Configural Weighting versus Prospect Theories of Risky Decision Making
Michael H. Birnbaum
California State University, Fullerton and Decision Research Center, Fullerton
Date: 01-24-03
Filename:BirnbaumReview01.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, SBR9410572, SES 99-86436, and BCS-0129453.
ABSTRACT
In recent years, prospect theory and its improved form, cumulative prospect
theory have replaced expected utility theory as the dominant descriptive theory of risky
decision making. However, empirical studies have now amassed considerable data
showing that the prospect theories are not descriptive. At the same time, these new data
are consistent with and in several cases, were predicted in advance of empirical studies
by configural weight models. These configural weight models assume that discrete
probability-consequence branches are weighted by a function that depends on the
probabilities and ranks of consequences on the discrete branches. Although these
configural weight models have some similarities to rank-dependent utility theories, they
violate stochastic dominance, coalescing, lower cumulative independence, upper
cumulative independence, ordinal (tail) independence, restricted branch independence,
and distribution independence. They also account for relationships between buying and
selling prices and their relations to choices.
1. INTRODUCTION
Following a period in which Expected Utility (EU) theory (Bernoulli, 1738/1954;
von Neumann & Morgenstern, 1947) dominated the study of risky decision making,
Prospect Theory ( PT; Kahneman & Tversky, 1979) became the focus of empirical
studies of decision making for the last two decades. PT was improved to include recent
improvements. The newer form, Cumulative Prospect Theory (CPT; Tversky &
Kahneman, 1992), has been so influential in the last ten years that it has been
acknowledged in the Nobel Prize in economics (2002) and has been recommended as the
new standard for economic analysis (Camerer, 1998; Starmer, 2000). The PT model
could account for choice paradoxes that were inconsistent with EU. CPT extended PT to
a wider domain and had other advantages over PT that will be discussed in a later section.
However, recent data have been accumulating that systematically violate both PT
and CPT. The accumulation of new evidence against CPT is now more substantial than
the case against EU as presented by Kahneman and Tversky (1979). This paper will
review evidence favoring configural weighting models (CWM) of risky decision making
over the class of rank-dependent expected utility models (RDU), rank-and signdependent utility (RSDU) models, and certain other modern theories as well.
There are two cases to be made in this paper. One is negative, which requires a
review of evidence that refutes PT and CPT as viable descriptions of risky decision
making. The other is positive, to show that configural weight models are available that
handle both the choices that are consistent with the prospect models and those that refute
those models.
Two Theories of Risk Aversion
Before proceeding to a more formal exposition, it will be helpful to consider two
ways of explaining “risk aversion,” the tendency to favor sure things over gambles with
the same or higher expected value. Consider the choice in Problem 1:
Problem 1. Which would you prefer?
F: $45 for sure
or
G:
.50 probability to win $0
.50 probability to win $100
Problem 1 is a choice between a sure thing and a two branch gamble. Each probabilityconsequence pair that is distinct in the presentation is called a branch of the gamble. The
lower branch of G is .5 to win $0 and the higher branch is .5 to win $100.
Most people prefer F ($45 for sure) rather than gamble G, even though G has a
higher expected value of $50; therefore, most people exhibit risk averse preferences in
this case.
Two distinct ways of explaining such risk aversion are illustrated in Figure 1. In
expected utility theory, we assume that people will choose F over G (denoted F  G) if
and only if EU(F) > EU(G). In the left side of Figure 1, there is a nonlinear
transformation from objective money to utility (subjective value). If this utility function,
u(x), is a concave downward function of money, x, then the expected utility of G can 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), this model can explain why people would
choose F over G. The left panel of Figure 1 shows that on the utility continuum, the
balance point (expectation) is at the utility of $33.3. Thus, a person should be indifferent
between a sure gain of $33.3 and gamble G (denoted $33.3 ~ G). The cash value having
the same utility as the gamble is known as the gamble’s certainty equivalent (CE). In this
case, CE(G) = $33.3.
Insert Figure 1 about here.
A second way to explain risk aversion is shown on the right side of Figure 1. In
the CWM illustrated, one third of the weight of the higher branch is taken and assigned to
the lower valued branch. In this case, the transformation from money to utility is linear.
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. Whereas the first theory
attributed risk aversion to the utility function, the second theory attributes risk aversion to
the transfer of weight from the higher to the lower valued branch.
2. NOTATION AND TERMINOLOGY
There are a few definitions used throughout this paper that should be stated in
advance: these are branches of a gamble and the properties of transitivity, consequence
monotonicity, coalescing, branch independence, and stochastic dominance.
Let G = (x, p; y, q; z, r) represent a gamble that yields monetary consequences of
x with probability p, y with probability q, and z otherwise (r = 1 – p – q). This gamble
may also be given the notation G = (x, p; y, q; z), when the context is clear. A branch of
a gamble is a probability (event)-consequence pair that is distinct in the gamble’s display
to the decision-maker. In this case, G is a three branch gamble, and F = (x, p; y) is a twobranch gamble. The two branches of F are (x, p) and (y, 1 – p).
Preferences are said to be transitive if and only if for all A, B, and C, A  B and B
 C implies A  C. Because people may be inconsistent in their preferences from one
choice to the next, the preference relation is usually given a probabilistic operational
definition. For example, Weak Stochastic Transitivity is defined as follows: if the
probability of choosing A over B is greater than 1/2, denoted P(A, B) > 1/2, and if P(B, C)
> 1/2, then P(A, C) > 1/2.
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
have been reported in judgment studies, they are rare in choice (Birnbaum, 1997;
Birnbaum & Sutton, 1992).
Coalescing is the assumption that if two branches of a gamble have the same
consequence, the two branches can be combined by adding their probabilities; and the
gamble’s utility will be unchanged. For example, coalescing holds that the three branch
gamble, G = ($100, .2; $100, .2; $0) should be indifferent to F = ($100, .4; $0). This
seemingly innocent property will play an important role in this paper. Combined with
transitivity, coalescing implies no “event-splitting effects,” as in Starmer and Sugden
(1993).
Branch independence is weaker than Savage’s (1954) “sure thing” axiom. It
holds that if two gambles have an identical branch (probability, consequence), then the
value of the common consequence can be changed without altering the order of the
gambles induced by the other branches. For example, for three-branch gambles, branch
independence requires
(x, p; y, q; z, r)  (x', p'; y', q'; z, r)
if and only if
(1)
(x, p; y, q; z', r)  (x', p'; y', q'; z', r).
where (z, r) is the common branch, the consequences (x, y, z, x', y', z') are all distinct, the
probabilities are not zero and sum to 1 in each gamble. This principle is weaker than
Savage’s independence axiom because it holds for branches of known probability and
also because it does not presume coalescing.
If we restrict the distributions and number of consequences in the gambles to be
the same in all four gambles, then the property is termed restricted branch independence.
In the case of three branch gambles, this requires p = p’, q = q’, and r = r’. The special
case of restricted branch independence in which corresponding consequences retain the
same ranks is termed comonotonic restricted branch independence. The case of branch
independence in which the probability distribution is systematically varied is termed
distribution independence.
Stochastic dominance (first 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 stochastically dominate gamble F. The statement that preferences satisfy
stochastic dominance means that if G dominates F, then F will not be preferred to G.
This paper will use an operational definition as follows: if G dominates F, then the
probability of choosing G over F should be greater than or equal 1/2. This is a very
conservative test, yet it is possible to reject this hypothesis statistically.
3. PROSPECT THEORIES OF RISKY DECISION MAKING
Subjectively Weighted Utility Theory and Prospect 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 models of the form,
n
SWU(G)   w(pi )u(x i )
(2)
i 1
Edwards (1954) discussed both S-shaped and inverse-S shaped functions as candidates
for the weighting function of probability, w(p). When w(p) = p, this model reduces to
EU. Edwards (1962) considered that weighting functions might differ for different
configurations of consequences; he made reference to a “book of weights” with different
pages for different configurations. Edwards (1962, p. 128) wrote:
“The data now available suggest the speculation that there may be exactly five
pages in that book, each page defined by a class of possible payoff arrangements. In
Class 1, all possible outcomes have utilities greater than zero. In Class 2, the worst
possible outcome (or outcomes, if there are several 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.”
Prospect theory (Kahneman & Tversky, 1979) was restricted to gambles with no
more than two nonzero consequences. This means that the 1979 theory is silent on many
of the tests that will be described below unless assumptions are made about how to
extend the model. One way to extend it would be to treat the theory as the special case of
Edwards (1962) model with Classes 1 and 5 and 2 and 4 combined.
Besides the restriction of domain 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, which was assumed to follow Expression 2.
The six principles of editing were as follows:
1. Combination: probabilities associated with identical outcomes are combined. This
principle corresponds to coalescing.
2. Segregation: a riskless component is segregated from risky components.
“the prospect (300, .8; 200, .2) is naturally decomposed into a sure gain of 200 and
the risky prospect (100, .8) (Kahneman & Tversky, 1979, p. 274).”
3. Cancellation: Components shared by both alternatives are discarded from the choice.
“For example, the choice between (200, .2; 100, .5; –50, .3) and (200, .2; 150, .5;
–100, .3) can be reduced by cancellation to a choice between (100, .5; –50, .3) and
(150, .5; –100, .3) (Kahneman & Tversky, 1979, p. 274-275).” If subjects cancel
common components, then they will satisfy branch independence and distribution
independence, which will be taken up in a later section.
4. Dominance: Transparently dominated alternatives are recognized and eliminated.
Without this principle, Equation 2 violates dominance in cases where people do not.
5. Simplification: probabilities and consequences are rounded off.
combined with cancellation, could produce violations of transitivity.
This principle,
6. Priority of Editing: Editing precedes and takes priority over evaluation. Kahneman
and Tversky (1979, p. 275) remarked, “Because the editing operations facilitate the task
of decision, it is assumed that they are performed whenever possible.”
These editing rules are unfortunately imprecise, self-contradictory, and conflict with
Equation 2. This means that PT often has the luxury of predicting opposite results,
depending on which principles are invoked or the order in which they are applied
(Stevenson, Busemeyer, & Naylor, 1991), which makes the theory easy to use as a post
hoc account and difficult to use as a scientific model.
RDU/RSDU/CPT
Cumulative Prospect Theory (Tversky & Kahneman, 1992) seemed to improve on
original prospect theory because it was assumed to apply to gambles with more than two
nonzero consequences, and because it removed the need for some the editing rules of
combination and stochastic dominance. CPT has the same representation as Rank and
Sign-Dependent utility theory (RSDU; Luce & Fishburn, 1991; 1995), though the two
theories are derived from different assumptions (Luce, 2000; Wakker & Tversky, 1993).
Many important papers contributed to the theoretical and empirical development of
these theories (Camerer, 1989; 1992; 1998; Diecidue & Wakker, 2001; Gonzalez & Wu,
1999; Karni & Safra, 1987; Luce, 2000; Luce & Narens, 1985; Machina, 1982; Prelec,
1998; Quiggin, 1982; 1985; 1993; Schmeidler, 1989; Starmer & Sugden, 1989; Tversky
& Wakker, 1995; Yaari, 1987; Wakker, 1994; 1996; Wakker, Erev, & Weber, 1994;
Weber & Kirsner, 1997; Wu, 1994; Wu & Gonzalez, 1996; 1998; 1999). For the sake of
the experiments reviewed here, RDU, RSDU, and CPT will be considered as one class.
The editing principles of PT will be considered separately.
For gambles of n strictly positive consequences, ranked such that
0  x1  x 2 
 xi 
x n , RDU, RSDU, and CPT use the same representation:
n
n
n
RDU (G)  [W ( pj )  W (  p j )]u(xi )
i 1


ji
(3)
j i 1
where RDU(G) is the rank-dependent expected utility of gamble G, W+(P) is the
n
weighting function of decumulative probability, Pi   p j , that monotonically relates
j i
total weight to total decumulative probability, which 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 ) (4)
i n 1


j
j i
j  i1
j
i
This theory satisfies transitivity, consequence monotonicity, coalescing, and
stochastic dominance. However, it violates restricted branch independence and
distribution independence (except when the weighting functions are linear) though it
satisfies restricted comonotonic branch independence.
4. CONFIGURAL WEIGHTING MODELS
Before describing CWMs and the evidence favoring these models over the class
of RDU/RSDU/CPT models, six preliminary clarifications are in order. First, the CWM
approach described here, like that of PT or CPT, is purely descriptive. The models are
intended to describe and predict humans’ decisions and judgments, rather than prescribe
how people should decide or judge.
Second, the approach is psychological; how stimuli are described, presented or
framed is part of the theory. Indeed, CWMs were originally developed as models of
judgment and perceptual psychophysics (Birnbaum, 1974; Birnbaum, Parducci, &
Gifford, 1971; Birnbaum & Stegner, 1979). Two situations that are objectively equivalent
(under someone’s normative analysis) 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, configural weight models, like PT and CPT, emphasize the role of
weighting of probability; however, it is in the details of this weighting that these theories
differ.
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 or RDU, nor should it be
imbued with excess meaning beyond its roles in reproducing empirical decisions and
judgments. [Footnote 1: The estimated functions in different models will in fact be quite
different. For modest consequences in the range of pocket money, $1 to $150, the utility
(value) function of the CWMs can be approximated with the assumption that utility is
proportional to objective cash value.]
Sixth, Birnbaum (1974, p. 559) remarked that in his configural weight models,
“the weight of an item depends in part on its rank within the set.” Thus, the configural
weight models have some similarities to rank-dependent weighting models. However, in
Birnbaum’s configural weight models, the weight of each branch depends on the ranks of
discrete branch consequences, whereas in the RDU models, weight is a function of
cumulative probability. 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).
Two configural weight models, the Rank-affected Multiplicative Weights (RAM)
and the Transfer of Attention Exchange (TAX) models have in common that utilities are
weighted by functions of probability and the ranks of branches in the gamble, divided by
total weight.
RAM and TAX
In the RAM model, the weight of each branch of a gamble is the product of a
function of the branch’s probability multiplied by a constant that depends on the rank and
augmented sign of the branch’s consequence. Augmented sign takes on three levels, –, +,
and 0. Rank refers to the rank of the branch’s consequence relative to the other discrete
branches in the gamble, ranked such that x1  x2 
 xn .
n
 a(i,s )t(p )u(x )
i
RAMU (G) 
i
i
i 1
n
 a(i,s )t(p )
i
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
(5)
branch’s consequence. For two, three, and four branch gambles on strictly positive
consequences, it has been found that the rank weights are approximately equal to their
branch ranks, t(p) can be approximated by a power function, t(p) = pb, with exponent, b <
1, and u(x) can be approximated by u(x) = x for 0 < x < $150.
For 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 p 100
, which gives a good


1 p  2 (1  p)
approximation of empirical data of Tversky & Kahneman (1992). A graph of their data,
with predictions of the RAM model are shown in Birnbaum (1997, Figure 9). Note that
although there are two rank weights (with weights 1 and 2, respectively) in these binary
gambles, there is only one free parameter in this situation since rank (and augmented
sign) weights could be multiplied by any constant and it would drop out of the equation.
One way to normalize rank weights is to force their sum to 1, which means that the
normalized weights would also describe the weights of the lower and higher of two
equally likely branches.
To fit Tversky and Kahneman’s (1992) data, normalized weights are .37 and .63
for the higher and lower ranked branches, respectively, and  = .56.
In the TAX model, weight is transferred from one branch to another. This
transfer can be thought of as representing the transfer of attention among branches,
reflecting the importance of each branch. In the simple version of TAX described here,
the amount of weight transferred between two branches is a fixed proportion of the
probability weighting of the branch losing weight. If there were no configural effects,
each branch would have a weight equal to a function of probability, t(p). However, if
lower-ranked branches have more importance (as they would to a risk-averse person),
then weight will be transferred from higher-ranked branches to lower-ranked branches.
For two branch gambles with positive consequences, it has been found that the
approximate transfer is 1/3 (as illustrated in the right side of Figure 1). In three branch
gambles with positive consequences, the approximate transfer is 1/4 of the weight of any
higher valued branch to each lower-valued branch.
n
TAXU(G) 
n
n
 t( pi )u(xi )    [u(x i )  u(x j )] (i, j)
i 1
i 1 j  i
n
(6)
 t( p )
i
i 1
where the
Both TAX and RAM satisfy transitivity and consequence monotonicity.
However, they systematically violate both restricted branch independence and coalescing.
The violations of coalescing can be used to create violations of stochastic dominance and
other properties.
5. VIOLATIONS OF STOCHASTIC DOMINANCE
Birnbaum (1997) deduced that his configural weight models of risky decision
making would violate stochastic dominance when choices were constructed from a
special recipe. Consider the following choice:
Problem 2. If you have a chance to reach into one of two urns, and win a prize
determined by the color of marble you draw blindfolded, from which urn would you
choose to draw?
I: 90 red marbles to win $96
J:
85 red marbles to win $96
05 blue marbles to win $14
05 blue marbles to win $90
05 white marbles to win $12
10 white marbles to win $12
Birnbaum and Navarrete (1998) tested this prediction and found that about 70%
of 100 undergraduates tested violated first stochastic dominance in four choices like
Problem 2.
In a recent study (Birnbaum, stochastic dominance), 74% of 330 participants
chose to draw from J, even though I dominates J. For an n = 330, the 95% confidence
interval for p = 1/2 is from 45% to 55%. The finding of 74% violations corresponds to z
= 6.23, which is extremely unlikely given the null hypothesis that people are just
guessing and are indifferent between I and J. Therefore, we can reject that null hypothesis
in favor of the hypothesis that the percentage of violations exceeds 50%.
Following Birnbaum and Navarrete (1998), a number of subsequent studies have
confirmed that these violations are observed when the gambles are presented in a variety
of formats. The violations are observed with or without branch juxtaposition, with or
without real monetary incentives, when branches are listed in order of their consequences
or reverse order. They are observed when probabilities are presented as pie charts,
percentages, probabilities, natural frequencies, or numerated lists, with or without event
framing (Tversky & Kahneman, 1986), and with other variations (Birnbaum, 1999b;
2000; Birnbaum, Patton, & Lott, 1999; Birnbaum & Martin, in press; Birnbaum,
submitted). Thus, these violations of first stochastic dominance been shown to be a
robust finding that needs explanation by any descriptive theory.
Birnbaum’s (1997) recipe is illustrated in the lower portion of Figure 1. Start with
a root gamble, G0, = ($96, .9; $12, .1). Next, split the lower-valued branch (.1 to win
$12) into two parts, one of which has a slightly better consequence (.05 to win $14 and
.05 to win $12), yielding G+ = ($96, .9; $14, .05; $12, .05). G+ dominates G0. However,
according to Birnbaum’s TAX and RAM models, G+ should seem worse than G0,
because the increase in total weight of the lower branches outweighs the increase in the
.05 sliver’s consequence from $12 to $14.
Starting again with G0,, we now split the higher valued branch of G0,,
constructing G– = ($96, .85; $90, .05; $12, .10), which is dominated by G0. According to
the configural weight models, this split increases the total weight of the higher branches,
which improves the gamble despite the decrease in the .05 sliver’s consequence from $96
to $90. The configural weight RAM and TAX models imply that people will prefer G–
over G+, in violation of stochastic dominance.
Insert Figure 1 about here.
Transitivity, coalescing, and consequence monotonicity imply satisfaction of
stochastic dominance in this recipe: G0, = ($96, .9; $12, .1).~ ($96, .9; $12, .05; $12, .05),
by coalescing. By consequence monotonicity, G+ =($96, .9; $14, .05; $12, .05).($96,
.9; $12, .05; $12, .05), by consequence monotonicity. G0, = ($96, .9; $12, .1).~ ($96, .85;
$96, .05; $12, .10), by coalescing, and ($96, .85; $96, .05; $12, .10) ($96, .85; $90, .05;
$12, .10) = G–, by consequence monotonicity. By transitivity, G+ = ($96, .90; $14, .05;
$12, .05)  G0  ($96, .85; $90, .05; $12, .10) = G–, so G+  G–. This derivation shows
that if people obeyed these three principles, they would not show this violation, except by
chance or error. Because people show systematic violations, at least one of these
assumptions must not be a correct descriptive principle.
Systematic violations of stochastic dominance refute the class of models that
includes RDU/RSDU/CPT, which imply stochastic dominance. Other descriptive
theories that assume or imply stochastic dominance are also refuted as by this finding
(Becker & Sarin, 1987; Lopes & Oden, 1999; Machina, 1982).
For the RAM model, let t(p) = p.7, and suppose the weights of branch ranks are 1, 2,
and 3 for the highest, middle, and lowest valued branches. For probabilities of (.9, .05, and
.05), the values of t(p) are (.929, .123, and .123). We multiply each of these by the rank
weights, (1.929,2 .123,3 .123) =(.929, .246,.369). Next, divide each product by the sum of
these products, 1.544, to find the relative weights (.602, .159, .239). For cash values less than
$150, we can let u(x) = x, so the RAM utility of G+ is .602 96 .159 14  .239 12 = 62.89.
For G–, the probabilities are (.85, .05, .10), the relative weights are (.514, .141, .345), and the
RAM utility is 66.20. Thus, the RAM model implies violations of stochastic dominance in
this case.
For the TAX model, we can also let t(p) = p, and u(x) = x. Suppose each branch gives
up 1/4 of its probability weight to any lower branch. For G+, the probability weights are
(.929, .123, and .123). The configural weight of the highest branch is .929 – (2/4)(.929) =
.464, since this branch gives up one fourth of its weight to the middle branch and one fourth
of its weight to the lowest branch. The configural weight of the middle branch is .123 +
(1/4)(.929) – (1/4)(.123) = .324, since this branch gains one fourth of the weight of the highest
branch and gives up one fourth of its weight to the lowest valued branch. Finally, the weight
of the lowest branch is .123 + (1/4)(.929) + (1/4)(.123) = .386. Again, we divide each weight
by the sum of the weights (1.174), giving weights of (.395, .276, .329). Calculating the TAX
utility of G+, we have 45.8. For G–, the weights are (.367, .260, .373), for a utility of only
63.1. Thus, the TAX model also violates stochastic dominance in this case.
Two calculators are freely available at the following URLs:
http://psych.fullerton.edu/mbirnbaum/cwtcalculator.htm
http://psych.fullerton.edu/mbirnbaum/taxcalculator.htm
(These calculators also allow calculations of the CPT model of Tversky and Kahneman, 1992,
and the revised version in Tversky and Wakker, 1995.)
6. 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; Birnbaum & Martin, in press). As
shown in the upper portion of Figure 1, one can split the lowest branch of G–, which
makes the split version, GS–, seem worse, and split the highest branch of G+, which
makes its split version, GS+ seem better, so in the split form, GS+ seems better than GS–.
As predicted by the configural weight RAM and TAX models, with parameters estimated
from previous data, most people prefer GS+ over GS–, and G– over G+, creating a
powerful event-splitting effect, apparently a violation of coalescing (Birnbaum, 1999b;
2000; 2001b; Birnbaum & Martin, in press). Consider Problems 3 and 4:
Problem 3 From which urn would you prefer to draw a ticket blindly at random?
I: 90
tickets to win $96
J: 85 tickets to win $96
05 tickets to win $14
05 tickets to win $90
05 tickets to win $12
10 tickets to win $12
Problem 4: From which urn would you prefer to draw a ticket blindly at random?
U: 85 tickets to win $96
V: 85 tickets to win $96
05 tickets to win $96
05 tickets to win $90
05 tickets to win $14
05 tickets to win $12
05 tickets to win $12
05 tickets to win $12
In a recent test (Birnbaum, 2003, tickets/lists), 342 participants were asked to
choose between I and J and between U and V, among other choices. There were 71%
violations in the coalesced form (Problem 3) and only 5.6% in the split form of the same
choice (Problem 4). It was found that 224 participants (65.5%) preferred J to I and U to
V, 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, giving z = 14.3. There were 16 (4.7%) who violated stochastic
dominance on both decisions, as if they coalesced before choosing, and only 27.8%
satisfied stochastic dominance on both choices.
According to the class of RDU/RSDU/CPT models, people should make the same
choice between I and J as they do between U and V; therefore, this event-splitting effect
(the reversal of preferences between Problems 3 and 4) refutes that class of models.
7. PREDICTED SATISFACTION OF STOCHASTIC DOMINANCE
Perhaps people are ignoring probability, and just averaging the values of the
consequences, or maybe they are counting the number of branches that favor one gamble
or the other. If so, they should continue to violate stochastic dominance on Problem 5.
If the configural weight models are correct, however, people should not always
violate stochastic dominance. For example, with previous parameters, we can calculate
from the TAX model that people should satisfy stochastic dominance in Problem 5:
Problem 5.
K: 90 red marbles to win $96
L: 25 red marbles to win $96
05 blue marbles to win $14
05 blue marbles to win $90
05 white marbles to win $12
70 white marbles to win $12
Birnbaum (stochastic dominance, study 2) found that of 232 participants, 72%
violated stochastic dominance in the comparison of I and J, but only 34% violated
stochastic dominance in the choice between K and L. 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).
Thus, people do not always violate stochastic dominance. They largely satisfy it
when splitting is used, as in Problem 4, and a 66% majority satisfy it when probability of
the highest consequence is reduced from 85% to 25%. Intermediate probabilities yielded
intermediate results, suggesting that people are attending to probability and utilizing it in
a fashion predicted by the configural weight models.
8. DISSECTION OF THE ALLAIS CONSTANT CONSEQUENCE PARADOX
The constant consequence paradox of Allais (1953, 1979) can be illustrated with
the following two choices:
A:
$1M for sure
B:
.10 to win $2M
.89 to win $1M
.01 to win $0
C:
.11 to win $1M
.89 to win $0
D:
.10 to win $2M
.90 to win $0
Expected Utility (EU) theory assumes that Gamble A is preferred to B if and only
if the EU of A exceeds that of B. This assumption is written, A  B EU(A) > EU(B),
where the EU of a gamble, G = (x1, p1, x2, p2;…xi, pi; …;xn, pn) can be expressed as
follows:
n
EU(G)   piu(x i )
(7)
i1
According to EU, A is preferred to B iff u($1M) > .10u($2M) + .89u($1M)
+.01u($0). Subtracting .89u($1M) from each side, it follows that .11u($1M) >
.10u($2M)+.01u($0). Adding .89u(0) to both sides, we have .11u($1M)+.89u($0) >
.10u($2M)+.90u($0), which holds if and only if C  D. Thus, from EU theory, one can
deduce that A
B C
D . However, many people choose A over B and prefer D over
C. This pattern of empirical choices violates the implication of EU theory, so such
results are termed “paradoxical.”
It is useful to decompose this type of paradox into simpler premises (Birnbaum,
1999a). Transitivity, coalescing, and restricted branch independence imply Allais
independence, as illustrated below:
A:
$1M for sure
B:
.10 to win $2M
.89 to win $1M
.01 to win $0
B:
.10 to win $2M
.89 to win $1M
.01 to win $0
B’:
.10 to win $2M
.89 to win $0
.01 to win $0
D:
.10 to win $2M
.90 to win $0
(coalescing & transitivity)
A’:
.10 to win $1M
.89 to win $1M
.01 to win $1M
 (restricted branch independence)
A”:
.10 to win $1M
.89 to win $0
.01 to win $1M
 (coalescing & transitivity)
C:
.11 to win $1M
.89 to win $0
The first step converts A to its split form, A’; A’ should be indifferent to A by coalescing,
and by transitivity, it should be preferred to B. From the third step, the consequence on
the common branch (.89 to win $1M) has been changed to $0 on both sides, so by
restricted branch independence, A” should be preferred to B’. By coalescing branches
with the same consequences on both sides, we see that C should be preferred to D.
This derivation shows that if people obeyed these three principles, they would not
show this paradox, except by chance or error. Because people show systematic
violations, at least one of these assumptions must be false. The term Allais independence
is meant to include all such derivations with arbitrary values for probabilities and
consequences that can be deduced from the premises of transitivity, coalescing, and
restricted branch independence. Systematic violations are termed Allais paradoxes.
Different theories attribute these paradoxes to different causes (Birnbaum, 1999a),
as listed in Table 1. SWU (including PT) attributes them to violations of coalescing. In
contrast, the class of RDEU, RSDU, and CPT explain the paradox by violations of
restricted branch independence. The configural weight, RAM and TAX models imply
that both coalescing and restricted branch independence can be systematically violated
Insert Table 1 about here.
Birnbaum (submitted, Allais1) tested among the theories in Table 1. Table 2 lists
Problems 6-10, which test each step in the derivation. According to EU, the choice in
each of these problems should be the same, because all of the choices have a common
branch (80 marbles to win $2 in Choices 6 and 7, $40 in Choice 8, or $98 in Choices 9
and 10. Birnbaum presented these choices, separated by others to 349 participants,
whose data in Table 2 are summarized 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. With n = 349, and p = 1/2, percentages less than 44.6%
or greater than 55.3% are significantly different from 50%, with  = .05. Thus, each
percentage, except for Problem 8 is significantly different from 50%. By the test of
correlated proportions, each successive comparison is significant.
Insert Table 2 about here.
The class of RDU/RSDU/CPT models and the TAX and RAM models, as fit to
previous data, all agree on the predicted choices in Problems 6, 8, and 10; that is, they all
predict the violations of Allais independence in those three problems. However, the class
of RDU/RSDU/CPT implies that Choice 6 and 7 should agree, except for error, and that
Choices 9 and 10 should agree as well, since those test coalescing; whereas, the RAM
and TAX models predict reversals between 6 and 7 and between 9 and 10. Put another
way, RDU/RSDU/CPT models attribute violations of Allais independence to violations
of branch independence, and not coalescing. Branch independence is tested between
Choices 7 and 9. However, the majority preference in 7 is opposite that of 6 and the
majority choice in 9 is opposite that of 10. Thus, as predicted by the CWMs violations of
branch independence are opposite of what is needed if branch independence is to explain
Allais paradoxes.
Now consider if the editing rule of cancellation can account for these results. If
people canceled the common branch in Choices 7 and 9, they would show no violations
of restricted branch independence. If they only cancelled on some proportion of the
trials, there would be violations of branch independence that would agree with Choices 6,
and 9. However, the observed violations are substantial and opposite the direction
needed to allow branch independence to account for Allais paradoxes. It might be
understandable if violations of branch independence in the transparent tests might be
attenuated, but they should still be in the same direction as the Allais paradox if they are
present.
According to the configural weight models, there are violations of both branch
independence and coalescing. Both RAM and TAX predict the model choice with
parameters estimated from previous studies. Furthermore, violations of branch
independence in these models are opposite from those predicted by the inverse-S
weighting of CPT. According to TAX/RAM, the cause of the Allais constant
consequence paradoxes is violation of coalescing, and violations of branch independence
actually reduce the magnitude of Allais paradoxes. Note that both splitting and
coalescing operations make R worse and S better as we proceed from Choice 6 to 10.
Splitting the lower branch makes R worse, and coalescing the upper branch makes it
worse. Similarly, splitting the higher branch from Problem 6 to 7 improves S as does
coalescing the lower branch from Problem 9 to 10.
9. EVENT-SPLITTING INDEPENDENCE
Consider Problems 11-18 in Table 3. These problems break down Allais
paradoxes further in order to test event-splitting independence (Birnbaum, Allais2).
Insert Table 3 about here.
Event-splitting independence is the assumption that if there are event-splitting
effects, they should operate in the same way. For example,
($50, .10; $50, .05; $7)  ($50, .15; $7)

($50, .10; $50, .05; $100)  ($50, .15; $100)
Note that in the first line, splitting .15 to win $50 produces an improvement in the
gamble. ESI requires that when the same branch is split in the context another gamble, in
this case, where the other consequence is $100 instead of $7, splitting should have the
same directional effect. ESI is implied by “stripped” prospect theory (Starmer & Sugden,
1993). See Birnbaum and Navarrete (1998).
Data in Table 3 are based on 203 participants. Note that the positions of S and R
are reversed with respect to Table 2, so the trend from Choice 11 to 18 agrees with the
trend in Table 2. The difference between Problems 11 and 13 is that the branch of 15
marbles to win $50 has been split into 10 marbles to win $50 and 5 more to win $50.
This splitting of the highest consequence in S produces a decrease in the percentage who
choose R from 78% to 52% (z = 4.12). (. This decrease is expected if S was improved by
splitting its highest branch. Now consider Problems 18 and 16. These also represent a
branch of 15 marbles to win $50, except that now this branch represents the lower branch
of the gamble, instead of the higher. The percentage choosing R now increases from
25% (Problem 18) to 37% (Problem16), as if the same splitting of the same branch made
S worse (z = 5). According to the TAX and RAM models, splitting the higher branch
should improve the gamble and splitting the lower branch should make the gamble worse.
Thus, RAM and TAX violate event-splitting independence.
The class of RDU/RSDU/CPT models implies no event-splitting effects (no
changes in choice percentage among Problems 11-14 and among 15-18. Clearly, those
models can be rejected. The class of SWU/PT models (aside from the editing rule of
combination) imply event-splitting independence, which is the assumption that splitting
the same branch into the same pieces should have the same effect, independent of
whether the branch is the lowest or highest branch in the gamble. The prior parameters
do predict the trends of these separate splitting operations fairly well. However, the
magnitude of splitting the upper branch of R has a greater effect than predicted in
Problem 17, which is the only case in Tables 2 and 3 in which the prior TAX model fails
to predict the modal choice.
10. RESTRICTED BRANCH INDEPENDENCE
Restricted branch independence should be satisfied by SWU (including the
extension to PT with the editing rule of cancellation). However, this property will be
violated according to RDU/RSDU/CPT and CWMs. The direction of violations,
however, is opposite in the two classes of models. A number of studies have been
conducted on this property. When violations have been observed, there are more
violations of the form SR’ than the opposite, RS’ (Birnbaum, 1999b; 2001; Birnbaum &
Beeghley, 1997; Birnbaum & Chavez, 1997; Birnbaum & McIntosh, 1996; Birnbaum &
Navarrete, 1998; Birnbaum & Veira, 1998; Birnbaum & Zimmermann, 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 (Wu and Gonzales,
1996; 1998; 1999; Tversky & Wakker, 1995).
The observed pattern is, however, consistent with the RAM and TAX models.
11. DISTRIBUTION INDEPENDENCE
Distribution independence is an interesting property because RDU/RSDU/CPT
models imply systematic violations, but the property should be satisfied, according to the
RAM model. It is violated, according to TAX, but in the opposite way from that
predicted by the inverse-S weighting function of CPT. Consider the choices in Problems
21 and 22:
Problem 21
Problem 22
Gamble S
Gamble R
S:
R:
.59 to win $4
.59 to win $4
.20 to win $45
.20 to win $11
.20 to win $45
.20 to win $97
.01 to win $110
.01 to win $110
S’: .01 to win $4
R’: .01 to win $4
.20 to win $45
.20 to win $11
.20 to win $45
.20 to win $97
.59 to win $110
.59 to win $110
S
R
.66
21.7
20.6
.49
49.8
50.0
Note that the trade-off between two branches of .1 is nested within a distribution
in which these two weights are either near the low end of cumulative probability or near
the upper end. According to CPT, this distribution should change the relative sizes of the
weights of these common branches, producing violations.
According to CPT model and parameters of Tversky and Kahneman (1992),
people should prefer R to S and S’ to R’. According to the RAM model there should be
no violations of this property. According to SWU and PT with cancellation, there should
be no violations. According to the TAX model and parameters of Birnbaum (1999a),
people should choose S over R and R’ over S’. Indeed, this pattern was the modal pattern
observed by Birnbaum and Chavez (1997) in a study with 100 participants. There were
23 who chose S over R and R’ over S’ against only 6 with the opposite reversal of
preferences (Birnbaum & Chavez, 1997).
12. TAIL INDEPENDENCE
Wu (1994) reported systematic violations of ordinal independence. The tests used
could also be called “tail” independence, since they test whether the “tail” of a
distribution can be used to reverse preferences. The property can be viewed as a
combination of transitivity, coalescing, and restricted comonotonic branch independence.
Ordinal independence should therefore be satisfied according to the class of
RDU/.RSDU/CPT models.
Birnbaum (2001) reported a test of upper tail independence based on an example
by Wu, with n = 1438 people who participated via the WWW. Note that people should
prefer t = ($92, .48; $0)  s = ($92, .43; $68, .07; $0) iff they prefer ($92, .43; $92, .05;
$0)  ($92, .43; $68, .07; $0), by coalescing and transitivity. Assuming restricted
comonotonic branch independence, we can change the common consequence on the .43
branch from $92 to $97, which means f = ($97; .43; $92, .05; $0)  e = ($97, .43; $68,
.07; $0). Instead, significantly more than half prefer f to e, and s to t, contradicting this
property.
First Gamble, S
s: .50 to win $0
.07 to win $68
.43 to win $92
Second Gamble, R
t: .52 to win $0
.48 to win $92
% t,f
TAX
34.0
32.3
29.8
e: .50 to win $0
f: .52 to win $0
.07 to win $68
.05 to win $92
.43 to win $97
.43 to win $97
62.0
33.4
36.7
13. UPPER AND LOWER CUMULATIVE INDEPENDENCE
Birnbaum (1997) noted that violations of branch independence 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 were to EU. Lower and Upper Cumulative Independence can be
deduced from transitivity, consequence monotonicity, coalescing, and comonotonic
restricted branch independence.
Upper cumulative Independence
pies*
TAX
n = 305
.10 to win $40
.10 to win $10
.10 to win $44
.10 to win $98
.80 to win $110
.80 to win $110
.20 to win $40
.10 to win $10
.80 to win $98
.90 to win $98
70*
65.03
69.59
42
68.04
58.29
pies*
TAX
Lower cumulative Independence
n = 305
.90 to win $3
.90 to win $3
.05 to win $12
.05 to win $48
.05 to win $12
.05 to win $52
62*
8.80
10.27
.95 to win $12
.90 to win $12
.05 to win $96
.10 to win $52
26
68.04
58.29
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Table 1. Comparison of Decision Theories
Branch Independence
Coalescing
Satisfied
Violated
Satisfied
EU (OP*/CPT*)
RDEU/RSDU/CPT*
Violated
SWU/OP*
RAM/TAX
Notes: Expected Utility Theory (EU) satisfies both properties. PT= Original Prospect
Theory and CPT = Cumulative Prospect Theory; these theories make different
predictions with and without their editing rules. The editing rule of combination
produces satisfaction of coalescing and the editing rule of cancellation implies
branch independence. CPT has the same representation as Rank Dependent
Expected Utility (RDU). With or without the editing rule of combination, CPT
satisfies coalescing. The Rank Affected Multiplicative (RAM) and Transfer of
Attention Exchange (TAX) models are configural weight models that violate both
branch independence and coalescing.
Table 2. Dissection of Allais Paradox into Branch Independence and Coalescing (Series
A).
Problem
Framed Version
Condition
Prior
No.
6
7
8
9
10
TAX model
First Gamble
Second Gamble
R
S
10 black marbles to win $98
20 black marbles to win $40
90 purple marbles to win $2
80 purple marbles to win $2
10 red marbles to win $98
10 red marbles to win $40
10 blue marbles to win $2
10 blue marbles to win $40
80 white marbles to win $2
80 white marbles to win $2
10 red marbles to win $98
10 red marbles to win $40
80 blue marbles to win $40
80 blue marbles to win $40
10 white marbles to win $2
10 white marbles to win $40
80 red marbles to win $98
80 red marbles to win $98
10 blue marbles to win $98
10 blue marbles to win $40
10 white marbles to win $2
10 white marbles to win $40
90 red marbles to win $98
80 red marbles to win $98
10 white marbles to win $2
20 white marbles to win $40
R
S
0.378
13.3
9.0
0.644
9.6
11.1
0.537
30.6
40.0
0.430
62.6
59.8
0.777
54.7
68.0
N=349
TABLE 3. Dissection of Allais Paradox for tests of Branch Independence, Coalescing
and Event-Splitting Independence ((Series B).
Choice
Choice (as in Condition FU)
Condition
No.
11
12
Prior
TAX model
First Gamble
Second Gamble
S
R
15 red marbles to win $50
10 blue marbles to win $100
85 black marbles to win $7
90 white marbles to win $7
15 red marbles to win $50
10 black marbles to win $100
85 black marbles to win $7
05 purple marbles to win $7
FU
UF S
R
0.783
13.6
18.0
0.704
13.6
14.6
0.517
15.6
18.0
0.443
15.6
14.6
0.695
68.4
69.7
0.367
68.4
62.0
0.702
75.7
69.7
0.249
75.7
62.0
85 green marbles to win $7
13
10 red marbles to win $50
10 blue marbles to win $100
05 blue marbles to win $50
90 white marbles to win $7
85 white marbles to win $7
14
15
16
10 red marbles to win $50
10 black marbles to win $100
05 blue marbles to win $50
05 purple marbles to win $7
85 white marbles to win $7
85 green marbles to win $7
85 red marbles to win $100
85 black marbles to win $100
10 white marbles to win $50
10 yellow marbles to win $100
05 blue marbles to win $50
05 purple marbles to win $7
85 red marbles to win $100
95 red marbles to win $100
10 white marbles to win $50
05 white marbles to win $7
05 blue marbles to win $50
17
85 black marbles to win $100
85 black marbles to win $100
15 yellow marbles to win $50
10 yellow marbles to win $100
05 purple marbles to win $7
18
85 black marbles to win $100
95 red marbles to win $100
15 yellow marbles to win $50
05 white marbles to win $7
Notes: Each entry is the percentage of people, in each condition, who chose the second
gamble,
Figure 1. Two explanations of risk aversion, nonlinear utility versus weighting. In each
case, the expected value (or utility) is the center of gravity. On the left, gamble G =
($100, .5; $0) is represented as a probability distribution with half of its weight at 0 and
half at 100. The expected value is 50. In the lower left panel, distance along the scale
corresponds to utility, by the function u(x) = x.63. Marginal differences in utility between
$20 increments decrease as one goes up the scale. The balance point on the utility axis
corresponds to $33.3, which is the certainty equivalent of this gamble. The right side of
the figure shows a configural weight theory of the same result: if one third of the weight
of the higher branch is given to the lower branch (to win $0), then the certainty
equivalent would also be $33.3, even when utility is proportional to cash.
Figure 2. Cultivating and weeding out violations of stochastic dominance. Starting at the
root, G0 = ($96, .9; $12,.1), split the upper branch to create G– = ($96, .85; $90, .05; $12,
.10), which is dominated by G0. Splitting the lower branch of G0, create G+ = ($96, .9;
$14, .05; $12, .05), which dominates G0. According to configural weight models, G– is
preferred to G+, because splitting increased the relative weight of the higher or lower
branches, respectively. Another round of splitting weeds out violations to low levels.