not sufficient enough to change the comparison relationship
with the TO results significantly. Third, given the observations during the eliciting
process, it implies that empirical deviations from the EU elicitation assumption in
the elicitation process might not be caused by probability distortions alone. Other
factors such as scale compatibility, sequencing effects might also play important
roles. Therefore, further research is recommended into the effect of interview structure
and framing.
Acknowledgment
We are grateful to Peter Wakker for helpful suggestions on the implementation of the
TradeOff approach. We thank colleagues in economic department and health economics
centre in Newcastle University for helpful and constructive comments during the
development of the paper. Special thanks go to WenHua Li and Fan Zhang for their
assistance in conducting the survey. Wei Pang gratefully acknowledges financial
support from Newcastle University.
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Notes
1.
2.
3.
4.
Hens and Bachman (2008), Provide a good comparison of Prospect Theory and the traditional
EU approach.
The study by Cohen & Jaffray (1988), Concluded that the possibility of a probability distortion
effect could be ruled out for probability 0.5.
In other words, some studies showed that the LE results with p = 0.75 were consistent with
those obtained when p = 0.5.
Theoretically, TO approach generates the same elicited value of results, independent of the use
of EU or of RDEU.
5.
The corresponding Risk-Risk method is developed by Viscusi et al (1991) for eliciting health
risk and it is assumed that to the extent individuals are equally averse to the uncertainty
associated with different types of health risks. Given this assumption, the method avoids the
analysis of risk attitudes towards different risks in the risk-monetary method.
6.
For detailed explanation of TO method, please refer to Wakker and Deneffe (1996).
7.
The value of probability p was chosen from trial exercises for different sets of (x 1...xn).
8.
The proof is by Jones-Lee (1989).
9.
Bleichrodt-Eechhoudt (2006) applied RDEU to the calculation of WTP for reductions in health
risk in the similar way.
10. The previous literatures applied TO approach to the choice between either monetary outcomes
or life duration.
11. Ideally, the set of states where respondents choose from should be continuous. However, a
discrete set is still applicable as long as the set is sufficiently rich.
12. The descriptions for all health states used in the questionnaire are available from the authors
on request.
13. The full description of questionnaire is available from the authors on request.
14. We can derive the results from the TO approach only if more than 3 states are elicited.
15. Due to the small difference between 0.06 and 0.04, it is not obvious to observe the slope changes
in the curves of LE and RDEU-LE in figure 2, 3, 4. However, a large-scaled figure will present
the subtle improvement between RDEU-LE and LE results using TO results as a benchmark.
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APPENDIX
Comparison among LE, TO and RDEU(LE)
Table 4: Results for those eliciting 5 Health States (obs. = 30). The figure in each cell is
the ratio multiplying the monetary value of health state x5. For instance, the mean
monetary value of health state x4, m(x4), is 0.66m(x5) using LE approach, and is 0.8m
(x5) using TO approach. After applying RDEU criteria to the LE results, the mean
monetary value of health state x4, m(x4), is 0.51m(x5). Similar interpretations apply to
the remaining results.
Table 4
Min
Max
LE
Median
Mean
TO value
Mean
x5
x4
0.33
0.79
0.67
0.66
0.8
0.51
0.50
0.69
0.21
x3
0.33
0.78
0.62
0.60
0.6
0.45
0.46
0.66
0.21
x2
0.20
0.75
0.60
0.54
0.4
0.39
0.43
0.61
0.11
x1
0.20
0.71
0.50
0.50
0.2
0.35
0.34
0.56
0.11
Elicited Heath
State
RDEU (LE)
Median
Max
Min
Table 5: Results for those eliciting 4 Health States (obs. = 12). The figure in each
cell is the ratio multiplying the monetary value of health state x4. For instance, the
mean monetary value of health state x3, m(x3) is 0.64m(x4) using LE approach, and is
0.75m(x4) using TO approach. After applying RDEU criteria to the LE results, the mean
monetary value of health state x3, m(x3), is 0.49m(x4). Similar interpretations apply to
results in all other cells.
Table 5
LE
Elicited Health
State
RDEU(LE)
Min
Max
Median
Mean
TO value
Mean
Median
Max
Min
x4
x3
0.43
7.00
0.70
0.64
0.75
0.49
0.54
0.66
0.28
x2
0.33
0.73
0.67
0.59
0.5
0.43
0.50
0.58
0.21
x1
0.33
0.67
0.55
0.53
0.25
0.37
0.38
0.50
0.21
Table 6: Results for those eliciting 3 Health States (obs. = 8). The figure in each cell
is the ratio multiplying the monetary value of health state x5 . For instance, the mean
monetary value of health state x4 , m(x4) is 0.55m(x5) using LE approach, and is 0.67m(x5)
using TO approach. After the correction using RDEU criteria, the mean monetary value
of health state x4 , m(x4) is 0.39m(x5). Similar interpretations apply to results in all
other cells.
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Table 6
LE
Elicited Health
State
x3
x2
x1
RDEU(LE)
Min
Max
Median
Mean
TO value
Mean
Median
Max
Min
0.33
0.33
0.67
0.67
0.62
0.46
0.55
0.46
0.6667
0.333
0.39
0.31
0.45
0.31
0.50
0.50
0.21
0.21
Figure 2: Results for Elicited 5 Health States (obs. = 30).
Figure 3: Results for Elicited 4 Health States (obs. = 12)
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Figure 4: Results for Elicited 3 Health States (obs. = 8)
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