
\documentclass{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Created=Tuesday, October 08, 2002 11:18:33}
%TCIDATA{LastRevised=Thursday, December 05, 2002 16:51:25}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="General\SW\Blank - Standard LaTeX Article">}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=LaTeX article (bright).cst}

\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}


December 05, 2002

\begin{center}
{\LARGE Empirical Evaluation of Theories for\medskip\ }

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

{\normalsize Michael H. Birnbaum and Sherry Yeary}

{\normalsize California State University, Fullerton and}

{\normalsize Decision Research Center}

{\normalsize R. Duncan Luce and Li Zhao}

{\normalsize University of California, Irvine\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.

This research was supported by National Science Foundation Grants,
SBR-9410572 and SES 9986436 to C.S.U.F. and SBR-9808057 to UCI.\medskip

File: BuySell21.tex\pagebreak

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

Judges estimated the values of gambles from viewpoints of buyers and
sellers. Consistent with previous results, selling prices exceed buying
prices by a substantial amounts, and these two judgments are not
monotonically related to each other. Two models based on loss aversion
combined with cumulative prospect theory (CPT) do not give accurate accounts
of the data. In particular, judgments violate complementary symmetry, which
is implied by the best-fitting of the CPT models. Models based on the theory
of joint receipts by Luce fit better than those based on CPT and loss
aversion, but the best-fitting of these does not give an adequate account of
judgments involving a lowest consequence that might be zero or positive. Two
configural weight models give better fits to the data using the same number
or fewer parameters estimated from the data. Some aspects of the data favor
one and some favor the other of these configural models.

\bigskip

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

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

When people are asked to judge the highest price they would pay to buy
something, they report a much lower value than when they are asked to judge
the least they would accept to sell the same thing. This finding has been
observed for both goods of uncertain and ambiguous value, such as used cars,
and for gambles with defined probabilities and consequences (Birnbaum \&
Stegner, 1979; Birnbaum \& Sutton, 1992; Coombs, Bezembinder, \& Goode,
1967; Johannesson, Liljas, \& O'Conor, 1997; Knetsch \& Sinden, 1984). The
finding also extends to people in the role of advisors. What is the most you
would advise a buyer to pay in order to play a lottery that has a
fifty-fifty chance to pay either \$0 or \$100? If your friend owned the same
lottery, what is the least you would advise your friend to accept to sell
it? In this study, it was found that the average judge advises the buyer to
pay no more than \$25 to purchase this gamble and the seller to accept no
less than \$54 to sell this same gamble.

The discrepancy between judged willingness to pay (WTP) and willingness to
accept (WTA) is surprisingly large from the perspective of classical
expected utility theory in economics. The same phenomenon has been discussed
in different segments of the scientific literature and attributed to
different sources. The wide difference between buying and selling prices has
been termed a \textit{contingent valuation} effect, an \textit{endowment}
effect, or a \textit{viewpoint} effect. This paper will review models that
have been proposed to explain the effect and compare them with respect to
relevant results of a new experiment. Other results are reported in Birnbaum
and Yeary (2002).

The term, \textit{contingent valuation} has been used most often in
economics, usually in reference to studies of the monetary value of
non-market goods, such as damage to the environment (Irwin, Slovic,
Lichtenstein, \& McClelland, 1993). For example, suppose an oil company
accidentally spilled oil on a favorite beach where you used to enjoy nature
in its pristine beauty. How much would you demand to be paid as compensation
for the loss of pleasure at your favorite beach? Now suppose that your beach
had been polluted by a natural disaster that caused the same damage; how
much would you be willing to pay to clean it up? People give quite different
answers when asked the two questions. The differences are great enough that
both sides in lawsuits know how to frame the questions in order to get
results that support their respective viewpoints in court. If you work for
the company that caused the spill, you ask people what they would be willing
to pay to clean up the beach (buying price), and if you work for the
beach-goers association, you ask what they would demand in compensation for
the damage (selling price).

Literature on the \textit{endowment effect}, summarized in Kahneman,
Knetsch, and Thaler (1991; 1992; Thaler, Kahneman, \& Knetsch, 1994),
suggests that mere possession of an object affects its value by virtue of
ownership, or endowment. This endowment is presumed to change the person's
frame of reference so that the seller views giving up the endowed good
(e.g., a coffee mug) as a loss and receiving payment as a gain. A buyer,
however, views the good as a potential gain, but the price paid as a loss.
In the endowment literature, the cause of the discrepancy between buying and
selling prices is attributed to this framing effect combined with loss
aversion, the notion that subjective ``losses loom larger'' than comparable
gains (Tversky \& Kahneman, 1991).

Literature on the judge's \textit{point of view} suggested that buying and
selling induce different patterns of weighting for people who combine
information from imperfect sources. It is assumed that buyers place more
weight on lower estimates of value and sellers put more weight on higher
estimates of value (Birnbaum \& Stegner, 1979).

A number of models are compared in this paper: three based on loss aversion
combined with cumulative prospect theory (Tversky \& Kahneman, 1992), six
based on Luce's (2000) work, and two based on configural weight ideas first
suggested by Birnbaum and Stegner (1979).

\section{Theoretical Approaches}

Before the psychological models are developed more fully, it is helpful to
review classical expected utility theory.

\subsection{Classical Expected Utility Theory}

Consider the gamble, $g=(x,p;y),$ which yields monetary prize $x$ with
probability $p$, and otherwise yields $y$, where $0\leq y<x$. According to
expected utility theory, a person should be willing to buy gamble $g$ for
price $b$ if 
\begin{equation}
pU(\mathcal{W}+x-b)+(1-p)U(\mathcal{W}+y-b)\geq U(\mathcal{W})  \label{EUbuy}
\end{equation}%
where $U(\mathcal{W})$ is the utility function of total wealth, which
assigns a real number representing the utility of having a total, numerical
wealth of $\mathcal{W}$. This expression states that if the expected utility
of buying the gamble for price $b$ exceeds the utility of not buying, then
the person should decide to buy. Note that utility is defined on final
wealth.

Similarly, the seller who owns $g$ should be willing to sell it for price $s$
if the utility of selling exceeds the utility of keeping the gamble and
playing it: 
\begin{equation}
pU(\mathcal{W}+x)+(1-p)U(\mathcal{W}+y)\leq U(\mathcal{W}+s)  \label{EUsell}
\end{equation}%
where the expected utility of playing the gamble is shown on the left and
the utility of selling at price $s$ is on the right in Expression (\ref%
{EUsell}). The limiting cases of Expressions (\ref{EUbuy}) and (\ref{EUsell}%
) are found by substituting equality for inequality in Expressions (\ref%
{EUbuy}) and (\ref{EUsell}), giving equations for the highest buying price
and lowest selling price.

A problem for expected utility theory is that for plausible utility
functions (e.g., logarithmic function), the difference between buying and
selling price of gamble $g$ quickly goes to zero as a person's wealth
increases above quite modest levels. For example, the difference between
buying and selling prices of a fifty-fifty chance to win $\$100$ or $\$0$
should be less than a nickel for a person with a total wealth of $\$75,000$.
Yet, the difference between buying and selling prices for such people is
much greater than the predictions of this model; which implies $s=\$49.95$
and $b=\$49.93.$

An alternative equation for selling, parallel to that of (\ref{EUbuy}) is as
follows 
\[
pU(\mathcal{W}+s\mathcal{-}x)+(1-p)U(\mathcal{W}+s-y)\geq U(\mathcal{W}). 
\]%
This formulation has the same problems; namely, buying and selling prices
should be nearly equal.

Markowtiz (1952) suggested that utility be defined on changes in wealth
rather than final wealth. Accordingly, we should set $\mathcal{W}=0$ in
Expressions 1 and 2. The idea of defining utility on changes from zero or
some reference level was incorporated into psychological theory by Edwards
(1954; 1962) and Kahneman and Tversky (1979), among others.

If we set $\mathcal{W}=0$ in Expressions (\ref{EUbuy}) and (\ref{EUsell}),
then the act of buying a gamble involves gains and losses. If the buyer pays 
$b$ for $g$, then the buyer wins $x-b$ with probability $p$ and $y-b$ (a
loss) with probability $1-p$. As we see in the next section, this idea
combined with other psychological theory can be used to develop models of
buying and selling prices.

\subsection{Cumulative Prospect Theory and Loss Aversion}

In the theories of Edwards (1954; 1962), Kahneman and Tversky (1979) and
Tversky and Kahneman (1992), probability weights are not necessarily equal
to objective probabilities. To model buying and selling prices in such
theories, one must incorporate both the weights of probabilities and also
characterize the relationship between the utilities of gains and losses.

Although Kahneman, Tversky, and their associates never proposed a specific
model of buying and selling prices (Kahneman, personal communication,
November, 1997), Birnbaum and Zimmermann (1998) specified models that
combine the idea of loss aversion in Tversky and Kahneman (1991; Kahneman,
et al., 1991; 1992) with the model of CPT used by Tversky and Kahneman
(1992).

The CPT model represents the value of a binary gamble, $g=(x,p;y),$ as
follows:%
\begin{equation}
\begin{tabular}{ll}
$U(x,p;y)=U(x)W^{+}(p)+U(y)[1-W^{+}(p)]$ & $(x\geq y\geq 0)$ \\ 
$U(x,p;-y)=U(x)W^{+}(p)+U(-y)W^{-}(1-p)$ & $(x\geq 0\geq -y)$ \\ 
$U(-x,p;-y)=U(-x)W^{-}(p)+U(-y)[1-W^{-}(p)]$ & $(0\geq -y\geq -x)$%
\end{tabular}%
.  \label{CPT'}
\end{equation}%
In these expressions $U(x)$ is the utility function, $U(-x)$ is the utility
of a loss of $x,$ and we assume $U(0)=0;$ thus, the utility function is a
ratio scale, not the interval scale of expected utility theory. $W^{+}(p)$
is the weighting function for a gain received with probability $p$, and $%
W^{-}(p)$ is the weight of a loss that occurs with probability $p$. \ It may
also be helpful to note that the third expression would be written as
follows for the case of $0\geq -x\geq -y$, which preserves the same rank
relationship as\ in the first two cases:

\[
U(-x,p;-y)=U(-x)[1-W^{-}(1-p)]+U(-y)W^{-}(1-p)\quad (0\geq -x\geq -y)
\]

The theory of loss aversion has been expressed as follows: 
\begin{equation}
U(-x)=-\lambda U(x),\text{ }x>0  \label{loss aversion}
\end{equation}%
where $U(-x)$ is the utility of a loss, and $\lambda $ is the factor by
which losses are said to ``loom larger'' than gains, when $\lambda >1$. If $%
\lambda =2$, for example, it means that a loss of $-x$ dollars has twice the
psychological impact as a gain of the same amount, $x$. We do not dispute
the fact that people would rather win than lose. In this paper, however, it
is important to keep in mind that the theory of loss aversion refers to
this\ utility representation of the concept (as opposed, for example, to the
theory that people are averse to losses because losses carry greater weight).

In the next section, we develop models that combine Tversky and Kahneman's
(1992) CPT\ model (\ref{CPT}) with the representation of loss aversion (\ref%
{loss aversion}). To do this, we need definitions, or models, of buying and
selling prices.

\subsubsection{Three Models of buying and selling prices}

Three natural definitions of buying prices and selling prices of $g$ with $%
x\geq y\geq 0$ come to mind. We assume that both $s$ and $b$ lie in the
interval $[y,x].$ In the first model, buying price is the largest $b$ and
selling price is the smallest $s$ for which the following expressions hold: 
\begin{eqnarray}
U(g)-U(b) &\geq &0  \label{buy a} \\
U(s)-U(g) &\geq &0  \label{sell a}
\end{eqnarray}%
Expression (\ref{buy a}) assumes that a person decides to buy if the utility
of the gamble exceeds the positive utility of the buying price, and (\ref%
{sell a}) says a person decides to sell if the utility of receiving the
sales price exceeds the positive utility of the gamble. This defintion when
combined with CPT and loss aversion model will be termed loss aversion model
1.

Good experimental reasons rule out both (\ref{buy a}) and (\ref{sell a}).
Expression (\ref{sell a}), which says that $U(s)$ equals the utility of the
gamble, implies that selling prices should be monotonically related to the
subjective values of gambles. In other words, if a person sets a higher
selling price on gamble $g$ than gamble $f$, then we know that the
subjective value of gamble $g$ exceeds that of $f$. That means that this
person should prefer gamble $g$ to $f$ in a direct choice. However, there
are gambles for which people systematically set a higher selling price on $g$
than $f$ and then prefer $f$ over $g$ in direct choice (Goldstein \&
Einhorn, 1987; Lichtenstein \& Slovic, 1971; Lindman, 1971; Mellers, Ord\'{o}%
\~{n}ez, \& Birnbaum, 1992; Mellers, Chang, Birnbaum, \& Ord\'{o}\~{n}ez,
1992; Tversky, Sattath, \& Slovic, 1988). Similarly, Expression (\ref{buy a}%
) says that buying price is also equal to the utility of the gamble, so if a
person sets a higher buying price on gamble $g$ than $f$, that person should
also prefer $g$ to $f$, which also has been shown to be violated.

Such results are called \textit{preference reversals} because one way of
comparing the utility of the gambles (direct choice) yields one preference
relation, and another method (selling or buying price) yields seeming
contradictions. Tversky and Kahneman (1991) noted that preference reversals
between choices and selling prices contradict this model of loss aversion
[Expression (\ref{sell a})].

Furthermore, if buying and selling prices are each equal to the gamble's
utility, then they should be equal to each other. In addition to the
numerical differences cited above, it has been reported that buying and
selling prices are not even monotonically related to each other (Birnbaum \&
Beeghley, 1997; Birnbaum \& Sutton, 1992; Birnbaum, Coffey, Mellers, and
Weiss, 1992). These are cases where gamble $g$ has a higher buying price
than $f$, but $f$ has a higher selling price than $g$. Therefore, this first
approach does not account for the empirical results we seek to explain.

Bostic, Herrnstein, and\ Luce (1990) and Tversky, Slovic, and Kahneman
(1990), using quite different designs (but both based purely on choices),
have provided evidence that such preference reversals are attenuated. Even
if purely choice methods were to completely eliminate preference reversals
(between two types of choice data), we would still have the problem of
explaining why judgments and choices do not agree. Because the first model
does not explain the empirical differences between judgments and choices, we
need another approach if we hope to account for such data.

In the second approach, the buying price is assumed to reflect an implicit
comparison between the positive utility of receiving the gamble against the
negative utility of\ (losing) the buying price, $U(-b)$. Similarly, the
selling price is represented as the result of an implicit comparison between
the positive utility of receiving the sales price compared against the loss
of the gamble, where the notation $U(-g)$ represents the utility of $%
-g=(-x,p;-y)$, where $-x<-y\leq 0.$ In contrast to the first approach, this
model involves comparisons between gains and losses, whereas the first
approach involved only comparisons of positive values.

\begin{eqnarray}
U(g)+U(-b) &\geq &0  \label{buy b} \\
U(s)+U(-g) &\geq &0  \label{sell b}
\end{eqnarray}

The third approach treats decisions to buy or sell as the result of an
integration of buying and selling prices into the consequences of the
gambles. In the case of buying prices, the person is presumed to evaluate a
new gamble, which we denote $g-b=(x-b,p;y-b)$, where $x-b$ is the profit if
the gamble wins and $y-b$ is the loss if the gamble yields only $y$.
Similarly, in selling, one considers it a ``loss''of $x-s$ if the gamble
wins $x$, since the seller lost the opportunity to win by selling; and when
the gamble pays only $y$, the seller considers that a ``win'' of $s-y$. In
decisions to sell, the integrated gamble is denoted as $s-g=(s-x,p;s-y).$

\begin{eqnarray}
U(g-b) &\geq &0  \label{buy c} \\
U(s-g) &\geq &0  \label{sell c}
\end{eqnarray}

Expression (\ref{buy c}) represents the decision to buy in terms of a
modified gamble in which the consequences are $x-b>0$ and $y-b<0$ compared
to the status quo. The interpretation of the selling expression (\ref{sell c}%
) is similar, except the seller \textit{wins} $s-y$\ when the buyer loses $%
y-b$.

\subsubsection{Loss Aversion Model 2: Balancing Costs and Benefits}

Recall that first first loss aversion model was immediately ruled out. The
second model of loss aversion, Expressions (\ref{buy b}) and (\ref{sell b})
appears more promising and it matches verbal statements made in the
endowment literature for the values of ambiguous goods (Tversky \& Kahneman,
1991). Combining CPT (\ref{CPT}), substituting loss aversion (Equation \ref%
{loss aversion}), and recalling that with $x\geq y$ then the order in $-g$
is $0\geq -y\geq -x,$ we have the following two expressions: 
\begin{eqnarray*}
\lambda U(b) &=&U(x)W^{+}(p)+U(y)[1-W^{+}(p)], \\
U(s) &=&\lambda U(x)[W^{-}(p)]+\lambda U(y)[1-W^{-}(p)].
\end{eqnarray*}%
If $W^{+}(p)=W^{-}(p)$, as assumed by Kahneman and Tversky (1979) and
reported as a good approximation by Tversky and Kahneman (1992), it follows
that 
\[
U(s)=\lambda ^{2}U(b). 
\]%
Therefore, 
\begin{equation}
s=U^{-1}[U(b)\lambda ^{2}];  \label{LAbs}
\end{equation}%
where $U^{-1}$ is the inverse of $U(x)$. Thus, this model implies that
selling and buying prices should be monotonically related to each other. If
we further suppose that $U(x)=x^{\beta }$, as postulated by Tversky and
Kahneman (1992); it follows that 
\begin{equation}
s=\lambda ^{2/\beta }b.  \label{LApower-bs}
\end{equation}%
For example, if $U(x)=x$ and $\lambda =2$, then $s/b=4.$ Table 1 of
Kahneman, et. al.\thinspace (1991) lists twelve values of the ratio of $s/b$%
, with the median between 4.0 and 4.2. If $\beta =0.88,$ as has been
estimated by Tversky and Kahneman, then $s/b=4$ requires $\lambda =1.84.$
With the power function for utility, the model implies that the ratio of
selling price to buying price should be constant within a dimension.

However, data from many studies in psychology have shown that selling prices
and buying prices are not monotonically related to each other, which
contradicts the general expression of Equation (\ref{LAbs}). For example,
Birnbaum and Sutton (1992) found that from the viewpoint of the seller, $%
g=(\$96,0.5;\$0)$ is judged higher than $f=(\$48;0.5;\$36)$; whereas, from
the viewpoint of the buyer, $g$ is judged lower than $f$. Birnbaum and
Stegner (1979) reported similar results for used cars, goods of ambiguous
value (see also Birnbaum, 1982).

Such reversals between buying and selling prices represent a second type of
preference reversal [in addition to reversals between prices and choices
discussed in Section 1.2.1, which refuted (\ref{buy a}) and (\ref{sell a})].
Such reversals have been reported in a number of papers (Birnbaum \& Sutton,
p.\thinspace 208; Birnbaum \& Stegner, 1979; Birnbaum, 1982, p.\thinspace
470-472; Birnbaum \& Beeghley, 1997; Birnbaum, et al., 1992; Birnbaum \&
Zimmermann, 1998). This second type of preference reversal rules out this
second model of loss aversion (\ref{buy b} and \ref{sell b}).

\subsubsection{Loss Aversion Model 3: Mixed Gambles}

The third approach assumes (\ref{buy c}) and (\ref{sell c}), which means
that both buyer and seller integrate the consequences of the gamble with the
buying or selling price. In (\ref{sell c}), when the gamble ``wins'' (i.e.,
pays the higher prize, $x$), the person who sold the gamble considers the
situation to be a subjective loss, since by selling, the seller ``lost'' the
opportunity to win. Similarly, if the gamble pays its lower prize, $y$, the
seller considers that a ``win'', since the seller was better off to have
sold the gamble rather than to have played it. This model is a simpler
version of another discussed in Section 1.3.1 below.

As in Kahneman, et al.\thinspace (1991), suppose that loss aversion is
represented by $U(-x)=-\lambda U(x)$. The highest buying prices of gambles, $%
g=(x,p;y),$ $x>y>0$, are then given by the following: 
\begin{eqnarray*}
0 &=&W^{+}(p)U(x-b)+W^{-}(1-p)U(y-b) \\
&=&W^{+}(p)U(x-b)-W^{-}(1-p)\lambda U(b-y)
\end{eqnarray*}%
i.e., 
\[
U(x-b)=\frac{W^{-}(1-p)\lambda U(b-y)}{W^{+}(p)} 
\]%
Following Tversky and Kahneman (1992), suppose that $U(x)=x^{\beta }$.
Define 
\[
T(p)=\left[ \frac{W^{+}(p)}{W^{-}(1-p)\lambda }\right] ^{\frac{1}{\beta }}, 
\]%
then 
\begin{equation}
b=\frac{T(p)x+y}{T(p)+1}  \label{LA3b}
\end{equation}

Similarly, if lowest selling prices are given by (\ref{sell c}), it follows
that, 
\begin{equation}
s=\frac{x+T(1-p)y}{T(1-p)+1}  \label{LA3s}
\end{equation}

For complementary gambles $g=(x,p;$ $y)$ and $g^{\prime }=(x,1-p;y)$, it
follows that: 
\begin{equation}
b(g)+s(g^{\prime })=x+y  \label{comp sym}
\end{equation}%
where $b(g)$ and $s(g^{\prime })$ represent the buying and selling prices
for gamble $g$ and its complement, $g^{\prime }$. Note that this expression
holds for all $x$, $y$, and $p$. We term this implication \textit{%
complementary symmetry}. This prediction of complementary symmetry produces
the horizontal line in Figure 5 (see Section 1.4.4). Intuitively, this
prediction follows from the presumed symmetry between how buyers and sellers
view the transaction. What is a loss to one is a gain to the other, and if
the buyer wins with probability $p$, then the seller wins with probability $%
1-p.$

Note that the derivations of Equations (\ref{LA3b}) and (\ref{LA3s}), and
thus of complementary symmetry [Equation (\ref{comp sym})] from the
cumulative prospect model do not assume anything about the forms of $W^{+}$
and $W^{-}$ or their relationship. They do depend upon the assumptions
concerning utility and loss aversion. Birnbaum and Zimmermann (1998) derived
the special case of this implication for fifty-fifty gambles, and noted that
the data of Birnbaum and Sutton (1992) are inconsistent with this prediction
when $x+y$ is fixed and $|x-y|$ is varied. The present study will test this
prediction as a function of $p$, holding $x+y$ and $|x-y|$ constant.

Predictions of the parameterized CPT model fit by Tversky and Kahneman
(1992) for gambles of the form, $g=(\$100,p;$ $\$0)$ are plotted in Figure 1
as a function of $p$, with separate curves for seller's (upper curve) and
buyer's prices (lowest curve). Also shown in Figure 1 are the calculated
certainty equivalents ($CE$s) for the same gambles. The $CE$ of a gamble is
the amount of cash that would be indifferent to the gamble; i.e., if $%
U(CE)=U(g)$, then $CE=U^{-1}[U(g)]$. In what follows, we will fit the more
general forms of Equations (\ref{LA3b}) and (\ref{LA3s}), which allow any $%
T(p)$ function. 
\[
\fbox{\textit{Insert Figure 1 about here.}} 
\]

\subsection{Rank- and Sign-Dependent Utility Models}

Luce (2000) deduced implications for buying and selling prices from his
theory of preferences\textbf{\ }that is based on joint receipts of gains,
losses, and gambles defined on those consequences. Joint receipt refers to
the psychological combination of two or more goods or gambles. One can view
the act of buying as the joint receipt of a loss of money and of the gain of
a gamble, and the act of selling as the joint receipt of cash and the loss
of a gamble. Different combinations of assumptions in the theoretical
framework of joint receipt give rise to different models as special cases.
Some of these models imply that buying and selling prices are identical,
contrary to evidence. However, the assumptions that buying and selling
prices are integrated into nonnegative consequences, producing mixed
gambles, can be used to derive models in which selling prices exceed buying
prices. In contrast with loss aversion theories, however, the utility
function for losses does not appear in the equations derived from this
approach; instead, these models attribute the discrepancy between buying and
selling prices to the pattern of weighting in mixed gambles.

This class of models, which is described in detail in Luce (2000), rests on
two primitives in addition to a weakly-ordered (reflexive, transitive, and
complete) preference relation, $\succsim ,$ over gambles. In the usual way,
define indifference, $\sim ,$ strict preference, $\succ ,$ and the converse
ordering, $\precsim $.

The first additional concept is a distinguished consequence, denoted $e,$
that is identified as \textit{no change from the status quo}. If $f$ is a
gamble (including, as a special case, pure consequences) and $f\succsim e,$
then $f$ is called a\textit{\ gain} and if $f\precsim e$, then $f$ is a%
\textit{\ loss}.

The second additional concept is a binary operation, denoted $\oplus $ and
called \textit{joint receipt.} If $f$ and $g$ are two gambles (again,
including pure consequences as special cases), then $f\oplus g$ means that
the decision maker holds or receives them at the same time. We assume that $%
\oplus $ is commutative in the sense that $f\oplus g\sim g\oplus f$ and that 
$e$ is its identity in the sense that $f\oplus e\sim f.$ It is useful below
to define a ``subtraction'' operation, $\ominus ,$ in terms of $\oplus $ by 
\begin{equation}
f\ominus g\sim h\Longleftrightarrow f\sim g\oplus h.  \label{sub}
\end{equation}%
We assume that such differences exist, which with money gambles is not
really an issue. We do not assume that for money amounts $x,y$ that $x\oplus
y=x+y.$ The reasons are discussed on pp.\thinspace 144 and 154-155 of Luce
(2000).

\subsubsection{Buying and selling prices}

The following models have some similarities to the third loss aversion, CPT\
model, in the sense that they replace actual subtraction with the
generalized subtraction operator $\ominus ,$ but the utility structure is
quite different. In terms of the operation $\ominus $, Expression (\ref{sub}%
), Luce (1991; see also Hazen \& Lee, 1991) defined (maximum) buying and
(minimum) selling prices, $b$ and $s,$ of a binary gamble $(x,p;y),$ with $%
x\succsim y,$ as solutions to the following indifferences 
\begin{equation}
(x\ominus b,p;y\ominus b)\sim e\text{\quad and\quad }(s\ominus x,p;s\ominus
y)\sim e.  \label{RDUbs}
\end{equation}%
Note that if $\ominus \equiv -,$ then this definition agrees with (\ref{buy
c}) and (\ref{sell c}) of the Loss Aversion Model 3 (\S\ 1.2.3).

The rationale is that if one buys the gamble for the amount $b,$ then the
net result is a gamble with the consequences reduced by the amount $b.$
Similarly, if one sells a gamble for $s,$ then one still ``has'' the gamble
but with, in effect, the consequence of $s$ less either $x$ or $y$ depending
upon the outcome of the chance phenomena underlying the gamble. Note: from
the assumptions that $\succsim $ is a weak order that satisfies consequence
monotonicity over gambles, it follows that:

\begin{itemize}
\item $x\ominus b\succsim e\succsim y\ominus b$ and $s\ominus x\precsim
e\precsim s\ominus y.$

\item Thus, no matter whether the gamble is all gains, as it is in the
present study, or all losses, or mixed gains and losses, the gambles
defining the buying and selling prices necessarily have mixed consequences.

\item For $b^{\prime }\prec b,$ $(x\ominus b^{\prime },p;y\ominus b^{\prime
})\succ e$ and for $b^{\prime }\succ b,$ $(x\ominus b^{\prime },p;y\ominus
b^{\prime })\prec e$

\item For $s^{\prime }\succ s,$ $(s^{\prime }\ominus x,p;s^{\prime }\ominus
y)\succ e$ and for $s^{\prime }\prec s,$ $(s^{\prime }\ominus x,p;s^{\prime
}\ominus y)\prec e.$
\end{itemize}

It is clear from the definitions we need to understand the form of the
utility of gambles of a mixed gain and a loss. To do this entails two steps.
The first is to decompose a mixed gamble into the joint receipt of two
gambles that are not mixed ones. The second is to see the form of the
utility of joint receipts. We take them up in that order.

\subsubsection{Decomposing mixed gambles}

For gambles of mixed gains and losses, two major hypotheses have been
explored that are designed to reduce the calculation of $U(x,p;y),$ where $%
x\succsim e\succsim y,$ to the joint receipt of gambles in which one of the
consequences is $e.$ Such simple gambles may be called \textit{unitary}. The
advantage of such decompositions is that if one can figure out how to
calculate utility of a joint receipt, then the calculation is reduced to
determining the form of $U$ for unitary gambles. And for most models that
have been proposed, they have the \textit{separable} form 
\[
U(x,p;e)=U(x)W^{(i)}(p),\quad \text{where }i=+\text{ if }x\succsim e,\text{ }%
i=-\text{ if }x\prec e. 
\]%
Qualitative conditions for separability to hold are well understood (Luce,
2000, \S\ 3.5.2).

One decomposition assumption is \textit{general segregation }(Luce, 1997),
for $x\succsim y,$ $x\succsim e,$ 
\[
(x,p;y)\sim \left\{ 
\begin{tabular}{ll}
$(x\ominus y,p;e)\oplus y,$ & if $(x,p;y)\succsim e$ \\ 
$(e,p;y\ominus x)\oplus x,$ & if $(x,p;y)\precsim e$%
\end{tabular}%
\ .\right. 
\]%
Note the normative quality of this condition in the sense that what one
receives in the decomposed gamble is exactly the same as in the original
gamble. This generalizes the property of segregation defined by Luce (1991)
and Luce and Fishburn (1991) for gambles of all gains (or all losses).

The other proposed rule is \textit{duplex decomposition} (Luce \& Fishburn,
1991; Slovic \& Lichtenstein, 1968) 
\[
(x,p;y)=(x,p;e)\oplus (e,p;y), 
\]%
where the two unitary gambles on the right are run independently. Note that
this is not normatively defensible because the possible outcomes on the
right are $x,$ $y,$ $x\oplus y,$ and $e\oplus e=e,$ whereas on the left they
are just $x$ or $y.$ Nevertheless there are considerable empirical data
indicating that it may hold (Luce, 2000, \S\ 6.2.3).\footnote{%
However, Cho, Luce, and Truong (2002) empirically tested duplex
decomposition and general segregation, and, depending upon the exact
criterion used to say that a rule can be rejected, they found that from one
third to one half the participants did not satisfy either}

\subsubsection{Utility of joint receipt}

So the task is\ reduced to determining the utility of joint receipt. In a
development that is too complex to summarize in detail here (see Luce, 2000,
\S\ 4.4.6), the following formula has been developed on the assumption that $%
\oplus $ is a commutative operation and that segregation holds and $f\ $and $%
g$ are gambles (or pure consequences) of both gains or both losses 
\[
U(f\oplus g)=U(f)+U(g)-\delta _{i}U(f)U(g), 
\]%
where $i$ distinguishes gains and losses. When $\delta _{i}=0,$ utility $U$,
is additive. For $\delta _{i}\neq 0,$ it is not difficult to show that, for
gains and for losses separately, there exists an additive representation $V,$
i.e., 
\[
V(f\oplus g)=V(f)+V(g), 
\]%
for which $U$ is a negative exponential function of $V$ (concave) when $%
\delta _{i}>0$ and an exponential function (convex) when $\delta _{i}<0$.
This means that $\oplus $ is also associative over gains and losses
separately.

Moreover, for money, invariance arguments imply that $V$ for money is a
power function (Luce, 2000, \S\ 4.5.3), 
\[
V(x)=\alpha x^{\beta },\quad x>0. 
\]
In the present context of buying and selling prices, negative consequences
will play no direct role in the calculation, although they definitely affect
the weights used and the forms of the equations derived. The cases where $U$
itself is additive will be ignored since they result in equal buying and
selling prices, which we know empirically is wrong. So there are four
combinations with the gains first and losses second: concave\TEXTsymbol{%
\backslash}concave (denoted C\TEXTsymbol{\backslash}C), convex\TEXTsymbol{%
\backslash}convex (V\TEXTsymbol{\backslash}V), concave\TEXTsymbol{\backslash}%
convex (C\TEXTsymbol{\backslash}V), convex\TEXTsymbol{\backslash}concave (V%
\TEXTsymbol{\backslash}C).

\subsubsection{Calculating the prices}

Formulas for buying and selling prices worked out in Chapter 7 of Luce (2000)%
\footnote{%
Note that one of the relevant propositions, 7.4.6(ii), is in error. This is
corrected in an errata listed in the references and it is reflected in the
equations of Appendix A of this paper.}, one arrives at equations for $%
U(b)=B,$ $U(s)=S$ that are stated in terms of $U(f)=F,$ $U(g)=G,$ and
various weighting functions over $p$ or $1-p$ that differ depending on
whether $p$ applies to a gain or loss. The (normed) utility expressions for
the concave cases are for the money amount $x>0$%
\begin{equation}
U(x)=1-\exp (-\kappa x^{\beta }),  \label{RDU-UC}
\end{equation}%
and in the convex cases 
\begin{equation}
U(x)=\exp (\kappa x^{\beta })-1.  \label{RDU-UV}
\end{equation}%
For the weighting functions, we used the form derived by Prelec (1998) and
in an improved way by Luce (2001), which has been indirectly supported
empirically (see Luce, 2000, \S\ 3.4.3), 
\begin{equation}
W^{(i)}(p)=\exp \left[ -\gamma (i)\left( -\ln p\right) ^{\eta (i)}\right] ,
\label{Prelec}
\end{equation}%
where $i$ is $+$ when the consequence associated with $p$ is a gain and $-$
when it is a loss.

Thus, there are a total of 6 parameters for each of these models, two from
the utility function of gains and 4 from the weights. And there are six
models that do not lead to $b\sim s,$ four based on duplex decomposition and
two based on general segregation. The expressions for $B$ and $S$ are shown
in Appendix A. Note that all of these require finding the ``best'' overall
solution by varying the 6 parameters. We used minimization of sum of squares
as the criterion of the fit.

In the duplex decomposition case the equations for $B$ and $S$ are
quadratics. Each of these equations has exactly one real solution in the
interval $(G,F).$ The proof is given in Appendix B for the $S$ equation for
Duplex Decomposition in the C\TEXTsymbol{\backslash}C and V\TEXTsymbol{%
\backslash}V cases; the other proofs are similar.

Figure 2 plots predictions of the best-fitting Duplex Decomposition, C%
\TEXTsymbol{\backslash}V model with parameters estimated from the data.
Figure 3 plots the test of Complementary Symmetry for four models, including
the two configural weight models of the next section. Only the loss aversion
CPT\ model implies complementary symmetry. The other models of this and the
next section violate this property, except in special cases. 
\[
\fbox{\textit{Insert Figures 2 and 3 about here.}} 
\]

\subsection{Configural Weighting and Point of View}

Birnbaum and Stegner (1979)\textbf{\ }gave an account of the difference
between buying and selling prices in terms of configural weights that change
as a function of the \textit{judge's viewpoint}. Whereas the concept of
endowment has only two levels, ownership or not, the concept of viewpoint
admits more than two positions. Besides the viewpoints of buyer or seller,
one can thus imagine viewpoints for sympathetic advisors to these parties.
One does not need actually to own a good in order to identify with the
seller. In addition, one can imagine the viewpoint of a ``neutral'' judge
who is instructed to judge the ``fair'' value of the gambles or goods, so
that neither buyer nor seller receives advantage from the judgment. Indeed,
in a lawsuit, juries and judges are often asked to estimate the value of
damages in a fashion that is intended to be fair to both sides in a dispute.

In contrast to the idea of loss aversion (in which utilities of gains and
losses are key), Birnbaum and Stegner (1979) proposed that the configural
weights of higher- or lower-valued information depend on the judge's point
of view. They asked judges to evaluate hypothetical used cars based on blue
book value and the opinion(s) of one (or two) source(s) who tested the cars.
The judges were put in three viewpoints; they were to advise the buyer, the
seller, or a neutral judge who was to evaluate the ``fair'' price of each
car. Birnbaum and Stegner fit their data with what they termed configural
weight models, because the weight of a component depends on the
configuration or relationships among the components. In such models, the
weight of a stimulus can be affected by the rank of that stimulus among the
stimuli to be integrated, so these models have some similarities to the
rank-dependent utility models that were developed at a later date. In both
of the models described by Birnbaum and Stegner (1979), the configural
weight of the higher or lower valued item was assumed to depend on the
judge's viewpoint.

Birnbaum, et al.\thinspace (1992) showed that if buyers are sensitive to the
cost of over-estimating value and sellers are sensitive to the cost of
underestimating value, then a configural weight model follows from a simple
assumption about the nature of how costs depend on errors of estimation (see
also Weber, 1994).\ In a sense, the theoretical concern for asymmetric
errors can be viewed as a new proposal for the carriers of utility; the
judge may be viewed as trying to select a price to minimize an asymmetric
function with asymmetric errors for over as opposed to under estimation.

In addition to the finding that buying and selling prices are not
monotonically related to each other (the second class of preference
reversals noted above), it has been found that judgments of buyer's seller's
or neutral's prices are not always a monotonic function of the consequences
(Birnbaum, 1997; Birnbaum, et al., 1992; Birnbaum \& Sutton, 1992; Mellers,
Weiss, \& Birnbaum, 1992; Mellers, Berrety, \& Birnbaum, 1995). As the
lowest consequence of a gamble is changed from zero to a small positive
consequence, it has been found that judgments can decrease. Mellers, et
al.\thinspace (1992) found a similar result with losses: changing the
highest consequence from zero to a small loss improved these negative
gambles. This violation of consequence monotonicity can be explained by the
assumption that the zero consequence receives less weight than nonzero
consequences. Although this type of violation has been observed in buyer's,
neutral's, and seller's prices and with indirect choices (Birnbaum, 1992;
1997; Birnbaum \& Thompson, 1996), such violations are rare in direct choice
(Birnbaum \& Sutton, 1992), which produces yet a third type of preference
reversals between judgment and choice.

Configural weight models known as the RAM and TAX models (see Sections 1.4.1
and 1.4.2) were applied to judged buying and selling prices of risky gambles
and uncertain investments in a series of papers by Birnbaum and his
colleagues (Birnbaum \& Sutton, 1992; Birnbaum, et al., 1992; Birnbaum \&
Beeghley, 1997; Birnbaum \& Veira, 1998; Birnbaum \& Zimmermann, 1998). In
both the RAM and TAX models described below it is assumed that consequences
are rank ordered in the sense that $(x,p;y)$ with $x>y$. In addition, both
models describe violations of consequence monotonicity by means of
configural weighting that depends on the rank and probabilities of the
consequences, as well as their augmented signs (augmented sign has 3 levels: 
$+$, $-$, and $0$).

For binary gambles of the form, $g=(x,p;$ $y)$, where $x>y\geq 0$, RAM and
TAX models are both relative weight averaging models of the form,%
\[
U[(x,p;y),j]=\left\{ 
\begin{tabular}{ll}
$U(x)W(p,j)+U(y)[1-W(p,j)]$ & $y>0$ \\ 
$U(x)W_{0}(p,j)+U(y)[1-W_{0}(p,j)]$ & $y=0$%
\end{tabular}%
\quad (j=B,S).\right. 
\]%
Assuming $U(0)=0$, the second equation can be simplified to $%
U[(x,p;0),j]=U(x)W_{0}(p,j)$ for $y=0.$ This model, like those of CPT and
rank-and sign-dependent utility for the utility of a gamble in the binary
case of gains, is bilinear. However, the relative weights of the more
preferred consequence, $W(p,j)$ and $W_{0}(p,j),$ depend on two things: the
probability, $p$, of that consequence $x$ and the judge's viewpoint, $j=$ $B$
or $S$ (buyer or seller). In addition, the weighting functions depend on
whether the smallest consequence is zero or not (i.e., $y>0$ or $y=0)$,
which provides a way to accomodate the observed non-monotonicities.

Unlike the previous models, this same equation (with different weights) is
applied to buying and selling prices as well as to choice. Unlike the other
models, therefore, the \textit{utility} of a gamble changes, depending on
the judge's point of view (though the utility of pure consequences is
invariant, as it is in the other models). Unlike the other models, this
model does not posit that losses play a role in buying and selling prices of
positive consequence gambles.

Birnbaum, et al.\thinspace (1992) estimated separate weighting functions of
probability for positive and zero-valued consequences. Birnbaum (1997)
argued that the effect can be fit with one degree of freedom. In this paper,
the relationship between relative weights $W(p,j)$ and $W_{0}(p,j)$ will be
fit with the following approximation:

\[
W_{0}(p,j)=\alpha _{0}W(p,j) 
\]%
where $\alpha _{0\text{ }}$is a constant reflecting greater relative weights
for positive consequences when the lowest consequence is zero than when the
lowest consequence is positive ($\alpha _{0}>1$).

The two models below will be fit with the approximation that utility is a
power function of monetary prizes, 
\begin{equation}
U(x)=x^{\beta }  \label{Upower}
\end{equation}%
In previous research, it has been found that for consequences in the domain, 
$0<x<\$150$, the exponent, $\beta $, can be set to $1$ with little loss of
predictive accuracy when fitting judgments and choices. Birnbaum considers
this proportionality between utility and cash for small sums to be an
attractive feature of models that tolerate it, based on the intuition that
the utility of small sums (on the order of pocket money) should be
approximately proportional to objective value.

All configural weighting models assume that if the utility function, $U(x),$
were estimated separately from buyer's, seller's, and neutral's judgments,
the three estimated functions should all agree, except for error. Birnbaum
and Stegner (1979), Birnbaum and Sutton (1992) and Birnbaum, et
al.\thinspace (1992) tested this implication and found data in good
agreement with it, despite the fact that the rank orders of the different
types of judgments are systematically different. In addition, this (single)
utility function for judgment should also agree with the utility function
used to predict choices. This prediction was reported as acceptable by
Birnbaum and Beeghley (1997), based on their review of judgment and choice
data (Birnbaum, 1982, p. 454-472; Birnbaum \& Sutton, 1992; Birnbaum, et
al., 1992; Birnbaum \& Beeghley, 1997; Birnbaum \& Chavez, 1997; Birnbaum \&
McIntosh, 1996).

In previous research, it has been found that the mapping from gambles'
inverse utilities to overt responses can be approximated by a constant of
proportionality, which is assumed to be the same for both viewpoints: 
\begin{eqnarray*}
s &=&\nu U^{-1}[U(g,\text{ }S)] \\
b &=&\nu U^{-1}[U(g,\text{ }B)]
\end{eqnarray*}%
where $s(g)$ and $b(g)$ represent the selling and buying prices of gamble $g$%
; and $\nu $ is a constant (Birnbaum \& Beeghley, 1997).

The RAM and TAX\ models differ in their assumptions concerning how
configural weights depend on the judge's viewpoint. They both violate
complementary symmetry, but do so in different ways.

\subsubsection{RAM Model}

In the Rank and Augmented sign-affected Multiplicative weights (RAM) model
of Birnbaum (1997), configural weights are represented as products of terms
involving the effects of viewpoint, rank, augmented sign, and probability.

In the RAM model the weight of the higher valued, probability-consequence
branch as displayed in the gamble (to win $x$ with probability $p)$ is given
by: 
\begin{equation}
W(p,j)=\frac{\alpha _{j}p^{\gamma }}{\alpha _{j}p^{\gamma }+(1-p)^{\gamma }}%
,\quad j=B,S,\;  \label{Tpower}
\end{equation}

When $\gamma $ $<1$, this expression implies that the \textit{relative}
weight of the higher consequence will be an inverse-\textsf{S} function of $%
p $, even though the \textit{absolute} weight of the higher consequence is
proportional to a power function of $p$. The inverse-\textsf{S} relationship
between the estimated value of a binary gamble and probability has been
reported by Tversky and Kahneman (1992), Gonzalez and Wu (1999), and others.
Birnbaum, et al.\thinspace (1992) showed how this model can be derived from
the assumption that the judge is minimizing an asymmetric loss function.%
\footnote{%
The denominator of Equation (\ref{Tpower}) is the sum of the absolute
weights of the higher and lower valued branches. In either the RAM or TAX\
model, if $\gamma $ $<1$, and a branch is split into two equal
probabilities, the sum of the absolute weights of the two pieces will be
greater than the weight of the original (coalesced) branch; because the sum
of the absolute weights increases as a result of the split, the relative
weights of unchanged branch are reduced by this event-splitting.}

The three configural weight parameters are $\alpha _{B}$, $\alpha _{S}$, and 
$\alpha _{0}.$ The first two represent the relative weights of the branch
with the higher valued consequence (relative to the branch with the
lower-valued consequence) in buyer's and seller's viewpoints, respectively,
and $\alpha _{0}$, reflects the relative weight of the higher valued branch
in the case of a lowest consequence of zero (relative to a positive lowest
consequence). The RAM model therefore has 6 parameters, $\nu ,$ $\beta ,$ $%
\gamma ,$ $\alpha _{B},$ $\alpha _{S},$ and $\alpha _{0}.$ Predictions of
the RAM model for buying and selling prices are shown in Figure 4. Notice
that the largest effect of viewpoint (vertical gap between buying and
selling prices) occurs when $p=1/2,$ and the curves converge as $%
p\rightarrow 1$. 
\[
\fbox{\textit{Insert Figure 4 about here.}} 
\]

\subsubsection{TAX Model}

In the Transfer of Attention Exchange (TAX) model weights are transferred
among (probability-consequence) branches in such a way that the sum of
absolute weights is preserved. The first idea to understand is that if there
were no configural effects, the absolute weight of a branch would be a
function of its probability, $p^{\gamma }$; and the relative weight would be
the ratio of this weight to total weight; i.e., $p^{\gamma }/[p^{\gamma
}+(1-p)^{\gamma }]$. When there are configural effects due to the judge's
viewpoint, weight (attention) is taken from one branch and given to another
branch. In the buyer's viewpoint, weight is taken from the higher-valued
consequence and transferred to the lower consequence, whereas in the
seller's viewpoint, weight is transferred from the lower branch to the
higher.

The relative weights of the branch to win $x$ with probability $p$ in the
buyer's and seller's viewpoints are as follows:

\begin{eqnarray}
W(p,B) &=&\frac{p^{\gamma }-\omega _{B}p^{\gamma }}{p^{\gamma
}+(1-p)^{\gamma }}  \label{TAXbuy} \\
W(p,S) &=&\frac{p^{\gamma }+\omega _{S}(1-p)^{\gamma }}{p^{\gamma
}+(1-p)^{\gamma }}.  \label{TAXsell}
\end{eqnarray}

The configural weight parameters, $\omega _{B}$ and $\omega _{S}$, represent
transfers of weight in the buyer's and seller's viewpoints, respectively. In
the buyer's viewpoint, the higher branch gives up a proportion of its weight
to the branch with the lower-valued consequence. The parameter, $\omega _{B}$%
\ , represents the proportion of weight of $x$ (which would otherwise have
been $p^{\gamma })$ that is taken from the higher ranked branch and given to
the lower-ranked branch in the buyer's point of view. The parameter, $\omega
_{S},$ represents the proportion of the lower-ranked branch's weight [which
would otherwise have been $(1-p)^{\gamma }$] that is transferred in the
seller's viewpoint from the lower to the higher-valued branch. It should be
noted that in previous papers, TAX models were written so that configural
weights could be positive or negative, and the signs indicated whether
weight was transferred from lower to higher consequences or vice versa. In
this presentation, the directions of weight transfers are assumed to be
known from previous research so the model is rewritten such that $\omega
_{B} $, and $\omega _{S}$ are both positive, to simplify the presentation.
The sign of $\omega _{B}$ must be reflected to compare it with values
reported in previous papers.

The TAX model thus has 6 free parameters, $\nu ,$ $\beta ,$ $\gamma ,\omega
_{B}$, $\omega _{S}$, and $\alpha _{0}.$ As in the RAM model, $\alpha _{0}$
reflects weighting when the lowest consequence is zero, and $\beta $ in this
model can be set to $1$ with little loss of predictive accuracy for $0\leq
x< $ $\$150$. Birnbaum (1999a) reviewed data for certainty equivalents,
choices, and judgments of value that can be well-approximated by the TAX
model with $U(x)$ proportional to $x$ and $\gamma =0.7$.

Predictions of the TAX model are shown in Figure 5, plotted as in Figure 4.
In the TAX model, as probability of the highest prize goes to $1$, the
asymptote for buyers is less than the value of the highest prize; and for
sellers, as $p\rightarrow 0$, the asymptote is greater than $0$. There are
discontinuities at $p=0$ and $p=1.$%
\[
\fbox{\textit{Insert Figure 5 about here.}} 
\]

Both RAM and TAX\ models violate complementary symmetry, as shown in Figure
3 (Section 1.3.4), but they each have a distinct pattern of violation as a
function of $p$.

In sum, in the configural weight models, the utilities of the consequences
are unaffected by buyer's or seller's viewpoint, but instead the configural
weights of the branches with lower or higher consequences are changed.
According to these models, the buyer places more weight on the less
attractive aspects of a gamble or good and the seller places relatively more
weight on the more attractive aspects. Different patterns of configural
weighting cause the rank orders of buyer's, neutral's, and seller's prices
to differ and yet the underlying utility function, $U(x),$ is assumed to be
the same. Choices between gambles are represented by the same model defined
on the same utility, with different configural weights.

\section{Properties Tested}

In addition to testing complementary symmetry, the present study examines
three other properties of binary gambles for which the predictions of the
theories compared here are understood. These are consequence monotonicity,
lower asymptotic independence, and upper asymptotic independence. Also, we
fit the several models to the binary data.

\subsection{Consequence Monotonicity}

Consequence monotonicity is the assumption that improving one consequence of
a gamble, holding everything else constant should improve the gamble.\ If $%
y^{+}$ is preferred to $y$, then $g^{+}=(x,p;y^{+})$ should be judged higher
than $g=(x,p;y)$. Birnbaum and Sutton (1992) found that this property is
systematically violated in judgments but rarely in choices; for example, $%
g^{+}=(\$96,.95;\$24)$ is judged systematically lower on average than $%
g=(\$96,.95;\$0)$, but most people choose $g^{+}$ over $g$. Birnbaum (1997)
reviewed evidence of violations and fits of configural models to these
violations. Luce (2002) explored\ this issue axiomatically by replacing an
earlier property--segregation--that involves gambles with a $0$ consequence
by a closely related property--distributivity--that does not. This gives
rise to a class of models that includes the configural weight models as a
special case. We have not yet determined what behavioral properties force
this general form to reduce to specific classes of configural weight models.

\subsection{Asymptotic Lower and Upper Independence}

Asymptotic lower independence is the assumption that no consequence is so
bad that it cannot be washed away as its probability approaches zero.
According to asymptotic lower independence, the limiting judgment of a
gamble approaches the value of the higher consequence as the probability of
that consequence goes to 1. The RAM model satisfies asymptotic lower
independence, but the TAX\ model of buying prices systematically violates it
(Birnbaum, 1997), because as $p\rightarrow 1$, $W(p,B)\rightarrow 1-\omega
_{B}$ $<1$. Asymptotic upper independence is the assumption that there is
nothing so good that the gamble cannot be made to approach the value of the
lowest consequence as the probability of the good consequence approaches
zero. RAM satisfies this property, but TAX violates it for seller's prices.

\section{Method}

\subsection{Instructions and Stimuli}

The judges were instructed to evaluate two- and three-outcome gambles from
the viewpoints of buyer and seller. Gambles were displayed as in the
following example 
\[
\frac{.25\text{ \ }.75\text{ }}{\$40\text{ }\$108} 
\]%
This display represents a binary gamble with a 0.25 probability to win \$40
and 0.75 probability 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 and 75 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 of which only two,
subdesigns 5 and 6, are directly relevant to the present paper.

Subdesign 5 was devised to test Asymptotic Lower Independence, Complementary
Symmetry, and Consequence Monotonicity. These 40 two-outcome gambles, of the
form $(\$100,p;y)$, were constructed using a 10 by 4 factorial design. The
probability $p$ to win $\$100$ was one factor and its 10 levels were $0.01$, 
$0.05$, $0.10$, $0.25$, $0.50$, $0.75$, $0.90$, $0.95$, $0.99$, and $0.999$.
The other factor was $y$, the lower outcome, with the 4 levels of $\$0$, $%
\$6 $, $\$24$, and $\$48$.

Subdesign 6 was devised to test Asymptotic Higher Independence and
Complementary Symmetry. These 40 two-outcome gambles, of the form $(x,p;0)$,
were constructed from a 10 by 4 factorial of probability to receive $x$, $p$%
, by $x$ design, in which the 10 levels of $p$ were the same as above; the 4
levels of the higher consequence, $x$, were $\$24$, $\$48$, $\$72$, and $%
\$100$.

A subset of data from Subdesigns 5 and 6 provide the data for binary gambles
that are used to fit the models.\ Judgments in cells that repeat in the two
designs were averaged, and only the 9 levels of $p$ from $0.01$ to $0.99$
were used, yielding a 9 by 7 factorial design of $p$ by $(x,y)$ in which the
7 levels of $(x,y)$ were $(\$100,\$0)$, $(\$72,\$0)$, $(\$48,\$0)$, $%
(\$24,\$0)$, $(\$100,\$6)$, $(\$100,\$24)$ and $(\$100,\$48)$.

The other subdesigns involved comparisons of two and three-consequence
gambles, reported in Birnbaum and Yeary, (2002)].

\subsection{Procedure}

The 166 gambles of all 8 subdesigns were printed in random order in
booklets, with the restriction that gambles from the same subdesign not be
presented on consecutive trials. Each person 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 for
the final paragraph of instructions in each booklet that reiterated the task
at hand. Each booklet included five practice trials for the task and 6
additional unlabeled warm-up trials, preceding the 166 experimental trials.
Half received the buyer's booklet first, followed by the seller's, and half
received the other task order. Judges worked at their own paces and
completed both tasks in two hours.

\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{Consequence Monotonicity}

Violations of monotonicity are shown in Figure 6 as the cross-over between
curves showing mean judgments of $(\$100,p;\$0)$, shown as open squares and
circles connected by dashed lines and $(\$100,p;\$6)$, shown as filled
circles and squares connected by solid lines. As reported previously (e.g.,
Birnbaum, 1992; 1997; Birnbaum \& Sutton, 1992; Birnbaum, et al., 1992),
increasing the lowest consequence from $y=0$ to a small positive consequence
increases judgments when the probability $(p)$ of the highest consequence is
small. However, when the probability of winning the highest consequence $(p)$
is large, increasing the lowest consequence from zero to a small positive
value actually makes the judgment lower.

Because the effect in Figure 6 is not as visually dramatic as in other
studies, we test the significance of the crossover representing violations
of consequence monotonicity. Contrasts were constructed for each judge by
computing the sum of six differences between judgments of $(\$100,p;\$6)$%
\textbf{\ }less those of $(\$100,p;\$0).$\textbf{\ }Summed over the three
smallest values of $p$ and over 2 viewpoints, 57 of 66 judges had positive
contrasts, indicating satisfaction of consequence monotonicity for small $p$%
. The mean contrast was $\$30.28$, or $\$5.05$ per difference, which is
significantly greater than $\$0$, $t(65)=7.00.$ However, at the highest
three levels of $p$, the mean contrast was negative, $-\$19$, or\textbf{\ }$%
\mathbf{-}\$3.17$ per difference, which is also significant, $t(65)=-3.33$.
Of the\textbf{\ }66 participants, 40 showed negative contrasts compared to
only 24 who had positive contrasts. Only 4 judges showed a crossover
opposite to that shown in the means against 38 who showed both contrasts
consistent with crossover in the means. Similarly, contrasts at $y=\$24$ $[$%
i.e., sum of differences in judgment between $(\$100,p;\$24)$ and $%
(\$100,p;\$0)]$ also showed the same crossover with a significant violation
of consequence monotonicity for the three highest levels of $p$ $%
[t(65)=-2.67]$ and significant satisfaction for the three lowest levels of $%
p $, $t(65)=16.11$. 
\[
\fbox{\textit{Insert Figure 6 about here.}} 
\]

\subsection{Point of View and Asymptotic Independence}

Figure 6 also shows the relationship between buyer's and seller's judgments.
Plotting seller's judgments against buyer's judgments produces a concave
downward curve, a shape that is predicted by the configural-weight RAM model
of Birnbaum, et al.\thinspace (1992, Figure 4). However, the RAM model
implies that the curves should approach the same limits as $p\rightarrow 0$
and $p\rightarrow 1$, as illustrated in Figure 4. Instead, there are large
vertical gaps at $p=0.01$ and $p=0.999$. These vertical gaps seem more
consistent with the TAX model (Figure 5), which violates asymptotic
independence, and with the Duplex Decomposition model of Fig.\thinspace 2.
Although, in fact, all of the joint-receipt models imply that $B=S=F$ as $%
p\rightarrow 1$, the figure establishes that one cannot really see that fact
in data because the slope near $1$ is very steep.

Lower asymptotic independence states that as the probability of the lowest
outcome goes to zero, the value of the lowest outcome becomes irrelevant
(Birnbaum, 1997). Figures 7 and 8 plot tests for lower asymptotic
independence in the seller's and buyer's viewpoints. In each case, judgments
are plotted for binary gambles of the form ($\$100,p;y$) as a function of $p$%
, with a separate curve for each level of $y$. The curves for the seller's
viewpoint appear to asymptote to a common value; note that $%
s(\$100,0.999;\$6)=\$81.08$ is slightly higher than $s(\$100,0.999;\$48)=%
\$80.80$; however, in the buyer's viewpoint, $b(\$100,0.999;\$6)=\$57.68$ is
less than $b(\$100,0.999;\$48)=\$68.24$, a difference of $\$10.56$. Thus,
buyer's prices appear to violate lower asymptotic independence. 
\[
\fbox{\textit{Insert Figures 7--8 about here.}} 
\]

Higher asymptotic independence implies that the higher outcome should become
irrelevant as its probability goes to zero. If the TAX model is correct,
then the seller's judgments should violate higher asymptotic independence
and the buyer's should satisfy it (Birnbaum, 1997). Figures 9 and 10 plot
judgments of $(x,p;y)$ as a function of $p$, with a separate curve for each
level of $x$. Note that in the buyer's viewpoint, the curves converge as $p$
approaches zero; $b(\$24,0.01;\$0)=\$2.32$ is not far below $%
b(\$100,0.01;\$0)=\$2.81$. In Figure 9, however, $s(\$24,0.01;\$0)=\$7.59$
is less than half of $s(\$100,0.01;\$0)=\$15.67$. 
\[
\fbox{\textit{Insert Figures 9--10 about here}} 
\]%
All of the models based on joint receipt give a decent fit to these gaps,
even though they satisfy asymptotic independence.

In summary, the duplex decomposition model and the TAX one, with configural
transfer of weights that go from higher to lower valued outcomes in buyer's
prices, and from lower to higher outcomes in seller's prices, both predict
the apparent pattern of violations of asymptotic independence in Figures 7,
8, 9, and 10.

\subsection{Complementary Symmetry}

Equation \ref{comp sym} (complementary symmetry) follows from the
assumptions of CPT, integration of prices and prizes, loss aversion, and the
power function. The sum of $b(x,p;y)+s(x,1-p;y)$ should be $x+y,$
independent of $p$. Figure 11 plots these sums for $g=(\$100,p;\$0)$ and $%
g=(\$72,p;\$0)$ as a function of $p$, as in Figure 5. Results in both cases
are similar, sharing four features in common. First, the sum of the
judgments falls below $x+y$ ($\$100$ or $\$72$) for all $p$. Second,
judgments decline as a function of $p$. Third, judgments show evidence of a 
\textsf{U}-shape, favoring RAM and Duplex Decomposition models. Fourth,
there is a ``kink'' at $p=0.5$, similar to one previously reported by
Birnbaum, et al.\thinspace (1992, p. 338-339). In sum, the data clearly
reject the complementary prediction of CPT Loss Aversion Model 3 and results
appear more consistent with the configural models and the best-fitting
joint-receipt model.

A graph similar to Figure 11 was made for each participant (each point
combines 4 judgments), and it was found that for all participants, the sum $%
b(\$100,p;\$0)+$ $s(\$100,1-p;\$0)$ was less than $\$100$; 47 participants
showed the decline, $b(\$100,0.01;\$0)+s(\$100,0.99;\$0)$ $%
<b(\$100,0.99;\$0)+s(\$100,0.01;\$0)$; in 44 participants the minimum of
this sum fell in the interval, $0.5<p<0.99$. Thus, each of these four trends
observed in the means are also characteristic of a majority of participants. 
\[
\fbox{\textit{Insert Figure 11 about here.}} 
\]

\subsection{Fit of Models to Binary Gambles}

The CPT Model with Loss Aversion (Model 3) was fit to the 126 mean judgments
of binary gambles in buyer's and seller's viewpoints of Subdesigns 5 and 6,
with the 9 levels of $p$ varying from 0.01 to 0.99. Recall that the
derivations of Expressions 11 and 12 did not assume anything about the forms
of $W^{+}$ and $W^{-}$ or their relationship. Because there are 9 levels of $%
p$ (or $1-p$) in the data matrix to be fit, there are 9 values of $T(p)$ to
estimate. The least-squares estimates of $T(0.01),$ $T(0.05),$ $T(0.10),$ $%
T(0.25),$ $T(0.50),$ $T(0.75),$ $T(0.9),$ $T(0.95),$ $T(0.99)$ are 0.1373,
0.1865, 0.2277, 0.3523, 0.5556, 1.0422, 1.5099, 1.8304, and 2.4614. The sum
of squared deviations for this model was 20,242.41, or a Root Mean Squared
Deviation ($RMS$) = 12.67. This model fails badly to fit the data, despite
using 9 free parameters. Forcing further restrictions on this model (e.g.,
adding assumptions concerning $W^{+}$ and $W^{-}$) would reduce the number
of parameters, but that would only make the fit worse.

The joint-receipt models were also fit to the same data. Some variations in
this class of models imply that buying and selling prices will be identical.
Of the six models that do not have this implication, the best-fitting case
is the Duplex Decomposition model with concave utility for gains and convex
utility for losses (Appendix A). These models were fit by parameter searches
for the six free parameters. The best-fit solution yielded parameter
estimates of $\kappa =0.2365,$ $\beta =0.5721,$ $\gamma (+)=0.6374,$ $\eta
(+)=0.4498,$ $\gamma (-)=0.3161,$ and $\eta (-)=0.3665.$ This model had a
sum of squared deviations of 8822.41, for a $RMS=8.37$.

For the same data array, the RAM model also has 6 parameters. The
least-squares fit to the same mean judgments yielded parameter values of $%
\nu =0.844$, $\gamma =0.505$, $\beta =1.250$, $\alpha _{B}=0.207$, and $%
\alpha _{S}=0.890$, and $\alpha _{0}=1.100$. These six parameters yield a
sum of squared deviations of $1,128.65$ or $RMS=2.99$. When $\beta $ is
fixed to $1$, the RAM(5) model fits nearly as well, with best-fit parameter
estimates of $0.823$, $0.478$, $1$ (fixed), $0.306$, $1.207$, and $1.113$,
respectively, for a sum of squares of $1,185.76$ or $RMS$ of $3.07$.

Fit to the same data, the least-squares estimates in the TAX(6) model are $%
\nu =0.788$, $\gamma =0.734$, $\beta =0.885$, $\omega _{B}=0.314$, $\omega
_{S}=0.186$, and $\alpha _{0}=1.094$. These six parameters yielded a sum of
squared errors of $1,051.1$, or $RMS=2.89$. Fixing $\beta $ = 1 gave
virtually the same parameter estimates and a sum of squares of $1,097.3$.

Clearly the two configural models fit best, the duplex decomposition one
next, and the CPT fits least well.

To check for individual differences, the models were also fit to every tenth
participant, and it was found that the best-fitting model was always the
same for individuals as it was for the averaged data. For each person, both
RAM and TAX fit better than the Duplex Decomposition model, with no clear
winner between these two configural models. The pattern of deviations was
also quite similar for each person as it was for the fit to the averaged
responses. In particular, the Duplex Decomposition C\TEXTsymbol{\backslash}V
model was the best fitting of the joint receipt models for each individual
fit, but for each person, it systematically failed to fit the relationship
between judgments of gambles in which the lowest valued consequence is zero
or positive.

To explore this source of deviations, the best-fitting, duplex decomposition
C\TEXTsymbol{\backslash}V model was fit to the data for these two cases ($%
y=0 $ and $y>0$) separately, using 12 parameters. It was found that the sum
of squared deviations for this version was $4,934.48$, for $RMS=6.26,$\
which is still worse than the fit of the RAM or TAX models with only 5
parameters, respectively. The parameters for the utility function changed
markedly between the two cases, which seems a complicated way to describe
the data. We have not tried to fit the RDU model using different weights for
buying and selling as is done in the configural weight models. This would,
of course, increase the number of parameters by 4, and it would also remove
the theoretical attraction of the joint receipt models, in which the
weighting functions of positive and negative consequences, which are already
estimated from the data, are supposed to explain the difference between
buying and selling prices according to the equations derived from the
theory. Table 1 summarizes the comparison of fit of the models. 
\[
\fbox{\textit{Insert Table 1 about here.}} 
\]

The TAX\ model was also fit to the cases of $y=0$ and $y>0$, separately,
with $\alpha _{0}$ set to 1, requiring 10 parameters. The increase from 5 to
10 parameters reduced the total sum of squared deviations from $1,185.8$ to $%
875.0$. The parameter estimates did not differ much, except in the case of $%
\omega _{B}$ and $\omega _{S}$, which were $0.231$ and $0.234$ for the case
of $y=0$, and $0.346$ and $0.191$ for the case of $y>0$, respectively.

The best-fit predictions for the TAX(5)\ model with $\beta $\ fixed to $1$
are shown in Figures 12 and 13. As shown in the Figures, the model correctly
predicts the violations of consequence monotonicity when lowest consequences
change from $y=0$ to $y>0$ as $p$ approaches $1$. The model also does a
fairly good job in describing the relationship between buying and selling
prices 
\[
\fbox{\textit{Insert Figures 12 and 13 about here.}} 
\]

To examine if TAX or RAM would be improved by the assumption that the
utility function changes between buyer and seller's viewpoint, $\beta $\ was
estimated separately in each viewpoint. These 7 parameter models did not fit
much better than the simpler 5-parameter models in which $\beta $ was fixed
to 1. For example, the sum-of-squares is 1018 for TAX(7), compared to 1097
for TAX(5). Examination of predictions and data, graphed as in Figures 12
and 13 did not reveal a systematic improvement. In summary, the 5-parameter
RAM and TAX\ models appeared to give the best fit to the data with the
fewest parameters.

\section{Discussion}

These data yield three clear results and a fourth, more tentative one.
First, comparisons of\textbf{\ }the fits of the models to the judgments of
binary gambles show that the configural weight RAM and TAX models fit better
than models of loss aversion combined with cumulative prospect theory.\
There may be other ways to try to save the concept of ``loss aversion'' as a
theoretical explanation for the discrepancy between buying and selling
prices, but we do not see how that would be done.

We do not question whether people are averse to losses; it is fairly obvious
that people would much rather win than lose money. However, the question is
whether we can attribute the difference between buying and selling prices to
a multiplier of the utility function.\ The most promising loss aversion
model (Model 3), implies complementary symmetry, a property that was
strongly refuted by the data. That loss aversion CPT\ model fit worse than
the other models to which it was compared here, despite CPT having the most
free parameters of the models compared. The problem for this model is that
the data do not show the proper symmetries between buying and selling prices
implied by the combination of CPT\ and the loss aversion representation.

Second, the joint receipt models, which do not imply complementary symmetry,
fit the data better than CPT. The best fitting of these models also appears
to describe the gaps in the data near probabilities of 0 and 1 that are
apparent violations of asymptotic independence.

Third, the configural weight RAM and TAX models, with the same number or
fewer parameters, fit the data better than the class of models that Luce
(2000) derived by decomposing gambles using joint receipts. The best-fitting
additive joint receipt model, duplex decomposition C\TEXTsymbol{\backslash}%
V, violates complementary symmetry in the proper way, but it does not do an
adequate job of accounting for judgments of gambles in which the lowest
consequence can be either zero or positive. Even when the best-fitting of
the joint receipt models was fit with separate parameters for these two
cases, using 12 parameters, it did not do as well as either RAM or TAX with
5 parameters.

In comparing these models the reader should keep in mind that other data
have established systematic discrepancies between judgments and choices.
Both CPT and RDU theories were formulated for choice, not judgment, but the
definitions given of buying and selling prices were an attempt to extend the
scope to at least these judgments. We are unaware of any parametric study
that has as yet reported choice-based estimates of buying and selling prices
even in the binary case, and so the conclusion about the inadequacy of these
theories must be tempered by the fact that these data represent judgments.
This distinction may well be important since in some situations one states
either a buying or selling price, but in others one simply makes choices
between a ``gamble'' (or a good) and a price, either buying or selling.

On the other hand, results of recent experiments using choices between
gambles have shown serious violations of important properties implied by the
class of RDU/RSDU/CPT models (Luce, 2000; Quiggin, 1993; Wakker \& Tversky,
1993): people systematically violate stochastic dominance, upper and lower
cumulative independence, and other properties implied by coalescing
(Birnbaum, 1999; Birnbaum \& Navarrete, 1998; Birnbaum, Patton, \& Lott,
1999; Wu, 1994). Event-splitting effects (Birnbaum, 1999; Starmer \& Sugden,
1993; Humphrey, 1995; Luce, 1998), which in choice are violations of
coalescing and transitivity, also violate the class of RDU/RSDU/CPT models.

For example, Birnbaum and Navarrete (1998) reported that about 70\% of 100
undergraduates tested choose $g^{-\text{ }}=(\$96,0.85;\$90,0.05;\$12)$ over 
$g^{+}=(\$96,0.90;\$14,0.05;\$12)$, even though $g^{+}$ dominates $g^{-}$.\
Birnbaum and Yeary (2002) reported that stochastic dominance and cumulative
independence are violated in judgment as well as choice. People set
significantly higher buying and selling prices on $g^{-}$\ than they do on $%
g^{+}$.

One might ask, why not allow different parameters within CPT or RSDU\ for
buying and selling prices and use the utility representations, as is done in
the configural weight models? The answer is that the violations of
stochastic dominance and upper and lower cumulative independence in
judgments of buying and selling prices indicate that within either
viewpoint, the numerical representation of RDU/RSDU/CPT would not describe
the data. Therefore, even allowing separate parameters for buying and
selling prices, and restricting to gambles with $y>0,$ these representations
of the utility of gambles would not describe judgments or choices between
gambles with more than two branches.

A simple interpretation is that the source of violations of the same
properties in choice and judgment are explained by the same mechanisms of
evaluation for the gambles.

Recall that Cho et al.\thinspace (2002) found that from one third to one
half the participants did not satisfy either general segregation or duplex
decomposition. This means that the joint receipt models discussed in this
paper may very well be inadequate. No one has yet suggested within Luce's
framework another way to deal with mixed gambles.

From the fit of the models derived from joint receipts, it appears that
violations of consequence monotonicity reported in previous judgment studies
as well as in this study are not well-described by the properties of general
segregation or duplex decomposition in additive joint receipt models.
Apparently, people treat the case with $\$0$ as a consequence differently
from cases with strictly positive consequences. This finding causes trouble
for the previously published models derived in Luce's approach\textbf{\ }%
because they are based on decomposing a general gamble to unitary ones
(i.e., with one consequence $0$), as in general segregation and duplex
decomposition. Some progress has been made on this issue in Luce (2002).

The RAM and TAX models both posit that when the worst outcome $y=\$0$, the
weight attached to the $x$ consequence is greater than when the worst
outcome $y$ is positive. Although the details of the configural weighting
differ between RAM and TAX\ models, both of those configural weight models
share this property and therefore describe violations of consequence
monotonicity.

The fourth finding is necessarily tentative, as it is a null result. The
data appear consistent with the assumption that the utility function is the
same for modest positive consequences to both buyers and sellers. In fact,
for consequences such as used in this study, it is possible to fit either
RAM or TAX to the data with the assumption that utility is proportional to
cash value (i.e., $\beta =1)$. That assumption has also worked well for
choices between gambles with similar consequences (Birnbaum, 1999). The
differences between choosing, buying and selling prices are explained in
terms of different patterns of configural weighting. This conclusion is in
sharp contrast to the work stemming from expected utility theory in which
the probabilities are veridical and all of the work is done by the utility
function. The invariance of utility between viewpoints is also assumed in
the Luce and CPT models; however, the best-fitting of these models uses a
utility function that is far from linear on the domain of positive
consequences used here.

The pattern of weighting in both the configural weight and Luce models does
not exhibit complementary symmetry, which is implied by the best of the CPT
models.\ In contrast, when configural models are fit to the data, the
buyer's viewpoint produces a pattern of giving the most weight to the lowest
consequence. Sellers, however, put only slightly more weight on the highest
consequence than the lowest consequence.

The RAM and TAX\ models make different predictions for violations of
complementary symmetry and for asymptotic independence. The empirical
results do not consistently favor one model over the other. Based on
examination of the graphs for these predictions and a small difference in
fit, we conclude that the TAX model fits the data slightly better than the
RAM model overall. The RAM model seems a bit better for the violation of
complementary symmetry, and the TAX\ model appears to describe the violation
of asymptotic independence better.\ 

It would be desirable to have a more direct statement of asymptotic
independence with a more precise way to distinguish its violation from
mis-specification of the probability weighting function.\ RAM and TAX models
also differ with respect to distribution independence, a property not tested
in this study but which has favored the TAX model (Birnbaum \& Chavez, 1997).

In principle, the theory of joint receipt could make predictions for choice,
buying, and selling prices with the same parameters. Even though it used an
equal or greater number of parameters to fit the present data as RAM or TAX,
it could have provided a more elegant representation of a wider array of
behavioral phenomena than the configural weight models, had it fit nearly as
well. Despite the seeming complexity of the equations in Appendix A, the
Ockham's razor argument would have appeared with the introduction of
choices, since no new parameters are required to account for choices, buying
prices, and selling prices. The CPT loss aversion model has the same
potential advantage. In addition, these approaches have the advantage that
gambles over losses should also be predicted from the same weighting
functions and utilities postulated to account for buying and selling prices.

The RAM and TAX models, in contrast, require new configural weights for
choice, but they would use the same utility function, and the same equation
for evaluating gambles presented for either choice or judgment. These models
also provide a way of interpolating to the neutral's viewpoint or
extrapolating to more extreme viewpoints than those of buyer and seller. But
the models impose no restriction on the representation of gambles over
losses.

Both the configural weight\ models and the joint-receipt ones have in common
that the utility function for consequences is assumed to be invariant with
respect to viewpoint, and that it should be the same in choice as in
judgment. It would be interesting to explore what additional behavioral
properties characterize a taxonomy of this class of models, and to determine
what behavioral properties would be sufficient to guarantee their
representations.

\pagebreak

\begin{center}
\textbf{Appendix A: Additive }$V$ \textbf{Models}
\end{center}

The notation is described in Section 1.3.4.

\noindent \textbf{Duplex Decomposition:}

\noindent \textbf{C\TEXTsymbol{\backslash}V }%
\begin{eqnarray*}
0 &=&B^{2}W^{-}(1-p)-B\left\{
W^{+}(p)+W^{-}(1-p)-G[W^{+}(p)-W^{-}(1-p)]\right\} \\
&&+FW^{+}(p)+GW^{-}(1-p)-FGW^{+}(p), \\
0 &=&S^{2}W^{+}(1-p)-S\left\{
W^{+}(1-p)+W^{-}(p)+G[W^{+}(1-p)-W^{-}(p)]\right\} \\
&&+FW^{-}(p)+GW^{+}(1-p)-FGW^{-}(p).
\end{eqnarray*}%
\textbf{V\TEXTsymbol{\backslash}C}%
\begin{eqnarray*}
0 &=&-B^{2}W^{-}(1-p)-B\left\{
W^{+}(p)+W^{-}(1-p)+G[W^{+}(p)-W^{-}(1-p)]\right\} \\
&&+FW^{+}(p)+GW^{-}(1-p)+FGW^{+}(p), \\
0 &=&-S^{2}W^{+}(1-p)-S\left\{
W^{+}(1-p)+W^{-}(p)-G[W^{+}(1-p)-W^{-}(p)]\right\} \\
&&+FW^{-}(p)+GW^{+}(1-p)+FGW^{-}(p).
\end{eqnarray*}%
\textbf{C\TEXTsymbol{\backslash}C}%
\begin{eqnarray*}
0 &=&B^{2}\left[ W^{+}(p)+W^{-}(1-p)-W^{+}(p)W^{-}(1-p)\right] \\
&&-B\left\{ (1+F)W^{+}(p)+(1+G)W^{-}(1-p)-(F+G)W^{+}(p)W^{-}(1-p)\right\} \\
&&+FW^{+}(p)+GW^{-}(1-p)-FGW^{+}(p)W^{-}(1-p). \\
0 &=&S^{2}W^{+}(1-p)W^{-}(p)-S\left[ (1-F)W^{+}(1-p)+(1-G)W^{-}(p)\right. \\
&&\qquad \left. +(F+G)W^{+}(1-p)W^{-}(p)\right] \\
&&+(1-F)GW^{+}(1-p)+(1-G)FW^{-}(p)+FGW^{+}(1-p)W^{-}(p)..
\end{eqnarray*}%
\textbf{V\TEXTsymbol{\backslash}V}%
\begin{eqnarray*}
0 &=&-B^{2}\left[ W^{+}(p)+W^{-}(1-p)-W^{+}(p)W^{-}(1-p)\right] \\
&&-B\left\{ (1-F)W^{+}(p)+(1-G)W^{-}(1-p)+(F+G)W^{+}(p)W^{-}(1-p)\right\} \\
&&+FW^{+}(p)+GW^{-}(1-p)+FGW^{+}(p)W^{-}(1-p). \\
0 &=&S^{2}W^{+}(1-p)W^{-}(p)+S\left[ (1+F)W^{+}(1-p)+(1+G)W^{-}(p)\right. \\
&&\left. -(F+G)W^{+}(1-p)W^{-}(p)\right] \\
&&-(1+F)GW^{+}(1-p)-(1+G)FW^{-}(p)+FGW^{+}(1-p)W^{-}(p).
\end{eqnarray*}

\noindent \textbf{General Segregation:\medskip }

\noindent \textbf{Concave Gains}%
\begin{eqnarray*}
B &=&FW^{+}(p)+G[1-W^{+}(p)], \\
S &=&\frac{G(1-F)W^{+}(1-p)+F(1-G)[1-W^{+}(1-p)]}{%
(1-F)W^{+}(1-p)+(1-G)[1-W^{+}(1-p)]}.
\end{eqnarray*}%
\medskip

\noindent \textbf{Convex Gains} 
\begin{eqnarray*}
B &=&FW^{+}(p)+G[1-W^{+}(p)], \\
S &=&\frac{G(1+F)W^{+}(1-p)+F(1+G)[1-W^{+}(1-p)]}{%
(1+F)W^{+}(1-p)+(1+G)[1-W^{+}(1-p)]}.
\end{eqnarray*}

\begin{center}
\textbf{Appendix B: Uniqueness of S Solutions in Duplex Decomposition C%
\TEXTsymbol{\backslash}C and V\TEXTsymbol{\backslash}V}\medskip
\end{center}

Define $d=sgn(\delta )1.$ In the C\TEXTsymbol{\backslash}C case we have $d=1$
and in the V\TEXTsymbol{\backslash}V, $d=-1,$ both cases $d^{2}=1$. Let $%
W^{+}=W^{+}(1-p)$ and $W^{-}=W^{-}(p).$ Then $S$ equations for these cases
are: 
\begin{eqnarray}
0 &=&S^{2}W^{+}W^{-}-dS\left\{ \left[ 1-dF\right] W^{+}+\left[ 1-dG\right]
W^{-}+d(F+G)W^{+}W^{-}\right\}  \nonumber \\
&&+d\left[ 1-dF\right] GW^{+}+d\left[ 1-dG\right] FW^{-}+FGW^{+}W^{-}.
\label{SAppB}
\end{eqnarray}%
For $0<p,1-p<1,$ divide by $W^{+}W^{-}$ and define 
\[
F^{\ast }=d\frac{1-dF}{W^{-}},\quad G^{\ast }=d\frac{1-dG}{W^{+}}, 
\]%
then the $S$ expression \ref{SAppB} becomes 
\[
0=S^{2}-S(F^{\ast }+G^{\ast }+F+G)+F^{\ast }G+FG^{\ast }+FG. 
\]%
So in the standard quadratic notation 
\[
A=1,\quad B=-(F^{\ast }+G^{\ast }+F+G),\quad C=F^{\ast }G+FG^{\ast }+FG. 
\]%
Thus, 
\begin{eqnarray*}
B^{2}-4AC &=&(F^{\ast }+G^{\ast })^{2}+(F+G)^{2}+2(F^{\ast }+G^{\ast })(F+G)
\\
&&-4(F^{\ast }G+FG^{\ast }+FG) \\
&=&(F^{\ast }+G^{\ast })^{2}+F^{2}+G^{2}+2FG-2F^{\ast }G-2FG^{\ast
}+2FF^{\ast } \\
&&+2GG^{\ast }-4FG \\
&=&(F^{\ast }+G^{\ast })^{2}+(F-G)^{2}+2(F^{\ast }-G^{\ast })(F-G).
\end{eqnarray*}

Now, consider $d=1$ and so $F^{\ast }>0,G^{\ast }>0$ and either $F^{\ast
}+G^{\ast }>F^{\ast }-G^{\ast }>0$ or $F^{\ast }+G^{\ast }>G^{\ast }-F^{\ast
}>0$ whence $(F^{\ast }+G^{\ast })^{2}>(F^{\ast }-G^{\ast })^{2}=(G^{\ast
}-F^{\ast })^{2}.$ So, 
\begin{eqnarray*}
B^{2}-4AC &>&(F^{\ast }-G^{\ast })^{2}+(F-G)^{2}+2(F^{\ast }-G^{\ast })(F-G)
\\
&=&(F^{\ast }-G^{\ast }+F-G)^{2}>0.
\end{eqnarray*}

Next, consider $d=-1$ and so $F^{\ast }<0,G^{\ast }<0.$ Therefore $F^{\ast
}-G^{\ast }>F^{\ast }+G^{\ast }.$ Since $F-G>0,$%
\begin{eqnarray*}
B^{2}-4AC &>&(F^{\ast }+G^{\ast })^{2}+(F-G)^{2}+2(F^{\ast }+G^{\ast })(F-G)
\\
&=&(F^{\ast }+G^{\ast }+F-G)^{2}>0.
\end{eqnarray*}
Thus, in both cases the solutions are real.

Let $f(S)$ denote the right side of Equation \ref{SAppB}. Then, since we are
only interested in solutions $G\leq S\leq F,$ we calculate $f(G)$ and $f(F):$%
\begin{eqnarray*}
f(G) &=&G^{2}W^{+}W^{-}-d(1-dF)GW^{+}-d(1-dG)GW^{-} \\
&&-(F+G)GW^{+}W^{-}+d(1-dF)GW^{+} \\
&&+d(1-dG)FW^{-}+FGW^{+}W^{-} \\
&=&W^{+}W^{-}G[G-(F+G)+F]+d(1-dF)(G-G)W^{+} \\
&&+d(1-dG)(F-G)W^{-} \\
&=&d(1-dG)(F-G)W^{-}.
\end{eqnarray*}

\begin{eqnarray*}
f(F) &=&F^{2}W^{+}W^{-}-d(1-dF)FW^{+}-d(1-dG)FW^{-} \\
&&-(F+G)FW^{+}W^{-}+d(1-dF)GW^{+}+d(1-dG)FW^{-}+FGW^{+}W^{-} \\
&=&W^{+}W^{-}F[F-(F+G)+G]-d(1-dF)(F-G)W^{+} \\
&&-d(1-dG)(G-G)W^{-} \\
&=&-d(1-dF)(F-G)W^{+}.
\end{eqnarray*}
Consider first $d=1.$ Because $1-G>0,$ $1-F>0,$ and $F-G>0,$%
\begin{eqnarray*}
f(G) &=&(1-G)(F-G)W^{-}>0, \\
f(F) &=&-(1-F)(F-G)W^{+}<0.
\end{eqnarray*}
So there is a unique solution $S$ between $G$ and $F.$

Next, consider $d=-1.$%
\begin{eqnarray*}
f(G) &=&-(1+G)(F-G)W^{-}<0, \\
f(F) &=&(1+F)(F-G)W^{+}>0
\end{eqnarray*}%
So, again, there is a unique solution $S$ between $G$ and $F.$

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

\item Birnbaum, M.\thinspace H. (1982). Controversies in psychological
measurement. In B.\thinspace Wegener (Ed.), \textit{Social attitudes and
psychophysical measurement} (pp.\thinspace 401-485). Hillsdale, NJ.: Erlbaum.

\item Birnbaum, M.\thinspace H. (1992). Violations of monotonicity and
contextual effects in choice-based certainty equivalents. \textit{%
Psychological Science}, 3, 310-314.

\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.\thinspace H., Coffey, G., Mellers, B.\thinspace A., \&
Weiss, R. (1992). Utility measurement: Configural-weight theory and the
judge's point of view. \textit{Journal of Experimental Psychology: Human
Perception and Performance}, 18, 331-346.

\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., \& Thompson, L.\thinspace A. (1996).
Violations of monotonicity in choices between gambles and certain cash. 
\textit{American Journal of Psychology}, 109, 501-523.

\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. (2002) Violations of stochastic
dominance, coalescing, lower and upper cumulative independence, and branch
independence in judgments of buying and selling prices of gambles.

\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 Bostic, R., Herrnstein, R.\thinspace J., \& Luce, R.\thinspace D.
(1990). The effect on the preference-reversal phenomenon of using choice
indifferences. \textit{Journal of Economic Behavior and Organization, }13,
198-212.

\item Cho, Y.-H., Luce, R.\thinspace D., \& Truong, L. (2002). Duplex
decomposition and general segregation of lotteries of a gain and a loss:\ An
empirical evaluation. \textit{Organizational Behavior and Human Decision
Processes.} In press.\textit{\ }

\item Coombs, C.\thinspace H., Bezembinder, T.\thinspace G., \& Goode,
F.\thinspace M. (1967). Testing expectation theories of decision making
without measuring utility or subjective probability. \textit{Journal of
Mathematical Psychology}, 4, 72-103.

\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 Goldstein, W., \& Einhorn, H.\thinspace J. (1987). A theory of
preference reversals. \textit{Psychological Review}, 94, 236-242.

\item Gonzalez, R. \& Wu, G. (1999). On the shape of probability weighting
function. \textit{Cognitive}

\textit{Psychology}, \textit{38}, 129-166.

\item Hazen, G.\thinspace B., \& Lee, J.-S. (1991). Ambiguity aversion in
the small and in the large for weighted linear utility. \textit{Journal of
Risk and Uncertainty, 4, }177-212.

\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 Irwin, J., Slovic, P., Lichtenstein, S., \& McClelland, G. (1993).
Preference reversals and the measurement of environmental values. \textit{%
Journal of Risk and Uncertainty}, 6, 5-18.

\item Johannesson, M., Liljas, B., \& O'Conor, R.\thinspace M. (1997).
Hypothetical versus real willingness to pay: Some experimental results. 
\textit{Applied Economics Letters}, 4, 149-151.

\item Kahneman, D., Knetsch, J.\thinspace L., \& Thaler, R.\thinspace H.
(1991). Experimental tests of the endowment effect and the coarse theorem.
In R.\thinspace H. Thaler (Ed.), \textit{Quasi rational economics}
(pp.\thinspace 167-188.). New York: Russel Sage Foundation.

\item Kahneman, D., Knetsch, J.\thinspace L., \& Thaler, R.\thinspace H.
(1992). The endowment effect, loss aversion, and status quo bias. In
R.\thinspace H. Thaler (Ed.), \textit{The winner's curse: Paradoxes and
anomalies of economic life} (pp.\thinspace 63-78). New York: The Free Press.

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

\item Knetsch, J.\thinspace L., \& Sinden, J. A. (1984). Willingness to pay
and compensation demanded: Experimental evidence of an unexpected disparity
in measures of value. \textit{The Quarterly Journal of Economics}, 99,
507-521.

\item Lichtenstein, S., \& Slovic, P. (1971). Reversals of preference
between bids and choices in gambling decisions. \textit{Journal of
Experimental Psychology}, 89, 46-55.

\item Lindman, H.\thinspace R. (1971). Inconsistent preferences among
gambles. \textit{Journal of Experimental Psychology}, 89, 390-397.

\item Luce, R.\thinspace D. (1991). Rank- and sign-dependent linear utility
models for binary gambles. \textit{Journal of Economic Theory, 53, }75-100.

\item Luce, R.\thinspace D. (1997). Associative joint receipts. \textit{%
Mathematical Social Sciences, 34,} 51-74.

\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. (2001). Reduction invariance and Prelec's
weighting functions. \textit{Journal of Mathematical Psychology,} 45,167-179%
\textit{.}

\item Luce, R.\thinspace D. (2002). Increasing increment generalizations of
rank-dependent theories. Submitted.

\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 Markowitz, H. (1952). The utility of wealth. \textit{Journal of
Political Economy}, 60, 151-158.

\item Mellers, B.\thinspace A., Berretty, P.\thinspace M., \& Birnbaum,
M.\thinspace H. (1995). Dominance violations in judged prices of two- and
three-outcome gambles. \textit{Journal of Behavioral Decision Making}, 8,
201-216.

\item Mellers, B.\thinspace A., Chang, S., Birnbaum, M.\thinspace H., \& Ord%
\'{o}\~{n}ez, L. (1992). Preferences, prices, and ratings in risky decision
making. \textit{Journal of Experimental Psychology: Human Perception and
Performance}, 18, 347-361.

\item Mellers, B.\thinspace A., Ord\'{o}\~{n}ez, L., \& Birnbaum,
M.\thinspace H. (1992). A change-of-process theory for contextual effects
and preference reversals in risky decision making. \textit{Organizational
Behavior and Human Decision Processes}, 52, 331-369.

\item Mellers, B.\thinspace A., Weiss, R., \& Birnbaum, M.\thinspace H.
(1992). Violations of dominance in pricing judgments. \textit{Journal of
Risk and Uncertainty}, 5, 73-90.

\item Prelec, D. (1998). The probability weighting function. \textit{%
Econometrica, 66, }497-527.

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

\item Slovic, P., \&\ Lichtenstein, S. (1968). Importance of variance
preferences in gambling decisions. \textit{Journal of Experimental
Psychology, 78, }646-654.

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

\item Thaler, R., Kahneman, D., \& Knetsch, J.\thinspace L. (1994). The
endowment effect, loss aversion, and status quo bias. In R. Thaler (Ed.), 
\textit{The winner's curse: Paradoxes and anomalies of economic life}
(pp.\thinspace 63-78). Princeton, NJ: Princeton University Press.

\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., Sattath, S., \& Slovic, P. (1988). Contingent weighting
in judgment and choice. \textit{Psychological Review}, 95, 371-384.

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

\item Weber, E.\thinspace U. (1994). From subjective probabilities to
decision weights: The effects of asymmetric loss functions on the evaluation
of uncertain outcomes and events. \textit{Psychological Bulletin}, 114,
228-242.

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

\pagebreak

Table 1. \ Comparison of Fit of Models to Binary Gambles

\bigskip

\begin{tabular}{ll}
Model (No. of parameters) & Sum of Squared Deviations \\ 
\multicolumn{2}{l}{\rule{3.7in}{0.02in}} \\ 
CPT Loss Aversion Model 2 (9) & \qquad 20,242 \\ 
Duplex Decomposition C\TEXTsymbol{\backslash}V(6) & \qquad\ \ 8,822 \\ 
Duplex Decomposition C\TEXTsymbol{\backslash}V(12) & \qquad\ \ 4,935 \\ 
RAM (6) & \qquad\ \ 1,129 \\ 
RAM (5) & \qquad\ \ 1,186 \\ 
TAX(6) & \qquad\ \ 1,051 \\ 
TAX(5) & \qquad\ \ 1,097 \\ 
TAX(10) & \qquad\ \ \ \ 875 \\ 
\multicolumn{2}{l}{\rule{3.7in}{0.02in}}%
\end{tabular}

\end{document}
