New Paradoxes of Risky Decision Making
Michael H. Birnbaum
California State University, Fullerton and Decision Research Center, Fullerton
Date: 10-04-04
Filename:BirnbaumReview26.doc
Note: Comments are welcome! Please note the questions to you addressed in red.
Thanks! Michael
Mailing address:
Prof. Michael H. Birnbaum,
Department of Psychology, CSUF H-830M,
P.O. Box 6846,
Fullerton, CA 92834-6846
Email address: mbirnbaum@fullerton.edu
Phone: 714-278-2102 or 714-278-7653
Fax: 714-278-7134
Author's note: Support was received from National Science Foundation Grants, SBR9410572, SES 99-86436, and BCS-0129453.
Case Against Prospect Theories 2
ABSTRACT
During the last twenty-five years, prospect theory and cumulative prospect theory
have replaced expected utility theory as the dominant descriptive theories of risky
decision making. However, eleven new decision paradoxes are reviewed to reveal where
prospect theories lead to self-contradiction or erroneous predictions. The new findings are
consistent with and, in several cases, were predicted in advance of experiments by simple
“configural weight” models in which people weight probability-consequence branches by
a function that depends on branch probability and ranks of consequence on the discrete
branches. Although they have some similarities to rank-dependent utility theories,
configural weight models do not satisfy coalescing, whereby branches leading to the
same consequence can be combined by adding their probabilities. These models correctly
predict results with the new paradoxes. This paper concludes that people do not frame
choices as prospects, but instead, that people represent decisions by trees with branches.
Key words: cumulative prospect theory, decision making, decision trees, expected
utility, rank dependent utility, risk, prospect theory
Case Against Prospect Theories 3
1. INTRODUCTION
Following a period in which Expected Utility (EU) theory (Bernoulli, 1738/1954;
von Neumann & Morgenstern, 1947; Savage, 1954) dominated the study of risky
decision making, prospect theory (PT; Kahneman & Tversky, 1979) became the focus of
empirical studies of decision making. PT was modified (Tversky & Kahneman, 1992) to
assimilate rank and sign-dependent utility (RSDU), and the newer form, cumulative
prospect theory (CPT) was able to describe the Allais paradoxes (Allais, 1953; 1979) that
were inconsistent with EU. They also describe patterns of risk-seeking and risk aversion
in the same person. CPT simplified and extended PT to a wider domain. The newer
form, Cumulative Prospect Theory (CPT; Tversky & Kahneman, 1992), has been so
successful in the last dozen years that it has been recommended as the new standard for
economic analysis (Camerer, 1998; Starmer, 2000) and was recognized in the 2002 Nobel
Prize in Economics (2002).
Many important papers contributed to the theoretical and empirical development
of these theories (Camerer, 1989; 1992; 1998; Diecidue & Wakker, 2001; Gonzalez &
Wu, 1999; Karni & Safra, 1987; Luce, 2000; Luce & Narens, 1985; Machina, 1982;
Prelec, 1998; Quiggin, 1982; 1985; 1993; Schmeidler, 1989; Starmer & Sugden, 1989;
Tversky & Wakker, 1995; Yaari, 1987; von Winterfeldt, 1997; Wakker, 1994; 1996;
Wakker, Erev, & Weber, 1994; Weber & Kirsner, 1997; Wu, 1994; Wu & Gonzalez,
1996; 1998; 1999). Add more here:
However, evidence has been accumulating in recent years that systematically
violates prospect theories. Authors have criticized CPT and pointed out various problems
with its descriptive accuracy (Baltussen, Post, & Vliet, 2004; Gonzalez & Wu, 2003;
Case Against Prospect Theories 4
Humphrey, 1995; Levy & Levy, 2002; Lopes & Oden, 1999; Luce, 2000; Starmer &
Sugden, 1993; Starmer, 1999, 2000; Wu, 1994; Wu and Gonzalez, 1999; Wu & Markle,
2004; …Add more here: .) Not all criticisms of CPT have been accepted (Wakker, 2003;
Baucells & Heukamp, 2004 More here), however, and some conclude that CPT as the
“best”, if imperfect, description of decision making under risk and uncertainty (Starmer,
2000; Harless & Camerer, 1994; More here). The purpose of this paper is to summarize
the case against PT and CPT as descriptions of risky decision making and to show that
other models are more accurate descriptively.
In addition to criticisms of CPT by others, my students and I have been testing
prospect theories against an older class of models known as “configural weight” models
(Birnbaum, 1974; Birnbaum & Stegner, 1979). In configural weighting models, the
weight of a stimulus(branch) depends on the relationship between that stimulus and
others that are presented in the same set. This class of models includes special cases that
will be compared in this paper.
In the “configural weight models, weights of branches depend on the probability
or event leading to the consequence and relationships between that branch’s consequence
and consequences of other branches in the gamble. These older models led me to reexamine old tests andto design new tests that are implications of either CPT or the
configural models, and which distinguish classes of models that do or do not satisfy each
property. These “new paradoxes,” create systematic violations of prospect theories.
They are also properties that can be derived from Expected Utility (EU) theory, so
evidence of systematic violation of these properties is also evidence against EU as a
descriptive model.
Case Against Prospect Theories 5
The mass of evidence has now reached the point where I believe that CPT can be
rejected as a descriptive model of decision making. The violations of CPT are largely
consistent with the transfer of attention exchange (TAX) model of Birnbaum and Chavez
(1997) which is a special case of a class of generic configural weighting models that
includes TAX, RDU, and CPT as special cases. Based on the growing case against
CPT/RSDU, Luce (2000) and Marley and Luce (2001; 2004) have recently developed
another subclass of the general model, Gains Decomposition Utility (GDU), which they
have shown has similar properties to TAX but is distinct from it.
The GDU models have in common with Birnbaum’s configural weight models the
following idea: people treat choices as trees with branches rather than as prospects or
probability distributions.
Some introductory examples will help to distinguish the characteristics of
prospect theory from the class of configural weight models. Consider the following
choice:
A:
.01 probability to win $100
B:
.01 probability to win $100
.01 probability to win $100
.02 probability to win $45
.98 probability to win $0
.97 probability to win $0
In prospect theory, people are assumed to simplify choices (Kahneman &
Tversky, 1979). Gamble A has two branches of .01 to win $100. In prospect theory, these
two branches could be combined to form a two-branch gamble, A’, with one branch of
.02 to win $100 and a second branch of .98 to win $0. If a person were to combine the
two branches leading to the same consequence, then A and A’ would be the same, so the
choice between A and B would be the same as that between A’ and B, as follows:
Case Against Prospect Theories 6
A’:
.02 probability to win $100
B:
.98 probability to win $0
.01 probability to win $100
.02 probability to win $45
.97 probability to win $0
In prospect theory, it is also assumed that people cancel common branches. In the
example, A and B share a common branch of .01 to win $100, and this branch might be
cancelled before a decision is reached. If so, then the choice should be same as the
following:
A’’:
.01 probability to win $100
.99 probability to win $0
B’:
.02 probability to win $45
.98 probability to win $0
These two editing principles, known as combination and cancellation, are features
of prospect theories that are violated by branch weighting theories such as TAX and
GDU.
In this class (including RAM, TAX, and GDU), different trees that represent the
same prospects can produce different psychological reactions. Thus, the three-branch
gamble, A, and the two-branch gamble A’ which are equivalent in prospect theory, are
different in RAM, TAX, and GDU, except in special cases. Furthermore, these models
do not assume that people “trim trees” by canceling branches common to both
alternatives of a choice.
There are two cases to be made in this paper. The easier case to make is the
negative one, which is to show that empirical data refute PT and CPT as descriptions of
risky decision making. The other, positive case, is necessarily more tentative; namely, to
show that particular rival models predicted the violations of CPT in advance of
Case Against Prospect Theories 7
experiments, and that these models also accurately describe previous data that were
successfully fit by CPT.
Because these models correctly predicted a series of new results, it does not
follow that they will continue to be correct for every new test that might be devised.
Therefore, the reader may decide to accept the negative case that CPT is false and reject
the positive case favoring the configural weight, TAX, model as a series of lucky
coincidences. In this case, the reader should seek a better theory that accounts for the
new phenomena as well as the classical ones and devise new tests to show that the new
model is more accurate than those presented here.
Before proceeding to exposition of the models, it will be helpful to consider three
issues that distinguish classes of descriptive decision theories: the source of risk aversion,
the effects of splitting of branches, and the roots of loss aversion.
Two Theories of Risk Aversion
The term “risk aversion” refers to the empirical finding that people often prefer a sure
thing over a gamble with the same or even higher expected value. Consider the following
choice:
Which would you prefer?
F: $45 for sure
or
G:
.50 probability to win $0
.50 probability to win $100
This is a choice between a “sure thing” and a two-branch gamble with equal chances of
winning $0 or $100. The lower branch of G is .5 to win $0 and the higher branch is .5 to
win $100.
Case Against Prospect Theories 8
Most people prefer F ($45 for sure) rather than gamble G, even though G has a
higher expected value of $50; therefore, these people are said to exhibit risk averse
preferences in this choice.
There are two distinct ways of explaining “risk aversion”, illustrated in Figure 1.
In expected utility theory, it is assumed that people choose F over G (denoted F  G) if
and only if EU(F) > EU(G), where
EU(F)   pi u(xi ) ,
where u(x) is the utility (subjective value) of the cash prize, x. In the left side of Figure 1,
there is a nonlinear transformation from objective money to utility (subjective value). If
this utility function, u(x), is a concave downward function of money, x, then the expected
utility of G will be less than that of F. For example, if u(x) = x.63, then u(F) = 11.0, and
EU(G) = .5u(0) + .5u(100) = 9.1.
Because EU(F) > EU(G), EU can reproduce the choice of F over G. The left
panel of Figure 1 shows that on the utility continuum, the balance point on the
transformed scale (the expectation) corresponds to the utility of $33.3. Thus, a person
should be indifferent between a sure gain of $33.3 and gamble G (denoted $33.3 ~ G).
This cash value having the same utility as the gamble is known as the gamble’s certainty
1
equivalent, CE(G)  u [EU (G)] . In this case, CE(G) = $33.3.
Insert Figure 1 about here.
A second way to explain risk aversion is shown on the right side of Figure 1. In
the TAX model illustrated, one third of the weight of the higher branch is taken from the
branch to win $100 and assigned to the lower-valued branch to win $0. The result is that
the weight of the lower branch is 2/3 and the weight of the higher branch is 1/3, so the
Case Against Prospect Theories 9
balance point corresponds to a CE of $33.3. In this example, the transformation from
money to utility is linear, illustrating that it is weighting rather than utility that is used to
account for risk aversion in this model.
Whereas EU, on the left of Figure 1, attributed risk aversion to the utility
function, the“configural weight” model on the right of Figure 1 attributes risk aversion to
the transfer of weight from the higher to the lower valued branch. Intuitively, the extra
weight applied to the lowest consequence represents a transfer of attention from the
highest to lowest consequence of the gamble. RDU, RSDU, CPT, GDU, RAM, and TAX
are all members of a generic class of configural weight models in that they can account
for risk aversion primarily by the assumption that branches with lower consequences
receive greater weight. In these models, it is this configural weighting that describes risk
aversion rather than the nonlinear transformation from money to utility. [Footnote: All
of these models allow a nonlinear utility function, so they include the possibility of
accounting for risk aversion in the same way as EU. In other words, these models
include EU as a special case.]
The next issue allows us to divide this class of generic “configural weight”
models into two groups.
Two Theories of Branch Splitting
Let G = (x, p; y, q; z, r) represent a three-branch gamble that yields monetary
consequences of x with probability p, y with probability q, and z otherwise (r = 1 – p –
q). A branch of a gamble is a probability (event)-consequence pair that is distinct in the
gamble’s display to the decision-maker.
Case Against Prospect Theories 10
Consider the two-branch gamble G = ($100, .5; $0, .5). Suppose we split the
branch of .5 to win $100 into two branches of .25 to win $100. That creates a three
branch gamble, G ($100,.25;$100,.25;$0,.5) , which is a “split” version of G, which is
called the coalesced version. It should be clear that there are many ways to split G, but
only one way to coalesce branches from G to G. According to upper coalescing,
G ~ G. Similarly, consider G ($100,.5;$0,.25;$0,.25) , which is another split version
of G with two lower branches of .25 to win $0. By lower coalescing, it is assumed that
G ~ G.
There are two classes of theories: those that satisfy coalescing (including RDU,
RSDU, CPT, and original prospect theory with the editing principle of combination) and
those that violate this property (including RAM, TAX, and GDU). According to the class
of RDU/RSDU/CPT models, a gamble has the same value (utility) whether branches are
split or coalesced, because the sum of the weights of the two split branches (in the
example, each branch has .25 to win $100) is the same as the weight of the single
coalesced branch with probability .5.
According to the class of models that violates coalescing, however, the sum of the
weights of the two splinters need not equal the weight of the coalesced branch. In
particular, these models (as fit to data) imply that the sum of weights of the splinters
exceeds the weight of the coalesced branch. RAM and TAX violate both lower and upper
coalescing. GDU violates lower coalescing, but satisfies upper coalescing.
Two Theories of Loss Aversion
Loss aversion refers to the finding that most people prefer to avoid “fair” (50-50)
bets to either win x or lose x with equal probability. In fact, it is often found that the
Case Against Prospect Theories 11
typical college undergraduate will only accept such bets if the amount to win is more than
twice the amount to lose. There are two ideas that have been proposed to account for this
finding.
In CPT, it is assumed that loss aversion is produced by the utility function for
gains and losses. The utility account of loss aversion assumes that |u(-x)| > u(x); i.e., that
“losses loom larger than gains.” Tversky and Kahneman (1992) theorize that the utility
for losses is a multiple of the utility for equal gains:
u(-x) = –Lu(x), x ≥ 0
Where L is the loss aversion coefficient. Suppose L = 2. If so, it follows from this model
of CPT that a fifty-fifty gamble to either win or lose $100 has a negative utility.
A second theory of loss aversion, in contrast, is that loss aversion is another
consequence of configural weighting. Suppose the weights of two equally likely
branches to lose $100 or win $100 are 2 and 1, respectively. Even with u(x) = x for both
positive and negative values of x, this model implies that people will avoid 50-50
gambles to win or lose equal amounts.
These two theories of loss aversion—utility versus weight—make different
predictions for a testable property known as gain-loss separability, as we see below. It is
important to distinguish the empirical phenomena of “risk aversion” and “loss aversion”
from theories of those phenomena. When the same terms are used for both the
phenomenon to be explained and for a particular account of that phenomenon, it could be
misleading if the theory is wrong.
2. NOTATION AND TERMINOLOGY
Case Against Prospect Theories 12
In addition to risk aversion, branch splitting/coalescing, and loss aversion, there
are a few other terms that need to be defined at the outset: these are the properties of
transitivity, consequence monotonicity, branch independence, and stochastic dominance.
Preferences are said to be transitive if and only if for all A, B, and C, A  B and B
 C implies A  C. Because people may be inconsistent in their preferences from one
choice to the next, the preference relation is usually given a probabilistic operational
definition. For example, Weak Stochastic Transitivity is defined as follows: if the
probability of choosing A over B is greater than 1/2, denoted P(A, B) > 1/2, and if P(B, C)
> 1/2, then P(A, C) > 1/2.Since P(A, B) represents the probability of choosing A over B,
the data analyst must show that the data allow rejection of the hypothesis that the
probabilities are 1/2; therefore, in many cases, the experimenter must stand undecided as
to whether P(A, B) = 1/2, and therefore whether A  B.
Consequence monotonicity is the assumption that if one consequence in a gamble
is improved, holding everything else constant, the gamble with the better consequence
should be preferred. For example, if a person prefers $100 to $50, then the gamble G =
($100, .5; $0) should be preferred to F = ($50, p; $0). Although systematic violations of
consequence monotonicity have been reported in judgment studies and in indirect choice,
they are rare in direct choice (Birnbaum, 1992; 1997; Birnbaum & Sutton, 1992).
Branch independence is weaker than Savage’s (1954) “sure thing” axiom. It
holds that if two gambles have an identical branch, then the value of the common
consequence can be changed without altering the order of the gambles induced by the
other branches. For example, for three-branch gambles, branch independence requires
(x, p; y, q; z, r)  (x', p'; y', q'; z, r)
Case Against Prospect Theories 13
if and only if
(1)
(x, p; y, q; z', r)  (x', p'; y', q'; z', r).
where (z, r) is the common branch, the consequences (x, y, z, x', y', z') are all distinct, the
probabilities are not zero and sum to 1 in each gamble, p + q = p' + q'. This principle is
weaker than Savage’s independence axiom because it holds for gambles with equal
numbers of branches of known probability and also because it does not presume
coalescing.
If we restrict the distributions and number of consequences in the gambles to be
the same in all four gambles, then the property is termed restricted branch independence.
In the case of three-branch gambles, this requires p = p', and q = q'. The special case of
restricted branch independence, in which corresponding consequences retain the same
ranks, is termed comonotonic restricted branch independence, also known as
comonotonic independence (Wakker, Erev, & Weber, 1994). Cases of (unrestricted)
branch independence in which only the probability distribution is systematically varied
are termed distribution independence.
Stochastic dominance (first order stochastic dominance) is the relation between
non-identical gambles, such that for all values of x, the probability of winning x or more
in gamble G is greater than or equal the probability of winning x or more in gamble F. If
so, G is said to stochastically dominate gamble F. The statement that preferences satisfy
stochastic dominance means that if G dominates F, then F will not be preferred to G.
This paper will use an operational definition as follows: if G dominates F, then the
probability of choosing G over F should be greater than or equal to 1/2. This is a very
Case Against Prospect Theories 14
conservative standard (lenient to stochastic dominance), yet it has been rejected by the
appropriate statistical test applied to empirical data.
It is important to distinguish first order stochastic dominance from other types of
“stochastic dominance” as defined in Levy and Levy (2002) and criticized by Wakker
(2003), for example. Although violations of some of these special forms of “stochastic
dominance” are consistent with CPT (Wakker, 2003; Baucells & Heukamp, 2004),
violations of first order stochastic dominance refute any version of RDU/RSDU/CPT.
3. SUBJECTIVELY WEIGHTED UTILITY AND PROSPECT THEORIES
1. Subjectively Weighted Utility Theory
Let G  (x1, p1; x 2, p2 ; ;x i , pi ; ;x n, pn ) represent a gamble to win xi with
probability pi, where the n outcomes are mutually exclusive and exhaustive. Edwards
(1954; 1962) considered subjectively weighted utility (SWU) models of the form,
n
SWU(G)   w(pi )u(x i )
(2)
i 1
Edwards (1954) discussed both S-shaped and inverse-S shaped functions as candidates
for the weighting function, w(p). When w(p) = p, this model reduces to EU.
Edwards (1962) theorized that weighting functions might differ for different
configurations of consequences; he made reference to a “book of weights” with different
pages for different configurations. Edwards (1962, p. 128) wrote:
“The data now available suggest the speculation that there may be exactly five
pages in that book, each page defined by a class of possible payoff
arrangements. In Class 1, all possible outcomes have utilities greater than zero.
In Class 2, the worst possible outcome (or outcomes, if there are several
Case Against Prospect Theories 15
possible outcomes all with equal utility), has a utility of zero. In Class 3, at
least one possible outcome has a positive utility and at least one possible
outcome has a negative utility. In Class 4, the best possible outcome or
outcomes has a utility of zero. And in Class 5, all possible outcomes have
negative utilities.”
2. Prospect Theory
Prospect theory (Kahneman & Tversky, 1979) is similar to that of Edwards,
except it was restricted to gambles with no more than two nonzero consequences, and it
reduced the number of pages in the book of weights to two—prospects (gambles) with
and without the consequence of zero. This means that the 1979 theory is silent on many
of the tests that will be described in this paper unless assumptions are made about how to
extend the model to three and four branch gambles.
One way to extend PT would be to treat the theory as the special case of Edwards
(1962) model with Classes 1 and 5 and 2 and 4 combined. A still simpler extension that
takes Equation 2 with different weighting functions for positive and negative
consequences has been called “stripped” prospect theory by Starmer and Sugden (1993).
The term “stripped” indicates that the editing principles and the application of differential
weighting for the consequence of $0 have been removed. The other way to extend PT is
to use CPT (Tversky & Kahneman, 1992), which is not equivalent to original prospect
theory for three-branch gambles and higher.
Besides the restriction in PT to prospects with no more than two nonzero
consequences, the other new feature of PT (retained by CPT) was the idea that an editing
phase precedes the evaluation phase. Kahneman (2003) described the development of the
Case Against Prospect Theories 16
editing rules in an article reviewing his collaborative work with Amos Tversky that led to
his winning a share of the 2002 Nobel Prize. He described how, when they were
completing their 1979 paper, they worried that their prospect model might be easily
tested and falsified. To rescue the model from its implications, which they thought were
implausible, they added editing rules that would provide excuses for potential refuting
evidence.
3. Editing Rules in Prospect Theory
The six principles of editing, which define “prospects,” are as follows:
1. Combination: probabilities associated with identical outcomes are combined. This
principle corresponds to coalescing.
2. Segregation: a riskless component is segregated from risky components.
“the prospect (300, .8; 200, .2) is naturally decomposed into a sure gain of 200 and
the risky prospect (100, .8) (Kahneman & Tversky, 1979, p. 274).”
3. Cancellation: Components shared by both alternatives are discarded from the choice.
“For example, the choice between (200, .2; 100, .5; –50, .3) and (200, .2; 150, .5;
–100, .3) can be reduced by cancellation to a choice between (100, .5; –50, .3) and
(150, .5; –100, .3) (Kahneman & Tversky, 1979, p. 274-275).” If subjects cancel
common components, then they will satisfy branch independence and distribution
independence, as we see below.
4. Dominance: Transparently dominated alternatives are recognized and eliminated.
Without this principle, Equation 2 violates dominance in cases where people do not.
5. Simplification: probabilities and consequences are rounded off. This principle,
combined with cancellation, could produce violations of transitivity.
Case Against Prospect Theories 17
6. Priority of Editing: Editing precedes and takes priority over evaluation. Kahneman
and Tversky (1979, p. 275) remarked, “Because the editing operations facilitate the task
of decision, it is assumed that they are performed whenever possible.”
These editing rules are unfortunately imprecise, contradictory, and conflict with the
equations of prospect theory. This means that PT often has the luxury (or misfortune) of
predicting opposite results, depending on which principles are invoked or the order in
which they are applied (Stevenson, Busemeyer, & Naylor, 1991). This makes the theory
easy to use as a post hoc account but makes it difficult to use as a predictive scientific
model.
In the 1979 paper there were 14 choice problems, none of which involved gambles
with more than two nonzero consequences. To fit these, there were two functions (value
and probability), a status quo point, and 6 editing principles. In addition, special biases
were postulated, such as consequence “framing”, which contradicts the editing principle
of segregation. Investigating this “theory” might seem like the struggle of Herakles
against the Hydra, because when each implication is refuted, two new heuristics, biases,
or editing rules can arise as excuses. Nevertheless, if we take each editing rule as a
separate scientific theory, we can isolate them and test them one by one. References to
other previous tests of these hypotheses would be very helpful here. Did Kahneman and
Tversky conduct experiments on these editing rules before publication in 1979? After
1979? Did others?
4. Rank Dependent Utility and Cumulative Prospect Theory
Rank dependent utility theory was proposed as a way to explain the Allais
paradoxes without having a theory that needed editing principles to cancel its implausible
Case Against Prospect Theories 18
predictions (Quiggin, 1982; 1985; 1993; More references here). Cumulative Prospect
Theory (Tversky & Kahneman, 1992) was considered an advance over original prospect
theory because CPT applied to gambles with more than two nonzero consequences, and
because it removed the need for the editing rules of combination and dominance
detection, which are automatically satisfied by the representation. CPT uses the same
representation as Rank and Sign-Dependent Utility theory (RSDU; Luce & Fishburn,
1991; 1995), though the two theories were derived from different assumptions (Luce,
2000; Wakker & Tversky, 1993). CPT was also a simplification over original prospect
theory, which had different equations for different types ofprospects, and it was more
general in that it allowed different weighting functions for positive and negative
consequences. More References here.
For experiments dealing with strictly nonnegative consequences, RDU, RSDU, and
CPT all they boil down to the same representation.
For gambles of n strictly positive consequences, ranked such that
0  x1  x 2 
 xi 
x n , RDU, RSDU, and CPT use the following representation:
n
n
n
i 1
ji
j i 1
RDU (G)  [W  ( pj )  W  (  p j )]u(xi )
(3)
where RDU(G) is the rank-dependent expected utility of gamble G; W+(P) is the
n
weighting function of decumulative probability, Pi   p j , which monotonically
j i
transforms decumulative probability to decumulative weight, and assigns W+(0) = 0 and
W+(1) = 1.
Case Against Prospect Theories 19
For gambles of strictly negative consequences, a similar expression applies,
except W–(P) replaces W+(P), where W–(P) is a function that assigns cumulative weight
to cumulative probability of negative consequences. For gambles with mixed gains and
losses, with consequences ranked such that x1  x2 
 x n  0  xn1 
 xm , rank-
and sign-dependent utility is the sum of two terms, as follows:
n
i
i 1
i 1
j 1
j 1
RSDU (G)  [W  ( pj )  W  ( p j )]u(xi ) 
m
m
m
i n 1
j i
j  i1
 [W  ( pj )  W  (  p j )]u(xi ) (4)
The representation in CPT is the same as that of RSDU.
This theory satisfies transitivity, consequence monotonicity, coalescing, and
stochastic dominance. It also satisfies restricted comonotonic branch independence.
However, it violates restricted branch independence and all forms of distribution
independence analyzed here (except when it reduces to EU).
4. RAM, TAX, and GDU MODELS
Before describing the class of “configural weight” models and evidence favoring
this class models over the class of RDU/RSDU/CPT models, six preliminary comments
are in order. First, the approach described here, like that of PT or CPT, is purely
descriptive. The models are intended to describe and predict humans’ decisions and
judgments, rather than prescribe how people should decide or judge.
Second, the approach is psychological; how stimuli are described, presented or
framed is part of the theory. Indeed, these models were originally developed as models of
social judgment and perceptual psychophysics (Birnbaum, 1974; Birnbaum, Parducci, &
Gifford, 1971; Birnbaum & Stegner, 1979). Two situations that are objectively equivalent
(under some normative analysis) but described differently to people must be kept distinct
in the theory if people respond differently to the different descriptions.
Case Against Prospect Theories 20
Third, as in PT and CPT, utility (or value) functions are defined on changes from
the status quo rather than total wealth states. See also Edwards (1954; 1962) and
Markowitz (1952).
Fourth, RAM, TAX, and GDU models, like PT and CPT, emphasize the role of
weighting of probability; however, it is in the details of this weighting that these theories
differ. Whereas CPT works with decumulative probabilities, these models work with
branch probabilities.
Fifth, although this paper uses the term “utility” and “value” interchangeably and
uses the notation u(x) to represent utility (value), rather than v(x) as in PT and CPT, the
u(x) functions of configural weight models are treated as psychophysical functions and
should not to be confused with “utility” as defined by EU, nor should these functions be
imbued with other excess meaning. The estimated utility functions in different models
will in fact be quite different. For modest consequences in the range of pocket money, $1
to $150, the utility (value) function of the configural weight models can be approximated
with the assumption that utility is proportional to objective cash value.
Sixth, Birnbaum (1974, p. 559) remarked that in his configural weight models,
“the weight of an item depends in part on its rank within the set.” Thus, his configural
weight models have some similarities to the models that later became known as “rankdependent” weighting models. However, in Birnbaum’s models, the weight of each
branch depends on the ranks of discrete branch consequences, whereas in the RDU
models, decumulative weight is a function of decumulative probability. In other words,
the definition of rank differs in the two approaches. For example, in two branch gambles,
Case Against Prospect Theories 21
there are exactly two ranks (lower and higher), and in three branch gambles, there are
exactly three ranks (lowest, middle, highest), no matter what their probabilities are.
Two configural weight models, the Rank-Affected Multiplicative Weights (RAM)
and the Transfer of Attention Exchange (TAX) models, have in common that utilities are
weighted by functions of probability and the ranks of branches in the gamble, divided by
total weight.
4.1 RAM Model
In the RAM model, the weight of each branch of a gamble is the product of a
function of the branch’s probability multiplied by a constant that depends on the rank and
augmented sign of the branch’s consequence. Augmented sign takes on three levels, –, +,
and 0. Rank refers to the rank of the branch’s consequence relative to the other discrete
branches in the gamble, ranked such that x1  x2 
 xn .
n
 a(i,n,s )t(p )u(x )
i
RAMU (G) 
i
i
i 1
n
 a(i,n,si )t(pi )
(5)
i 1
where RAMU(G) is the utility of gamble G in the RAM model, t(p) is a strictly
monotonic function of probability, i and si are the rank and augmented sign of the
branch’s consequence, and a(i,n,sj ) are the rank and augmented sign affected branch
weights. Rank takes on levels of 1,or 2 in two-branch gambles; 1, 2, or 3 in three-branch
gambles. The augmented sign of a consequence, si, takes on three levels, negative, zero,
and positive. The function, t(p), describes how a branch’s weight depends on its
probability, apart from these configural effects.
Case Against Prospect Theories 22
For two, three, and four branch gambles on strictly positive consequences, it has
been found that the rank weights are approximately equal to their branch ranks. In other
words, the rank weights in a three-branch gamble are 3 for the lowest branch, 2 for the
middle branch, and 1 for the branch with the highest consequence. In practice, t(p) can
be approximated by a power function, t(p) = p, with exponent,  = .6, and u(x) can be
approximated by u(x) = x for $1 < x < $150. These parameters were chosen to roughly
approximate the data of Tversky and Kahneman (1992), and the model with these
parameters will be called the “prior” RAM model. The term “prior” here has meaning
because the model was estimated from previous data and used to predict what would
happen in new choices that had not been previously tested.
For positive valued, two-branch gambles, this model implies that CEs are an
inverse-S function of probability to win the higher consequence. For example, consider
G = ($100, p; $0). The CEs of these gambles are given by CE(G) 
1 pg 100
,
1 p g  2 (1 p)g
which gives a good approximation of empirical data of Tversky & Kahneman (1992). A
graph of their data and predictions of the RAM model are shown in Birnbaum (1997,
Figure 9).
Note that although there are two rank weights (with weights 1 and 2 in the “prior”
model for higher and lower consequence, respectively) in these two-branch gambles,
there is only one free weight parameter in this situation since rank (and augmented sign)
weights could be multiplied by any constant with no loss of generality. For two-branch
gambles, one could constrain them to sum to 1, for example.
Birnbaum and McIntosh (1996) fit Tversky and Kahneman’s (1992) data,
reporting best-fit normalized weights are .37 and .63 for the higher and lower ranked
Case Against Prospect Theories 23
branches, respectively, and  = .56. Birnbaum later simplified these parameters by letting
the branch rank weights be equal to the ranks, so that the weight of the higher of two
equally likely branches would be 1/3. He rounded  to .6, and then used this simple
“prior” model to make new predictions for properties not previously tested. Although the
RAM model agrees with Tversky and Kahneman’s (1992) data, it makes very different
predictions for new tests, as we see below.
To describe gambles with consequences that are strictly less than or equal to zero,
one can use the same equations, inserting absolute values in the equations and then
multiplying the result by –1. This means that the order for strictly nonnegative gambles
A, B,
A
and for strictly nonpositive gambles  A, B,
B  A
will show “reflection”; that is,
B where –A is the same as gamble A, except that each consequence has
been converted from a gain to a loss by multiplication by –1.
To describe gambles with mixed consequences, the consequences are ranked in
the same fashion as for strictly positive gambles. In this simplified model applied to a
four branch gamble with two negative and two positive branches, the rank weights are 4,
3, 2, and 1 for the lowest (worst negative consequence), the next-lowest negative
consequence, medium high, and highest consequences, respectively. This means that
negative consequences get greater weight than equally probable positive consequences in
mixed gambles. This approach contrasts with CPT, where the utilities of negative
consequences are changed by a “loss aversion” parameter, but the weighting functions for
positive and negative consequences are approximately the same for equal probabilities.
In the RAM model, Equation 5 allows for different weighting for positive and negative
branches with the same rank; however, the simplification appears to provide a rough
Case Against Prospect Theories 24
approximation for the (meager) data available on mixed gambles. In Edwards’ (1962)
language, this additional simplification treats nonnegative gambles and mixed gambles
with rank-affected weights on the same page.
4.2 TAX Model
Intuitively, a decision maker deliberates by considering the possible consequences
of an action. Those that are more probable deserve more attention; however, lower
valued consequences also deserve more attention. In the TAX model, weight is
transferred from branch to branch. If there were no configural effects, each branch would
have weights as a function of probability, t(p). However, depending on the participants
point of view (analogous to “risk attitude”) weight is transferred from branch to branch,
where ( i, j) represents the weight transferred from branch i to branch j (j ≥ i, so x j  xi ).
n
TAXU(G) 
n
j
 t( pi )u(xi )    [u(x j )  u(xi )] (i, j)
i 1
j 1 i1
n
(6)
 t( p )
i
i 1
This model is fairly general and includes several interesting special cases (Marley &
Luce, 2001; 2004). A special case of this generic model (“special TAX”) assumes that
all weight transfers are the same fixed proportion of the weight of the branch giving up
weight, as follows:
 (i, j) 
 t( pj )
(n  1)
if  < 0, and  (i, j) 
 t( pi )
(n  1)
if  ≥ 0.
In this “special” TAX model, the amount of weight transferred between any two branches
is a fixed proportion of the probability weighting of the branch losing weight. If lowerranked branches have more importance (as they would to a “risk-averse” person), it is
theorized that weight is transferred from branches with better consequences to branches
Case Against Prospect Theories 25
with lower-valued consequences; i.e.,  > 0. This model is equivalent to that in
Birnbaum and Chavez (1997); however, the ordering of the branches has been changed to
make it compatible with that in CPT. Therefore,  > 0 in this paper corresponds to  < 0
in previous papers (Birnbaum, 1999b; Birnbaum & Chavez, 1997; Birnbaum &
Navarrete, 1998).
When lower valued branches receive greater weight, this special TAX model can
be written as follows for G  (x1 , p1; x 2, p2 ; ;x n, pn ) , where x1  x2 
n
TAX(G) 
[t( pi ) 
i 1

i1
 t(p j ) 
n  1 j 1

 xn  0 :
n
 t(pi )]u(xi )
n  1 j  i1
n
 t(p )
i
i 1
One can give a rough approximation of the data of Tversky and Kahneman (1992)
with  = 1; in other words, the approximate proportion of weight transferred in a twobranch gamble is 1/3 (as illustrated in the right side of Figure 1). In three-branch
gambles with positive consequences, 1/4 of the weight of any higher valued branch is
transferred to each lower-valued branch.
With these restrictions, the TAX model has three parameters, one to represent the

psychophysical function for probability, t(p)  p , one to represent the utility function of

monetary consequences, u(x)  x , and one, , to represent the configural transfers of
weight (risk aversion or risk seeking). In this paper, predictions of the model will be
calculated with  = 1,  = .7, and  = 1. This model, with parameters estimated to
roughly approximate the data of Tversky and Kahneman (1992) will be termed the
“prior” TAX model. This model makes similar predictions to RAM in many studies, but
it differs from RAM in tests of distribution independence.
Case Against Prospect Theories 26
Both TAX and RAM satisfy transitivity and consequence monotonicity. They
both satisfy comonotonic restricted branch independence. They also satisfy certain
distribution independence properties that are violated by CPT. However, they
systematically violate coalescing and they violate restricted branch independence in the
opposite way from the violations predicted by CPT.
For gambles composed of strictly positive, strictly negative, or mixed
consequences, different values of  would be allowed in this simple model. However, a
still simpler model (similar to that used by RAM) appears to give a good first
approximation. In this model, the value of  is simply reflected for gambles with purely
negative consequences and the same value is assumed to apply to gambles with all
positive consequences as to gambles with mixed consequences. These assumptions
account for reflection, for the four-fold pattern of risk seeking and risk aversion, and for
violations of gain loss separability.
4.3 GDU Model
Luce (2000, p. 200) proposed a “less restrictive theory” that satisfies a property
known as (Lower) Gains Decomposition but does not satisfy coalescing. Marley and
Luce (2001; 2004) present the axioms and representation theorem and show that the
model is similar with respect to the new paradoxes to TAX. The key idea is to analyze a
complex gamble as a sequence of two-branch gambles. The decomposition can be
viewed as a tree in which the decision maker breaks down a 3 branch gamble into two
stages: first, the chance to win the lowest consequence, and otherwise to win a second,
binary gamble to win one of the two higher prizes. Binary gambles are represented by
binary RDU, as follows:
Case Against Prospect Theories 27
GDU(x, p; y)  W(p)u(x)  [1  W(p)]u(y)
(6)
where GDU(G) is the utility of G according to this model.
For a three-branch gamble, G  (x, p; y,q;z,1  p  q) , where x  y  z  0 , the
lower gains decomposition rule (Luce, 2000, p. 200-202) can be written as follows:
GDU(G)  W( p  q)* GDU(x, p/( p  q); y) [1  W(p  q)]u(z)
(7)
Note that this utility is decomposed into the gamble to win the worst outcome, z, or to
win a binary gamble between x and y otherwise. Marley and Luce (2001) have shown
that RDU is a special case of GDU in which the weights take on a special form.

To illustrate this model, let u(x)  x , and let the weighting function be
approximated by the expression developed by Prelec (1998) and by Luce (2000):
W(p)  exp[ ( ln p) ]
(8)
With these assumptions for the functions, this model has a total of three parameters, two
for the weighting function, and one for the utility function. These functions match those
used by Luce (2000, p. 200-202) in his illustration of how this model could account for
phenomena that violate CPT.
Marley and Luce (2001; 2004) have explored the theoretical underpinnings of
GDU models and have shown that this GDU model is similar to TAX in that it can
violate coalescing and properties derived from coalescing, but it is distinct from TAX.
Birnbaum (submitted-a) noted that this model satisfies upper coalescing, though it can
violate lower coalescing. Luce (personal communication) is currently working with
several colleagues on more general models that satisfy gains decomposition but do not
necessarily satisfy binary RDU.
5. THE CASE AGAINST PROSPECT THEORIES
Case Against Prospect Theories 28
1. Violations of Coalescing: Splitting Effects
Because RDU, RSDU, and CPT models satisfy coalescing and transitivity, they
cannot explain “event-splitting” effects. Starmer and Sugden (1993) and Humphrey
(1995) found that preferences depend on how branches are split or coalesced. Although
Luce (2000) expressed reservations concerning these tests because they were conducted
between groups of participants, subsequent research has determined that “event-splitting
effects” (violations of coalescing combined with transitivity) are robust and can be
demonstrated consistently within subjects. Consider the following two choices:
Problem 1: In each choice below, a marble will be drawn from an urn, and the color of
marble drawn blindly and randomly will determine your prize. You can choose the urn
from which the marble will be drawn. In each choice, which urn would you choose?
A: 85 red marbles to win $100
B: 85 black marbles to win $100
10 white marbles to win $50
10 yellow marbles to win $100
05 blue marbles to win $50
05 purple marbles to win $7
A’: 85 black marbles to win $100
B’: 95 red marbles to win $100
15 yellow marbles to win $50
05 white marbles to win $7
Note that A is the same as A’, except for coalescing, and B’ is the same prospect as B. So
if a person follows coalescing, that person should make the same choice between A and B
as she or he does between A’ and B’. Birnbaum (2004a) presented these two choices,
included among 19 others, to 200 participants, finding that 63% (significantly more than
half) chose B over A and that only 20% (significantly less than half) chose B’ over A’. Of
the 200 participants, 96 switched from B to A’ against only 13 who switched from A to
B’,
z = 7.95. According to CPT, no one should switch except by random error, but both
prior RAM and prior TAX predict this reversal, because splitting the branch to win $100
in B makes it better and splitting the lower branch in A (to win $50) makes it worse.
Case Against Prospect Theories 29
By coalescing the two branches to win $100 in B’, we have made that gamble
relatively worse, and by coalescing the two lower branches in A’, we have made that
gamble relatively better. According to prior TAX, the predicted certainty equivalents of A
and B are $68.4 and 69.7, respectively; however, the certainty equivalents of A’ and B’
are 75.7 and 62.0, respectively. So prior TAX predicts a reversal of preferences in this
case.
According to prior CPT people should choose A over B and A’ over B’; however,
any CPT model implies A
B  A B. According to prior TAX, people should
choose B over A and A’ over B’. This effect is large in magnitude and is significant even
by the conservative standard: significantly more than half of all participants chose B over
A
(63%), and significantly more than half (80%) of participants chose A’ over B’. Luce
(1998) provided a list of other theories that satisfy coalescing that are also refuted by
significant violations.
2. Violations of First Order Stochastic Dominance
Birnbaum (1997) deduced that his configural weight models of risky decision
making would violate stochastic dominance when choices were constructed from a
special recipe, illustrated in Problem 2.
Problem 2. From which urn would you prefer to draw?
Choice
I: 90 red marbles to win $96
J:
TAX CEs
85 red marbles to win $96
05 blue marbles to win $14
05 blue marbles to win $90
05 white marbles to win $12
10 white marbles to win $12
45.8
63.1
After Birnbaum’s (1997, p. 93-94) prediction had been set in print, Birnbaum and
Navarrete (1998) tested this prediction empirically, reporting that about 70% of 100
Case Against Prospect Theories 30
undergraduates tested violated first stochastic dominance in four choices like Problem 2.
Birnbaum, Patton, & Lott (1999) found similar results with five new variations of the
recipe and a new sample of 110 participants.
In a recent study (Birnbaum, submitted-d), 74% of 330 participants chose to draw
from J, even though I dominates J. For n = 330, the 95% confidence interval for p = 1/2
ranges from 45% to 55%. The finding of 74% violations is well outside this interval, z =
6.23. It would be extremely unlikely given the null hypothesis that people are just
guessing. Therefore, we can reject that null hypothesis in favor of the hypothesis that the
percentage of violations exceeds 50%.
The calculated certainty equivalents of the gambles according to prior TAX are
shown in the right portion of Problem 2. According to prior TAX, people should choose
the dominated gamble. This prediction was made before any data were collected
(Birnbaum, 1997).
Birnbaum’s (1997) recipe was constructed as in the lower portion of Figure 2.
Start with a root gamble, G0, = ($96, .9; $12, .1). Next, split the lower-valued branch (.1
to win $12) into two parts, one of which has a slightly better consequence (.05 to win $14
and .05 to win $12), yielding G+ = ($96, .9; $14, .05; $12, .05). G+ dominates G0.
However, according to Birnbaum’s TAX and RAM models, G+ should seem worse than
G0, because the increase in total weight of the lower branches outweighs the increase in
the .05 sliver’s consequence from $12 to $14.
Starting again with G0,, we now split the higher valued branch of G0,,
constructing G– = ($96, .85; $90, .05; $12, .10), which is dominated by G0. According to
the configural weight models, this split increases the total weight of the higher branches,
Case Against Prospect Theories 31
which improves the gamble despite the decrease in the .05 sliver’s consequence from $96
to $90. The configural weight RAM and TAX models imply that people will prefer G–
over G+, in violation of stochastic dominance.
Insert Figure 2 about here.
Transitivity, coalescing, and consequence monotonicity imply satisfaction of
stochastic dominance in this recipe: G0, = ($96, .9; $12, .1).~ ($96, .9; $12, .05; $12, .05),
by coalescing. By consequence monotonicity, G+ =($96, .9; $14, .05; $12, .05).($96,
.9; $12, .05; $12, .05), by consequence monotonicity. G0, = ($96, .9; $12, .1).~ ($96, .85;
$96, .05; $12, .10), by coalescing, and ($96, .85; $96, .05; $12, .10) ($96, .85; $90, .05;
$12, .10) = G–, by consequence monotonicity. By transitivity, G+ = ($96, .90; $14, .05;
$12, .05)  G0  ($96, .85; $90, .05; $12, .10) = G–, so G+  G–. This derivation
shows that if people obeyed these three principles, they would not show this violation,
except by chance or error. Because people show systematic violations, at least one of
these assumptions must not be a correct descriptive principle.
Following the early studies cited above, a number of subsequent studies have
confirmed that violations of stochastic dominance are observed in a variety of different
conditions and variations of this recipe. Majority violations have been observed with or
without branch juxtaposition, and whether branches are listed in increasing order of their
consequences or in reverse order. They are observed when probabilities are presented as
decimal fractions, as pie charts, as percentages, as natural frequencies, or as lists of
equally likely consequences (Birnbaum, 2004b). They are observed with or without the
event framing used by Tversky & Kahneman (1986).
Case Against Prospect Theories 32
They are observed among men, women, undergraduates, college graduates, and
holders of doctoral degrees (Birnbaum, 1999b). Although the rate of violation varies
with education, the rate of violation among doctorates who have studied decision making
is still substantial. Majority violations have been observed when participants are tested in
class, in the lab, or via the WWW, and with consequences that are purely hypothetical or
offer real chances at modest prizes (Birnbaum & Martin, 2003). They are found with
three variations of Savage’s (1974) display showing association of tickets with
consequences in aligned and unaligned tables and with text representation (Birnbaum,
submitted-b), with huge or small consequences (Birnbaum, submitted-a), and with both
positive and negative consequences (Birnbaum, submitted-b).
Perhaps the violations are due to a counting heuristic, in which people choose the
gamble with the greater number of high consequences. To test this heuristic, the
following choice was employed by Birnbaum (submitted-d):
Choice
I’: 90 black to win $97
05 yellow to win $15
05 purple to win $13
TAX CEs
J’: 85 red to win $90
46.8
05 blue to win $80
10 white to win $10
Here, the dominant gamble (I’) has greater consequences on two of the three
branches, yet a significant majority of 57% of 394 participants continued to choose the
dominated gamble, J’ instead of the dominant gamble I’, as predicted by prior TAX.
According to CPT, the certainty equivalents of I’ and J’ are $73.3 and $66.6,
respectively, satisfying dominance.
Birnbaum and Yeary (submitted) obtained judgments of the highest buying price
for each of 166 gambles and judgments of the lowest selling prices of the same gambles,
57.6
Case Against Prospect Theories 33
presented separately. Mixed in among the 166 trials in each task were eight gambles
separated by many other trials that were devised to create four tests of stochastic
dominance in this recipe. They found that both buying prices and selling prices were
significantly higher for the dominanted gambles (like I) than for the dominant gambles
(like J). This finding suggests that these violations are not due to processes of
comparison or choice, but rather to the process of evaluating the gambles.
As of 10-04-04, my students and I had completed 31 choice studies with a total of
9486 participants testing first order stochastic dominance. In these studies, violation of
stochastic dominance has been a very robust finding (Birnbaum, 1999b; 2000; 2001;
Birnbaum, et al, 1999; Birnbaum, 2004a; 2004b; Birnbaum, submitted-a, -b, -c, -d). I
conclude that any theory that proposes to be descriptive must reproduce these violations.
Have others replicated these results or failed to replicate? Are there other studies
preceding Birnbaum & Navarrete (1998) that reported significantly more than 50%
violations of stochastic dominance? Is there anyone who still thinks that some
modification of CPT can account for these violations? If so, how would it do this?
Systematic violations of stochastic dominance refute the class of models that
includes RDU/RSDU/CPT. Other descriptive theories that assume or imply stochastic
dominance are also refuted as by this finding (Becker & Sarin, 1987; Lopes & Oden,
1999; Machina, 1982). Theories that satisfy consequence monotonicity, transitivity, and
coalescing imply satisfaction of stochastic dominance in this recipe. Luce (1998; 2000)
lists additional models that satisfy coalescing and stochastic dominance.
Three calculators are freely available from the following URLs:
http://psych.fullerton.edu/mbirnbaum/calculators/
Case Against Prospect Theories 34
These calculators also allow calculations of the CPT model of Tversky and Kahneman, 1992,
and the revised version in Tversky and Wakker, 1995. A calculator by Veronika Köbberling
that calculates CPT predictions is also linked from the same URL.
It is important to distinguish first order stochastic dominance, which must be satisfied
by RDU/RSDU/CPT, with other types of “stochastic dominance” discussed by Levy and Levy
(2002), Wakker (2003), and Baucells & Heukamp (2004). Although CPT may be able to
account for the data of Levy and Levy (2002), it cannot account for violations of first order
stochastic dominance, as defined in this section.
Interestingly, Tversky and Kahneman (1986) reported a case of violation of first order
stochastic dominance. In their example, colors of marbles were framed so that for each color
of marble, the urn with the dominated gamble gave an equal or higher prize than the prize for
the same color in the dominant gamble. The numbers of marbles of each color were slightly
different in the two urns, so people would be mistaken if they treated “events” as colors of
marbles drawn from the urns. This “event framing” was intended to make the worse gamble
look better. They reported that 58% of the participants violated dominance, which was not
significantly different from 50%. Tversky and Kahneman (1986) noted that several
manipulations were confounded in their study, including not only framing and coalescing, but
also the number of branches in each gamble with positive or negative consequences. The
skeptic might conclude of their study that their framing and other manipulations merely
confused participants, so they were choosing at random, and that is why the choice percentage
in the “framed” condition was not significantly different from 50%. Was this one result
published the only test, or were there other tests, from which this case was selected as giving
Case Against Prospect Theories 35
the “best” results? Is there other experimental work by them or by others that followed up on
their 1986 example?
Birnbaum (2004a; 2004b; submitted-a) manipulated the marble color framing
independently of coalescing, number of branches, and other manipulations, and concluded
that this type of event-framing played no detectable role in violations of stochastic dominance
in his recipe.
By running studies in which each person makes each decision more than once, it is
possible to estimate “random error” rates. This analysis allows one to estimate the proportion
of participants who “truly” violate stochastic dominance from those who might do so by
“error” alone. In Study 3 of Birnbaum (2004b), for example, each of 156 participants
completed four choices in the stochastic dominance recipe. Of these, there were 79 with four
violations, 31 with three violations, 21 with two, 13 with one, and 12 with zero violations.
Suppose there are two types of participants: those who truly violate stochastic
dominance in this recipe and those who truly satisfy it, apart from error. Then the probability
that a person satisfies stochastic dominance on the first three repetitions of the choice and
violates it on the fourth is given by the following:
P(SSSV)  a(1 e)3 e  (1 a)e3 (1  e)
Where a is the probability that a person “truly” satisfies stochastic dominance, e is the
probability of making an “error” in reporting one’s “true” preference. In this case, the person
who truly satisfies stochastic dominance has correctly reported the preference three times and
3
made one error ( (1  e) e ), whereas the person who truly violates it has made three “errors”
and one correct report.
Case Against Prospect Theories 36
When this true and error model was fit to the response sequences of Birnbaum (2004b;
Study 3), it indicated that 83% of participants “truly” violated stochastic dominance on all
four choices (a = .17), except that people made “errors” on 15% of their decisions on these
choices. Similar results have been obtained in analyses of other sets of data (Birnbaum,
2004b).
Even more extreme results were obtained when this model was fit to an experiment
(Birnbaum, 2004b, Study 4) in which the choice was presented in decumulative probability
format, as follows:
Which do you choose?
I: .90 probability to win $96 or more
.95 probability to win $14 or more
1.00 probability to win $12 or more
OR
J: .85 probability to win $96 or more
.90 probability to win $90 or more
1.00 probability to win $12 or more
This format was used in an attempt to help participants see that the probability to win $96 or
more his higher in I than J, that the probability to win $90 or more in J is the same as that of
winning $96 or more in I, that the probability of winning $14 or more is higher in I, and that
the probability of winning $12 or more is the same in both gambles. Indeed, CPT is written as
a theory of decumulative probabilities, so this format should “help” CPT, by saving the work
of adding probabilities. However, the true and error model indicated that 92% of the 445
participants in this condition “truly” violated stochastic dominance, and that the error rate in
this condition is 12%. The finding of a lower error rate might be expected by CPT, but the
finding of a higher rate of true violation is surprising, since this condition was thought to be
one in which detection of stochastic dominance and use of CPT might be facilitated.
Case Against Prospect Theories 37
3. Event-Splitting and Stochastic Dominance
If the configural-weight models are correct, then splitting can not only be used to
cultivate violations of stochastic dominance, but splitting can also be used to weed out
violations of stochastic dominance (Birnbaum, 1999b). As shown in the upper portion of
Figure 2, one can split the lowest branch of G–, which makes the split version, GS–, seem
worse, and split the highest branch of G+, which makes its split version, GS+ seem
better, so in the split form, GS+ seems better than GS–. As predicted by the configural
weight RAM and TAX models, with parameters estimated from previous data, most
people prefer GS+ over GS–, and G– over G+, creating a powerful event-splitting effect,
apparently a violation of coalescing (Birnbaum, 1999b; 2000; 2001b; Birnbaum &
Martin, 2003). Consider the two choices in Problem set 3:
Problem 3.
Choices
3.1
3.2
I: 90
tickets to win $96
Prior TAX
J: 85 tickets to win $96
05 tickets to win $14
05 tickets to win $90
05 tickets to win $12
10 tickets to win $12
U: 85 tickets to win $96
V: 85 tickets to win $96
05 tickets to win $96
05 tickets to win $90
05 tickets to win $14
05 tickets to win $12
05 tickets to win $12
05 tickets to win $12
45.8
< 63.1
53.1 >
51.4
In a recent test (Birnbaum, 2004b), 342 participants were asked to choose
between I and J and between U and V, among other choices. There were 71% violations
in the coalesced form (Problem 3.1) and only 5.6% in the split form of the same choice
(Problem 3.2). It was found that 224 participants (65.5%, significantly more than half)
preferred J to I and U to V, violating stochastic dominance on the choice between G– and
Case Against Prospect Theories 38
G+, but satisfying it in the choice between GS– and GS+ (U versus V). Only 3
participants had the opposite reversal of preferences, z = 14.3. There were 16 (4.7%) who
violated stochastic dominance on both decisions, which might happen if people coalesced
before choosing, and only 27.8% satisfied stochastic dominance on both choices.
According to the class of RDU/RSDU/CPT models, people should make the same
choice between I and J as they do between U and V; therefore, this event-splitting effect
(the reversal of preferences between the two Problems 3) refutes that class of models.
Similar results have been obtained in 25 studies with 7809 participants (Birnbaum,
1999b; 2000; 2001; 2004a; 2004b; submitted-b; Birnbaum & Martin, 2003).
4. Predicted Satisfactions of Stochastic Dominance
If a model is to be considered accurate, it should be able to predict when
stochastic dominance will or will not be violated. Birnbaum (submitted –d) conducted a
series of five studies manipulating features of the choice to determine whether models
that can describe violation of stochastic dominance correctly predict where stochastic
dominance will be satisfied. This series of studies also tested the ideas is that people
ignore probability, and just average the values of the consequences, or that they are
counting the number of branches that favor one gamble or the other. If such consequence
counting heuristics are correct, people should continue to violate stochastic dominance in
Problem 4.
If the configural weight models are correct, however, people should not always
violate stochastic dominance. For example, with previous parameters, we can calculate
from the TAX model that the majority should satisfy stochastic dominance in Problem 4:
Problem 4.
Prior TAX
Case Against Prospect Theories 39
K: 90 red marbles to win $96
L: 25 red marbles to win $96
05 blue marbles to win $14
05 blue marbles to win $90
05 white marbles to win $12
70 white marbles to win $12
45.8
35.8
Birnbaum (submitted-d, study 2) found that of 232 participants, 72%
(significantly more than half) violated stochastic dominance in the comparison of I and J
(Problem 3.1), but only 34% (significantly less than half) violated stochastic dominance
in the choice between K and L in Problem 4. There were 103 who violated stochastic
dominance in the choice between I and J and satisfied it in the choice between K and L,
compared to only 18 who had the opposite reversal of preferences (z = 7.7). TAX
correctly predicted this reversal.
Therefore, people do not always violate stochastic dominance in variations of this
recipe. They largely satisfy it when splitting is used, as in Problem 3.2, and a 66%
majority satisfy it when probability of the highest consequence is reduced from 85% to
25%. We can therefore reject the hypothesis that people are ignoring probability.
Intermediate probabilities yielded intermediate results, suggesting that people are
attending to probability and utilizing it in a fashion predicted by quantitative models.
The series of five studies found that prior TAX gave the best predictions to the data,
followed by RAM. CPT was the worst, since it never predicts majority violations of
stochastic dominance and therefore cannot give an explanation of why the majority
sometimes does and sometimes does not violate stochastic dominance (Birnbaum,
submitted-d). The heuristics were systematically violated because people changed
behavior as the probabilities and consequences were manipulated.
Domain of Violation of SD. To understand how “often” the TAX model implies
violations of stochastic dominance, Birnbaum (2004a) simulated “random” choices
Case Against Prospect Theories 40
between “random” three-branch gambles. The “random” gambles were selected by
choosing three “random” numbers uniformly distributed between 0 and 1, and dividing
each by their sum to produce branch probabilities. Next, three consequences were
independently drawn from a uniform distribution between $0 and $100. Finally, a
“random” choice was generated by selecting two such random gambles independently.
For each of 1,000,000 choices thus constructed, it was determined whether a stochastic
dominance relationship was present in the choice, and whether the prior TAX model
predicted a violation. It was found that 33.3% of such choices involve stochastic
dominance, but only 1.8 per 10,000 are predicted to be violated according to prior TAX.
This means that the model only rarely violates stochastic dominance in this “random”
environment.
Prior TAX and prior CPT made the same predictions in 94% of these “random”
choices. It would not be efficient to compare theories based on haphazardly or randomly
devised choices.
One might ask, is special TAX just so flexible that it can account for anything?
The answer is no. Given the following two properties, both present in the data of
Tversky and Kahneman (1992), TAX is forced to violate stochastic dominance in
Birnbaum’s (1997) original recipe. With the assumption that the utility function is linear,
if people are (1) risk averse for 50-50 gambles, then  < 0, and if people are
simultaneously (2) risk-seeking for positive consequence gambles with small p, then  <
1. With  < 0 and  < 1, TAX is forced to violate stochastic dominance in Birnbaum’s
original recipe. So, special TAX is not flexible enough to satisfy stochastic dominance in
Case Against Prospect Theories 41
Birnbaum’s test, if it is parameterized to account for the Tversky and Kahneman (1992)
data.
5. Upper Tail Independence
Wu (1994) reported systematic violations of “ordinal” independence. These were
not full tests of ordinal independence as defined by Green and Julian (1988); however,
the property tested by Wu is implied by RDU/RSDU/CPT. These tests can be called
“upper tail” independence, since they test whether the upper tail of a distribution can be
used to reverse preferences induced by other aspects of the choice. The property can be
viewed as a combination of transitivity, coalescing, and comonotonic restricted branch
independence. Tail independence should therefore be satisfied according to the class of
RDU/.RSDU/CPT models. In the original form tested by Wu (1994), the property made
use of a lowest consequence of $0. Wu reported systematic violations, which he
reasoned might be due to violations of CPT or might be attributed to the editing rule of
cancellation, which contradicts implications of the CPT model.
Birnbaum (2001) reported a test of upper tail independence based on one from
Wu (1994). In Problem 5, that people should prefer t = ($92, .48; $0)  s = ($92, .43;
$68, .07; $0) iff they prefer ($92, .43; $92, .05; $0)  ($92, .43; $68, .07; $0), by
transitivity, comonotonic restricted branch independence, and upper coalescing.
Assuming restricted comonotonic branch independence, we can change the common
consequence on the .43 branch from $92 to $97, which means f = ($97; .43; $92, .05; $0)
 e = ($97, .43; $68, .07; $0). There were n = 1438 people who participated via the
WWW and had a chance to win a small cash prize. Significantly more than half preferred
Case Against Prospect Theories 42
s to t and significantly more than half preferred f to e, contradicting this property.
Because this implication rests on upper coalescing, this result also contradicts GDU.
Problem 5
First Gamble, S
5.1
s: .50 to win $0
.07 to win $68
Second Gamble, R
t: .52 to win $0
TAX
% R
34*
32.3
29.8
62*
33.4
36.7
.48 to win $92
.43 to win $92
5.2
e: .50 to win $0
f: .52 to win $0
.07 to win $68
.05 to win $92
.43 to win $97
.43 to win $97
6. Violations of Upper Tail Independence without Zero Consequences
A new test of upper tail independence (Birnbaum, Submitted-c, Exp. 2) is
displayed below in Problem 6. In this example, the lowest consequence is not zero in all
four gambles, as it was in earlier tests. Note that the second choice is created from the
first by reducing the consequence on the common branch of 80 marbles from $110 to $96
on both sides, and then coalescing it with the 10 marbles to win $96 branch on the right
side. In this test, there were 503 participants. As predicted by TAX, the majority was
significantly reversed, contradicting CPT.
Problem 6
S
6.1
R
80 red to win $110
80 black to win $110
10 yellow to win $44
10 purple to win $96
%R
67.3*
TAX
65.0
69.0
Case Against Prospect Theories 43
6.2
10 blue to win $40
10 green to win $10
80 red to win $96
90 green to win $96
10 white to win $44
10 yellow to win $10
33.1*
60.3
10 blue to win $40
A caveat should be noted with respect to both versions of upper tail
independence; namely, the test involves a comparison of two three-branch gambles in
one case and a three-branch against a two-branch gamble in the other. If people prefer
gambles with more (positive) branches, it might bias the results of the second
comparison. This concern is addressed in the next sections, which uses choices between
gambles with equal numbers of branches.
7. Upper Cumulative Independence
Birnbaum (1997) noted that violations of restricted branch independence reported
by Birnbaum and McIntosh (1996) contradict the implications of the inverse-S weighting
function of CPT, which is needed to account for CEs of binary gambles and the Allais
paradoxes. He was able to characterize this contradiction more precisely in the form of
two new paradoxes that are to CPT as the Allais paradoxes are to EU. Lower and Upper
Cumulative Independence can be deduced from transitivity, consequence monotonicity,
coalescing, and comonotonic restricted branch independence. Thus, violations of these
properties are paradoxical to any theory that implies these properties, including EU,
RDU, and CPT. Consider Problem 7.
Choice between S and R
pies*
TAX
n = 305
S’ vs. R’
.10 to win $40
.10 to win $10
.10 to win $44
.10 to win $98
70*
65.03
69.59
57.2
Case Against Prospect Theories 44
S”’ vs. R”’
.80 to win $110
.80 to win $110
.20 to win $40
.10 to win $10
.80 to win $98
.90 to win $98
42*
68.04
58.29
Case Against Prospect Theories 45
Upper cumulative independence can be written ( z x x  y  y z  0 ):
S  ( z,1  p  q;x, p; y,q)
R (z,1  p  q; x, p;y,q)
 S (x ,1 p  q; y, p  q)
R ( x,1  q; y, q)
Violations of upper cumulative independence, as in Problem 7, can be interpreted as a
contradiction in the weighting function, if one assumes CPT (Birnbaum, et al, 1999,
Appendix). According to the class of RDU/RSDU/CPT models, R S 
W(.8)u(110) [W(.9)  W(.8)]u(98) [1  W(.9)]u(10) >
W(.8)u(110) [W(.9)  W(.8)]u(44)  [1  W(.9)]u(40)
 [W(.9)  W(.8)][u(98)  u(44)]  [1  W(.9)][u(40)  u(10)]

u(98)  u(44) W(.9)  W(.8)

u(40)  u(10)
1  W(.9)
Similarly, in this class of models, R S 
W(.8)u(98)  [W(.9)  W(.8)]u(98)  [1  W(.8)]u(10)
< W(.8)u(98) + [W(.9) - W(.8)]u(40) + [1- W(.9)]u(40)

W(.9)  W(.8) u(98)  u(40) u(98)  u(44)


.
1  W(.9)
u(40)  u(10) u(40)  u(10)
Thus, CPT leads to self-contradiction when it attempts to handle this result.
Birnbaum and Navarrete (1998) investigated 27 variations of this test with 100
participants; Birnbaum, et al. (1999) explored 6 new variations with 110 new
participants. Birnbaum (1999b) compared results from 1224 people recruited from the
Web against 124 undergraduates tested in the lab. Birnbaum (2004b) investigated this
property with two previously used variants and with a dozen different procedures for
representing the gambles; that paper summarized results from 3440 participants.
Case Against Prospect Theories 46
Birnbaum (submitted-b) replicated two tests with another 663 participants and three
variations of Savage’s presentation format and with strictly positive and strictly negative
consequences. These results were also replicated among “filler trials” in a series of
studies of stochastic dominance (Birnbaum, submitted-d) with 1467 participants. There
have now been 26 studies with 33 problems and 7186 participants. The cumulative results
are strongly in violation of upper cumulative independence and CPT.
For example, in the pies* condition of Birnbaum (2004b), probability was
displayed by pie charts in which slices of the pie had areas proportional to probabilities.
There were 305 participants who responded to Problem 7 in this condition. Significantly
more than half chose R S  and significantly more than half chose S  R, contrary
to CPT: 112 changed preferences in the direction violating the property against only 28
who switched in a manner consistent with the property, z = 7.10.
8. Lower Cumulative Independence
Lower cumulative independence, also deduced by Birnbaum (1997) is the
following:
S  (x, p; y, q;z,1 p  q)
R  ( x, p; y,q;z,1  p  q)
 S (x, p  q; y,1  p  q)
R ( x, p; y,1 p) .
Like upper cumulative independence, lower cumulative independence also follows from
transitivity, coalescing, consequence monotonicity, and comonotonic restricted branch
independence. This property was also developed and predictions made before the
experiments were carried out. Problem 8 illustrates a test of this property.
Problem 8
Case Against Prospect Theories 47
Risky
Safe
pies*
TAX
n = 305
8.1
8.2
.90 to win $3
.90 to win $3
.05 to win $12
.05 to win $48
.05 to win $96
.05 to win $52
.95 to win $12
.90 to win $12
.05 to win $96
.10 to win $52
62*
8.8
10.3
26*
68.0
58.3
In the pies* condition of Birnbaum (2004b), it was found that 62% (significantly
more than half) chose S over R, but only 26% (significantly less than half) chose S” over
R”;
significantly more participants switched in the direction in violation of the property
than in the direction consistent with it. As in the case of upper cumulative independence,
lower cumulative independence can be interpreted as a contradiction in the weighting
function if one assumes CPT. By this writing, there had been 26 studies with 33
problems and 7186 participants, creating very strong combined case against this property.
9. The Allais common consequence paradox
The common consequence paradox of Allais (1953, 1979) can be illustrated with
the two choices in Problem 9:
Problem 9: Allais Paradox
A:
$1M for sure
B:
.10 to win $2M
TAX Model
1M >
0.81M
0.13M
< 0.24M
.89 to win $1M
.01 to win $0
C:
.11 to win $1M
.89 to win $0
D:
.10 to win $2M
.90 to win $0
Case Against Prospect Theories 48
Expected Utility (EU) theory assumes that Gamble A is preferred to B if and only
if the EU of A exceeds that of B. This assumption is written, A  B EU(A) > EU(B),
where the EU of a gamble, G = (x1, p1, x2, p2;…xi, pi; …;xn, pn) can be expressed as
follows:
n
EU(G)   piu(x i )
(7)
i1
According to EU, A is preferred to B iff u($1M) > .10u($2M) + .89u($1M)
+.01u($0). Subtracting .89u($1M) from each side, it follows that .11u($1M) >
.10u($2M)+.01u($0). Adding .89u(0) to both sides, we have .11u($1M)+.89u($0) >
.10u($2M)+.90u($0), which holds if and only if C  D. Thus, from EU theory, one can
deduce that A
B C
D . However, many people choose A over B and prefer D over
C. This pattern of empirical choices violates the implication of EU theory, so such
results are termed “paradoxical.” RDU, CPT, RSDU, RAM, TAX, and GDU can all
account for this finding.
10. Decomposition of the Allais Common Consequence Paradox
It is useful to decompose this Allais paradox into transitivity, coalescing, and
restricted branch independence, as illustrated below:
A:
$1M for sure
B:
.10 to win $2M
.89 to win $1M
.01 to win $0
B:
.10 to win $2M
.89 to win $1M
.01 to win $0
(coalescing & transitivity)
A’:
.10 to win $1M
.89 to win $1M
.01 to win $1M
 (restricted branch independence)
Case Against Prospect Theories 49
A”:
.10 to win $1M
.89 to win $0
.01 to win $1M
B’:
.10 to win $2M
.89 to win $0
.01 to win $0
D:
.10 to win $2M
.90 to win $0
 (coalescing & transitivity)
C:
.11 to win $1M
.89 to win $0
The first step converts A to its split form, A’; A’ ~ A by coalescing, and by transitivity,
it should be preferred to B. From the third step, the consequence on the common branch
(.89 to win $1M) has been changed to $0 on both sides, so by restricted branch
independence, A’’ should be preferred to B’. By coalescing branches with the same
consequences on both sides, we see that C should be preferred to D.
This derivation shows that if people obeyed these three principles, they would not
show this paradox, except by chance or error. Systematic violations are termed Allais
paradoxes, and imply that at least one of these principles must be false.
Different theories attribute these paradoxes to different causes (Birnbaum, 1999a),
as listed in Table 1. SWU (including PT) attributes them to violations of coalescing. In
contrast, the class of RDU, RSDU, and CPT attribute the paradox to violations of
restricted branch independence.
Insert Table 1 about here.
Birnbaum (2004a) tested among the theories in Table 1. Table 2 lists Problems
10.1-10.5, which tests each step in the derivation. According to EU, the choice in each of
these problems should be the same, because all of the choices have a common branch (80
marbles to win $2 in Choices 10.1 and 10.2, $40 in Choice 10.3, or $98 in Choices 10.4
and 10.5. Birnbaum presented these choices, separated by other choices to 349
participants, whose data are summarized in Table 2 [Data have been aggregated over
Case Against Prospect Theories 50
choices that were framed, meaning that the same marble colors were used on
corresponding branches, or unframed, with different marble colors on all branches.
There were no systematic effects of this framing manipulation, and the findings were
replicated with different problem sets and in a new study with different participants.]
With n = 349, and p = 1/2, percentages less than 44.6% or greater than 55.3% are
significantly different from 50%, with  = .05. Thus, each percentage, except for
Problem 8 is significantly different from 50%. By the test of correlated proportions, each
successive comparison is significant.
Insert Table 2 about here.
The class of RDU/RSDU/CPT models and the TAX and RAM models, as fit to
previous data, all agree on the predicted choices in Choices 10.1, 10.3, and 10.5 of Table
2; that is, they all predict the violations of Allais independence in those three problems.
However, the class of RDU/RSDU/CPT implies that Choices 10.1 and 10.2 should agree,
except for error, and that Choices 10.4 and 10.5 should agree as well, since those test
coalescing; whereas, the RAM and TAX models predict reversals. Put another way,
RDU/RSDU/CPT models attribute violations of Allais independence to violations of
branch independence, and not coalescing. Branch independence is tested between
Choices 10.2 and 10.4. However, the majority preference in 10.2 is opposite that of 10.4.
Thus, as predicted by the RAM and TAX violations of branch independence are opposite
of what is needed if branch independence is to explain Allais paradoxes.
Now consider the editing rule of cancellation. If people canceled the common
branch in Choices 10.2 and 10.4, they would show no violations of restricted branch
independence. If they only cancelled on some proportion of the trials, there would be
Case Against Prospect Theories 51
violations of branch independence that would be in the direction of Choices 10.1 and
10.5. However, the observed violations are substantial and opposite the direction needed
to allow branch independence to account for Allais paradoxes. It might be
understandable if violations of branch independence in the transparent tests might be
attenuated, but they should still be in the same direction as the Allais paradox if we adopt
CPT plus the partial use of cancellation.
According to the prior RAM and TAX models, violations of both branch
independence and coalescing should show the pattern they do. Furthermore, violations of
branch independence in these models are opposite from those predicted by the inverse-S
weighting of CPT. GDU can also account for these results.
According to TAX/RAM, the cause of the Allais constant consequence paradoxes
is violation of coalescing, and violations of branch independence actually reduce the
magnitude of Allais paradoxes. Note that all splitting and coalescing operations make R
worse and S better as we proceed from Choice 10.1 to 10.5. Splitting the lower branch
makes R worse, and coalescing the upper branch makes it worse. Similarly, splitting the
higher branch from Problem 10.1 to 10.2 improves S as does coalescing the lower
branch from Problem 10.4 to 10.5.
11. True and Error Analysis of Splitting in the Allais Problem
Table 3 shows the number of participants who showed each preference pattern on
two replications of the following Allais Paradox choices.
11.1
11.2
11 blues to win $1,000,000
10 reds to win $2,000,000
89 whites to win $2
90 whites to win $2
10 blues to win $1,000,000
10 reds to win $2,000,000
76
77
125 K
236 K
35
39
241 K
172 K
Case Against Prospect Theories 52
01 blues to win $1,000,000
01 whites to win $2
89 whites to win $2
89 whites to win $2
In Table 3, S indicates preference for a “safe” gamble (with 11 marbles to win $1
Million, and 89 to win $2) rather than the “risky” (R) gamble (with 10 marbles to win $2
Million, and 89 to win $2). Here SRSS refers to a case where the participant chose S on
the first replicate of Choice 6, where the branches were coalesced, R on Choice 9 where
the branches were split, and chose S on both of these trials in the second replicate.
A three parameter true and error model was fit to these data (See Birnbaum,
2004b). In this model, the three parameters are the probability that a person truly prefers
R over S in Choice 6, the conditional probability that this person has the same true
preference in Choice 9, and the probability of making an “error” expressing true
preferences. The estimates are that 89.3% of participants truly prefer R over S in Choice
6 and that only 33.8% of these have the same true preference in Choice 9, where the
gambles have been split to apparently make the S gamble better and the R gamble worse.
2
The error probability is estimated to be 0.185, and  (12) = 14.3.
Insert Table 3 about here.
Put another way, this model indicates that 59% of participants truly switch from R
in Choice 11.1 to S in Choice 11.2 and that no one switched in the other direction, even
though the observed data show 46% made this switch and that 12% made the opposite
switch, averaged over replicates. Table 3 shows that 56 people (28%) switched in this
fashion on both replicates, compared to 4 who made the opposite switch both times. A
2
two-parameter model that assumes no one truly switched fits much worse,  (13) =
Case Against Prospect Theories 53
236.0. In sum, these results violate RDU/RSDU/CPT, but are consistent with RAM,
TAX, and GDU. In the next section, we can test GDU, which implies upper coalescing.
12. Violations of Branch Splitting Independence
Consider Choices 12.1-12.8 in Table 4. These problems break down Allais
paradoxes into still finer steps in order to test branch-splitting independence (Birnbaum,
submitted-a).
Insert Table 4 about here.
Event-splitting independence is the assumption that splitting a branch with the
same probability and same consequence in the same way should have the same effect,
independent of whether that consequence is the highest or lowest in the gamble. For
example,
($50, .10; $50, .05; $7)  ($50, .15; $7)

($50, .10; $50, .05; $100)  ($50, .15; $100)
Note that in the first line, splitting .15 to win $50 produces an improvement in the
gamble. BSI requires that when the same branch is split in the context another gamble, in
this case, where the other consequence is $100 instead of $7, splitting should have the
same directional effect. BSI is implied by SWU and by “stripped” prospect theory
(Starmer & Sugden, 1993). See Birnbaum and Navarrete (1998).
Data in Table 4 are based on 203 participants. Note that the positions of S and R
are reversed with respect to Table 2, so the trend from Choice 12.1 to 12.8 agrees with
the trend in Table 2. The difference between Choices 12.1 and 12.3 is that the branch of
15 marbles to win $50 has been split into 10 marbles to win $50 and 5 more to win $50.
Case Against Prospect Theories 54
This splitting of the highest consequence in S produces a decrease in the percentage who
choose R from 78% to 52% (z = 4.12). This decrease is expected if S was improved by
splitting its highest branch.
Now compare Choices 12.8 and 12.6. These also contain a branch of 15 marbles
to win $50, except that now this branch represents the lower branch of the gamble,
instead of the higher. The percentage choosing R now increases from 25% (Choice 12.8)
to 37% (Choice 12.6), as if the same splitting of the same branch made S worse (z = 5.0).
According to the TAX and RAM models, splitting the higher branch should improve the
gamble and splitting the lower branch should make the gamble worse. Thus, the data
appear to show evidence of violations of branch-splitting independence as predicted by
RAM and TAX.
The class of RDU/RSDU/CPT models implies no event-splitting effects (no
changes in choice percentage among Problems 12.1-12.4 and among 12.5-12.8. Clearly,
those models can be rejected. The class of SWU/PT models (aside from the editing rule
of combination) imply event-splitting independence, which is the assumption that
splitting the same branch into the same pieces should have the same effect, independent
of whether the branch is the lowest or highest branch in the gamble.
GDU implies upper coalescing, so it is not consistent with the data in Table 4.
Consider, for example, Choices 12.7 and 12.8, where only the upper branch of 95
marbles to win $100 in the gamble on the right has been either coalesced (Choice 12.8) or
split (Choice 12.7). The preference changes from 25% to 70% in response to this
splitting of the upper branch, contradicting GDU.
Case Against Prospect Theories 55
The prior TAX model predicts the trends of these separate splitting operations
fairly well. However, the magnitude of splitting the upper branch of R has a greater
effect than predicted in Choice 12.7, which is the only case in Tables 2 and 4 in which the
prior TAX model fails to predict the modal choice. Perhaps there is a bias favoring
gambles with more branches leading to positive consequences.
13. Violations of Gain-Loss Separability
According to CPT/RSDU, the overall utility of a gamble can be represented as the
sum of two terms, one for the gain part of a gamble and the other for the loss part of the
gamble. CPT therefore implies the gain-loss separability, which can be written as
follows:

If A  (y, p; x,q;0,1  p  q)
B  ( y , p; x , q;0,1  p q )
and
A   (0,1  r  s;z,r;v,s)
B  (0,1  r s; z , r; v , s)
Then A  (y, p;x,q; z,r;v,s)
B  ( y, p; x, q ; z, r ; v, s)
Where r  s 1  p  q and r  s 1  p q .
Wu and Markle (2004) devised the following test, which was modeled after a
choice in Levy and Levy (2002, Exp 2):
Problem 13.
A  : .25 to win $2000
B : .25 to win $1600
.25 to win $800
.25 to win $1200
.50 to get $0
.50 to get $0
72% B
497
552
Case Against Prospect Theories 56
A :
.50 to get $0
B : .50 to get $0
.25 to lose $800
.25 to lose $200
.25 to lose $1000
.25 to lose $1600
A : .25 to win $2000
B : .25 to win $1600
.25 to win $800
.25 to win $1200
.25 to lose $800
.25 to lose $200
.25 to lose $1000
.25 to lose $1600
72% B
-358
-276
38% B
-280
-300

In this case, based on 60 participants, the majority prefers B

majority prefers B
A  ; however, the majority does NOT prefer B
A  and the
A, contrary to
CPT/RSDU and any model that satisfies gain loss separability. This example is
consistent with prior TAX model with the following assumptions: (1) rank-affected
weighting is the same for mixed consequences as it is for purely positive consequences,
and (2) weighting for gambles with purely nonpositive consequences is applied by
absolute value. This means that the predicted order for nonpositive gambles and
nonnegative gambles obeys the “reflection” property. This model assumes u(x) = x, and
there is no “loss aversion” coefficient.
This point deserves emphasis: this model explains mixed gambles (and “loss
aversion”) by greater weight assigned to branches with negative consequences, instead of
by assigning a greater disutility to negative consequences, as is done in CPT. Few
experiments have been conducted on gain loss separability, consequently, it is not clear
that the simple assumptions above should suffice.
Birnbaum (1997) described a more complex model in which branch weights
depend not only on probability and rank, but also on augmented sign of the
consequences. (Augmented sign has three levels, +, 0, and –). Thus, a branch’s weight in
Case Against Prospect Theories 57
the more complex model depends on whether the branch leads to a positive, zero, or
negative consequence, and also depends on the rank. That model uses more parameters
than this simple version of TAX, and it would also account for violation of gain loss
separability.
14. Violations of Restricted Branch Independence
Restricted branch independence should be satisfied by EU, SWU (including the
extension of PT with the editing rule of cancellation). However, this property will be
violated according to RDU/RSDU/CPT (apart from the editing rule of cancellation) and
by RAM, TAX, and GDU. The direction of violations, however, is opposite in the two
classes of models. A number of studies have been conducted on this property. Wakker,
Erev, & Weber (1994) failed to find systematic violations predicted by CPT, but their
study was not designed to test RAM or TAX.
Many experiments have subsequently studied a special form of restricted branch
independence (RBI) that can be written as follows ( 0  z  y y  x  x  z):
S  (x, p; y, p; z,1 2 p)
R  ( x, p; y, p;z,1  2p)

S  ( z,1  2p; x, p; y, p)
R  ( z,1 2 p; x, p; y, p)
In this version of RBI, two branches have equal probabilities. The consequence on the
common branch is the only change from the first line to the third.
When the number of branches, their ranks, and the probability distribution are all
fixed, as above, CPT, RAM, TAX, and GDU models reduce to what Birnbaum and
McIntosh (1996, p. 92) called the “generic rank-dependent configural weight” theory,
also known as the rank weighted utility model (Luce, 2000; Marley & Luce, 2004). This
generic model can be written for the choice between S and R as follows:
Case Against Prospect Theories 58
S
R  w1u(x)  w2u(y)  w3u(z)  w1u(x' )  w2u(y' )  w3u(z)
Where w1,w2 and w3 are the weights of the highest, middle, and lowest ranked branches,
respectively which depend on the value of p (differently in different models). The
generic model allows us to subtract the term w3u(z) from both sides, which yields,
S
R 
w2 u(x )  u(x)
.

w1 u(y)  u(y )
There will be an SR’ violation of RBI (i.e., S  R and R’  S’) if and only if the
following holds:
w2 u( x )  u(x) w

 3
w1 u(y)  u( y ) w
2
where w1 and w2 are the weights of the highest and middle branches, respectively (when
both have probability p in Gambles S and R), and w2 and w3 are the weights of the
middle and lowest branches with probability p in Gambles S  and R, respectively. Both
RAM and TAX models imply this type of violation, SR . With their prior parameters,
the weight ratios are 2/1 > 3/2 in both RAM and TAX.
CPT also systematically violates RBI; however, CPT with its inverse-S weighting
function violates it in the opposite way from that of RAM and TAX. Define a weakly
inverse-S function as one satisfying the following, for all p < p*:
W(2 p)  W(p)  W( p)  W(0) and W(1  p)  W(1 2 p)  W(1)  W(1 p) . Such a
function is illustrated in Figure 3. Therefore, w2  w1 and w2  w3. It follows that
w2
w
1  3 .
w1
w2
Case Against Prospect Theories 59
Therefore, CPT with its inverse-S weighting function implies that violations of
restricted branch independence in this situation, if they are observed, should have the
opposite ordering; that is, S  R and R’  S’, called the RS  pattern of violation of RBI.
[A strongly inverse-S function is one that is weakly inverse- S and also satisfies the
following: W(p)  p p  p * and W(p)  p p  p *. It should be clear that if we can
reject the weakly inverse- S, then we can reject the stronger form.]
Insert Figure 3 about here.
Consider the choices in Problem 14 (Birnbaum & Chavez, 1997). Significantly
more than half choose S
R , and significantly more than half choose R S . Thus,
violations are significant, but opposite the predictions of the inverse-S weighting
function.
Case Against Prospect Theories 60
Problem 14.
No.
14.1
14.2
Gamble S
Gamble R
%R
S
R
S:
R:
40
20.0 >
19.2
62
57.2
< 60.7
.50 to win $5
.50 to win $5
.25 to win $40
.25 to win $10
.25 to win $44
.25 to win $98
S’: .25 to win $40
R’: .25 to win $10
.25 to win $44
.25 to win $98
.50 to win $111
.50 to win $111
When violations are observed, there are more of the form SR  than the opposite,
(Birnbaum, 1999b; 2001; Birnbaum & McIntosh, 1996; Birnbaum & Navarrete, 1998;
Weber & Kirsner, 1997). This pattern of violations is opposite that predicted by the
inverse-S weighting function used in CPT to account for the Allais paradoxes. This same
pattern of violations has been observed to be more frequent when p = 1/3 (Birnbaum &
McIntosh, 1996), p = .25 (Birnbaum & Chavez, 1997; Birnbaum & Navarrete, 1998), p =
.2 (Birnbaum, et al., 1999), p = .1 (Birnbaum, 1999b; 2004b; submitted-a,b, c, d;
Birnbaum & Navarrete, 1998), and p = .05 (Birnbaum, 1999b; 2004b; submitted-b, d). In
addition, the same pattern as been found to be more frequent in judged buying and selling
prices of gambles (Birnbaum & Beeghley, 1997; Birnbaum & Veira, 1998; Birnbaum &
Zimmermann, 1998). There have been 31 studies with 7774 participants, the results of
which indicate that violations of branch independence are more often in opposition to
predictions of CPT than in agreement with them. Birnbaum (2004b) analyzed replicated
choices to show that these violations cannot be attributed to random “error.”
Case Against Prospect Theories 61
The results of tests of RBI do not appear to depend strongly on how gambles are
presented, financial incentives, positive or negative consequences, or event-framing
(Birnbaum, 2004b; submitted-b, d).
15. 4-Distribution Independence
4-Distribution independence is an interesting property because RDU/RSDU/CPT
models imply systematic violations, but the property should be satisfied, according to the
RAM model. It is violated, according to TAX, but in the opposite way from that
predicted by the inverse-S weighting function of CPT. Consider Problem 15:
No.
Gamble S
Gamble R
15.1
S:
R:
15.2
.59 to win $4
.59 to win $4
.20 to win $45
.20 to win $11
.20 to win $45
.20 to win $97
.01 to win $110
.01 to win $110
S’: .01 to win $4
R’: .01 to win $4
.20 to win $45
.20 to win $11
.20 to win $45
.20 to win $97
.59 to win $110
.59 to win $110
%R
S
R
34
21.7>
20.6
51
49.8
<50.0
Note that the trade-off between two branches of .2 is nested within a distribution
in which these two weights are either near the low end of decumulative probability or
near the upper end. According to CPT, this distribution should change the relative sizes
of the weights of these common branches, producing violations.
According to CPT model and parameters of Tversky and Kahneman (1992),
people should prefer R to S and S’ to R’. According to the RAM model there should be
no violations of this property. According to SWU and PT with cancellation, there should
Case Against Prospect Theories 62
be no violations. According to the TAX model and parameters of Birnbaum (1999a),
people should choose S over R and R’ over S’. Indeed, this pattern of reversal was more
frequent than the opposite as observed by Birnbaum and Chavez (1997) in a study with
100 participants and 12 tests. In Problems 15.1 and 15.2 there were 23 who chose S over
R and R’ over S’ against only 6 who showed the opposite reversal of preferences.
16. 3-Lower Distribution Independence
Let G = (x, p; y; q; z, 1 – p – q) represent a gamble to win x with probability p, y
with probability q, and z otherwise; x > y > z > 0. Define 3- Lower Distribution
Independence (3-LDI) as follows:
S  (x, p; y, p; z,1 2 p)
R  ( x, p; y, p;z,1  2p)
 S2  (x, p ;y, p;z,1  2 p )
R2  ( x, p; y, p;z,1  2 p )
Note that the first comparison involves a distribution with probability p to win x
or y, whereas the second involves a different probability, p’.
Consider Problem 16:
16.1
16.2
S
R
60 blue to win $2
60 green to win $2
20 red to win $56
20 black to win $4
20 white to win $58
20 purple to win $96
10 black to win $2
10 white to win $2
45 green to win $56
45 red to win $4
45 purple to win $58
45 blue to win $96
n = 503
Prior TAX Model
23.6*
21.7
13.8
18.7*
36.9
22.9
According to 3-LDI, people should choose S over R if and only if they choose S2 over
R2, apart from error. The prior CPT model predicts that people should violate this
Case Against Prospect Theories 63
property by choosing R over S and S2 over R2, whereas prior TAX and RAM predict that
people should choose both S and S2.
We can again apply the generic model to the choice between S and R as follows:
S
R  w1u(x)  w2u(y)  w3u(z)  w1u(x' )  w2u(y' )  w3u(z)
Where w1,w2 and w3 are the weights of the highest, middle, and lowest ranked branches,
respectively which depend on the value of p (differently in different models). Subtracting
the term w3u(z) from both sides yields,
S
R 
w2 u(x )  u(x)
.

w1 u(y)  u(y )
Suppose there is a violation of 3-LDI, in which R2  S2. By a similar derivation, this
means,
S2
R2 
u( x )  u(x) w2

u(y)  u(y ) w1
where the (primed) weights now depend on the new level of probability, p . Therefore,
there can be a preference reversal from S  R to S2  R2 if and only if the ratio of weights
changes as a function of probability and “straddles” the ratio of differences in utility, as
follows:
w2 u( x )  u(x) w

 2
w1 u(y)  u( y ) w1
A reversal from S  R to S2  R2 can occur with the opposite ordering. But if the ratio of
weights is independent of p or p  (e.g., as in EU), then there can be no violations of this
property.
According to the RAM model, this ratio of weights is given by the following:
Case Against Prospect Theories 64
w2 a(2,3)s( p) a(2,3)s( p) w


 2
w1 a(1,3)s( p) a(1,3)s( p) w1
Therefore, RAM satisfies 3- LDI. With the further assumption that branch rank weights
equal their objective ranks, this ratio will be 2/1, independent of the value of p (or p’).
According to the special TAX model, this ratio of weights can be written as
follows:
 t(p) )  ( t(p) )
w2 t( p)  (
t( p)
w 
4
4 

 2
2
w1
t(p)  ( 4) t( p)
t( p)[1 2 4] w1
In addition, if  = 1, as in the prior model, then this ratio will be 2/1, as in the case of
RAM. Therefore, special TAX, like the general RAM model, implies 3-LDI.
According to CPT, there should be violations of 3-LDI, if the decumulative
weighting function, W(P), is nonlinear. The relation among the weights will be as
follows:
w2 W(2p)  W( p) W(2 p)  W( p) w2



w1
W( p)
W( p )
w1
For the model and parameters of Tversky and Kahneman (1992), the ratios of weights for
p = .2 and p’ = .45 are .1093/.2608 = 0.4191 and .3165/.3952= 0.80, respectively. Note
that the inverse-S weighting function of CPT implies that both of these ratios are less
than 1, but they differ by a factor of almost two. The middle branch has less weight than
the highest branch of the same probability and that the ratios are not independent of p. In
contrast, both prior RAM and TAX hold that these ratios are equal and greater than 1; in
particular, the middle branch has twice the weight of the highest.
Case Against Prospect Theories 65
In these tests, unlike previous ones, CPT could go on the offense by predicting
violations, leaving TAX and RAM to defend the null hypothesis. Birnbaum (submittedc) conducted three studies with 1578 participants testing these predictions. The results, as
in Problem 16, showed that predictions of TAX were more accurate than CPT. In
particular, prior CPT predicted a reversal Problem 16 that failed to materialize.
17. 3-2 Lower Distribution Independence
The property of 3-2 LDI requires:
S0  (x, 12 ; y, 1 2)
R0  (x , 12 ; y, 1 2)
 S  (x, p; y, p;z,1  2p)
R  ( x, p; y, p; z,1 2 p)
By a derivation similar to that used to prove lower distribution independence, prior RAM
predicts the same choices between S0 and R0 as between S and R. This conclusion
follows when a(2,2)/ a(1,2)  a(2,3)/ a(1,3) , as it does in the prior RAM model, where
this ratio is 2/1 in both cases.
Prior TAX also implies that people should make the same choices in S0 and R0
as between S and R, since the weight ratio in a 50-50, two branch gamble is
w2 t( 12 )   3 t(1 2)

, which also equals 2/1, so prior TAX model also satisfies 3-2 LDI.
w1 t( 1 )   t(1 )
2
3 2
In contrast, the prior CPT model predicts that people should choose S0 over R0
and choose R over S in Problems 17.3 and 17.4 in Table 5. This prediction follows
because the ratio of weights (second to first in the two-branch case) is .58/.47 = 1.38, but
in the three branch case, the weights of third, second, and highest branches are .18, .40,
and .41, so the ratio of the second (middle) branch to the highest has been reduced to only
0.99. Thus, CPT violates 3-2 LDI.
Case Against Prospect Theories 66
The data in Table 1 are based on 1075 participants in Birnbaum (submitted-c,
Exp. 1). As shown in the Table, CPT makes the wrong prediction in three of the four
rows, so CPT again predicts an effect that failed to materialize.
18. 3-Upper Distribution Independence
The property, 3-UDI, can be written as follows:
S  ( z,1  2p; x, p; y, p)
 S2  ( z,1  2 p ;x, p; y, p)
R  ( z,1 2 p;x, p; y, p)
R2  ( z,1 2 p; x, p; y, p)
For example, consider Problem 18:
Choice
S’:
.10 to win $40
R’:
%R
.10 to win $4
.10 to win $44
.10 to win $96
.80 to win $100
.80 to win $100
S2’: .45 to win $40
R2’: .45 to win $4
.45 to win $44
.45 to win $96
.10 to win $100
.10 to win $100
Prior TAX
56.0
61.6
63.4
33.2
45.9
43.9
This property will be violated, with S’  R’ and S2’  R2’ if and only if
w3 u( x )  u(x) w

 3
w2 u(y)  u( y ) w2
The opposite reversal of preference would occur when the order above is reversed.
The RAM model implies that
w3 a(3,3)s( p) a(3,3)s( p ) w3



w2 a(2,3)s( p) a(2,3)s( p) w
2
Therefore, the RAM model implies no violations of 3-UDI.
In the special TAX model, however, this ratio of weights is as follows:
w3 t( p)  ( 4)t(p)  ( 4)t(1  2p) w3


w2 t( p)  ( )t(p)  ( )t(1  2p) w
2
4
4
Case Against Prospect Theories 67
which shows that this weight ratio is not independent of p. For prior TAX, this weight
ratio increases from 1.27 to 1.60, which straddles the ratio of differences (96-44)/(40-4) =
1.44; therefore, this model predicts that S’  R’ and S2’  R2’. The prior TAX implies
this pattern (R’S2’) of violation of 3-UDI.
CPT allows violations of 3-UDI because the ratios of weights are as follows:
w3
1 W(1  p)
1  W(1  p)
w 


 3
w2 W(1 p)  W(1  2p) W(1 p)  W(1 2 p) w
2
In particular, for the inverse-S weighting function of Tversky and Kahneman (1992), this
ratio of weights increases and then decreases, with a net decrease from 1.99 to 1.71.
Therefore, the prior model implies violations of the opposite pattern from that predicted
by TAX, but CPT predicts no violation in Problem 18, given its prior parameters:
Birnbaum (submitted-c) found that significantly more than half of the 1075
participants in Experiment 1 chose R’ over S’ and significantly more than half chose S2’
over R2’. This result violates RAM, was not predicted by CPT, and was correctly
predicted by TAX. Other cases where prior CPT was predicted to show violations of 3LDI and 3-UDI were tested in Experiment 2 of Birnbaum (submitted-c) and failed to
materialize in the modal choices.
6. SUMMARY AND DISCUSSION
The violations of prospect theories are summarized in Tables 6 and 7. Table 6
contains properties that should be satisfied according to CPT and should be violated by
prior RAM and TAX. Violations of these properties cannot be explained by any version
of RSDU/CPT with any weighting and utility functions. They require a revision in the
theory.
Case Against Prospect Theories 68
Table 7 lists properties that should be violated according to CPT, but where the
data either fail to confirm predicted violations or show the opposite pattern of violations
predicted by the inverse-S weighting function. All of the models except EU, SWU, and
stripped PT violate restricted branch independence. With the principle of cancellation,
either PT or CPT would imply no violations. RAM, TAX, and GDU imply the SR’
pattern of violation that is observed in a number of empirical studies, but which
contradicts the inverse-S weighting function.
RAM satisfies all of the distribution independence properties, so violations of 4DI and 3-Upper DI refute the RAM model. Prior TAX and GDU satisfy 3- Lower DI and
violate 3-Upper DI, consistent with the data. CPT violates all of the properties, but its
predicted violations either failed to materialize or were refuted by patterns in opposition
to its predictions.
One important property not listed in the tables is branch-splitting independence.
Since CPT cannot violate coalescing, this property is moot under CPT. TAX and RAM
imply violations of this property such that splitting the higher branch should improve a
gamble and splitting the lower branch should make it worse. GDU implies upper
coalescing, so splitting the upper branch should have no effect. Original prospect theory,
stripped of its editing principles, implies violations of coalescing and satisfaction of
branch-splitting independence. Data in Table 4 show violations of upper coalescing,
contrary to GDU. They also show a significant violation of BSI, consistent with TAX
and RAM.
Considering all of the evidence, there is a strong case against CPT. There is also
strong evidence against the editing principles of prospect theory of cancellation (which
Case Against Prospect Theories 69
implies branch independence and distribution independence) and combination (which
implies coalescing). If there is a dominance detector, it fails to work effectively for cases
predicted to fail according to RAM and TAX.
Two models are partially consistent with the data, RAM and GDU. Despite
considerable success, prior RAM cannot describe violations of 4-distribution
independence and 3-Upper distribution independence. Therefore, the special TAX model
is a better description than RAM, since it succeeds in these two cases where RAM fails.
GDU gives a fairly good account of results except for those based purely on upper
coalescing. This means GDU cannot account for violations of Wu’s upper tail
independence, nor can it account for separate tests of upper coalescing in Table 4. There
was a rounding error in Luce (2000, p.200-202) that made it seem capable of accounting
for this effect. GDU can, however, account for violations of stochastic dominance, lower
and upper cumulative independence, and the pattern of violations of branch independence
and distribution independence. Luce is currently exploring Gains Decomposition without
the assumption of binary RDU, which provide a more accurate description as well as an
axiomatic representation.
The model that best describes the evidence reviewed is prior TAX, which
correctly predicted all of the modal choices reviewed here, except for the modal choice in
Table 4, Choice 12.7. It is worth repeating that prior TAX was calculated based on
previous data that did not involve tests of lower cumulative independence, upper
cumulative independence, stochastic dominance, 3-lower DI, 3-2 lower DI, and 3-upper
DI. Therefore, its predictions for these six properties and for the breakdowns in Tables 2
and 4 (analysis of Allais paradoxes) were indeed calculated in advance of the collection
Case Against Prospect Theories 70
of data. Thus, the model’s predictions in these cases were true predictions and not
calculations based on post hoc fitting to data in hand.
In my opinion, the empirical evidence now exceeds the critical threshold where
prospect theories can be relegated to join EU in the repository of failed descriptive
theories. The best description of empirical results is TAX, followed closely by RAM and
GDU.
Questions:
1. Please suggest your works that should be cited, and where I should best cite your
related work.
2. Do you know of a situation where CPT successfully predicts in a case where RAM,
TAX, GDU fail?
3. Can one modify CPT to “save the phenomena” without making an implausibly
complicated formulation?
4. Is there any case where CPT has been successful in predicting choices among threebranch gambles?
5. GDU is a theory that is similar to TAX, but which has been axiomatized, unlike TAX.
This theory is more accurate than CPT. Is there any reason to favor CPT over GDU?
6. This draft has focused on the evidence from my own lab. It would be helpful to me to
point out important findings and results that have not been reviewed, with suggestions
as to where they best fit in this paper.
7. I would like to conduct a long-term experiment where intelligent people study
decision making for at least a semester, and continue to perform these tasks. What
would their decisions look like as the semester progresses? I suspect that after one
Case Against Prospect Theories 71
week, they would adopt EV; however, I suspect that after three weeks, they will
abandon EV, but what would the ending policy be? EU? If we can teach people how
to coalesce and split, what model would fit their choices?
8. Should a longer section be added to review work on judgments of buying, selling,
neutral prices, and of judged certainty equivalents? This would bring in much more
data and theory, but certainly adds to the case against CPT.
9. Should a section be added on the “constant ratio” paradox of Allais? Constant ratio
hypothesis can be tested in binary gambles, where CPT and TAX give similar
predictions.
Choice
A: .8 win $4000
KT 79
N=95
Prior TAX
B: $3000 for sure
80
1934
3000
C: .2 to win $4000
D: .25 win $3000
35
733
633
.8 to win $0
.75 win $0
.2 win $0
This effect was replicated with a number of variations in Birnbaum (2001). Are there
data that test the detailed predictions of inverse-S for this paradox?
10. Should there be a section on the “reflection” effect? I find that purely positive
gambles (those studied in Birnbaum, 1999b) show good fit to strong reflection, which
can be expressed as follows:
P(A, B) = 1 – P(-A, -B),
where P(A, B) is the probability of choosing Gamble A over B, gambles with
strictly positive consequences, and where –A and –B are gambles in which positive
consequences are converted to negative ones. Reflection CANNOT hold for all gambles
Case Against Prospect Theories 72
(Birnbaum, submitted-b). However, it can hold for gambles with strictly positive
consequences.
Case Against Prospect Theories 73
References
Abdellaoui, M. (2000). Parameter-free elicitation of utility and probability weighting
functions. Management Science, 46, 1497-1512.
Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: Critique des
postulats et axiomes de l'école Américaine. Econometrica, 21, 503-546.
Allais, M. (1979). The foundations of a positive theory of choice involving risk and a
criticism of the postulates and axioms of the American School. In M. Allais & O.
Hagen (Eds.), Expected utility hypothesis and the Allais paradox (pp. 27-145).
Dordrecht, The Netherlands: Reidel.
Allais, M., & Hagen, O. (Ed.). (1979). Expected utility hypothesis and the Allais
paradox. Dordrecht, The Netherlands: Reidel.
Baltussen, G., Post, T., & van Vliet, P. (2004). Violations of CPT in mixed gambles.
Working Paper, July, 2004., Available from Pim van Vliet, Erasmus University
Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands.
Baucells, M., & Heukamp, F. H. (2004). Stochastic dominance and cumulative prospect
theory. Working paper, dated June, 2004., Available from Manel Baucells, IESE
Business School, University of Navarra, Barcelona, SPAIN.
Bernoulli, D. (b. L. S. (1954). Exposition of a new theory on the measurement of risk.
Econometrica, 22, 23-36. (Translation of Bernoulli, D. (1738). Specimen theoriae
novae de mensura sortis. Commentarii Academiae Scientiarum Imperialis
Petropoliannae, 5, 175-192.)
Birnbaum, M. H. (1974). The nonadditivity of personality impressions. Journal of
Experimental Psychology, 102, 543-561.
Case Against Prospect Theories 74
Birnbaum, M. H. (1992). Violations of monotonicity and contextual effects in choicebased certainty equivalents. Psychological Science, 3, 310-314.
Birnbaum, M. H. (1999a). Paradoxes of Allais, stochastic dominance, and decision
weights. In J. Shanteau, B. A. Mellers, & D. A. Schum (Eds.), Decision science
and technology: Reflections on the contributions of Ward Edwards (pp. 27-52).
Norwell, MA: Kluwer Academic Publishers.
Birnbaum, M. H. (1999b). Testing critical properties of decision making on the Internet.
Psychological Science, 10, 399-407.
Birnbaum, M. H. (2000). Decision making in the lab and on the Web. In M. H. Birnbaum
(Eds.), Psychological Experiments on the Internet (pp. 3-34). San Diego, CA:
Academic Press.
Birnbaum, M. H. (2001). A Web-based program of research on decision making. In U.-D.
Reips & M. Bosnjak (Eds.), Dimensions of Internet science (pp. 23-55).
Lengerich, Germany: Pabst.
Birnbaum, M. H. (2004a). Causes of Allais common consequence paradoxes: An
experimental dissection. Journal of Mathematical Psychology, 48(2), 87-106.
Birnbaum, M. H. (2004b). Tests of rank-dependent utility and cumulative prospect theory
in gambles represented by natural frequencies: Effects of format, event framing,
and branch splitting. Organizational Behavior and Human Decision Processes,
xxx, in press.
Items listed here as “in press” or “submitted” can be downloaded from the following
URL:
http://psych.fullerton.edu/mbirnbaum/birnbaum.htm#inpress
Case Against Prospect Theories 75
Birnbaum, M. H. (submitted-a). Tests of Branch Splitting and Branch-Splitting
Independence in Allais Paradoxes. Submitted for publication.
Birnbaum, M. H. (submitted-b). Tests of prospect theories in gambles displayed in matrix
and text formats with positive and negative consequences. Submitted for
publication.
Birnbaum, M. H. (submitted-c). Three new tests among models of risky decision making:
3-Upper and lower distribution independence and 3-2 Distribution Independence.
(tentatively accepted, pending revision).
Birnbaum, M. H. (submitted-d). Violations of first order stochastic dominance in risky
decision making. Submitted for publication.
Birnbaum, M. H., & Beeghley, D. (1997). Violations of branch independence in
judgments of the value of gambles. Psychological Science, 8, 87-94.
Birnbaum, M. H., & Chavez, A. (1997). Tests of theories of decision making: Violations
of branch independence and distribution independence. Organizational Behavior
and Human Decision Processes, 71(2), 161-194.
Birnbaum, M. H., & Martin, T. (2003). Generalization across people, procedures, and
predictions: Violations of stochastic dominance and coalescing. In S. L. Schneider
& J. Shanteau (Eds.), Emerging perspectives on decision research (pp. 84-107).
New York: Cambridge University Press.
Birnbaum, M. H., & McIntosh, W. R. (1996). Violations of branch independence in
choices between gambles. Organizational Behavior and Human Decision
Processes, 67, 91-110.
Case Against Prospect Theories 76
Birnbaum, M. H., & Navarrete, J. B. (1998). Testing descriptive utility theories:
Violations of stochastic dominance and cumulative independence. Journal of Risk
and Uncertainty, 17, 49-78.
Birnbaum, M. H., Patton, J. N., & Lott, M. K. (1999). Evidence against rank-dependent
utility theories: Violations of cumulative independence, interval independence,
stochastic dominance, and transitivity. Organizational Behavior and Human
Decision Processes, 77, 44-83.
Birnbaum, M. H., & Thompson, L. A. (1996). Violations of monotonicity in choices
between gambles and certain cash. American Journal of Psychology, 109, 501523.
Birnbaum, M. H., & Yeary, S. (submitted). Tests of stochastic dominance and cumulative
independence in buying and selling prices of gambles. Submitted for Publication.
Birnbaum, M. H., & Zimmermann, J. M. (1998). Buying and selling prices of
investments: Configural weight model of interactions predicts violations of joint
independence. Organizational Behavior and Human Decision Processes, 74(2),
145-187.
Bleichrodt, H., & Luis, P. J. (2000). A parameter-free elicitation of the probability
weighting function in medical decision analysis. Management Science, 46, 14851496.
Camerer, C. F. (1989). An experimental test of several generalized utility theories.
Journal of Risk and Uncertainty, 2, 61-104.
Case Against Prospect Theories 77
Camerer, C. F. (1992). Recent tests of generalizations of expected utility theory. In W.
Edwards (Eds.), Utility theories: Measurements and applications (pp. 207-251).
Boston: Kluwer Academic Publishers.
Camerer, C. F. (1998). Bounded rationality in individual decision making. Experimental
Economics, 1, 163-183.
Camerer, C. F., & Hogarth, R. M. (1999). The effects of financial incentives in
experiments: A review and capital-labor-production theory. Journal of Risk and
Uncertainty, 19, 7-42.
Carbone, E., & Hey, J. D. (2000). Which error story is best? Journal of Risk and
Uncertainty, 20(2), 161-176.
Diecidue, E., & Wakker, P. P. (2001). On the intuition of rank-dependent utility. Journal
of Risk and Uncertainty, 23, 281-298.
Green, J. R., & Jullien, B. (1988). Ordinal Independence in Non-Linear Utility Theory.
Journal of Risk and Uncertainty, 1, 355-387.
Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function.
Cognitive Psychology, 38, 129-166.
Gonzalez, R., & Wu, G. (2003). Composition rules in original and cumulative prospect
theory. Working Manuscript dated 8-14-03.
Harless, D. W., & Camerer, C. F. (1994). The predictive utility of generalized expected
utiliity theories. Econometrica, 62, 1251-1290.
Humphrey, S. J. (1995). Regret aversion or event-splitting effects? More evidence under
risk and uncertainty. Journal of risk and uncertainty, 11, 263-274.
Case Against Prospect Theories 78
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under
risk. Econometrica, 47, 263-291.
Karni, E., & Safra, Z. (1987). "Preference reversal" and the observability of preferences
by experimental methods. Econometrica, 55, 675-685.
Leland, J., W. (1994). Generalized similarity judgments: An alternative explanation for
choice anomalies. Journal of Risk and Uncertainty, 9, 151-172.
Levy, M., & Levy, H. (2002). Prospect Theory: Much ado about nothing. Management
Science, 48, 1334–1349.
Lopes, L. L. (1990). Re-modeling risk aversion: A comparison of Bernoullian and rank
dependent value approaches. In G. M. v. Furstenberg (Eds.), Acting under
uncertainty (pp. 267-299). Boston: Kluwer.
Lopes, L. L., & Oden, G. C. (1999). The role of aspiration level in risky choice: A
comparison of cumulative prospect theory and SP/A theory. Journal of
Mathematical Psychology, 43, 286-313.
Luce, R. D. (1990). Rational versus plausible accounting equivalences in preference
judgments. Psychological Science, 1, 225-234.
Luce, R. D. (1998). Coalescing, event commutativity, and theories of utility. Journal of
Risk and Uncertainty, 16, 87-113.
Luce, R. D. (2000). Utility of gains and losses: Measurement-theoretical and
experimental approaches. Mahwah, NJ: Lawrence Erlbaum Associates.
Luce, R. D., & Fishburn, P. C. (1991). Rank- and sign-dependent linear utility models for
finite first order gambles. Journal of Risk and Uncertainty, 4, 29-59.
Case Against Prospect Theories 79
Luce, R. D., & Fishburn, P. C. (1995). A note on deriving rank-dependent utility using
additive joint receipts. Journal of Risk and Uncertainty, 11, 5-16.
Luce, R. D., & Narens, L. (1985). Classification of concatenation measurement structures
according to scale type. Journal of Mathematical Psychology, 29, 1-72.
Machina, M. J. (1982). Expected utility analysis without the independence axiom.
Econometrica, 50, 277-323.
Markowitz, H. (1952). The utility of wealth. Journal of Political Economy, 60, 151-158.
Marley, A. A. J., & Luce, R. D. (2001). Rank-weighted utilities and qualitative
convolution. Journal of Risk and Uncertainty, 23(2), 135-163.
Marley, A. A. J., & Luce, R. D. (2004). Independence properties vis-à-vis several utility
representations. Draft Manuscript, March 3, 2004.
Prelec, D. (1998). The probability weighting function. Econometrica, 66, 497-527.
Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior and
Organization, 3, 324-345.
Quiggin, J. (1985). Subjective utility, anticipated utility, and the Allais paradox.
Organizational Behavior and Human Decision Processes, 35, 94-101.
Quiggin, J. (1993). Generalized expected utility theory: The rank-dependent model.
Boston: Kluwer.
Savage, L. J. (1954). The foundations of statistics. New York: Wiley.
Schmeidler, D. (1989). Subjective probability and expected utility without additivity.
Econometrica, 57, 571-587.
Starmer, C. (1992). Testing new theories of choice under uncertainty using the common
consequence effect. Review of Economic Studies, 59, 813-830.
Case Against Prospect Theories 80
Starmer, C. (1999). Cycling with rules of thumb: An experimental test for a new form of
non-transitive behaviour. Theory and Decision, 46, 141-158.
Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a
descriptive theory of choice under risk. Journal of Economic Literature, 38, 332382.
Starmer, C., & Sugden, R. (1989). Violations of the independence axiom in common
ratio problems: An experimental test of some competing hypotheses. Annals of
Operations Research, 19, 79-101.
Starmer, C., & Sugden, R. (1993). Testing for juxtaposition and event-splitting effects.
Journal of Risk and Uncertainty, 6, 235-254.
Stevenson, M. K., Busemeyer, J. R., & Naylor, J. C. (1991). Judgment and decisionmaking theory. In M. Dunnette & L. M. Hough (Eds.), New handbook of
industrial-organizational psychology (pp. 283-374). Palo Alto, CA: Consulting
Psychologist Press.
Tversky, A., & Fox, C. R. (1995). Weighing Risk and Uncertainty. Psychological
Review, 102(2), 269-283.
Tversky, A., & Kahneman, D. (1986). Rational choice and the framing of decisions.
Journal of Business, 59, S251-S278.
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative
representation of uncertainty. Journal of Risk and Uncertainty, 5, 297-323.
Tversky, A., & Wakker, P. (1995). Risk attitudes and decision weights. Econometrica,
63, 1255-1280.
Case Against Prospect Theories 81
Viscusi, K. W. (1989). Prospective reference theory: Toward an explanation of the
paradoxes. Journal of risk and uncertainty, 2, 235-264.
von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior
(2nd ed.). Princeton: Princeton University Press.
von Winterfeldt, D. (1997). Empirical tests of Luce's rank- and sign-dependent utility
theory. In A. A. J. Marley (Eds.), Choice, decision, and measurement: Essays in
honor of R. Duncan Luce (pp. 25-44). Mahwah, NJ: Erlbaum.
Wakker, P. (1994). Separating marginal utility and probabilistic risk aversion. Theory
and decision, 36, 1-44.
Wakker, P. (1996). The sure-thing principle and the comonotonic sure-thing principle:
An axiomatic analysis. Journal of Mathematical Economics, 25, 213-227.
Wakker, P. P., & Tversky, A. (1993). An axiomatization of cumulative prospect theory.
Journal of Risk and Uncertainty, 7, 147-176.
Wakker, P., Erev, I., & Weber, E. U. (1994). Comonotonic independence: The critical
test between classical and rank-dependent utility theories. Journal of Risk and
Uncertainty, 9, 195-230.
Weber, E. U., & Kirsner, B. (1997). Reasons for rank-dependent utility evaluation.
Journal of Risk and Uncertainty, 14, 41-61.
Wu, G. (1994). An empirical test of ordinal independence. Journal of Risk and
Uncertainty, 9, 39-60.
Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighting function.
Management Science, 42, 1676-1690.
Case Against Prospect Theories 82
Wu, G., & Gonzalez, R. (1998). Common consequence conditions in decision making
under risk. Journal of Risk and Uncertainty, 16, 115-139.
Wu, G., & Gonzalez, R. (1999). Nonlinear decision weights in choice under uncertainty.
Management Science, 45, 74-85.
Wu, G. (in press). Testing prospect theories using tradeoff consistency. Journal of Risk
and Uncertainty, 00, 000-000.
Wu, G., & Markle, A. B. (2004). An empirical test of gain-loss separability in prospect
theory. Working Manuscript, 06-25-04.
Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55, 95-115.
Case Against Prospect Theories 83
Table 1. Comparison of Decision Theories
Branch Independence
Coalescing
Satisfied
Violated
Satisfied
EU (PT*/CPT*)
RDU/RSDU/CPT*
Violated
SWU/PT*
RAM/TAX/GDU
Notes: EU = Expected Utility Theory; PT= Original Prospect Theory and CPT =
Cumulative Prospect Theory. Prospect theories make different predictions with
and without their editing rules. The editing rule of combination implies
coalescing and cancellation implies branch independence. With or without the
editing rule of combination, CPT satisfies coalescing. The Rank Affected
Multiplicative (RAM) and Transfer of Attention Exchange (TAX) models violate
both branch independence and coalescing. GDU = Gains decomposition utility.
Case Against Prospect Theories 84
Table 2. Dissection of Allais Paradox into Branch Independence and Coalescing (Series
A).
Problem
Framed Version
Condition
Prior
No.
10.1
10.2
10.3
10.4
10.5
TAX model
First Gamble
Second Gamble
R
S
10 black marbles to win $98
20 black marbles to win $40
90 purple marbles to win $2
80 purple marbles to win $2
10 red marbles to win $98
10 red marbles to win $40
10 blue marbles to win $2
10 blue marbles to win $40
80 white marbles to win $2
80 white marbles to win $2
10 red marbles to win $98
10 red marbles to win $40
80 blue marbles to win $40
80 blue marbles to win $40
10 white marbles to win $2
10 white marbles to win $40
80 red marbles to win $98
80 red marbles to win $98
10 blue marbles to win $98
10 blue marbles to win $40
10 white marbles to win $2
10 white marbles to win $40
90 red marbles to win $98
80 red marbles to win $98
10 white marbles to win $2
20 white marbles to win $40
N = 349
R
S
0.378
13.3
9.0
0.644
9.6
11.1
0.537
30.6
40.0
0.430
62.6
59.8
0.777
54.7
68.0
Case Against Prospect Theories 85
Table 3. Numbers of Participants showing each choice combination in tests of Event
Splitting.
Choice Pattern
Series A
Series B
Observed
Fitted
Observed
Fitted
RR’RR’
31
29.24
19
18.65
RR’RS’
17
17.89
11
15.24
RR’SR’
4
6.72
5
3.90
RR’SS’
7
4.51
6
3.86
RS’RR’
25
17.89
18
15.24
RS’RS’
56
53.88
62
60.48
RS’SR’
3
4.51
7
3.86
RS’SS’
9
14.21
17
16.17
SR’RR’
3
6.72
3
3.90
SR’RS’
3
4.51
2
3.86
SR’SR’
4
1.98
1
1.59
SR’SS’
1
3.04
1
4.84
SS’RR’
5
4.51
3
3.86
SS’RS’
14
14.21
15
16.17
SS’SR’
3
3.04
2
4.84
SS’SS’
14
12.14
28
23.52
Notes: S = Chose “safe” gamble with higher probability to win smaller prize, R = chose
“risky” gamble, S, S’ and R’ represent the same gambles in split form, respectively. The
first two letters indicate the choice pattern on the first replicate, respectively, and the
second two indicate the choice pattern on the second replication.
Case Against Prospect Theories 86
TABLE 4. Dissection of Allais Paradox for tests of Branch Independence, Coalescing
and Event-Splitting Independence (Series B).
Choice
Choice
Prior
TAX model
No.
12.1
12.2
First Gamble
Second Gamble
S
R
15 red marbles to win $50
10 blue marbles to win $100
85 black marbles to win $7
90 white marbles to win $7
15 red marbles to win $50
10 black marbles to win $100
85 black marbles to win $7
05 purple marbles to win $7
FU
UF S
R
0.783
13.6
18.0
0.704
13.6
14.6
0.517
15.6
18.0
0.443
15.6
14.6
0.695
68.4
69.7
0.367
68.4
62.0
0.702
75.7
69.7
0.249
75.7
62.0
85 green marbles to win $7
12.3
10 red marbles to win $50
10 blue marbles to win $100
05 blue marbles to win $50
90 white marbles to win $7
85 white marbles to win $7
12.4
12.5
12.6
10 red marbles to win $50
10 black marbles to win $100
05 blue marbles to win $50
05 purple marbles to win $7
85 white marbles to win $7
85 green marbles to win $7
85 red marbles to win $100
85 black marbles to win $100
10 white marbles to win $50
10 yellow marbles to win $100
05 blue marbles to win $50
05 purple marbles to win $7
85 red marbles to win $100
95 red marbles to win $100
10 white marbles to win $50
05 white marbles to win $7
05 blue marbles to win $50
12.7
85 black marbles to win $100
85 black marbles to win $100
15 yellow marbles to win $50
10 yellow marbles to win $100
05 purple marbles to win $7
12.8
85 black marbles to win $100
95 red marbles to win $100
15 yellow marbles to win $50
05 white marbles to win $7
Case Against Prospect Theories 87
Table 5. Tests of 3-Lower Distribution Independence and 3-2 Lower Distribution Independence (From Birnbaum, submitted-c, Exp.
1).
No.
S
17.1
17.2
17.3
17.4
%R
Choice
R
.80 to win $2
.80 to win $2
.10 to win $40
.10 to win $4
.10 to win $44
.10 to win $96
.10 to win $2
.10 to win $2
.45 to win $40
.45 to win $4
.45 to win $44
.45 to win $96
.04 to win $2
.04 to win $2
.48 to win $40
.48 to win $4
.48 to win $44
.48 to win $96
.50 to win $40
.50 to win $4
.50 to win $44
.50 to win $96
Prior TAX
n = 1075
S
Prior CPT
R
S
R
42.4
11.4
9.79
10.4
14.5
30.2
27.1
22.9
30.5
35.9
34.0
29.1
24.5
34.7
37.7
31.4
41.3
34.7
41.7
39.3
Case Against Prospect Theories 88
Table 6. Summary of 6 New Paradoxes that Refute Cumulative Prospect Theory
Property Name
Coalescing & Transitivity
Expression
A  (x, p;z,1  p)
Implications
B  (y,q; z,1  q)
 A (x, p  r;x,r;z,1  p)
B (y,q; z,s; z,1 q  s)
Violate
SS93, H95, B99b, B00, B01, BM03,
RDU,RSDU/CPT
B04a; B04b;
TK86; BN98; B99b; BPL99; B00; B01
Stochastic Dominance
P(x  t | G )  P(x  t | G )t  G
Upper Tail Independence
(x, p; y, q;z,1 p  q) (x, p; y, q; z,1  p  q )
Violate GDU
 ( y, p; y,q;z,1  p  q) ( y, p  q; z,1 p  q )
RDU/RSDU/CPT*
Lower Cumulative
Independence
Upper Cumulative
Independence
Gain-Loss Separability
S  (x, p; y, q;z,1 p  q)
R  ( x, p; y,q;z,1  p  q)
 S (x, p  q; y,1  p  q)
S  ( z,1  p  q;x, p; y,q)
B and A
B  A
Violate RDU/ /CPT
W94,B01,B04c
Violate RDU/RSDU/ CPT
BN98,B99b,BPL99;B04b;B-s-b;B-s-d;
Violate any model of CPT
BN98,B99b,BPL99;B04b;B-s-b;B-s-d;
Violate any RSDU/CPT
WM04
R ( x, p; y;1 p  q)
R (z,1  p  q; x, p; y,q)
 S (x ,1 p  q; y, p  q)
A
G
References
R ( x,1  q; y, q)
B
Notes: B99b = Birnbaum (1999b); B00 = Birnbaum (2000); B01 = Birnbaum (2001); B04b = Birnbaum (2004b); B-s-a = Birnbaum (submitted-a); B-s-d =
Birnbaum (submitted-d); BC97 = Birnbaum & Chavez (1997); H95 = Humphrey (1995); SS93 = Starmer & Sugden (1993); WM04 = Wu & Markle (2004).
Case Against Prospect Theories 89
Table 7. Summary of 5 Tests that Contradict or Fail to Confirm Predictions of Cumulative Prospect Theory.
Property Name
Branch
Independence
4-DI
Expression
S  (x, p; y, q;z,1 p  q)
R  ( x, p; y,q;z,1  p  q)
 S (z ,1 p  q; x, p; y, q)
R ( z,1 p  q; x, p; y, q)
S  ( z,r;x, p; y, p;z,1  r  2p)
R  ( z,r;x, p; y, p;z,1  r  2p)

S  ( z, r ; x, p; y, p;z,1  r 2p)
3-Lower DI
S  (x, p; y, p; z,1 2 p)
R  ( x, p; y, p;z,1  2p)
 S2  (x, p ;y, p;z,1  2 p )
3-2 Lower DI
S0  (x, 12 ; y, 1 2)
R2  ( x, p; y, p;z,1  2 p )
R0  (x , 12 ; y, 1 2)
 S  (x, p; y, p;z,1  2p)
3-Upper DI
R (z, r ;x, p; y, p;z,1  r 2p)
S  ( z,1  2p; x, p; y, p)
R  ( x, p; y, p; z,1 2 p)
R  ( z,1 2 p;x, p; y, p)
 S2  ( z,1  2 p ;x, p; y, p)
R2  ( z,1 2 p; x, p; y, p)
Empirical Results
Violate inverse-S
References
BMc96,BC97,BN98,B99b;B04a;
B04b;B-s-a;B-s-b;B-s-c;B-s-d
Violate RAM
BC97
inverse-S
Failed to
B-s-c
materialize
Failed to
B-s-c
materialize
Violate RAM;
B-s-c
inverse- S
Notes: B99b = Birnbaum (1999b); B00 = Birnbaum (2000); B01 = Birnbaum (2001); BB97 = Birnbaum & Beeghley (1997); BC97 = Birnbaum & Chavez
(1997); BMcI96 = Birnbaum & McIntosh (1996); BN98 = Birnbaum & Navarrete (1998); BPL99 = Birnbaum, et al. (1999); H95 = Humphrey (1995); SS93 =
Starmer & Sugden (1993); W94 = Wu (1994). The gamble, S = (x, p; y, q; z, r), represents a gamble to win a cash prize of x with probability p, y with probability
q and z with probability r = 1 – p – q. The consequences are ordered such that 0 < z < x' < x < y < y' < z'.
Case Against Prospect Theories 90
Figure 1. Two explanations of risk aversion, nonlinear utility versus weighting. In each
case, the expected value (or utility) is the center of gravity. On the left, gamble G =
($100, .5; $0) is represented as a probability distribution with half of its weight at 0 and
half at 100. The expected value is 50. In the lower left panel, distance along the scale
corresponds to utility, by the function u(x) = x.63. Marginal differences in utility between
$20 increments decrease as one goes up the scale. The balance point on the utility axis
corresponds to $33.3, which is the certainty equivalent of this gamble. The right side of
the figure shows a configural weight theory of the same result: if one third of the weight
of the higher branch is given to the lower branch (to win $0), then the certainty
equivalent would also be $33.3, even when utility is proportional to cash.
Case Against Prospect Theories 91
Figure 2. Cultivating and weeding out violations of stochastic dominance. Starting at the
root, G0 = ($96, .9; $12,.1), split the upper branch to create G– = ($96, .85; $90, .05; $12,
.10), which is dominated by G0. Splitting the lower branch of G0, create G+ = ($96, .9;
$14, .05; $12, .05), which dominates G0. According to configural weight models, G– is
preferred to G+, because splitting increased the relative weight of the higher or lower
branches, respectively. Another round of splitting weeds out violations to low levels.
Figure 3. Inverse-S weighting function. In such a function, the weight of the highest
branch in (x, p; y, p; z, 1 – 2p) is greater than the weight of the middle branch; w1  w2 ;
Case Against Prospect Theories 92
i.e., 1 
w2
. Similarly, the weight of the middle branch in (z’, 1 – 2p; x, p; y, p) is less
w1
than the weight of the lowest branch; therefore, w2  w3 1 
w3
. Together, these two
w2
conditions imply that violations of branch independence will be of the form S
R and
S  R; that is, the RS  pattern of violations. Instead, the opposite pattern is more
common.