An Experimental Test of Theories of Behavior
in Allais-Type Tasks
Elif Incekara-Hafalir∗
Jack Douglas Stecher†
November 2012
Abstract
This article tests the predictions of expected utility theory and of
several of its most prominent alternatives and assesses their ability to
explain behavior in Allais-type tasks. We find that, given a transparent presentation, expected utility theory performs surprisingly well,
and that the alternative theories perform poorly except inasmuch as
they make the same predictions as expected utility theory. We also
contribute to methodology, and study the consequences of using hypothetical versus real incentives. Although hypothetical incentives do
not affect compliance with expected utility, they have a large effect on
attitude toward risk.
1
Introduction
Critiques of expected utility theory all start with Allais (1953). The choice
pattern in Allais’s lottery tasks has led to discussion whether expected utility
theory is a reasonable normative theory (see, e.g., Slovic and Tversky, 1974)
and, more often, whether it is descriptive of choice behavior. There are
numerous proposed alternatives to expected utility theory, many of which
were developed with the explicit aim of explaining behavior in Allais-type
tasks. Starmer’s (2000) survey article gives a good overview.
∗
†
[email protected]
[email protected]
1
Our aim in this article is to persuade you that expected utility theory
performs surprisingly well as a descriptive theory, and that the proposed
alternatives to expected utility are not descriptive, except inasmuch as they
coincide with expected utility. Our argument is in the spirit of Borch (1968)
and Savage (1972), who maintain that subjects who see an Allais-type task
clearly will be less likely to violate expected utility theory.
We present experimental evidence that supports these claims. Our design uses the presentation Savage (1972, pp. 101–103) suggests to make the
common consequence explicit. See Table 1.1 In the table, m is the middle
reward, which the safe lottery gives with probability 11%, and h represents
the high reward, which the risky lottery gives with probability 10%. The
common consequence is denoted by c. Additionally, we ran a session with
identical prize levels but with hypothetical lotteries to contribute to the discussion about the effects of salient rewards (e.g., Conlisk, 1989; Harrison,
1994; Burke et al., 1996; Beattie and Loomes, 1997; Binmore, 1999; Camerer
and Hogarth, 1999; Fan, 2002; Harrison, 2006). We ran several additional
sessions as robustness checks, which we describe in Sections 4 and 5.
Table 1: Savage’s presentation of an Allais-type lottery choice
89%
10%
Safe lottery
c
m
Risky lottery
c
h
1%
m
0
In contrast to most other tests of expected utility theory and its rivals,
our design keeps the probabilities fixed. Instead, we manipulate the common
consequence c from a low value of 0 (giving one of the standard Allais lottery
pairs) up to a value of 2h.2 This design has several advantages. First, as we
show in Section 3, it enables us to get predictions that distinguish several
of the most prominent rivals to expected utility theory. In particular, Gul’s
(1991) disappointment aversion is known to be difficult to test with fixed
prizes and variations in probability.3 We show later that if the probabili1
MacCrimmon (1968) and Slovic and Tversky (1974) use the same presentation in the
discussion parts of their experiments. Moskowitz (1974) also uses this presentation as one
of the ways to structure decision problems. Keller (1985) has subjects convert a choice
task from an Allais-style narrative to this presentation.
2
Most other studies do not consider values of c above m. Two exceptions are Camerer
(1989) and Starmer (1992), who include cases where c = h.
3
See, for example, Harless and Camerer (1994). More recently, Gill and Prowse (2012)
2
ties are fixed and the common consequence varies, disappointment aversion
makes much stronger predictions. Machina’s (1982) fanning out hypothesis, Quiggin’s (1981; 1982) rank-dependent utility, and cumulative prospect
theory with different proposed weighting functions (Tversky and Kahneman,
1992; Prelec, 1998) are also testable.
Second, our design is inherently more transparent than designs that manipulate probabilities, such as Camerer (1989), or those that illustrate independence by decomposing lotteries into compound lotteries, as in Conlisk
(1989). A prize of $10 is self-explanatory; communicating probabilities is
not. On this point, see Davidson et al. (1957), Gigerenzer (1991), Carlin
(1992), Kadane (1992), and Wakker and Deneffe (1996), or consider your
own reaction the last time a stranger told you that he or she was “99% sure”
that you could park somewhere and not get towed.
Our first question is whether presenting the lottery pairs in the Savage
format reduces violations of expected utility theory. To test this, we ran two
treatments with the same payoff structure: one with the Allais presentation
and one with the Savage presentation. Using the Savage presentation sharply
increases the frequency of expected utility conformity. Around 60% of subjects who face the Savage presentation behave consistently with expected
utility theory, compared with around 40% of those who face the Allais presentation.
Our second question is whether those subjects who violate expected utility
theory behave in accordance with any of its major alternatives. In addition
to the theories mentioned earlier, we test for a certainty effect and for a zero
effect. The certainty effect, pointed out in Kahneman and Tversky (1979),
is a change in behavior when the safe lottery is certain. In our design, this
is when c = m. The zero effect, raised in Birnbaum et al. (1992) and Mellers
et al. (1992), is a change in behavior when c = 0.
We find that almost all of the violators of expected utility theory fall
into one of two groups. The larger consists of those whose behavior violates
every theory we test. The smaller consists of those who show a zero effect.
All other choice patterns occur with negligible frequency. Accordingly, the
ability of an alternative to expected utility to explain the Allais paradox rests
entirely on the zero effect.
Is the zero effect a real phenomenon? It seems doubtful. The zero effect
developed a test of disappointment aversion, but in a tournament setting rather than a
decision problem.
3
choice pattern occurs when the first lottery presented has a common consequence c = 0. Changing presentation order appears to make the zero effect
vanish.
The structure of the remainder of this paper is as follows. Section 2
describes our main task and our auxiliary session used to select the values
of the prizes in our lottery choices. Section 3 provides the predictions of the
theories we test. Section 4 describes our data. Section 5 gives our results and
robustness checks, and section 6 concludes. All proofs are in an appendix.
2
2.1
Details on the Lottery Choice Task
Baseline Session: Structure of the Task
To establish the prize levels for our lottery tasks, we ran a baseline session,
where subjects made six choices between certain lotteries and risky lotteries
with a payoff structure matching that in Table 1. The purpose of this baseline
session was to make sure there would be a reasonable chance of some subjects
choosing the safe lottery, at least when a certainty effect is present and when
the Allais presentation is used. Conlisk (1989) and Fan (2002) both find
that almost all subjects prefer risky lotteries if the payoffs are in the same
proportions as the original Allais payoffs. A typical choice task was as follows:
Which of the following two alternatives do you prefer? Please circle one.
(a) $8 for sure
(b) A lottery that has an 89% chance of paying $8, a 10% chance of
paying $16, and a 1% chance of paying nothing.
Choose: A
B
I’m indifferent
The other choices were similar. On each pair of choices, the subject would
choose between receiving m for sure or a lottery that has an 89% chance of
paying m, a 10% chance of paying h, and a 1% chance of paying nothing.
The numbers we chose, and the corresponding differences in expected value
between the safe and risky lotteries, are shown in Table 2.
We rolled a six-sided die to determine which of the subjects’ choices would
count and rolled the six-sided die a second time to determine which lottery to
assign to subjects who had expressed indifference on the selected choice. This
4
m
$10
$8
$12
$10
$8
$12
Table 2: Outcomes in the baseline experiment
h
E[R] − E[S]
$12
$0.10
$10
$0.12
$15
$0.18
$15
$0.40
$16
$0.72
$24
$1.08
procedure and its justification are discussed in Starmer and Sugden (1991).
We then rolled a 10-sided die twice to get a two-digit number (interpreting a
roll of 00 as drawing a 100). We described this procedure in our instructions,
which we read aloud to the subjects, and we had an arbitrarily chosen subject
roll the dice in front of all the subjects.
The subjects made their choices using pencils and saw all six choices on
a single sheet of paper; this was to make it easy for them to change their
decisions if they so desired. The sheets with the instructions and the task
were at the subjects’ desks before they were seated. After the last subject
was seated, we read the instructions aloud and then gave the subjects an
additional five minutes to make their decisions. This meant that the subjects
typically had 10–15 minutes from the time they were seated until they made
all their decisions.
2.2
Baseline Session: Results
We ran our baseline session with 34 subjects from the Pittsburgh Experimental Economics Laboratory (PEEL) subject pool, recruited through the PEEL
online system. Average earnings were $17 total, which included a $5 show-up
fee. Among the 34 subjects, there were three violations of monotonicity.4
Four patterns described the remaining 31 subjects. These are shown in
Table 3. The table shows that even if the gap between h and m was as small
as $2, just over 60% of subjects preferred the risky lottery to the certain one.
If the gap increased to $3, around 3/4 of subjects made all risky choices.
4
Two subjects expressed indifference between $8 for sure and the following two lotteries:
(8, .89; 10, .1; 0, .01) and (8, .89; 16, .1; 0, .01). A third was indifferent between $10 and
(10, .89; 12, .1; 0, .01) and was also indifferent between $8 and (8, .89; 10, .1; 0, .01).
5
Table 3: Frequency of patterns in baseline task
Pattern
All R
S if h − m = 2, else R
S if h − m ≤ 3, else R
S if h − m ≤ 5, else R
n (%)
19 (61%)
4 (13%)
6 (19%)
2 (6%)
The results from the baseline session suggest that in our main treatment,
we should choose either m = 10 and h = 12 or m = 8 and h = 10. We
selected the latter and address robustness checks in Section 5.
2.3
Main Treatment
For our main test, we provided two presentations of the Allais-type lotteries.
The Savage presentation follows Table 1, with m = 8 and h = 10. For the
common consequence c, we chose values in {0, 5, 8, 10, 16, 20}. We show this
here as Table 4.
Table 4: Savage’s presentation, our main treatment
89%
10%
Safe lottery
c
8
Risky lottery
c
10
1%
8
0
The Allais presentation had the same choices as the Savage presentation,
in the same order. We displayed the Allais presentation in the same way as
in our baseline treatment. As with our baseline treatment, we used rolls of
dice to select lotteries, settle indifferences, and determine payments.
3
Predictions: Nonexpected Utility
Our design enables us to test the predictions of expected utility theory, the
fanning out hypothesis, Gul’s (1991) disappointment aversion, and various
forms of rank-dependent utility, including cumulative prospect theory with
different proposed weighting functions. We present only a cursory overview;
Wakker (2010) provides a comprehensive treatment. In deriving our predic-
6
tions, we assume throughout that the Bernoulli utility u(·) is monotone, and
we choose the normalization u(0) = 0.
Table 5 shows the patterns of choices that are consistent with at least one
of the theories we test. For subjects who did not express any indifferences,
there were 26 = 64 possible patterns of choices. Table 5 shows that only 20
of these are consistent with at least one of the theories we consider.5 From
the first two lines in Table 5, it is easy to see that predictions of expected
utility theory are special cases of all the models we test.
Table 5: Choice patterns for c ∈ {0, 5, 8, 10, 16, 20}
Choice 0 5 8 10 16 20 EU FO DA RDU CPT PW
1
R R R R R R
X
X
X
X
X
X
2
S S S S
S
S
X
X
X
X
X
X
3
R R S S
S
S
X
X
X
X
X
4
R S S S
S
S
X
X
X
X
X
5
R R R S
S
S
X
X
X
X
6
R R S R R R
X
X
X
7
R S S R R R
X
X
X
8
S S S R R R
X
X
X
9
R R R R R S
X
10
R R R R S
S
X
11
R R S S R R
X
12
R R S S
S R
X
13
R S S S R R
X
14
R S S S
S R
X
15
S S S S R R
X
16
S S S S
S R
X
17
S R R R R R
X
18
S R R S
S
S
X
19
S S R R R R
X
20
S S R S
S
S
X
Note. Here S = safe lottery; R = risky lottery; EU = expected utility; FO =
fanning out; DA = disappointment aversion; RDU = rank-dependent utility;
CPT = cumulative prospect theory; PW = Prelec’s probability weighting
model.
5
Henceforth, we ignore indifferences. These were rare in observations and were straightforward to analyze. Details are available from the authors on request.
7
The standard Allais choices consist of a subject choosing the risky lottery
if c = 0 and the safe lottery if c = 8; the reverse Allais choices consist of
the opposite behavior. There are 16 possible patterns consistent with Allais
choices (and 16 consistent with reverse Allais choices). Among these, eight
Allais patterns and four reverse Allais patterns appear in Table 5. Rankdependent utility is the only theory that justifies reverse Allais choices.
Four choice patterns are worth special attention. Lines 4 (RSSSSS) and
17 (SRRRRR) consist of identical choices in all lotteries, except when c =
0. Lines 6 (RRSRRR) and 20 (SSRSSS) consist of identical choices in all
lotteries, except when the safe lottery has a certain outcome (i.e., when
c = 8). Birnbaum et al. (1992), Mellers et al. (1992), and Wu et al. (2005)
observe a zero effect, that is, a change in behavior in lotteries where c = 0.
Conversely, Kahneman and Tversky (1979), Dickhaut et al. (2003), Wu et al.
(2005), and many others observe a certainty effect. We have more to say
about these later, but for now, we note that every theory except expected
utility is consistent with the pattern RSSSSS, which we call the zero effect
pattern, while only rank-dependent utility is consistent with SRRRRR, which
we call the reverse zero effect pattern. Similarly, several theories justify the
usual certainty effect (RRSRRR), while only rank-dependent utility justifies
the reverse (SSRSSS).
Disappointment aversion and rank-dependent utility are the least falsifiable theories in the sense that they justify the largest set of choice patterns:
13 for disappointment aversion and 12 for rank-dependent utility, compared
with 8 for cumulative prospect theory, 7 for fanning out, 5 for Prelec’s probability weighting, and 2 for expected utility. Rank-dependent utility, unlike its
special cases, can justify the reverse of the patterns that have been of most
interest (the Allais choices, the certainty effect, and the zero effect). However, even disappointment aversion and rank-dependent utility are consistent
with only around 20% of the possible choice patterns.
We now describe the justification for the predictions in Table 5. In describing each theory, we will use U S (c) for the value of the safe lottery and
U R (c) for the value of the risky lottery, given the common consequence of c.
R
Thus UDA
(c) denotes the disappointment averse utility of the risky lottery
when the common consequence is c. We will also write ∆(c) := U S (c)−U R (c).
8
3.1
Fanning Out
For our purposes, the crucial hypothesis of the fanning out theory of Machina
(1982, p. 300) is Hypothesis II, which states that as a lottery improves
in the sense of first-order stochastic dominance, a decision maker becomes
more risk averse. The term refers to a graphical depiction of the decision
maker’s indifference curves for three-outcome lotteries with fixed prizes, in
the triangle diagram of Marschak (1950) and Machina (1987).
Figure 1: Marschak–Machina triangle diagram, under expected utility (left)
and fanning out (right). The lines represent indifference curves.
In Figure 1, the triangle gives the probability mass for a lottery with three
fixed outcomes, {x1 , x2 , x3 }, where without loss of generality, we assume that
u(x1 ) ≤ u(x2 ) ≤ u(x3 ). The x-axis is the probability of a lottery giving
outcome x1 , and the y-axis is the probability of a lottery giving outcome x3 .
A consequence of the independence axiom of von Neumann and Morgenstern
(1944) is that the indifference curves for an expected utility maximizer must
be parallel lines, as in Figure 1 (left).
Figure 1 (right) shows indifference curves that fan out. Lottery a1 gives
the middle-ranked prize x2 with certainty as p1 = p3 = 0. Lottery a2 has a
small probability of giving the lowest prize x1 (as a2 is slightly to the right
of the origin) and a somewhat larger probability of giving the highest prize
x3 (as a2 is slightly above the origin). In terms of our task, if c = 8, then
9
a1 is our safe lottery and a2 is our risky lottery. Lottery a4 is a modification
of a1 in which the probability of receiving the lowest prize is increased by a
fixed amount (the distance on the x-axis from a4 to a1 ) and the probability of
receiving the middle-ranked prize is decreased by the same amount. Lottery
a3 is an analogous modification of a2 . In terms of our task, if c = 0, then a3
is our risky lottery and a4 is our safe lottery.
The parallel indifference curves in Figure 1 (left) illustrate why an expected utility maximizer who prefers a1 to a2 must prefer a4 to a3 . By
contrast, the fan-shaped pattern of the indifference curves depicted in Figure
1 (right) justifies the choice of a1 over a2 and also of a3 over a4 . This fanshaped pattern follows from Machina’s Hypothesis II. As lotteries improve in
the sense of first-order stochastic dominance, the indifference curves become
steeper, reflecting greater risk aversion. This implies the following result:
Proposition 3.1.1. Let c be the common consequence in an Allais-type lottery. If a decision maker satisfies the fanning out hypothesis, then either
1. the decision maker’s choices are indistinguishable from those of an expected utility maximizer, or
2. there exists some c∗ ∈ R+ such that the decision maker chooses the
risky lottery when c < c∗ and chooses the safe lottery when c > c∗ .
The patterns RRSSSS and RSSSSS (lines 3 and 4 of Table 5) are the
only cases in which fanning out is consistent with the Allais choices; the
latter of these two is the zero effect pattern. The reverse Allais choices are
incompatible with fanning out, as are the reverse zero, the certainty, and the
reverse certainty effects.
3.2
Disappointment Aversion
Theories of disappointment aversion go back to Bell (1985) and Loomes and
Sugden (1986). They differ from other models in which utility depends on the
set of outcomes in that theories of disappointment aversion leave the agent’s
Bernoulli utility fixed. Instead, the agent’s utility is a weighted average of
“elating” and “disappointing” outcomes, where the weight depends on both
the probability of a given outcome and whether that outcome will seem good
or bad compared with other possibilities.
10
Grant et al. (2001) provide a good overview of several models of disappointment averse utility. We choose here to focus on the model of Gul (1991,
p. 667), which aims to be parsimonious, as his opening indicates:
The purpose of this paper is to develop an axiomatic model of decision making under uncertainty that (i) includes expected utility
theory as a special case, (ii) is consistent with the Allais Paradox,
and (iii) is the most restrictive possible model that satisfies (i)
and (ii) above.
Gul’s model works as follows: to calculate the utility of a lottery p with
finite support, one starts by decomposing the lottery into a weighted average
of two lotteries, q and r, such that
1. every prize in q is weakly better than the certainty equivalent of p
2. every prize in r is weakly worse than the certainty equivalent of p
The probability of receiving each prize in q is the conditional probability
of getting the same prize under p, given that a prize in the support of q is
drawn. Similarly, the probabilities associated with the prizes in the support
of lottery r are their conditional probabilities under p. Putting that together,
(∃α ∈ [0, 1])
p = αq ⊕ (1 − α)r,
(1)
where ⊕ represents the sum of two lotteries and where α is the probability
that a draw from p yields a prize in q. If the certainty equivalent of p is not
in the support of p, then the decomposition is unique.
A disappointment-averse or elation-seeking decision maker has a utility
representation similar to equation (1) but where the weighted average uses
a transformation of γ(·) of α. A single parameter, β > −1, determines the
weights:
γ(α) =
α
1 + (1 − α)β
1 − γ(α) =
(1 + β)(1 − α)
,
1 + (1 − α)β
yielding
p
UDA
=
α
1 + (1 − α)β
EU(q) +
11
(1 + β)(1 − α)
1 + (1 − α)β
EU(r).
(2)
To derive implications for disappointment aversion, we make the following
assumptions. First, β > 0; this means that the decision maker is not elation
seeking. Second, if the outcome c is disappointing in the safe lottery, then
it is disappointing in the risky lottery; this means that the decision maker
determines which outcomes are disappointing before making his or her choice.
In our setting, suppose that the common consequence c < 8. Then q
must be the lottery that pays 8 for sure, and r must be the lottery that pays
c for sure. This means that α = 0.11. Conversely, suppose that c > 8. Then
q must be the lottery that pays c for sure, and r must be the lottery that
pays 8 for sure, giving α = 0.89.
For our safe lottery with c < 8,
S
UDA
(c) =
.89(1 + β)u(c)
.11u(8)
+
.
1 + .89β
1 + .89β
For c > 8,
.11(1 + β)u(8)
.89u(c)
+
.
1 + .11β
1 + .11β
For the risky lottery R, the decomposition into disappointment and elation lotteries is straightforward in two special cases: c = 0 and c = 10. In
each of these cases, R is a nondegenerate binary lottery. For c = 0,
S
UDA
(c) =
.1u(10)
.9(1 + β)u(0) + .1u(10)
=
.
1 + .9β
1 + .9β
(3)
.01(1 + β)u(0) + .99u(10)
.99u(10)
=
.
1 + .01β
1 + .01β
(4)
R
UDA
(0) =
For c = 10,
R
UDA
(10) =
Thus, at c = 0, the common consequence c is disappointing, and at
c = 10, the common consequence c is elating. For c ∈ (0, 10), there are two
possibilities:
R
q = {c, 10} r = {0} ⇒ UDA
(c) =
.89u(c) + .1u(10)
1 + .01β
.89(1 + β)u(c) + .1u(10)
.
1 + .9β
We now derive restrictions on behavior for c > 10 and for c < 8.
R
q = {10} r = {0, c} ⇒ UDA
(c) =
12
Proposition 3.2.1. If c > 10, then either ∆DA (c) < 0 or ∆DA (c) is decreasing in c. Consequently, there is at most one switch as c increases above 10,
which must be from the safe lottery to the risky lottery.
Proposition 3.2.2. If 0 < c < 8, then ∆DA (c) is increasing in c and is
greater than ∆DA (0). Consequently, for c between 0 and 8, there can be at
most one switch, which must be from the risky lottery to the safe lottery.
To obtain restrictions at c = 8, we need to consider the right and left
limits. That is because 8 is disappointing if c > 8 and elating if c < 8, leading
to a jump discontinuity in ∆DA (c) at 8. We have the following results.
Proposition 3.2.3. For every c ∈ [0, 8), ∆DA (8) > ∆DA (c). Consequently, if
a disappointment-averse decision maker chooses the risky lottery when c = 8,
then the decision maker also chooses the risky lottery when c < 8.
Proposition 3.2.4. A disappointment-averse decision maker who prefers
the risky lottery if c = 8 must also prefer the risky lottery if c = 10.
In Table 5, we show that disappointment can explain 13 choice patterns.
Six of these are patterns of mixed fanning. Only two of the other theories
we consider can justify mixed fanning, rank-dependent utility and cumulative
prospect theory, and even these can only explain two special cases (RRSRRR
and RSSRRR), both of which disappointment aversion also explains.
3.3
Rank-Dependent Utility
In general, one calculates the rank-dependent utility of a lottery as follows:
suppose lottery L has possible outcomes {x1 , . . . , xn }, ordered from worst to
best; that is, u(x1 ) ≤ . . . ≤ u(xn ). Denote the probability of prize xi by pi .
The rank-dependent utility of L is then
!
!
n
n
n
X
X
X
URDU (L) =
w(pi )u(xi ), where w(pi ) = π
pj − π
pj .
i=1
j=i
j=i+1
Here π(·) represents the weight attached to probability pi . The utility of
each outcome is weighted by the decumulative events: w(·) gives the difference between the probability of getting a prize at least as good as xi and the
probability of getting a prize that is better than xi . Aside from requiring that
13
π(0) = 0 and π(1) = 1, we do not impose any restrictions on the probability
weighting function π(·).
The rank ordering of the prizes in our task depends on whether c is below
8, between 8 and 10, or above 10. As we do not offer lottery choices with
c ∈ (8, 10), we focus on the other regions.6 We consider first the case in
which c ∈ [0, 8). The rank-dependent utility of the safe lottery is then
S
URDU
(c) = [1 − π(.11)] u(c) + π(.11)u(8).
As c increases to 8,
S
S
lim URDU
(c) = u(8) = URDU
(8).
c%8
The risky lottery, given that c ∈ (0, 8], has rank-dependent utility
R
URDU
(c) = [π(.99) − π(.1)] u(c) + π(.1)u(10).
As c decreases to 0,
R
R
lim URDU
(c) = π(.1)u(10) = URDU
(0).
c&0
We can therefore consider the entire interval [0,8]. The difference in rankdependent utility over this range is
∆RDU (c) = [1 − π(.99) − π(.11) + π(.1)] u(c) + π(.11)u(8) − π(.1)u(10). (5)
We therefore obtain the following restrictions.
Proposition 3.3.1. For c ∈ [0, 8], a rank-dependent utility maximizer can
switch between choosing the safe lottery and the risky lottery at most once.
Next, consider the case in which c ≥ 10. The rank-dependent utilities are
S
URDU
(c) = [1 − π(.89)] u(8) + π(.89)u(c)
R
URDU
(c) = [π(.99) − π(.89)] u(10) + π(.89)u(10).
The difference in rank-dependent utilities for c ≥ 10 is therefore
∆RDU (c) = [1 − π(.89)] u(8) − [π(.99) − π(.89)] u(10).
(6)
Consequently, we have the following.
6
It is easy to show that the rank-dependent utility of the safe lottery is continuous,
even at c ∈ {8, 10}.
14
Proposition 3.3.2. A rank-dependent utility maximizer makes the same
choice at every c ≥ 10.
There are six possible patterns for c ∈ {0, 5, 8} consistent with Proposition 3.3.1: SSS, SSR, SRR, RSS, RRS, RRR. There are two possible patterns
for c ∈ {10, 16, 20} consistent with Proposition 3.3.2: SSS, RRR. Together,
these give the patterns shown in Table 5.
3.3.1
Cumulative Prospect Theory
In our setting, cumulative prospect theory gives identical predictions for any
reference point in the set {0, 5, 8, 10, 16, 20}. This means that cumulative
prospect theory is a special case of rank-dependent utility in our task, with
additional restrictions arising from the probability weighting function that
the theory imposes. Tversky and Kahneman (1992) suggest the following
functional form, which gives the probability weighting function an inverse-S
shape:
pγ
,
π(p) =
1/γ
(pγ + (1 − p)γ )
where γ ∈ (0, 1].7
This form implies overweighting of extreme events, specifically the worst
and the best outcomes. For 0 ≤ c ≤ 8, the common consequence c is overweighted in the safe lottery, as the worst outcome, but not in the risky lottery.
Therefore a cumulative prospect theoretic decision maker who chooses the
safe lottery when c = 0 would also choose the safe lottery for c ∈ {5, 8}.
Proposition 3.3.3. The cumulative prospect theory probability weighting
function restricts the choice patterns for c ∈ {0, 5, 8} to one of the following:
{SSS, RSS, RRS, RRR}.
A consequence of Proposition 3.3.3 is that only 8 of the 12 patterns that
are consistent with rank-dependent utility are also consistent with cumulative
prospect theory.
7
Some authors allow more general ranges for the curvature parameter γ. For instance,
Camerer and Ho (1994) provide several estimates of γ, ranging from 0.28 to 1.87. Some
values of γ, however, can make π(·) nonmonotone. See Ingersoll (2008).
15
3.4
Prelec Probability Weighting Function
Prelec (1998) provides axioms for the following probability weighting function, called the compound invariant probability weighting function:
π(p) = exp(−(− ln p)α )
α ∈ (0, 1).
This form imposes the same restrictions as those implied by Proposition 3.3.3.
An additional restriction is that the decision maker cannot become more risk
tolerant as c increases from 8 to 10.
Proposition 3.4.1. The compound invariant probability weighting function
restricts the choice patterns to one of the following:
{SSSSSS, RSSSSS, RRSSSS, RRRSSS, RRRRRR}.
4
Data
The data for our main tests came from subjects in the PEEL subject pool, as
for the baseline treatment described in Section 2, and from an undergraduate
economics course at Carnegie Mellon University. All subjects were at least
18 years old. The PEEL subjects all received financial compensation; the
Carnegie Mellon subjects took part in experiments with hypothetical lotteries. Among the PEEL subject pool, there were 215 enrolled subjects across
all robustness checks, including the 34 in the baseline treatment. Average
earnings were $14.24. Among the Carnegie Mellon students, there were 35
subjects, giving us a total of 250 subjects.
For our main questions, we had three conditions:
• Allais presentation, real rewards (n = 29)
• Savage presentation, real rewards (n = 32)
• Savage presentation, hypothetical rewards (n = 35)
In each of the presentations, we ordered the lotteries on a single sheet of
paper in one of three different orders. One order had c = 0 on the lottery
pair on the top of the page. Another had c = 8 first, and a third had c = 20
first.
16
To test whether our results were sensitive to the number alternatives
presented, we ran two sessions in which only the two standard Allais choices
(c ∈ {0, 8}) were presented. This enabled us to check whether the frequency
of Allais and reverse Allais choices was affected by there being six rather than
two decisions. One of these sessions used the Allais presentation (n = 32);
another used the Savage presentation (n = 29).
We had two tests of the sensitivity of our results to the size of the rewards.
In one session, we used an Allais presentation with two choices (n = 28),
with (m, h) = ($12, $24) and with the common consequence c ∈ {0, $12}. In
another session, we used the Savage presentation with six choices (n = 31).
For this session, we chose (m, h) = ($12, $24) for comparability with the
Allais larger reward session and chose c ∈ {0, 6, 12, 18, 24, 30}.
5
Results and Robustness Checks
5.1
Main Treatment
Table 6 shows the frequency of subjects complying with each tested theory.
This is depicted graphically in Figure 2.
Several aspects of the data are immediately apparent. First, the Savage
presentation dramatically increases the frequency with which subjects behaved as expected utility theory predicts (p = 0.034 on a one-tailed test).8
Subjects who saw the Savage presentation were 1.53 times more likely to behave like expected utility maximizers than were subjects who saw the Allais
presentation. This result is true even if the Savage presentation is given with
hypothetical lotteries. Overall, this is in keeping with Savage’s view that the
Allais paradox is about the common consequence being difficult to see and
not about the size of the rewards.
Second, behavior that is unexplained by any theory is much more common
with the Allais presentation than with the Savage presentation. Unexplained
choices are 2.31 times more likely among subjects who face the Allais presentation compared with those who face the Savage presentation (p = 0.021
on a two-tailed test; we use a two-tailed test here because we did not have
8
Because the Savage presentation had similar rates of conformity with expected utility,
we mostly focused on the combined results in the Savage presentation and distinguish
behavior with hypothetical versus real rewards when there is an important feature of the
data to highlight. See Table 6.
17
an initial prediction about the frequency of unexplained choices.
Third, nearly all the departures from expected utility theory are either explained by the zero effect patterns or not explained by any theory. Expected
utility theory, unexplained behavior, and the zero effect explain a combined
total of 93% of the data, in both the Allais and the Savage presentations.9
Table 6: Frequency of compliance with each theory
Allais
Savage
Savage
Savage
real
total
real
hypothetical
(n = 29) (n = 67) (n = 32) (n = 35)
Expected utility
38% (11) 58% (39) 56% (18) 60% (21)
Unexplained
38% (11) 16% (11) 22% (7)
11% (4)
Zero effect
17% (5)
18% (12) 19% (6)
17% (6)
All others (incremental) 7% (2)
7% (5)
3% (1)
11% (4)
Figure 2: Patterns of choices in Allais and Savage presentations.
To put this in context, Table 7 shows the most commonly observed choice
patterns. Only three choice patterns were selected by more than 5% of the
subjects: the two expected utility patterns and the ordinary zero effect pattern. Two patterns were chosen by 4% of subjects: the reverse zero effect
and a rank-dependent pattern that was consistent with cumulative prospect
9
The departures from these three patterns were higher with hypothetical choices than
with real choices, but the total number of observations was too small for all meaningful
testing.
18
theory but inconsistent with the Prelec weighting function; this pattern was
also consistent with disappointment aversion. This last pattern was chosen
by one subject who faced the Allais presentation and by three subjects who
faced hypothetical choices in the Savage presentation.
Table 7: Most common patterns
Pattern
Theories
Frequency (n = 96)
RRRRRR
All
35% (34)
SSSSSS
All
17% (16)
RSSSSS
Zero effect, all non-EU
13% (13)
SRRRRR
Reverse zero effect, RDU
4% (4)
RSSRRR
DA, RDU, CPT
4% (4)
Note. Among the choices that were consistent with any theory, these five
patterns account for 90% (71/79).
None of the unexplained patterns was chosen by more than three subjects. By comparison, three subjects made their choices by alternating as
they proceeded down the page. It would hence seem difficult to explain the
patterns that occur with very low frequencies.
Summarizing, we have the following.
Claim 1. The Savage presentation dramatically increases consistency with
expected utility theory and dramatically decreases behavior that is inconsistent
with any theory.
Claim 2. Essentially all the behavior that is not explained by expected utility
theory is attributable to the zero effect.
Claim 2 implies that the robustness of the zero effect is critical if any of
the tested nonexpected utility theories is to explain behavior in an Allaistype task. Some additional features of the data are germane to the point in
Claim 2. No subjects showed a certainty effect. It is thus exceedingly unlikely
that the certainty effect could be driving the Allais paradox. Additionally,
excluding subjects making the zero effect choices, one paid subject and four
subjects making hypothetical choices made decisions that were consistent
with at least one theory and included the standard Allais pattern. The
lone paid subject and two of the subjects making hypothetical choices were
consistent with disappointment aversion and no other theories. The other two
subjects in the hypothetical treatment were consistent with disappointment
19
aversion, rank-dependent utility, and cumulative prospect theory with the
Kahneman–Tversky weighting function but not with Prelec weighting. All
the subjects who made the reverse Allais choices were either unexplained by
any theory or were those who chose the reverse zero effect pattern. Summing
up, we have the following claim.
Claim 3. The evidence for the standard Allais pattern of choices and for the
reverse Allais pattern of choices rests entirely on the robustness of the zero
effect.
To understand the zero effect better, we provide a breakdown of subject
choice patterns by presentation order. Table 8 summarizes this information.
Table 8: Choice patterns by presentation order
0 first
8 first
20 first
(n = 32)
(n = 34)
(n = 30)
EU
56%
(18)
41%
(14)
60%
Unexplained
13%
(4)
29%
(10)
27%
Zero effect
28%
(9)
18%
(6)
7%
Others
3%
(1)
12%
(4)
7%
(18)
(8)
(2)
(2)
The most striking feature of Table 8 is that the zero effect appears to be
an artifact of the 0-first presentation order. The zero effect occurs in 28% of
the observations with 0 first and in 13% of the other observations (p = 0.029
on a one-tailed test). The level of conformity with expected utility does not
decrease in the 0-first order. This suggests that subjects who choose the zero
effect pattern are comparable to those who, in a different presentation order,
behave inconsistently with any theory. Putting the zero common consequence
first can push subjects toward a particular, seemingly systematic, pattern of
behavior. Yet this pattern appears to be illusory.
Figure 3 shows the effects of presentation order split by the presentation
format (i.e., by Allais vs. Savage presentation). The 0-first order, shown
in Figure 3 (top left), and the 20-first order, shown in Figure 3 (bottom),
show comparable levels of subject compliance with expected utility theory—
around 40% for the Allais presentation and between 60% and 70% in the
Savage presentation. The main effect of presenting the zero common consequence first is, in both formats, seen among those who do not comply with
expected utility theory: they make the zero-effect choices, while in the 20first order, they do not, and they simply make a choice that is inconsistent
20
with every theory. Presenting the common consequence of 8 first is now seen
to have the greatest effect on the subjects who saw the Savage presentation:
it makes them behave more like those who saw the Allais presentation.
To summarize, Table 8 and Figure 3 show that switching from a 0-first
to a 20-first order does not make the subjects significantly more compliant
with expected utility theory. Instead, a 0-first presentation moves subjects
from being inconsistent to choosing in accordance with the zero effect, while
an 8-first presentation order makes the subject decisions appear more erratic
(but, again, does not produce a zero effect). We therefore make the following
observation.
Claim 4. The zero effect is a result of presentation.
Figure 3: Presentation order, split by presentation format.
5.2
Auxiliary Results and Robustness Tests
Our main tests included six choices, whereas the standard Allais task involves
two choices, with c ∈ {0, m}. As a robustness check, we investigated the
consequences of running our experiment with only the two standard Allais
choices. Our null hypothesis is that the number of choices would not affect the
frequency of compliance with expected utility, irrespective of the presentation
and irrespective of whether the rewards were real or hypothetical.
21
To test this, we ran two additional sessions, where the subjects faced two
lottery pairs. In the Allais presentation, they were as follows:
1. A lottery paying $8 for sure versus a lottery with an 89% chance of
paying $8, a 10% chance of paying $10, and a 1% chance of paying
nothing
2. A lottery with an 11% chance of paying $8 and an 89% chance of paying
nothing versus a lottery with a 10% chance of paying $10 and a 90%
chance of paying nothing
The Savage presentation was analogous.
As in the six-choice tasks, we explained how the rolls of dice would determine payouts, which kept the two presentations equivalent in terms of
regret. There were 32 subjects in the session with two choices and the Allais
presentation, and there were 29 subjects in the session with two choices and
the Savage presentation. All subjects in these two sessions played for real
rewards.
We compared the frequency of choosing either SS, RR, or indifference on
both choices—that is, of satisfying the predictions of expected utility theory
in the two-choice task—with the frequency of subjects in the six-choice tasks
making similar choices when c ∈ {0, 8}. The results are in Table 9.
Table 9: Frequency of making expected utility-consistent choices when c ∈
{0, 8}, by presentation and by total number of choices
No. choices
Presentation
EU | c ∈ {0, 8}
2
Allais
56% (18/32)
6
Allais
62% (18/29)
2
Savage
66% (19/29)
6
Savage (combined)
64% (43/67)
6
Savage (real only)
63% (20/32)
6
Savage (hypothetical only)
66% (23/35)
There is no difference in the frequency of expected utility violations on
the comparable decisions when there are two or six lottery choices. We infer
that it is safe to conclude that extending the number of choices to six lotteries
in our main treatment did not introduce any confounds.
Our next tests involved the size of the rewards. Other researchers (e.g.,
Conlisk, 1989; Harrison, 1994; Fan, 2002) have found that as real rewards
22
increase, Allais-type behavior vanishes. This could be due to salience—see
Binmore (1999) on this point—or to the increase in the gap in expected value
between the risky and safe lotteries.
To test whether our results were robust to larger rewards, we ran two
sessions with (m, h) = ($12, $24). The first session used an Allais presentation with only two choices (this sufficed for our purposes, given the results in
Table 9) and had 28 subjects. The second used the Savage presentation with
six choices: c ∈ {0, 6, 12, 18, 24, 30}; this session had 31 subjects. Our null
hypothesis was that there would be no decrease in expected utility violations.
In the high-rewards Allais presentation session, 93% of subjects (26/28)
chose consistently with expected utility theory. Among these, all but one
subject selected the risky lottery in both choices; a lone subject expressed
indifference on both choices. The remaining two subjects both expressed
indifference on the lottery where c = 0 and a preference for the safe lottery
where c = 12.
In the high-rewards Savage presentation, 71% of the subjects (22/31)
were consistent with expected utility on all six choices. All of these selected
the risky lottery on every choice. Among the subjects who did not satisfy
expected utility theory, no choice pattern occurred more than once. There
were no zero effect or certainty effect choices. Two subjects had the standard Allais pattern, and one had the reverse Allais pattern on c ∈ {0, 12}.
Overall, these results are consistent with prior literature and suggest that
the relatively strong performance of expected utility theory in our main task
was not due to the small size of the rewards.
We saw earlier that the use of real versus hypothetical rewards had no
significant effects on compliance with expected utility theory. However, this
does not mean that the choice of hypothetical rewards is inconsequential.
Figure 4 gives two comparisons of real and hypothetical rewards, holding the
presentation fixed (as the Savage presentation).
In Figure 4 (top), it is easy to see that the frequency of choosing consistently with expected utility theory is indistinguishable between the hypothetical and real rewards treatments. Figure 4 (bottom) tells a different part of
the story. With hypothetical rewards, the subjects are much more willing to
take risks. Subjects who faced the Savage presentation and real rewards were
more likely to comply with expected utility theory by choosing exclusively
safe lotteries, by roughly a 3:2 ratio. With hypothetical rewards, nearly all
of the subjects who complied with expected utility chose only risky lotteries.
The last column of Figure 4 (bottom) underscores this point: roughly 70%
23
Figure 4: Real and hypothetical rewards, by expected utility compliance and
by risk preference.
24
of hypothetical choices were risky lotteries, compared with just over 40% of
real choices.
This brings us back to Binmore (1999). Transparency of the decision is
one crucial feature for laboratory experiments. The common consequence is
equally transparent in subjects facing real and hypothetical choices, given the
Savage presentation. The crux of Savage’s argument is that seeing the common consequence is enough to make a decision maker avoid the Allais paradox
and comply with expected utility theory. But for other tasks, understanding
the question is not enough; stable risk preferences may be important. The
Figure 4 (bottom) shows what we might expect: when there is nothing at
stake, subjects will be more willing to gamble.
6
Discussion and Conclusion
After reviewing the major proposed explanations for behavior in Allais-type
tasks, we have argued, we hope persuasively, that expected utility is reasonably descriptive. The chief rivals of expected utility theory, given a transparent test, do not show much promise. All of their incremental explanatory
power is due to the zero effect. We find that the zero effect appears most
prominently when a zero common consequence is presented first. If a certain lottery is presented first, the zero effect diminishes. More importantly,
when the first choice presented has neither a zero common consequence nor
a contant lottery, the zero effect vanishes. So once more, transparency is the
key.
Suppose we have convinced you, and that you are now in full agreement
with Savage. His initial choices conflicted with his own theory because Allais’s presentation misdirected his attention. What questions remain open
on the Allais paradox? A natural agenda is to view standard Allais behavior
in the same way that game theorists view off-equilibrium play. It may be
reasonable to ask how an opaque presentation might introduce bias in noisy
expected utility maximization.
From a methodological viewpoint, we contribute to the discussion of
salience of rewards. Similar to Camerer and Hogarth (1999); Slovic (1969);
Harrison (2006), we find that hypothetical rewards make little difference in
expected utility compliance but that hypothetical rewards do alter behavior.
Our interpretation is that subjects who face hypothetical rewards respond
as they imagine themselves. Sometimes this matches their actually behavior.
25
In an Allais-type task, compliance with expected utility theory is a matter
noticing the common consequence, which does not depend on differences between self-image and actual choices. In other respects, how subjects imagine
themselves differs from how they are. We see this in our setting by a sharply
increased willingness to take risks when nothing is at stake.
Appendix: Proofs
Proof of Proposition 3.1.1. As the common consequence c increases, both
the safe and the risky lottery improve in the sense of first-order stochastic
dominance (and are constant in terms of expected utility). Hence increasing
risk aversion can only result in a switch from the risky lottery to the safe
one.
To prove Proposition 3.2.1, we first prove the folliwing lemma:
Lemma .0.1. Disappointment aversion implies u(8)/u(10) < 10/11.
Proof of Lemma .0.1. Let c∗ < 10 be a point where c < c∗ is disappointing
and c > c∗ is elating. Then
R
lim∗ UDA
(c) =
c%c
.89(1 + β)u(c∗ ) + .1u(10)
1 + .9β
R
lim∗ UDA
(c) =
c&c
.89u(c∗ ) + .1u(10)
.
1 + .01β
R
R
R
(c):
(c∗ ) ≤ limc&c∗ UDA
(c) ≤ UDA
Monotonicity requires that limc%c∗ UDA
⇒
⇒
.89(1 + β)u(c∗ ) + .1u(10)
.89u(c∗ ) + .1u(10)
≤
1 + .9β
1 + .01β
.89(1 + β)(1 + .01β)u(c∗ ) + .1(1 + .01β)u(10)
≤ .89(1 + .9β)u(c∗ ) + .1(1 + .9β)u(10)
⇒ .89(1 + β + .01β + .01β 2 )u(c∗ ) + .1(1 + .01β)u(10)
≤ .89(1 + .9β)u(c∗ ) + .1(1 + .9β)u(10).
Combining like terms,
.89(.11β + .01β 2 )u(c∗ ) ≤ .1(.89β)u(10).
26
If β ≤ 0, the decision maker is not disappointment averse. Otherwise, we
can divide both sides by .89β and preserve the inequality:
(.11 + .01β)u(c∗ ) ≤ .1u(10) ⇒ 11 + β ≤ 10
β≤
u(10)
u(c∗ )
10u(10) − 11u(c∗ )
.
u(c∗ )
It is clear that c∗ < 10 because c∗ = 10 would imply β <= −1. Also, because
disappointment aversion implies β > 0, we have
0<β≤
10u(10) − 11u(c∗ )
u(c∗ )
⇒ 11u(c∗ ) < 10u(10) ⇒
10
u(c∗ )
< .
u(10)
11
For any c < 8, the prize c must be disappointing in the safe lottery. By
assumption, c < 8 cannot be elating in the risky lottery, which means that
c∗ ≥ 8. Hence
u(8)
u(c∗ )
10
≤
< .
u(10)
u(10)
11
Proof of Proposition 3.2.1. For c > 10, the outcome 0 is always disappointing, but the outcome 10 may be either elating or, if c is sufficiently high,
disappointing.
Suppose first that 10 is always elating. Then (∀c ≥ 10),
R
UDA
(c) =
S
UDA
(c) =
.1u(10) + .89u(c)
1 + .01β
.11(1 + β)u(8) + .89u(c)
,
1 + .11β
and hence
.11(1 + β)u(8)
.1u(10)
∆DA (c) =
−
+ .89u(c)
1 + .11β
1 + .01β
27
1
1
−
1 + .11β 1 + .01β
.
Because β > 0 for a disappointment-averse decision maker,
∂∆DA (c)
1
1
0
= .89u (c)
−
< 0.
∂c
1 + .11β 1 + .01β
Hence the decision maker becomes less risk averse as c increases, allowing at
most one switch, and this switch must be from safe to risky.
Now suppose instead that for some ĉ > 10, the outcome 10 is disappointing for c > ĉ. Then, for c > ĉ,
R
UDA
(c) =
.1(1 + β)u(10) + .89u(c)
1 + .11β
S
UDA
(c) =
.11(1 + β)u(8) + .89u(c)
.
1 + .11β
By Lemma .0.1, u(8)/u(10) < 10/11, so
S
UDA
(c)
.11(1 + β) 10u(10)
+ .89u(c)
11
<
1 + .11β
=
.11(1 + β)u(8) + .89u(c)
R
= UDA
(c).
1 + .11β
This implies ∆DA (c) < 0 so that the decision maker always chooses the risky
lottery when c > ĉ.
Proof of Proposition 3.2.2. For c ∈ (0, 8),
.11u(8) + .89(1 + .89β)u(c) .1u(10) + .89(1 + β)u(c)
−
1 + .89β
1 + .9β
.11u(8)
.1u(10)
1
1
=
−
+ .89(1 + β)u(c)
−
1 + .89β 1 + .9β
1 + .89β 1 + .9β
1
1
= ∆DA (0) + .89(1 + β)u(c)
−
.
1 + .89β 1 + .9β
∆DA (c) =
The last term is positive because β > 0, so this is above ∆DA (0). This means
the first derivative with respect to c is a positive coefficient times u0 (c), which
is also positive due to monotonicity.
28
Proof of Proposition 3.2.3. The limit from the left is
lim ∆DA (c) = u(8) −
c%8
.1u(10) + .89(1 + β)u(8)
,
1 + .9β
which by Proposition 3.2.2 is larger than ∆DA (c) for any c ∈ [0, 8). Rearranging,
lim ∆DA (c) =
c%8
u(8) + .9βu(8) − .1u(10) − .89u(8) − .89βu(8)
1 + .9β
.11u(8) + .01βu(8) − .1u(10)
=
.
1 + .9β
The limit from the right is
.1u(10) + .89u(8)
1 + .01β
u(8) + .01βu(8) − .89u(8) − 1u(10)
=
1 + .01β
.11u(8) + .01βu(8) − .1u(10)
=
1 + .01β
1 + .9β
=
lim ∆DA (c).
1 + .01β c%8
lim ∆DA (c) = u(8) −
c&8
For a disappointment-averse decision maker, β > 0, so the limit of ∆DA (c)
from the right is strictly greater than the limit from the left as c → 8.
Therefore
∆DA (8) ≥ lim ∆DA (c),
c%8
which implies that for all c < 8, ∆DA (8) > ∆DA (c).
Proof of Proposition 3.2.4. If the decision maker chooses the risky lottery
when c = 8, then ∆DA (8) < 0. Because ∆DA (8) ≤ lim c & 8∆DA (c), a
necessary condition for the decision maker to choose the risky lottery at
c = 8 is
lim ∆DA (c) ≤ 0.
c&8
Expanding,
u(8) −
.89u(8) + .1u(10)
.11u(8) + .01βu(8) − .1u(10)
≤0 ⇒
≤0
1 + .01β
1 + .01β
29
⇒ .11u(8) + .01βu(8) ≤ .1u(10)
⇒ (11 + β)u(8) ≤ 10u(10).
Rearranging,
10
u(8)
≤
.
u(10)
11 + β
(7)
If a disappointment-averse decision maker chooses the safe lottery at c =
10, then ∆DA (10) > 0. This implies
.11(1 + β)u(8) + .89u(10)
.99u(10)
>
1 + .11β
1 + .01β
⇒ .11(1 + β)(1 + .01β)u(8) > [.99(1 + .11β) − .89(1 + .01β)]u(10)
= [.1 + (.99)(.11)β − (.89)(.01)β]u(10)
= .1(1 + β)u(10).
Dividing both sides by 1 + β gives
.11(1 + .01β)u(8) > .1u(10).
Rearranging,
10
u(8)
>
.
u(10)
11(1 + .01β)
(8)
To choose the risky lottery at c = 8 and the safe lottery at c = 10, the
decision maker would need to satisfy inequalities (7) and (8). This requires
10
10
<
11(1 + .01β)
11 + β
⇒ 11 + β < 11 + .11β
⇒ β < .11β,
which implies β < 0.
Proof of Proposition 3.3.1. Equation (5) states that for c ∈ [0, 8],
∆RDU (c) = [1 − π(.99) − π(.11) + π(.1)] u(c) + π(.11)u(8) − π(.1)u(10).
30
Differentiating with respect to c,
d∆RDU (c)
= u0 (c) [1 − π(.99) − π(.11) + π(.1)] .
dc
Because u0 (c) > 0 by the monotonicity assumption, it follows that ∆RDU (c)
is either increasing, constant, or decreasing on [0,8].
Proof of Proposition 3.3.2. Equation (6) is independent of c.
Proof of Proposition 3.3.3. For 0 ≤ c ≤ 8, we found that
d∆RDU (c)
= − [π(.99) − π(.1) − 1 + π(.11)] u0 (c).
dc
If we plug the Tversky–Kahneman probability weighting function into the
previous expression,
d∆RDU (c)
> 0.
dc
Proof of Proposition 3.4.1. Recall that
∆RDU (8) = [1 − π(.99) + π(.1)] u(8) − π(.1)u(10)
∆RDU (10) = [1 − π(.89)] u(8) − [π(.99) − π(.89)] u(10).
Subtracting and rearranging,
∆RDU (8) − ∆RDU (10) = [u(8) − u(10)] [π(.1) − π(.99) + π(.89)] < 0.
This implies that the choices on {0, 5, 8, 10} must be one of SSSS, RSSS,
RRSS, RRRS, or RRRR. Because these are predictions for a rank-dependent
theory, the choice at c = 10 determines the choice at c ∈ {16, 20}.
References
M. Allais. Le comportement de l’homme rationnel devanti le risque, critique
des postulates et axiomes de l’école Americaine. Econometrica, 21(4):503–
546, 1953.
31
J. Beattie and G. Loomes. The impact of incentives upon risky choice experiments. Journal of Risk and Uncertainty, 14(2):155–168, 1997.
D. E. Bell. Disappointment in decision making under uncertainty. Operations
Research, 33(1):1–27, 1985.
K. Binmore. Why experiment in economics? Economic Journal, 109(453):
F16–F24, 1999.
M. H. Birnbaum, G. Coffey, B. A. Mellers, and R. Weiss. Utility measurement: Configural-weight theory and the judge’s point of view. Journal
of Experimental Psychology: Human Perception and Performance, 18(2):
331–346, 1992.
K. Borch. The Allais paradox: A comment. Behavioral Science, 13(6):488–
489, 1968.
M. S. Burke, J. R. Carter, R. D. Gominiak, and D. F. Ohl. An experimental
note on the Allais paradox and monetary incentives. Empirical Economics,
21(4):617–632, 1996.
C. Camerer and R. Hogarth. The effects of financial incentives in experiments: a review and capital-labor framework. Journal of Risk and Uncertainty, 19(1–3):7–42, 1999.
C. F. Camerer. An experimental test of several generalized utility theories.
Journal of Risk and Uncertainty, 2(1):61–104, 1989.
C. F. Camerer and T.-H. Ho. Violations of the betweenness axiom and
nonlinearity in probability. Journal of Risk and Uncertainty, 8(2):167–
196, 1994.
P. S. Carlin. Violations of the reduction and independence axioms in Allaistype common-ratio effect experiments. Journal of Economic Behavior and
Organization, 19(2):213–235, 1992.
J. Conlisk. Three variants on the Allais example. American Economic Review, 79(3):392–407, 1989.
D. Davidson, P. Suppes, and S. Siegel. Decision Making: An Experimental
Approach. Stanford University Press, 1957.
32
J. Dickhaut, K. McCabe, J. C. Nagode, A. Rustichini, K. Smith, and J. V.
Pardo. The impact of the certainty context on the process of choice. Proceedings of the National Acadamy of Sciences, 100(6):3536–3541, 2003.
C.-P. Fan. Allais paradox in the small. Journal of Economic Behavior and
Organization, 49(3):411–421, 2002.
G. Gigerenzer. How to make cognitive illusions disappear: Beyond heuristics
and biases. European Review of Social Psychology, 2(1):83–115, 1991.
D. Gill and V. Prowse. A structural analysis of disappointment aversion in a
real effort competition. American Economic Review, 102(1):469–503, 2012.
S. Grant, A. Kajii, and B. Polak. Different notions of disappointment aversion. Economics Letters, 70(2):203–208, 2001.
F. Gul. A theory of disappointment aversion. Econometrica, 59(3):667–686,
1991.
D. W. Harless and C. F. Camerer. The predictive utility of generalized
expected utility theories. Econometrica, 62(6):1251–1289, 1994.
G. W. Harrison. Expected utility theory and the experimentalists. Empirical
Economics, 19(2):223–253, 1994.
G. W. Harrison. Hypothetical bias over uncertain outcomes. In J. A. List,
editor, Using Experimental Methods in Environmental and Resource Economics, pages 41–69. Edward Elgar, 2006.
J. Ingersoll. Non-monotonicity of the Tversky-Kahneman probability weighting function: A cautionary note. European Financial Management, 14(3):
385–390, 2008.
J. B. Kadane. Healthy skepticism as an expected-utility explanation of the
phenomena of Allais and Ellsberg. Theory and Decision, 32(1):57–64, 1992.
D. Kahneman and A. Tversky. Prospect theory: An analysis of decision
under risk. Econometrica, 47(2):263–292, 1979.
L. R. Keller. The effects of problem representation on the sure-thing and
substitution principles. Management Science, 31(6):738–751, 1985.
33
G. Loomes and R. Sugden. Disappointment and dynamic consistency in
choice under uncertainty. Review of Economic Studies, 53(2):271–282,
1986.
K. R. MacCrimmon. Descriptive and normative implications of the decisiontheory postulates. In K. H. Borch and J. Mossin, editors, Risk and Uncertainty, pages 3–23. St. Martin’s Press, 1968.
M. J. Machina. “Expected utility” analysis without the independence axiom.
Econometrica, 50(2):277–324, 1982.
M. J. Machina. Choices under uncertainty: Problems solved and unsolved.
Journal of Economic Perspectives, 1(1):121–154, 1987.
J. Marschak. Rational behavior, uncertain prospects, and measurable utility.
Econometrica, 18(2):111–141, 1950.
B. Mellers, R. Weiss, and M. Birnbaum. Violations of dominance in pricing
judgments. Journal of Risk and Uncertainty, 5(1):73–90, 1992.
H. Moskowitz. Effects of problem representation and feedback on rational
behavior in Allais and Morlat-type problems. Decision Sciences, 5(2):225–
242, 1974.
D. Prelec. The probability weighting function. Econometrica, 66(3):497–527,
1998.
J. Quiggin. Risk perception and the analysis of risk attitudes. Australian
Journal of Agricultural Economics, 25(2):160–169, 1981.
J. Quiggin. A theory of anticipated utility. Journal of Economic Behavior
and Organization, 3(4):323–343, 1982.
L. J. Savage. The Foundations of Statistics. Dover Publications, 2nd revised
edition, 1972. Revised republication of 1954 edition, published posthumously in 1972.
P. Slovic. Differential effects of real vs. hypothetical payoffs upon choices
among gambles. Journal of Experimental Psychology, 80(3):434–437, 1969.
P. Slovic and A. Tversky. Who accepts Savage’s axiom? Behavioral Science,
19(6):368–373, 1974.
34
C. Starmer. Developments in non-expected utility theory: The hunt for a
descriptive theory of choice under risk. Journal of Economic Literature,
38(2):332–382, 2000.
C. Starmer. Testing new theories of choice under uncertainty using the common consequence effect. Review of Economic Studies, 59(4):813–830, 1992.
C. Starmer and R. Sugden. Does the random-lottery incentive system elicit
true preferences? An experimental investigation. American Economic Review, 81(4):971–978, 1991.
A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative
representation of uncertainty. Journal of Risk and Uncertainty, 5(4):297–
323, 1992.
J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton University Press, 1944.
P. Wakker and D. Deneffe. Eliciting von Neumann-Morgenstern utilities
when probabilities are distorted or unknown. Management Science, 42(8):
1131–1150, 1996.
P. P. Wakker. Prospect Theory for Risk and Ambiguity. Cambridge University
Press, New York, 2010.
G. Wu, J. Zhang, and M. Abdellaoui. Testing prospect theories using probability tradeoff consistency. Journal of Risk and Uncertainty, 30(2):107–131,
2005.
35