The effects of losses and event splitting on the Allais paradox

Judgment and Decision Making, Vol. 2, No. 2, April 2007, pp. 115–125
The effects of losses and event splitting on the Allais paradox
Bethany J. Weber?
Brain Imaging and Analysis Center
Duke University
Abstract
The Allais Paradox, or common consequence effect, has been a standard challenge to normative theories of risky
choice since its proposal over 60 years ago. However, neither its causes nor the conditions necessary to create the effect
are well understood. Two experiments test the effects of losses and event splitting on the Allais Paradox. Experiment
1 found that the Allais Paradox occurs for both gain and mixed gambles and is re?ected for loss gambles produced by
re?ection across the origin. Experiment 2 found that the Allais Paradox is eliminated by splitting the outcomes even
when the probabilities used do not increase the salience of the common consequence. The results of Experiment 1
are consistent with Cumulative Prospect Theory, the current leading theory of risky choice. However, the results of
Experiment 2 are problematic for Cumulative Prospect Theory and suggest that alternate explanations for the Allais
Paradox must be sought.
Keywords: risk, Allais Paradox, certainty effect, preference reversals, mixed gambles, event splitting, Prospect Theory,
Cumulative Prospect Theory.
1 Introduction
The Allais Paradox has been demonstrated under many
different conditions (e.g., Birnbuam 2004; Camerer,
When expected utility theory (EU) was ?rst proposed
1989; Conlisk, 1989; Kahneman and Tversky 1979;
(e.g., Savage, 1954) it was assumed that EU was not only
Slovic and Tversky, 1974; Wu & Gonzalez, 1996). It is
the normative theory of risky decision making, but a de-
a real and robust phenomenon, at least when it involves
scriptive theory as well. However it quickly became ap-
a standard presentation of large monetary gains. How-
parent that EU did not work as a descriptive theory of
ever, the literature on the results of varying the presen-
risky choice. One of the ?rst and most famous challenges
tation of the paradox is mixed. The present experiments
to EU was presented by Allais in 1953. Suppose one is
investigate two circumstances in which the leading expla-
given a choice between the following gambles:
nation of the Allais Paradox, Cumulative Prospect The-
$1 million for sure
10% chance of $5 million
ory (CPT), predicts an Allais common consequence ef-
fect should occur, but in which the effect has not always
89% chance of $1 million
previously been found. Experiment 1 addresses the is-
1% chance of $0
sue of the Allais Paradox and losses, while Experiment 2
Decision-makers, Allais proposed, will generally choose
investigates the effects of event splitting on the paradox.
the safer, certain gamble. However, when asked to choose
between
1.1 Explanations of the Allais Paradox
11% chance of $1 million
10% chance of $5 million
89% chance of $0
90% chance of $0
A number of explanations for the Allais Paradox have
been advanced. These theories include fanning-out the-
decision-makers will generally chose the riskier gamble.
ories, which explain the paradox via the shape of indif-
Closer inspection reveals that the second set of gambles
ference curves in the unit triangle (Machina, 1982), and
are obtained from the ?rst by removing a common con-
expected cardinality-speci?c utility theories, which spec-
sequence, an 89% chance of winning $1 million, whose
ulate that the utility function varies with the number of
presence or absence should have no effect on preference.
outcomes (Neilson, 1992; Humphrey, 1998, 2001). How-
Thus, subjects’ preference reversals are nonnormative.
ever, for the present discussion I will concentrate on two
?This research was supported by an NSF graduate fellowship to
particular classes of theories: probability weighting theo-
the author. The author would also like to thank Dr. Gretchen Chap-
ries and con?gural weight theories.
man for her comments on the manuscript and suggestions on the
The most common explanation for the paradox is that
experimental design and Dr. Scott Huettel for his comments on the
decision-makers weight the probabilities of outcomes via
manuscript. Address: Brain Imaging and Analysis Center, Duke Uni-
versity Medical Center, PO Box 3918, Durham, NC 27710, E-mail:
a ? function that overweights small values of p and under-
weber@biac.duke.edu
weights large values of p, as in original Prospect Theory
115

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
116
(Kahneman & Tversky 1979). In the ?rst pair of gam-
bles than for the CC-low gambles — for losses obtained
bles, when the common consequence (CC) is $1 million
by subtracting a common amount from all the outcomes
(“CC-high” gambles), the 1% greater chance of winning
of small-magnitude gains gambles. Neither probabil-
the safe gamble is overweighted because it falls in the
ity weighting theories nor the TAX model predict a re-
very steep section of the ? function near ?(1). Decision-
verse paradox under such conditions. However, Birn-
makers are thus drawn to the safe gamble. When the com-
baum (2007) obtained an Allais Paradox for mixed gam-
mon consequence is changed to $0 (“CC-low” gambles),
bles obtained in the same manner. Three other studies
both the 10% chance of winning the riskier gamble and
found no paradox for either losses or similarly sized gains
the 11% chance of winning the safer gamble fall in a ?at
(Camerer 1989; Chew & Waller, 1986; Harless 1992).
portion of the ? function. The difference between them
Thus, the literature is con?icting on the existence of
seems small, and decision-makers choose the more valu-
Allais Paradox for losses. Moreover, no studies that I am
able, riskier gamble.
aware of have examined the effect of re?ecting the para-
Under Cumulative Prospect Theory (Tversky & Kah-
dox across the origin rather than shifting it. The question
neman, 1992) the ? function is not applied directly to
of the Allais Paradox for re?ected loss gambles will be
the probability of an outcome but rather to the cumula-
addressed by Experiment 1.
tive probabilities: the utility of the outcome is multiplied
by ?(the probability of obtaining an outcome at least as
1.3 Event splitting and the Allais Paradox
good as X) minus ?(the probability if an outcome strictly
better than X). Thus, the weight given to an outcome de-
Suppose we take the standard Allais Paradox CC-low
pends not only on the probability and utility of the out-
gambles and split each of the gambles into three out-
come itself, but also on how good the outcome is relative
comes instead of two:
to the other possible outcomes of the gamble.
10% chance of $1 million 10% chance of $5 million
Rather than using cumulative probabilities, con?gu-
ral weight models (e.g., Birnbaum, 1997, 1999) directly
1% chance of $1 million
1% chance of $0
weight the outcome according to its rank in the out-
89% chance of $0
89% chance of $0
come set, with the smallest outcomes given the highest
Normatively, the split gambles are identical to the stan-
weight. By weighting smaller outcomes more heavily
dard ones. Because CPT incorporates the rank of an out-
than larger ones, a con?gural weight model captures the
come using cumulative probability, the two sets of gam-
intuition that people are more interested in avoiding the
bles are also identical under CPT, which therefore pre-
worst outcomes than they are in obtaining the best out-
dicts that the paradox should be unaltered by the split.
comes. For example, in the transfer of attentional ex-
Under original PT, the effects of the split depend on
change (TAX) model (Birnbaum 1997, 1999; Birnbaum
whether the decision-maker chooses to coalesce the gam-
& Chavez, 1997; Birnbaum & Stegner, 1979), each lower
bles during the editing stage. If so, the split gambles
outcome “taxes” probability weight from each higher out-
should be treated identically to the standard ones. If not,
come. The TAX model therefore explains the Allais Para-
the split should serve to increase the desirability of the
dox primarily via the transfer of probability weights: in
safe CC-low gamble (as 10% and 1% considered sep-
the three-outcome CC-high risky gamble, the highest out-
arately seem larger than 11%) and decrease that of the
come is weighted less than its probability alone would
risky CC-low gamble, thus eliminating the paradox. Un-
suggest, the middle outcome weighted somewhat less,
der the TAX model, the split will also serve to increase
and the lowest outcome more. The decision-maker thus
the value of the safe CC-low gamble: the outcomes with
prefers the safe gamble in the CC-high pair, which, as
the highest payout lose less weight to the lowest outcome
a single-outcome gamble, has an unaltered probability
when split into two outcomes than when coalesced as one
weight. In the CC-low gambles, both gambles have two
outcome. At the same time, the value of the CC-low
outcomes. Thus, the probability weights undergo simi-
risky gamble decreases when split, as the highest out-
lar changes in both gambles and decision-makers simply
come loses more weight to the lowest outcome when the
choose the better, higher-paying risky gamble.
lowest outcome is split. Thus the TAX model predicts
that such a split should reduce or eliminate the paradox.
1.2 Losses and the Allais Paradox
Such a manipulation is known as event splitting
(Starmer & Sugden, 1993). Several studies have found
Although the Allais Paradox has been demonstrated for
that event splitting, or presentation formats that constitute
a wide variety of monetary gains, the few studies that
event splitting, reduce violations of EU in Allais Paradox.
have used losses or mixed gambles have had con?ict-
(e.g., Birnbaum, 2004, 2007; Carlin, 1990; Humphrey,
ing results. Camerer (1989) found a reverse Allais Para-
2000; Keller, 1985; Slovic & Tversky, 1974; Starmer,
dox — that is, greater risk-seeking for the CC-high gam-
1992; Starmer & Sugden, 1991), although other studies

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
117
have failed to ?nd an effect of presentation format (e.g.,
ing gain gambles in one of two ways: either by shifting
Moskowitz, 1974). The extent to which event splitting
the outcomes across the origin by subtracting a common
disrupts something fundamental to the paradox is unclear.
amount from all the gambles (as in Camerer, 1989) or by
It is possible that it simply makes the common conse-
re?ecting the gambles across the origin.
quence more obvious by separating out the shared prob-
ability of obtaining the lowest outcome, a possibility that
2.1 Method
could tested by splitting the gambles in a fashion that does
not make the common consequence more apparent. Birn-
2.1.1 Subjects
baum (2007) partially accomplished this by examining
the effects of splitting only one of the two CC-low gam-
229 Rutgers University undergraduates participated in the
bles. He found that splitting only the risky gamble (which
experiment as part of a class requirement for an introduc-
makes the common consequence most evident) did not
tory psychology class.
eliminate the Allais Paradox, while splitting only the safe
gamble did. This result argues against the suggestion that
2.1.2 Design
event splitting makes the common consequence more ev-
Each subject saw both levels of the Allais Paradox: gam-
ident. However, splitting one of the two CC-low gambles
ble pairs with both the common consequence present
and not the other still means at least one of the outcomes
(“CC-high” gambles) and those with common conse-
is easily comparable across gambles. The effect of split-
quence absent (“CC-low” gambles). Each subject also
ting the Allais Paradox CC-low gambles so that none of
saw three sign conditions: gain gambles, shifted loss
the outcomes may be easily compared is not known and
gambles, and re?ected loss gambles. This resulted in
will be examined in Experiment 2.
a 2 (CC-high vs. CC-low) × 3 (sign condition) within-
subjects design, for a total of 6 gambles. The gain gam-
1.4 Choice and the Allais Paradox
bles used a middle outcome of $250 and a lowest outcome
All studies that have examined the Allais Paradox for
of $0, with the highest outcome varying as part of the
losses or event splitting in the past have used a simple
choice titration matching technique (described below).
choice technique: subjects demonstrate the paradox by
The shifted loss gambles were obtained from the gain
choosing the risky CC-high gamble and the safe CC-low
gambles by subtracting $249 from all outcomes. This left
gamble. However, what is important about the Allais
a middle outcome of $1 and a lowest outcome of -$249.
Paradox is not the preference reversal itself, but rather the
($249 was chosen rather than $250 because, in pilot stud-
increase in risk seeking when the common consequence
ies, subjects had expressed the opinion that being asked to
is removed. A set of two single-choice pairs can detect
make choices that included a 100% chance of $0 seemed
a shift in risk preference only if the presented gambles
strange.) The re?ected loss gambles were obtained from
happen to span the shift — that is, if the shift causes to
the gain gambles by negating all outcomes, for a middle
decision-maker to prefer the safe CC-high gamble but the
outcome of -$250 and a “lowest” (in absolute value) out-
risky CC-low gamble. A decision-maker who chose the
come of $0. The gambles used in Experiment 1 are shown
risky CC-high gamble might well be more risk-seeking
in Table 1.
for the CC-low gambles — and therefore be experienc-
To facilitate comparison with the literature, the six
ing the paradox – but be unable to demonstrate this using
pairs of gambles were also presented as single choice
the single-choice technique. This limitation of the choice
questions. In these questions the highest outcome of both
technique poses challenges for experimenters: if a ma-
the CC-high and CC-low gambles was derived from the
nipulation produces a reduction in the number of subjects
corresponding choice titration questions: it was equal to
making the Allais Paradox pattern of choices, does it ac-
the mean of the indifference points for the CC-high and
tually indicate a reduction in the common consequence
CC-low conditions. The results of the single choice ques-
effect, or have risk preference merely been changed over-
tions did not differ from the choice titration results ex-
all? One aim of the present experiments was to examine
cept where noted in the footnotes. Subjects also saw one
the effects of sign and event splitting on the paradox us-
choice titration question and one single choice question
ing a matching technique more sensitive than the choice
that was not related to gains and losses and is not reported
technique used in previous studies.
here.
2 Experiment 1: Losses
2.1.3 Obtaining indifference points
The present experiments used a matching technique,
The purpose of Experiment 1 was to examine the Allais
which is more sensitive than a choice technique and there-
Paradox for loss gambles obtained from the correspond-
fore better able to detect changes in the Allais Paradox.

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
118
Table 1: Experiment 1 gambles.
Risky option
Safe option
Gains
CC-high
10% chance of $___
100% chance of $250
89% chance of $250
1% chance of $0
CC-low
10% chance of $___
11% chance of $250
90% chance of $0
89% chance of $0
Re?ected Losses CC-high
10% chance of –$___
100% chance of –$250
89% chance of –$250
1% chance of $0
CC-low
10% chance of –$___
11% chance of –$250
90% chance of $0
89% chance of $0
Shifted Mixed
CC-high
10% chance of $___
100% chance of $1
89% chance of $1
1% chance of –$249
CC-low
10% chance of $___
11% chance of $1
90% chance of –$249
89% chance of –$249
Subjects’ indifference points for the largest outcome of
responses or until the question had been presented three
Allais Paradox gamble pairs were elicited using a com-
times. Questions for which subjects failed to give a con-
puterized choice titration procedure. The subject made
sistent series of responses after three repetitions were not
a repeated series of choices between the risky and safe
used in the analysis.
gambles of an Allais Paradox pair. The size of the largest
outcome of the risky gamble was adjusted towards the
2.1.4 Materials
subject’s indifference point in response to the subject’s
previous choices using a bisection algorithm, described
Questions were presented to the subjects on their own
in greater detail in Chapman and Weber (2006), until the
computers via the World Wide Web.
Subjects read
indifference point was obtained to the desired degree of
through a series of instruction pages before starting the
accuracy: to within $11.72 for gains and re?ected losses,
experiment. The instructions were available to the sub-
and $1.53 for the shifted losses. At this point, the mid-
jects at all times during the experiment. The choices were
point of the interval was considered the indifference point
presented one at a time in a separate window, with the or-
for the subject. The starting interval was (250, 1000) for
der of the gambles randomly determined.
the gains gambles, (-250, -1000) for the re?ected gam-
bles, and (1, 50) for shifted gambles. The fact that the
shifted loss interval is not simply the gains interval mi-
2.2 Results
nus $249 is the result of a programming error, and the
The shape of the distribution of the individual cells var-
implications are discussed in the results, below.
ied with the common consequence condition. For the
To eliminate incorrect indifference points caused by
CC-high gambles, all distributions were bimodal, with a
subject inattention or an incorrect choice, the subjects
large peak at responses representing the largest possible
were presented with two check choices after their in-
absolute value of the indifference point and a small peak
difference point was obtained. One check choice pre-
at responses representing the smallest possible absolute
sented a value for the largest outcome slightly higher
value of the indifference point. For the CC-low gambles,
than the inferred indifference point, the other an amount
the responses were simply skewed, with the mode at the
slightly lower than the inferred indifference point. If
smallest possible absolute value of the indifference point.
a subject’s response to either check choice was incon-
Providing the largest possible indifference point indicates
sistent with the estimated indifference point, a message
extreme risk aversion: the subjects are indicating that
box was displayed informing the subject that he had re-
they prefer the safer option regardless of the payouts of
sponded inconsistently, and the series of choices was pre-
the risky option. Providing the smallest possible indiffer-
sented again from the beginning. Each question was re-
ence point indicates extreme risk seeking. The shapes of
peated until the subject produced a consistent series of
the distributions suggest that subjects are more prone to

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
119
Table 2:
a. Mean (SD) indifference values and mean difference between indifference values for highest outcome,
Experiment 1.
Common consequence
Gains
Re?ected Loss
Shifted Loss
CC-high
434 (281)
–407 (256.)
16 (19)
CC-low
327 (169)
–331 (196)
6.8 (12)
CC-high minus CC-low
107 (282)
–78 (241)
9.3 (21)
b. Percentage (N) of subjects displaying the Allais Paradox and the reverse Allais Paradox, Experiment 1.
Response Pattern
Gains
Re?ected Loss
Shifted Loss
Allais Paradox
36% (73)
30% (51)
47% (96)
Reverse Allais Paradox
15% (31)
11% (19)
13% (26)
EU consistent
48% (97)
59% (102)
39% (81)
extreme risk aversion when certainty is present (the CC-
× 2 (gains vs. re?ected losses) ANOVA on the absolute
high gambles) than when it is not, a pattern consistent
values of the indifference points found that the main ef-
with PT, CPT, and the TAX model.
fect of common consequence was signi?cant, (F(1,215)
I re-analyzed the indifference point data after eliminat-
= 31.78, p < 0.0001), indicating the Allais Paradox was
ing all responses at either ?oor or ceiling. These results
signi?cant.1 However, neither the main effect of gains vs.
did not differ from the results of the analyses using the
re?ected losses nor the interaction between outcome sign
full data set except where noted and are not reported. Be-
and common consequence was signi?cant. (F(1,199) =
cause the indifference point distributions were not nor-
0.53, p=0.47; F(1,156) = 0.98, p=0.32) The lack of a main
mal, in addition to ANOVAs, I also ran nonparametric
effect of sign indicates that subjects were not more risk-
Friedman and Wilcoxon-Mann-Whitney tests on the size
seeking for losses than they were risk-averse for gains.
of the Allais Paradox (the difference between the indif-
The lack of an interaction indicates that the re?ected Al-
ference points for the CC-high and CC-low gambles for
lais Paradox for re?ected losses was not smaller than the
each subject). Results from the nonparametric tests did
normal Allais Paradox for gains.
not differ from the parametric tests except where noted
Because of the programming error that resulted in
and are not reported.
a smaller range of possible indifference points for the
The mean indifference points for the gambles used in
shifted losses than for the other two sign conditions, the
Experiment 1 are shown in Table 2a. Each of 229 subjects
shifted losses could not be compared to the other loss
provided six indifference points. Of the resulting 1374
conditions using an ANOVA. Although the shifted losses
indifference points, 111 (8%) were missing due to failed
had a much smaller response range than the other two
check questions. The missing indifference points were
gambles, additional analyses (data not shown) gave no in-
produced by a total of 83 subjects. These observations
dication of an enhanced ceiling effect for shifted losses,
were not included in the analyses.
allowing the subtracted loss condition to be analyzed for
the presence of the Allais Paradox. A paired-sample t-test
Because the highest outcome is found only in the
showed that the effect of common consequence was sig-
riskier gamble, higher indifference points for the high-
ni?cant, (t(202) = 6.41, p < 0.0001), indicating an Allais
est outcome indicate greater preference for the safe gam-
Paradox.2
ble and higher levels of risk aversion. If the indifference
points are higher for the CC-high gambles then the CC-
1For single choice, a 2 (CC-high vs. CC-low) × 2 (gains vs. re?ected
low gambles, it indicates that subjects are showing the
losses) logistic regression analysis found the interaction between out-
Allais Paradox.
come sign and common consequence was signi?cant (?2(2, N=229) =
20.50, p < .0001). The Allais Paradox for the re?ected condition was
As shown in Table 2a, the indifference points are
signi?cant, ?2(2, N=229) = 9.62, p=0.002, while for the gain condi-
higher for the CC-high gambles than the CC-low gam-
tion the paradox was only marginally signi?cant ?2(2, N=229) = 3.37,
bles for both the gain and subtracted loss outcomes. For
p=0.067.
2
the re?ected loss outcomes, the absolute values of the in-
When responses at ceiling and ?oor were removed, the paired-
sample t-test was no longer signi?cant, (t(38) = 1.46, p = 0.15). This
difference points are higher for the CC-high gambles than
may be due to the large reduction in the N when all ceiling and ?oor
the CC-low gambles as well. A 2 (CC-high vs. CC-low)
responses were removed. It may also be due to the differences between

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
120
To compare shifted losses to the other two sign con-
berling, 2002). The TAX model can incorporate a util-
ditions, I calculated whether the subject showed or did
ity function of varying shape, however Birnbaum (2004)
not show the Allais Paradox for each pair of gambles,
suggests the utility function u(x) = x. Under this utility
and a 1 × 3 (sign condition) logistic regression analysis
function, TAX predicts no difference between the size of
was performed on the result. (The percentage of subjects
the paradox in the shifted and gain conditions.
showing the Allais Paradox for each sign condition is pre-
None of the theories discussed predicts a reverse Allais
sented in Table 2b. For purposes of this analysis, the re-
Paradox (as found by Camerer, 1989, for shifted gam-
verse paradox and EU-consistent categories shown in Ta-
bles), nor was one found in the present experiment. Al-
ble 2b were combined.) The logistic regression analysis
though the limited response range of the shifted gambles
found that the main effect of sign was signi?cant (?2(1,
did not allow for direct comparison of the size of the para-
N=229) = 16.92, p = .0002), indicating that the percent-
dox between gains and shifted losses, a higher percentage
age of subjects showing the Allais Paradox varied across
of subjects showed the paradox for shifted losses than for
the three conditions.3 Planned comparisons showed that
gains. This suggests that the shift does not diminish the
the difference between the percentage of subjects show-
size of the Allais Paradox (as CPT predicts) and may even
ing the Allais Paradox for gains and re?ected losses was
serve to increase it.
not signi?cant (?2(1, N=229) = 46.50, p = .16). However,
signi?cantly more subjects showed the Allais Paradox for
shifted losses than for re?ected losses (?2(1, N=229)=
3 Experiment 2: Event Splitting
6.97, p = .0083).
As discussed in the introduction, CPT and TAX make
2.3 Discussion
very different predictions about the effects of event split-
ting on the size of the Allais Paradox: CPT predicts no
The present experiment shows that re?ecting the Allais
effect, while TAX predicts elimination of the effect. The
Paradox gambles across the origin produces a re?ected
purpose of Experiment 2 was to use a matching technique
Allais common consequence effect, as predicted by both
to determine whether the paradox is eliminated when the
probability weighting theories and the TAX model.
CC-low gambles are split, and if so whether this elim-
I also returned to the question of shifting the Allais
ination occurs only for splits that ease comparisons of
Paradox gambles downwards across the origin. The be-
outcomes by highlighting the presence of the common
havior of the paradox predicted by PT, CPT or TAX in
consequence.
this situation depends upon the shape of the utility func-
tion. A downwards shift does not change the probabilities
of the outcomes, the shape of the ? function, or the num-
3.1 Method
ber and rank order of the outcomes. However, because the
3.1.1 Subjects
utility function proposed by PT and CPT is steepest near
the origin, moving the Allais Paradox gambles across the
174 Rutgers University undergraduates participated in the
origin changes the relative utilities of the outcomes. This
experiment as part of a class requirement for an introduc-
affects indifference points obtained by matching as well
tory psychology class. No subject participated in both
as single choice: because the utility of the value the sub-
Experiment 1 and Experiment 2.
jects match on (the highest outcome) is larger relative
to the other outcomes for shifted gambles than for gains
3.1.2 Design
ones, they will tend to provide smaller indifference points
for the shifted gambles than for the gains gambles.
As in Experiment 1, in Experiment 2 each subject saw
The effect of this change on the Allais Paradox depends
both gamble pairs with both the common consequence
on the exact parameters of the utility function. Accord-
present and those with common consequence absent.
ing to the parameters suggested by Tversky and Kahne-
Each subject also saw three types of split gambles: the
man (1992), the shift used in Experiment 1 should have
standard, unsplit gambles, gambles split in a manner that
slightly decreased the size of the Allais Paradox (Köb-
highlighted the common consequence, and gambles split
in a manner that obscured the common consequence.
the number of subjects at ceiling and ?oor in the Cc-high and CC-low
gambles mentioned above. These differences diminish when the ceiling
These splits involved only the CC-low gambles. For
and ?oor are removed, tending to reduce the size of the Allais Paradox.
both the standard and nonstandard splits the risky CC-
3The Friedman test indicated that the effect of sign condition on the
high gamble did not differ from the unsplit risky CC-high
Allais Paradox size was not signi?cant, (?2(2, N=576) = 0.076, p =
gamble, as, having three outcomes already, it could not
0.96). When responses at ceiling and ?oor are removed, the effect of
sign condition on the Allais Paradox size was likewise not signi?cant,
be split further without adding additional outcomes. The
(?2(2, N=152) = 0.22, p = 0.90).
safe CC-high gamble was split into three outcomes along

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Allais paradox
121
Table 3: Experiment 2 gambles.
Risky option
Safe option
Normal
CC-high
10% chance of $___
100% chance of $25,000
89% chance of $25,000
1% chance of $0
CC-low
10% chance of $___
11% chance of $25,000
90% chance of $0
89% chance of $0
Standard Split CC-high
10% chance of $___
10% chance of $25,000
89% chance of $25,000
89% chance of $25,000
1% chance of $0
1% chance of $25,000
CC-low
10% chance of $___
10% chance of $25,000
1% chance of $0
1% chance of $25,000
89% chance of $0
89% chance of $0
Odd Split
CC-high
10% chance of $___
10% chance of $25,000
89% chance of $25,000
89% chance of $25,000
1% chance of $0
1% chance of $25,000
CC-low
10% chance of $___
5% chance of $25,000
45% chance of $0
6% chance of $25,000
45% chance of $0
89% chance of $0
the standard 10%, 89%, 1% lines to parallel the risky CC-
gambles, skewed with the mode at ?oor for the CC-low
high gamble. The same CC-high responses were used for
gambles. The standard and nonstandard splits had very
both the standard and nonstandard splits. Thus there were
weakly bimodal distributions (with very little peak at the
only 5 questions in the 2 (CC-high vs. CC-low) × 2 (split
ceiling and a large peak at the ?oor) for all conditions.
type: unsplit, standard, or nonstandard). The gambles
As in Experiment 1, I re-analyzed the data with the ceil-
used in Experiment 2 are shown in Table 3.
ing and ?oor removed. Results did not differ from the
Subjects also saw six single choice questions obtained
analysis of the full data set except where noted and are
from their indifference points for the choice titration
not reported. Because the indifference point distributions
questions, in the manner described in Experiment 1. The
were not normal, in addition to parametric ANOVAs, I
results of the single choice questions did not differ from
also ran nonparametric Friedman and Wilcoxon-Mann-
the choice titration results and are therefore not reported
Whitney tests on the size of the Allais Paradox (the dif-
further.
ference between the indifference points for the CC-high
The choice titration procedure used was conducted in
and CC-low gambles for each subject). Results from the
exactly the same manner as described in Experiment 1.
latter did not differ from the parametric tests except where
Indifference points for the highest outcome were obtained
noted and are not reported.
using a bisection algorithm as described above. The bi-
The mean indifference points for the highest outcome
section algorithm process continued for 5 choices, after
are shown in Table 4a. The mean difference between the
which subjects saw two check choices.
CC-high and CC-low indifference points are also shown
in Table 4a. (Note that these are the means of the dif-
3.1.3 Materials
ferences, not the differences between the means.) Larger
indifference points indicate greater risk aversion. The Al-
The materials and procedure were analogous to those in
lais Paradox is present when the indifference points for
Experiment 1.
the CC-high gambles are higher than those for the CC-
low gambles, with larger differences indicating a larger
3.2 Results
paradox.
Each of 174 subjects provided ?ve indifference points.
As in Experiment 1, the indifference point distributions
Of the resulting 870 indifference points, 94 (11%) were
were not normal. For the unsplit gambles, the distribu-
missing due to failed check questions. These observa-
tions showed the same pattern as in Experiment 1: bi-
tions were produced by 44 subjects. The missing obser-
modal with modes at the ?oor and ceiling for the CC-high
vations were not included in the analyses.

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
122
Table 4:
a. Mean (SD) indifference values and mean difference between indifference values for highest outcome,
Experiment 2.
Common consequence
Unsplit
Standard Split
Nonstandard Split
CC-high
49,007 (30,039)
42,652 (23,867)
42,652 (23,867)
CC-low
33,275 (17,511)
39,950 (20,878)
45,024. (23,108)
CC-high minus CC-low
14,866 (28,754)
2498 (22,073)
–2438 (25,528)
b. Percentage (N) of subjects displaying the Allais Paradox and the reverse Allais Paradox, Experiment 1.
Response Pattern
Unsplit
Standard Split
Nonstandard Split
Allais Paradox
44% (61)
30% (45)
28% (41)
Reverse Allais Paradox
12% (17)
22% (33)
38% (56)
EU Consistent
44% (62)
49% (74)
35% (52)
As shown in Table 4a, the indifference points are
3.3 Discussion
higher for the CC-high gambles than the CC-low gam-
bles for the unsplit gambles, only slightly higher for the
Experiment 2 tested the possibility that event splitting
standard splits, and lower for the nonstandard splits. Be-
eliminates the Allais Paradox by increasing the salience
cause the standard and nonstandard splits had the CC-
of the common consequence by splitting the risky and
high gambles in common, an ANOVA on the mean in-
safe CC-low gambles such that no two outcomes share
difference points could not be performed. Instead, a one-
a common probability. The results of the present study
way ANOVA on split type (unsplit, standard split, or non-
indicate that even a nonstandard splitting of the CC-low
standard split) was performed on the difference between
gambles — one that does not make the common con-
the indifference points for the CC-high gambles and the
sequence more salient — dramatically increases consis-
CC-low gambles. It found that the main effect of split
tence with EU. The paradox was not signi?cant for either
type was signi?cant (F(2,274) = 17.52, p < 0.0001), indi-
the standard or nonstandard splits, and the two splits did
cating that the Allais Paradox differed in size for the three
not differ from each other. This indicates that the split
types of splits. Planned comparisons found that the stan-
itself, not highlighting the common consequence, elim-
dard and nonstandard splits did not differ signi?cantly
inates the paradox. These results tend to support Birn-
from each other, (F(1,274) = 2.82, p = 0.09)4, but they
baum’s TAX model and Kahneman and Tversky’s orig-
did differ from the unsplit condition, (F(1, 274) = 32.33,
inal Prospect Theory over Cumulative Prospect Theory,
p < .0001). A one-sample t-test indicated that the dif-
which predicts that splitting the gambles should have no
ference between the CC-high and CC-low gambles was
effect on the paradox.
signi?cantly larger than 0 for the unsplit gambles (t(139)
Past literature on event splitting and the Allais Paradox
= 6.12, p < .0001), indicating a signi?cant Allais Paradox.
has concentrated on the effects of splitting the CC-low
However, the Allais Paradox size (that is, the difference
gambles (e.g., Birnbaum, 2004, 2007). Thus, in Exper-
between the indifference points for the CC-high and CC-
iment 2 I used a nonstandard split only for the CC-low
low gambles) was not signi?cantly different from 0 for
gambles. It is possible that the split increased the salience
the standard and nonstandard splits. (t(151) = 1.40, p =
of the common consequence of the CC-high gamble. This
0.17; t(148) = -1.17, p = 0.25)5
possibility cannot be ruled out by the present data. It
is true that the indifference point was higher for the un-
4The Friedman test, however, indicated that the difference between
split CC-high gambles than for the split CC-high gambles
the size of the Allais Paradox in the standard and nonstandard split con-
(t(136) = 2.26, p = 0.026). However, this is predicted by
ditions was signi?cant, (?2(1, N=301) = 5.89, p = 0.15).
original PT (although not the TAX model) even without
5As in Experiment 1, the percentage of subjects showing the Allais
Paradox was calculated for each split condition and are presented in
any change in the salience of the common consequence.
Table 4b. The results of this analysis are the same as the results of
Moreover, the elimination of the Allias Paradox in Exper-
the ANOVA with one exception: when responses representing ?oor and
iment 2 cannot be fully explained by increased salience of
ceiling are removed, the main effect of split type on the percentage of
the common consequence in the CC-high gambles, as the
subjects showing the Allais Paradox did not attain signi?cance (?2(2,
N=127)= 4.69, p = 0.096). The results of this analysis are not discussed
paradox is not eliminated by splitting the CC-high gam-
further.
bles alone (F(1, 151) = 16.21, p < .0001).

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
123
4 General discussion
4.2 Event splitting and the Allais Paradox
One possible confound with earlier studies of the Allais
The present experiments investigated the effects of two
Paradox and event splitting is that the split may facilitate
manipulations that previous literature suggests may af-
comparison between the risk and safe gambles, in par-
fect the Allais Paradox in a manner inconsistent with
ticular making the presence of the common consequence
CPT. Experiment 1 examined the transformation of gains
extremely salient. The present Experiment 2 eliminates
into losses via both re?ection and shifting of the gam-
this confound, at least for the CC-low gambles, by us-
bles, while Experiment 2 investigated the transformation
ing a nonstandard split that conceals the common con-
of gambles into three-outcome gambles via event split-
sequence, in addition to the standard increased-salience
ting. Both experiments used a choice titration technique,
split. The fact that the Allais Paradox is eliminated for
which has greater sensitivity than the single choice tech-
both the standard and nonstandard splits, and that the two
nique that has been used in past studies of the paradox.
splits are not signi?cantly different from each other, indi-
This is especially useful when testing a manipulation that
cates that the effects of event splitting on the paradox are
is thought to remove the paradox, as it precludes the pos-
not due to the increased salience of the common conse-
sibility that the paradox has not been eliminated but sim-
quence, but are a property of the split itself.
ply shifted out of the range in which single choice can
detect it. In Experiment 1, the change from gains into
What does the ?nding that event splitting eliminates
losses had a minimal effect on the Allais Paradox (ex-
the Allais Paradox say about the underlying mechanism
cept to re?ect it where appropriate), but in Experiment 2
of the paradox? It is inconsistent with CPT’s explana-
the splitting of the CC-low gambles rendered the paradox
tion of the paradox, as splitting an outcome leaves its cu-
nonsigni?cant.
mulative probability unchanged. However, PT is consis-
tent with these results assuming the decision maker does
not coalesce the outcomes of the gambles in the editing
phase, as a 10% chance of $25,000 and a 1% chance of
4.1 The Allais Paradox and losses
$25,000 receive a larger decision weight separately than
combined. This increases the value of the safe CC-low
Unlike earlier results of Camerer (1989), the results of
gamble and decreases that of the risky CC-low gamble
Experiment 1 are compatible with CPT. CPT predicts
eliminating the paradox for the split gambles. A simi-
an Allais Paradox for both the normal and shifted loss
lar explanation applies to the nonstandard split in Exper-
gambles: using the parameters suggested by Tversky and
iment 2.
Kahneman (1992) CPT predicts a difference of $123 be-
The TAX model also easily accounts for the results of
tween the mean indifference points for the normal CC-
Experiment 2. Under TAX, splitting the lowest outcome
high and CC-low gambles, and a difference of $83 be-
of the risky CC-low gamble decreases the value of the
tween the shifted CC-high and CC-low gambles (Köbber-
gamble because now the probability weight of the high-
ling, 2002). Thus, CPT predicts that the paradox should
est outcome is taxed by two separate low outcomes, rather
be slightly smaller for the shifted gambles than for the
than one. This means the probability of losing is treated
normal ones. The predicted $83 change between the CC-
as though it is higher relative to the probability of win-
high and CC-low gambles in the shifted condition was
ning after the gamble is split, and the value of the gamble
larger than the overall range of possible responses, so the
therefore decreases. At the same time, splitting the high-
failure to ?nd such a large change in Experiment 1 does
est outcome of the safe CC-low gamble increases the safe
not indicate a failing of CPT. The actual change in mean
gamble’s value.
indifference points for the normal CC-high and CC-low
There is also a very simple theory that may explain
gambles was $107.25, roughly comparable with CPT’s
the results of Experiment 2. Subjects may see the split
prediction. The TAX model predicts that the difference
CC-low gambles and notice that the split safe CC-low
between the indifference points for the CC-high and CC-
gamble offers “more ways to win” than the risky gam-
low gambles should be about $785 for both the normal
ble, while the split risky gamble appears to offer “more
and shifted gambles, much larger than was actually found
ways to lose.” If the probabilities of winning and los-
for even the normal gambles (Birnbaum & Bailey, 1998).
ing were very dissimilar across the two gambles, perhaps
Nonetheless, it is clear that both CPT and TAX predict a
subjects would combine the probabilities and note that
robust Allais Paradox for the shifted gambles in Exper-
“more ways to win” does not necessarily translate into
iment 1, and a robust Allais Paradox was found. Thus,
a signi?cantly increased probability of winning. How-
while the results of Experiment 1 contradict the ?ndings
ever, lacking an obvious dissimilarity in the probabilities,
of Camerer (1989), they agree with CPT, and to a lesser
the subjects may simply be choosing according to a sim-
extent with the TAX model.
ple, “more ways to win is good, more ways to lose is

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Judgment and Decision Making, Vol. 2, No. 2, April 2007
Allais paradox
124
bad” heuristic. There is some evidence that subjects pay
of Ward Edwards (pp. 27–52). Norwell, MA: Kluwer
attention to the number of ways to win or lose a gam-
Academic Publishers.
ble (e.g., Humphrey, 1999; Lopes & Oden, 1999; Payne,
Birnbaum, M. H. (2004). Causes of Allais common
2005) and if this is the case, event splitting may not be
consequence paradoxes: An experimental dissection.
disrupting the basic mechanism of the Allais Paradox so
Journal of Mathematical Psychology, 48, 87–106.
much as it is overriding it. It may be that the decision pro-
Birnbaum, M. H. (2007). Tests of branch splitting and
cess that decision-makers normally use is one that, like
branch-splitting independence in Allais paradoxes with
CPT, should cause the subjects to choose the risky CC-
positive and mixed consequences. Organizational Be-
low gamble even after the split. However, CPT cannot
havior and Human Decision Processes, 102, 154–173
cause subjects to choose the risky gamble if it is never
Birnbaum, M.H. & Bailey, R. (1998).
Con?gu-
invoked. If subjects choose between the split CC-low
ral weight, TAX model and cumulative prospect
gambles according to a “more ways to win good, more
model [Computer software].
Retrieved from
ways to lose bad” heuristic before even evaluating the
http://psych.fullerton.edu/mbirnbaum/
gamble according to CPT, then CPT cannot in?uence the
calculators/taxcalculator.htm.
outcome of the decision.
Birnbaum, M. H. & Chavez, A. (1997). Tests of theo-
If the latter explanation is correct, then the failure to
ries of decision making: Violations of branch indepen-
?nd the Allais Paradox for split gambles does not re-
dence and distribution independence. Organizational
?ect on the mechanism of the paradox at all, but rather
Behavior and Human Decision Processes, 71, 161–
identi?es a heuristic that we invoke to make our deci-
194.
sions simpler. Such an explanation is consistent with a
Birnbaum, M. H. & Stegner, S. E. (1979). Source cred-
bounded rationality perspective of decision-making (e.g.,
ibility in social judgment: Bias, expertise, and the
Simon, 1955, 1956; Gigerenzer, et al., 1999) in which de-
judge’s point of view. Journal of Personality and So-
cision making is driven by a set of heuristics, which are
cial Psychology, 37, 48–74.
automatically invoked whenever situationally appropriate
Carlin, P. S. (1990). Is the Allais paradox robust to a
(and sometimes when inappropriate).
seemingly trivial change of frame? Economics Letters,
Thus, although the results of Experiment 2 are sug-
34, 241–244.
gestive, they are not in and of themselves suf?cient to
Camerer, C. F. (1989). An experimental test of several
demonstrate that CPT does not offer an adequate explana-
generalized utility theories. Journal of Risk and Un-
tion for the Allais Paradox. Further research is required to
certainty, 2, 61–104.
investigate the possibility that subjects are simply choos-
Chapman G.B. & Weber, B.J. (2006). Decision biases
ing according to a “more ways to win is good” heuris-
in intertemporal choice and choice under uncertainty:
tic and are not invoking a deeper analysis of the gambles
testing a common account. Memory & Cognition, 34,
at all. Assuming that subjects are not invoking such a
589–602.
heuristic, further research is also necessary to determine
Chew, S. H. & Waller, W. (1986). Empirical tests of
whether PT or TAX provides a better explanation of the
weighted utility theory. Journal of Mathematical Psy-
paradox. Ultimately it may be that no single theory can
chology, 30, 55–72.
provide an adequate explanation of all the phenomena re-
Conlisk, J. (1989). Three variants on the Allais example.
lated to the Allais Paradox.
The American Economic Review, 79, 392–407.
Gigerenzer, G., Todd, P. and the ABC Research Group.
(1999). Simple heuristics that make us smart. New
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