1. No category

How to solve the St Petersburg Paradox in Rank-Dependent
Models?
Marie Pfiffelmann∗
Abstract
Cumulative Prospect Theory, as it was specified by Tversky and Kahneman (1992)
does not explain the St Petersburg Paradox. This study shows that the solutions proposed in the literature (Blavatskyy 2005; Rieger and Wang 2006) to guarantee, under
rank dependent models, finite subjective utilities for any prospects with finite expected
values, have to cope with many limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggest replacing the weighting
functions generally proposed in the literature by another specification which respects
the following properties: 1) to guarantee finite subjective values for all prospects with
finite expected values, the slope at zero has to be finite. 2) to account for the fourfold pattern of risk attitudes, the probability weighting has to be strong enough to
overcome the concavity of the value function.
Keywords : St Petersburg Paradox, Cumulative Prospect Theory, Gambling,
Probability Weighting.
JEL Classification : D81, C91.
∗
LaRGE, Faculty of Business and Economics, Louis Pasteur University. Pˆ
ole Europ´een de Gestion et
d’Economie, 61 avenue de la Forˆet Noire, 67000 Strasbourg, FRANCE. Tel. + 33 (0)3.90.24.21.47 / Fax
+ 33 (0)3.90.24.20.64. E-mail: [email protected].
1
1
Introduction
The St. Petersburg Paradox developed in 1713 by Nicholas Bernoulli shows that for
prospects with infinite expected monetary value decision makers are not willing to pay
an infinite sum of money. This observation can be taken as an evidence against expected
value theory. In fact, according to this theory introduced by Blaise Pascal, individuals
evaluate risky prospects by their expected value. So any decision-maker should accept
to pay an infinite amount of money for prospects with infinite expected value. In 1738,
Daniel Bernoulli proposed a solution to the paradox by introducing the idea of diminishing
marginal utility. He postulated that individuals evaluate prospects not by their expected
value but by their expected utility, where utility is not linearly related to outcomes but
increases at a decreasing rate. So if we consider that individual’s preferences are represented by a strictly increasing and concave utility function this paradox can be solved.
Since this date, the expected utility (EU) theory has been considered as a benchmark for
describing decision making under risk.
However, the Allais paradox (1953) and many other experimental studies (Slovic et
Lichtenstein 1968; Kahneman and Tversky 1979) have reported persistent violations of
EU theory’s axioms. Moreover its behavioral predictions are seriously questioned. On the
one hand, individual preferences for insurance lead to a risk-averse behavior. On the other
hand, acceptance of gambling indicates risk-seeking behavior. Two conflicting behavioral
choices are therefore observed. Cumulative prospect theory (CPT) developed by Tversky
and Kahneman (1992) was proposed as an alternative model to the well established expected utility model. CPT is based on 4 important features: 1) Utility is defined over gains
and losses rather than over final wealth levels. So risky prospects are evaluated relatively
to a reference point. This reference point corresponds to the asset position one expects
to reach. 2) The sensitivity relative to the reference point is decreasing; the value function is then concave over gains and convex over losses. 3) Individuals have asymmetric
perception of gains and losses: they are loss-averse, hence the slope of the value function is steeper for losses than for gains. 4) Individuals do not use objective probabilities
when evaluating risky prospects. They transform probabilities via a weighting function.
2
They overweight the small probabilities of extreme outcomes (events at the tails of the
distribution). Conversely, they underweight outcomes with average probabilities.
CPT is one of the most accepted alternatives to expected utility theory because its
predictions of behavior are consistent with the recently accumulated empirical evidence
on individual’s preferences. Although its predictions are superior to the EU predictions
of behavior, this model must cope with other types of difficulties. Blavatskyy (2005)
emphasized an intrinsic limitation of that model. The overweighting of small probabilities
can lead to the re-occurrence of the St Petersburg Paradox. He showed that the valuation of
a prospect (the subjective utility) by cumulative prospect theory can be infinite. However
it is important to notice that this dilemma is not specific to rank dependent models such
as CPT. It is now well known that even in the expected utility framework the Bernoulli’s
resolution of the paradox is unsatisfactory. Actually, in the EU framework, that is to
say when individuals do not transform probabilities, the introduction of a concave utility
function can resolve the paradox as it was introduced by N. Bernoulli. But the game
can be modified (by making prizes grow sufficiently fast) so that the concavity of the
utility function is not sufficient to guarantee a finite expected utility value1 . Arrow (1974)
proposed to resolve this problem by only considering distributions with finite expected
value. In that case, the concavity of the utility function is sufficient to guarantee, under
the EU framework, a finite valuation. This assumption is quite realistic because neither
individuals nor organizations can offer a prospect with infinite expected value. However,
we cannot implement it in the CPT framework. In fact, in cumulative prospect theory, a
prospect with finite expected value can have an infinite subjective value (Rieger and Wang
2006).
The purpose of the present paper is to determine how we can solve this paradox.
Blavatskyy (2005) and Rieger and Wang (2006) have already proposed some solutions to
guarantee finite subjective values for all prospects with finite expected value. In this study,
we explore the behavioral implications of these propositions. We establish that if we take
them into account, the modified cumulative prospect theory is not anymore consistent with
some behavior observed on the market. There are situations where the CPT modified by
1
This super paradox was first illustrated by Menger (1934).
3
these propositions cannot anymore accommodate both gambling and insurance behavior.
Our aim consists then in developing a new solution to this problem that can take into
account this behavior. Our paper makes thus an important contribution since it proposes
a new specification for CPT that solves the St Petersburg Paradox without introducing
new difficulties and limitations.
The paper is structured in six sections. Section II reviews the cumulative prospect
theory model paying particular attention to the functional form of the value and weighting
functions. In section III, we present how the paradox occurs under CPT and report the
solutions derived by Blavatskyy (2005) and Rieger and Wang (2006) to resolve it. Section
IV analyses the behavioral implications of these propositions. In section V, we propose
a more appropriate solution to this paradox. We illustrate our findings by applying the
different versions of CPT to a famous european lottery game. Section VI concludes with
a summary of our findings.
2
Cumulative Prospect Theory
In this section we briefly present cumulative prospect theory formalized by Tversky and
Kahneman in 1992. Consider a prospect X defined by:
X = ((xi , pi )i = −m, ....n)
with x−m < x−m+1 < .... < x0 = 0 < x1 < x2 < ... < xn .
We have mentioned previously that gains and losses are evaluated differently by individuals. In order to take into account this assumption, the evaluation function V of a prospect
X is defined by:
V (X) = V (X + ) + V (X − )
where X + = max(X; 0) et X − = min(X; 0).
We set:
4
V (X + ) =
n−1
X


n
n
X
X
w+ (
pj ) − w + (
pj ) v(xi ) + w+ (pn )v(xn )
i=0
0
X
V (X − ) =
i=−m+1
j=i
j=i+1

w − (
i
X
pj ) − w − (
j=−m
i−1
X

pj ) v(xi ) + w− (p−m )v(x−m )
j=−m
where v is a strictly increasing value function defined with respect to a reference point
satisfying v(x0 ) = v(0) = 0 and with w+ (0) = 0 = w− (0) and w+ (1) = 1 = w− (1).
The value function captures the three first features exposed in the introduction (reference
point, decreasing sensitivity, asymmetric perception of gains and losses). Tversky and
Kahneman (1992) proposed the following functional form for the function v:
v(x) =
xα
−λ(−x)β
if x > 0
if x < 0
For 0 < α < 1 and 0 < β < 1 the value function v is concave over gains and convex
over losses. The parameter λ defines the degree of loss aversion (K¨obberling and Wakker
2005). Based on experimental evidence, Tversky and Kahneman estimated the values of
the parameters α, β and λ. They found α = β = 0.88 and λ = 2.25.
The last feature (overweighting the small probabilities) is captured by the weighting function. Tversky and Kahneman proposed the following functional form for this function:
w+ (p) =
pγ+
[pγ+ + (1 − p)γ+ ]1/γ+
w− (p) =
pγ−
[pγ− + (1 − p)γ− ]1/γ−
For γ < 1, this form integrates the overweighting of low probabilities and the greater
sensitivity for changes in probabilities for extremely low and extremely high probabilities.
The weighting function is concave near 0 and convex near 1. Tversky and Kahneman
estimated the parameters γ + and γ − as 0.61 and 0.69.
5
3
The St Petersburg Paradox
The St Petersburg Paradox is usually explained by the following example: consider a
gamble L in which the player gets $2n when the coins lands heads for the first time at the
nth throw. L has an infinite expected value.
E(L) =
1
2
× 2 + ( 12 )2 × 22 + ........ + ( 12 )n × 2n + ...... =
∞
X
1 = +∞
(1)
k=1
According to a number of experiments, the maximum price an individual is willing to
pay for this gamble is around $3. This observation can be taken as an evidence against
expected value theory. Daniel Bernoulli (1738) proposed to replace the monetary value of
each outcome in (1) by its subjective utility (where the subjective utilities are represented
by a strictly concave utility function). In that framework the introduction of a strictly
concave utility function u(x) = ln x leads to:
U E(L) =
∞
X
( 12 )k × ln(2k ) = 2 ln 2 < +∞
(2)
k=1
This solution solves this particular paradox. Since this resolution, expected utility theory
became, for more than 200 years, the major model of choices under risk. However, it is
now challenged by a lot of empirical evidence. There is a broad consensus on the fact that
EU theory fails to provide a good explanation of individual behavior under risk. These
observations have motivated the development of alternative models of choices. One of the
most famous and well accepted is the cumulative prospect theory developed by Tversky
and Kahneman (1992).
Even if this model is deemed to be one of the best alternatives to EU theory, it also
has some difficulties to deal with. Actually, Blavatskyy (2005) established that, under
CPT, the overweighting of small probabilities restores the St Petersburg Paradox. Under
CPT, the subjective utility (V ) of the game described above (L) is given by:
V (L) =
+∞
X
n=1
=
+∞
X
#
+∞
+∞
X
X
−i
−i
v(2 ) × w(
2 ) − w(
2 )
"
n
i=n
i=n+1
v(2n ) × w(21−n ) − w(2−n )
n=1
6
(3)
In section 2, we underline that the functional forms proposed for v and w by Tversky and
Kahneman are v(x) = xα with α > 0 and w(p) = pγ /(pγ + (1 − p)γ )1/γ with 0 < γ < 1.
As
lim
n−→+∞
2−n = 0 and
lim
n−→+∞
21−n = 0, [pγ + (1 − p)γ ]1/γ converges to unity. In that
case, the function w, specified by Tversky and Kahneman, can be approximated by pγ .
According to these remarks we, can rewrite V (L) as:
V (L) ≈ (2γ − 1)
+∞
X
2(α−γ)n
n=1
≈ 0, 526 ×
+∞
X
20,27n −→ +∞
(4)
n=1
One can object that this paradox does not involve a real problem in as far as the expected
value of this game is infinite. Actually, it is not realistic to assume that an institution can
offer a prospect with unlimited expected value (Arrow 1974). However, Rieger and Wang
(2006) pointed out that, under CPT, a prospect with finite expected value can have infinite
subjective utility. Such a result is possible because the weighting function has an infinite
slope at zero. Therefore, the lower the probability is, the more the overweighting. An
extremely small probability can thus be ”infinitely” overweighted. As the value function is
unbounded, there are situations for which the subjective value of a consequence weighted
by its decision weight can be infinitely high.
Rieger and Wang (2006) characterized situations where this problem can be resolved2 . They focused on fitting parameterized functional forms to CPT’s functions and
determined which sets of parameters lead to finite subjective values for all lotteries with
finite expected values.
Let’s consider V the subjective utility of a prospect X under CPT3 :
2
De Giorgi and Hens (2006) proposed to solve this paradox by considering a bounded value function. As
the problem comes from the overweighting of small probabilities, we are interested in the solutions related
to the weighting function.
3
For more details on the demonstration see Rieger and Wang (2006).
7
Z
0
d
v(x) (w+ (F (x)))dx +
V (X) =
dx
−∞
Z
+∞
v(x)
0
d
(w− (F (x)))dx
dx
(5)
where the value function v is continuous, monotone, convex for x < 0 and concave for x
> 0. Assume that there exist constants α, β > 0 such that:
lim
x−→+∞
lim
v(x)
xα
|v(x)|
x−→−∞
|x|β
= v1
∈ (0, +∞)
= v2
∈ (0, +∞)
Assume that the weighting functions w are continuous and strictly increasing from [0,1]
to [0,1] such that w(0) = 0 and w(1) = 1. Moreover, assume that w is continuously
differentiable on ]0,1[ and that there are constants γ + , γ − such that:
lim
0 (y)
w−
−
y−→0 y γ −1
0 (y)
1 − w+
+
y−→1 (1 − y)γ −1
lim
= w1
∈ (0, +∞)
= w2
∈ (0, +∞)
Consider a prospect X for which E(X) < ∞. If all the conditions described above are
satisfied V (X) is finite if α < γ + et β < γ − .
Thus if we consider Tversky and Kahneman’s specification for the value and weighting
functions, the valuation of any prospect by CPT will be finite only if α < γ + and β <
γ − . The estimates of α, β, γ + and γ − are usually obtained from parametric fitting to
experimental data. The estimated parameters obtained by Camerer and Ho (1994) and
8
Wu and Gonzales (1996) are the only ones that are consistent with these conditions4 . In
that case, the concavity of the value function is sufficiently strong relative to the probability
weighting function to avoid the St Petersburg Paradox.
Rieger and Wang (2006) proposed another solution to avoid the paradox under
CPT. They suggested considering a polynomial of degree three as a weighting function.
The specification is given by:
w(p) =
3 − 3b
× (p3 − (a + 1)p2 + ap) + p
−a+1
a2
(6)
with a ∈ (0, 1) et b ∈ (0, 1).
As its slope at zero is finite, this weighting function permits the avoidance of infinite
subjective utilities for all prospects with finite expected value.
4
The behavioral implications of the solutions proposed in
the literature
If we take into account the propositions described above, the St Petersburg Paradox will
not occur under CPT. But at the same time, this theory looses a major part of its descriptive power: it fails to accommodate both gambling and insurance behavior. However,
the possibility of explaining several well-established anomalies such as gambling and insurance is one of the reasons why we should consider CPT instead of classical models such
as expected utility theory (Camerer 1998).
4.1
General considerations
Rieger and Wang (2006) established that prospects with finite expected value will not
have infinite subjective utility if the power coefficient of the value function is lower than
the power coefficient of the probability weighting function. Two parameterized versions of
4
If we look at the estimates obtained by Tversky et Fox (1995), Abdellaoui (2000), Bleichrodt et Pinto
(2000) and Kilka et Weber (2001), α is always superior to γ.
9
CPT are consistent with this condition (Camerer and Ho 1994; Wu and Gonzales 1996).
Accepting them solves the paradox. However, they generate other difficulties. Actually,
these parameterizations of CPT can no longer accommodate the four-fold pattern of risk
attitude (risk aversion for most gains and low probability of losses, and risk seeking for
most losses and low probability of gains). Therefore, these versions of CPT often fail to
capture the gambling behavior observed on the market and more precisely the tendency
of individuals to bet on unlikely gains (Neilson and Stowe 2002)5 . This pattern is still
one of the most fundamental contributions of CPT as it was developed by Tversky and
Kahneman. But it can emerge only if the probability weighting over-rides the curvature
of the value function (for low probabilities). In fact, in rank dependent models, gambling
behavior can be captured only if the overweighting of probabilities is strong enough to
compensate for the concavity of the value function. When α is lower than γ 6 , the convexity
of the weighting function cannot overcome the concavity of the value function. In that case,
optimism generated by the weighting function does not offset risk aversion resulting from
the value function. It is thus impossible to account for gambling behavior. If we consider
a value function whose power coefficient is lower than the coefficient of the probability
weighting function (as suggested by Rieger and Wang), CPT does not restore the St
Petersburg Paradox but in return it does not provide a good description of individual
behavior under risk.
The same problem can be observed with the polynomial specification proposed by
Rieger and Wang (2006). As the slope at zero and unity is finite in this model, the
subjective utility of any finite expected value prospects will be finite. However, as in the
solution proposed above, the behavioral implication of this new specification of CPT does
not allow for betting on unlikely gains. The highest slope of this function at zero (when
b = 0 and a = 1) is actually equal to 4, which is too low. The small probabilities are
not overweighted enough. Therefore the model modified by Rieger and Wang’s weighting
function does not imply that individuals insure against unlikely losses or bet on unlikely
gains.
5
Neilson and Stowe (2002) showed that these two estimates imply almost ”no gambling on unlikely
events”.
6
For all usual estimates of γ.
10
4.2
Illustration: the case of Euromillions
In order to illustrate this evidence, we apply CPT (the version estimated by Tversky and
Kahneman and the ones modified by Rieger and Wang) to the most popular European
lottery: Euromillions. We find that with the modification operated by Rieger and Wang,
CPT can no longer explain the popularity of this game.
Euromillions is a unique lottery played every Friday by millions of players throughout
Europe. One can play by using a playslip that contains six sets of main boards and lucky
star boards. Each player selects five numbers on a main board and two numbers on the
associated lucky star board to make one entry. A Euromillions lottery ticket costs 2e. 50%
of the money paid for a ticket goes directly to the operator selling it. The remaining 50%
of ticket fees goes into the ”Common Prize Fund” out of which all prizes are paid. Table 1
represents the percentage share of the fund allocated to each prize with the corresponding
probabilities7 .
We apply the different versions of CPT (the original and the ones modified by Rieger
and Wang’s propositions) to this European game. In order to determine the average
monetary game for each winner at each rank, we assume the number of participants to
be 40 millions (so the prize fund will be e40 millions)8 . The winning probability at the
1
, so on average there are 7 winners at this rank. The average
5 448 240
40 000 000 × 7.4%
gain per winner is then equal to:
= 422 857.14. Tables 2, 3, 4 and 5
7
second rank is
display the results of the game’s valuation by CPT and modified CPT9 .
The valuation of the euromillions game with cumulative prospect theory, as it was
developed by Tversky and Kahneman, is positive (table 2). Thus, this version of CPT can
explain the popularity of this widely played lottery. But if we substitute the estimates
obtained by Tversky and Kahneman with those of Camerer and Ho and Wu and Gonzales
(table 3 and 4), we obtain a negative valuation. These two estimates do not represent a
particular case. Since the power coefficient of the value function is smaller than the power
coefficient of the weighting function, the subjective utility of this lottery game is always
7
We assume that all the booster fund is devoted to the first rank.
The prize fund reached e38 734 739 in October 12, 2007 and e50 916 665 in September the 20.
9
We consider a reference point of 2e.
8
11
negative for any values of γ superior to 0.38510 . Therefore if we consider this new version
of CPT, this theory fails to explain the popularity of state lottery with very long odds.
The same result is obtained if we substitute the inverse S- shape probability weighting
function specified by Tversky and Kahneman with the one proposed by Rieger and Wang.
In that case (table 5)11 the subjective utility of the lottery is also negative. So decision
makers who transform probabilities via the polynomial weighting function would prefer
to keep the price of the lottery ticket rather than to participate in the lottery. This new
version of CPT is therefore also not able to explain the popularity of public lotteries. This
limitation is quite severe, since betting on unlikely gains is one of the most important
stylized facts that cumulative prospect theory aims to explain.
These negative results cannot be explained by Rabin’s criticism (2000) and are not
due to the size of the lottery payments12 . We can show that the same kind of results can be
obtained with medium payoffs. Consider these two pairs of prospects: A = (6000; 0.001),
B = (3000; 0.002) and A0 = (5000; 0.001), B 0 = (5; 1). In their experiments, Kahneman
and Tversky (1979) showed that a majority of subjects choose prospect A (respectively A0)
rather than B (respectively B0). This (gambling) behavior is consistent with cumulative
prospect theory as it was developed in 1992 but not with the modified versions proposed
by Rieger and Wang (tables 6 and 7 illustrate this fact). Actually, with Camerer and
Ho’s and Wu and Gonzales’ estimates or with the polynomial weighting function, the
subjective utility of B (respectively B0) is always greater than the subjective utility of A
(respectively A0)13 . As previously, these new versions cannot explain why individuals tend
to bet on unlikely gains because the convexity of the weighting function does not overcome
the concavity of the value function.
10
So for all usual estimates of γ.
We took for the computation of π: a = 0.4 and b = 0.5. Nevertheless, the same result (a negative
valuation) is obtained with any other combinations of a and b.
12
According to Rabin, a utility function calibrated for low payoffs gambles implies unreasonable behavior
for high payoffs gambles.
13
If we focus on the choice between A0 and B0: when γ is greater than α, there is no combination of α
and γ which guarantee a greater subjective utility for A0. Concerning the choice between A and B, when
γ is significantly greater than α (so more than 0.05), A will be preferred to B only for values of γ inferior
to 0.42.
11
12
5
An alternative weighting function
5.1
Specification
Section III underlines that CPT needs to be remodelled in order to be applied to problems
of choices. The occurrence of the paradox under CPT comes from the overweighting of
small probabilities. As the slope of the weighting function at zero is infinity, an extremely
small probability can be infinitely overweighted. And as the slope of the value function does
not decrease for high values of outcomes, the subjective value of a consequence weighted by
its decision weight can be infinitely high. In order to overcome this difficulty, we propose
an alternative weighting function that avoids infinite values for subjective utility. This
function should respect the following properties:
1. w(0) = 0 and w(1) = 1
2. w strictly increasing on [0, 1].
3. w continuously differentiable on [0, 1], with w0 (0) et w0 (1) 6= ∞. We underlined
previously that if the slope at zero is too low14 , CPT cannot accommodate the
gambling and insurance behavior observed on the market. Therefore w0 (0) and
w0 (1) should be sufficiently large.
The evidence presented previously criticizes the way probabilities are transformed
near 0 and 1 with the weighting functions generally proposed in the literature. The
difficulties related to the use of the Tversky and Kahneman’s weighting function only
concerns its slope at 0 and 1. Concerning the remainder of the interval, their specification
perfectly characterizes attitudes towards probabilities15 . We can thus keep the global
framework of the Tversky and Kahneman’s functional form and modify it lightly in such
a way that the slopes at the bounds of the interval are not anymore infinite.
14
If we use the ”bad news” approach proposed by Quiggin (1982), the weighting function is applied to
the cumulative distribution of the prospect. In that case, when studying gambling behavior, we should not
focus on the slope at zero but at the slope at one.
15
We agree that an inverse S shape weighting function, first concave and then convex really represents
the way individuals transform probabilities.
13
Let this functional form be given by:
w(p) = w1 (p)1[0;h] + w2 (p)1[h;1]
(7)
with
w1 (p) =
(p + ε1 )γ
1
[(p + ε1 )γ + (1 − (p + ε1 ))γ ] γ
− ρ1
(8)
w2 (p) =
(p − ε2
)k
1
[(p − ε2 )k + (1 − (p − ε2 ))k ] k
+ ρ2
The parameters ε1 , ε2 , ρ1 , ρ2 , and h satisfy the following conditions:



w1 (0) = 0
(a)














w2 (1) = 1
(b)












0

(c)

 w1 (0) = S

(9)






w20 (1) = S
(d)













w1 (h) = w2 (h)
(e)












 w0 (h) = w0 (h)
(f)
1
2
The two first conditions correspond to the first property presented above. The third and
fourth conditions refer to the last property exposed above: the slope at the extremities
of the interval has to be finite and large. Thus S refers to a finite and important value
which should be experimentally estimated. ε1 , ε2 are obtained by solving the following
equations:
14

(1−ε1 )−1+γ −ε1−1+γ

γ
γ −1
γ
γ
γ

S = ε1 × [(1 − ε1 ) + ε1 ] × ε1 +


(1−ε1 )γ +εγ1 )









− 1+k
γ
1
2
−1+k
k
k
(10) S =
(1 − ε2 )
× (1 − ε2 ) + ε2
× εk2 + (−1 + k) × (1 − ε2 )k × ε2
ε2










i



+ ε1+k
2
Conditions 5 and 6 imply that w is strictly continuous and differentiable on [0,1]. h is thus
the solution16 of:



























(11)


























(h + ε1 )γ
0=
[(h + ε1 )γ + (1 − (h + ε1 ))γ ]
0 = (ε1 +
γ
+
ε1 + h
h)γ
× [(1 − ε1 −
h)γ
1
γ
− ρ1 − ρ2 −
+ (ε1 +
h)γ ]−1/γ
(p − ε2 )k
1
[(p − ε2 )k + (1 − (p − ε2 ))k ] k
×
(1 − ε1 − h)−1+γ − (ε1 + h)1−γ
(1 − ε1 − h)γ + (ε1 + h)γ
−1/k
(−ε2 + h)k × (1 + ε2 − h)k + (−ε2 + h)k
×k 1
+
+
(−ε2 + h)k
ε2 − h
k
−1−1/k
k(−ε2 + h)k
k
k
−1+k
× (1 + ε2 − h) + (−ε2 + h)
× −(1 + ε2 − h)
×k−
ε2 − h
This new specification of w is steepest near zero and one and shallower in the middle. It
satisfies the properties of subadditivity and is first concave then convex. The condition
w1 (0) = 0 and w2 (1) = 1 is satisfied. Finally, its slope at the extremities is finite and can
be strong.
5.2
Illustration: the case of Euromillions
In this section, we compute the parameters of the new weighting function for different
values of S. Table 8 represents these data for S equals 500, 1 000 and 2 000.
16
For γ we take the estimates realized by Tversky and Kahneman (so 0.61 for gains). We cannot do
the same for k which should be smaller than γ. If γ = k, w becomes concave again when w2 replaces w1 ,
whatever the value of h. So we use the relation (11) to obtain the value of k relative to γ.
15
As the slope of this function at the extremities is finite (500, 1 000 or 2000), the subjective
utility of ay finite expected value prospect will be finite. Moreover unlike the specifications
described above, the slope of this function is sufficiently strong to account for gambling behavior. To illustrate this observation, we can apply this new version of CPT to the lottery
game presented in the previous section. Table 9 displays the results of this application.
The valuation of the gamble is positive (in all three cases)17 . So individuals whose preferences are represented by this version of CPT will be attracted by this lottery. In that
case, CPT explains gambling behavior.
6
Conclusion
The aim of this study was to determine how we could solve the St Petersburg Paradox in
rank dependent models. First, we established that the solutions proposed in the literature
lead to other kinds of difficulties. We underlined that if we took them into account, the
probability weighting wouldn’t be strong enough to compensate for the concavity of the
value function. In that case, CPT cannot accommodate both gambling and insurance
behavior. This theory then looses a major part of its descriptive power. In a second
part, we proposed an alternative way to fix the infinite subjective utility’s problem. Like
Rieger and Wang (2006), we suggested an alternative weighting function whose slope at
zero is not infinite. In this case, the subjective value of any prospect wouldn’t be infinitely
high. Nevertheless, in order to preserve the fourfold pattern of risk attitudes, we set a
specification whose shape dominates (for low probabilities) the value function. Thanks
to this requirement, the overweighting of small probabilities reverses risk averse behavior
for gains (and risk seeking behavior for losses) generated by the value function. CPT still
provides a good description of individual behavior under risk.
17
When S increases the new valuation V converges to the valuation obtained with Tversky and Kahneman’s estimates (V T K ).
16
References
Allais, M. (1953): “Le Comportement de l’Homme Rationnel Devant Le Risque, Critiques des Postulats et Axiomes de l’Ecole Am´ericaine,” Econometrica, 21, 503–546.
Arrow, K. J. (1974): “The Use of Unbounded Functions in Expected-Utility Maximization: Response,” The Quaterly Journal of Economics, 88, 136–138.
Bernoulli, D. (1738): “Specimen Theoriae Novae de Mensura Sortis,” Commentarii
Academiae Scientiarum Imperialis Petropolitanae, 5, 175–192.
(1954, Original Edition, 1738): “Exposition of a new Theory of the Measurement
of Risk,” Econometrica, 22, 123–136.
Blavatskyy, P. (2005): “Back to the St Petersburg Paradox?,” Management Science,
51, 677–678.
Camerer, C. (1998): “Bounded Rationality in Individual Decision Making,” Experimental
Economics, 1, 163–183.
Camerer, C. F., and T. Ho (1994): “Violations of the Betweenness Axiom and Nonlinearity in Probability,” Journal of Risk and Uncertainty, 8, 167–196.
De Giorgi, E., and S. Hens (2006): “Making Prospect Theory Fit for Finance,” Financial Market Portfolio Management, 20, 339–360.
Kahneman, D., and A. Tversky (1979): “Prospect Theory : An Analysis of Decision
under Risk,” Econometrica, 47, 263–291.
K¨
obberling, V., and P. Wakker (2005): “An Index of Loss Aversion,” Journal of
Economic Theory, 122, 119–131.
Menger, K. (1934): “Das Unsicherheitsmoment in der Wertlehre,” Zeitschrift National¨
okonomie, 51, 459–485.
Neilson, W., and J. Stowe (2002): “A further Examination of Cumulative Prospect
Theory Parameterizations,” Journal of Risk and Uncertainty, 24, 31–46.
17
Quiggin, J. (1982): “A Theory of Anticipated Utility,” Journal of Economic Behavior
and Organization, 3, 323–343.
Rabin, M. (2000): “Risk Aversion and Expected-Utility Theory: a Calibration Theorem,”
Econometrica, 68, 1281–1292.
Rieger, M., and M. Wang (2006): “Cumulative Prospect Theory and the St Petersburg
Paradox,” Economic Theory, 28, 665–679.
Slovic, P., and S. Lichtenstein (1968): “The Relative Importance of Probabilities and
Payoffs in Risk Taking,” Journal of Experimental Psychology, 78, 1–18.
Tversky, A., and D. Kahneman (1992): “Advances in Prospect Theory : Cumulative
Representation of Uncertainty,” Journal of Risk and Uncertainty, 5, 297–323.
Wu, G., and R. Gonzalez (1996): “Curvature of the Probability Weighting Function,”
Management Science, 42, 1676–1690.
7
Tables
Table 1: Prizes and probabilities associated to Euromillions
Gains rank
% Prize Fund
Probability
1
st
1 rank
32%+6%
76 275 360
1
2nd rank
7.4%
5 448 240
1
th
3 rank
2.1%
3 632 160
1
th
4 rank
1.5%
339 002
1
5th rank
1%
24 214
1
th
6 rank
0.7%
16 143
1
7th rank
1%
7 705
1
th
8 rank
5.1%
550
1
th
9 rank
4.4%
538
1
10th rank
4.7%
367
1
th
11 rank
10.1%
102
1
12th rank
24%
38
18
Table 2: Euromillions Valuation - Tversky and Kahneman’s estimates
πi
Gains rank Probabilities
Gains
v(xi )
CPT
1
st
1 rank
15 200 000 2 089 403.44 1.55501×10−5
76 275 360
1
nd
422 857.14
89 340.55
6.55653×10−5
2 rank
5 448 240
1
76 363.63
19 012.08
5.72386×10−5
3th rank
3 632 160
1
th
4 rank
5 128.2
1839.16
0.00032463
339 002
1
242.27
124.46
0.00175136
5th rank
24 214
1
th
6 rank
113.03
63.09
0.00153935
16 143
1
7th rank
77.05
44.7
0.00232386
7 705
1
28.05
17.61
0.0160782
8th rank
550
1
th
9 rank
23.67
14.98
0.0101514
538
1
10th rank
17.25
10.99
0.0115014
367
1
th
10.30
6.44
0.0290591
11 rank
102
1
9.12
5.62
0.048556
12th rank
38
No gain
0.9572
0
-4.14
0.86336
T
K
V (X)
37.94
Table 3: Euromillions Valuation - Camerer and Ho’s estimates
v(xi )
πi
Gains rank Probabilities
Gains
Camerer Ho
Camerer Ho
1
st
1 rank
15 200 000
454.236
3.85332×10−5
76 275 360
1
nd
2 rank
422 857.14
120.695
0.000136992
5 448 240
1
th
3 rank
76 363.63
64.07134
0.000111006
3 632 160
1
4th rank
5
128.2
23.58438
0.000581451
339 002
1
th
5 rank
242.27
7.60087
0.002775051
24 214
1
6th rank
113.03
5.7125
0.002260728
16 143
1
7th rank
77.05
4.94189
0.003262787
7 705
1
th
8 rank
28.05
3.340792
0.020489824
550
1
9th rank
23.67
3.120927
0.01197572
538
1
10th rank
17.25
2.740316
0.01302709
367
1
th
11 rank
10.30
2.188255
0.0311254
102
1
9.12
2.067367
0.04819050
12th rank
38
No gain
0.9572
0
-2.90779
0.7638842
VCH (X)
-1.80
19
Table 4: Euromillions Valuation - Wu and Gonzales’ estimates
v(xi )
πi
Gains rank Probabilities
Gains
Wu Gonzales
Wu Gonzales
1
1st rank
15 200 000
5426.9851
2.5322×10−6
76 275 360
1
2nd rank
422 857.14
842.59648
1.478610−5
5 448 240
1
th
76 363.63
346.01515
1.4926×10−5
3 rank
3 632 160
1
th
5 128.2
84.936411
9.935×10−5
4 rank
339 002
1
242.27
17.296997
0.0006838
5th rank
24 214
1
th
6 rank
113.03
11.578418
0.0006947
16 143
1
th
77.05
9.4449954
0.0011434
7 rank
7 705
1
8th rank
28.05
5.4477971
0.0095197
550
1
9th rank
23.67
4.9507266
0.0069054
538
1
th
10 rank
17.25
4.1237036
0.0084206
367
1
10.30
3.0059076
0.0235416
11th rank
102
1
12th rank
9.12
2.7751710
0.0451533
38
No gain
0.9572
0
-3.226399
0.8742871
-2.43
VW G (X)
Table 5: Euromillions Valuation - Polynomial weighting function
π polynomial
Gains rank Probabilities
Gains
v(xi )
weighting function
1
st
1 rank
15 200 000 2 089 403.44
2.34607×10−8
76 275 360
1
nd
2 rank
422 857.14
89 340.55
3.2845×10−7
5 448 240
1
3th rank
76 363.63
19 012.08
4.92674×10−7
3 632 160
1
th
4 rank
5 128.2
1839.16
5.27862×10−6
339 002
1
5th rank
242.27
124.46
7.38969×10−5
24 214
1
th
6 rank
113.03
63.09
0.000110825
16 143
1
7th rank
77.05
44.7
0.000232125
7 705
1
th
8 rank
28.05
17.61
0.003242095
550
1
9th rank
23.67
14.98
0.003295609
538
1
10th rank
17.25
10.99
0.004796966
367
1
th
11 rank
10.30
6.44
0.016926816
102
1
9.12
5.62
0.042932219
12th rank
38
No gain
0.9572
0
-4.14
0.912226886
VP W F (X)
-3.14
20
A
73%∗
Table 6: Medium payoffs Valuation by CPT and CPT modified
Choices
CPT
Polynomial
p
x
π
v
V
π
v
0.999
0
0.986
0
0.998
0
0.001
6000
0.014 2112 30.53
0.02
2112
B
27%
0.998
0.002
0
3 000
0.978
0.022
0
1147
A0
72%∗
0.999
0.001
0
5000
0.986
0.014
B0
28%
0
1
0
5
0
1
A
73%∗
3.77
25.02
0.996
0.04
0
1147
4.09
0
1799
26
0.998
0.02
0
1799
3.24
0
4.12
4.12
0
1
0
4.12
4.12
Table 7: Medium payoffs Valuation by CPT modified
Choices
Camerer and Ho
Wu and
p
x
π
v
V
π
v
0.999
0
0.98
0
0.993
0
0.001
6000
0.02
25
0.5
0.007
92
B
27%
0.998
0.002
0
3 000
0.971
0.029
0
19.34
A0
72%∗
0.999
0.001
0
5000
0.98
0.02
0
23.36
B0
28%
0
1
0
5
0
1
0
1.81
S
500
1000
2000
V
0.68
0.56
0.988
0.012
0
64
0.77
0.47
0.993
0.007
0
83.8
0.61
1.81
0
1
0
2.30
2.30
Table 8: Parameters of the new weighting funtion (for gains)
ε1
ρ1
ε2
ρ2
h
−8
−5
−7
−5
3.38085×10
2.77132×10
1.20554×10
9.88766×10
0.9464296
5.7177×10−9
9.3734×10−6
2.0332×10−8
3.334×10−5
0.941749
9.6688×10−10 3.1701×10−6
3.4357×10−9 1.12683×10−5 0.973276
21
Gonzales
V
k
0.60985
0.60995
0.60998
Gains rank
1st rank
2nd rank
3th rank
4th rank
5th rank
6th rank
7th rank
8th rank
9th rank
10th rank
11th rank
12th rank
No gain
V(X)
Table 9: Euromillions Valuation - New weighting function
πi
πi
Probabilities
Gains
v(xi )
S = 500
S = 1000
1
−6
15 200 000 2 089 403.44 6.13×10
1.01×10−5
76 275 360
1
−5
422 857.14
89 340.55
5.55×10
6.31×10−5
5 448 240
1
76 363.63
19 012.08
5.49×10−5
5.68×10−5
3 632 160
1
5 128.2
1839.16
0.0003214
0.0003240
339 002
1
242.27
124.46
0.0017495
0.0017510
24 214
1
113.03
63.09
0.00153905 0.00153930
16 143
1
77.05
44.7
0.00232366 0.00232382
7 705
1
28.05
17.61
0.0160779
0.0160781
550
1
23.67
14.98
0.01015141
0.010151462
538
1
17.25
10.99
0.0115013
0.01150142
367
1
10.30
6.44
0.029059
0.0290591
102
1
9.12
5.62
0.0485567
0.0485568
38
0.9572
0
-4.14
0.8633634
0.8633637
17.31
26.16
22
πi
S = 2000
1.31×10−5
6.51×10−5
5.71×10−5
0.0003245
0.0017513
0.00153934
0.00232385
0.0160782
0.010151469
0.01150143
0.0290591
0.04855681
0.8633638
32.72