
\documentclass{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphicx}
\usepackage{amsmath}

%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Created=Tue Aug 15 08:58:43 2000}
%TCIDATA{LastRevised=Wednesday, January 01, 2003 03:45:08}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="General\Blank Document">}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=LaTeX article (bright).cst}

\renewcommand{\baselinestretch}{1.5}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}

\begin{document}


Dec 31, 2002

\begin{center}
{\LARGE Tests of Stochastic Dominance and Cumulative Independence in}

{\LARGE Buying and Selling Prices of Gambles\bigskip }

{\normalsize Michael H. Birnbaum and Sherry Yeary}

{\normalsize California State University, Fullerton and}

{\normalsize Decision Research Center, Fullerton}

\bigskip
\end{center}

Mailing Address: \ \qquad Michael H. Birnbaum

\qquad \qquad \qquad \qquad \qquad Department of Psychology H-830M

\qquad \qquad \qquad \qquad \qquad California State University, Fullerton

\qquad \qquad \qquad \qquad \qquad P. O. Box 6846

\qquad \qquad \qquad \qquad \qquad Fullerton, CA 92834-6846

Phones: (714)-278-2102\ or (714)-278-7653 (Psychology Dept.)

fax:\qquad \qquad (714) 278-7134

e-mail: mbirnbaum@fullerton.edu\medskip

Running Head:buying and selling prices\medskip

Footnotes

Author's address: Department of Psychology, C.S.U.F., P.O. Box 6846,
Fullerton, CA 92834-6846. Email address: mbirnbaum@fullerton.edu.

Thanks are due to Teresa Martin and R. Duncan Luce for useful discussions of
these ideas and comments on an earlier draft. This research was supported by
National Science Foundation Grants, SBR-9410572 and SES 9986436 to C.S.U.F.

\pagebreak

\begin{center}
\bigskip Abstract
\end{center}

Judges estimated the values of gambles from the viewpoints of\ both buyers
and sellers. Sellers prices exceed buyers prices and these two judgments are
not monotonically related to each other. Both types of judgments
systematically violate stochastic dominance, lower and upper cumulative
independence, similar to violations reported previously with choices between
gambles. Judged buying and selling prices also showed event-splitting
effects, violations of branch independence, and violations of
event-splitting independence. Configural weight models developed to describe
previous judgment and choice data give reasonable fits to the new data.

\bigskip

Keywords: branch independence, buying price, choice, coalescing, configural
weighting, contingent valuation, cumulative independence, cumulative
prospect theory, endowment effect, event-splitting, expected utility,
gambles, judge's point of view, loss aversion, preference reversals, rank
and sign-dependent utility theory, selling price, stochastic dominance,
utility, viewpoint effects, willingness to accept (WTA), willingness to pay
(WTP).

\begin{center}
\pagebreak
\end{center}

This study was designed to test whether judgments of buying and selling
prices show similar characteristics to those observed in recent studies of
choices between gambles. Some properties of choice have been attributed to
processes that are specific only to choice. This paper tests those theories
by asking if the same results are obtained when the gambles are evaluated
separately, as they are in judgment tasks, rather than compared directly. \
If the same results are obtained in judgment, then theories that make use of
contrasts or comparisions of dimensions between gambles lose attraction,
compared to explanations that attribute these effects to the process of
evaluation of the gambles. 

Birnbaum, Yeary, Luce, \& Zhou (2002) reported results of a comparison of
fit of models to judgments of buying and selling prices of binary gambles. \
That study compared three distinct approaches to the analysis of buying and
selling prices. \ They noted that three models based on Cumulative Prospect
Theory (Tversky \& Kahneman, 1992) combined with a theory of Loss Aversion
(Tversky \& Kahneman, 1991) failed to give a good account of judged values.
\ Six models based on Luce's (2000) theory of joint receipts did better, but
did not fit as well as configural weight models derived from Birnbaum and
Stegner's (1979) models. That study analyzed four properties: complementary
symmetry, lower and upper asymptotic independence, and consequence
monotonicity. The data systematically violated complementary symmetry, which
refutes the best-fitting of the loss aversion models combined with CPT.

This paper analyzes properties of buying and selling prices that involve
three branch gambles (gambles with three distinct outcomes): stochastic
dominance, coalescing, lower and upper cumulative independence, restricted
branch independence, and event-splitting independence. Recent studies of
these properties in choice experiments have shown that choices between
gambles violate properties that are assumed or implied by Rank Dependent
Expected Utility (RDEU) Theory (Quiggin, 1993), Rank and Sign Dependent
Utility (RSDU) Theory (Luce \& Fishburn, 1991; Luce, 2000), and Cumulative
Prospect Theory (CPT) (Tversky \& Kahneman, 1992; Wakker \& Tversky, 1993).
\ These violations in choice are to this class of Rank Dependent models as
the Allais Paradoxes (Allais \& Hagen, 1979) are to Expected Utility (EU):
violation of the properties represent contradictions within the theories. 

There are two types of explanations for these violations: The first type of
explanation describes the violations in terms of the process by which
gambles are evaluated. \ Birnbaum (1999a; 1999b) described these results by
configural weight models which assign an overall utility to each gamble and
assume that people tend to choose the gamble with the higher computed
evaluation. \ The second type of explanation attributes the violations of
processes of choice between the gambles. \ An example of this type of
explanation are the editing rules of prospect theory (Kahneman \& Tversky,
1979) such as simplification and cancellation of (nearly) equal components
of the gambles, prior to evaluation. See also Leland (1994), and Starmer
(1999). Models of the comparison processes can explain intransitivities of
preference (Tversky, 1969; Starmer, 1999; Birnbaum, Patton, \& Lott, 1999).

The properties tested in this paper rest on two key ideas. \ First is the
definition of a \textit{branch} in a gamble, which is a
probability-consequence pair that is distinct in the gamble's presentation.
\ The same gamble can be presented in different ways (with different numbers
of branches), and it is important to keep these (objectively) equivalent
ways of presenting the same gamble distinct. \ 

The second key to this paper is a very simple property known as \textit{%
coalescing} (Birnbaum, 1997; Birnbaum \& Navarrete, 1998; Luce, 1998). \
According to this property, a three branch gamble with two branches having
identical consequences can be simplified to a two-branch gamble by adding
the probabilities of the branches. \ Let $g=(x,p;y,q,z,r)$ represent a
three-branch gamble to win cash prizes of $x,$ $y,$ or $z$, with
probabilities of $p$, $q$, or $r=1-p-q$, respectively. \ Suppose that $y=z$.
\ If coalescing holds, $g=(x,p;y,q;y,r)$ should be indifferent to $g^{\prime
}=(x,p;y,q+r).$ \ For example, if $g=(\$0,.5;\$100,.2;\$100,.3)$, then
people should treat $g^{\prime }=(\$0,.5;\$100,.5)$ as equal in utility to $g
$ Coalescing was assumed as an editing principle of \textit{combination} by
Kahneman and Tversky (1979) and it is automatically satisfied by CPT and the
other rank-dependent models (Luce, 2000; Quiggin, 1993). Violation of this
property may be the cause of the violations of rank-dependent models in
previous studies of choice. \ Configural weight models systematically
violate coalescing in a particular way, which allowed those models to
successfully predict violations that have been observed in choice
experiments.

Three other definitions used in this paper agree with common useage. \
Denote gamble $f$ is preferred to $g$, as follows: $f\succ g$, where $\succ $
is the preference relation in a choice study. \textit{Transitivity} means
that if $f\succ g$ and $g\succ h$, then $f\succ h.$ \textit{Consequence
monotonicity} means that if a gamble is modified by increasing the value of
any consequence, holding everything else constant in the gamble, the
modified gamble (with the better consequence) should be preferred to the
original gamble. If there are two nonidentical gambles, $f$ and $g$, such
that the probability of getting prize $x$ or better in gamble $f$ is always
better than or equal to the probability of getting $x$ or better in gamble $g
$, then $f$ is said to stochastically dominate $g$. This relation is
sometimes called first stochastic dominance. \ If choices \textit{satisfy
stochastic dominance}, a person should not prefer the dominated gamble $g$
over $f$ in a direct choice. These definitions, as well as others, described
below, are listed in Table 1.

\begin{equation*}
\fbox{\textit{Insert Table 1 about here.}}
\end{equation*}

\section{Properties Tested}

Birnbaum (1997) noted that his configural weight models imply violations of
stochastic dominance in specially constructed choices. Birnbaum's recipe for
constructing violations of stochastic dominance starts with a root gamble,
such as $g=(\$12,0.10;\$96)$, a two-branch gamble to either win $\$12$ with
probability of one-tenth, and otherwise win $\$96$. Next, split the lower
branch of $g$, $(\$12,0.10)$ into two branches of $0.05$ to win $\$12$. \
Now, increase the consequence on one of the splinters to $\$14$, which
produces gamble $g^{+}=(\$12,0.05;\$14,0.05;\$96,0.90)$ that dominates $g.$
Next, split the higher branch of $g$ and lower the consequence on the
splinter to produce gamble $g^{-}=(\$12,0.10;\$90,0.05;\$96,0.85)$, which is
dominated by $g$. By transitivity, if $g^{+}$ is preferred to $g$\ and $g$\
is preferred to $g^{-}$\ then by transitivity, $g^{+}$\ should be preferred
to $g^{-}$. The choice between $g^{-}$ and $g^{+}$ was proposed as a test
between Birnbaum's configural weight models which predict that people will
violate stochastic dominance in this case by preferring $g^{-}$ to $g^{+}$
in this case, against the class of rank-dependent models (including CPT)
which do not violate dominance. Note that the implication that $g^{+}$ is
preferred to $g^{-}$\ can be deduced from coalescing, consequence
monotonicity, and transitivity. \ 

Birnbaum and Navarrete (1998) tested these interesting predictions, and
found that about 70\% of undergraduates tested violated dominance by
choosing $g^{-}$over $g^{+}$ in choices constructed from this recipe.
Similar results were reported by Birnbaum, Patton, \& Lott (1999) with new
participants and other choices constructed from the same recipe. Violations
appear substantial with highly educated participants (Birnbaum, 1999b) and
have been observed with a number of different ways of displaying this choice
(Birnbaum \& Martin, in press).

Such choices violate the predictions of a number of theories of decision
making, including those of Cumulative Prospect Theory (Tversky \& Kahneman,
1992; Wakker \& Tversky, 1993), Rank-and Sign-Dependent Utility theory (Luce
\& Fishburn, 1991; 1995; Luce, 2000), and Rank-Dependent Utility (Quiggin,
1993; Lopes, 1990).

Birnbaum (1999a; Birnbaum \& Navarrete, 1998) argues that the violation is
due to violation of coalescing, because people largely satisfy consequence
monotonicity in matched choices. \ However, the violation may be due to
systematic violation of transitivity. Perhaps such violations occurs because
people cancel those probability-consequence branches that are approximately
the same in both gambles, and base their choices on what remains (cf.
Leland, 1994; Wu, 1994; Kahneman \& Tversky, 1979; Starmer, 1999). 

According to this theory, people see that the highest and lowest branches in 
$g^{+}$ and $g^{-}$ are approximately the same, but that $g^{-}$ has the
higher-valued middle branch. If people simplify and cancel the
(approximately) equal branches and choose based on what is left, such a
choice strategy might explain why people choose the dominated alternative.

However, it is difficult to see how such an account would produce higher 
\textit{judgments} for $g^{-}$ than $g^{+}$ when the two gambles are
presented for judgment on separate trials, separated by many other gambles
to judge. Because judgments must satisfy transitivity, this theory cannot
account for violations of stochastic dominance in judgment.

Therefore, a major purpose of this study is to examine whether violations of
stochastic dominance are also observed in judgment. \ A similar analysis can
also be made of other phenomena observed in recent choice experiments that
refute the class of RDU/RSDU/CPT mdoels. According to the configural weight
models, violations of these properties are produced by the mechanism by
which gambles are evaluated, not by editing rules or processes unique to
choice such as simplification and cancellation. \ According to the choice
account, violations of these crucial properties must be explained by editing
(e.g., cancellation of similar branches) or other processes specific to
choice, so violations should not occur in judgment, which is necessarily
transitive.

\subsection{Stochastic Dominance}

Applied to price judgments, this property means that if gamble $f$
stochastically dominates $g$ then people will not assign a higher price to $g
$ than to $f$. \ This property should hold for any theory that assumes
coalescing, and consequence monotonicity. \ Configural weight models violate
coalescing and therefore imply violations when the gambles are constructed
from the recipe described above.

\subsection{Branch Independence}

Restricted branch independence is a weaker version of Savage's sure thing
axiom. If two gambles with the same number of distinct branches and
probability distribution share a common branch, then the consequence on that
common branch can be changed without changing the preference order between
the gambles. Applied to buying prices, it would require that $%
b(x,p;y,q;z,r)>b(x^{\prime },p;y^{\prime },q;z,r)$ $\Leftrightarrow $ $%
b(x,p;y,q;z^{\prime },r)>b(x^{\prime },p;y^{\prime },q;z^{\prime },r),$
where $b(g)$ is the buying price for gamble $g$.  The same implication can
be tested for selling prices, denoted $s(g)$, as well. 

Violations of branch independence in choice were analyzed by Birnbaum and
McIntosh (1996) with respect to a generic model of which RAM, TAX, RSDU, and
CPT are special cases. All of the configural weight and rank-dependent
models considered here, including CPT, can systematically violate branch
independence when the change from $z$ to $z^{\prime }$ alters the ranking of
the consequences. \ However, as noted in a number of papers, the empirical
violations of branch independence in both choice and judgment have been
opposite those predicted by the inverse-\textsf{S} weighting function used
in CPT (Birnbaum, 1999a; Birnbaum \& McIntosh, 1996; Birnbaum \& Chavez,
1997; Birnbaum \& Beeghley, 1997).

\subsection{Lower and Upper Cumulative Independence}

In this case, let $0<z<x^{\prime }<x<y<y^{\prime }<z^{\prime }$ and $r+p+q=1$%
. Lower Cumulative Independence requires that if $S=(z,r;x,p;y,q)$ is judged
higher than $R=(z,r;x^{\prime },p,y^{\prime },q)$, then $S^{\prime \prime
}=(x^{\prime },r;y,p+q)$ should be judged higher than $R^{\prime \prime
}=(x^{\prime },r+p;y^{\prime },q)$. Upper cumulative independence requires
that if $S^{\prime }=(x,p;y,q;z^{\prime },r)$ is judged lower than $%
R^{\prime }=(x^{\prime },p,y^{\prime },q;z^{\prime },r)$, then $S^{\prime
\prime \prime }=(x,p+q;y^{\prime },r)$ should be judged lower than $%
R^{\prime \prime \prime }=(x^{\prime },p;y^{\prime },q+r)$.\ This property
should hold for any model of prices that satisfies transitivity, consequence
monotonicity, comonotonic branch independence and coalescing. For more
information, see Birnbaum, et al. (1999).

\subsection{Coalescing and Event-Splitting Independence}

Coalescing requires that if a gamble has two identical consequences, they
can be combined by adding their probabilities without changing the utility
of the gamble.\ For buying prices, for example, coalescing would require
that $b(x,p;x,q;z,r)$ $=b(x,p+q;z,r),$ with a similar expression for selling
prices. When coalescing is violated, it is called an event-splitting effect
(Birnbaum, 1997; 1999b; Humphrey, 1995; Luce, 1998; Starmer \& Sugden, 1993).

It also helps to keep in mind that coalescing $p$ and $q$ into $p+q$ is
determinate; whereas, splitting probability $r$ into two parts, say $%
p^{\prime }$ and $q^{\prime }$, so that $r=p^{\prime }+q^{\prime }$ is far
from determinate. It may take ingenuity to see a split that converts
stochastic dominance into consequence monotonicity and thus makes dominance
obvious. This asymmetry is not caught in the mathematics. For example, $%
g^{-}=(\$10,0.10;\$90,0.05;\$96,0.85)$ and $g^{+}=(\$10,0.05;\$12,0.05;%
\$96,0.90)$ can be split into $g_{s}^{-}=(\$10,0.05;\$10,0.05;90,0.05;%
\$96,0.85)$ and $g_{s}^{+}=(\$10,0.05;$ $\$12,0.05;\$96,0.05;\$96,0.90)$;
which makes it clear that the $g_{s}^{+}$ dominates $g_{s}^{-}$. It is then
easy to see that $g_{s}^{+}$ and $g_{s}^{-}$ can be coalesced to produce\ $%
g^{+}$ and $g^{-}$, respectively. Birnbaum (1999b) found that most
undergraduates violate stochastic dominance when choosing between these
gambles in the coalesced form ($g^{+}$ versus $g^{-}$ ) \textit{and} satisfy
it in the split form of this same comparison ($g_{s}^{+}$ versus $g_{s}^{-}$
); this reversal of preference is a violation of coalescing. The
rank-dependent utility models, including CPT, imply coalescing when applied
to choice.

If coalescing is violated, is it always violated in the same way?\textit{\
Event-splitting independence} requires that for positive consequences,

\begin{equation*}
b(x,p;x,q;z,r)>b(x,p+q;z,r)\Leftrightarrow b(x,p;x,q;z^{\prime
},r)>b(x,p+q;z^{\prime },r)
\end{equation*}

Another formulation of the concept is as follows: 
\begin{equation*}
b(x,p;x,q;z,r)>b(x,p+q;z,r)\Leftrightarrow b(x^{\prime },p;x^{\prime
},q;z,r)>b(x^{\prime },p+q;z,r),\text{ \ }x>x^{\prime }>0.
\end{equation*}

The configural weight RAM and TAX models imply violations of coalescing 
\textit{and} violations of event-splitting independence: If splitting a
branch with the highest consequence improves a gamble, then splitting an
equally likely lowest branch of a gamble should make it worse, according to
those models. Models such as that used by Edwards (1954; 1962), called 
\textit{stripped} prospect theory by Starmer and Sugden (1993), in which the
utility of a gamble is represented by $\sum\limits_{i=1}^{n}\pi
(p_{i})u(x_{i})$ [where $\pi (p_{i})$ is the weight function, which assigns
positive weights to the probabilities] imply violations of coalescing
(except in special cases), but they imply event-splitting independence
(Birnbaum \& Navarrete, 1998). So, if violations of coalescing
(event-splitting effects) are observed, one can test event-splitting
independence to distinguish the configural weight models against the
stripped prospect models.

\section{Method}

\subsection{Instructions and Stimuli}

The judges were instructed to evaluate two- and three-branch gambles from
the viewpoints of buyer and seller. Gambles were displayed as in the
following example: 
\begin{equation*}
\frac{.25\text{ \ }.25\text{ }.50}{\$40\text{ }\$44\text{ }\$108}
\end{equation*}

This display represents a three-branch gamble with a 0.25 probability to win
\$40, 0.25 probability to win \$44, and a probability of 0.50 to win \$108.
Probability was described in terms of relative frequency by analogy to a can
containing 100 otherwise identical slips that have the monetary consequences
printed on them. Slips would be mixed, and one would be chosen blindly, at
random, to determine the prize won. In this gamble, 25 slips say \$40, 25
say \$44, and 50 say \$108.

In the buyer's point of view, judges were asked to imagine that they were
``deciding the most that a cautious buyer should pay to buy the chance to
play the gamble.'' They were told that the buyer exchanges money for the
chance to play the lottery.

In the seller's point of view, judges were asked to imagine they were
``deciding the least that a seller should accept to sell the lottery.'' They
were told the seller receives money and gives up the chance to play the
lottery. They were told that this seller was ``venturesome, and ``would
rather take a chance on the gamble than sell for too low of a price.''
Additional instructions are given in Birnbaum and Sutton (1992).

\subsection{Designs}

There were 166 gambles, composed of 8 subdesigns. Subdesigns 1--4 were
constructed to test cumulative independence and branch independence. In all
cases in Subdesigns 1--4, $0<z<x^{\prime }<x<y<y^{\prime }<z^{\prime }$ and
probabilities sum to one, $r+p+q=1$.

There were 16 trials in each of Subdesigns 1 and 3. These were of the forms, 
$S=(z,r;x,p;y,q)$, $R=(z,r;x^{\prime },p,y^{\prime },q)$, $S^{\prime \prime
}=(x^{\prime },r;y,p+q)$, and $R^{\prime \prime }=(x^{\prime },r+p;y^{\prime
},q)$. There were 7 levels of $(x,y)=$ (\$50, \$54), (\$48, \$52), (\$44,
\$48), (\$40, \$44), (\$36, \$40), (\$32, \$36), and (\$28, \$32); $%
(x^{\prime },y^{\prime })=$ (\$10, \$98). In Subdesign 1, $r=0.5$, $p=0.25$,
and $q=0.25$; in Subdesign 3, $r=0.8$, $p=0.1$, and $q=0.1$; $z=\$2$.

The 16 trials in each of Subdesigns 2 and 4 were of the forms, $S^{\prime
}=(x,p;y,q;$ $z^{\prime },r)$, $R^{\prime }=(x^{\prime },p,y^{\prime
},q;z^{\prime },r)$, $S^{\prime \prime \prime }=(x,p+q;y^{\prime },r)$, and $%
R^{\prime \prime \prime }=(x^{\prime },p;y^{\prime },q+r)$. There were 7
levels of $(x,y)=$ (\$50, \$54), (\$48, \$52), (\$44, \$48), (\$40, \$44),
(\$36, \$40), (\$32, \$36), and (\$28, \$32); $(x\prime ,y\prime )$ = (\$10,
\$98). In Subdesign 2, $r=0.5$, $p=0.25$, and $q=0.25$; in Subdesign 4, $%
r=0.8$, $p=0.1$, and $q=0.1$; $z^{\prime }=\$108$.

Subdesigns 5 and 6 consisted of 80 binary gambles that were analyzed in
Birnbaum, Yeary, Luce, and Zhou (submitted). \ Those data are not presented
or used here, except in the review of the parameters estimated.

Subdesign 7 consisted of 8 three-branch gambles created to test stochastic
dominance. These gambles were composed of four pairs of the forms, $%
g^{+}=(x^{\prime },p-q;x,q;y^{\prime },1-p)$ and $g^{-}=(x^{\prime
},p;y,r;y^{\prime },1-p-r)$, where $0<x^{\prime }<x<y<y^{\prime }$. The
values of $(x^{\prime },x,y,y^{\prime })$ in the four pairs were $%
(\$12,\$14,\$90,\$96)$, $(\$3,\$5,\$92,\$97)$, $(\$6,\$8,\$91,\$99)$, and $%
(\$4,\$7,\$89,\$95)$; the values of $(p,q,r)$ were $(0.10,0.05,0.05)$, $%
(0.12,0.06,0.04)$, $(0.05,0.03,0.03)$, and $(0.02,0.01,0.02)$, respectively.
Note that $g^{+}$ stochastically dominates $g^{-}$. These are the same four
pairs used by Birnbaum and Navarrete (1998) in a study of choice.

Subdesign 8 consisted of 14 two- and three-branch gambles that were included
to allow tests of coalescing, combinations of coalescing and monotonicity,
and event-splitting independence. The three-branch gambles contained two
consequences that were equal or nearly equal. For example, four of these
trials were as follows: $(\$3,0.8;\$97,0.2)$, $(\$3,0.8;\$97,0.17;\$97,0.03)$
, $(\$3,0.8;\$97,0.1;\$97,0.1)$, and $(\$3,0.8;\$95,0.1,\$97,0.1)$. Note
that the first three of these describe the same gamble, splitting the
probability two win \$97 into two ''events,'' in which the probabilities are
equal (0.1 each) or unequal (0.17 and 0.03). The fourth case splits the
probability equally, but makes the consequences slightly unequal (\$95 and
\$97). Another four of these trials are as follows: $(\$2,0.2;\$98,0.8)$, $%
(\$2,0.04;\$2,0.16;\$98,0.8)$, $(\$2,0.1;\$2,0.1;\$98,0.8)$, and $%
(\$2,0.04;\$6,0.16;\$98,0.8)$. Again, the first three are equivalent
gambles, assuming coalescing, and the fourth also involves monotonicity.

\subsection{Procedure}

The 166 gambles were printed in random order in booklets, with the
restriction that gambles from the same subdesign would not be presented on
consecutive trials. Each person tested judged all 166 gambles from
viewpoints of both buyer and seller, with viewpoints performed in a
``blocked'' and counter-balanced order. Buyers' and sellers' booklets were
identical, with instructions that described both points of view, except that
the final paragraph of instructions in each booklet reiterated the task at
hand.

\subsection{ Judges}

Judges (participants) were 66 undergraduates at California State University,
Fullerton, who participated as one option toward an assignment in lower
division psychology courses.

\section{Results}

\subsection{Stochastic Dominance}

Median judgments are given in Table 1 for each pair of gambles in Subdesign
7. In all eight comparisons (four pairs by two viewpoints), the dominated
gamble received the higher mean and median judgment, violating stochastic
dominance in each case. The mean judgment was $\$63.31$ for the four
dominated gambles, compared to $\$55.11$ for the dominant gambles, averaged
over viewpoints. This difference was significant, $F(1,65)=20.56$. The
difference was greater in the buyer's point of view, where the means were $%
\$53.02$ and $\$41.18$, than it was in the seller's viewpoint, where the
means were $\$73.61$ and $\$69.05$, respectively. This interaction between
viewpoint and magnitude of dominance violations was significant, $%
F(1,65)=11.62.$%
\begin{equation*}
\fbox{\textit{Insert Table 2 about here.}}
\end{equation*}

The data of each judge were examined separately, and it was found that 51 of
66 (77\%) assigned higher mean judgments to the 4 dominated gambles than to
the 4 dominant gambles in this design (averaged over viewpoint).\textbf{\ }%
Only 15 assigned higher mean judgments to the dominant gambles.
Significantly more individuals violated than satisfied stochastic dominance
in these judgments.

The gambles 
\begin{equation*}
g^{+}=(\$12,0.05;\$14,0.05;\$96,0.90)\text{ and }g^{-}=(\$12,0.10;\$90,0.05;%
\$96,0.85)
\end{equation*}%
exhibited the largest violation. Although the former dominates the latter,
the $g^{-}$ received a mean judgment of $\$53.52$ in the buyer's viewpoint,
compared to a mean judgment of only $\$34.52$ for $g^{+}$. Smaller
violations were found for pairs with smaller values of $p$ and $q$. For
example, $(\$6,0.02;$ $\$8,0.03;\$99,0.95)$ dominates $(\$6,0.05;\$91,0.03;%
\$99,0.92)$, but the mean judgment in the buyer's viewpoint for the
dominated gamble was $\$53.32$ compared to $\$45.50$ for the dominant
gamble. Different gamble pairs showed significantly different magnitudes of
violation, $F(3,195)=5.73$; but the three-way interaction of Gambles by
Dominance by Viewpoint was not significant, $F(3,195)=2.31$.

These judgment results agree with direct choices between the same gambles.
Birnbaum and Navarrete (1998) found that 73, 61, 73, and 73 judges (out of
100) chose the dominated gamble $g^{-}$ over the dominant gamble, $g^{+}$,
in direct choices between the pairs of gambles of Table 1, respectively.
Thus, median (and mean) judgments of both buying and selling prices of the
gambles as well as direct choices between gambles all violate stochastic
dominance for these gambles.

Both RAM and TAX (configural weight) models predicted these violations of
stochastic dominance, based on parameters estimated from previous
experiments that did not test stochastic dominance. 

These data refute the theory that buying and selling prices can be
represented by RDU/CPT representation in which the weighting and utility
functions are allowed to be different for buying and selling prices. It can
also be shown that three other CPT models of buying and selling prices also
satisfy stochastic dominance for both buying and selling prices, so they can
be rejected based on these results. \ We do not yet know if violations of
stochastic dominance are consistent with Luce's (2000) joint receipt models
of buying and selling prices. See Appendix.

\subsection{Branch Independence}

Figures 1 and 2 show tests of branch independence in the seller's and
buyer's viewpoints, respectively. In both figures, mean judgments of $%
S=(z,r;x,p;y;p)$, $R=(z,r;x^{\prime },p;y^{\prime },p)$, $S^{\prime
}=(x,p;y,p;z^{\prime },r)$, and $R^{\prime }=(x^{\prime },p;y^{\prime
},p;z^{\prime },r)$ are plotted as separate functions of $y$. In these
figures, $r$ $=0.5$, $p=0.25$, $x=y-\$4$; $z=$ $\$2$, and $z^{\prime }=$ $%
\$108$. The values of $x^{\prime }$ and $y^{\prime }$ are fixed at $%
x^{\prime }=\$10$ and $y^{\prime }=\$98$, respectively, so these judgments
plot as horizontal lines. According to branch independence ($S\succ
R\Leftrightarrow S^{\prime }\succ R^{\prime }$), i.e., the pairs of curves
should cross at the same abscissa value. Instead, Figure 1 for the seller's
viewpoint shows that the mean judgments $s(S)$ exceed those of $s(R)$ for $y$
$\geqslant \$48$ but the means for $s(S^{\prime })$ (filled circles) are
strictly below those of $s(R^{\prime })$(filled circles) for all values of $y
$ tested. Similarly, for buyer's prices (Figure 7), the data show that mean
judgments of $b(S)$ (filled squares) exceed mean judgments of $b(R)$ (open
squares) for all values of $y$ tested; however, mean judgments of $%
b(S^{\prime })$ (filled circles) are below mean judgments of $b(R^{\prime })$
(open circles) for $y$ $\leqslant $ \$48. Thus, both buying and selling
prices show the $SR^{\prime }$ pattern of violations of branch independence. 
\begin{equation*}
\fbox{\textit{Insert Figures 1 and 2 about here.}}
\end{equation*}

Figures 1 and 2 also show preference reversals between viewpoints. For
example, Figure 1 shows that $s(S)$ exceed those of $s(R)$ only for $%
y\geqslant \$48$, whereas, Figure 2 shows that means of $b(S)$ exceed mean
judgments of $b(R)$ for all values of $y$ tested.

Although the violations occur for different gambles in each viewpoint, the
particular pattern in both Figures 1 and 2 is consistent with the patterns
reported for both buyer's and seller's viewpoints by Birnbaum and Beeghley
(1997) and Birnbaum and Veira (1998); namely, the pattern of violations in
both figures is the same, $SR^{\prime }:$ $S\succ R$ and $R^{\prime }\succ $ 
$S^{\prime }$. This pattern also agrees with the pattern reported in choice
experiments (Birnbaum, 1999a;1999b; Birnbaum \& Chavez, 1997; Birnbaum \&
McIntosh, 1996; Birnbaum \& Navarrete, 1998; Birnbaum, et al., 1999). The
particular inverse-\textsf{S} weighting functions used by several authors
(Tversky \& Kahneman, 1992; Tversky \& Wakker, 1995; Wu and Gonzalez, 1996)
imply the opposite pattern of violation. Because the $SR^{\prime }$ pattern
has been observed with different values of $p$ and $r$, it is becoming more
and more difficult for any inverse-\textsf{S} function to fit all of the
data reported.

\subsection{Cumulative Independence}

Figures 3 and 4 show tests of lower cumulative independence. $%
S=(\$2,0.5;x,0.25;y,0.25)$ $\succ $ $R=(\$2,0.5;\$10,0.25;\$98,0.25)$ $%
\Rightarrow $ $S^{\prime \prime }=(\$10,0.5;y,0.5)$ $\succ $ $R^{\prime
\prime }=(\$10,0.75;\$98,0.25)$; in these tests, $y=x+\$4$. Lower cumulative
independence requires that if the filled squares exceed the unfilled squares
for any $y$ ($S$ $\succ $ $R$), then the filled circle should exceed the
unfilled circles at that same value ($S^{\prime \prime }$ $\succ $ $%
R^{\prime \prime }$). Instead, Figure 8 shows that mean judgments selling
prices of $S$ (filled squares) exceed those of $R$ (open squares) only for $%
y\geqslant \$44$; however, mean judgments of $S^{\prime \prime }$ (filled
circles) are below those of $R^{\prime \prime }$ (open circles) for all
values of $y$, in violation of lower cumulative independence. 
\begin{equation*}
\fbox{\textit{Insert Figures 3 and 4 about here.}}
\end{equation*}

In Figure 4, for buyer's prices, mean judgments of $S$ (filled squares)
exceed those of $R$ (open squares) for all values of $y$ tested. However,
mean judgments of $S^{\prime \prime }$ (filled circles) are less than those
of $R^{\prime \prime }$ (open circles) for $y$ $\leqslant $ \$36, in
violation of lower cumulative independence.\ Comparing Figures 3 and 4,one
can see again that the rank order of the judgments is different in buying
and selling prices.

Figures 5 and 6 show the corresponding tests of upper cumulative
independence, plotted as a function of $x$. According to upper cumulative
independence, if $S^{\prime }=(x,0.25;y,0.25;\$108,0.5)$ $\prec $ $R^{\prime
}=(\$10,0.25;\$98,0.25;\$108,0.5)$ then $S^{\prime \prime \prime
}=(x,0.50;\$98,0.50)$ $\prec $ $R^{\prime \prime \prime
}=(\$10,0.25;\$98,0.75)$. The property can be satisfied if the vertical gap
between the two curves connecting filled symbols exceeds the vertical gap
between the two curves connecting unfilled symbols, and again, the data show
smaller gaps between the curves connecting filled symbols. In Figure 5, the
filled squares ($S^{\prime }$) are below the unfilled squares ($R^{\prime }$%
) for all values of $x$; however, the filled circles ($S^{\prime \prime
\prime }$) exceed the open circles ($R^{\prime \prime \prime }$) for $%
x\geqslant \$48$. In Figure 6, the filled squares are below the open squares
for $x\leqslant \$44$; however, the filled circles exceed the open circles
for all values of $x$ tested. 
\begin{equation*}
\fbox{\textit{Insert Figures 10 and 11 about here}}
\end{equation*}

Figures 1-6 have two properties in common: in all cases: First, the vertical
gap between the curves connecting filled symbols (judgments of $S$ gambles)
is less than the corresponding vertical gap between curves connecting
unfilled symbols (judgments of $R$ gambles). Second, the effect of point of
view (changing from buyer's to seller's viewpoint) produces a greater
vertical shift for the wide range gambles ($R$) than for the small range
gambles ($S$). These two trends were also observed in corresponding figures
(not shown) for the tests of branch independence, lower, and upper
cumulative independence with $r=0.8$, $p=q=0.1$. 

These results replicate patterns observed in choice (Birnbaum \& Navarrete,
1998; Birnbaum, et al., 1999). As in the case of violations of stochastic
dominance, these data refute the theory that the RDU representation with
different weighting functions and different utility functions in buyer's and
seller's prices could reproduce the data. 

\subsection{Coalescing and Event-Splitting Independence}

The splitting of probability of a lower or higher valued branch with a
probability of 0.2 into two equal pieces produced minimal effects. $%
b(\$2,0.2;\$98,0.8)$ $=\$36.97$ and $b(\$2,0.1;\$2,0.1;\$98,0.8)$ $=36.24$;
similarly, $s(\$2,0.2;\$98,0.8)$ $=\$64.98$ and $s(\$2,0.1;\$2,0.1;\$98,0.8)$
$=64.67$. Splitting the higher-valued branch into equal pieces also had
minimal effects, $b(\$3,0.8;\$97,0.2)$ $=\$9.96$ and $b(\$3,0.8;\$97,0.1;%
\$97,0.1)$ $=\$8.98$; similarly, $s(\$3,0.8;\$97,0.2)$ $=\$25.46$ and $%
s(\$3,0.8;\$97,0.1;\$97,0.1)$ $=\$26.94$.

However, when the probability of the branch is split into two unequal
probabilities, or into two unequal consequences, splitting the lower valued
event makes the gamble worse and splitting the higher valued branch makes
the gamble better. Splitting the probability of the lower consequence
produces lower prices, $b(\$2,0.04;$ $\$2,0.16;\$98,0.8)$ $=\$32.50$ and $%
b(\$2,0.1;\$6,0.1;\$98,0.8)$ $=\$34.20$, both less than $b(\$2,0.2;\$98,0.8)$
= $\$36.97$; $s(\$2,0.04;\$2,0.16;\$98,0.8)$ $=\$57.68$ and $s(\$2,0.1;$ $%
\$6,0.1;\$98,0.8)$ $=\$62.96$, both less than $s(\$2,0.2;\$98,0.8)$ $=\$64.98
$. Splitting the probability of the higher consequence increases the prices, 
$b(\$3,0.8;\$97,0.17;\$97,0.03)$ $=\$12.89$ and $b(\$3,0.8;\$95,0.1;\$97,0.1)
$ $=\$12.80$, both higher than $b(\$3,0.8;\$97,0.2)$ $=\$9.96$; similarly, $%
s(\$3,0.8;\$97,0.17;$ $\$97,0.03)$ $=\$26.73$ and $s(\$3,0.8;\$95,0.1;%
\$97,0.1)$ $=\$31.98$, both greater than $s(\$3,0.8;$ $\$97,0.2)$ $=\$25.46$%
. In all 8 of these 8 cases, prices shift in the directions predicted by the
configural weight models.

The changes in judgments violate event-splitting independence. Note that in
11 of 12 comparisons, splitting a 0.2 probability of the lower consequence
lowers the mean judgment, and splitting the 0.2 probability of the higher
consequence improves the mean judgments. Thus, the results give a score of
11 out of 12 for the averaging models (by a binomial sign test, this result
is significant, $p<0.01$). 

These results violate event-splitting independence, and show that event
splitting can even override the effects of consequence monotonicity. In each
case, consequence monotonicity would have produced the opposite result to
that observed. Because event-splitting independence is implied by
``stripped'' prospect theory, these data violate the subjectively weighted
equation of that prospect theory, extended to three-branch gambles and
stripped of its editing principles (Birnbaum \& Navarrete, 1998; Starmer \&
Sugden, 1993).

\begin{equation*}
\fbox{\textit{Insert Figure 12 about here.}}
\end{equation*}

\subsection{Fit of RAM Model}

For $n$- branch gambles, $g=(x_{1},p_{1},\ldots ,x_{i},p_{i},,\ldots
,x_{n},p_{n})$ with $0<x_{1}<x_{2}<\ldots <x_{n}$, the Rank-Affected,
Multiplicative (RAM) model can be written:%
\begin{equation*}
P(j,g)=\nu \left[ \frac{\sum\limits_{i=1}^{n}p_{i}^{\gamma }w(j,i,n)u(x_{i})%
}{\sum\limits_{i=1}^{n}p_{i}^{\gamma }w(j,i,n)}\right] 
\end{equation*}%
where $P(j,g)$, $j=buy,sell,$ is the predicted judgment of buyer's or
seller's price of the gamble; $w(i,n)$ is the weight of the branch at rank $i
$, which has probability $p_{i}$ and consequence $x_{i}$. The highest ranked
branch is numbered $n$ and the lowest ranked branch is numbered $1$. For any
number of branches, $n$, the weights can be normalized to sum to $1$ without
loss of generality. \ Note that the weights of branches of different rank
depend on the judge's viewpoint (buyer or seller).

Although the weights of the branches are affected by the ranks of the
consequences of the branches, note that the branch weights are independent
of probability. Unlike RDU, this model violates coalescing and therefore
violates stochastic dominance. \ If a branch is split into two branches, the
sum of the weights can exceed the weight of the coalesced branch.

For the case, $n=2$, Birnbaum, et al. (submitted) fit the model with $%
w(buyer,2,2)=0.234$, (hence, $w(buyer,1,2)=.766=1-0.234$) reflecting greater
weight for lower valued consequences and $w(seller,2,2)=0.547$, indicating
greater weight on higher-valued consequences in the seller's viewpoint. The
parameter $\gamma $ describes the weighting of probability, which was
estimated to be $0.478$, so judgments of binary gambles are predicted to be
an inverse-\textsf{S} function of probability, holding the consequences
fixed. The constant $\nu $ was estimated to be $0.823$. For small positive
consequences, $0<x_{i}<\$150$, it has been found that one can fit the data
well with the approximation that utility is proportional to cash value
(Birnbaum \& Naverette, 1998; Birnbaum \& Beeghley, 1998; Birnbaum, et al.,
submitted).

Birnbaum and Beeghley (1997) fit a generic model that is consistent with RAM
or TAX, as well as CPT/RSDU/RDEU, to judgments of gambles consisting of
three equally likely branches with positive consequences. The experimental
design of Birnbaum and Beeghley did not permit one to distinguish these
models, nor did it permit estimation of $\gamma $ in the RAM model. However,
it does allow one to estimate the weights of rank positions in three branch
gambles. They reported that in the buyer's viewpoint, the weights are $0.52$%
, $0.39$, and $0.09$ for $w(buyer,1,3)$, $w(buyer,2,3)$, and $w(buyer,3,3)$,
respectively, and in the seller's viewpoint, they are $0.25$, $0.53$, $0.22$.

Putting together the parameters from these two studies, we can predict that
judgments of both buying and selling prices will violate stochastic
dominance in both buyer's and seller's viewpoints. It is important to
realize that the data used to estimate the parameters do not themselves
include or imply violations of stochastic dominance, except if the model is
assumed. The prediction follows from the configural weight models, given the
parameters implied by the previous data.

The TAX model makes nearly identical predictions to the RAM in this study,
so we present only those of RAM, which fit a tiny bit better and is a
simpler model to describe.

There are 8 parameters in the RAM model, assuming that utility is
proportional to cash value. The parameters were estimated to minimize the
sum of squared deviations between predicted and obtained mean judgments of
buying and selling prices. The estimated values of $\nu $ and $\gamma $ were 
$0.843$ and $0.560$, respectively. The estimated weights of lower and
higher-valued branches in two-branch gambles are $w(buyer,1,2)=0.787$ (and $%
w(buyer,2,2)=0.213)$ in the buyer's viewpoint, and $w(seller,1,2)=0.471$
(and $w(seller,2,2)=0.529$) in the seller's viewpoint, respectively. For
three-branch gambles the weights of lowest, middle, and highest valued
branches were $0.498$, $0.386$, and $0.115$ in the buyer's viewpoint, and $%
0.245$, $0.393$, and $0.363$ in the seller's viewpoint, respectively. (The
weights are not all independent parameters, since they were constrained to
sum to $1$ within each $n$ and viewpoint. Note how similar they are to the
values reported by Birnbaum \& Beehgley, 1997).

The total sum of squared deviations was $1072.6$, summed over $144$ means ($2
$ viewpoints $\times 72$ cells in Subdesigns 1-4 and 7), with a root mean
squared deviation of $2.73$. Correlations between predicted means and
obtained means were $0.99$ and $0.99$ for buyer's and seller's judgments. 

Such correlations between models and data can seem high despite systematic
deviations (Birnbaum, 1973); therefore, we calculated more sensitive indices
that allow more precise assessment of models. Ranking the obtained means of
buyer's and seller's judgments, we computed the differences in ranks between
buyer's and seller's judgments. These differences represent preference
reversals between buyer's and seller's judgments. \ According to the second
model of loss aversion and CPT, buyer's and seller's judgments should be
monotonically related (see Birnbaum, et al, submitted; Birnbaum \&
Zimmermann, 1998; Appendix). Therefore, if buyer's and seller's judgments
are ranked, the rank orders of the judgments should be the same except for
random error, and therefore rank differences should be unpredictable.
Instead, predicted and obtained differences in rank order had a correlation
of $0.74$, showing that the changes in rank order (preference reversals)
were systematic and predictable from the RAM model. The numerical
differences in judgment between buying and selling prices were even more
highly predictable from the numerical differences between predictions, with
a correlation of $0.93$. This should not be surprising since the model was
fit to a criterion of numerical fit.

According to the second model of loss aversion with the parameterized model
of CPT (which includes power functions for the utility functions), ratios of
seller's to buyer's prices should be constant except for error, and
therefore unpredictable. However, ratios of predictions and observed ratios
of seller's to buyer's prices were correlated with a coefficient of $0.90$.
The predictability of these ratios and differences in value (and rank order)
refutes the second model of CPT\ and loss aversion because it shows that
these rank differences, ratios, and differences are systematic and
predictable.

The third model of CPT, which assumes that people integrate the buying and
selling prices into the gambles (Appendix), does not require the ranks of
buyer's and seller's judgments to be identical. This third model was used to
calculate predictions, based on the model and parameters of Tversky and
Kahneman (1992). This model provides a frame of reference with which the
accuracy of the RAM model can be compared. This model had a sum of squared
deviations of $50,722$, with a root mean squared deviation of $18.77$.
Correlations between predicted and observed mean judgments were $0.97$ and $%
0.98$ in the buyer's and seller's viewpoints respectively. \ However, the
rank differences in prediction had a correlation of only.$0.23$ with the
observed rank differences between seller's and buyer's prices. The
correlations with numerical differences and ratios were $0.51$ and $0.79$,
substantially lower than the corresponding values for the RAM model.

This third CPT\ model satisfies stochastic dominance. Dominant gambles, $%
g^{+}$ were predicted to receive judgments about $\$0.30$ higher than the
dominated gambles, $g^{-}$. Indeed, all versions of CPT combined with or
without loss aversion imply that buyer's and seller's judgments should
satisfy stochastic dominance, even when these models are allowed to have
different parameters for buying and selling prices, which would grant them
double the free parameters of competing models. (See Appendix). 

In contrast, the RAM model correctly predicts violations of stochastic
dominance in all 8 cases in Table 1. It predicts violations of stochastic
dominance with parameters from previous studies, and it also does so with
parameters estimated from these data. \ In the buyer's viewpoint, the
average predicted violation was $-\$9.16$ compared to an observed mean
difference of $-\$11.84$. In the seller's viewpoint, the average difference
was predicted to be $-\$6.26$, and the observed difference was $-\$4.56$. 

The estimated weights in both buyer's and seller's viewpoints satisfy the
following relation:

\begin{equation*}
\frac{w(j,1,3)}{w(j,2,3)}<\frac{w(j,2,3)}{w(j,3,3)}
\end{equation*}

This relation implies that for equal ratios of probability, systematic
violations of restricted branch independence in this study will be of the
form,

\begin{equation*}
S\succ R\text{ and }S^{\prime }\prec R^{\prime },
\end{equation*}
which is the observed pattern shown in Figures 1 and 2, and the pattern
reported in previous research.

The observed pattern of violation of branch independence is opposite that
predicted from CPT\ if we assume that the equations of Tversky and Kahneman
for certainty equivalents also apply to judgments of buying and selling
prices. \ The third version CPT model, which assumes integration of prices
and prizes, with the prior parameters of Tversky and Kahneman (1992) also
predicts the opposite pattern of violation.

The pattern is also opposite that predicted by the anchoring with
insufficient adjustment model (Birnbaum \& Zimmermann, p. 183-184). This
model assumes that participants place the greatest weight on the anchor and
smaller but equal weights on the two other values. The buyer is presumed to
anchor on the lowest consequence and the seller to anchor on the highest
consequence. Anchoring and adjustment gives a rough way to describe the
pattern of configural weighting in two branch gambles (``rough'' because the
weights are not symmetric in buyer's and seller's prices). However, it
predicts the opposite pattern of violation of branch independence in three
branch gambles from what is observed. The fit of anchoring and adjustment
model yielded a sum of squared deviations of $2156$, about twice that of the
RAM model. The weight of the ``anchor'' in two branch gambles was estimated
to be $0.677$, and in three-branch gambles it was estimated to be $0.547$.

\begin{equation*}
\fbox{\textit{Insert Table 3 about here.}}
\end{equation*}

\section{Discussion}

There are five\ main findings of this research, all of which are consistent
with the configural weight models of buying and selling prices.

First, judgments of both buying and selling prices violate stochastic
dominance. Following the publication of Birnbaum's (1997) predictions based
on the fit of the RAM and TAX\ models to previous data, a series of studies
have now been conducted (including this one) to see whether those
predictions would be observed empirically in choice and judgment. A number
of studies of choice, starting with Birnbaum and Navarrete (1998), have
shown that choices systematically violate stochastic dominance with choices
constructed from Birnbaum's (1997) recipe.

The present study obtained buying and selling prices of those same gambles
presented for choice by Birnbaum and Navarrete (1998). Results show that
most undergraduates assign higher buying and selling prices to the dominated
gambles, $g^{-}$, than they assign to the dominant gamble, $g^{+}$.

Second, judgments of buying and selling prices also violate lower and upper
cumulative independence in the same manner as has been reported in studies
of choice, as predicted by Birnbaum (1997).

As noted by Birnbaum and Navarrete (1998; Birnbaum, et al, 1999), violations
of coalescing may lie at the root of these three phenomena, as well as
violations of ordinal independence reported by Wu (1994). Because RAM and
TAX models violate coalescing, they predict violations of stochastic
dominance, lower and upper cumulative independence and ordinal independence.

Third, violations of branch independence in this study show the same pattern
as has been previously observed with both choice and judgment.

Because the same types of violations of stochastic dominance, branch
independence, lower and upper cumulative independence occur with judgments
as have been observed in choice experiments, it seems unattractive to
theorize that previous results with choice were produced by something unique
to choice, such as editing with cancellation of approximately equal
branches. Because the gambles are judged separately on different trials in
this study, it appears that violations of these properties do not require
direct comparison of two gambles on each trial. 

RAM and TAX models would explain all three sets of results with the same
evaluation model and utility function, but with different configural weights
in choice, buying prices, and selling prices. Although there may be certain
other phenomena, such as systematic violations of transitivity, that may
require choice-specific explanations (Birnbaum, et al., 1999; Starmer, 1999;
Leland, 1994), the phenomena studied here do not appear to require such
interpretations, nor do those theories provide any way we can see to explain
numerical judgments.

Fourth, although violations of branch independence, cumulative independence,
and stochastic dominance are similar to patterns obtained with choice,
buying prices, selling prices, and choices all yield different orderings on
the gambles--that is, there are preference reversals between any pair of
methods.\ In this study, the reversals between buying and selling prices
were predictable from the RAM model. \ This model assumes that the utility
of money is the same, but the pattern of weighting of the ranked branches
differs: in the buyer's viewpoint, more weight is given to lower-valued
branches, and in the seller's viewpoint, relatively more weight is assigned
to the middle and higher valued branch.

However, the patterns of violations are similar across methods, even if the
ordering of certain gambles differ. For example, violations of branch
independence consistently show that the $SR^{\prime }$ pattern of violation
is more frequent than the opposite pattern, $RS^{\prime .}$ (Incidentally,
the observed pattern is opposite that predicted by CPT using the inverse-%
\textsf{S} cumulative weighting functions estimated in previous research).
Although the exact orders change between buying, selling, and choice, the
pattern\ observed, $SR^{\prime }$, has been reported for buying prices,
selling prices, and choices. \ However, even though the weights differ for
judgments of buying and selling prices and choice, they maintain relations
among their ratios that produce the $SR^{\prime }$ pattern (Birnbaum \&
Beeghley, 1997). 

Fifth, these data find evidence of violations of event-splitting
independence. Configural weight models, including both RAM and TAX\ ,
violate coalescing, which is the reason that these models predict violations
of stochastic dominance, cumulative independence, and other violations of
the class of CPT/RSDU/RDU models (Birnbaum, 1999a). These configural weight
models (unlike Starmer and Sugden's Stripped Prospect Theory, which also
violates coalescing), imply violation of event-splitting independence:
splitting the highest branch of a gamble should make that (re-written)
gamble better and splitting an equally probable lowest consequence branch
should make that (rewritten) gamble worse. This implication of RAM and TAX
was observed in 11 of 12 tests. The effects of splitting seemed greater when
the probability was split into unequal pieces or into unequal consequences,
even when the change in consequence value was in the opposite direction to
the predicted effect.

All five of these findings are consistent with the idea that the evaluation
mechanism and utility function are the same in choice and in judgment, but
that configural weights differ between choosing, buying, and selling.

\section{Appendix: CPT Models of Buying and Selling Prices}

\pagebreak This paper considers four ways of applying the model of Tversky
and Kahneman (1992) to buying and selling prices. Let $g=(x,p;y)$ represent
a two-branch gamble to win $x$ with probability $p$ and otherwise win $y$.
Let $U(g)$ represent the cumulative prospect theory value (or utility) of
gamble $g$. According to the first approach, a person decides to buy gamble $%
g$ at price $b$ and to sell at price $s$ whenever 
\begin{eqnarray*}
U(g) &>&u(b) \\
U(g) &<&u(s)
\end{eqnarray*}

This model implies that buying and selling prices will be equal to each
other, which is contradicted by all of the data, which show that selling
prices exceed buying prices.

The second model assumes that a person sees the price paid as a loss and the
gamble purchased as a gain; however, when selling, a person sees the gamble
as a loss and the price received as a gain. \ A person would then decide to
buy and sell at $b$ and $s$, respectively whenever:

\begin{eqnarray*}
U(g) &>&-u(-b) \\
-U(-g) &<&u(s)
\end{eqnarray*}%
where the gamble $-g$ is interpreted as gamble $g$ with negative
consequences, $-g=(-x,p;-y)$, and $-u(-b)$ is the negative disutility of
paying $b$. This model, combined with loss aversion, defined as $%
u(-x)=-\lambda u(x)$, $x>0,$ implies that buying and selling prices will be
monotonically related to each other. \ Assuming that utility of money is a
power function of cash value, this model implies that the ratio of selling
to buying price will be a constant. \ This model is disproved by the fact
that buying and selling prices show systematic preference reversals
(Birnbaum \& Stegner, 1979; Birnbaum \& Sutton, 1992; Birnbaum, et al,
submitted).

The third CPT\ model assumes that prices are integrated into the prizes, and
a person views either buying or selling as a mixed gamble, as follows:

\begin{eqnarray*}
U(g-b) &>&0 \\
U(s-g) &>&0
\end{eqnarray*}

Here, the gamble $g-b=(x-b;p,y-b)$ and $s-g=(s-x;p,s-y)$. This theory sees
both buying and selling as mixed gambles in which one makes a profit if one
buys for a lower price than the gamble pays or selling a gamble for more
than the gamble pays, and one loses if one buys for more than the gamble
pays or selling a gamble that pays more than the price received. This model,
coupled with loss aversion and the model of CPT with power functions for
utility implies complementary symmetry, that $b(g)+s(g^{\prime })=k$, where $%
g^{\prime }=(y,p;x)$ and $k$ is a constant. This property is systematically
violated (Birnbaum \& Zimmermann, 1998; Birnbaum, et al., submitted).

This third version of CPT also implies satisfaction of stochastic dominance
in the recipe of Birnbaum (1997). CPT, like RSDU, partitions consequences
into positive and negative cases. \ On each side of zero, the model
satisfies coalescing, consequence monotonicity, and transitivity. Consider
gambles $g=(x,p;y)$, $g^{+}=(x,p-q;x^{+},q;y)$, and $g^{-}=(x,p;y^{-},r;y)$
and buying price, $b$, such that $0<x<x^{+}<b<y^{-}<y.$ For any such value,
it follows from CPT\ that $U(g^{+}-b)>U(g-b)>U(g^{-}-b).$ Choose $b$ such
that $U(g-b)=0$. A person would be indifferent between paying $b$ for gamble 
$g$ or not. \ However, a person would prefer to buy $g^{+}$ at price $b$ and
would prefer not to buy $g^{-}$ at price $b$. \ This shows that buying
prices will satisfy stochastic dominance. \ A similar argument can be made
in the case of selling prices.

The fourth version of CPT would take the representation of Tversky and
Kahneman (1992) and use different parameters for buying and selling prices.
\ This version implies that buying and selling prices will satisfy
stochastic dominance, upper and lower cumulative independence, and
coalescing (Birnbaum \& Navarrete, 1998).

Luce's (2000) models of buying and selling prices derived from a joint
receipt operation are considered in Birnbaum, et al. (submitted). \ The
implications of those models for stochastic dominance and cumulative
independence in judgments of buying and selling prices have not yet been
determined (Luce, 2002, personal communication).

\begin{description}
\item[{\protect\Large References}] \medskip

\item Allais, M., \& Hagen, O. (Eds.). (1979). \textit{Expected utility
hypothesis and the Allais paradox}. Dordrecht, The Netherlands: Reidel.

\item Birnbaum, M. H. (1973). The Devil rides again: Correlation as an index
of fit. \textit{Psychological Bulletin}, 79, 239-242. 

\item Birnbaum, M.\thinspace H. (1997). Violations of monotonicity in
judgment and decision making. In A.\thinspace A.\thinspace J.\thinspace
Marley (Ed.), \textit{Choice, decision, and measurement: Essays in honor of
R.\thinspace Duncan Luce} (pp.\thinspace 73-100). Mahwah, NJ: Erlbaum.

\item Birnbaum, M.\thinspace H. (1999a). Paradoxes of Allais, stochastic
dominance, and decision weights. In J. Shanteau, B.\thinspace A.\thinspace
Mellers, \& D.\thinspace A.\thinspace Schum (Eds.), \textit{Decision science
and technology: Reflections on the contributions of Ward Edwards}
(pp.\thinspace 27-52). Norwell, MA: Kluwer Academic Publishers.

\item Birnbaum, M.\thinspace H. (1999b). Testing critical properties of
decision making on the Internet. \textit{Psychological Science}, 10, 399-407.

\item Birnbaum, M.\thinspace H., \& Beeghley, D. (1997). Violations of
branch independence in judgments of the value of gambles. \textit{%
Psychological Science}, 8, 87-94.

\item Birnbaum, M.\thinspace H., \& Chavez, A. (1997). Tests of theories of
decision making: Violations of branch independence and distribution
independence. \textit{Organizational Behavior and Human Decision Processes},
71(2), 161-194.

\item Birnbaum, M. H., \& Martin, T. (in press). Generalization across
People, Procedures, and Predictions: Violations of Stochastic Dominance and
Coalescing. In S. L. Schneider \& J. Shanteau (Eds.), \textit{Emerging
perspectives on decision research}. New York: Cambridge University Press.

\item Birnbaum, M.\thinspace H., \& McIntosh, W.\thinspace R. (1996).
Violations of branch independence in choices between gambles. \textit{%
Organizational Behavior and Human Decision Processes}, 67, 91-110.

\item Birnbaum, M.\thinspace H., \& Navarrete, J. (1998). Testing
descriptive utility theories: Violations of stochastic dominance and
cumulative independence. \textit{Journal of Risk and Uncertainty}, 17, 49-78.

\item Birnbaum, M.\thinspace H., Patton, J.\thinspace N., \& Lott,
M.\thinspace K. (1999). Evidence against rank-dependent utility theories:
Tests of cumulative independence, interval independence, stochastic
dominance, and transitivity. \textit{Organizational Behavior and Human
Decision Processes}, 77, 44-83

\item Birnbaum, M.\thinspace H., \& Stegner, S.\thinspace E. (1979). Source
credibility in social judgment: Bias, expertise, and the judge's point of
view. \textit{Journal of Personality and Social Psychology}, 37, 48-74.

\item Birnbaum, M.\thinspace H., \& Sutton, S.\thinspace E. (1992). Scale
convergence and utility measurement. \textit{Organizational Behavior and
Human Decision Processes}, 52, 183-215.

\item Birnbaum, M.\thinspace H., \& Veira, R. (1998). Configural weighting
in judgments of two- and four-outcome gambles. \textit{Journal of
Experimental Psychology: Human Perception and Performance}, 24(1), 216-226.

\item Birnbaum, M. H., Yeary, S., Luce, R. D., \& Zhou, L. (2002,
submitted). Contingent Valuation, Endowment, or Viewpoint Effects: Testing
properties in judgments of buying and selling prices of lotteries. 

\item Birnbaum, M.\thinspace H., \& Zimmermann, J. M. (1998). Buying and
selling prices of investments: Configural weight model of interactions
predicts violations of joint independence. \textit{Organizational Behavior
and Human Decision Processes}, 74(2), 145-187.

\item Edwards, W. (1954). The theory of decision making. \textit{%
Psychological Bulletin}, 51, 380-417.

\item Edwards, W. (1962). Subjective probabilities inferred from decisions. 
\textit{Psychological Review}, 69, 109-135.

\item Humphrey, S.\thinspace J. (1995). Regret aversion or event-splitting
effects? More evidence under risk and uncertainty. \textit{Journal of risk
and uncertainty}, 11, 263-274.

\item Kahneman, D., \& Tversky, A. (1979). Prospect theory: An analysis of
decision under risk. \textit{Econometrica}, 47, 263-291.

\item Leland, J.\thinspace W. (1994). Generalized similarity judgments: An
alternative explanation for choice anomalies. \textit{Journal of Risk and
Uncertainty}, 9, 151-172.

\item Lopes, L. (1990). Re-modeling risk aversion: A comparison of
Bernoullian and rank dependent value approaches. In G.\thinspace M.
v.\thinspace Furstenberg (Ed.), \textit{Acting under uncertainty}
(pp.\thinspace 267-299). Boston: Kluwer.

\item Luce, R.\thinspace D. (1998). Coalescing, event commutativity, and
theories of utility. \textit{Journal of Risk and Uncertainty}, 16, 87-113.

\item Luce, R.\thinspace D. (2000). \textit{Utility of Gains and Losses:
Measurement-Theoretical and Experimental Approaches. }Mahwah,\ NJ: Lawrence
Erlbaum Associates. Errata: see Luce web page at http://socsci.uci.edu.

\item Luce, R.\thinspace D., \& Fishburn, P.\thinspace C. (1991). Rank- and
sign-dependent linear utility models for finite first-order gambles. \textit{%
Journal of Risk and Uncertainty, 4, }29-59.

\item Luce, R. D., \& Fishburn, P. C. (1995). A note on deriving
rank-dependent utility using additive joint receipts. Journal of Risk and
Uncertainty, 11, 5-16.

\item Quiggin, J. (1993). \textit{Generalized expected utility theory: The
rank-dependent model}. Boston: Kluwer.

\item Starmer, C. (1999). Cycling with rules of thumb: an experimental test
for a new form of non-transitive behaviour , \textit{Theory and Decision},
46, 141-158.

\item Starmer, C., \& Sugden, R. (1993). Testing for juxtaposition and
event-splitting effects. \textit{Journal of risk and uncertainty}, 6,
235-254.

\item Tversky, A. (1969). Intransitivity of Preferences. \textit{%
Psychological Review}, \textit{76}, 31-48.

\item Tversky, A., \& Kahneman, D. (1986). Rational choice and the framing
of decisions. \textit{Journal of Business}, 59, S251-S278.

\item Tversky, A., \& Kahneman, D. (1991). Loss aversion in riskless choice:
A reference-dependent model. \textit{Quarterly Journal of Economics},
106(4), 1039-1061.

\item Tversky, A., \& Kahneman, D. (1992). Advances in prospect theory:
Cumulative representation of uncertainty. \textit{Journal of Risk and
Uncertainty}, 5, 297-323.

\item Tversky, A., \& Wakker, P. (1995). Risk attitudes and decision
weights. \textit{Econometrica}, 63, 1255-1280.

\item Wakker, P., \& Tversky, A. (1993). An axiomatization of cumulative
prospect theory. \textit{Journal of Risk and Uncertainty}, 7, 147-176.

\item Wu, G. (1994). An empirical test of ordinal independence. \textit{%
Journal of Risk and Uncertainty}, 9, 39-60.

\item Wu, G., \& Gonzalez, R. (1996). Curvature of the probability weighting
function. \textit{Management Science}, 42, 1676-1690.
\end{description}

\bigskip \pagebreak

Table 1. Testable Properties of Decision Making

\begin{tabular}{ll}
Property Name & Expression \\ 
\multicolumn{2}{l}{\rule{7in}{0.02in}} \\ 
Coalescing & $(x,p;x,q;z,r)\thicksim (x,p+q;z,r)$ \\ 
Consequence Monotonicity & $(x,p;y,q;z)\succ (x^{\prime
},p;y,q;z)\Leftrightarrow x\succ x^{\prime }$ \\ 
Stochastic Dominance & $P(x>t|f)\geq P(x>t|g)\forall t\Rightarrow
f\succcurlyeq g$ \\ 
Restricted Branch Independence & $(z,r;x,p;y,q)\succ (z,r;x^{\prime
},p;y^{\prime },q)\Leftrightarrow (x,p;y,q;z^{\prime },r)\succ (x^{\prime
},p;y^{\prime },q;z^{\prime },r)$ \\ 
Lower Cumulative Independence & $(z,r;x,p;y,q)\succ (z,r;x^{\prime
},p;y^{\prime },q)\Rightarrow (x^{\prime },r;y,p+q)\succ (x^{\prime
},r+p;y^{\prime },q)$ \\ 
Upper Cumulative Independence & $(x,p;y,q;z^{\prime },r)\prec (x^{\prime
},p;y^{\prime },q;z^{\prime },r)\Rightarrow (x,p+q;y^{\prime },r)\prec
(x^{\prime },p;y^{\prime },q+r)$ \\ 
Event-Splitting Independence & $(z,r;x,p;x,q)\succ
(z,r;x,p+q)\Leftrightarrow (x,p;x,q;z^{\prime },r)\succ (x,p+q;z^{\prime },r)
$%
\end{tabular}

\rule{7in}{0.02in}

Notes: $0<z<x^{\prime }<x<y<y^{\prime }<z^{\prime },$ $p+q+r=1$ The
relation, $\succ $, represents the preference relation in choice, which may
or may not be transitive. \ In judgment, the relation represents the
numerical relation, $>$, applied to numerical judgments of buying or selling
prices, which must be transitive. 

\bigskip \pagebreak 

Table 2. Tests of Stochastic Dominance in Judgments of Buying and Selling
Prices

\begin{tabular}{lllll}
Buying Prices & $\quad \;g^{+}$ & Median & $\quad \;g^{-}$ & Median \\ 
\multicolumn{5}{l}{\rule{3.7in}{0.02in}} \\ 
1 & $\frac{.05\text{ }.05\text{ }.90}{\$12\text{ }\$14\text{ }\$96}$ & \$30.0
& $\frac{.10\text{ }.05\text{ }.85}{\$12\text{ }\$90\text{ }\$96}$ & \$60.0
\\ 
2 & $\frac{.06\text{ }.06\text{ }.88}{\$3\text{ \ }\$5\text{ }\$97}$ & \$22.5
& $\frac{.05\text{ }.03\text{ }.92}{\$6\text{ }\$91\text{ }\$99}$ & \$50.0
\\ 
3 & $\frac{.02\text{ }.03\text{ }.95}{\$6\text{ \ }\$8\text{ }\$99}$ & \$40.0
& $\frac{.05\text{ }.03\text{ }.92}{\$6\text{ }\$91\text{ }\$99}$ & \$54.0
\\ 
4 & $\frac{.01\text{ }.01\text{ }.98}{\$4\text{ \ }\$7\text{ }\$97}$ & \$50.0
& $\frac{.02\text{ }.02\text{ }.96}{\text{ }\$4\text{ \ }\$89\text{ }\$97}$
& \$62.5 \\ 
\multicolumn{5}{l}{\rule[0.04in]{3.7in}{0.02in}} \\ 
Selling Prices &  &  &  &  \\ 
\multicolumn{5}{l}{\rule{3.7in}{0.02in}} \\ 
1 & $\frac{.05\text{ }.05\text{ }.90}{\$12\text{ }\$14\text{ }\$96}$ & \$73.5
& $\frac{.10\text{ }.05\text{ }.85}{\$12\text{ }\$90\text{ }\$96}$ & \$81.5
\\ 
2 & $\frac{.06\text{ }.06\text{ }.88}{\$3\text{ \ }\$5\text{ }\$97}$ & \$68.0
& $\frac{.05\text{ }.03\text{ }.92}{\$6\text{ }\$91\text{ }\$99}$ & \$80.0
\\ 
3 & $\frac{.02\text{ }.03\text{ }.95}{\$6\text{ \ }\$8\text{ }\$99}$ & \$82.5
& $\frac{.05\text{ }.03\text{ }.92}{\$6\text{ }\$91\text{ }\$99}$ & \$83.5
\\ 
4 & $\frac{.01\text{ }.01\text{ }.98}{\$4\text{ \ }\$7\text{ }\$97}$ & \$81.0
& $\frac{.02\text{ }.02\text{ }.96}{\text{ }\$4\text{ \ }\$89\text{ }\$97}$
& \$87.5 \\ 
\multicolumn{5}{l}{\rule{3.7in}{0.02in}}%
\end{tabular}

\bigskip

\bigskip \pagebreak

Figure 1. Tests of branch independence in the seller's viewpoint. Branch
independence requires $S=(z,1-2p;x,p;y;p)$ judged above $R=(z,1-2p;x^{\prime
},p;y^{\prime },p)\Leftrightarrow $ $S^{\prime }=(x,p;y;p;z^{\prime },1-2p)$
judged above $R^{\prime }=(x^{\prime },p;y^{\prime };p;z^{\prime },1-2p)$.
Therefore, if the unfilled circles ($S^{\prime }$) exceed filled circles ($%
R^{\prime }$) for a given value of y, then unfilled squares ($S$) should
exceed should exceed filled squares ($R$) for the same value of $y$. In
other words, the curves should cross at the same value of $y$. The data show
that for $y>\$45$, $s(S)>s(R)$ but $s(R^{\prime })>s(S^{\prime })$.

\FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F00.wmf';tempfile-properties "XPR";}}

\bigskip \pagebreak

Figure 2. Tests of Branch Independence in the Buyer's Viewpoint. Note that
for $y<\$50$, $b(S)>b(R)$ and $b(S^{\prime })<b(R^{\prime })$, showing the
same pattern of violation as in Figure 1. However, note that the rank order
of these judgments differs from that in Figure 1. Here, the upper two curves
cross, whereas in Figure 1, the lowest two curves cross.

\FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F01.wmf';tempfile-properties "XPR";}}

\bigskip \bigskip \pagebreak

Figure 3. Test of Lower Cumulative Independence in the Seller's Viewpoint, $%
S=(z,q;x,p;y,p)$ judged above $R=$ $(z,q;x^{\prime },p;y^{\prime },p)$ $%
\Rightarrow $ $S^{\prime \prime }=(x^{\prime },q;y,2p)$ judged above$%
R^{^{\prime \prime }}=(x^{\prime },p+q;y^{\prime },p)$. Note that for $%
y>\$44 $, filled squares exceed unfilled squares $s(S)>s(R)$; however, open
circles exceed filled circles for all y, so $s(R^{\prime \prime
})>s(S^{\prime \prime })$, violating lower cumulative independence.

\FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F02.wmf';tempfile-properties "XPR";}}

\bigskip \pagebreak

\bigskip Figure 4. Test of lower cumulative independence in the Buyer's
Viewpoint, as in Figure 3. \FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F03.wmf';tempfile-properties "XPR";}}

\bigskip

\bigskip \pagebreak

Figure 5. Test of upper cumulative independence: $S\prime =(x,p;y,q;z\prime
,r)$ judged below $R^{\prime }=(x^{\prime },p;y^{\prime },q;z^{\prime },r)$ $%
\Rightarrow $ $S^{\prime \prime \prime }=(x,p+q;y^{\prime },r)$ judged below 
$R^{\prime \prime \prime }=(x^{\prime },p;y^{\prime },q+r)$. Note that
filled circles ($S^{\prime }$) are below unfilled circles ($R^{\prime }$)
for all values of $x$, but that filled squares ($S^{\prime \prime \prime }$)
exceed unfilled squares ($R^{\prime \prime \prime }$) for $x>\$44$.

\FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F04.wmf';tempfile-properties "XPR";}}

\bigskip \bigskip \bigskip \pagebreak

Figure 6. Test of upper cumulative independence for mean buying prices, as
in Figure 5. Note that filled circles ($S^{\prime }$) are below unfilled
circles ($R^{\prime }$) for $x,<\$44$, but that filled squares ($S^{\prime
\prime \prime }$) exceed unfilled squares ($R^{\prime \prime \prime }$) for
all values of $x$.

\FRAME{ftbpF}{305pt}{226.6875pt}{0pt}{}{}{Figure}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 305pt;height 226.6875pt;depth
0pt;original-width 529.3125pt;original-height 392.4375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'H80G5F05.wmf';tempfile-properties "XPR";}}

\end{document}
