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\pagestyle{myheadings} \markright{\protect\small \href{http://journal.sjdm.org}{Judgment and
  Decision Making}, Vol.~7, No.~4, July 2012 \hfill Individual true and error theory \qquad}

\title{Separating response variability from structural inconsistency to test models of risky decision making}

\author{Michael H. Birnbaum\thanks{Department of Psychology, California State University, Fullerton, CSUF
H-830M, Box 6846, Fullerton, CA 92834--6846, USA.  Email:
mbirnbaum@fullerton.edu.}
\and 
Jeffrey P. Bahra\thanks{Department of Psychology, California State University, Fullerton}
}

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\href{http://journal.sjdm.org}{Judgment and Decision Making}, vol. 7, no. 4, July 2012, pp.
402-426
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{\small \href{http://journal.sjdm.org}{Judgment and Decision Making}, Vol.~7, No.~4, July 2012, pp.\ 402--426}
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Individual true and error theory assumes that responses by the same
person to the same choice problem within a block of trials are based
on the same true preferences but may show preference reversals due to
random error.  Between blocks, a person{}'s true preferences may
differ or stay the same. This theory is illustrated with studies
testing two critical properties that distinguish models of risky
decision making: (1) restricted branch independence, which is implied
by original prospect theory and violated in a specific way by both
cumulative prospect theory and the priority heuristic; and (2)
stochastic dominance, which is implied by cumulative prospect theory.
Corrected for random error, most individuals systematically violated
stochastic dominance, ruling out cumulative prospect
theory. Furthermore, most people violated restricted branch
independence in the opposite way predicted by that theory and the
priority heuristic.  Both violations are consistent with the transfer
of attention exchange model.  No one was found whose data were
compatible with cumulative prospect theory, except for those that were
also compatible with expected utility, and no one satisfied the
priority heuristic.

\smallskip
\noindent
Keywords:
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\section{Introduction}

When offered a choice, different individuals express different
preferences.  For example, given a choice between \$40 for sure and a
fifty-fifty gamble to win either \$0 or \$100, some people choose the
sure cash and others prefer the gamble.  When the same person is
presented with the same choice problem on two occasions, that
individual may give two different responses.  Given these two
phenomena (between and within-person variability), it is possible that
part or all of what are called individual differences may actually be
attributable to variability or ``instability'' of choice behavior.
Furthermore, when we test structural properties of theories, apparent
violations of a theory may be due to instability of responses rather
than to true flaws in a theory.

Therefore, to study individual differences, we need to separate true
differences between people from differences that might be due to
variation within a person.  Furthermore, to study behavioral
properties of theories, we need to determine whether violations are
``real'' or might instead be attributed to instability in the response.

This report investigates two critical properties that distinguish
models of risky decision making: first order stochastic dominance (SD)
and restricted branch independence (RBI).  We apply individual true
and error theory (\textit{i}TET), which permits separation of random
variability from true violations of models of risky decision making.
These tests allow us to test original prospect theory (OPT) (Kahneman
\& Tversky, 1979), cumulative prospect theory (CPT) (Tversky \&
Kahneman, 1992), the priority heuristic (PH) (Brandst\"atter,
Gigerenzer, \& Hertwig, 2006), and transfer of attention exchange
(TAX) model (Birnbaum, 2008b).

\section{Critical tests that distinguish models of risky decision making}

Critical properties are theorems that are implied by certain models
and which can be violated by rival models.  By testing critical
properties that hold true for all functions and parameters within a
theory, it is possible to refute models without the need to specify
functions, to estimate parameters, or to calculate an index of fit.

\subsection{First order stochastic dominance}

If the probability of receiving a prize greater than or equal to
$x$ in Gamble $G$ is always greater than or equal to the
corresponding probability in Gamble $F$, for all $x$,
and if it is strictly greater for at least one value of \textit{x,} we
say that $G$ dominates $F$ by \textit{first order
  stochastic dominance} (SD).  If choices satisfy SD, a person would
not prefer $F$ to $G$, except by random error.  That is,\\

\noindent $P(x>t | G) \geq P(x > t | F) \forall t \Rightarrow G \succ F$\\

Rank-dependent utility models, including CPT (Tversky \& Kahneman,
1992) imply that choices must satisfy SD, apart from error.  This
property is implied by CPT for any strictly monotonic utility function
of money and any strictly monotonic probability weighting function.

Birnbaum{}'s configural weight models violate SD in specially designed
choices (Birnbaum, 1997; 2005; 2008b; Birnbaum \& Navarette, 1998;
Luce, 2000).  For example, these models predicted that people should
violate SD in the following
choice:\\

\noindent $F$ = (\$96, 0.85; \$90, 0.05; \$12, 0.10) $\succ$ \\
\rule{1cm}{0mm} $G$ = (\$96, 0.90; \$14, 0.05; \$12, 0.05) \\

Following publication of the predictions (Birnbaum, 1997), Birnbaum
and Navarrete (1998) tested this choice and found that indeed, more
than half of undergraduates tested violated dominance as predicted by
the models. Birnbaum and Navarrete (1998) also confirmed the
predictions by Birnbaum (1997) of violation of upper and lower
cumulative independence.

\subsection{Restricted branch independence}

A \textit{branch} of a gamble is a probability-consequence pair that
is distinct in the presentation of a gamble.  For example, \textit{A}
= (\$100, .01; \$100, .01; \$2, .98) is a 3- branch gamble in which
there are two branches having probability .01 to win \$100 and a
branch with probability of .98 to win \$2.  $A^\prime$ = (\$100, .02;
\$2, .98) is a two-branch gamble.  The assumption of
\textit{coalescing} holds that if two branches lead to the same
consequence, they can be combined by adding their probabilities.
Coalescing implies that \textit{A} $\sim A^\prime$, where $\sim$
indicates indifference.

\textit{Restricted branch independence} (RBI) is the assumption that,
if two gambles have a common consequence on a common branch, the
consequence can be changed to another value in both gambles without
changing the preference between the two gambles. The term
``restricted'' refers to the requirements that the number of branches
is the same and the probability distribution is the same in all cases.
This property is a special case of Savage's ``sure thing'' axiom
(Savage, 1954).

For three-branch gambles, RBI can be written as follows: \\

\noindent$S$ = (\textit{x, p; y, q; z, }1\textit{--p--q}) $\succ R$ = (\textit{$x^\prime$, p; $y^\prime$, q; z, }1\textit{--p--q})

\noindent${\Leftrightarrow}$

\noindent$S^\prime$ = (\textit{x, p; y, q; $z^\prime$, }1\textit{--p--q}) $\succ R^\prime$ = (\textit{$x^\prime$, p; $y^\prime$, q; $z^\prime$,} 1\textit{--p--q})

Note that both $S$ and $R$ share a common branch,
(\textit{z}, 1\textit{--p--q}).  The consequence on the common branch
has been changed from \textit{z} (in the choice problem, $S$
versus $R$) to $z^\prime$ ($S^\prime$ versus $R^\prime$),
but the two choice problems are otherwise the same.  If a person
satisfies RBI, then they should choose either $S$ and
$S^\prime$ or $R$ and $R^\prime$ but should not switch,
except by random error.

This behavioral property, RBI, is implied by subjectively weighted
utility (SWU) models, including EU and original prospect theory (next
section).  It is also implied by the editing rule of ``cancellation'',
proposed by Kahneman and Tversky (1979), and certain other heuristics.
RBI is violated by CPT, PH, and TAX, as we see in the next section.

\section{Models of risky decision making}

\subsection{Subjectively weighted utility models}

Let $G$ = (\textit{x\textsubscript{1}, p\textsubscript{1};
  x\textsubscript{2}, p\textsubscript{2}; \dots ; x\textsubscript{n},
  p\textsubscript{n}} denote an \textit{n}-branch gamble with monetary
consequences of $x_i$ with probabilities
\textit{p\textsubscript{i}}, where the sum of the probabilities is 1.
Subjectively weighted utility (SWU) can be written as follows:
\begin{equation}
U(G) = \sum w(p_i)u(x_i)
\end{equation}
where the summation is over all \textit{n branches} of a gamble
[probability consequence pairs ($x_i$,
\textit{p\textsubscript{i}}) that are distinct in the presentation to
the judge].  It is assumed that $G \succ F$
$\Leftrightarrow$ \textit{U}($G$) {\textgreater}
\textit{U}($F$), apart from random error.

SWU models have been studied by Edwards (1962), and by Kahneman and
Tversky (1979), among others.  Equation 1 is sometimes called
``stripped'' original prospect theory (PT) because it is stripped of
the restriction to gambles with no more than two nonzero branches as
well as the editing rules that were postulated to precede evaluation
via Equation 1 (Kahneman \& Tversky, 1979).  Without these
restrictions, PT violates SD in cases where people do not; for that
reason, an editing rule was added to original prospect theory to say
that when people detect dominance, they will satisfy it, though the
theory did not specify when people would or would not detect
dominance.

The subjectively weighted average utility (SWAU) model can be written:
\begin{equation*}
U(G) = \sum w(p_i)u(x_i) / \sum w(p_i)
\end{equation*}
Although this model does not always satisfy SD, it does not show the
kind of violation of transparent dominance that Equation 1 can imply,
and which was described by Fishburn (1978) as a criticism of that
model (Birnbaum, 1999a). See Karmarkar (1978, 1979), and Viscusi (1989)
for variants of this model.

Expected utility (EU) theory is a special case of both SWU and of
SWAU, in which $w(p) = p$.  Expected value (EV) is a special case of
EU in which $u(x) = x$.

Violations of RBI constitute evidence against both SWU and SWAU
models, including PT.  Kahneman and Tversky (1979) further specified
the editing rule of ``cancellation'' which also implies RBI.
Therefore, violations of RBI are critical to EU, SWU, SWAU, and to PT.

\subsection{Cumulative Prospect Theory}

According to CPT (Tversky \& Kahneman, 1992; Wakker, 2011), the value of gamble \textit{G,} with strictly
nonnegative consequences, is as follows:
\begin{equation}
U(G)=\sum[W(P_i)-W(Q_i)]u(x_i)
\end{equation}
where \textit{U}($G$) is the value (utility) of the gamble, $u(x)$ is
the value (utility) of the consequence, $x_i$; $W(P)$ is a strictly
increasing weighting function such that $W(0) =0$ and $W(1)=1$;
$P_{i}$ is the (decumulative) probability of winning a prize greater
than or equal to $x_i$, and $Q_{i}$ is the probability of receiving a
prize strictly greater than $x_i$.  EU is a special case of CPT in
which $W(P) = P$.

When consequences are greater than or equal to zero, the
representation of CPT is the same as that of rank-dependent utility
(RDU) (Quiggin, 1985, 1993; Quiggin \& Wakker, 1994), and rank and
sign dependent utility (RSDU) (Luce, 2000; Luce \& Fishburn, 1991,
1995).  All of these models imply coalescing and first order SD, but
they can violate RBI, as can other forms of configural weighting,
described next.

\subsection{Configurally weighted utility models violate RBI}

\textit{Configurally weighted} models differ from Equation 1 in that
the weight of a branch may depend not only on its probability, but
also on the relationship between the value of a branch's consequence
and consequences of other branches of a gamble (Birnbaum \& Stegner,
1979).  Equation 2 (CPT) is an example of a configural weighting
model, so too are earlier models of Birnbaum (1974a), Birnbaum and
Stegner (1979), the rank affected multiplicative weights (RAM) model
(Birnbaum \& McIntosh, 1996), the transfer of attention exchange (TAX)
model (Birnbaum \& Chavez, 1997), and gains decomposition utility
(GDU) (Luce, 2000). These models are special cases of what Birnbaum
and McIntosh (1996, p.\ 92) called the ``generic rank-dependent
configural weight'' model, also known as the ``rank weighted utility''
model (Luce, 2000; Marley \& Luce, 2001; 2005).  Johnson and Busemeyer
(2005) present a dynamic, computational theory to account for
configural weighting.

A number of studies tested a special form of RBI in which two branches
had equal probability, $p = q$, and using consequences as follows: $0
< z < y^\prime < y < x < x^\prime < z^\prime$.  In this case, two
branches have equal probabilities and the common branch, $(z, 1-2p)$
or $(z^\prime, 1-2p)$, changes from $z$ smallest in the first choice
to $z^\prime$ highest in the second choice (Birnbaum, 2008b).

The \textit{generic configural weight model} can be written for
these choices as follows:\\
\[ S \succ R \Leftrightarrow w_1 u(x) + w_2 u(y) + w_3 u(z) > \]\\[-6.5ex]
\[ w_1 u(x^\prime ) + w_2 u(y^\prime ) + w_3 u(z) \]
\[ S^\prime \succ R^\prime \Leftrightarrow w^\prime_1 u(z^\prime ) +
w^\prime_2 u(x)+ w^\prime_3 u(y) > \]\\[-6.5ex]
\[w^\prime_1 u(z^\prime ) + w^\prime_2 u(x^\prime ) + w^\prime_3
u(y^\prime ) \] where $w_1$, $w_2$, and $w_3$ are the weights of the
highest, middle, and lowest ranked branches, respectively, which
depend on the value of $p$ (differently in different models) in the
choice between $S$ and $R$.  Note that the configural weights in the
second choice may differ from those in the first.  The generic model
allows us to subtract the common term, $w_3 u(z)$, from both sides of
the expression for $S \succ R$, which yields,
\begin{equation}
S \succ R \Leftrightarrow \frac{w_2}{w_1} > \frac{u(x^\prime )-u(x)}{u(y)-u(y^\prime )}
\end{equation}

Similarly, we can subtract the common term, $w_1 u(z^\prime )$, in the
expression for the choice between $S^\prime$ and $R^\prime$. There
will be an $SR^\prime$ violation of RBI (i.e., $S \succ R$
\textit{and} $R^\prime \succ S^\prime$) if and only if the following:
\begin{equation}
\frac{w_2}{w_1} > \frac{u(x^\prime )-u(x)}{u(y)-u(y^\prime )} > \frac{w^\prime_3}{w^\prime_2}
\end{equation}
where $w_1$ and $w_2$ are
the weights of the highest and middle branches, respectively (when
both have probability $p$ in $S$ and $R$), and
$w_2^\prime$ and $w^\prime_3$ are the
weights of the middle and lowest branches (with probability
$p$) in $S^\prime$ and $R^\prime$, respectively.  This pattern
is referred to as the $SR^\prime$ pattern of violation.

The opposite type of violation, \textit{R $\succ$ S} and $S^\prime
\succ R^\prime$, denoted the $RS^\prime$ pattern, will occur whenever
\begin{equation}
\frac{w_2}{w_1} < \frac{u(x^\prime )-u(x)}{u(y)-u(y^\prime )} < \frac{w^\prime_3}{w^\prime_2}
\end{equation}

We can use violations of RBI to test among models of risky
decision-making. In SWU and SWAU models, including PT (Equation 1),
there can be no violations of RBI, except due to random error, because
$w_1 = w_2 = w^\prime_2 = w^\prime_3 = w(p)$,
so both ratios are 1; in EU the weights are all equal to $p$,
so again the ratios are 1.  As shown in the next sections, CPT, PH and
TAX can imply violations of RBI, which should be of different types.

\subsection{CPT violates RBI}

The $W(P)$ function of CPT (Tversky \& Kahneman, 1992; Tversky \& Fox, 1995; Wakker, 2011; Wu \&
Gonzalez 1996, 1998) has an inverse-S shape (it is steeper near 0 and 1 than in the middle).  This shape is needed to
describe the classic Allais paradoxes and other well-established phenomena.  Appendix A shows that CPT with any
$W(P)$ function with an inverse-S shape implies violations of RBI of the $RS^\prime$ pattern.

This study employs three branch gambles with equally likely consequences, $G$ = ($x$, $y$,
\textit{z}).  Given the parameters of Tversky and Kahneman (1992), which imply an inverse-S weighting function, CPT
predicts the following $RS^\prime$ pattern of choices, for example:\\

\noindent $R$ = (\$2, \$5, \$95) $\succ S$ = (\$2, \$35, \$40) 

\noindent  $R^\prime$ = (\$5, \$95, \$98) $\prec S^\prime$ = (\$35, \$40, \$98)

\subsection{Priority Heuristic violates RBI}

The priority heuristic (PH) of Brandst\"atter, Gigerenzer, and Hertwig (2006) assumes that people begin by comparing the
lowest consequences of two gambles; if these differ by at least 10\% of the largest consequence, the decision is based
on lowest consequences only.  If not, then the probabilities to receive the lowest consequences are examined; if these
differ by 0.1 or more, the decision is based on this factor alone.  If not, people compare the highest consequences and
make their decision based on that, if the difference exceeds 10\% of the highest consequence.  If not, they compare
probabilities to win the highest consequences and decide based on that.  If none of these four features is decisive,
the person is assumed to choose randomly. 

In the choice between $S$ and $R$, the lowest consequences are the same and the probabilities are the
same; therefore, people should choose $R$, which has a sufficiently higher best consequence.  In the choice
between $S^\prime$ and $R^\prime $, however, the person should choose $S^\prime$, which has a sufficiently better
lowest consequence.  Therefore, PH implies the $RS^\prime$ pattern of violation of RBI, as does CPT:\\

\noindent $R$ = (\$2, \$5, \$95) $\succ S$ = (\$2, \$35, \$40) 

\noindent $R^\prime$ = (\$5, \$95, \$98) $\prec$  $S^\prime$ = (\$35, \$40, \$98)\\

Now suppose different people followed the PH model, but they used
different thresholds for comparing the consequences.  For example,
suppose a person required a difference of 50\% (instead of 10\%) of
the largest outcome in order to reach a decision.  In that case, the
person would choose $R$ in the first choice, since the
difference \$95$-$\$40 exceeds 50\%, but this person would choose
randomly between $S^\prime$ and $R^\prime$ because \$35$-$\$5 is
less than 50\% of the highest consequence.  This model could then
account for $R^\prime \sim S^\prime$ where $\sim$ indicates
indifference.  Next, suppose someone started by comparing the highest
consequences first and then compared the lower ones, with variable
thresholds.  That reverse order also leads to either the $RS^\prime$
pattern or indifference, depending on the free parameters representing
thresholds.

% is this correct?

In response to evidence against the priority heuristic,
Brandst\"atter, et al.\ (2008) proposed an ``adaptive toolbox,'' in
which people mimic EV or use other heuristics, such as ``toting up'',
cancellation, or a similarity heuristic before using the PH. These
heuristics are analyzed in Appendix B, where it is shown that these
preliminary heuristics imply RBI. Therefore, systematic evidence of
the $SR^\prime$ pattern refutes the particular ``toolbox'' assembly
of these postulated heuristics, followed by the PH.

\subsection{Special TAX Model violates RBI}

In EU, risk-aversion is represented by the shape of the utility function. In the TAX model, however, risk attitudes
are represented in part by transfers of weight (attention) from branch to branch, according to a person{}'s point of
view.  

Birnbaum and Stegner (1979) tested this theory by instructing people
to take on the viewpoints of buyer, seller, or neutral judge. They
found the predicted interactions and changes of rank order implied by
the theory that viewpoint affected the configural weights: Buyers
place greater configural weight on lower estimates of value and
sellers place greater weight on higher estimates of value.  Birnbaum,
Coffey, Mellers, and Weiss (1992) and Birnbaum and Beeghley (1997)
showed that this model describes evaluations of gambles as well. The
findings of Birnbaum and Stegner (1979), and others that followed,
contradict the implications of the ``loss aversion'' interpretation
(Tversky \& Kahneman, 1991) proposed to describe the so-called
``endowment'' effect (Birnbaum \& Zimmermann, 1998).

In the ``special'' TAX model, all transfers of weight are proportional
to the probability weight of the branch losing weight. Let ${\omega}$
represent the proportion of weight transferred from highest
consequence to lowest.  Appendix C shows that as long as this value is
not zero, violations of restricted branch independence should be
opposite the pattern predicted by CPT and PH; namely, $SR^\prime$.

The ``prior'' TAX model refers to non-optimal parameters that were
originally chosen to calculate predictions for a new experiment
(Birnbaum \& Chavez, 1999).  These parameters held up fairly well
predicting the results of other new experiments as well (Birnbaum,
2008b; 2010).  For choices between three-branch gambles (\textit{n} =
3), ${\omega}$ = $\frac{\delta}{(n + 1)}$, where ${\delta} = 1$, so
${\omega}$ = \textonequarter, $u(x)$ = $x$ for $0 < x
< \$150$, and $t(p) = p^{0.7}$.  With ${\delta}= 1$
(${\omega} = \frac{1}{4}$) and $p = \frac{1}{3}$, for example, it
follows that $w_2$/$w_1 = 2 > \frac{3}{2} =
\frac{w^\prime_3}{w^\prime_2}$. With these parameters, the model predicts the following: \\

\noindent $R$ = (\$2, \$5, \$95) $\prec S$ = (\$2, \$35, \$40) 

\noindent  $R^\prime$ = (\$5, \$95, \$98) $\succ S^\prime$ = (\$35, \$40, \$98)\\

\noindent When ${\omega}$ = 0, the TAX model reduces to SWAU, in which case there would be no violations of RBI; when ${\omega}$ =0 and $t(p)$ = $p$, the model reduces to EU, in which case there would be no violations
of either RBI or SD.  

\subsection{Unsettled issues from previous studies}

Previous research (see Birnbaum, 2008b for a summary) reported that the frequency of violations of the pattern
$SR^\prime$ predicted by special TAX exceeds that of the pattern $RS^\prime$ predicted by CPT. For example, Birnbaum
(1999b) tested the following choices: \\

\noindent $S$ = (\$44, 0.10; \$40, 0.10; \$2, 0.80) versus \\
\noindent $R$ = (\$96, 0.10; \$4, 0.10; \$2, 0.80)\smallskip

\noindent and\smallskip

\noindent $S^\prime$ = (\$100, 0.80; \$44, 0.10; \$40, 0.10) versus \\
\noindent $R^\prime$ = (\$100, 0.80; \$96, 0.10; \$4, 0.10)\\

In one case where these two choices were presented to 124
undergraduates, 19 showed the $SS^\prime$ pattern, 32 had the pattern
$SR^\prime$, 13 were $RS^\prime$ and 60 were $RR^\prime$.  Because 32
is significantly greater than 13, these results were interpreted as
evidence against CPT, SWU, and EU and in favor of TAX.

Three related issues remain unsettled by results like these, however.
First, there were 19 + 60 = 79 people out of 124 (63\%) whose response
patterns, $SS^\prime$ or $RR^\prime$, are compatible with EU (and
SWU/SWAU/PT models).  Perhaps this means that this class of models is
compatible with the data of most people.  On the other hand, the same
person who satisfied RBI in one test might show violations if we
repeated the same choices or if we presented other choice problems
that were more appropriately selected for that person's individual
parameters; therefore, it would be premature to conclude that people
who satisfied the property in a single test also satisfy EU (or SWU,
SWAU or PT) in general.

Note that Expressions 4 and 5 depend on two factors that might easily
show individual differences: configural weights and the utility
function.  The ``prior'' TAX model predicts the $SR^\prime$ violation
in this choice problem, but if a person had, for example,
$u(x)$ = $x$\textsuperscript{0.6} instead of
$u(x)$ = $x$, with the same configural weights,
then that person would not show a violation in these choice problems
(the person would instead choose the safe gamble in both cases,
$SS^\prime$).  However, that same person would show the $SR^\prime$
violation if we tested RBI with (\textit{x,} $y$) = (\$31,
\$27) instead of (\$44, \$40) in otherwise identical choice problems.
So unless we used this choice, we might undercount the proportion of
people who have this pattern compatible with TAX.

Second, significantly more people showed the $SR^\prime$ pattern of
violation than showed the $RS^\prime$ pattern.  Thus, the more
frequent pattern agrees with TAX and contradicts CPT. Because SWAU and
EU are special cases of the TAX model, a person enamored with TAX
might argue that TAX is the best representation because 79 + 32 = 111
of 124 are compatible with this model.  However, this summary might
overestimate the success of TAX because some of those 79 who satisfied
RBI in this test might have shown the pattern of violations predicted
by CPT or PH, had they been presented with different choice problems,
so unless we test a greater variety of choice problems, we might
underestimate the frequency of people who contradict TAX and perhaps
satisfy CPT or PH.

Third, 13 people showed the $RS^\prime$ pattern predicted by CPT with
the inverse-S function or by PH.  These people might actually be
consistent with CPT or PH models, in which case we might conclude that
different people apparently use different models.  Because EU is a
special case of CPT, there might be 13 + 79 = 92 who were consistent
with CPT, for example.  On the other hand, these 13 people showing
$RS^\prime$ might simply show this pattern by chance, as a result of
random error.

A person who truly preferred $R$ and $R^\prime$ on these choices might show the $RS^\prime$ pattern by making
an error on the second choice, and a person whose true pattern was $SS^\prime$ could show this pattern by making 
an error on the first choice.  If we presented
this same choice problem repeatedly, these people might show their true patterns when retested.  To address these three
issues we need a way to distinguish whether responses compatible with or in violation of a model are true or due
instead to error.  Otherwise, we cannot determine whether some or none of these individuals are obeying the predictions
of a given model. 

\subsection{Purposes of the present studies}

This paper presents individual true and error theory (\textit{i}TET) and applies it to illustrate how it can address the
above questions left open by a typical group analysis.  Are there indeed any individuals whose data can be best
represented by PH or CPT models?  In two studies, each participant is presented with a number of repetitions of each
choice problem, and \textit{i}TET is used to separate variation due to error from true violations. 

\pagebreak

%Table 1. 
\begin{table*}[t]
\caption{Predictions of models and heuristics for the properties tested.  EU/EV
= Expected Utility theory or expected value; SWU/PT = subjectively
weighted utility and prospect theory; CPT = Cumulative Prospect
theory; TAX = transfer of attention exchange model; PH = priority
heuristic.  Critical = implied with all functions and parameters;
parametric violations = the model may satisfy or violate the property.}

\begin{tabular}{L{2.2in}L{2in}L{2.0in}}\toprule
Model & Stochastic Dominance & Restricted Branch Independence\\\midrule
EU/EV & Critical, satisfies & Critical, satisfies\\
SWU/SWAU/PT & Parametric violations & Critical, satisfies\\
CPT & Critical, satisfies & Parametric violations; Inverse-S implies $RS^\prime$\\
TAX & Parametric violations & Parametric violations; Special TAX implies $SR^\prime$\\
PH  & Critical, satisfies & Critical $RS^\prime$ \\
PH with free parameters, reverse order & Parametric: satisfies or undecided & $RS^\prime$ or undecided.\\\bottomrule\end{tabular}\vspace{-2ex}
\end{table*}

Table 1 summarizes the implications of the models for tests of SD and RBI.  As noted in the table, SD is a critical
property of EU and CPT, meaning that there are no functions or parameters that can allow either of these models to
violate it.  Similarly, RBI is a critical property of EU, SWU and SWAU models including PT, with or without the editing
rule of cancellation.  People who satisfy CPT or PH should show two properties: (a) they should not systematically
violate SD and (b) they should show the predicted $RS^\prime$ pattern of violation of RBI. A person
satisfying the special TAX model might violate SD and show the $SR^\prime$ pattern of violation of RBI.  As
shown in Appendix B, the ``adaptive toolbox'' model that includes EV, EU, toting up (with variable monotonic utility
functions), cancellation, similarity, and the PH with variable thresholds can either satisfy RBI or violate it with the
pattern $RS^\prime$, but this toolbox cannot fit the pattern $SR^\prime$, except by random error.

\section{Individual true and error theory}

A major problem in analyzing empirical data is the fact that variation in the response might produce apparent
violations of critical properties, even if a person would be otherwise consistent with a model.  A number of papers
have dealt with this issue, and there remains disagreement about how to solve it (Birnbaum, 2004b; 2011; Birnbaum \&
Schmidt, 2008; Carbone \& Hey, 2000; Harless \& Camerer, 1994; Hey \& Orme, 1994; Loomes \& Sugden, 1995; Morrison,
1963; Regenwetter, Dana, \& Davis-Stober,  2010, 2011; Regenwetter, Dana, Davis-Stober, \& Guo, 2011; Sopher \&
Gigloitti, 1993; Tversky, 1969; Wilcox, 2008).  

The \textit{true} \textit{and} \textit{error} model applied to group data has had reasonable success
(Birnbaum, 2008c, 2010, 2011; Birnbaum \& Gutierrez, 2007; Birnbaum \& Schmidt, 2008).  This model allows each person
to have a different pattern of ``true'' preferences, allows each choice problem to have a different ``error'' rate,
and it can also allow people to differ in their levels of noise (Birnbaum \& Gutierrez, 2007, p.\ 103).  

However, to address the issues raised in the previous section, we need a theory that can describe the behavior of an
individual.  To distinguish the theory proposed here from that in previous work, we designate \textit{i}TET to refer to
the individual true and error theory, and \textit{g}TET to refer to the version of true and error theory used in
previous studies of group data.

The next sections illustrate how \textit{i}TET can be applied to test critical properties based on one choice (SD) and
two choices (RBI).  It is allowed that each person might have different ``true'' preferences in different blocks of
trials, and preference reversals within a block of trials are attributed to random error.  An important special case of
this model is one in which a person's ``true'' preferences stay the same throughout the study (Birnbaum, 2011).

\subsection{True and error analysis of stochastic dominance}

CPT implies that people should never violate SD, except by error.  In
contrast, the TAX model implies that people
will violate SD in choice problems like the following: \\

\noindent $G$ =  (\$96, 0.90; \$14, 0.05; \$12, 0.05) versus \\
\noindent $F$ =  (\$96, 0.85; \$90, 0.05; \$12, 0.10).\\ \\
\noindent Note that $G$ dominates $F$ because the
probability to win \$96 or more is larger in $G$ than
$F$, the probability to win \$90 or more is the same; the
probability to win \$14 or more is greater in $G$ than
$F$, and there is no value $x$ such that the probability
to win $x$ or more is greater in $F$ than in $G$.

Within each block of trials, each person receives two choice problems
testing SD, the \textit{GF} problem above, and
the following choice between $F^\prime$ and $G^\prime$:\\

\noindent  $F^\prime$ = (\$89, 0.70; \$88, 0.10; \$11, 0.20) versus \\
\noindent $G^\prime$ = (\$90, 0.80; \$13, 0.10; \$12, 0.10).\\ \\
Note that this is a similar test, except positions of the dominant and dominated gambles are reversed.  

Assume that in a given block of trials, an individual either truly
satisfies or truly violates SD (in both problems).  Over blocks
containing both choices, embedded among other trials, the probability
of showing two violations of SD is assumed to obey the following:
\begin{equation}
p(FF^\prime )=p_F(1-e_1)(1-e_2)+(1-p_F)e_1e_2
\end{equation}
where $p(FF^\prime )$ is the probability of showing two
violations of SD; $p_F$ is the probability that
this individual ``truly'' violates SD in a block of the experiment,
$e_1$ and $e_2$ are the error rates for the two choice problems, which
are assumed to be independent and less than {\textonehalf}.  Note
that, if the person truly violates SD, he or she can show two
violations only by making no error on either trial, and if the person
truly satisfies SD, she or he can show two violations only by making
two errors. The probability of showing two satisfactions of SD is
then:
\begin{equation*}
p(GG^\prime )=p_F e_1 e_2+(1-p_F)(1-e_1)(1-e_2); 
\end{equation*}
\noindent it follows that $p(FF^\prime )$ \textgreater $p(GG^\prime )
\Leftrightarrow p_F > \frac{1}{2}$.

Error rates can be estimated from preference reversals between
presentations of the same choice within blocks.  If we assume the
error rates for these two choice problems are equal, $e_1 = e_2 =
e_3$, the probability of reversals within a block of trials is given
as follows:
\[ p(FG^\prime )=p(GF^\prime )= \]\vspace{-4.5ex}
\begin{equation}
p_F(1-e)e+(1-p_F)e(1-e)=e(1-e)
\end{equation}
Therefore, the probability of the union of both types of preference
reversals is $2e(1-e)$.  For example, if there are 32\% reversals
between two versions of the same choice within blocks, $2 e (1 - e) =
0.32$, from the quadratic equation, $e = 0.2$ for this choice problem.

If $e = 0.2$, and if the probability of ``true'' violations is 0 (i.e.,
$p_F = 0$), as implied by models that satisfy SD, then we should
expect to see two violations in a block only 4\% of the time, because
Equation 6 reduces to: $p (FF^\prime ) = e^2$ when $p_F = 0$.
Therefore, we can test if $p_F > 0$ by estimating the error rate, $e$,
from the proportion of preference reversals, and then comparing the
observed proportion of two violations to the squared error rate.

For example, suppose Jane was presented with 20 blocks of trials with
two tests of SD in each block and showed two violations ($FF^\prime$)
in 2 blocks, 4 blocks with $FG^\prime$, 5 blocks with $GF^\prime$, and
9 blocks with two satisfactions ($GG^\prime$).  She violated SD 13
times out of 40 choice problems, so she violated SD in 33\% of the
tests.  From the 9 reversals ($FG^\prime$ \& $GF^\prime$) out of 20 we
estimate, $2 e (1 - e) = .45$; from the quadratic equation, $e = .34$;
therefore, $e^2 = 0.12$, so if $p_F = 0$, we expect (20)(0.12) = 2.34
blocks with $FF^\prime$; but we observed only 2 blocks with
$FF^\prime$; therefore, we can retain the hypothesis that her ``true''
rate of violation is $p_F = 0$.  In this case, \textit{i}TET indicates
that an observed rate of violation of 33\% could occur when the true
rate was 0.

Now consider the case of Joe who had 4 blocks with $FF^\prime$, 2
blocks with $GF^\prime$, 2 blocks with $FG^\prime$, and 12 blocks with
$GG^\prime$. Although Joe violated stochastic dominance only 30\% of
the time (less often than Jane), we conclude that Joe truly violated
SD.  In his case, $2 e (1 - e) = .2$, so $e = 0.11$.  We can reject
the hypothesis that $p_F = 0$, because $e^2 = 0.013$; the binomial
probability to observe 4 or more cases out of 20 with this probability
is small (\textit{p} {\textless} .001), so we can reject the
hypothesis that Joe truly satisfied stochastic dominance in favor of
the hypothesis that $p_F = 0.24$.

When responses are independent, $p(FF^\prime ) = p(F) p(F^\prime )$;
however, independence does not hold in \textit{i}TET unless $p_F$ is
either 0 or 1, even though the errors are independent. Note that in
this case, the choice proportions are as follows: $p(F) = p(F^\prime )
= 0.3$, but $p(FF^\prime ) = 0.20 > p(F) p(F^\prime )=.09$.  Thus, in
the case of Joe, we can reject independence via the two-tailed, Fisher
exact test of independence on his observed frequencies (12, 2, 2, 4),
which is significant (\textit{p} = .037).

This case violates the assumptions of Regenwetter, et al.\ (2011),
whose approach requires that probabilities are identically distributed
and independent (people do not learn or change during a study).  If a
person has different ``true'' preferences in different blocks, $p_F$
will fall between 0 and 1 and \textit{i}TET implies that independence
will be violated, such that $p(FF^\prime ) > p(F) p(F^\prime )$.  In
Joe's case, he may have learned to satisfy dominance during the study.

Now consider Jill, who made two violations on 13 blocks, one violation
on 6 blocks (reversals), and had zero violations in one block.  Six
reversals out of 20 trials is 0.30, so $e$ is estimated to be 0.19.
Furthermore, 13 $FF^\prime$ out of 20 is 0.65; from Equation 6, the
estimated probability of ``true'' violations is $p_F = 1$.  That is, her
results are consistent with the hypothesis that she consistently
violated SD throughout the experiment.

Finally, consider a case that would be in violation of \textit{i}TET.
As shown in Equation 7, the two types of preference reversals should
be equally likely.  In addition, error rates are assumed to be less
than {\textonehalf}; since the total probability of preference
reversals is $2e(1-e)$, it follows that this total must be
less than {\textonehalf}.  For example, suppose Jake had 2 blocks with
$FF^\prime$, 2 blocks with $FG^\prime$, 14 blocks with $GF^\prime$,
and 2 blocks with $GG^\prime$, we could reject the \textit{i}TET model
for two reasons: First, 14 is significantly greater than 2, since the
binomial probability to observe 14 or more hits out of 16 trials with
probability of {\textonehalf} is .002.  Second, the number of
preference reversals (14 + 2 = 16) is significantly more than half the
20 cases, since the binomial probability to observe 16 or more out of
20 is .006.  Thus, it is possible to test \textit{i}TET, and if data
so indicate, to reject it, as in the case of Jake.

\subsection{True and error model of RBI}

The property of RBI involves two choice problems: a choice between $S$
and $R$ and a choice between $S^\prime$ and $R^\prime$.  To test this
property, we present each choice twice within each block of trials.
RBI is the assumption that $S \succ R \Leftrightarrow S^\prime \succ
R^\prime$, apart from error.  Appendix D shows that we can again use
reversals within blocks to estimate the error rates and this allows us
to test the two-choice property of restricted branch independence by
analyzing four choice problems per block.  Appendix D presents both
analysis and results.

\section{Method}

Each participant made choices between gambles in two sessions separated by one week.  Each choice was evaluated multiple
times by each person, separated by many filler trials.  Each gamble was described in terms of an urn containing equally
likely tickets, which differed only in the prize values printed on them.  The participant could choose the urn from
which a ticket would be drawn randomly to determine the cash prize. 

They were informed that 10 people would be selected randomly to play one of their chosen gambles for real cash prizes. 
Participants were told that any of their decisions might be selected for play, so they should choose carefully.  Prizes
were awarded following completion of the study as promised. 

\subsection{Choice formats}

Choices were displayed in one of two formats.  In one group of trials, gambles were described in terms of urns
containing exactly 100 tickets.  These trials were displayed as in the following example:\\

\pagebreak

\noindent \textbf {First gamble:}

\noindent 90 tickets to win \$96\newline
5 tickets to win \$14\newline
5 tickets to win \$12\smallskip

\noindent OR\smallskip

\noindent \textbf{Second gamble:}\\
85 tickets to win \$96\newline
5 tickets to win \$90\newline
10 tickets to win \$12\\

In the other group of trials, each urn contained exactly two or three equally likely tickets, whose values were listed. 
These trials were displayed as in the following example:\\

\noindent  \textbf{First Gamble:} 
(\$2, \$40, \$45)\smallskip

\noindent OR \smallskip

\noindent  \textbf{Second Gamble:} 
(\$2, \$5, \$95)

Instructions and materials can be viewed via the following URL:\newline
{\small \href{http://ati-birnbaum-2009.netfirms.com/Spr\_2010/thanks3.htm}
{http://ati-birnbaum-2009.netfirms.com/Spr\_2010/thanks3.htm}}

\subsection{RBI subdesigns}

In each of two sub-designs testing RBI, the urns were described as holding exactly two or exactly three equally
likely tickets.  Again, the participant could choose the urn from which a randomly selected ticket would be drawn to
determine the prize.  Subdesign RBI1 consisted of 24 trials of the form, $S$ = (\textit{z,}\textit{
}\textit{x,} $y$) versus $R$ = (\textit{z},  \$5, \$95).  The trials were constructed from an 8
X 3, (\textit{x,} $y$) by \textit{z}, factorial design, in which the 8 levels of (\textit{x,}\textit{
}$y$) in the safe gamble were (\$15, \$20), (\$20, \$25), (\$25, \$30), (\$30, \$35), (\$35, \$40), (\$40,
\$45), (\$45, \$50), or  (\$50, \$55), and the 3 levels of \textit{z} were \$2, \$98, or \textit{none}. When \textit{z}
was not presented, the trial was a choice between 2, two-branch gambles.  

Subdesign RBI2 consisted of 27 trials of the form, $R$ = (\textit{z}, \$4, \$96) versus $S$ =
(\textit{z,} \textit{x,} $y$), constructed from a 3 X 9, \textit{z} by (\textit{x,}\textit{
}$y$), factorial design in which the 3 levels of \textit{z} were \$3, \$97, or none, and the 9 levels of
(\textit{x,} $y$) were (\$14, \$18), (\$27, \$31), (\$30, \$34), (\$33, \$37), (\$36, \$40), (\$39,
\$43), (\$42, \$46), (\$45, \$49), and (\$48, \$52).  The two sub-designs counterbalanced the positions of the
``safe'' and ``risky'' gambles but are otherwise quite similar.  

Stochastic dominance was tested by two choice problems embedded in a block of 31 trials like those in Birnbaum (2008b):
$G$ =  (\$96, 0.90; \$14, 0.05; \$12, 0.05) versus $F$ =  (\$96, 0.85; \$90, 0.05; \$12, 0.10) and
$G^\prime$ = (\$90, 0.80; \$13, 0.10; \$12, 0.10) versus $F^\prime$ = (\$89, 0.70; \$88, 0.10; \$11, 0.20).  Position
of the dominant gamble was counterbalanced. 

\subsection{Procedure of Study 1}
Each participant was tested via computer in a lab and served in two
sessions of 1.5 hours each, separated by one week between sessions.
Each replication consisted of either 131 or 134 trials, and included
one of the two blocks testing RBI (either RBI1 or RBI2), each preceded
by 6 warmup trials, as well as the two trials testing SD embedded
among other choice problems.  Trials were blocked in sets of 25 to 31
choice problems; each block tested different properties.  Most of
these blocks involved choices between two-branch gambles, designed to
test transitivity.  For the analyses in this article, these blocks of
trials can be regarded as ``fillers'' that served to separate the
blocks of experimental trials described here; results of our tests of
transitivity are described elsewhere (Birnbaum and Bahra, submitted).

\subsection{Procedure of Study 2}
Study 2 was the same as Study 1, except for the manner in which trials were blocked and procedures for instructions and
warmups. The blocking scheme in Study 2 for the tests of RBI allowed us to apply the \textit{i}TET model to this
property.

All trials involving choices between urns containing 100 tickets, including tests of SD and transitivity, were
intermixed to form large blocks with 107 choice problems.

The two designs testing restricted branch independence (RBI1 and RBI2) were intermixed and preceded by six warmups to
create blocks of 6 + 24 + 27 = 57 trials. The two blocks were alternated (total = 107 + 57 = 164), so any two
presentations of the same choice problem were always separated by at least 107 ``filler'' trials.  Within the block
of 57 RBI trials, most trials had a common probability-consequence branch; if a person wanted to cancel common
branches, it would be easy to do so in this study. 

Each participant was given additional printed instructions and completed several preliminary trials using paper and
pencil that included tests of transparent dominance.  If a participant violated dominance on these trials, the
experimenter asked the participant to explain the decision.  If the participant did not spontaneously identify the
error, the experimenter asked the participant to re-read the printed instructions and complete the warm-ups again. 
When the participant appeared to at least superficially understand the task, she or he was then directed to begin the
study via computer.  Each participant was tested individually in a lab via computer in two sessions of 1.5 hours each,
separated by one week between sessions.  

\subsection{Participants}

Participants in Studies 1 and 2 were 43 and 59 undergraduates enrolled in lower division psychology at California State
University, Fullerton.  Because they were free to work at their own paces, some participants completed more repetitions
than others. Because each replicate in Study 1 included only one of RBI1 or RBI2 designs, whereas each replicate in
Study 2 included both RBI1 and RBI2 intermixed, participants in Study 2 completed relatively more tests of RBI relative
to the tests of SD, but they completed fewer blocks on average.  All participants in Study 1 completed at least 10
replicates and 42 of 59 participants in Study 2 completed 10 or more (10 replicates means 20 choice problems testing
SD).

\section{Results}

Tables 2 and 3 show aggregate results of the tests of RBI, averaged
over participants in both studies, which gave comparable
results. Median percentages choosing the ``risky'' gamble, $R =
(z, \$5, \$95)$ over $S = (z, x, y$) are shown, with separate
rows for different values of $(x, y)$, in both tables.
Columns represent choices with different common consequences; e.g., $z
= \$2$, $z^\prime = \$98$, or without $z$ (none) in Table 2.

%Table 2
\begin{table}[t] \centering
%\raggedright

\caption{Median proportions for choosing $R$=(\textit{z}, \$5, \$95) over $S$=(\textit{z}, \textit{x, y}).  The common consequence, \textit{z}, was \$2, \$98, or none was presented, which
designates choices between $R$=(\$5, \$95) and $S$=(\textit{x, y}).  (Both studies)}
\begin{tabular}{lrrr}\toprule
Safe Gamble & \multicolumn{1}{l}{ \textit{z} = \$2} & \multicolumn{1}{l}{\textit{z} = (none)} & \multicolumn{1}{l}{
$z^\prime$ = \$98}\\\midrule
(\textit{z}, 15, 20) & 80 & 94 & 100\\
(\textit{z}, 20, 25) & 50 & 78 & 100\\
(\textit{z}, 25, 30) & 80 & 94 & 100\\
(\textit{z}, 30, 35) & 24 & 63 & 94\\
(\textit{z}, 35, 40) & 15 & 31 & 89\\
(\textit{z}, 40, 45) & 06 & 15 & 87\\
(\textit{z}, 45, 50) & 00 & 17 & 77\\
(\textit{z}, 50, 55) & 00 & 03 & 67\\\bottomrule
\end{tabular}
\end{table}


If people satisfied RBI, as they should do according to EU, SWU, SWAU
and original PT, or by ``canceling'' common branches (or using toting
up, as in Appendix B), there would be no effect of columns.  Instead,
the percentages choosing the risky gamble are higher when the common
consequence, \textit{z} = \$98 than when \textit{z} = \$2.  For
example, the median percentage is 24\% choosing $R$ = (\$2, \$5, \$95)
over $S$ = (\$2, \$30, \$35), and the percentage is 94\% choosing
$R^\prime$ = (\$5, \$95, \$98) over $S^\prime$ = (\$30, \$35, \$98).
With no common consequence, the choice percentage is 63\% choosing $R$
= (\$5, \$95) over $S$ = (\$30, \$35).

The medians in both Tables 2 and 3 show the $SR^\prime$ pattern of
violation of RBI that is opposite the predictions of both CPT with its
inverse-S probability weighting function, and it is opposite the
predictions of the PH.

\begin{table}[t] \centering
\caption{Median percentage choosing $R$=(\textit{z}, \$4, \$96) over $S$=(\textit{z}, \textit{x, y}).  The common consequence was either \$3 or \$97, or none was presented.}
\begin{tabular}{lrrr}\toprule
Safe Gamble & \multicolumn{1}{l}{\textit{z}=\$3} & \multicolumn{1}{l}{\textit{z}=(none)} & \multicolumn{1}{l}{$z^\prime$=\$97}\\\midrule
(\textit{z}, 14, 18) & 88 & 100 & 100\\
(\textit{z}, 27, 31) & 29 & 50 & 100\\
(\textit{z}, 30, 34) & 22 & 48 & 100\\
(\textit{z}, 33, 37) & 14 & 31 & 89\\
(\textit{z}, 36, 40) & 12 & 33 & 82\\
(\textit{z}, 39, 43) & 13 & 29 & 82\\
(\textit{z}, 42, 46) & 00 & 12 & 78\\
(\textit{z}, 45,49) & 00 & 14 & 71\\
(\textit{z}, 48,52) & 00 & 08 & 63\\\bottomrule
\end{tabular}

\end{table}

The PH implies that the majority should choose $S$=($x$, $y$) over $R$=(\$5, \$95)
whenever $x {\geq}$ \$15, since the lowest consequence of $S$ is at least \$10 better than the lowest
consequence of $R$.  Therefore, all of the percentages (choosing $R$) in the middle column of Table 2
should be less than 50.  Instead, only the last four entries of Table 2 satisfy this prediction.  When the smallest
consequence is the same in both $S$ and $R$ (\textit{z} = \$2, in the first column), the majority should
choose the $R$ gamble, which has the better highest consequence; instead, only the first three entries in this
column of Table 2 satisfy this prediction.  In the third column (\textit{z} = \$98), people should choose $S$
because of the better lowest consequences; instead, more than half of the sample chose $R$ more than half the
time in all rows.  

In summary, the PH was correct in predicting the majority choice in only 7 of 24 choice problems in Table 2.  Similarly,
the PH was correct in predicting only 9 of 27 modal choices in Table 3.  Overall, the PH was correct only 16 times out
of 51 choice problems, which is significantly fewer than half the cases-{}-a random coin toss would have a probability
of .99 to outperform this model!  As shown in Appendix B, the adaptive toolbox model proposed by Brandst\"atter, et al.
(2008) including EV, toting up, similarity, and cancellation, as well as the PH either implies RBI or the opposite
pattern of violations from what we observe.  In addition, allowing different thresholds for different people either
creates random responding or the pattern of violation that we do not observe.  That particular toolbox does not contain
the tool needed to fit these data. Two other heuristics that can violate RBI are described there.

In addition to the effect of columns in Tables 2 and 3, there is an
effect of rows: as ($x$, $y$) in the ``safe'' gambles
increase, the proportion choosing the risky gamble decreases.  This
indicates that people are not simply choosing the gamble with the
higher median (Birnbaum \& McIntosh, 1996).\footnote{One could improve
  the median model by averaging median and midpoint, as in
  range-frequency theory (Parducci, 1995).}  In addition, there is a
small but systematic violation of 3--2 lower distribution
independence; that is, the entries in the middle column are
consistently larger than those in the first column (Birnbaum, 2008b).

According to the TAX model with prior parameters (${\delta}=1$, so
${\omega}=0.25$, $u(x) = x$,
$t(p) = p^{0.7}$) there should
be just one $SR^\prime $ violation of RBI in Table 2, in the fifth row
where $(x, y) = (\$35, \$40)$, Rows 1 to 4 should show
the pattern $RR^\prime$ and Rows 6--8 should show the response pattern
$SS^\prime$.  But if different people have different parameters, TAX
allows other violations.  For example, if ${\omega} = 0.45$ and
$u(x) = x$, there should be $SR^\prime$
violations for that person in the first four rows of Table 2.  As
shown in Appendix C, special TAX model with free parameters implies
that either RBI should be satisfied or that violation should be of the
$SR^\prime$ type.

Although the aggregate results in Tables 2 and 3 violate both original PT, CPT with the inverse-S weighting function and
the PH, aggregated results leave open the possibility that some individuals might satisfy the predictions of these
models.  

%Figure 1.
\begin{figure}[t]\centering
\caption{Individual results for participants in Study 1 (filled circles) and
Study 2 (triangles).  Differences between two types of violations of
RBI (as a proportion of the number of tests) are shown on the ordinate
against the proportion of violations of stochastic dominance on the
abscissa.  EU (expected utility) theory implies no violations of
either property; CPT (cumulative prospect theory with the inverse-S
weighting function) implies no violations of stochastic dominance and
predicts the $RS^\prime$ pattern of violation of RBI (restricted
branch independence).  PH (priority heuristic) predicts that people
should show 100\% $RS^\prime$ violations of RBI, apart from error,
and should satisfy SD.  TAX predicts violations of both properties
with $SR^\prime$ violations of RBI.}
\includegraphics[width=\columnwidth]{fig1}%{fig1.eps}
\end{figure}

Each pair of tests of SD (two tests per block) for each person in each
repetition block can be scored as \textit{GG',} $FG^\prime$,
$GF^\prime$, or $FF^\prime$.  CPT, PH, and EU imply SD, so these
theories predict we should see few choices including $F$ or
$F^\prime$ (violating dominance), and the pattern $FF^\prime$ should
be rare and limited by the frequency of preference reversals
($FG^\prime$ and $GF^\prime$). Of the 102 participants, 67 had a
greater frequency of $FF^\prime$ (two violations) than of $GG^\prime$
(none), meaning that 67 are estimated to have $p_F > \frac{1}{2}$. The
median proportion of violations of stochastic dominance was 0.69.
Only two people had no violations of SD.

Similarly, each test of RBI in each replicate can be scored as one of
four patterns: $SS^\prime$, $SR^\prime$, $RS^\prime$, or
$RR^\prime$. Whereas CPT (with the inverse-S probability weighting
function) implies RBI violations of the type $RS^\prime$, the special
TAX model implies violations only of the type $SR^\prime$.  The PH
also predicts violations of RBI of the $RS^\prime$ type, but it
predicts violations in every row of Tables 2 and 3.

By adding response patterns across blocks and tests, we can compare
the number of $SR^\prime$ violations with $RS^\prime$ violations for
each person. Most participants (90 out of 102) had more violations of
$SR^\prime$ than of the type predicted by CPT or PH ($RS^\prime$).  A
\textit{z} test was computed for each person, with the result that 75
of the 102 people had \textit{z} {\textgreater} 2 (significant at the
5\% level). Only 4 out of 102 participants showed more $RS^\prime$
violations than $SR^\prime$ violations; and only one of these showed
significantly more $RS^\prime$ than $SR^\prime$.

Figure 1 plots the observed proportion of violations of SD on the
abscissa and the difference between the number of $SR^\prime$ and
$RS^\prime$ violations of RBI as a proportion of the number of tests
for each participant.  Filled circles and triangles represent
individuals in Studies 1 and 2, respectively. The label ``EU'' in the
figure is placed near the coordinates (0, 0), because EU theory
implies no violations of either RBI or SD.  The label ``CPT'' is
placed to show that CPT implies no violations of SD and it predicts
RBI violations of the $RS^\prime$ type.  ``PH'' is placed to
indicate that it violates RBI more strongly than does CPT but both
imply the same $RS^\prime$ pattern, and both CPT and PH should
satisfy SD in these tests.  SWU, SWAU, and PT allow violations of SD
but imply no violations of RBI, so cases with ordinate values close to
zero are compatible with these models.\textsuperscript{2} The special
TAX model implies violations of RBI of the $SR^\prime$ type and it
violates SD with its prior parameters.  Most of the points in the
figure are in the vicinity of the TAX model; that is, most people show
violations of RBI more frequently of the $SR^\prime$ type and most
show substantial violations of SD.\footnote{There were 8 out of 102
  people who showed a difference of zero between $SR^\prime$
  and\textit{ }$RS^\prime$; however, in 7 of these 8 cases all or
  almost all of the response patterns were either $SS^\prime$ or
  $RR^\prime$.  Although such cases do not violate SWU, SWAU, and PT,
  these models have not really been tested by the RBI designs in these
  8 cases.  These models imply that a person would switch from
  $RR^\prime$ to $SS^\prime$ as (\textit{x,} $y$) is
  increased (in Table 2 or 3), but these cases do not show the
  switch.}

% no labels on any figure I have.

\subsection{True and error analysis of stochastic dominance}

According to \textit{i}TET, $p(FF^\prime )$ \textit{{\textgreater}}
$p(GG^\prime )$ if and only if the probability of ``truly'' violating
SD exceeds {\textonehalf}.  There were 67 who showed this relation. We
can estimate the ``error'' rate from preference reversals within
blocks and use it to test whether the probability of violating SD
exceeds 0.\footnote{The estimated error rate was uncorrelated with the
  number of blocks completed, \textit{r} = 0.05.}  From these
calculations we estimate that in 86 of 102 cases, people were
``truly'' violating SD; that is, 86 people had estimated $p_F > 0$.

When \textit{i}TET was fit to each person's data, to minimize the sum
of squared discrepancies between observed proportions and predictions,
the median estimates of $p_F$ and $e$ are 0.88 and 0.17; 39 people
were estimated to violate SD with $p_F = 1$.

The mean proportion of violations of SD showed a slight decrease as
trial blocks increased over the first ten blocks.  Among those
participants who completed at least 10 blocks, the mean proportion of
violations decreased from 0.69 averaged over the first two blocks to
0.60 averaged over the 9\textsuperscript{th} and
10\textsuperscript{th} blocks.  No further reduction was found among
those who completed 20 blocks or more.  There were 48 participants who
showed no change between their first and last blocks, 25 showed
increases, and 29 showed decreases in the percentage of violations of
SD.

For those cases for which $0.01 < p_F < .99$, independence was tested
by comparing $P(FF^\prime )$ with $P(F)P(F^\prime )$.  It was
found that in 34 cases, $P(FF^\prime ) > P(F)P(F^\prime )$, and in 12
cases the opposite relation held.  This difference is significant
(\textit{z} = 3.24), consistent with the pattern of non-independence
predicted by the \textit{i}TET and TAX model when $0 < p_F < 1$.

In 7 of 102 cases, \textit{i}TET appeared to give a poor account of
the observed data. The model implies that $p(FG^\prime ) = p(GF^\prime )
= e(1-e)$, $p(FG^\prime ) + p(GF^\prime ) = 2e(1-e) < \frac{1}{2}$, so
$p(F) =p(F^\prime )$.  However, one case violated SD in the $FG$
choice problem 20 times and violated it on the $F^\prime G^\prime$
choice problem only 3 times out of 21 blocks.  Similarly, another case
violated SD in the $FG$ choice 16 times but violated it in the
$F^\prime G^\prime$ choice problem just 3 times out of 20. It is
unclear if these deviations are a fault of \textit{i}TET or of the
simplifying assumption that the two choice problems testing SD are
equivalent.  Note that in the \textit{FG} choice problem, the dominant
gamble $G$ has two features that favor it over $F$, but in the
$F^\prime G^\prime$ problem, the dominant gamble has four features
favoring $G^\prime$ over $F^\prime$.  Perhaps a few people attended to
these features.

%Figure 2.
\begin{figure}%\centering
\caption{Estimated violations of restricted branch independence (ordinate)
showing the $SR^\prime$ pattern rather than the $RS^\prime$ pattern,
plotted against estimated probability of violating stochastic
dominance, corrected for unreliability by individual true and error
theory (Study 2). EU=expected utility, CPT=cumulative prospect
theory, PH=priority heuristic, SWU=subjectively weighted utility,
TAX=transfer of attention exchange.}
\includegraphics[width=\columnwidth]{fig2}
%\includegraphics{fig2}
\end{figure}

\subsection{True and error analysis of RBI}

Individual analysis of RBI is described in Appendix D.  Each test by each person was based on four choice problems per
block and was analyzed separately.

Figure 2 shows the effect of using \textit{i}TET to ``correct'' both
results for unreliability of response.  Each triangle represents an
individual from Study 2, where the \textit{i}TET model could be
applied to the tests of RBI.  The ordinate shows for each case, the
difference between the maximal estimated $p_{SR^\prime}$
minus the maximal estimate of $p_{RS^\prime}$; the
abscissa shows the estimated probability of violating stochastic
dominance, $p_F$.  The same outlier case as in Figure 1 appears in the
lower right of Figure 2.

The results in Figures 1 and 2 show that not even one person fit the
pattern predicted by either CPT or PH.  Only one person showed
frequent $RS^\prime$ violations of RBI compatible with CPT and its
inverse-S weighting function (triangle in lower right of Figures 1 and
2), but that person violated SD in 30 of 32 tests, including violating
both problems in 14 of 16 blocks.  Therefore, this case does not fit
either CPT or PH.  One might try to argue that this person was
confused or indifferent in the tests of SD (and thus choosing
randomly), but this person's data are so extreme that they also refute
this argument, since the binomial probability to observe 30 or more
hits out of 32 trials with $e = \frac{1}{2}$ is less than one in a
million.

When the configural weighting parameter in TAX is zero (${\omega}=0$),
the TAX model reduces to SWAU. This model implies no violations of
RBI, but it can still violate SD.  There appear to be several cases in
each study with ordinate value close to 0 in Figure 1, who might be
consistent with this special case of TAX or perhaps original PT
(Equation 1), which satisfies RBI and can violate SD.  When
${\omega}=0$ and $t(p) = p$, TAX reduces to EU.  There are a few cases
close enough to EU that EU cannot be rejected for those
individuals. Because EU is also a special case of CPT, when $W(P) =
P$, cases that satisfy EU also remain consistent with CPT.  Appendix E
presents tests of RBI, showing the number of response patterns of each
type for each individual.\footnote{Independence was tested by
  comparing the choice proportion repeating the $SR^\prime$ pattern,
  $P(SR^\prime , SR^\prime )$, with the product of the constituent
  marginal proportions.  The \textit{i}TET model implies that the
  conjunction will be more frequent than predicted by independence
  when a response pattern is ``real'' but not perfect; e.g., when $0 <
  p_{SR^\prime} < 1$.  According to independence, the difference,
  $P(SR^\prime, SR^\prime ) - P(S) P(R^\prime ) P(S) P (R^\prime)$,
  should be equally likely to be positive or negative.  Instead this
  difference was more often positive than negative; the frequencies of
  positive and negative differences were (13, 11), (24, 8), (21, 6),
  (26, 6), (24, 11), (28, 4), and (22, 8) in tests corresponding to
  Rows 1, 3, 4, 5, 6, 7, and 8 (of Tables 1), respectively.  All
  except the first are significant ($z$ = 0.41, 2.83, 2.89, 3.54,
  2.20, 4.24, and 2.56, respectively).}$^{,}$\footnote{To search for
  systematic effects of trial blocks, we compared the modal response
  pattern in the first and last blocks for each participant in Study
  2. There were 21, 19, and 19 who showed the $SS^\prime$,
  $SR^\prime$, and $RR^\prime$ patterns most often on the first
  block. Of the 21 who began with the $SS^\prime$ pattern, 14 ended on
  that pattern, 4 shifted to the $SR^\prime$ pattern, and 1 to the
  $RS^\prime$ pattern on the last block.  Of the 19 who began with the
  $RR^\prime$ pattern, 14 ended with the same pattern, 2 switched to
  $SS^\prime$, 2 switched to $SR^\prime$ and 1 to $RS^\prime$; of the
  19 who started on the $SR^\prime$ pattern, 10 ended on that pattern,
  4 switched to $SS^\prime$ and 5 to $RR^\prime$.  Other comparisons
  of first and tenth, and third and last trial blocks showed the same
  picture; namely, about an equal number of people switched in
  different directions so that the marginal distributions of response
  patterns did not show a systematic trend.}

\section{Discussion}

These studies illustrate how \textit{i}TET can be used as a referee to
evaluate models of risky decision-making in individuals.  It allows
the investigator to determine whether violations of a critical
property are real or attributable to random error in response.  The
results of the studies are quite clear: no individual was consistent
with predictions of the PH, and no one was consistent with CPT, except
for those who satisfied EU, which is a special case of both CPT and of
TAX.  Because CPT was intended as a theory to account for violations
of EU, finding cases that satisfy EU would not be much consolation to
a supporter of CPT.

Many cases were observed with the $SR^\prime$ pattern of violation of
RBI and who strongly violated SD; these cases (depicted in the upper
right portion of Figures 1 and 2) are not consistent with EU, SWU,
SWAU, or original PT with or without its editing rule of cancellation.
This pattern of violation of RBI and SD are consistent with the
special TAX model.

Only one case was an outlier to the special TAX model (and to all of
the other models); this one person showed strong violations of SD and
also showed a prevalence of violations of RBI of the type
$RS^\prime$. This person might be compatible with some more general
form of configural weighting (Birnbaum, 2004a; Johnson \& Busemeyer,
2005), but is not compatible with either CPT or PH because of the
systematic violations of SD.

From previous results, it was entirely possible that a subgroup of
people might have been found who were consistent with stochastic
dominance and the $RS^\prime$ pattern of RBI violation characteristic
of CPT with the inverse-S weighting function.  Had the results been
different, one might have concluded that different groups of people
use different models or that we should seek a more general model in
which people differ in their parameters with respect to that more
general model.  But such conclusions were not dictated by our results,
which failed to find any person consistent with CPT or PH who showed
violations of EU.

These conclusions are much stronger than those possible from previous
research, as reviewed by Birnbaum (2008b), for example.  By
application of \textit{i}TET to multiple tests of RBI, we were able to
establish for almost all individual what type of true violations they
had.  [In fact, we might have done still better with our experimental
design, because we found some cases that were too risk-averse (e.g.,
showed only $SS^\prime$ patterns), who might have been better
diagnosed with ($x$, $y$) smaller than (\$14, \$18); and there were
others who were too risk-seeking (e.g., who showed only $RR^\prime$
responses) who might have been better assessed using ($x$, $y$)
greater than (\$50, \$55).  Although these cases show no violations of
RBI and therefore remain consistent with EU and PT, we think it
possible that with different choice problems, some of them may have
shown violations.  The adaptive technique recently proposed by
Cavagnaro, Pitt, and Myung (2011) might be useful in future studies to
find out whether there are indeed any people consistent with RBI.

Had we investigated only one property, we could not have reached these
stronger conclusions.  Had we studied only SD, we would have found 16
people who satisfy this property (in 16 cases, estimated $p_F$ = 0),
which might have been consistent with CPT.  Had we studied only RBI,
we would have found one person whose violations of RBI satisfied the
predictions of CPT and PH.  Because we tested both properties, we are
able to rule out CPT and PH because no person satisfied both critical
tests of these models.

\subsection{Comparing fit of models versus crititcal tests}

Our conclusions are also much stronger and clearer than are possible
when models are fit to data and compared by an index of fit.  Indices
of fit are susceptible to many problems: they depend on functional
assumptions, parameter estimation methods, response error,
experimental design, and a host of other issues besides the accuracy
of the model.  In fact, when experimenters start with erroneous
assumptions about parameters, a seriously wrong model can even achieve
a better score on an index of fit than the model used to calculate
perfect data (Birnbaum, 1973, 1974b).

Certain models cannot be distinguished in certain experimental
designs; for example, there would be no point in comparing the fit of
the TAX model and CPT in choices between binary gambles, because these
two models are virtually identical for such cases. Similarly, these
models can make almost identical predictions inside the probability
simplex useful for testing EU theory, but not useful for comparing CPT
and TAX (Birnbaum, 2008b).  An index of fit in such cases might yield
conclusions that are worse than worthless since they likely say more
about the assumptions made by the analyst when tweaking models than
about the true behavior of the participants.

By testing critical properties, one can evaluate models without having
to make particular assumptions concerning functions and parameters.
For example, CPT must satisfy SD for any choice of monotonic $u(x)$
and $W(P)$ functions.  We need not compare logarithmic functions,
power functions, exponentials or a polynomials for $u(x)$, nor do we
need to compare various parametric $W(P)$ functions to see which
combination of functions and parameters gives the ``best'' overall
index of fit.  When a participant violates SD, we know that no version
of Equation 2 can ``fit'' this result.

Certain ideas are useful for comparing the ``fit'' of models with
differing ``flexibility'' in non-experimental data.  For example, one
approach is to use rival models to fit half of the data and then see
which model does best in predicting the other half.  Another approach
is to correct a model for its space of possible predictions
(Davis-Stober \& Brown, 2011).  Although these approaches are useful,
especially with limited data, they do not necessarily apply in
experimental research where critical tests are possible.

For example, there would be no need to fit both TAX and CPT to half of
our data and then to use these fitted models to predict the other half
of the data, with the ``winner'' being the model that had the highest
cross validation index of fit.

Because we used critical tests, the outcome of such a cross-validation
contest is a foregone conclusion. We know that if people violate
stochastic dominance in each half of their data (which they do) and
because no CPT model can fit either half of data (since no CPT model
predicts violations of SD), we know CPT cannot fit the other half of
the data.  Because TAX (for the tests of SD in this paper) predicts
violations with a fairly wide range of parameters (Birnbaum, 2004a),
it would be hard for TAX to miss.  So the TAX model already ``won''
this contest when its ``prior'' parameters informed us how to design a
new test that in fact yields strong violations of a critical property.

Now it may seem that critical tests, such as this test of SD, are not
equally ``fair'' to both models because TAX can either violate or
satisfy SD and CPT must satisfy it.  But who said experiments can be
fair?  Indeed, the test of SD is a test of CPT and not a test of TAX,
which was never in danger, except of having assumed a poor choice of
parameters.  Tests of critical properties put one class of models in
danger, whereas the rival models can stand on the sidelines and watch
the demolition derby.  Of course, it was not a foregone conclusion
that so many people would violate SD with this test, but once it has
happened, we don{}'t need to confirm the comparison of models by
cross-validation.

In contrast, when testing the critical property of transitivity of
preference, both TAX and CPT are in danger and the family of
lexicographic semiorder (LS) models, including PH, have the advantage
that with suitable parameters, these models can predict violations of
transitivity, whereas TAX and CPT (as well as other transitive models
like EU, SWU, SWAU, original PT) would be refuted by systematic
violations.  The family of LS models are not refuted by transitive
data, however, because with suitable parameters, these models can
describe transitive data.  So a test of transitivity is ``unfair'' (if
we want to use such terms) to transitive models, where the LS models
can describe either transitive or intransitive results, and models
like TAX or CPT are in danger of refutation.

Birnbaum and Bahra (submitted) performed a series of tests designed to search for systematic violations of transitivity.
 Just as the TAX model{}'s ``prior'' parameters were used here to design experiments testing RBI and SD, Birnbaum and
Bahra (submitted) used the PH and its parameters to design their tests of transitivity.  These tests used a feature
called ``linked designs,'' that allow one to test a family of models in which different people might have different
priority orders for examining the attributes, and they were allowed to have individual differences in thresholds, where
a fairly wide range of parameters would imply intransitive preferences.  Only one person showed a pattern of
intransitive preferences consistent with use of a LS model, but that person also violated interactive independence.  No
person was found by Birnbaum and Bahra (submitted) whose data satisfied the predictions of the PH.  

The test of RBI is an interesting case because CPT, TAX, and PH all predict violations (but of different types) and the
family of EU, SWU, SWAU, PT, and certain heuristics like toting up imply no violations.  So this class of models
including EU could complain that the test is not ``fair,'' and they could not win a cross-validation contest if both
halves of the data show systematic violations of RBI (which they do).

In the case of RBI, however, CPT is the most flexible model with respect to this property because with a free
$W(P)$ function, it could predict satisfaction, or either type of violations. But CPT needs its
inverse-S shaped weighting function if it plans to account for the Allais paradoxes, so CPT{}'s ``flexibility'' is
limited if we also require it to predict Allais paradoxes.  And the special TAX model, the form in which the amount of
weight transferred is always a fixed proportion of the branch losing weight allows only one type of violation, or none.
 So, special TAX is less ``flexible'' than CPT with free parameters.  Should CPT be punished for being more flexible?
 If one experimenter required CPT to have the inverse-S and another did not, then this same model might be declared to
be of different levels of flexibility.

Because both TAX and CPT can ``fit'' violations of RBI, does it make sense to compare them by cross-validation from
half of the RBI data to the other half?  If CPT is free to choose any $W(P)$ function, it can fit
better because it can pick up not only the vast majority who showed the $SR^\prime$ pattern of violation (but using an
S-shaped weighting function), it can also fit the one case who had the $RS^\prime$ pattern (with the more usual
inverse-S).  Does this mean that CPT should be favored by such a contest of ``fit?''  We say, no, because the
S-shaped weighting function that would be required for the vast majority of cases has other implications that are not
tested here and which could easily be tested by experiment.  

In particular, choices among binary gambles show that CPT requires an inverse-S function.  So the inverse-S required for
binary gambles is contradicted by the S{}-shape required for three branch gambles (Birnbaum \& Navarrete, 1998).  

Instead of calculating an index of fit, and cross-validating the CPT model fit to data, which in this case might lead to
victory for CPT on cross-validation (because of the one person), or trying to ``correct'' the fit of these models by
computing flexibility, which might favor the special TAX model (because it does not allow the pattern of that one
outlying case), the consistency implied by CPT can be formalized as a critical property, which we think is a more
useful way to analyze the connection between these phenomena than simply fitting data to models trying to use a formula
to do the thinking for us.  

The critical properties in which CPT is required to have both an inverse-S function and an S{}-shaped function are known
as upper and lower cumulative independence, which Birnbaum (1997) deduced and which Birnbaum and Navarrete (1998)
tested, with the results that CPT needs both an inverse-S and S{}-shaped weighting functions; in other words, CPT leads
to self-contradiction. Parameters played their role when we designed the study. So, using critical properties, there is
no need to estimate parameters or compute an index of fit. 

The case of the PH provides a good example of how conclusions based on an index of fit fall apart under deeper
examination.  The argument for PH over models like CPT and TAX was based on its supposedly better accuracy in
``predicting'' modal choices, when fit was calculated for certain previously published data (Brandst\"atter, et al.,
2006).  However, the winners and losers in the contest of fit are reversed when both PH and its rivals are allowed to
estimate their parameters from the same data (Birnbaum, 2008a).  The conclusions were also reversed when PH was
evaluated for previously published data that had not been considered when the model was devised (e.g., Birnbaum \&
Navarrete, 1998). 

But the most important problem for PH has been its failure to predict new results when its implications have been tested
in new experiments (Birnbaum, 2008c, 2010; 2011; Fiedler, 2010; Gl\"ockner \& Betsch, 2008; Gl\"ockner \& Herbold, 2011;
Hilbig, 2008, 2010, Rieskamp, 2008).  The present data contribute an even stronger conclusion; namely, we found no one
who conformed to the predictions of this theory, even when it is expanded to include a toolbox of other processes and
free parameters, except for those cases compatible with EU, whose data might also be described by the ``toting up''
heuristic and not the PH.

Two recent studies have been cited as compatible with the PH.  Arieli, Ben-Ami, \& Rubinstein (2011) examined eye
movements and found that decision makers sometimes moved their eyes as if they were comparing alternatives
attribute-wise, and in other cases, moved their eyes as if they were examine attributes within an alternative.  It is
unclear, however, that the order in which people read in information is necessarily related to the order in which they
use it.  A naive view of attention is that a person must look at something in order to attend to it; however, a large
body of evidence has shown that attention may be directed to different parts of a visual image, independent of where
the eye fixates (see review by Sperling, Reeves, Blaser, Lu,  \& Weichselgartner, 2001 and papers cited there).  In the
days when astronomers looked through telescopes, they looked away from the objects on which they focused their
attention, in order to see them better.  Consequently, eye fixations do not identify unambiguously where attention is
directed within a display.

%\renewcommand{\baselinestretch}{1.04} \large\normalsize

Because there are multiple interpretations, we think that eye movements and related co-variables should not be naively
interpreted as tests of models of decision-making. See Johnson, Schulte-Mecklenbeck, and Willemsen (2008) and
Schulte-Mecklenbeck, K\"uhberger \& Ranyard (2011) for additional discussions of evidence and issues involved.

Brandst\"atter and Gussmack (2012) found that people often report
comparisons of attributes as ``reasons'' for their choices in
``think-aloud'' and ``predict-aloud'' tasks that these authors declare
are not varieties of ``introspection.'' We find it doubtful that
decision-making is mediated by language because animals make decisions
without language, and because language is too slow, imprecise, and
demanding to mediate important life-and-death decisions that people
make, for example, while driving.  Ben Franklin (1771--1788) remarked,
``So convenient a thing it is to be a reasonable Creature, since it
enables one to find or make a Reason for everything one has a mind to
do.''

The idea of an ``adaptive toolbox'' that contains different strategies
does not seem controversial, in our opinion.  What we find odd is not
what has been imagined inside the toolbox, but what has so far been
excluded from it---banished without due process.  Why does such a
toolbox exclude addition, multiplication, and all of psychophysics?

It seems reasonable to suppose that with sufficient training, people could be trained to apply the priority heuristic. 
One could then test the trainees using choice problems such as used here.  If training were successful, those
instructed to use PH should satisfy SD and show the $RS^\prime$ pattern of violation of RBI.  They should also show
the predicted violations of transitivity implied by the model, and they should satisfy integrative independence,
interactive independence, and priority dominance.  However, people who have not yet been trained do not appear to
exhibit such behavior (Birnbaum, 2010), and we consider it more important to predict what people do, rather than where
they look or what they say. 

\subsection{Individual true and error theory}

We consider that \textit{i}TET provides a reasonable way to separate variability of response due to ``error'' from
``true'' violations of the model.  Wilcox (2008) noted that certain theories of variability of response interact with
the models to be evaluated.  For example, representing error as an additive random effect attached to utility of
gambles can force the implication that true preferences are transitive.  That approach is therefore not well-suited for
the evaluation of transitivity (Birnbaum \& Schmidt, 2008).  Similarly, the assumption that error rates are equal for
all choice problems or for all people can make it too easy to refute EU theory, in cases where \textit{i}TET would
indicate that this model is acceptable. 

%\renewcommand{\baselinestretch}{1.06} \large\normalsize

The key idea in \textit{i}TET is that preference reversals within a block of trials provide an estimate of the error
rate for that person and that choice problem.  The model assumes that within a block of trials, a person has a fixed
set of ``true'' preferences.  In any given block of trials, for example, the person is assumed to truly satisfy or
truly violate stochastic dominance in both of the tests treated as equivalent.  This model has testable implications,
and in fact, it did not provide a satisfactory fit to all of the individual tests. 

A rival method intended to separate the variability of response from structural inconsistency of data relative to theory
is that of Regenwetter, et al.\ (2011).  This approach requires stronger assumptions than those of \textit{i}TET.  It
assumes that repeated responses to the same item can be treated as an independent and identically distributed sample
from a theoretical mixture of true preferences.  In our present SD and RBI data, we found that the observed proportions
of repeated response patterns frequently exceeded the product of the marginal choice proportions, contrary to these iid
assumptions.

Our data testing transitivity in linked designs (Birnbaum and Bahra, submitted) also show systematic violations of iid
(see also Birnbaum, 2012).  Therefore, we think that the assumption of independence and identical distribution of
responses by the same person to repeated presentations of the same choice problem is not empirically descriptive. In
iTET, the iid assumptions could hold if a person had a fixed set of preferences, but it appears instead that people
change their true preferences during a long study. 

The \textit{i}TET model assumes that a person does not change true preferences within a block of trials, but only
between blocks.  A more realistic assumption might be that people update parameters throughout the study from trial to
trial (Birnbaum, 2011), so this assumption may also be incorrect; however, we think it more realistic to assume that
response patterns between blocks are independent than to assume that individual trials within blocks are independent. 

\renewcommand{\baselinestretch}{1.0} \large\normalsize

\section*{References}

\begin{hangparas}{1em}{1}

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\end{hangparas}

%\end{hangparas}
%\newpage
%\begin{hangparas}{1em}{1}

\bigskip

\section*{Appendix A.  Analysis of Restricted Branch Independence in CPT}

Equation 2 (RDU, RSDU, and CPT) could in principle accommodate either pattern of violation of RBI when the decumulative
probability weighting function is free.  However, in order to account for the Allais paradoxes, these models assume an
inverse-S weighting function of decumulative probability.  In these models, weights are assumed to be as follows:\\

\noindent $w_1=W(p)$

\noindent $w_2=W(2p)-W(p)$

\noindent $w_3=1-W(2p)$\\

\noindent and in the case where the common branch has the highest consequence, \textit{$z^\prime$:}\\

\noindent $w_1 ^\prime=W(1-2p)$

\noindent $w_2 ^\prime=W(1-p)-W(1-2p)$

\noindent $w_3 ^\prime=1-W(1-p)$\\

Birnbaum (2008b) noted that such a function satisfies, for all \textit{p} {\textless} \textit{p}*: \\

\noindent $W(2p)-W(p) < W(p)-W(0) and W(1-p)-W(1-2p) < W(1)-W(1-p)$; that is,$w_2<w_1$ and $w_2 ^\prime<w_3 ^\prime$. It follows that \\

\noindent (A.1)~~~~~~$w_2/w_1<1<w_3 ^\prime/w_2 ^\prime.$\\  

Comparing Expression A.1 with Expressions 4 and 5, we see that CPT with any inverse-S implies that violations of RBI, if
they are observed, should be of the form, \textit{R$\succ$S} and \textit{S'$\succ$R'}, denoted the $RS^\prime$ pattern. CPT with an S-shaped weighting
function could not account for the Allais paradoxes but it could predict the opposite pattern of violation, and if
$W(P) = P$, CPT reduces to EU and implies RBI. Therefore, CPT is flexible with respect to RBI,
depending on its parameters. 

\section*{Appendix B. Analysis of Adaptive Toolbox and Restricted Branch Independence}

Brandst\"atter, et al.\ (2008) replicated a portion of a study by
Birnbaum and Navarrete (1998), in which the PH had failed to predict
even half of the modal choices; their replication confirmed that the
PH performed worse than chance in describing their new data (Birnbaum,
2008d).  In an attempt to fit their results, Brandst\"atter, et al.\
proposed that people use an ``adaptive toolbox'' of heuristics that
precede or may circumvent the PH.  In particular, they proposed that
people search each choice problem for ``triggering'' conditions that
help them decide which ``tools'' in the toolbox to use.  Do the
gambles differ in EV by a ratio exceeding 2?  If so, choose the gamble
with higher EV.  Are the probabilities of two branches equal?  If so,
add their outcomes.  Do two branches have the same consequences?  If
so, add their probabilities.  Do two alternatives have common
branches? If so, cancel them.

They proposed that people perform a similarity analysis in order to
``search for a no-conflict solution.'' Only if no decision is
determined by one of these higher priority heuristics in the toolbox,
do people use the PH.

This appendix shows that none of these particular heuristics accounts
for the systematic $SR^\prime$ pattern that is more frequent in this
study than the $RS^\prime$ pattern predicted by PH, nor do two other
ways of expanding the PH model (by introducing individual threshold
parameters or allowing a different order of examining the attributes)
accomplish the goal of fitting these results.

\subsection*{Toting up heuristic}

The toting up heuristic assumes that people add the consequences
within each gamble and compare the sum.  Suppose that people compare
$S = (x, y, z)$ and $R = (x^\prime, y^\prime, z)$ as follows:
if $T(S) = u(x) + u(y) + u(z) > T(R) = u(x) + u(y) + u(z)$, then
choose $S$; if $T(S) < T(R$), choose $R$; otherwise, choose randomly.
To allow for individual differences, each person might have a
different utility function, $u(x)$.

Brandst\"atter, et al.\ (2006, 2008) argued against psychophysical
functions, so they might not allow each person to have a different
utility function. They proposed only the special case, $u(x)$ = $x$,
which implies that a person should choose the ``risky'' gamble in the
first seven rows of Table 2 and in the first eight rows of Table 3,
with the pattern $RR^\prime$.  A person should choose the ``safe''
gambles in all cases in the last row of Table 2, where $S$ = (\$2,
\$50, \$55) and $R$ = (\$2, \$5, \$95).  Thus, the ``toting up''
hypothesis with $u(x)$ = $x$ implies that the response patterns should
all be $RR^\prime$, except in the last row of Table 2, where it should
be $SS^\prime$, and the person should be undecided in the last row of
Table 3, where $S$ = (\$3, \$48, \$52) and $R$ = (\$3, \$4, \$96).  No
one fit this pattern of responses exactly, and only one person gave
data even close to this predicted pattern, which is the same as EV.

If different people have different utility functions, however, then
people might switch from the pattern $RR^\prime$ in the first rows to
$SS^\prime$ in the last rows.  For example, suppose $u(x) = x^{0.6}$.
In that case, the person would switch from preferring the ``risky''
gambles in the first five rows of Table 2, $RR^\prime$, to $SS^\prime$
in the last three rows.  As shown in Appendix D, we did not find cases
that switched from $RR^\prime$, to $SS^\prime$, without showing
violations of RBI.

The ``toting up'' heuristic, with any function \textit{u}, implies RBI. \\\\
\noindent\textit{\textbf{Proof:}}\newline
\noindent $S \succ R {\Leftrightarrow} T(S) > T(R) {\Leftrightarrow}
u(z)+u(x)+u(y) > u(z) + u(x^\prime ) + u(y^\prime ) \Leftrightarrow
u(x) + u(y) > u(x^\prime ) + u(y^\prime ) \Leftrightarrow u(z^\prime
) + u(x) + u(y) > u(z^\prime ) + u(x^\prime ) + u(y^\prime )
\Leftrightarrow S^\prime \succ R^\prime$ \\\\
\noindent Thus, RBI is a critical test of ``toting up'' and systematic
violations of RBI disprove this heuristic.  Note that when the
probabilities are equal, as in this study, ``toting up'' is equivalent
to EU, and EV is a special case of ``toting up,'' where $u(x) = x$.
Therefore, the toting up heuristic is equivalent to the EV heuristic;
neither accounts for systematic violation of RBI.

\subsection*{Cancellation}

Cancellation was proposed as an editing rule by Kahneman and Tversky
(1979), who gave an example showing that it implies RBI.  The idea of
cancellation is that if two gambles in a choice problem have a common
probability-consequence branch, the branch can be canceled.
Therefore, $S =(x, y, z)$ versus $R=(x^\prime , y^\prime , z)$ is
equivalent to $S =(x, y)$ versus $R = (x^\prime , y^\prime)$, which is
equivalent to $S = (x, y, z^\prime )$ versus $R = ( x^\prime , y^\prime
, z^\prime )$.  Therefore, RBI is a critical test of the cancellation
heuristic because anyone who uses cancellation, followed by any fixed
decision rule will satisfy RBI.  If people are inclined to follow
cancellation, this experiment would certainly make it easy for them to
do so because so many trials had common branches.

\subsection*{Similarity evaluation}

Similarity evaluation has been discussed by Rubinstein (1988) and in a
modified form by Leland (1988).  Brandst\"atter, et al.\ (2008) argued
for this evaluation as a prelude to the priority heuristic, even
though it employs psychophysical transformation of consequences and
probability.  Leland{}'s similarity model can handle certain phenomena
that have been shown to contradict CPT and the PH (Birnbaum, 2008c).
In Leland{}'s version, the decision maker first evaluates EU, and if
the difference in EU is not decisive, the decision is based on
similarities evaluated on components of the gambles.  EU satisfies
RBI; therefore, the response pattern must be either $RR^\prime$ or
$SS^\prime$ whenever EU is decisive.

If EU is not decisive, the person compares consequences and
probabilities. It is assumed that when probabilities or consequences
are equal they are similar and thus drop out of the process.  Suppose
there is a similarity function, $v(x)$. Two versions of similarity are
as follows: (1) \textit{Largest dimension difference}: Choose $R$
($S$) if $v (x^\prime ) - v(x) > (<) v(y) - v(y^\prime )$ and
otherwise be indecisive.  Because the common consequence has been
canceled, this comparison is the same in the two remainder choices, so
this model implies RBI.  (2) \textit{Dissimilarity}
\textit{threshold}: Choose $R$ ($S$) if $v (x^\prime ) - v(x) > d (< d)$
\textit{and} $v(y) - v(y^\prime ) < d (> d)$, where $d$ is a
similarity threshold; otherwise indecisive.  Either the person
chooses$S$ and $S^\prime$ or $R$ and $R^\prime$, or is indecisive in
both cases, but there is no reason why a person would systematically
choose $R$ and $S^\prime$ or $S$ and $R^\prime$.  Therefore, RBI is a
critical test of either form of this similarity hypothesis, whether
preceded by EU evaluation or not.

A ``greatest advantage'' heuristic that violates RBI is as follows:
choose the gamble with the greatest consequence difference compared to
the least value in the other gamble.  In this case, $R$=(\$2, \$5,
\$95) $\succ S$=(\$2, \$35, \$40) because \$95--\$2 is the greatest
advantage, and $S^\prime$=(\$35, \$40, \$98) $\succ R^\prime$=(\$5,
\$95, \$98) because \$98--\$5 is the greatest advantage.  Like the PH,
this heuristic also implies the opposite pattern of violation from the
pattern more frequently observed.

One could modify the similarity heuristic as follows to produce the
$SR^\prime$ pattern: Introduce a different similarity function on each
ranked branch.  In other words, in the choice between $S$ and $R$, use
$u(x^\prime$)--$u(x)$, $v(y) - v(y^\prime )$, and $w(z) - w(z)$; in
the choice between $S^\prime$ and $R^\prime$, use $u(z^\prime ) -
u(z^\prime )$, $v(x^\prime ) - v(x)$, and $w(y) - w(y^\prime )$, and
map differences in value on the middle branch, $v$, into a
larger range of values, effectively making differences in the middle
ranked branch more important.  With that modification, this heuristic
with three similarity functions (or one function with three weights)
can produce the $SR^\prime$ pattern of violation.

\subsection*{Priority heuristic with free parameters}

According to the priority heuristic, a person starts by comparing the
lowest consequences of the two gambles, and chooses a gamble when the
consequences differ by more than 10\% of the highest consequence.  In
this study, that threshold is \$10; therefore, the person should
always choose always choose $R = (\$2, \$5, \$95) \succ S = (\$2, x,
y)$, because $\$95 - y {\geq} \$10$ in all rows of Table 2.
Similarly, $S^\prime = (x, y, \$98) \succ R^\prime = (\$5, \$95,
\$98)$ in all rows Table 2 because $x - \$5 {\geq} \$10$, so the
response pattern should be $RS^\prime$ in every row.

However, suppose that different individuals have different thresholds.
For example, a person with a threshold of \$20 would continue to
prefer $R \succ S$ in all rows, but would be undecided between
$S^\prime$ and $R^\prime$ in the first two rows where the differences
in lowest outcomes are less than \$20.  The response patterns in the
remaining rows are $RS^\prime$. Increasing the threshold increases the
number of undecided cases, where people respond randomly, but it would
not produce the opposite response pattern, except by chance.

Suppose some people start with the highest consequences first and
compare the lowest consequences later.  Such people would still choose
$R \succ S$ in all rows of Table 2 because the highest consequences
differ by more than \$40 in all cases.  When comparing $S^\prime$ and
$R^\prime$ the person would again choose $S^\prime$ in all cases
because the highest consequences are equal and the lowest one differ
by \$10 or more.  Increasing the threshold would again create
undecided cases, but it would not produce the $SR^\prime$ response
pattern, except by chance.

As noted in Birnbaum and McIntosh (1996), the theory that people
compare only the middle branches (they called this heuristic
``median'' theory) implies the $SR^\prime$ pattern of violations in
every row.  Despite such evidence in the literature, however,
Brandst\"atter, et al.\ (2006) proposed that people do not even
consider any middle branches of gambles.  The best way to improve the
fit of the priority heuristic to the results of this paper would be to
assume that people start by examining the middle branches of gambles.
Of course, that would not solve other critical violations of the
family of Lexicographic Semiorder models for properties such as
interactive independence and integrative independence (Birnbaum,
2010).

\section*{Appendix C.  Analysis of the RBI in the TAX model}

RAM, TAX and GDU models, with their typical parameters, imply the
opposite type of violation of RBI; i.e., $SR^\prime$ instead of the
pattern predicted by CPT. According to the ``special'' TAX model,
weight transfers among branches are always the same proportion of the
branch losing weight. In this special TAX model, only the
$SR^\prime$pattern is possible (Birnbaum, 2008b, p.\ 486) or the
person will satisfy RBI.

In the TAX model, weight (attention) is drawn from branches leading to
higher consequences to branches leading to lower consequences, leading
to $w_3$ (the weight of the lowest branch) gaining weight at the
expense of $w_1$.  The weights in this case, where
$t(p)$ is the probability weighting function and
${\omega} {\geq} 0$ is the weight transfer (configural weighting)
parameter, are given as
follows:\\

\noindent(C.1a)~~~~~~$w_1=[t(p)(1-2\omega)]/D$

\noindent (C.1b)~~~~~~$w_2=t(p)/D$

\noindent (C.1c)~~~~~~$w_3=[t(1-2p)+2\omega t(p)]/D$ \\

\noindent where $D = t(p) + t(p)+ t(1 - 2p)$ is the sum of the
weights. Note that the highest consequence gave up ${\omega} t(p)$ to
each of the other branches and that the middle branch gave up that
amount to the lowest branch but received the same amount from the
highest branch.
In the choice between $S^\prime$ and $R^\prime$, the weights are as follows:\\

\noindent(C.2a)~~~~~~$w_1^\prime=[t(1-2p)(1-2\omega)]/D$

\noindent(C.2b)~~~~~~$w_2 ^\prime=[t(p)(1-\omega)+\omega t(1-2p)]/D$

\noindent(C.2c)~~~~~~$w_3 ^\prime=[t(p)+\omega t(p)+\omega t(1-2p)]/D$\\

\noindent where \textit{D} is the same as in Equations C.1; it follows that\\

\noindent(C.3)~~~~~~$w_2/w_1=1/(1-2\omega) > w_3 ^\prime/w_2
^\prime=$\newline
$[t(p)(1+\omega)+\omega t(1-2p)]/[t(p)(1-\omega)+\omega t(1-2p)]$\\

Therefore, people should show the $SR^\prime$\textsubscript{ }pattern of violation of RBI, unless ${\omega}$ = 0, in
which case there would be no violations.  

In the case of three equally likely consequences, as in this study, condition C.3 simplifies to,\\

\noindent(C.4)~~~~~~$w_2/w_1=1/(1-2\omega)>w_3^\prime/w_2 ^\prime=(1+2\omega)/1$\\

In the case where weight is transferred from the lowest consequences to the highest, the same relation holds because,\\

\noindent(C.5)~~~~~~$w_2/w_1 = 1/(1 + 2\omega)>w_3 ^\prime/w_2 ^\prime=(1-2\omega)/1$\\

\noindent Therefore, whether weight is transferred from highest to lowest or
lowest to highest, the special TAX model implies the same
$SR^\prime$ pattern of violations; only when
${\omega}$=0, does this model imply RBI.  Birnbaum and Beeghley
(1997) found this pattern in both buyer's and seller's prices,
which are theorized to have opposite directions of weight transfer.

\section*{Appendix D: True and error analysis of RBI}

There are four possible response patterns in each pair of choices (in
each test): $SS^\prime$, $SR^\prime$, $RS^\prime$, and
$RR^\prime$.  We have the following for the probabilities of
observing each response pattern:\\

\noindent (D.1a)
$p(SS^\prime )=p_{SS^\prime}(1-e)(1-e^\prime )+$\newline
$p_{SR^\prime}(1-e)e^\prime+p_{RS^\prime} e(1-e^\prime )+p_{RR^\prime}ee^\prime$\\

\noindent (D.1b) $p(SR^\prime )=p_{SS^\prime}(1-e)e^\prime+$\newline
$p_{SR^\prime}(1-e)(1-e^\prime )+p_{RS^\prime}ee^\prime+ p_{RR^\prime}e(1-e^\prime )$\\

\noindent (D.1c) $p(RS^\prime )=p_{SS^\prime}e(1-e^\prime )+$\newline
$p_{SR^\prime}ee^\prime+p_{RS^\prime}(1-e)(1-e^\prime )+p_{RR^\prime}(1-e)e^\prime$\\

\noindent (D.1d)
$p(RR^\prime )=p_{SS^\prime}ee^\prime+p_{SR^\prime}e(1-e^\prime )+$\newline
$p_{RS^\prime}(1-e)e^\prime+p_{RR^\prime}(1-e)(1-e^\prime )$\\

\noindent where $p_{SS^\prime}$, $p_{SR^\prime}$, $p_{RS^\prime}$, and
$p_{RR^\prime}$ are the ``true'' probabilities of each response
pattern in a block of trials (which sum to 1), $e$ and $e^\prime$ are
the error rates in the two choice problems; these are assumed to be
independent and less than {\textonehalf}.  The assumption of RBI is
the assumption that $p_{SR^\prime} = p_{RS^\prime} = 0$.

Because each choice problem is presented twice in each block, we can
estimate the error rates from preference reversals within blocks,
$p(SR) = p(RS) = (1 - e)e$.  For the two
choice problems testing $S$ versus $R$, we have:\\

\noindent (D.2a) $p(SR)+p(RS)=2(1-e)e$\\

\noindent (D.2b) $p(SS)=(p_{SS^\prime}+p_{SR^\prime})(1-e)^2+(p_{RS^\prime}+p_{RR^\prime})e^2$\\

\noindent (D.2c) $p(RR)=(p_{RS^\prime}+p_{RR^\prime})(1-e)^2 + (p_{SS^\prime}+p_{SR^\prime})e^2$\\

\noindent
where $p(SS)$ is the probability that the person selected the response $S$ in both choices
between $S$ and $R$ in a block; $p_{S} = 1 - p_{R} = p_{SS^\prime} + p_{SR^\prime}$;
$p_{R} = p_{RS^\prime} + p_{RR^\prime}$
and $e$ is the error term in the choice between $S$ and $R$.  Expressions D.2 involve no
additional parameters beyond those in Expressions D.1.

The preference reversals in the two choices between $S^\prime$ and
$R^\prime$ can also be used to estimate the error rate in this choice
because $p(S^\prime R^\prime) = p(R^\prime
S^\prime) = (1 - e^\prime ) e^\prime $, where $e^\prime$ is the error rate in this choice.\\

\noindent (D.3a) $p(S^\prime R^\prime )+p(R^\prime S^\prime )=2(1-e^\prime )e^\prime$\\

\noindent (D.3b) $p(S^\prime
S^\prime )=(p_{SS^\prime}+p_{RS^\prime})(1-e^\prime )^2 + $\newline $ (p_{SR^\prime}+p_{RR^\prime})(e^\prime )^2$\\

\noindent (D.3c) $p(R^\prime R^\prime ) = (p_{SR^\prime} + p_{RR^\prime})(1 - e^\prime )^{2} +$\newline $ (p_{SS^\prime}
+ p_{RS^\prime})(e^\prime )^{2}$\\

In addition, because each pair of choice problems is presented twice
within each block, there are 16 possible, 4 choice, response patterns
within each block.  In particular, we can analyze the probabilities of
repeating each choice pattern
on both tests in a block: \\

\noindent (D.4a) $p(SS^\prime,
SS^\prime )=p_{SS^\prime}(1-e)^2(1-e^\prime )^2+ $\newline $p_{SR^\prime}(1-e)^2 (e^\prime )^2+
p_{RS^\prime}e^2 (1-e^\prime )^2+p_{RR^\prime}e^2(e^\prime )^2$\\

\noindent (D.4b)\ $p(SR^\prime,
SR^\prime )=p_{SS^\prime}(1-e)^2(e^\prime )^2 + $\newline $ p_{SR^\prime}(1-e)^2(1-e^\prime )^2+
p_{RS^\prime}e^2(e^\prime )^2+p_{RR^\prime}e^2(1-e^\prime )^2$\\

\noindent (D.4c) $p(RS^\prime, RS^\prime )=
p_{SS^\prime}e^2(1-e^\prime )^2 +
p_{SR^\prime}e^2(e^\prime )^2 +
p_{RS^\prime}(1-e)^2(1-e^\prime )^2 +
p_{RR^\prime}(1-e)^2(e^\prime )^2$\\

\noindent (D.4d) $p(RR^\prime, RR^\prime )=
p_{SS^\prime}e^2(e^\prime )^2 +
p_{SR^\prime}e^2(1-e^\prime )^2  +
p_{RS^\prime}(1-e)^2(e^\prime )^2 +
p_{RR^\prime}(1-e)^2(1-e^\prime )^2$\\

\noindent where $p(SR^\prime$, $SR^\prime )$ is the probability of
showing the same $SR^\prime$ response pattern on both tests of RBI
within a block.  These probabilities in Expressions D.4 are the same
as those in Expressions D.1, except that each of the components
involving error terms are squared; these four terms do not sum to
1.\footnote{It may be helpful to keep in mind that because $e <
  \frac{1}{2}$, it follows that $ee < e(1 - e) < (1 - e)(1 - e)$; for
  example, with $e=.2$, these three terms are .04, .16, and .64,
  respectively; the corresponding squared terms are even smaller.}

The assumption that RBI is satisfied corresponds to the assumption
that $p_{SR^\prime} = p_{RS^\prime}$=0, which means that
$p_{RR^\prime} = 1 - p_{SS^\prime}$.  If so, the probability of
showing a repeated violation of the type predicted by the TAX model is
$p(SR^\prime$, $SR^\prime) = p_{SS^\prime} (1 - e)^2 (e^\prime)^2 +
p_{RR^\prime} e^2 (1 - e^\prime)^2$.  It follows that the maximal rate
of showing two violations of this type in a block is 0.0625.  If a
person completed 20 blocks and showed 4 blocks in which a particular
violation (i.e., $SR^\prime$) is repeated, we can reject the
hypothesis that $p_{SR^\prime} = 0$, because the binomial probability to
show 4 or more repeated $SR^\prime$ violations in 20 blocks is only
.033.  This upper bound follows from the conservative assumption that
errors cannot exceed \textonehalf.

We can impose a still smaller upper bound by estimating the error
terms using preference reversals to each type of choice problem to, as
in the previous section.  Suppose we find 32\% preference reversals
for both $S$ versus $R$ and $S^\prime$ versus
$R^\prime$; it follows that $e = e^\prime = 0.2$; therefore, the
maximal rate of repeated violations of one type is 0.0256, if RBI
holds.  In this case, 3 repeated $SR^\prime$\textit{ }violations out
of 20 blocks would suffice to reject the hypothesis that
$p_{SR^\prime}$ = 0, since the binomial
probability to show 3 or more out of 20 with \textit{p}=.0256 is only
.014.

We can again explore independence, which is the assumption that the
probability of any response combination is the product of the marginal
probabilities of the components; for example, $p(SR^\prime ) =
p(S) p(R^\prime)$ and $p(SR^\prime , SR^\prime) = p(S)^2
p(R^\prime)^2$.  Independence does not follow from the \textit{i}TET
model unless one of the four ``true'' probabilities of a response
combination is 1; for example, if $p_{SR^\prime}$=1.  If a person
changed ``true'' preferences from block to block, as might be expected
under CPT or TAX if a person{}'s parameters changed from block to
block, for example, then independence will be violated. Again, the
probability of a repeated pattern that is ``real'' but imperfect will
exceed the product of the marginal probabilities according to
\textit{i}TET.

There are 16 equations in Expressions D.1 through D.4 that express
predictions of \textit{i}TET for probabilities of response patterns.
One can estimate five parameters in order to fit the 16 observed
proportions corresponding to these expressions: $e$, $e^\prime$,
$p_{SS^\prime}$, $p_{SR^\prime}$, $p_{RS^\prime}$, and $p_{RR^\prime}
= 1 - p_{SS^\prime} - p_{SR^\prime} - p_{RS^\prime}$.  The models of
risky decision making to be compared are then special cases of
\textit{i}TET: the prior TAX model implies $p_{SR^\prime} > 0$; CPT
with an inverse-S weighting function and PH imply instead that
$p_{RS^\prime} > 0$; EU and SWU (including original prospect theory)
imply RBI, which means that $p_{SR^\prime} = p_{RS^\prime}=0$.

\subsection*{True and error analysis of RBI: Results}

In Study 2, the two RBI designs were intermixed in the same blocks;
this procedure allows us to apply \textit{i}TET to each person{}'s
data.  The tests in Row 5 of Table 2 [$S$ = (\$2, \$35, \$40), $R$ =
(\$2, \$5, \$95), $S^\prime$ = (\$35, \$40, \$98), $R^\prime$ = (\$5,
\$95, \$98)] and Row 5 of Table 3 [$S$ = (\$3, \$36, \$40), $R$ =
(\$3, \$4, \$96), $S^\prime$ = (\$36, \$40, \$97), $R^\prime$ = (\$4,
\$96, \$97)] were treated as equivalent. These two choice problems
also counterbalanced position, so to repeat the same response pattern
on both tests, a participant would have to press opposite buttons
appropriately on four trials.

In the same fashion, Rows 1 \& 1, 3 \& 2, 4 \& 3, 6 \& 6, 7 \& 8, and
8 \& 9 (in Tables 2 and 3, respectively) were paired to create a total
of seven tests of RBI for each of the 59 participants in Study 2,
making 413 tests. The model has five parameters per test: $e$,
$e^\prime $, $p_{SS^\prime}$, $p_{SR^\prime}$, $p_{RS^\prime}$, and
$p_{RR^\prime} = 1 - p_{SS^\prime} - p_{SR^\prime} - p_{RS^\prime}$.
Each test for each person was fit separately to \textit{i}TET to
minimize the sum of squared differences between the 16 predicted and
obtained proportions corresponding to the probabilities in Equations
D.1 through D.4.

The results of the seven tests for participant \#216 are shown in
Table D.1.  Parameter estimates are listed in the last six rows.  Note
that in the first test (first row of Table 2), this person was
estimated to have had a perfect true preference for $R$ and
$R^\prime$; i.e., $p_{RR^\prime} = 1$.  As the ``safe'' gambles are
improved, $p_{RR^\prime}$ decreased to zero and is replaced in the
last four columns by perfect adherence to the $SR^\prime$ pattern of
violation of RBI.

\begin{table}[t]\centering
 \parbox{\columnwidth}{Table D.1. Observed choice proportions and estimated parameters of
  the individual true and error model in seven tests of RBI for
  participant \#216, who completed 21 blocks.  Tests are numbered
  according to the row number in Table 2.}\smallskip

\begin{tabular}{lrrrrrrr}\toprule
 & \multicolumn{7}{c}{Tests}\\\cmidrule{2-8}
 & 1 & 3 & 4 & 5 & 6 & 7 & 8\\
\midrule
$P (SS)$ & 0.00 & 0.52 & 0.86 & 0.76 & 0.95 & 1.00 & 0.95\\
$P (SR)$ & 0.10 & 0.05 & 0.05 & 0.14 & 0.05 & 0.00 & 0.05\\
$P (RS)$ & 0.00 & 0.14 & 0.05 & 0.10 & 0.00 & 0.00 & 0.00\\
$P (RR)$ & 0.90 & 0.29 & 0.05 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (S^\prime S^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (S^\prime R^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.05 & 0.00 & 0.00\\
$P (R^\prime S^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.05 & 0.00 & 0.00\\
$P (R^\prime R^\prime )$ & 1.00 & 1.00 & 1.00 & 1.00 & 0.90 & 1.00 & 1.00\\
$P (SS^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.05 & 0.00 & 0.00\\
$P (SR^\prime )$ & 0.05 & 0.62 & 0.90 & 0.88 & 0.93 & 1.00 & 0.98\\
$P (RS^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (RR^\prime )$ & 0.95 & 0.38 & 0.10 & 0.12 & 0.02 & 0.00 & 0.02\\
$P (S S^\prime , S S^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (S R^\prime , S R^\prime )$ & 0.00 & 0.52 & 0.86 & 0.76 & 0.86 & 1.00 & 0.95\\
$P (R S^\prime , R S^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (R R^\prime , R R^\prime )$ & 0.90 & 0.29 & 0.05 & 0.00 & 0.00 & 0.00 & 0.00\\
$e$ & 0.05 & 0.11 & 0.05 & 0.13 & 0.03 & 0.00 & 0.02\\
$e^\prime$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.05 & 0.00 & 0.00\\
$p_{SS^\prime }$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$p_{SR^\prime }$ & 0.00 & 0.65 & 0.95 & 1.00 & 1.00 & 1.00 & 1.00\\
$p_{RS^\prime }$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$p_{RR^\prime }$ & 1.00 & 0.35 & 0.05 & 0.00 &
0.00 & 0.00 & 0.00\\
\bottomrule
\end{tabular}
\end{table}

Results for participant \#234 are shown in Table D.2, as in Table D.1.  This person's data for the first two columns
(Tests 1 and 3) do not fit \textit{i}TET.  Note that $P (SR )$ = 0.53 {\textgreater}
$P (RS )$  = 0.05.  The estimated error terms for these two tests are also relatively large and unequal. 
Nevertheless, the results for Tests 4--8 appear to fit the model and provide reasonable parameter estimates.

\begin{table}[t]\centering
\parbox{\columnwidth}{Table D.2. Observed choice proportions and estimated parameters of the
individual true and error model in seven tests of
RBI for participant \#234, who completed 19 blocks, as in Table
D.1.}\smallskip

\begin{tabular}{lrrrrrrr}
\toprule
 & \multicolumn{7}{c}{Tests}\\
\midrule
 & 1 & 3 & 4 & 5 & 6 & 7 & 8\\
\midrule
$P (SS )$ & 0.00 & 0.42 & 0.53 & 0.79 & 0.89 & 0.79 & 0.89\\
$P (SR )$ & 0.53 & 0.11 & 0.11 & 0.05 & 0.05 & 0.05 & 0.05\\
$P (RS )$ & 0.05 & 0.37 & 0.21 & 0.11 & 0.05 & 0.16 & 0.05\\
$P (RR )$ & 0.42 & 0.11 & 0.16 & 0.05 & 0.00 & 0.00 & 0.00\\
$P (S^\prime S^\prime )$ & 0.00 & 0.05 & 0.05 & 0.16 & 0.11 & 0.16 & 0.26\\
$P (S^\prime R^\prime )$ & 0.05 & 0.00 & 0.11 & 0.05 & 0.05 & 0.05 & 0.26\\
$P (R^\prime S^\prime )$ & 0.05 & 0.11 & 0.21 & 0.11 & 0.16 & 0.05 & 0.05\\
$P (R^\prime R^\prime )$ & 0.89 & 0.84 & 0.63 & 0.68 & 0.68 & 0.74 & 0.42\\
$P (SS^\prime )$ & 0.03 & 0.11 & 0.16 & 0.21 & 0.18 & 0.18 & 0.39\\
$P (SR^\prime )$ & 0.26 & 0.55 & 0.53 & 0.66 & 0.76 & 0.71 & 0.55\\
$P (RS^\prime )$ & 0.03 & 0.00 & 0.05 & 0.03 & 0.03 & 0.03 & 0.03\\
$P (RR^\prime )$ & 0.68 & 0.34 & 0.26 & 0.11 & 0.03 & 0.08 & 0.03\\
$P (S S^\prime , S S^\prime )$ & 0.00 & 0.05 & 0.00 & 0.11 & 0.11 & 0.16 & 0.21\\
$P (S R^\prime , S R^\prime )$ & 0.00 & 0.37 & 0.26 & 0.53 & 0.63 & 0.63 & 0.37\\
$P (R S^\prime , R S^\prime )$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
$P (R R^\prime , R R^\prime )$ & 0.42 & 0.11 & 0.11 & 0.05 & 0.00 & 0.00 & 0.00\\
$e$ & 0.31 & 0.35 & 0.21 & 0.09 & 0.05 & 0.10 & 0.06\\
$e^\prime $ & 0.06 & 0.05 & 0.21 & 0.09 & 0.11 & 0.04 & 0.20\\
$p_{SS^\prime }$ & 0.00 & 0.07 & 0.02 & 0.17 & 0.11 & 0.18 & 0.36\\
$p_{SR^\prime }$ & 0.00 & 0.91 & 0.78 & 0.77 & 0.88 & 0.82 & 0.64\\
$p_{RS^\prime }$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.00\\
$p_{RR^\prime }$ & 1.00 & 0.02 & 0.21 & 0.06 & 0.00 & 0.00 & 0.00\\
\bottomrule
\end{tabular}
\end{table}

Examining tables corresponding to Tables D.1 (and D.4) for each of the
59 participants, we found the majority showed two common trends: most
people showed evidence of the $SR^\prime$ pattern of violation of RBI
and most showed either decreasing
$p_{RR^\prime }$or increasing
$p_{SS^\prime }$ (or both) as the values of
($x$, $y$) increased.

Over all 413 tests, the mean estimated error terms, $e$ and
$e^\prime$ were 0.07 and 0.07, with 88\% of the estimated error terms
less than 0.2 and 94\% less than 0.3.  Slightly more than half of all
estimated error terms were 0.  These error terms are smaller than
those estimated in the tests of SD.  Perhaps the lower error rates
occur because the choice problems testing RBI are simpler than those
testing SD: In this analysis, they all have three equally likely
consequences without probabilities to read or to process.

Of the 413 tests, 171 cases had estimated
$p_{SR^\prime }$
{\textgreater} 0.2 including 72 for which
$p_{SR^\prime }$
{\textgreater} .9, whereas in only 9 cases, estimated
$p_{RS^\prime }$
{\textgreater} 0.2, all from just three participants.  Summed over
seven tests for each person, 49 of 59 participants had
${\Sigma} p_{SR^\prime }$
{\textgreater} ${\Sigma} p_{RS}$; only
one person (\#241) had the opposite relation, and there were 9 people
who showed a difference of zero.  These 9 might be candidates for EU
(or another model that implies RBI, such as PT); however, 4 of these 9
had $p_{RR^\prime }=1$ in
all seven tests, 4 had
$p_{SS^\prime }=1$ in all
seven tests, and one had $p_{SS^\prime }$
{\textgreater} 0.85 in all tests; therefore, these cases that
satisfied RBI could be better described as ``moot'', or ``untested''
by the design.

\section*{Appendix E. Individual tests of restricted branch independence}

Table E.1 shows the number of response patterns for each individual,
summed over the tests of restricted branch independence.  RBI is
implied by EU, SWU, SWAU, original PT, cancellation, toting up, and
two similarity models in Appendix B.  It should be violated with the
pattern $RS^\prime$ according to CPT with an inverse-S weighting
function and by PH.  The special TAX model implies the pattern
$SR^\prime$, as long as ${\omega}$ is not zero, and
implies RBI when ${\omega}$ = 0.  Random responding does not imply
violations, except by chance.

Cases where one type of violation is significantly more frequent than
the other are shown in bold.  For cases where the sum of violations
was less than 25, the binomial distribution was used, and in other
cases, the \textit{z}{}-test was performed.  Most participants show
significantly more violations of the type $SR^\prime$,
predicted by TAX than of $RS^\prime$, predicted by CPT and PH.
Only one person shows significantly more violations of the
$RS^\prime$ type, but that person also systematically violated SD,
contrary to both CPT and PH.

%Table E.1.
\begin{table}[H]\centering
\parbox{\columnwidth}{Table E.1. Summary of individual tests of restricted branch
independence, summed across tests.  Cases numbered 101--143 are from
Study 1 and cases starting at 201 from Study 2. Bold entries show
significant differences between $SR^\prime$ and $RS^\prime$
patterns, with ${\alpha}$ = .05.}\smallskip

\begin{tabular}{crrrrr}\toprule
\multicolumn{1}{l}{Case No.} & \multicolumn{1}{l}{$SS^\prime$} &
\multicolumn{1}{l}{$SR^\prime$} & \multicolumn{1}{l}{$RS^\prime$} &
  \multicolumn{1}{l}{$RR^\prime$} & \multicolumn{1}{l}{SUM}\\\midrule
101 & 163 & \bfseries 7 & 0 & 0 & 170\\
102 & 21 & \bfseries 80 & 0 & 18 & 119\\
103 & 0 & 0 & 0 & 170 & 170\\
104 & 0 & \bfseries 36 & 0 & 65 & 101\\
105 & 1 & 4 & 0 & 216 & 221\\
106 & 0 & \bfseries 27 & 0 & 91 & 118\\
107 & 38 & \bfseries 48 & 1 & 15 & 102\\
108 & 26 & \bfseries 86 & 0 & 6 & 118\\
109 & 6 & \bfseries 76 & 1 & 19 & 102\\
110 & 0 & 3 & 0 & 211 & 214\\
111 & 5 & \bfseries 180 & 0 & 1 & 186\\
112 & 1 & \bfseries 19 & 0 & 81 & 101\\
113 & 9 & \bfseries 111 & 2 & 55 & 177\\
114 & 15 & \bfseries 70 & 1 & 23 & 109\\
115 & 139 & 2 & 5 & 16 & 162\\
116 & 1 & \bfseries 170 & 0 & 7 & 178\\
117 & 30 & \bfseries 49 & 6 & 84 & 169\\
118 & 36 & 4 & 4 & 49 & 93\\
119 & 143 & 1 & 0 & 0 & 144\\
120 & 101 & 0 & 0 & 0 & 101\\
121 & 0 & 5 & 1 & 79 & 85\\
122 & 0 & \bfseries 14 & 2 & 85 & 101\\
123 & 235 & 2 & 1 & 0 & 238\\\bottomrule
\end{tabular}\\[.3ex]
Continued on next page.
\end{table}

\begin{table}[t]
Table E.1, continued.\\[.3ex]
\begin{tabular}{crrrrr}\toprule
\multicolumn{1}{l}{Case No.} & \multicolumn{1}{l}{$SS^\prime$} &
\multicolumn{1}{l}{$SR^\prime$} & \multicolumn{1}{l}{$RS^\prime$} &
  \multicolumn{1}{l}{$RR^\prime$} & \multicolumn{1}{l}{SUM}\\\midrule
124 & 11 & \bfseries 93 & 0 & 6 & 110\\
125 & 144 & \bfseries 82 & 0 & 3 & 229\\
126 & 36 & \bfseries 64 & 0 & 17 & 117\\
127 & 12 & \bfseries 93 & 0 & 39 & 144\\
128 & 3 & \bfseries 142 & 0 & 16 & 161\\
129 & 73 & \bfseries 59 & 0 & 2 & 134\\
130 & 34 & \bfseries 39 & 9 & 11 & 93\\
131 & 6 & \bfseries 85 & 2 & 8 & 101\\
132 & 22 & \bfseries 8 & 0 & 97 & 127\\
133 & 180 & \bfseries 14 & 1 & 0 & 195\\
134 & 145 & \bfseries 13 & 2 & 1 & 161\\
135 & 80 & \bfseries 40 & 16 & 8 & 144\\
136 & 20 & \bfseries 36 & 10 & 27 & 93\\
137 & 2 & \bfseries 19 & 0 & 183 & 204\\
138 & 95 & 2 & 2 & 2 & 101\\
139 & 1 & \bfseries 109 & 0 & 0 & 110\\
140 & 0 & 0 & 0 & 246 & 246\\
141 & 0 & 3 & 0 & 201 & 204\\
142 & 27 & \bfseries 91 & 0 & 0 & 118\\
143 & 2 & \bfseries 42 & 0 & 58 & 102\\
201 & 119 & \bfseries 57 & 2 & 9 & 187\\
202 & 0 & \bfseries 44 & 1 & 210 & 255\\
203 & 76 & 11 & 10 & 22 & 119\\
204 & 5 & \bfseries 240 & 0 & 27 & 272\\
205 & 88 & \bfseries 11 & 1 & 2 & 102\\
206 & 3 & \bfseries 24 & 1 & 125 & 153\\
207 & 108 & \bfseries 20 & 1 & 24 & 153\\
208 & 131 & 5 & 0 & 0 & 136\\
209 & 2 & \bfseries 103 & 0 & 116 & 221\\
210 & 0 & \bfseries 23 & 0 & 11 & 34\\
211 & 0 & \bfseries 9 & 2 & 142 & 153\\
212 & 114 & 7 & 4 & 28 & 153\\
213 & 2 & \bfseries 88 & 3 & 94 & 187\\
214 & 29 & \bfseries 11 & 3 & 42 & 85\\
215 & 0 & \bfseries 14 & 1 & 257 & 272\\
216 & 2 & \bfseries 270 & 0 & 85 & 357\\
217 & 2 & \bfseries 21 & 0 & 62 & 85\\
218 & 8 & 5 & 3 & 103 & 119\\
219 & 198 & 4 & 2 & 0 & 204\\
220 & 15 & \bfseries 119 & 0 & 2 & 136\\\bottomrule
\end{tabular}
\end{table}

\begin{table}[t]
Table E.1, continued.\\[.3ex]
\begin{tabular}{crrrrr}\toprule
\multicolumn{1}{l}{Case No.} & \multicolumn{1}{l}{$SS^\prime$} &
\multicolumn{1}{l}{$SR^\prime$} & \multicolumn{1}{l}{$RS^\prime$} &
  \multicolumn{1}{l}{$RR^\prime$} & \multicolumn{1}{l}{SUM}\\\midrule
220 & 15 & \bfseries 119 & 0 & 2 & 136\\
221 & 3 & \bfseries 104 & 0 & 29 & 136\\
222 & 21 & \bfseries 127 & 3 & 18 & 169\\
223 & 95 & \bfseries 39 & 5 & 14 & 153\\
224 & 141 & \bfseries 102 & 1 & 11 & 255\\
225 & 32 & \bfseries 70 & 14 & 122 & 238\\
226 & 198 & \bfseries 18 & 5 & 0 & 221\\
227 & 27 & \bfseries 86 & 26 & 116 & 255\\
228 & 93 & \bfseries 7 & 1 & 1 & 102\\
229 & 169 & 1 & 0 & 0 & 170\\
230 & 103 & 8 & 7 & 1 & 119\\
231 & 2 & \bfseries 84 & 0 & 50 & 136\\
232 & 391 & 0 & 0 & 0 & 391\\
233 & 61 & \bfseries 120 & 0 & 6 & 187\\
234 & 58 & \bfseries 185 & 10 & 70 & 323\\
235 & 1 & \bfseries 16 & 1 & 50 & 68\\
236 & 14 & \bfseries 138 & 1 & 0 & 153\\
237 & 116 & \bfseries 127 & 8 & 38 & 289\\
238 & 51 & \bfseries 29 & 0 & 5 & 85\\
239 & 9 & \bfseries 61 & 3 & 63 & 136\\
240 & 40 & \bfseries 190 & 0 & 25 & 255\\
241 & 66 & 1 & \bfseries 149 & 39 & 255\\
242 & 0 & 0 & 4 & 115 & 119\\
243 & 3 & \bfseries 154 & 1 & 63 & 221\\
244 & 0 & 0 & 0 & 204 & 204\\
245 & 7 & 9 & 4 & 184 & 204\\
246 & 12 & 15 & 12 & 12 & 51\\
247 & 83 & \bfseries 129 & 3 & 6 & 221\\
248 & 11 & \bfseries 29 & 7 & 259 & 306\\
249 & 181 & \bfseries 6 & 0 & 0 & 187\\
250 & 20 & \bfseries 28 & 0 & 3 & 51\\
251 & 82 & \bfseries 97 & 28 & 48 & 255\\
252 & 0 & 0 & 1 & 118 & 119\\
253 & 29 & \bfseries 16 & 7 & 84 & 136\\
254 & 24 & \bfseries 27 & 1 & 16 & 68\\
255 & 2 & \bfseries 20 & 4 & 127 & 153\\
256 & 0 & 3 & 3 & 266 & 272\\
257 & 0 & \bfseries 11 & 2 & 344 & 357\\
258 & 19 & \bfseries 46 & 2 & 1 & 68\\
259 & 68 & 6 & 3 & 8 & 85\\\bottomrule
\end{tabular}
\end{table}
\end{document}
