Case Against Prospect Theories 1
New Paradoxes of Risky Decision Making
Michael H. Birnbaum
California State University, Fullerton and
Decision Research Center, Fullerton
Date: 03-16-05 a
Filename:BirnbaumReview34.doc
Mailing address:
Prof. Michael H. Birnbaum,
Department of Psychology, CSUF H-830M,
P.O. Box 6846,
Fullerton, CA 92834-6846
Email address: [email protected]
Phone: 714-278-2102 or 714-278-7653
Fax: 714-278-7134
Author's note: Support was received from National Science Foundation Grants, SBR-9410572,
SES 99-86436, and BCS-0129453. Thanks are due to Eduard Brandstaetter, R. Duncan Luce,
and Peter Wakker for their comments on an earlier draft.
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 paradoxes are reviewed to demonstrate 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 consequences on discrete branches. Although they have some
similarities to rank-dependent utility theories, configural weight models do not satisfy
coalescing, the assumption that 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
them 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) became the focus of empirical studies of decision making (Kahneman &
Tversky, 1979). PT was later modified (Tversky & Kahneman, 1992) to assimilate rank and
sign-dependent utility (RSDU). The newer form, cumulative prospect theory (CPT) was able to
describe the classic Allais paradoxes (Allais, 1953; 1979) that were inconsistent with EU. CPT
simplified and extended PT to a wider domain.
CPT also describes the “four-fold pattern” of risk-seeking and risk aversion in the same
person. In this pattern, the typical participant shows risk-seeking for binary gambles with small
probabilities to win large prizes, and risk-aversion for gambles with medium to high probability
to win moderate to small prizes. For gambles with strictly nonpositive consequences, this
pattern is reversed. Such reversal is known as the “reflection” effect. Finally, CPT describes
risk aversion in mixed gambles, also known as “loss aversion,” a tendency to prefer sure things
over mixed gambles with the same or higher expected value.
CPT 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 (Abdellaoui, 2000; Brandstaetter & Kuehberger, 2002; Camerer, 1989; 1992;
1998; Diecidue & Wakker, 2001; Gonzalez & Wu, 1999; Karni & Safra, 1987; Luce, 2000;
2001; Luce & Narens, 1985; Machina, 1982; Prelec, 1998; Quiggin, 1982; 1985; 1993;
Schmeidler, 1989; Starmer & Sugden, 1989; Tversky & Wakker, 1995; Yaari, 1987; von
Case Against Prospect Theories 4
Winterfeldt, 1997; Wakker, 1994; 1996; Wakker, Erev, & Weber, 1994; Weber & Kirsner,
1997; Wu, 1994; Wu & Gonzalez, 1996; 1998; 1999).
However, evidence has been accumulating in recent years that systematically violates
both versions of prospect theory. Authors have criticized CPT (Baltussen, Post, & Vliet, 2004;
Gonzalez & Wu, 2003; Humphrey, 1995; Neilson & Stowe, 2002; Levy & Levy, 2002; Lopes
& Oden, 1999; Luce, 2000; Starmer & Sugden, 1993; Starmer, 1999, 2000; Wu, 1994; Wu and
Gonzalez, 1999; Wu & Markle, 2004) Not all criticisms of CPT have been without controversy
(Wakker, 2003; Baucells & Heukamp, 2004), however, and some conclude that CPT as the
“best”, if imperfect, description of decision making under risk and uncertainty (Camerer, 1998;
Starmer, 2000; Harless & Camerer, 1994). This paper summarizes the case against both
versions of prospect theory as descriptions of risky decision making and shows that other
models are more accurate.
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 these
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 simple special cases that
will be compared in this paper against CPT.
In the “configural weight models, weights of branches depend on the probability or
event leading to a consequence and relationships between that consequence and consequences
of other branches in the gamble. These older models led me to re-examine old tests and devise
new properties that can be used to test among classes of models. These “new paradoxes,” create
systematic violations of prospect theories. The properties can also be derived from Expected
Case Against Prospect Theories 5
Utility (EU) theory, so evidence of systematic violation of these properties is also evidence
against EU as a descriptive model.
The mass of evidence has now reached the point where I conclude that prospect theories
should be rejected as descriptive models of decision making. The violations of CPT are largely
consistent with a simple case of transfer of attention exchange (TAX) model of Birnbaum and
Chavez (1997). Also more accurate than CPT the rank affected multiplicative weights (RAM)
model, which is a bit less accurate than TAX (Birnbaum & Chavez, 1997). Based on the
growing evidence against CPT/RSDU, Luce (2000) and Marley and Luce (2001; 2004) have
recently developed a new subclass of configural weight models, Gains Decomposition Utility
(GDU), which they have shown has similar properties to the special TAX model 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 distinguish 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 such 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
Case Against Prospect Theories 6
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:
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, RAM, and GDU.
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 in 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 strongly refute PT and CPT as accurate descriptions.
The other, positive case, is necessarily more tentative; namely, that the TAX model which
correctly predicted the violations of CPT in advance of experiments, gives a better description
of old and new data.
Because a model correctly predicted results in a series of new tests, it does not follow
that it will continue to be correct for every new test that might be devised. Therefore, the reader
Case Against Prospect Theories 7
may decide to accept the negative case (that CPT is false) and dismiss the positive case favoring
the TAX model as a series of lucky coincidences. If so, the reader should seek a better way to
describe the new phenomena as well as the classical ones and devise new tests to compare it
with those given here.
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 origins 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:
Which would you prefer?
F: $45 for sure
or
G:
.50 probability to win $0
.50 probability to win $100
This represents a choice between F, a “sure thing” to win $45, and a two-branch gamble, G,
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. Most people prefer $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 Figures 1 and 2.
In expected utility (EU) 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 ) ,
(1)
Case Against Prospect Theories 8
where u(x) is the utility (subjective value) of the cash prize, x. In 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 preference for F over G. Figure 1
shows that on the utility continuum, the balance point on the transformed scale (the expectation)
corresponds to a utility of $33.3. Thus, a person should be indifferent between a sure gain of
$33.3 and gamble G (denoted $33.3 ~ G). The cash value with the same utility as the gamble is
1
known as the gamble’s certainty 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 in Figure 2. 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 balance point corresponds to a
CE of $33.3. In this case, the transformation from money to utility is linear, and it is weighting
rather than utility that accounts for risk aversion. Intuitively, the extra weight applied to the
lowest consequence represents a transfer of attention from the highest to lowest consequence of
the gamble.
Insert Figure 2 about here.
Whereas EU (Figure 1) attributed risk aversion to the utility function, the “configural
weight” TAX model (Figure 2) attributes risk aversion to the transfer of weight from the higher
to the lower valued branch.
Quiggin’s (1993) rank dependent utility (RDU), Luce and Fishburn’s (1991; 1995) rank
and sign dependent utility (RSDU), Tversky and Kahneman’s (1992) cumulative prospect
Case Against Prospect Theories 9
theory (CPT), Marley and Luce’s (1991) lower gains decomposition utility (GDU), Birnbaum’s
(1997) rank affected, multiplicative weights (RAM), and transfer of attention exchange (TAX)
models are all members of a generic class of configural weight models in that they can account
for risk aversion 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
(or apart from) the nonlinear transformation from money to utility.
The next issue allows us to sub-divide these 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.
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” form of G, which is called the coalesced
form. 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.
We can divide theories in two classes: 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 LGDU, and “stripped” prospect theory,
without the editing rule). According to the class of RDU/RSDU/CPT models, a gamble has the
Case Against Prospect Theories 10
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. LGDU 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 (Schmidt & Zank, 2002). In fact, it is often found
that the typical college undergraduate will only accept such bets if the amount to win is more
than twice the amount to lose (Tversky & Kahneman, 1992). 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, defined on
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
(2)
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 number of
other expressions have been suggested to describe how utilities of gains relate to losses of equal
Case Against Prospect Theories 11
absolute value; these are reviewed and compared in Abdellaoui, Bleichrodt, & Paraschiv
(2004). All of these variations attribute loss aversion to the utility function.
A second theory of loss aversion, in contrast, is that loss aversion is another
consequence of configural weighting. As in Figure 2, suppose the weights of two equally likely
branches to lose $100 or win $100 are 2/3 and 1/3, respectively. Even when u(x) = x for both
positive and negative values of x, this model implies that this gamble has a value of –$33, so
people would avoid such 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. Schmidt and Zank
(2002) note that 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 can lead to
theoretical confusion.
2. NOTATION AND TERMINOLOGY
There are a few other terms that should be defined at the outset: transitivity,
consequence monotonicity, branch independence, and stochastic dominance. Preferences are
said to be transitive if for all A, B, and C, A  B and B  C implies A  C.
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).
Case Against Prospect Theories 12
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
S  (x, p; y, q;z,r)
R  ( x, p; y, q;z,r)
if and only if
S  (x, p; y,q;z , r)
(3)
R ( x, p; y, q; z,r)
where the consequences (x, y, z, x', y', z') are all distinct, the probabilities are not zero and sum
to 1 in each gamble, and p + q = p' + q'. The branch (z, r) is known as the common branch in
the first choice, and ( z,r) is the common branch in the second choice. 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.
When we restrict the distributions and number of consequences in the gambles to be the
same in all four gambles ( p  p; q  q), then the property is termed restricted branch
independence. 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 nonidentical 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
Case Against Prospect Theories 13
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.
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 )
(4)
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, each of which
would describe a weighting function, wC ( p) .
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 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
Case Against Prospect Theories 14
Prospect theory (Kahneman & Tversky, 1979) is similar to that of Edwards (1962),
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 and
properties that will be described in this paper unless assumptions are made about how to extend
it to gambles with more than two nonzero consequences. One way to extend PT is to treat the
theory as the special case of Edwards (1962) model.
A still simpler extension with different weighting functions for positive and negative
consequences has been called “stripped” prospect theory by Starmer and Sugden (1993). The
term “stripped” was used to indicate that the editing principles and the use of differential
weighting for the consequence of $0 have been removed. A third way to extend PT is to use
CPT (Tversky & Kahneman, 1992), which is not equivalent to original prospect theory.
Besides the restriction in PT to prospects with no more than two nonzero consequences,
the other new feature of PT was the idea that an editing phase precedes the evaluation phase.
Kahneman (2003) described the development of the 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 model
could be easily falsified. To rescue the model from implications they thought were implausible,
they added editing rules that would provide preemptive excuses for potential refuting evidence.
3. Editing Rules in Prospect Theory
The six principles of editing are as follows:
a. Combination: probabilities associated with identical outcomes are combined. This
principle implies coalescing.
Case Against Prospect Theories 15
b. 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).”
c. 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.
d. Dominance: Transparently dominated alternatives are recognized and eliminated.
Without this principle, Equation 4 violates dominance in cases where people do not.
e. Simplification: probabilities and consequences are rounded off. This principle,
combined with cancellation, could produce violations of transitivity.
f. 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.” The 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 14 problems, there were two functions
(value and probability), a status quo point, and 6 editing principles. In addition, special biases
Case Against Prospect Theories 16
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. The following defense of a theory is provided: The theory is mostly correct,
except that an additional principle must be added to account for this seemingly contrary result.
Had it not been for that exception, the theory would have been correct. Nevertheless, if we take
each editing rule as a separate scientific theory, we can isolate them and test them one by one.
Equation 2, SWU, is sometimes called “stripped” prospect theory, in reference to the
theory of Kahneman and Tversky (1979) after Edwards (1962), stripped of the editing
principles, and extrapolated to any number of branches (cf. Starmer and Sugden, 1993).
4. Rank Dependent Utility and Cumulative Prospect Theory
Rank dependent utility theory was proposed as a way to explain the Allais paradoxes
without violating transparent dominance (Quiggin, 1982; 1985; 1993). 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 used different equations for different types of prospects, and it
was more general in that it allowed different weighting functions for positive and negative
consequences.
Case Against Prospect Theories 17
For gambles with strictly nonnegative consequences, RDU, RSDU, and CPT all boil
down to the same representation. With 0  x1  x 2 
 xi 
x n , RDU, RSDU, and CPT use
the following representation:
n
n
n
RDU (G)  [W ( pj )  W (  p j )]u(xi )

i 1

ji
(5)
j i 1
where RDU(G) is the rank-dependent expected utility of gamble G; W+(P) is the weighting
n
function of decumulative probability, Pi   p j , which monotonically transforms decumulative
j i
probability to decumulative weight, and assigns W+(0) = 0 and W+(1) = 1.
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
RSDU (G)  [W ( pj )  W ( p j )]u(xi ) 
i 1

j 1

j 1
m
m
m
 [W ( p )  W (  p )]u(x ) (6)
i n 1


j
j i
j  i1
j
i
The representation in CPT is the same.
4. RAM, TAX, and GDU 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 to 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
Case Against Prospect Theories 18
and perceptual psychophysics (Birnbaum, 1974; Birnbaum, Parducci, & Gifford, 1971;
Birnbaum & Stegner, 1979). Two situations that are objectively equivalent but described
differently to people must be kept distinct in the theory if people respond differently to the
different descriptions.
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 SWU, PT and CPT, emphasize the role of
weighting of probability; 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 of risky decision making
will in fact be quite different, but models preserving scale convergence should be preferred over
those that do not (Birnbaum & Sutton, 1992).
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 “rank-dependent” 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
Case Against Prospect Theories 19
approaches. For example, in two branch gambles, 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.
4.1 RAM Model
In RAM, 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,s )t(p )
i
(7)
i
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, and so on. The function, t(p),
describes how a branch’s weight depends on its probability, apart from these configural effects.
For two, three, and four branch gambles on strictly positive consequences, it has been
found that 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
Case Against Prospect Theories 20
Kahneman (1992) and the model with these parameters will be called the “prior” RAM model.
The term “prior” here means the model was estimated from previous data and used to predict
what would happen in new properties of choice 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
, which gives a good
g
g
1 p  2 (1 p)
approximation of empirical data of Tversky & Kahneman (1992). A graph of their data and fit
of the RAM model are shown in Birnbaum (1997, Figure 9).
To describe gambles with consequences that are strictly less than or equal to zero, one
can use the same equations, inserting absolute values and then multiplying the result by –1. This
means that the order for strictly nonnegative gambles A, B,
gambles  A, B,
will show “reflection”; that is, A
and for strictly nonpositive
B  A
B where –A is the same as
gamble A, except that each positive consequence has been converted from a gain to a loss.
To describe gambles with mixed consequences, the consequences are ranked in the same
fashion as for strictly positive gambles. 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 approximation for the (meager) data available on
mixed gambles. In Edwards’ (1962) terminology, this additional simplification treats
nonnegative gambles and mixed gambles with weights on the same page.
Case Against Prospect Theories 21
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 if a person is risk averse. In the TAX model, weight
transferred from branch to branch represents these transfers of attention. 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 (“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
n
 t( p )u(x )    [u(x
i
TAXU(G) 
j
i 1
i
j
)  u(xi )] (i, j)
j 1 i1
n
(8)
 t( p )
i
i 1
This model is fairly general and includes several interesting special cases (Marley & Luce,
2001; 2004). A simple special case 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 lower-ranked
branches have more importance (as they would do in a “risk-averse” person), it is theorized that
weight is transferred from branches with better consequences to those 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 match the conventions in CPT.
Therefore,  > 0 in this paper corresponds to  < 0 in previous papers.]
Case Against Prospect Theories 22
When lower valued branches receive greater weight ( > 0), this special TAX model can
be written for three-branch gambles, G  (x1, p1; x 2 , p2 ;x 3, p3) , where x1  x2  x3  0 :
TAXU(G) 
Au(x1 )  Bu(x 2 )  Cu(x3 )
A  B C
(9)
A  t( p1 )  2t( p1 ) / 4
B  t( p2 )  t(p2 )/ 4  t(p1 )/ 4
where
C  t(p3 )  t(p1 )/ 4  t(p2 )/ 4
One can approximate the data of Tversky and Kahneman (1992) with  = 1. In threebranch gambles with positive consequences, 1/4 of the weight of any higher valued branch is
transferred to each lower-valued branch. In two-branch gambles, 1/3 of the weight of the higher
branch is given to the lower branch (as in Figure 2).
With these restrictions, the TAX model has three parameters,  represents the

psychophysical function for probability, t(p)  p ,  represents the utility function of monetary

consequences, u(x)  x , and , represents the configural transfer of weight (which affects risk
aversion or risk seeking). In this paper, predictions of the model will be calculated with  = 1, 
= .7, and  = 1. This model, which roughly approximates the data of Tversky and Kahneman
(1992), will be termed the “prior” TAX model.
For gambles composed of negative or mixed consequences, different values of  would
be allowed in this simple model. However, a still simpler model 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 mixed
consequences. With these assumptions, the model accounts for reflection, for the four-fold
pattern of risk seeking and risk aversion, for loss aversion, and for violations of gain loss
separability.
Case Against Prospect Theories 23
Three calculators are freely available from the following URLs:
http://psych.fullerton.edu/mbirnbaum/calculators/
These calculators also allow calculations of RAM, TAX, 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.
4.3 GDU Model
Luce (2000, p. 200) proposed a “less restrictive theory” that satisfies a property known
as (Lower) Gains Decomposition but which does not satisfy coalescing. Marley and Luce
(2001; 2004) present the axioms and representation theorem for this form of GDU and show
that this model is similar with respect to many of the new paradoxes to TAX. The key idea that
a complex gamble can be decomposed into a series of two-branch gambles. The decomposition
can be viewed as a tree in which a three-branch gamble is broken 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:
GDU(x, p; y)  W(p)u(x)  [1  W(p)]u(y)
(10)
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)
(11)
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.
Case Against Prospect Theories 24

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) ]
(12)
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 certain 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 violates lower coalescing. Luce
(personal communication) is currently working with colleagues on more general models that
satisfy gains decomposition but do not necessarily satisfy binary RDU.
5. THE CASE AGAINST PROSPECT THEORIES
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 within subjects (Birnbaum,
1999b; 2004a; Humphrey, 1998; 2000; 2001a; 2001b).
Consider the following two choices in Problem 1:
Case Against Prospect Theories 25
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 obeys coalescing, that person should make the same choice between A and B as between
A’ and B’,
apart from random “error.” Birnbaum (2004a) presented these two choices mixed
among 19 other choices to 200 participants, finding that 63% (significantly more than half)
chose B over A and 80% (significantly more than half) chose A’ over B’. Of 200 participants, 96
switched from B to A’ but only 13 switched from A to B’, z = 7.95.
According to CPT, no one should switch except by chance. However, both RAM and
TAX, with parameters estimated from previous data, predicted 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. Table 1 shows the calculations for the TAX and CPT models in Choices 1.1
and 1.2, with prior parameters for these choices. CPT must predict no difference between the
rows.
Insert Table 1 about here.
However, according to TAX, coalescing the two branches to win $100 in B’ 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
Case Against Prospect Theories 26
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 correctly predicts this reversal of preferences.
According to prior CPT people should choose A over B and A’ over B’; but, any CPT
model implies A
B  A B. These findings and others (Birnbaum, 2004a) refute CPT and
other theories that satisfy coalescing, such as those of Lopes and Oden (1999), Becker and Sarin
(1987), Chew (1983), Chew, Epstein, and Segal (1991), among others (Luce, 2000).
2. Violations of First Order Stochastic Dominance
Birnbaum (1997) deduced that RAM and TAX would violate stochastic dominance
when choices were constructed from a special recipe, illustrated in Problem 2 of Table 1. The
calculated certainty equivalents of the gambles according to TAX are shown in the right portion
of the Table. According to TAX, the dominant gamble (I, shown on the left) has a certainty
equivalent of only $46 and the dominated gamble (J) has a value of $63.
After Birnbaum’s (1997, p. 93-94) prediction had been set in print, Birnbaum and
Navarrete (1998) tested it empirically, finding that 70% of 100 undergraduates violated first
order stochastic dominance in four choices like Problem 2. Birnbaum, Patton, & Lott (1999)
found 73% violations with five new variations of the recipe and a new sample of 110
undergraduates. In a recent study (Birnbaum, submitted-d), 74% of 330 participants chose to
draw from J, even though I dominates J. In all of these studies, significantly more than half of
the participants violated stochastic dominance.
Birnbaum’s (1997) recipe is illustrated in the lower portion of Figure 3. 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
Case Against Prospect Theories 27
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.
Insert Figure 3 about here.
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, which improves the gamble
despite the decrease in the .05 sliver’s consequence from $96 to $90. RAM and TAX models
predict that people will prefer G– over G+, in violation of stochastic dominance.
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
Case Against Prospect Theories 28
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).
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 PhDs who have studied decision making is nearly 50%. Majority
violations have been observed when undergraduates are tested in class, in the lab, or via the
WWW, with consequences that are purely hypothetical or when real chances at modest prizes
are possible (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
57.6
05 blue to win $80
10 white to win $10
Here, the dominant gamble (I’) has higher consequences on two of the three branches,
yet a significant majority of 57% of 394 participants continued to choose the dominated gamble,
Case Against Prospect Theories 29
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, 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 dominated
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.
By 03-04-05, my students and I had completed 34 choice studies with a total of 9854
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; Fall 04, explain and
2A_mixed_reps). I conclude that any theory that proposes to be descriptive must reproduce
these violations and should account for manipulations that increase or decrease their incidence.
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.
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),
Case Against Prospect Theories 30
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 here.
[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 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
framing, coalescing, and 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%. 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 for each choice problem. This analysis allows one to distinguish
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
Case Against Prospect Theories 31
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
(SSSV) is given by the following:
P(SSSV)  a(1 e) e  (1 a)e (1  e)
3
3
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 made one error
3
( (1  e) e ), whereas the person who truly violates it has made three “errors” and one correct report.
Similar expressions can be written for each of the other 15 response patterns.
When this true and error model was fit to the observed frequencies of the 16 possible
response sequences in 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 results 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
Case Against Prospect Theories 32
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 might “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.
In Fall of 2004, Birnbaum collected data from 92 participants who made choices between
gambles framed as losing or mixed gambles. These were framed by subtracting either $25, $50, or
$100 from each consequence in the gambles (in the 20 choices used by Birnbaum, 2004b,
Experiment 3) and adding that constant as a separate “gift”. This meant that participants could not
lose their “own” money (they were playing with “house” money). There were 4 tests of stochastic
dominance in each framing condition; because there were three ways to frame each of the 20
choices, and each participant made each of these 60 choices twice (separated by about 25 minutes),
there were 24 tests of stochastic dominance. The average rate of violation was 76% (including 12
who violated it on all 24 tests), and rates did not differ much as a function of the constant subtracted.
Case Against Prospect Theories 33
In another test, Birnbaum asked 276 undergraduates to make 6 choices between gambles and
to write paragraphs explaining why they made each choice. One was a test of stochastic dominance.
Although some wrote that being asked to justify their decisions caused them to have to think about
their choices, the rate of violation in this study was 80%, higher than the rate obtained in longer
experiments where people did not have to justify their choices. The “reasons” people gave were
sorted into 18 categories, of which only 6 were used by more than 10 people. Among those who
violated dominance, the most frequent “reason” was given by 78 people who said they could win
more money with the dominated gamble. “More money” was also cited by 8 of those who satisfied
dominance. Among the 66 who cited “better probability” as their reason, 22 chose the dominant
gamble and 43 violated dominance. Of the 42 people who mentioned both probability and money in
their reasons, 40 violated dominance and 2 satisfied it. There were 17 who provided reasoning that
was leniently scored as a “dominance” argument; of these 17, only four violated stochastic
dominance. There were 15 who described a tradeoff, among whom 13 violated dominance. And of
the 11 who cited contrasts between consequences on corresponding branches, 10 violated
dominance. The bottom line is that the request to justify choices did not reduce the incidence of
violations.
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 3, 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
Case Against Prospect Theories 34
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 Choices 3.1 and 3.2 in Table 1. In a recent test (Birnbaum, 2004b), 342
participants were asked to choose between I and J and between M and N, 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 M to N, violating stochastic dominance on the choice between G–
and 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 M and N; 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. These studies also tested heuristic models that people ignore probability and just
Case Against Prospect Theories 35
average the values of the consequences, or that they count the number of branches favoring one
gamble or the other. If such heuristics are correct, people should continue to violate stochastic
dominance in Problem 4 of Table 1. If the TAX model is correct and its parameters generalize
to a new test, however, people should satisfy stochastic dominance in this case.
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), and only
34% (significantly less than half of the same group) 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 ignore 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.
Domain of Violation of SD. To understand how “often” the TAX model implies
violations of stochastic dominance, Birnbaum (2004a) simulated choices between “random”
three-branch gambles. Gambles were constructed 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
Case Against Prospect Theories 36
between $0 and $100. “Random” choices were generated by independently constructing pairs
of such gambles. Among 1,000,000 choices thus constructed, it was found that 33.3% of the
choices involved stochastic dominance, but only 1.8 per 10,000 are predicted to be violated
according to TAX with its parameters. This means that the model only rarely violates stochastic
dominance in such an environment. TAX and CPT, with their prior parameters, made the same
predictions in 94% of these “random” 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. Assuming that utility is proportional to x, 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 Problem 2. So, special TAX is not flexible enough to satisfy stochastic
dominance in Birnbaum’s test, if participants show the typical pattern of risk aversion and risk
seeking.
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. This property can be called “upper tail
independence,” because Wu tested whether the upper tail of a distribution can be used to reverse
preferences induced by other aspects of the choice. The property follows from transitivity,
upper coalescing, and comonotonic restricted branch independence; it should therefore be
satisfied according to the class of CPT models. Wu reported systematic violations, which he
Case Against Prospect Theories 37
reasoned might be due to violations of CPT or to the editing rule of cancellation, which
contradicts implications of the CPT model.
Birnbaum (2001) reported a test of upper tail independence similar to one by Wu (1994).
In Problem 5 of Table 1, people should prefer t = ($92, .48; $0)  s = ($92, .43; $68, .07; $0) iff
they prefer v = ($92, .43; $92, .05; $0)  u = ($92, .43; $68, .07; $0). There were n = 1438
people tested via the WWW who had a chance to win small cash prizes. Significantly more than
half preferred 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 lower GDU.
6. Violations of Upper Tail Independence without Zero Consequences
A new test of upper tail independence (Birnbaum, in press-c, Exp. 2) is displayed in
Problems 6.1 and 6.2. The lowest consequence is not zero in all four gambles, in contrast with
earlier tests. 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. There were 503 participants. As predicted by
TAX, the majority was significantly reversed, contradicting CPT and GDU.
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.
Upper cumulative independence can be written ( z x x  y  y z  0 ):
Case Against Prospect Theories 38
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)
Consider Problems 7.1 and 7.2 in Table 2. Violations of upper cumulative independence, 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 analyze this result. Table 2 about
here.
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 (including highly
educated ones) against 124 undergraduates tested in the lab. Birnbaum (2004b) investigated this
property with two variants and with a dozen different procedures for representing the gambles;
that paper summarized results from 3440 participants. Birnbaum (submitted-b) replicated two
Case Against Prospect Theories 39
tests with another 663 participants and three variations of Savage’s presentation format and with
strictly positive and strictly negative consequences. Finally, 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 results consistently violate upper cumulative independence.
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 and only 28 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) .
Problems 8.1 and 8.2 in Table 2 illustrate a test of this property.
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”; in
addition, significantly more participants switched in the direction violating 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 7,186
participants, creating very strong case against this property.
Case Against Prospect Theories 40
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. These properties were developed and predictions made
(Birnbaum, 1997) before the experiments were carried out.
9. The Allais common consequence paradox
The common consequence paradox of Allais (1953, 1979) can be illustrated with the two
choices in Problem 9.1 and 9.2 of Table 2.
Expected Utility (EU) theory holds that A is preferred to B if and only if the EU of A
exceeds the EU of B, A  B EU(A) > EU(B), where EU is defined in Equation 1.
According to EU, A is preferred to B iff u($1M) > .10u($2M) + .89u($1M) +.01u($2).
Subtracting .89u($1M) from each side, it follows that .11u($1M) > .10u($2M)+.01u($2).
Adding .89u(2) to both sides, we have .11u($1M)+.89u($2) > .10u($2M)+.90u($2), 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.” However, SWU, CPT,
RAM, TAX, and GDU can all account for this finding, but they do so in different ways.
10. Decomposition of the Allais Common Consequence Paradox
In order to test among the models, we decompose this Allais paradox into transitivity,
coalescing, and restricted branch independence, as illustrated below:
A:
$1M for sure
(coalescing & transitivity)
B:
.10 to win $2M
.89 to win $1M
.01 to win $2
Case Against Prospect Theories 41
A’:
.10 to win $1M
.89 to win $1M
.01 to win $1M
B:
.10 to win $2M
.89 to win $1M
.01 to win $2
B’:
.10 to win $2M
.89 to win $2
.01 to win $2
D:
.10 to win $2M
.90 to win $2
 (restricted branch independence)
A”:
.10 to win $1M
.89 to win $2
.01 to win $1M
 (coalescing & transitivity)
C:
.11 to win $1M
.89 to win $02
The first step converts A to its split form, A; A ~ A by coalescing; by transitivity, A 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
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 3. SWU (“stripped” 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 3 about here.
Birnbaum (2004a) tested among the theories in Table 3 by means of Problems 10.1-10.5
in Table 2, which teases out two properties that are confounded in the original Allais paradox.
According to EU, people should make the same choice in each case, because all of the choices
are the same, except for a common branch (80 marbles to win $2 in Choices 10.1 and 10.2, 80
to win $40 in Choice 10.3, or 80 to win $98 in Choices 10.4 and 10.5. Birnbaum presented
Case Against Prospect Theories 42
these choices, among others to 349 participants whose data are summarized in Table 2. [Footnote
]
Each percentage, except for Problem 10.3, is significantly different from 50%. By tests of
correlated proportions, each successive contrast is also significant.
CPT, RSDU, SWU, TAX and RAM models, as fit to previous data, correctly fit the
predicted choices in Choices 10.1, 10.3, and 10.5 of Table 2; that is, they all predict the three
types of Allais paradoxes in those three problems. 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, RAM, TAX, and GDU models predict
reversals in those two cases. SWU implies that the choices should be the same in 10.2 and 10.4,
since it satisfies branch independence. However, the majority preference in 10.2 is opposite
that of 10.4, and opposite of what is required by CPT to explain the Allais paradoxes. RAM and
TAX correctly predict this pattern of violations of branch independence.
Suppose people used the editing rule of combination on Choices 10.2 and 10.4. If so,
they would have made the same decision in 10.1 and 10.2 and within 10.4 and 10.5.
Suppose people used the editing rule of cancellation in Choices 10.2 and 10.4; if so, they
would have made the same decision in 10.2 and 10.4. If they only cancelled on some
proportion of the trials, with CPT governing the rest, there would be violations of branch
independence that should be in the same direction as Choices 10.1 and 10.5. However,
observed violations are substantial and opposite the direction needed to allow branch
independence to account for Allais paradoxes. So, these results (as well as others in Birnbaum,
2004a) refute both original and cumulative prospect theory, with or without the editing rules of
cancellation and combination.
Case Against Prospect Theories 43
TAX, RAM, and GDU are consistent with these results, which violate RDU, RSDU,
CPT, and SWU.
According to TAX/RAM, the cause of the Allais constant consequence paradoxes is
violation of coalescing, and violations of branch independence actually reduce their magnitude.
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 of R (left side) from 10.1 to 10.2 makes R worse
and coalescing the upper branches of R from 10.4 to 10.5 makes R worse again. Similarly,
splitting the higher branch in S (right side) from Problem 10.1 to 10.2 improves S as does
coalescing the lower branches of S from Problem 10.4 to 10.5.
11. True and Error Analysis of Splitting in the Allais Problem
Table 4 shows the number of participants who showed each preference pattern on two
replications of Allais Paradox choices in Problems 9.2 and 9.3, which differ only by coalescing.
In Table 4, 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 11.1, where the branches were coalesced, R on Choice 11.2 where the branches were
split, and chose S on both of these trials in the second replicate. Insert Table 4 about here.
A three parameter “true and error” model was fit to these data (cf. Birnbaum, 2004b). In
this model, the three parameters are the probability that a person truly prefers R over S in
Choice 9.2, the conditional probability that this person has the same true preference in Choice
9.3, and the probability of making an “error” expressing “true” preferences. The estimates are
that 89.3% of participants truly prefer R over S (right over left) in Choice 9.2 but that 66.2% of
these have the opposite true preference in 9.3, where gambles have been split to apparently
Case Against Prospect Theories 44
make S (left side) better and R (right side) worse. The error rate is estimated to be 18.5%,
 2 (12) = 14.3.
Put another way, this model indicates that 59% of participants truly switch from R in
Choice 9.2 to S in Choice 9.3 and that no one switched in the other direction. The observed data
show 46% made the former switch and that 12% made the opposite switch, averaged over
replicates. Table 4 shows that 56 people (28%) switched in this fashion on both replicates,
compared to 4 who made the opposite switch both times. A two-parameter model that assumes
2
no one truly switched fit much worse,  (13) = 236.0. In sum, these results violate
RDU/RSDU/CPT, but are consistent with RAM, TAX, and GDU.
12. 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)
Then A  (y, p;x,q; z,r;v,s)

B  (0,1  r s; 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 a test modeled after a choice in Levy and Levy (2002,
Exp 2), shown in Choices 12.1, 12.2, and 12.3 in Table 5. Insert Table 5 about here.

Wu and Markle (2004) found that the majority prefers B

prefers B

A ; however, the majority does NOT prefer B

A and the majority
A, contrary to CPT/RSDU and
any model that satisfies gain loss separability. This example is consistent with the TAX model
Case Against Prospect Theories 45
with the following simplifying 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 are calculated using “reflection” property; that is, calculate the
values using absolute values and then multiply the result by –1. The model was fit with the
assumption that u(x) = x, for both positive and negative consequences near zero.
This point deserves emphasis: this model describes “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 if these simple assumptions will suffice.
In the model of Birnbaum (1997), branch weights depend not only on probability and
rank, but also on augmented sign of the consequences. Thus, a branch’s weight in that 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 the simple version of TAX used to
fit the Wu and Markle result, but it also predicts violation of gain loss separability.
13. Violations of Restricted Branch Independence
Restricted branch independence should be satisfied by EU, SWU and any theory that
uses prospect theory’s editing principle of cancellation. However, this property will be violated
according to RDU/RSDU/CPT (without the editing rule of cancellation) and by RAM, TAX,
and GDU. The direction of violations, however, is opposite in the two groups 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.
Case Against Prospect Theories 46
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 case, two branches have equal probabilities. The consequence on the common branch is
the only change from the first line to the last.
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, 2001; 2004). This generic model can be
written for 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). 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) w3


w1 u(y)  u( y ) w
2
Case Against Prospect Theories 47
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 the same probability p in Gambles S  and R, respectively. RAM, TAX
and GDU models imply this type of violation, SR . With previously estimated 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. The W(P) function
estimated by Tversky and Kahneman (1992) is shown as the solid curve in Figure 4. Their
function crosses the identity line, also shown in Figure 2. Define a weakly inverse-S function as
any strictly increasing monotonic function from zero to 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 by the dashed curve in Figure 4. Therefore, w2  w1 and w2  w3. It
follows that
w2
w
1  3 .
w1
w
2
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.
Case Against Prospect Theories 48
Consider the choices in Problem 13.1 in Table 5 (from Birnbaum & Chavez, 1997).
Significantly more than half chose S
R , and significantly more than half chose R S .
Thus, violations are significant, and in opposite the predictions of the inverse-S weighting
function.
There are significantly more of the form SR  than the opposite (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. The same pattern of violations has been observed 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 RBI are significantly more often in opposition to predictions of CPT than in agreement.
Birnbaum (2004b) applied the “true and error” model to replicated choices which
indicated that the SR violations cannot be attributed to random “error” but the opposite pattern
can be set to zero with no loss of fit. The results of tests of RBI do not appear to depend
strongly on how gambles are presented, on financial incentives, whether tests are composed of
strictly positive or negative consequences, event-framing, or other procedural manipulations
(Birnbaum, 2004b; submitted-b, d).
14. 4-Distribution Independence
Case Against Prospect Theories 49
4-Distribution independence is an interesting property because RDU/RSDU/CPT models
imply systematic violations, but the property should be satisfied, according to RAM.
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)
R (z, r ;x, p; y, p;z,1  r 2p)
It is violated, according to TAX, but in the opposite way from that predicted by the inverse-S
weighting function of CPT. Consider Problems 14.1 and 14.2 in Table 5.
Note that the trade-off between two branches of equal probability (.2 in this example) 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 with it estimated parameters, R
S and S  R; this is the
RS  pattern of violations. The RAM model allows no systematic violations of this property.
According to SWU and any theory with prospect theory’s cancellation editing, there should be
no violations. According to the TAX model and parameters of Birnbaum (1999a), however,
people should show the SR  pattern of violations. In a study with 100 participants and 12 tests,
Birnbaum and Chavez (1997) found that the pattern predicted by TAX was more frequent than
the opposite. In Problems 14.1 and 14.2 there were 23 who had the SR  pattern 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 (3LDI) as follows:
S  (x, p; y, p; z,1 2 p)
R  ( x, p;y, p;z,1  2p)
Case Against Prospect Theories 50
 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 Problems 16.1 and 16.2 in Table 6.: According to 3-LDI, people should choose
S over R in 16.1 if and only if they choose S2 over R2 in 16.2, apart from error. The CPT model
as fit to previous data predicts that people should violate this 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
Case Against Prospect Theories 51
A reversal from S  R to S2  R2 can occur with the opposite ordering. But if the ratio of
weights is independent of 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:
w2 a(2,3)s( p) a(2,3)s( p) w2



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
2

w1
t(p)  ( 4) t( p)
t( p)[1 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
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. In contrast, both RAM and TAX hold that these ratios are
equal, so there can be no systematic violations.
In these tests, unlike previous cases, CPT goes on offense by predicting violations,
leaving TAX and RAM to defend the null hypothesis. Birnbaum (in press-c) conducted three
studies with 1578 participants testing these predictions. The results, as in Problems 16 and 17,
Case Against Prospect Theories 52
showed that predictions of TAX were more accurate than CPT, which predicted a reversal in 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. The results are based on 1075 participants in Birnbaum (in press-c, Exp. 1). 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)
R  ( z,1 2 p;x, p; y, p)
Case Against Prospect Theories 53
 S2  ( z,1  2 p ;x, p; y, p)
R2  ( z,1 2 p; x, p; y, p)
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 ) w
2
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
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, CPT
implies violations of the opposite pattern from that predicted by TAX, but CPT predicts no
violation in Problem 18, given its prior parameters:
Case Against Prospect Theories 54
Birnbaum (in press-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 3-LDI and 3-UDI were tested in
Experiment 2 of Birnbaum (in press-c) and failed to materialize in the modal choices.
6. SUMMARY AND DISCUSSION
Violations of prospect theories are summarized in Tables 7 and 8. Table 7 contains
properties that must be satisfied according to CPT and are violated by prior RAM and TAX.
Violations of these properties cannot be explained by any version of RSDU/CPT.
Table 8 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 from that
predicted. All of the models except EU, SWU (stripped PT), and Viscusi’s Prospective
Reference Theory 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 and which contradicts the inverse-S weighting function.
RAM satisfies all of the distribution independence properties, so violations of 4-DI and
3-Upper DI refute the RAM model. Prior TAX and GDU satisfy 3- Lower DI but violate 3Upper DI, consistent with the data. CPT violates all of these properties, but its predicted
violations either failed to materialize or were refuted by violations showing the opposite pattern
from that predicted.
One important property not listed 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
Case Against Prospect Theories 55
branch should make it worse. GDU implies upper coalescing, so splitting the upper branch
should have no effect. SWU and stripped prospect theory imply violations of both upper and
lower coalescing and satisfaction of branch-splitting independence. Birnbaum (submitted) has
recently reported small but significant violations of branch-splitting independence.
Considering all of the evidence, there is a very strong case against CPT. There is also
strong evidence against the editing principles of prospect theory of cancellation (which implies
branch independence and distribution independence) and combination (which implies
coalescing). If there is a dominance detector, it fails to work effectively for cases as in Table 1
that are predicted to fail according to RAM and TAX.
Two models are partially consistent with the data, RAM and GDU. Despite some
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 is a rounding error in
Luce (2000, p.200-202) that made GDU appear 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 and Marley are currently exploring Gains Decomposition without the
assumption of binary RDU, which may provide a more accurate description as well as an
axiomatic representation.
Case Against Prospect Theories 56
The model that best describes the evidence reviewed is prior TAX, which correctly
predicted all of the modal choices reviewed here. It is worth repeating that the parameters for
TAX was estimated 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 breakdown of the
Allais paradoxes were indeed calculated in advance of the collection of data. Thus, the model’s
predictions in these cases were true predictions and not calculations based on post hoc fitting to
data already in hand.
In my opinion, the empirical evidence now exceeds the critical threshold where prospect
theories should join EU in the repository of inaccurate descriptive theories.
Questions:
1. 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.
2. 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 w $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?
Case Against Prospect Theories 57
3. 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 (Birnbaum, submittedb). However, it can hold for gambles with strictly positive consequences.
Case Against Prospect Theories 58
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Case Against Prospect Theories 69
Table 1. Choice Problems, Choice Percentage, TAX and CPT predictions in tests of branchsplitting, stochastic dominance, and upper tail independence.
Choice
1.1
1.2
TAX
A: 85 to win $100
B: 85 to win $100
62
68.4
69.7
82.2
79.0
26
75.7
62.0
82.2
79.0
85 to win $96
70,
45.8
63.1
70.3
69.7
05 to win $14
05 to win $90
71
05 to win $12
10 to win $12
06
53.1
51.4
70.3
69.7
34
45.8
35.8
34
32.3
29.8
62
33.4
36.7
67
65.0
69.0
33
60.3
57.2
10 to win $50
10 to win $100
05 to win $50
05 to win $7
A’: 85 to win $100
B’: 95 to win $100
15 to win $50
2,
3.1
3.2
4
5.1
CPT
I: 90 to win $96
05 to win $7
J:
M: 85 win $96
N: 85 win $96
05 win $96
05 win $90
05 win $14
05 win $12
05 win $12
05 win $12
K: 90 to win $96
L: 25 to win $96
05 to win $14
05 to win $90
05 to win $12
70 to win $12
s: 50 to win $0
07 to win $68
t: 52 to win $0
48 to win $92
43 to win $92
5.2
6.1
6.2
u: 50 to win $0
v: 52 to win $0
07 to win $68
05 to win $92
43 to win $97
43 to win $97
w: 80 to win $110
x: 80 to win $110
10 to win $44
10 to win $96
10 to win $40
10 to win $10
y: 80 to win $96
z: 90 to win $96
10 to win $44
10 to win $10
10 to win $40
1. B(2004a, n = 349); 2. B & N (1998, n = 100); B, P, & L (1999, n = 110). 3. B (2004b, n =
342 tickets); 4. B(d, study 2, n = 232); B(2001, n = 1438); 6 B (in press-c, study 2, n = 503).
Case Against Prospect Theories 70
Table 2. Choice Problems testing TAX and CPT
Choice
7.1
7.2
8.1
8.2
9.1
TAX
M: 10 to win $40
N: 10 to win $10
10 to win $44
10 to win $98
80 to win $110
80 to win $110
O: 20 to win $40
P: 10 to win $10
80 to win $98
90 to win $98
Q: 90 to win $3
R: 90 to win $3
05 to win $12
05 to win $48
05 to win $96
05 to win $52
S: 95 to win $12
T: 90 to win $12
05 to win $96
10 to win $52
A: $1M for sure
B: 10 to win $2M
70
65.0
69.6
42
68.0
58.3
62
8.8
10.3
26
68.0
58.3
42
1000K
810K
76
125K
236K
37
241 K
172 K
38
13.3
9.0
64
9.6
11.1
54
30.6
40.0
43
62.6
59.8
78
54.7
68.0
89 to win $1M
01 to win $2
9.2
C: 11 to win $1M
89 to win $2
9.3
10.1
X: 10 to win $1M
10.3
10.4
10.5
90 to win $2
Y: 10 to win $2M
01 to win $1M
01 to win $2
89 to win $2
89 to win $2
E: 10 to win $98
90 to win $2
10.2
D: 10 to win $2M
G: 10 to win $98
F: 20 to win $40
80 to win $2
H: 10 to win $40
10 to win $2
10 to win $40
80 to win $2
80 to win $2
I: 10 to win $98
J: 10 to win $40
80 to win $40
80 to win $40
10 to win $2
10 to win $40
K: 80 to win $98
L: 80 to win $98
10 to win $98
10 to win $40
10 to win $2
10 to win $40
M: 90 to win $98
10 to win $2
N: 80 to win $98
20 to win $40
7. B(2004b, pies*, n = 305); 8 B(2004b, pies*, n = 305).9. B (Allais, study 1, n = 200 X 2 reps).
10. B (2004a, combined, n = 349).
Case Against Prospect Theories 71
Table 3. 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 72
Table 4. Numbers of Participants showing each choice combination in tests of Event Splitting
in Problems 9.2 and 9.3. The pattern RR’RS’, for example, represents choice of the “risky”
gamble (right side) in both 9.2 and 9.3 (Table 2) on the first replicate, and the choice of “risky”
in 9.2 and “safe” in 9.3 on the second replicate. Series B tested the same properties using
different gambles (Birnbaum, 2004a).
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.
Table 5.
Choice
12.1
TAX
A

: 25 win $2000
25 win $800

B
: 25 win $1600
25 win $1200
72
497
552
Case Against Prospect Theories 73
50 win $0
12.2
12.3
13.1
13.2
14.1
14.2
A

: 50 win $0
50 win $0

B
: 50 lose $0
25 lose $800
25 lose $200
25 lose $1000
25 lose $1600
A:
25 win $2000
B:
25 win $1600
25 win $800
25 win $1200
25 lose $800
25 lose $200
25 lose $1000
25 lose $1600
S: 50 to win $5
R: 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
S: 59 to win $4
R: 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
20 to win $45
R’: 01 to win $4
72
-358
-276
38
-280
-300
40
20.0
19.2
62
57.2
60.7
34
21.7
20.6
51
49.8
50.0
20 to win $11
20 to win $45
20 to win $97
59 to win $110
59 to win $110
12. Wu & Markle (2004, n = 60). 13. B&C (1997, n = 100); 14. B & C (1997, n = 100).
Case Against Prospect Theories 74
Table 6. Tests of 3-Lower Distribution Independence and 3-2 Lower Distribution Independence
(From Birnbaum, submitted-c, Exp. 1).
No.
S
16.1
16.2
17.1
17.2
17.3
17.4
18.1
18.2
%R
Choice
R
60 to win $2
60 to win $2
20 to win $56
20 to win $4
20 to win $58
20 to win $96
10 to win $2
10 to win $2
45 to win $56
45 to win $4
45 to win $58
45 to win $96
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
.0 to win $4
50 to win $44
50 to win $96
S’: 10 to win $40
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
Prior CPT
S
S
R
R
23.6*
21.7
13.8
18.7*
36.9
22.9
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
56.0
61.6
63.4
33.2
45.9
43.9
Case Against Prospect Theories 75
Table 7. 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, H
RDU,RSDU/CPT
B04a; B
TK86; B
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
Violate RDU/RSDU/ CPT
BN98,B9
Violate any model of CPT
BN98,B9
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
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 76
Table 8. Summary of 5 Tests that Refute SWU and 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
Contradict inverse-
BMc
S
B04b
Violate RAM &
BC9
inverse-S
Failed to
materialize
Failed to
B-s-
materialize
Violate RAM &
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'.
B-s-
B-s-
Case Against Prospect Theories 77
Table 9. Summary of Tests that Distinguish GDU, TAX, SWU
Property Name
Upper Coalescing
Expression
S  (x, p; y, q;z,1 p  q)
Empirical Results
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)
Lower Coalescing
Branch Splitting
Independence
Notes: 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 78
Figure 1. Utility theory of risk aversion. Expected value (or utility) is the center of
gravity. The 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 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.
Case Against Prospect Theories 79
Figure 2. A configural weight theory of risk aversion: 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 80
Figure 3. 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 4. The solid curve is strongly Inverse-S and the dashed curve is weakly inverseS. In either case, the weight of the highest branch in (x, p; y, p; z, 1 – 2p) is greater than
Case Against Prospect Theories 81
the weight of the middle branch; w1  w2 ; i.e., 1 
w2
. Similarly, the weight of the
w1
middle branch in (z’, 1 – 2p; x, p; y, p) is less than that of the lowest branch; therefore,
w2  w3 1 
w3
. Together, these two conditions imply that violations of branch
w2
independence can only be of the form S
R and S  R; that is, the RS  pattern of
violations. Instead, the SR  pattern is more significantly more frequent.
Footnotes:
Case Against Prospect Theories 82
1. [Papers that find departures from the parameters of the published version of
CPT do not count as evidence against CPT, in my opinion, because they may only
indicate that individual differences, contextual effects, or experimental manipulations
affect the parameters of the theory. For example, Battalio, Kagel, and Jiranyakul (1990)
found that the majority of 32 economics students showed risk seeking instead of the usual
risk aversion in cases where prior CPT implies risk aversion. Humphrey and Verschoor
(2004) found data consistent with an S-shaped weighting function instead of inverse-S
shaped in common consequence problems.]
2. [Data have been aggregated over 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.]
Footnote to Section 6. A caveat can be expressed with respect to these tests 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.