Willingness to pay and willingness to accept for a
lottery and reference point in prospect theory
Michal Lewandowski
EUI Florence, Italy
[email protected]
June 23, 2009
Abstract
The paper proposes a behavioral decision rule for choice under risk.
This rule is designed to explain patterns of behavior in two paradoxes of
classic decision theory i.e. Asian disease paradox and preference reversal
puzzle. The loss aversion which is the driving force of this explanation is
captured in the W T A/W T P disparity. The proposed decision rule is then
showed to be equivalent to decision-making in the Cumulative Prospect
Theory with appropriately chosen quasi-endogenous reference points.
W T A, W T P and reference point
1
Michal Lewandowski
Introduction and Motivation
In reference dependent models in general, two questions are crucial:
• What is to be regarded as losses and what as gains?
• How losses should be evaluated as compared to gains?
The first issue above is a problem of reference point determination, and the
second is an issue of modeling attitudes to gains and losses. This paper is
aimed at both of these issues, but it addresses them in the opposite order than
they appear above.
The problem of modeling attitudes to gains and losses is treated here along
the lines of prospect theory of Kahneman and Tversky (1979) or a later cumulative version of Tversky and Kahneman (1992). According to these theories,
a reference dependent utility function exhibits loss aversion (losses loom larger
than gains), risk aversion for gains, risk loving for losses and diminishing sensitivity (the further away a given outcome is from a reference point, the smaller
is the marginal utility of this outcome). In this paper we focus on loss aversion
which is captured in one constant parameter λ. The goal is to design a behavioral decision rule that accommodates certain patterns of behavior, which
were not accommodated in the classic Expected Utility Theory. These behavior patterns were analyzed in the experimental literature and they are usually
called choice paradoxes. We analyze two paradoxes i.e. Asian disease paradox
and preference reversal puzzle and use them to design a decision rule. This rule
is based on Willingness-To-Pay (W T P ) and Willingness-To-Accept (W T A),
measures which are often used in the experimental literature. We model them
formally, derive their properties and show that in our model the existence of
disparity between W T P and W T A is equivalent to loss aversion. Therefore,
apart from the rule itself, we propose an explanation for the two paradoxes,
which is based solely on loss aversion. To sum up the main idea, the goal
here is to use two choice paradoxes to design a decision rule which accommodates behavior patterns from these paradoxes. Additionally, we form testable
predictions which can be used to evaluate the decision rule empirically.
The problem of reference point determination is also addressed in this paper. As a matter of fact, this paper grew out of a quest for a good reference
point in decision-making under uncertainty. The importance of this issue stems
from the fact that in reference dependent models losses and gains are treated
differently and reference point serves as a simple device to determine what is
to be regarded as losses and what is to be treated as gains. In the literature,
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it is often implicitly or explicitly assumed that a reference point is given and
hence exogenous. In some applications, where the reference point is naturally
defined from the context, it is an innocuous assumption. Then the reference
point is just a status quo or an endowment. However there are situations in
which there is more than one candidate for a reference point or even situations
in which it is hard to think about a good natural reference point. For example,
when a person decides upon buying a fire insurance for her house, then what
should be taken as a reference point? The status quo here is an uncertain situation with at least two states, either a house gets burnt and loses its value or
it doesn’t. The question is whether a good reference point should be random
here or it should be one of the possible outcomes of a status quo or maybe it
should be an expected value of a house, given possibility of damage. This is
a case in which it is hard to decide which candidate is a ”natural” reference
point. A case in which it is hard to think about a good reference point is for
instance, when a decision maker decides whether she will go to a cinema. In
this situation there is no natural status quo or endowment. Some authors assume a zero reference point in such cases, but this is probably not correct as a
decision maker may have expectations about the movies she considers watching
and these expectations seem to define what is considered a gain here and what
is considered a loss. Both these examples suggest that reference determination
is an important part of reference dependent models. There were only few attempts in the literature to model it formally. One of them is Koszegi and Rabin
(2003). They identify a person’s reference point with her recent expectations
about outcomes and introduce the concept of personal equilibrium, which states
that a person maximizes utility given her rational expectation about outcomes.
These expectations in turn depend on her own anticipated behavior. While this
approach provides a disciplined way to determine reference points, it assumes
that people have rational expectations depending on their anticipated behavior
which does not have an intuitive appeal.
This paper suggests an alternative way to define reference points in decisionmaking under risk or uncertainty. The idea is to use W T A and W T P measures
to define reference points. We show equivalence between the decision rule derived from choice paradoxes, which I mentioned in the earlier paragraph, and
prospect theory decision making with appropriately chosen reference points,
based on W T A and W T P measures. Hence, reference points in this paper follow from behavioral decision rule I propose which in turn is designed especially
to explain some well known paradoxes in individual decision-making. Thus, ref-
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W T A, W T P and reference point
Michal Lewandowski
erence points are not endogenous here as they are in Koszegi and Rabin (2003).
The advantage of the current approach lies hopefully in the intuitive appeal of
the proposed reference points.
This paper is organized as follows. First, I introduce a model - main assumptions, notation, basic definitions. Then I define two variables, Willingness-ToPay and Willingness-To-Accept, which will be key in my proposal and I state
and prove couple of properties of these variables. Second, after introducing the
necessary formalism, I will analyze two examples from experimental literature
which show patterns of choices, inconsistent with the traditional theory i.e.
Expected Utility Theory. On the basis of these two examples I will state two
hypotheses concerning individual decision-making. These hypotheses will then
be used to design a behavioral decision rule. This decision rule is designed in a
way that it accommodates patterns of choices described in two examples from
the previous section. Then I will show that this rule is capable of capturing
behavior also in other situations, such as status quo bias and it yields testable
predictions in such situations. Finally, I show that the decision rule which I
proposed is equivalent to the decision-making in prospect theories (Prospect
Theory and the later versions of it) with appropriately chosen reference points.
The last section concludes. A brief introduction to the Cumulative Prospect
Theory can be found in the Appendix.
2
The Model
The model presented here is aimed at capturing the effect of loss aversion.
Therefore, I decided to focus on a simplest possible set-up: the case of risk
instead of uncertainty, no probability weighting, clear separation of different
elements of individual preferences such as loss aversion and risk aversion.
2.1
Assumptions and notation
In the model I propose outcomes are monetary, the outcome set S is bounded
and belongs to R. Given a probability space (Ω, A, P ), a lottery on Ω is a
measurable function x : (Ω, A) → S with associated cumulative distribution
function F ∈ F over the outcome space, where F denotes a family of cumulative
distributions over S. Hence lottery is a random variable.
Given an exogenous outcome R ∈ S (the reference point) and a lottery
x, prospect is defined as y = x − R. In general when a reference is allowed
to be a random variable R : (Ω, A) → S, a prospect is a lottery y = x − R
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and the distribution of y can be derived from a joint distribution of x and R.
One way to think about a lottery and a prospect is that a lottery is something
objective and given to the decision maker and it is outside of his/her control.
The decision maker can either accept it or reject it. A prospect on the other
hand is a lottery where all the outcomes are defined relative to some subjective
reference point. In this way all the outcomes of the lottery are coded as gains
and losses. A reference is therefore individual-specific.
To save on notation I will sometimes identify a lottery or a prospect with the
cdf that describes it. Let’s denote F , G as typical elements of F. EF or EG
shall denote expectation taken with respect to F or G respectively. When the
context is unambiguous I will skip the subscripts. I assume that preferences
over prospects can be represented as in prospect theory. However, since I want
to focus on reference dependence I will exclude probability weighting. I define
a utility function U : S → R which represents decision maker’s preferences over
prospects. It assigns a real valued index to a given realization of a random
variable x.
Assumption 1 Utility function is of the following form:
(
u(x)
if x ≥ 0
U (x) =
−λu(−x) if x < 0
(1)
where λ > 0 denotes the loss attitude parameter. Furthermore, a function
u(.) is absolutely continuous, bounded, strictly increasing with u(0) = 0 and
concave on its domain.
The above assumption has several implications. First, a utility function is
defined over prospects and not lotteries. Negative prospect’s outcomes are interpreted as losses and positive prospect’s outcomes as gains. Second, I assume
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Michal Lewandowski
that it is possible to separate risk and loss/gain attitudes of a decision maker.
Risk attitudes are embodied in the curvature of a utility function and are symmetric for gains and losses. Symmetric means two things:
• diminishing sensitivity, i.e. the further from a reference point, the lower
the marginal effect of an increase in either gain or loss on the utility
function, this implies that a decision maker is risk averse in the positive
outcomes of a prospect and risk loving in the negative outcomes of a
prospect
• equal diminishing sensitivity for gains and losses, i.e. a decision maker is
equally risk averse for a given gain x > 0 as she is risk loving for a the
same loss −x
Loss/gain attitudes are represented by a λ parameter in Assumption 1. For a
given prospect outcome x, the utility of this outcome U (x) is equal to the basic
utility of this outcome u(|x|) times λ in case the outcome x is negative. If λ is
greater than one, then a decision maker is loss averse. If λ equals to one, then a
decision maker is loss neutral. The rationale for this assumption is Köbberling
and Wakker (2005).
2.2
Willingness to pay and willingness to accept - definition
and properties
Let’s define two scalar measures which will be used in the paper.
Definition 1 For a given lottery x ∼ F , willingness to accept for x is a scalar
denoted by W T A(x), willingness to pay for x is a scalar denoted by W T P (x)
and the following holds:
W T A(x) ∈ R :
s.t.
EF [U (W T A(x) − x)] = 0
(2)
W T P (x) ∈ R :
s.t.
EF [U (x − W T P (x))] = 0
(3)
Notice that these measures are well defined and uniquely determined. This is
so because the utility function is continuous, strictly increasing and its range
covers positive as well as negative values. Below we will state and prove some
important properties of these measures.
Property 1 Willingness to pay and willingness to accept for a lottery x on
[x, x] are in the interior of the support of this lottery:
W T A(x), W T P (y) ∈ (x, x)
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Michal Lewandowski
Proof: straightforward from definition, by inspection of (2) and (3).
Property 2 Given lottery x, the following equivalence relations hold:
λ S 1 ⇔ W T A(x) S W T P (x)
(4)
Proof:
Using Assumption 1 and combining equations (2), (3), which define W T P (x)
and W T A(x) gives:
Z
W T A(x)
x
Z
u(W T A(x) − x)dF (x) +
x
"Z
W T P (x)
W T P (x)
Z
x
u(x − W T A(x))dF (x)
u(W T P (x) − x)dF (x) +
=λ
u(x − W T P (x))dF (x)
#
(5)
W T A(x)
x
Observe first that if W T A(x) = W T P (x), then the LHS of equation (5) and
the term in brackets on the RHS are the same. So λ = 1 in this case because
otherwise equation (5) would be violated. Conversely, if λ = 1 then by Assumption 1 utility function is symmetric relative to the origin and therefore
EF [U (W T A(x) − x)] = EF [U (x − W T P (x))] = 0 only if W T A(x) = W T P (x).
If W T A(x) > W T P (x), then by strict monotonicity of u(.) we have:
W T A(x)
Z
Z
W T P (x)
u(W T A(x) − x)dF (x) >
x
Z
u(W T P () − x)dF (x) (6)
x
x
Z
x
u(x − W T P (x))dF (x) >
W T P (x)
u(x − W T A(x))dF (x) (7)
W T A(x)
Therefore, equation (5) will be satisfied only if λ > 1. Conversely, if λ > 1,
then equation (5) can only be satisfied if W T A(x) > W T P (x).
Similarly, W T A(x) < W T P (x) holds if, and only if λ < 1. 2
Property 2 tells us in particular that in our model there is an equivalence
between the existence of loss aversion (λ > 1) and willingness-to-accept for a
given lottery being greater than willingness-to-pay. This is obviously attributed
to the specific assumption we made on the shape of the utility function. This
assumption however allows us to separate different forces driving the preferences
and as such it shouldn’t be regarded as causing the loss of generality.
Property 3 Given lottery x and a constant A ∈ R , the following holds:
W T X(x + A) = W T X(x) + A
where X ∈ {A, P }.
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Michal Lewandowski
Proof:
Directly from definition:
0 = E[U (x − W T P (x))]
= E[U (x + A − (W T P (x)) + A)]
= E[U ((x + A) − W T P (x + A))]
Similarly for W T A(x + A). 2
This property tells us that shifting all the outcomes of a lottery by a constant
should change the measures exactly by this constant.
Property 4 Given lottery x, the following holds:
W T X(x) = −W T Y (−x)
where X, Y ∈ {A, P } and X 6= Y.
Proof:
Directly from definition:
0 = E[U (W T A(x) − x)]
= E[U (−x − (−W T A(x)))]
= E[U ((−x) − W T P (−x))]
Similarly for W T P (x). 2
This property informs us that minus willingness-to-pay for a given lottery
is equal to willingness-to-accept for a minus this lottery. And similarly, minus
willingness-to-accept for a given lottery is equal to willingness-to-pay for a minus
this lottery. Hence W T A and W T P are not resistant to reversing the outcomes
of a lottery around zero.
Property 5 Given lottery x and lottery y = −x, the following holds:
E[U (W T A(y) − y)|y < W T A(y)] = E[U (W T P (x) − x)|x > W T P (x)]
E[U (y − W T A(y))|y > W T A(y)] = E[U (x − W T P (x))|x < W T P (x)]
Proof:
Follows the same simple logic as the proofs above. 2
Property 6 Given a lottery x and a scalar θ 6= 0, for utility function u homogeneous of any positive degree α, the following holds:
if θ > 0 then :W T X(θx) = θW T X(x)
where X, Y ∈ {A, P }
if θ < 0 then :W T X(θx) = θW T Y (x)
where X, Y ∈ {A, P } and X 6= Y.
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Michal Lewandowski
Proof:
From definition and the fact that the utility function is homogeneous of degree
α:
0 = E[U (W T A(θx) − θx)]
1
= θα E[U (W T A(θx) − x)]
θ
= θα E[U (W T A(x) − x)]
Similar proof holds also for the rest of the above relations. 2
Important to notice is that the above property holds only for homogeneous utility functions, but for homogeneous utility functions of any positive
degree.
2.3
A connection to the dual theory of choice under risk
In the classic formulation we take expected utility of outcomes. However, it
is only one possibility. We could alternatively and equivalently take expected
value of outcomes using an appropriately chosen probabilities, which is the idea
of Yaari (1987). We hope that it will show a useful way to look at W T A
and W T P measures. Let’s work on a simple example. We are given a lottery
x = (−y, 1 − p; x, p) and u(x) = axα , where it is supposed that Assumption 1
holds. Let’s write a definition of Willingness-To-Pay:
pu(x − W T P (x)) − λ(1 − p)u(W T P (x) + y) = 0
Using the specific form of the utility functions we can solve for W T P (x):
W T P (x) =
p1/α x − [λ(1 − p)]1/α y
p1/α + [λ(1 − p)]1/α
Let’s define the following variables:
p1 =
p1/α
,
p1/α + [λ(1 − p)]1/α
p2 =
[λ(1 − p)]1/α
p1/α + [λ(1 − p)]1/α
It is not hard to see that they sum up to one and therefore they can be regarded
as probabilities. Now we can write W T P as:
W T P (x) = Ep1 ,p2 x
where Ep1 ,p2 denotes an expectation operator with respect to probabilities p1
and p2 . This shows that we can interpret W T P as expected value of the outcomes of a lottery where probabilities are adjusted appropriately by ”sticking
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W T A, W T P and reference point
Michal Lewandowski
loss aversion and risk aversion” into them. We can repeat the same exercise for
W T A:
−λpu(x − W T A(x)) + (1 − p)u(W T A(x) + y) = 0
W T A(x) =
(λp)1/α x−(1−p)1/α y
(λp)1/α +(1−p)1/α
We define again two variables:
p01 =
(λp)1/α
,
(λp)1/α + (1 − p)1/α
p02 =
(1 − p)1/α
(λp)1/α + (1 − p)1/α
And we can write:
W T A(x) = Ep01 ,p02 x
It is clear from above that W T A and W T P for a given lottery can be interpreted
as expected values of outcomes of this lottery, where probabilities are subjective
and different for both measures. The probability assigned to the high outcome
is higher in the case of W T A than in the case of W T P , and the opposite for
the low outcome.
This relation holds for general lotteries with many outcomes as shown in Yaari
(1987)1 , but the closed form solution can be found only in particular cases as
one described in this section.
3
Two examples
To give motivation for the choice rule I am about to introduce I will describe
two paradoxes of choice patterns observed systematically in experiments which
are not explained by the classical theory.
The first example comes from Tversky and Kahneman (1981) and it is called
an Asian disease paradox. It shows how framing issues may affect choices
between lotteries. It will motivate the first hypothesis we state that people
make different choices depending whether they are to choose among lotteries
with non-positive outcomes or among lotteries with non-negative outcomes.
The second example comes from Grether and Plott (1979) and it is called a
preference reversal paradox. It shows how valuing and choosing tasks differ. In
the classical theory valuing and choosing tasks are indistinguishable and hence
the observed difference is treated as a paradox. This example will motivate the
second hypothesis concerning the difference in valuing and choosing tasks.
1
The argument has to be adapted slightly to account for the fact that we consider W T A
and W T P measures and not expected utility or prospect theory evaluation function.
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W T A, W T P and reference point
3.1
Michal Lewandowski
Asian disease paradox
This example is motivated by experimental results such as in (Tversky and
Kahneman, 1986). The following paradox illustrates the role of framing in decision problems and suggests a simple way of dealing with it.
Suppose there are 600 people who are touched by some serious disease or epidemic. The decision makers are asked to choose among two treatments which
lead to certain consequences:
• treatment A: 200 people saved for sure
• treatment B: 1:1 chance of either 400 or 0 people saved
The well documented experiments reveal that when faced with such a choice,
the majority of respondents commonly chooses treatment A. Imagine now that
we are asked to choose among yet the other two options which are as follows:
• treatment A’: 400 people die for sure
• treatment B’: 1:1 chance of either 600 or 200 people dying
Now the same experiments reveal that when faced with such a choice, the
majority of respondents commonly chooses treatment B’.
If we look at the above two situations, it is not hard to notice that treatment
A is equivalent to treatment A’ and treatment B is equivalent to treatment
B’. The only difference comes from framing differently the two options we are
supposed to choose from. In case of choice among option A and option B
the implicit reference point was that all people die. The choice was about
how many people we can save. In case of choice among option A’ and B’ the
implicit reference point was that all people survive. The choice then was how
many people will be lost. Obviously, many people from the experiments did
not realize this equivalence, because otherwise the minimal rationality requires
voting consistently for the equivalent alternative.
From this example we can deduce several implications. First, the individual
reference point can be influenced by the induced or framing-related reference
point if we agree on the fact that people make choices relative to their individual
reference points. Although it is a very interesting issue the analysis of a framingrelated reference points lies outside the scope of this paper and hence I will not
elaborate on this here. Second, people take risk in the domain of losses but
refrain from risk in the domain of gains. This clearly supports our choice of the
utility function (see assumption 1). Third, maybe there is a mental difference
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W T A, W T P and reference point
Michal Lewandowski
between considering lotteries with only positive outcomes versus lotteries with
only negative outcomes. If something has some positive value to us, we call it a
good and we are willing to pay for it. We want to buy goods. If something has
some negative value to us, we call it a bad and we want to get rid of this. We
can think of buying a negative lottery as selling a positive one. And conversely,
selling a negative lottery is buying a positive one. We will try to show later on
that we can make the same kind of thought exercise for the case of valuing vs.
choosing tasks.
Let us now turn to formalizing this example and then we can draw some further
conclusions. Consider the first choice situation. We have to choose between the
two lotteries: x1 = (200, 1) and y1 = (0, 0.5; 400, 0.5). From definition of W T A,
W T P and using assumption 1, we have the following equalities:
1
1
u(400 − W T P (y1 )) − λu(W T P (y1 )) = 0
2
2
1
1
− λu(400 − W T A(y1 )) + u(W T A(y1 )) = 0
2
2
Since λ > 1 and the function u(.) is strictly increasing, it follows that:
W T P (y1 ) < 200 = W T P (x1 )
(8)
W T A(y1 ) > 200 = W T A(x1 )
(9)
In the second choice situation we have to choose between two lotteries: x2 =
(−400, 1) and y2 = (−200, 0.5; −600, 0.5). As already pointed out, the following
relations hold:
x2 = x1 − 600
y2 = y1 − 600
By property 3 we know that shifting all outcomes in some lottery shifts the
willingness-to-pay and willingness-to accept for this lottery by the same amount.
In our case it means that:
W T P (y2 ) < −400 = W T P (x2 )
(10)
W T A(y2 ) > −400 = W T A(x2 )
(11)
Suppose that people make choices on the basis of W T P and W T A. Suppose also
that the majority of people in the experiments internalized a framing related
reference points in both choice situations. The above equations (8), (9), (10) and
(11) suggest that in the choice between treatment A and B people were choosing
on the basis of W T P (W T P (y1 ) < W T P (x1 )) and in the choice between
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Michal Lewandowski
treatment A’ and B’ the same people were choosing on the basis of W T A
(W T A(y1 ) > W T A(x1 )). Based on this observation, we state the following
hypothesis:
Hypothesis 1 Among lotteries with non-negative outcomes people choose
on the basis of W T P , whereas among lotteries with non-positive outcomes
people choose on the basis of W T A.
It is a hypothesis coming straight from experiments. It doesn’t have any
other motivation. However it has an intuition. There are two ways to understand this intuition. One simply comes from words. We are willing to accept
a loss and willing to pay for a gain. The other way comes from Cumulative
Prospect Theory. This theory suggests that decisions observed in experiments
are reflexive around zero. Most of the patterns observed for non-negative lotteries have an equivalent reflexive patterns observed for non-positive lotteries.
And according to property 4 which says that W T A(x) = −W T P (−x), W T A
and W T P measures are also reflexive around zero.
When we already discuss the bridge between the properties of W T P and W T A
measures and experimentally observed patterns of choice, it is tempting to check
whether the implication of property 6 is also appealing as considered in relation to the experimental data. Property 6 states in particular that for a utility
function homogenous of any positive degree and any scalar θ > 0 and lottery x
we have that: W T A(θx) = θW T A(x) and the same for W T P (x). This clearly
implies that scaling down all the outcomes of all the lotteries to be compared
either in terms of choice or in terms of value should not affect the comparison.
This is obviously true only if we are willing to accept the above hypothesis as
well as the homogeneity assumption regarding utility function.
3.2
Preference reversal example
Preference reversal arises in the following way. Consider two lotteries: x = (x, p)
2
and y = (y, q), where y > x > 0 and 1 > p > q > 0. It is important to notice
that all the outcomes of these lotteries are non-negative. Many experiments
(e.g. Slovic and Lichtenstein (1983), Tversky et al. (1990), Tversky and Thaler
(1990), Loomes et al. (2003)) reveal that people often choose a lottery x over
y, but assign higher willingness-to-accept to a lottery y. Additionally, which is
important part of a paradox, the opposite pattern happens less frequently. This
2
This notation means that with probability p outcome x is generated and with probability
1 − p outcome 0 is generated.
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Michal Lewandowski
additional condition is important because it shows that the asymmetry observed
in experiments is not balanced by the opposite asymmetry. This phenomenon is
called preference reversal. There were many explanations of this phenomenon
in the literature. My explanation is closest to that of Schmidt et al. (2005)
and is as follows. When people are to choose a lottery which has nonnegative
outcomes they think how much I am willing to pay for this. So my intuition
would be to use W T P in a choosing task. When people are asked to value a
lottery, they often think in terms of how much reward they are going to accept
for a loss of this lottery. This is like with selling. That’s why my intuition
would be to use W T A for a valuing task.3 The second hypothesis is therefore
as follows:
Hypothesis 2 Among lotteries with non-negative outcomes people choose/buy
on the basis of W T P and ”value”/sell on the basis of W T A.
To get testable predictions out of this hypothesis it is instructive to conduct a
calibration exercise. I will try to show on the basis of the specific example what
are the conditions for the preference reversal to hold. This can be then used to
test whether these conditions indeed hold in reality.
I assume that the decision maker originally does not participate in any nondegenerate lottery and he has to choose whether to participate in x or in y.
This is basically equivalent to assuming that a decision problem in question can
be separated from other decision problems of an individual. Hence, suppose
that people choose lotteries on the basis of their willingness-to-pay and they
value lotteries on the basis of their willingness-to-accept. To obtain preference
reversal we need the following pattern of preferences: W T Ax < W T Ay and
W T Px > W T Py . We also assume that agents are loss averse (λ > 1), and hence
by property 2 from the second section W T Ax > W T Px and W T Ay > W T Py .
From the definition, we obtain:
pu(W T Ax − x) + (1 − p)u(W T Ax ) = 0 ⇒
pu(x − W T Px ) + (1 − p)u(−W T Px ) = 0 ⇒
u(x−W T Ax )
u(W T Ax )
u(x−W T Px )
u(W T Px )
=
(1−p)
λp
=
λ(1−p)
p
Let’s assume that u(x) = axα , for x > 0. This defines U (x) using Assumption
1. Additionally we assume without loss of generality4 that p, q < 1/2. We can
3
In many experiments illustrating preference reversal the subjects were even asked to state
directly their certainty equivalent which in our case corresponds to Willingness-to-accept.
4
The same can be shown with p, q < 1/2 and q < 1/2 < p.
14
W T A, W T P and reference point
Michal Lewandowski
calculate Willingness-To-Pay and Willingness-To-Accept for this example:
W T Ax =
W T Px =
(λp)1/α
x
(1−p)1/α +(λp)1/α
(p)1/α
x
(λ(1−p))1/α +(p)1/α
In the same way, we obtain W T Py and W T Ay . It follows that to obtain
preference reversal we need that the following condition happens more often
than the opposite condition:
1
1
(λq) α
1
1
(1 − q) α + (λq) α
(λp) α
y>
x
1
1
>
1
1
(1 − p) α + (λp) α
(p) α
1
1
(λ(1 − p)) α + (p) α
x>
(q) α
1
1
(λ(1 − q)) α + (q) α
y
Combining these inequalities gives:
1
1
1
1
(1 − p) α + (λp) α
(1 − q) α + (λq) α
1
>
pα x
1
>
qαy
1
1
1
1
(λ(1 − p)) α + p α
(λ(1 − q)) α + q α
I will consider two cases. One in which E[x] = E[y], so that expected values of
both lotteries are the same and one in which E[U (x)] = E[U (y)], so that the
expected utility of both lotteries are the same.
In the first case we assume that px = qy and
we obtain the condition that:
1
pα x
1
qαy
1
pα x
1
qαy
= ( pq )
1−α
α
and in the second
= 1 Let’s consider first the case with equal
expected values. We obtain the following two conditions on a loss aversion
parameter.
1
1
λα
>
1
1
1
p α −1 (1 − q) α − q α −1 (1 − p) α
1
(pq) α −1 (p − q)
1
1
λα
>
(pq) α −1 (p − q)
1
1
1
1
p α −1 (1 − q) α − q α −1 (1 − p) α
Observe that the first inequality implies the second as shown below:
1
1
1
1
p α −1 (1 − q) α − q α −1 (1 − p) α
1
1
= (p(1 − q)) α −1 (1 − q) − (q(1 − p)) α −1 (1 − p)
1
> ((p(1 − q)) α −1 )(p − q)
1
> ((pq) α −1 )(p − q) > 0
where the first inequality follows from the fact that p > q and then obviously
1 − q > 1 − p, and the second inequality follows from our additional assumption
that q < 1/2.
15
W T A, W T P and reference point
Michal Lewandowski
The case with equal expected utilities is analyzed similarly. I summarize it
below:
Case 1: E(x) = E(y): Preference reversal occurs when the first condition
below happens more often than the second condition below:
!α
1
1
1
1
p α −1 (1 − q) α − q α −1 (1 − p) α
>1
λ >
1
(pq) α −1 (p − q)
!α
1
(pq) α −1 (p − q)
λ <
<1
1
1
1
1
p α −1 (1 − q) α − q α −1 (1 − p) α
Case 2: E(U (x)) = E(U (y)): Preference reversal occurs when the first condition below happens more often than the second condition below:
!α
1
1
(1 − q) α − (1 − p) α
λ >
>1
1
1
pα − q α
!α
1
1
pα − q α
λ <
<1
1
1
(1 − q) α − (1 − p) α
It is immediately clear that if the decision maker is loss averse (λ > 1), then
the second condition in both case 1 and case 2 cannot be satisfied. Hence, to
show whether and for which parameters is preference reversal possible in our
model it is sufficient to consider only the first condition in both cases.
We focus here on the second case but the results for the first case can be found
in the appendix. In the second case we assume that lotteries and preferences
are such that the Expected Utility decision maker should be indifferent between
the two lotteries. In such setting we are looking for the values of parameters
(p, q, λ, α) for which preference reversal occurs. In the appendix in figure 1 we
present tables with cut-off values for some values of parameters α, p, q.
5
Below
we summarize few observations concerning values given in these tables.
• There is preference reversal for relevant values of parameters - rough results
• More risk aversion and less loss aversion makes preference reversal more
pronounced - testable prediction
• Lotteries with both probabilities p and q close to zero or both close to one
generate preference reversal less likely - testable prediction
5
We no longer use an assumption that p, q < 1/2. It is straightforward to modify the
conditions in cases of p, q > 1/2 and q < 1/2 < p. The table presents results for these cases
as well.
16
W T A, W T P and reference point
Michal Lewandowski
To make the above inferences we used a couple of empirical facts. We consider
cases with constant relative risk aversion equal to: α = 0.68, 0.78, 0.88, 0.98.
Kahneman and Tversky (1979) find that, on average and assuming that people
are CRRA, people have a coefficient of α = 0.88. The average loss aversion is
reported to be around λ = 2.
The conclusion and testable predictions we stated above are also valid for the
case of E[x] = E[y]. The only difference is that for this case lotteries with
both probabilities p and q close to zero generate preference reversal more likely,
however for lotteries with probabilities both close to one it is already not the
case. Figure 2 in the appendix presents cut-off values for λ in the case of
E[x] = E[y].
4
A decision rule based on W T A and W T P measures
The two examples from the previous section and the hypotheses which were
stated on the basis of them will be now used to design a rule of decision making
under risk. The Asian disease example shows that people make different choices
for negative vs. positive lotteries. We call this difference positive-negative.
The preference reversal suggests that people use different logic when valuing
and when choosing lotteries. We call this phenomenon valuing-choosing difference. Let me start with the case of non-negative lotteries.
4.1
The case of non-negative lotteries
I will first consider a very simple setting. Suppose we have two non-degenerate
lotteries x, y distributed on a non-negative support according to F, G ∈ F,
respectively and two periods. In the first period the decision maker decides
upon participation in one of the lotteries. In the second period the uncertainty
is resolved and outcomes are realized. In our setting with loss aversion it is
crucial to distinguish between three situations:
• a decision maker does not originally participate in any non-degenerate
lottery and has to choose between participating in x and y
• a decision maker originally participates in lottery y and considers switching to a different lottery x, otherwise she stays with y
• a decision maker originally participates in both lotteries and has to choose
which lottery to give up
17
W T A, W T P and reference point
Michal Lewandowski
The first situation will correspond to what we will call ”choosing decision”, the
third situation will correspond to what we will call ”valuing decision” and the
second situation will be a mixture of ”valuing” and ”choosing” decisions.
Decision Rule 1 The case of non-negative lotteries6
• Rule 1+ Choose x over y if: U (W T P (x)) > U (W T P (y))
• Rule 2+ Switch from y to x if: U (W T P (x)) > U (W T A(y))
• Rule 3+ Value x more than y if: U (W T A(x)) > U (W T A(y))
Obviously, since utility function is strictly increasing, we could write these rules
only in terms of arguments of the utility functions and drop the functions. We
keep them because they will be important in the case of mixed lotteries.7
What is the intuition underlying the above rules. For instance, let’s consider
the second situation in which a decision maker considers switching from one
lottery to the other. We can think of this as follows. A decision maker first
asks herself a question what is the least price she demands in exchange for
the lottery she participates in (lottery W T A(y)). Then she thinks what is the
maximal price she is willing to pay for the other lottery i.e. W T P (x). If the
first price is smaller than the second one then she should switch. It means that
she is paying W T A(y) for the object which is worth to her W T P (x). And the
condition W T P (y) > W T A(y) ensures that she wants to do it. Such condition
may be interpreted as a condition for trade.
The above behavioral decision rule can be used to yield testable predictions
of the model underlying it. In the next subsection I am going to introduce
an example. This example is aimed at illustrating a status quo bias - another
well known paradox in decision theory. This term, status quo bias, was first
mentioned in Samuelson and Zeckhauser (1988). A version of this paradox
which I present in this paper shows the difference between switching from one
lottery to the other versus choosing between two lotteries. The key issue here
is the departure point or a status quo i.e. whether a decision maker originally
participates in one of the two lotteries or not.
6
7
We refer to non-negative lottery as to a lottery which does not have negative outcomes.
We refer to mixed lotteries as to lotteries which have both positive and negative outcomes.
18
W T A, W T P and reference point
4.2
Michal Lewandowski
Status quo bias
Status quo bias arises in a following way.8 Consider two lotteries and two decision makers. Assume that preferences are identical for both agents and there
is no uncertainty regarding lotteries. One decision maker (agent 1) originally
participates in one of the lotteries, whereas the other (agent 2) does not participate in any of them. The traditional theory i.e. Expected Utility Theory
does not distinguish between these decision makers since they share the same
preferences. However, experimental evidence such as in Kahneman et al. (1991)
or Samuelson and Zeckhauser (1988) shows that it often happens that agent 1
chooses to keep the lottery she has participated in before but agent 2 chooses
to participate in the other lottery. Such a situation is called status quo bias,
because agent 1 chooses to remain in her status quo, although she would have
chosen the other lottery, had she not participated in any lottery originally.
Let’s use the decision rule 1 above to see what are the conditions implied
by our model for status quo bias to occur. Let me write once more some of
the conditions written earlier in the paper which are necessary to analyze our
example:
• Rule 1+ Choose x over y if: U (W T P (x)) > U (W T P (y))
• Rule 2+ Switch from y to x if: U (W T P (x)) > U (W T A(y))
• Property 2 λ S 1 ⇔ W T AF S W T PF
Since we focus on the case with loss aversion, where λ > 1, by property 2 we
have W T A(y) > W T P (y). Because the utility function is strictly increasing we
see that the condition in Rule 2+ implies the condition in Rule 1+. Therefore,
we arrive at the following prediction of our model:
• Status quo bias occurs for W T P (x) ∈ (W T P (y), W T A(y))
The above condition is a testable prediction and can be checked empirically.
I should stress once more here, that all explanations to paradoxical behavior
in this model are based on loss aversion, because the difference in W T A and
W T P measures in our model stems from loss aversion, captured in a parameter
λ.
8
This version of the status quo bias is tailored to fit the purposes of this paper. The
original version of the status quo bias was analyzed among others in Kahneman et al. (1991)and
Samuelson and Zeckhauser (1988). The underlying idea of the original version and the current
one is however the same.
19
W T A, W T P and reference point
4.3
Michal Lewandowski
An extension to the negative and mixed lotteries
I will now offer an extension of the decision rule 1 proposed above to the case of
lotteries with non-positive outcomes and lotteries with both positive and negative outcomes. The following simple table summarizes the main idea underlying
the rules proposed in this paper:
Positive lotteries
Negative lotteries
Choose/Buy
WTP
WTA
Value/Sell
WTA
WTP
Basically, the rule is that for negative lotteries everything is the opposite
of the positive lotteries. As already mentioned, it is very intuitive since selling
positive is buying negative and buying positive is selling negative. It follows
that we can rewrite the Decision Rule 1 above for the case of non-positive
lotteries just by exchanging W T P and W T A everywhere. Because this relation
is so straightforward (it mirrors the rule for non-negative lotteries) we will not
state it explicitly, it should rather be treated as a special case of mixed lotteries
which we now turn to. Suppose we have an ordered lottery x(or prospect) in
which 0 ∈ [xk , xk+1 ]. We can define a positive part of this lottery (denoted x+ )
and a negative part of this lottery (denoted x− ):
x = (x1 , p1 ; ...; xk , pk ; xk+1 , pk+1 ; ...; xn , pn )
x+ = (0, p1 + ... + pk ; xk+1 , pk+1 ; ...; xn , pn )
x− = (x1 , p1 ; ...; xk , pk ; 0, pk+1 + ... + pn )
As before, there are three rules corresponding to three different situations:
choosing, switching, ”valuing”:
Decision Rule 2 The case of mixed lotteries
• Rule 1+/- Choose x over y if:
U (W T P (x+ )) + U (W T A(x− )) > U (W T P (y+ )) + U (W T A(y− ))
• Rule 2+/- Switch from y to x if:
U (W T P (x+ )) + U (W T A(x− )) > U (W T A(y+ )) + U (W T P (y− ))
• Rule 3+/- Value x more than y if:
U (W T A(x+ )) + U (W T P (x− )) > U (W T A(y+ ) + U (W T P (y− )))
20
W T A, W T P and reference point
Michal Lewandowski
These rules may be rewritten also in a different form, for instance a Rule 1+/can be written as:
U −1 u(W T P (x+ )) − λu(−W T A(x− ))
> U −1 u(W T P (y+ )) − λu(−W T A(y− ))
The idea behind these rules is to separately evaluate the loss and the gain part
of a lottery and then to combine them together in one measure. We combine
the two parts additively since it is in line with Cumulative Prospect Theory,
which also treats separately gains and losses and combines them additively.
This additivity in gains and losses is however a questionable part. In general
we could evaluate a mixed lottery on a basis of a function T : R2 → R. For
instance Rule 1+/- would then be written as:
T (W T P (x+ ), W T A(x− )) > T (W T P (y+ ), W T A(y− ))
5
A connection to the Cumulative Prospect Theory
and the determination of a reference point
It should be stressed here that we present here a very restricted version of the
Cumulative Prospect Theory since we analyze decisions under risk and not uncertainty and we do not distort probabilities. In this case Cumulative Prospect
Theory is equivalent to Prospect Theory. For the brief introduction to Prospect
Theory, Cumulative Prospect Theory and more about restrictions and simplifying assumptions we make here see Appendix. The Cumulative Prospect Theory
decision rule can be written as:
CPT decision rule
point R ∈ R if:
Choose lottery x ∼ F over y ∼ G, given a reference
EF [U (x − R)] > EG [U (y − R)]
As mentioned before, one of the important things in Prospect Theory is the
reference point determination. Most authors take it exogenously and depending on the context they usually use current wealth position or status quo outcome as a reference point. There are however also situations in which it is
hard or impossible to find a good reference point. In this paper I am neither
proposing an endogenous reference point nor give I a reference point that is
better than the existing ones. What I suggest is merely a reference point which
makes Cumulative Prospect Theory equivalent to the decision rules based on
21
W T A, W T P and reference point
Michal Lewandowski
W T A/W T P measures described in the former section. Unfortunately, this
equivalency holds only for the case of non-negative lotteries and for the case of
non-positive lotteries but it doesn’t work for the general case of mixed lotteries.
Proposition 1 Given two non-negative lotteries x and y distributed according
to cdf F and G, respectively, the following equivalences hold9 :
U (W T P (x)) ≥ U (W T P (y)) ⇔ EF [U (x − W T P (y))] ≥ 0
(12)
U (W T P (x)) ≥ U (W T A(y)) ⇔ EF [U (x − W T A(y))] ≥ 0
(13)
U (W T A(x)) ≥ U (W T A(y)) ⇔ EF [U (W T A(y) − x)] ≤ 0
(14)
Proof:
We prove only condition (12). Conditions (13) and (14) are proved analogously. The definition of W T P (x) is: EF [U (x − W T P (x))] = 0. Since U (.)
is strictly increasing, W T P (x) is determined uniquely. Furthermore, the strict
monotonicity of U (.) implies: U (W T P (x)) ≥ U (W T P (y)) ⇔ W T P (x) ≥
W T P (y) ⇔ EF [U (x − W T P (y))] ≥ 0. 2
What this proposition tells us is that the Rules 1+, 2+ and 3+, which I proposed before are equivalent to Cumulative Prospect Theory decision-making
with appropriately chosen reference points. To be more precise, it says that if
we decide upon choosing lottery x versus lottery y and we agree on the rule
1+ then we could use the CPT evaluation function for a lottery x with reference point being equal to willingness-to-pay for a lottery y. If we decide upon
switching from lottery y to lottery x, we could use the CPT evaluation function
with W T A(y) being a reference point.
Notice that W T A and W T P measures which are used as reference points are derived from preferences and therefore we can call them quasi-endogenous. They
are not entirely endogenous since the W T A/W T P rules we propose do not follow from the theoretical model but they are merely suggestions inspired by the
experimental literature. If there are grounds to believe that the decision rules
based on W T P /W T A describe well the behavior of human beings then there is
also support for W T P /W T A reference points in the Cumulative Prospect Theory. Beyond demonstrating the connection between W T A/W T P decision rules
and the CPT theory, there is yet another aspect of the above equivalences which
casts some light on the reference point determination. For instance, we may ask
a question when it is appropriate to take the decision maker’s wealth as a reference for evaluating lotteries. The above proposition answers this question. It is
9
The same equivalences hold for non-positive lotteries.
22
W T A, W T P and reference point
Michal Lewandowski
appropriate namely then, when the original lottery (the point of departure) is
degenerate and equal to wealth for all states of the world. More precisely, given
some level of wealth in the first period, call it W , it is appropriate to take W as
a reference point to evaluate lotteries, if this wealth level will remain unchanged
in the second period almost surely, i.e. with probability one. Notice that if x
is a degenerate lottery (x = W ∈ R a.s), then W T P (x) = W T A(x) = W .
6
Conclusion
In this paper I proposed a behavioral rule which is designed to accommodate
certain patterns of behavior analyzed in experimental literature. This rule
is constructed using Willingness-To-Pay and Willingness-To-Accept. In my
model, disparity between these two measures comes solely from loss aversion.
Therefore, the patterns of behavior which I analyze here are explained by loss
aversion, although other explanations are also conceivable. Since I make specific assumptions in the model, testable predictions are obtained which can be
used to justify the decision rule. The conjecture is also, that the decision rule
accommodates patterns of behavior in many situations, apart from these analyzed here, in which loss aversion plays crucial rule. This also can be tested
empirically.
The second contribution of this paper is to show the equivalence of making
decisions according to the proposed decision rule and according to reference
dependent theories such as Cumulative Prospect Theory with appropriately
chosen reference points. Technically, this equivalence is almost trivial as represented in the simplicity of the proof, but conceptually it is an important
relationship. First, it suggests what reference point should be set to describe
behavior. This is a relevant problem especially in cases described in the introduction, in which reference points cannot be simply derived from context.
Second, the equivalence links reference dependent theories with the decision
rule which is propose and hence it is possible to test these theories by testing
the decision rule and vice versa. So the rationale for reference points constructed on the basis of Willingness-To-Pay and Willingness-To-Accept lies in
the conjectured strength of the proposed decision rule.
23
W T A, W T P and reference point
7
Michal Lewandowski
Appendix: A Brief Introduction to Cumulative
Prospect Theory
To maintain sufficient level of generality I will sketch the Cumulative Prospect
Theory by Tversky and Kahneman (1992) (CPT) for the case of choice under
uncertainty as opposed to choice under risk. It is easy to retreat to the less
general case starting from the more general one but the opposite process is more
difficult.
There are two components of Cumulative Prospect Theory:
• Prospect Theory (Kahneman and Tversky, 1979)
• Rank Dependence (Schmeidler (1989), Gilboa and Schmeidler (1989),
Quiggin (1982))
Both these components were theories born out of the need to improve on
Expected Utility Theory. Expected Utility Theory generated many well documented choice paradoxes which left the economists unsatisfied. These paradoxes
fall into three main themes:
• violations of independence axiom e.g. common consequence effect such as
Allais paradox (Allais, 1953) or common ratio effect
• violations of descriptive and procedural invariance e.g. framing effects or
preference reversal
• source dependence e.g. ambiguity aversion and Ellsberg paradox (Ellsberg, 1961)
Roughly speaking, prospect theory as a positive theory of choice motivated
by psychological experiments was aimed at explaining paradoxes which fall
under first two of the above three themes. On the other hand, normative
models which examined rank dependence were motivated mainly by the third
of the above three themes.
Before proceeding to sketch the two theories underlying Cumulative Prospect
Theory one remark concerning terminology. Under uncertainty the object of
choice is called an act and it is a collection of monetary outcomes and events
associated with them. These events partition a state space. Under risk the
object of choice is called a lottery and it is a collection of monetary outcomes,
events associated with them and additionally probability measure defined over
these events. These probabilities sum up to one. Prospect is a term used in
24
W T A, W T P and reference point
Michal Lewandowski
the reference dependent models only and depending on the context it can be a
reference dependent act or a reference dependent lottery. Prospect Theory was
originally developed for situation under risk and therefore I will use lotteries
and probabilities to describe it. Rank Dependent model of Gilboa and Schmeidler (1989) deal with the case of uncertainty and therefore I will use acts and
events to describe it.
Prospect Theory contains several elements which differ from the benchmark
Expected Utility model. Outcomes of a lottery are all coded as losses or gains,
where the exogenous cut-off outcome is called a reference point. Utility for
losses is convex and for gains it is concave. This characteristic S-shape of the
utility function under prospect theory is termed diminishing sensitivity in
utility as going away from the reference point diminishes marginal utility in
both directions. It means that the decision maker is willing to take risk in the
domain of losses and dislikes to take risk in the domain of gains. Additionally
the utility for losses is steeper than for gains which is termed loss aversion.
It reflects the fact that people dislike losses more than they like gains. Additionally probabilities are weighted in prospect theory. Small probabilities are
overweighted and high probabilities underweighted. This gives a characteristic
inverse-S shape of the weighting function and is called diminishing sensitivity in probability weighting, as going away from probability zero upwards
or going away from probability one downwards diminishes the marginal probability weight.
This is a very short description of the main elements in prospect theory. The
problem with prospect theory is the implicit assumption that the probability
weight for a given state depends only on this state or only on the probability of
this state. This assumption together with nonlinear weighting function causes
violation of monotonicity10 which is hard to reconcile with any kind of reasonable choice pattern. This problem is solved however under rank dependent
models, which we discuss here for the more general case of uncertainty. Imagine
we define a rank ordered act (x1 , E1 ; ...; xn , En ); x1 < x2 < ... < xn , where
xi , for i = 1, ...n are outcomes and Ei , for i = 1, ..., n are associated events
which form a partition of a state space. Rank dependent models assume that
the decision weight associated with a given event Ei should depend on the event
itself and on the rank of this event which is defined as Di ≡ E1 ∪E2 ∪...∪Ei . This
assumption is called rank dependence and together with the monotonicity
10
Monotonicity assumption states that first order stochastically dominating acts are pre-
ferred to acts which they dominate
25
W T A, W T P and reference point
Michal Lewandowski
assumption imply the so called Choquet expected utility VCEU =
Pn
i=1 πi u(xi ),
where πi = W (Ei ∪ ... ∪ En ) − W (Ei+1 ∪ ... ∪ En ). A function W (.) is called
a capacity and it generalizes the concept of a probability measure. Notice that
with probability measure one could not account for nonadditive decision weights
because probability measure is additive i.e. if µ is a probability measure and
A, B any disjoint events from the state space then µ(A ∪ B) = µ(A) + µ(B).
Capacity is not additive and hence allows nonadditive weights. The additivity
assumption is replaced by much weaker monotonicity requirement, i.e. if A ⊂ B
then W (A) ≤ W (B).
I sketched briefly prospect theory and rank dependence. Let me now show
how they come together and form Cumulative Prospect Theory. Notice that
while describing rank dependent model I used the term act and not prospect.
Prospect in this context would be an act rescaled by the exogenous reference
point. If reference point is a constant as with the original formulation and is
denoted R ∈ R and y : σ(Ω) → S is an act then f ) = y − R is a prospect
derived from this act. Suppose we are given a rank ordered prospect:
f = y − R ≡ (x1 , E1 ; ...; xk , Ek ; xk+1 , Ek+1 ; ...; xn , En )
where x1 < ... < xk < 0 < xk+1 < ... < xn
Define:
f + ≡ (0, E1 ∪ ... ∪ Ek ; xk+1 , Ek+1 ; ...; xn , En )
πi+ = W + (Ei ∪ ... ∪ En ) − W + (Ei+1 ∪ ... ∪ En ) for i = k + 1, ..., n
πn+ = W + (En )
f − ≡ (x1 , E1 ; ...; xk , Ek ; 0, Ek+1 ∪ ... ∪ En )
πi− = W − (E1 ∪ ... ∪ Ei ) − W − (E1 ∪ ... ∪ Ei−1 ) for i = 1, ..., k
π1− = W − (E1 )
where f + and f − is a positive and a negative part of the prospect, respectively.
πi+ and πi− are weights associated with the positive and negative part of the
prospect, respectively. We also have two different capacities, denoted W + and
W − , for losses and for gains. Having defined all this, the Cumulative Prospect
Theory evaluates prospects according to the following evaluation function:
VCP T (f ) =
k
X
πi− u(xi ) +
i=1
VCP T (y − R) =
n
X
πi+ u(xi )
i=k+1
k
X
i=1
πi− u(yi − R) +
n
X
πi+ u(yi − R)
i=k+1
26
W T A, W T P and reference point
Michal Lewandowski
The first equation above is written directly in terms of a prospect and the
second equation is written in terms of an act and a reference point to stress
the reference dependence aspect of Cumulative Prospect Theory. Obviously
the Cumulative Prospect Theory can be easily presented for the case of risk as
well. Given a lottery x, reference point R and the associated cdf F the CPT
evaluation function for the general outcome spaces is as follows:
VCP T (x, R, F ) =
Z 0
Z
=
u(x − R)d[w− (F (x − R))] +
−∞
Z 0
u(x − R)d[w+ (1 − F (x − R))]
∞
Z ∞
u(x − R)d[w+ (F (x − R))]
u(x − R)d[w− (F (x − R))] +
=
0
−∞
where F (x − R) =
0
Rx
−∞ dp
is a cumulative distribution function for outcomes of
a prospect.
Discrete case is a special example of this formulation. We can just set
P
p(x − R) = i δxi pi , where δx is a Dirac probability mass at x and probabilities
satisfy the usual requirements.
As I hope is visible from the above formulation, Cumulative Prospect Theory
combines Prospect Theory with Rank Dependent models. Additionally, there
is a novel feature which is called sign dependence, which merely means that
the model allows to use different weighting functions for losses and for gains.
What I sketched is a very general setup. In the paper I make many simplifications to focus on the relevant issue. First, I analyze a decisions under risk
instead of uncertainty. Second, my model does not contain probability weighting i.e. weighting functions are identity functions. Third, I am using a very
special kind of utility function. This utility is very useful because it separates
different individual characteristics in a clear-cut manner. Risk attitudes intensities are the same for losses and gains (but still: risk loving for losses and risk
aversion for gains). Loss aversion is a constant and is embedded in one parameter. All these simplifications serve one purpose. I want to analyze reference
dependence aspect of the theory and hence I reduce the complexity of all other
aspects such as decision weights, nonexistence of objective probabilities and so
on.
27
W T A, W T P and reference point
Michal Lewandowski
for alpha=0.68
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.5
0
2.0350
1.7660
1.5857
1.4492
1.3380
1.2428
1.1575
1.0782
1.0000
0.15
0
0
1.5609
1.4272
1.3196
1.2279
1.1466
1.0717
1.0000
1.0782
0.25
0
0
0
1.3123
1.2211
1.1412
1.0684
1.0000
1.0717
1.1575
0.35
0
0
0
0
1.1388
1.0668
1.0000
1.0684
1.1466
1.2428
0.45
0
0
0
0
0
1.0000
1.0668
1.1412
1.2279
1.3380
0.55
0
0
0
0
0
0
1.1388
1.2211
1.3196
1.4492
0.65
0
0
0
0
0
0
0
1.3123
1.4272
1.5857
0.75
0
0
0
0
0
0
0
0
1.5609
1.7660
0.85
0
0
0
0
0
0
0
0
0
2.0350
0.25
0
0
0
1.2056
1.1477
1.0957
1.0471
1.0000
1.0502
1.1114
0.35
0
0
0
0
1.0936
1.0456
1.0000
1.0471
1.1007
1.1678
0.45
0
0
0
0
0
1.0000
1.0456
1.0957
1.1538
1.2287
0.55
0
0
0
0
0
0
1.0936
1.1477
1.2119
1.2976
0.65
0
0
0
0
0
0
0
1.2056
1.2784
1.3794
0.75
0
0
0
0
0
0
0
0
1.3587
1.4838
0.85
0
0
0
0
0
0
0
0
0
1.6329
0.25
0
0
0
1.1075
1.0782
1.0514
1.0256
1.0000
1.0277
1.0618
0.35
0
0
0
0
1.0500
1.0247
1.0000
1.0256
1.0546
1.0911
0.45
0
0
0
0
0
1.0000
1.0247
1.0514
1.0820
1.1218
0.55
0
0
0
0
0
0
1.0500
1.0782
1.1113
1.1555
0.65
0
0
0
0
0
0
0
1.1075
1.1439
1.1943
0.75
0
0
0
0
0
0
0
0
1.1822
1.2421
0.85
0
0
0
0
0
0
0
0
0
1.3077
0.25
0
0
0
1.0172
1.0127
1.0084
1.0043
1.0000
1.0046
1.0104
0.35
0
0
0
0
1.0082
1.0041
1.0000
1.0043
1.0090
1.0150
0.45
0
0
0
0
0
1.0000
1.0041
1.0084
1.0133
1.0197
0.55
0
0
0
0
0
0
1.0082
1.0127
1.0178
1.0247
0.65
0
0
0
0
0
0
0
1.0172
1.0227
1.0303
0.75
0
0
0
0
0
0
0
0
1.0283
1.0370
0.85
0
0
0
0
0
0
0
0
0
1.0458
for alpha=0.78
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
1.6329
1.4838
1.3794
1.2976
1.2287
1.1678
1.1114
1.0569
1.0000
0.15
0
0
1.3587
1.2784
1.2119
1.1538
1.1007
1.0502
1.0000
1.0569
for alpha=0.88
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
1.3077
1.2421
1.1943
1.1555
1.1218
1.0911
1.0618
1.0324
1.0000
0.15
0
0
1.1822
1.1439
1.1113
1.0820
1.0546
1.0277
1.0000
1.0324
for alpha=0.98
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
1.0458
1.0370
1.0303
1.0247
1.0197
1.0150
1.0104
1.0056
1.0000
0.15
0
0
1.0283
1.0227
1.0178
1.0133
1.0090
1.0046
1.0000
1.0056
Figure 1: The case of EU (x) = EU (y). Cut-off values for λ for some values of
parameters α, p, q.
28
W T A, W T P and reference point
Michal Lewandowski
for alpha=0.68
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
18.7467
12.6985
9.7587
7.9824
6.7760
5.8931
5.2119
4.6639
4.2055
0.15
0
0
6.6503
5.2647
4.3943
3.7833
3.3224
2.9561
2.6521
2.3878
0.25
0
0
0
3.8792
3.2663
2.8276
2.4905
2.2173
1.9857
1.7789
0.35
0
0
0
0
2.6535
2.3018
2.0276
1.8019
1.6070
1.4289
0.45
0
0
0
0
0
1.9502
1.7146
1.5180
1.3454
1.1839
0.55
0
0
0
0
0
0
1.4790
1.3019
1.1438
1.0077
0.65
0
0
0
0
0
0
0
1.1247
1.0244
1.2046
0.75
0
0
0
0
0
0
0
0
1.2082
1.4645
0.85
0
0
0
0
0
0
0
0
0
1.8587
0.25
0
0
0
2.4843
2.2020
1.9906
1.8213
1.6781
1.5508
1.4295
0.35
0
0
0
0
1.9197
1.7438
1.6002
1.4765
1.3641
1.2537
0.45
0
0
0
0
0
1.5679
1.4405
1.3288
1.2252
1.1205
0.55
0
0
0
0
0
0
1.3131
1.2093
1.1109
1.0086
0.65
0
0
0
0
0
0
0
1.1054
1.0098
1.1024
0.75
0
0
0
0
0
0
0
0
1.0938
1.2377
0.85
0
0
0
0
0
0
0
0
0
1.4251
0.25
0
0
0
1.6396
1.5298
1.4445
1.3739
1.3120
1.2547
1.1968
0.35
0
0
0
0
1.4199
1.3470
1.2853
1.2301
1.1777
1.1230
0.45
0
0
0
0
0
1.2741
1.2180
1.1668
1.1171
1.0636
0.55
0
0
0
0
0
0
1.1619
1.1132
1.0648
1.0110
0.65
0
0
0
0
0
0
0
1.0645
1.0163
1.0409
0.75
0
0
0
0
0
0
0
0
1.0329
1.1004
0.85
0
0
0
0
0
0
0
0
0
1.1773
0.25
0
0
0
1.0874
1.0737
1.0628
1.0535
1.0451
1.0370
1.0285
0.35
0
0
0
0
1.0601
1.0505
1.0422
1.0345
1.0270
1.0186
0.45
0
0
0
0
0
1.0410
1.0333
1.0260
1.0187
1.0103
0.55
0
0
0
0
0
0
1.0255
1.0185
1.0113
1.0027
0.65
0
0
0
0
0
0
0
1.0115
1.0041
1.0050
0.75
0
0
0
0
0
0
0
0
1.0033
1.0133
0.85
0
0
0
0
0
0
0
0
0
1.0236
for alpha=0.78
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
7.9328
5.7864
4.6857
3.9942
3.5089
3.1430
2.8519
2.6097
2.3977
0.15
0
0
3.6399
3.0621
2.6813
2.4030
2.1850
2.0051
1.8492
1.7058
for alpha=0.88
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
3.4256
2.7409
2.3738
2.1353
1.9631
1.8295
1.7202
1.6262
1.5399
0.15
0
0
2.0561
1.8479
1.7052
1.5974
1.5103
1.4360
1.3692
1.3042
for alpha=0.98
p\q
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
0.05
0
1.2821
1.2092
1.1686
1.1414
1.1214
1.1054
1.0920
1.0801
1.0686
0.15
0
0
1.1362
1.1118
1.0946
1.0812
1.0700
1.0603
1.0512
1.0419
Figure 2: The case of E[x] = E[y]. Cut-off values for λ for some values of
parameters α, p, q.
29
W T A, W T P and reference point
Michal Lewandowski
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