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\title{The opportunity-threat theory of decision-making under risk}

\author{Mohan Pandey\thanks{Bristol-Myers Squibb, E3140, Route 206 \& Province Line Road, 
Princeton, NJ 08543, USA. Email: mohan.pandey@bms.com.}
}

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\newcommand{\jhead}{Judgment and Decision Making, Vol.~13, No.~1}
\newcommand{\jdate}{January 2018}
\pagestyle{myheadings} \markright{\protect\small \href{\jref}{\jhead}, \jdate \hfill The opportunity-threat theory \qquad}
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\begin{onecolabstract}
%\end{latexonly}
A new theory of decision-making under risk, the Opportunity-Threat 
Theory is proposed. Analysis of risk into opportunity and threat 
components allows description of behavior as a combination of 
opportunity seeking and threat aversion. Expected utility is a special 
case of this model. The final evaluation is an integration of the 
impacts of opportunity and threat with this expectation. The model can 
account for basic results as well as several ``new paradoxes'' that 
refuted cumulative prospect theory in favor of configural weight models. 
The discussion notes similarities and differences of this model to the 
configural weight TAX model, which can also account for the new paradoxes. 

% that -> which

\smallskip
\noindent
Keywords: decision, risk, opportunity, threat, expected utility, 
prospect theory, transfer of attention exchange, behavior
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{\renewcommand{\thefootnote}{}

\footnotetext{

I thank Bruce Car for reviewing an earlier version of this paper. I 
greatly benefitted from extensive feedback on subsequent versions from 
Konstantinos Katsikopoulis and Jonathan Baron. I am indebted to Michael 
Birnbaum for his significant guidance that strongly influenced this 
work. Support from Bristol-Myers Squibb is gratefully acknowledged. Most 
of the work was conducted during my tenure at the Biocon BMS R\&D Center 
in Bangalore, India.

Copyright: \copyright\ 2018.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

\section{Introduction}

Expected Utility Theory (EUT) (Von Neumann \& Morgenstern, 1944) is
the most widely accepted normative theory of decision-making under
risk. However, as demonstrated by the Allais Paradox (Allais, 1953),
EUT does not accurately describe how people decide when presented
choices between risky prospects. Many theories have been proposed to
account for the Allais paradoxes. Two classes of models that have been
the focus of recent experimental work are original and cumulative
prospect theory (Kahneman \& Tversky, 1979; Tversky \& Kahneman, 1992)
and configural weight models (Birnbaum, 1974; Birnbaum \& Stegner,
1979), including the Transfer of Attention Exchange (TAX) model of
Birnbaum \& Chavez (1997). This paper proposes an alternative, called
the Opportunity-Threat Theory (OTT) and shows how a simplified special
model of OTT (SSOT) can explain the classic fourfold pattern of risk
attitude as well as key features of the ``new paradoxes'' (Birnbaum,
2008) that refute the prospect models in favor of the configural
weight models. In particular, the new model can account for
event-splitting effect, violation of stochastic dominance and
violation of restricted-branch independence.

The intuitions behind OTT are rather simple. People are influenced not 
only by the expected results of their actions, but also are affected by 
two components of risk, opportunity and threat. A model in which 
expectation, opportunity, and threat aggregate to form the evaluation of 
a risky prospect will be presented first in a simple form, to show that 
it can account for empirical phenomena. Comparisons with alternative 
approaches and ways in which the simple model might be generalized will 
be taken up in the discussion.

\section{A Special OTT (SOT) Model of Risky Decision Making}

Let M refer to a risky gamble of the form\textbf{ }$M= 
(x_{1},p_{1};x_{2},p_{2}\ldots x_{i},p_{i}\ldots x_{n},p_{n})$, which 
represents a lottery in which there are exactly \textit{n} possible 
mutually exclusive and exhaustive consequences. Consequence $x_{i}$ 
occurs with probability $p_{i}$ and the sum of the probabilities is 1. 
The expected value of such a gamble, $EV$, is given by Equation 1:
\[ EV= \sum_{i=1}^{n}{p_{i}x_{i}} \tag{1} \]

Expected utility theory (EUT) allows that utilities (subjective values) 
of the consequences may be a nonlinear function of the monetary values 
of the consequences, $u_{i}=u(x_{i})$. The expected utility of the 
gamble is $\mu $:
\[\mu = \sum_{i=1}^{n}{p_{i}u_{i}} \tag{2.1} \]

Under SOT, $\mu =EU$ can be viewed as a \textit{reference}. With at 
least two unequal consequences, there will be at least one that will be 
preferred to $\mu $, and at least one over which $\mu $ will be 
preferred. The present model adds two risk components to expected 
utility; the first of these components is a risk factor due to asymmetry 
and the second is a risk factor due to variation. 

It is useful to define the average utility of the consequences in a 
gamble, $\bar{u}$\, computed as if each consequence is equally 
likely:
\[\bar{u}= \frac{\sum_{i=1}^{n}{u_{i}}}{n} \tag{2.2} \]

When there are two or more consequences, the difference between the 
average utility of the consequences and the expected utility of 
consequences is used to define the asymmetry component of the model, $
\theta $, as follows:
\[\theta =\frac{\bar{u}-\mu }{(n-1)/n} \tag{2.3} \]

A negative deviation is perceived as \textit{threat} (the possibility
of landing below the
\textit{reference}). A positive deviation is perceived as
\textit{opportunity} (the possibility of landing above
the \textit{reference}).

The variation component of the model, $\psi $, is defined as follows:
\[\psi = \lbrace \sigma ^{2}+ (\alpha \theta )^{2}\rbrace ^{1/2} \tag{2.4} \]
where $\alpha $ is a parameter representing the weight of the $\theta$
component and $\sigma $ is the standard deviation, 
\[\sigma = \left\{ \sum_{i=1}^{n}{p_{i}(u_{i}-\mu )^{2}} \right\}^{1/2} \tag{2.5} \]

It has been found that people are typically risk averse for positive 
valued gambles and typically risk seeking for negative valued gambles; 
therefore, to account for this empirical finding, a multiplier of $\psi 
$, \textit{b}, reflecting the sign of $\mu $ is used, as follows: 
\[b= +1, if \mu <0, else-1 \tag{2.6} \]

When all the $u$ values are non-negative ($\mu \ge 0)$, spread is 
perceived as \textit{threat} (the possibility of landing below the 
\textit{reference} ); $b=-1$. On the other hand, when all the 
\textit{u} values are negative ($\mu <0)$, spread is perceived as 
\textit{opportunity} (the possibility of landing above the \textit{
reference} ): $b=+1$. For mixed cases with both positive and 
negative outcomes, sign of $\mu $ determines the sign of \textit{b}. 

The overall evaluation of a gamble, $V$, for gambles with two or more 
possible consequences is a linear combination of all three components, 
as follows: 
\[ V= \mu + \alpha \theta + \beta b \psi + \varepsilon \tag{2.7} \] % removed dots
where coefficients $\alpha $ and $\beta $ represent psychological
weights assigned to $\theta $ and $\psi $ respectively; and
$ \varepsilon $ is an error term. For the case of $n=1$, $V=u$. The
SOT model is idempotent, reducing to $V=u$, when all outcomes are
equal. When given a choice between two gambles, the decision maker
chooses the option with the higher evaluation, $V, $ apart from error.

Equation 2.7 shows that SOT reduces to EU when $\alpha $ = $\beta $ = 
0, or when $\alpha \theta + \beta b \psi =0$. When $\theta $ = 0, $
V= \mu + \beta b \sigma + \varepsilon $, which is a special case of 
the TAX model when \textit{n} = 2.\footnote{The Appendix shows the
  derivation of $\theta$ and $\psi$.}

\subsection{Simplified special opportunity-threat (SSOT) model}

For some situations, it may be possible to use a simplified version of
SOT. Consider cases where outcomes are monetary and within a
relatively narrow range allowing $u(x)=x$. Further, assume that values
of $p$ are non-extreme allowing the approximation, $\psi $ =
$\sigma $.\footnote{For simple binary gambles $(x,p;0,1-p)$, with
  $x>0$ and typical $ \alpha =\frac{1}{8}$,
  $(\frac{\alpha \theta }{\sigma })^{2}=\frac{(1-2p)^{2}}{64p(1-p)}$
  has its minimum value at $ p=\frac{1}{2}$, where it equals 0. The
  expression $(\frac{\alpha \theta }{\sigma })^{2}$ increases in value
  as $p$ moves towards 0 or 1. Even at $p=0.05 (or, 0.95)$,
  $(\frac{\alpha \theta }{\sigma })^{2}=0.26$, only.}  For simplicity,
it will be assumed that there are no errors. Equation 3 represents the
simplified SOT (SSOT) model
\[V= \mu + \alpha \theta + \beta b\sigma \tag{3} \]

For simplification, it is assumed that coefficients $\alpha $ and $
\beta $ do not change due to change in domains (positive to negative or 
vice versa). Examples in this paper use this SSOT model unless otherwise 
mentioned. It is noted that SSOT model reduces to Expected Value when $
\alpha $ = $\beta $ = 0, or when $\alpha \theta + \beta b\sigma =0
$.

\subsection{Constraints and coefficients}

\renewcommand{\baselinestretch}{1.1} \large\normalsize

Consider the case of binary gambles $(x,p;0,1-p)$, with $ x>0$; 
here, $n=2 $ and $ \mu =p x$. From Equation (2.3), $\theta 
=\frac{\frac{x}{2}-px}{\frac{2-1}{2}}=(1-2p)x$ and from Equation (2.5), 
$\sigma ^{2}=p (x-xp)^{2}+(1-p) (0-xp)^{2}=\lbrace p(1-p)\rbrace x^{2}
$. Since $\mu $ is non-negative, $b= -1$. Therefore,
\[V=px+ \alpha (1-2p)x-\beta \lbrace p(1-p)\rbrace ^{1/2}x \tag{4.1} \]
transforming to:
\[ \frac{V}{x}=p+ \alpha (1-2p)-\beta \lbrace p(1-p)\rbrace ^{1/2} \tag{4.2} \]

%Table 1
\begin{table*}[t]\centering
 \caption{Fourfold pattern shown with experimental dataset for 
gambles of type $(x,p;0,1-p)$ from Tversky \& Kahneman (1992). SSOT, 
CPT and TAX, all three are able to explain the fourfold pattern. 
However, SSOT analyzes risk into \textit{opportunity} and \textit{
threat} components.}
 \begin{tabular}{cC{1in}R{.3in}R{.3in}R{.2in}R{.4in}R{.4in}R{.4in}}\toprule
 Gamble & Observed & \mc{3}{c}{SSOT components} & \mc{3}{c}{Calculated cash equivalents (V)} \\
 \cmidrule(lr){3-5}\cmidrule(lr){6-8}
 x,p & cash equivalent  & \mc{1}{C{.6in}}{$\mu$} & \mc{1}{C{.4in}}{$\alpha\theta$} & 
 \mc{1}{C{.4in}}{$b\beta\sigma$} & \mc{1}{C{.7in}}{Prior SSOT} & \mc{1}{C{.7in}}{Prior CPT} & \mc{1}{C{.7in}}{Prior TAX} \\\midrule
 $-$\$100,0.95 & $-$84 & $-$95 & 11 & 7 & $-$77 & $-$83 & $-$59 \\
 $-$\$100,0.05 & {\ } $-$8 & $-$5 & $-$11 & 7 & $-$9 & $-$8 & $-$8 \\
 \$100,0.95 & {\ } 78 & 95 & $-$11 & $-$7 & 77 & 77 & 59 \\
 \$100,0.05 & {\ } 14 & 5 & 11 & $-$7 & 9 & 10 & 8 \\\bottomrule
\mc{8}{p{5.8in}}{Note: Data presented only to demonstrate the pattern and not to show 
accuracy of prediction. OTT does not preclude the coefficients from 
taking different values for gains and losses domain. However, it is not 
necessary to explain the pattern here. Observed and calculated cash 
equivalents are in \$, in this table as well as the 
subsequent tables.}
\end{tabular}
\end{table*}

Now, constraints $0<\frac{V}{x}<1$ and $0<p<1$ are applied. They set
the boundary conditions such that even for the smallest probability of
smallest positive value of $x$, $V$ does not reduce to zero.  Further,
even for the smallest probability of $x$ not obtaining, magnitude of
$V$ remains positive under certain $x$. Then, at $ p=\frac{1}{2}$, from
Equation 4.2, $ \frac{V}{x}=\frac{1}{2}-\frac{\beta }{2} $. Thus,
$ 0<\frac{1}{2}-\frac{\beta }{2}<1 $, which implies $-1<\beta <1$.
Also, for $p\sim 0$, $\frac{V}{x} \approx \alpha $, thus
$0<\alpha <1$.

For estimation of $\beta $, consider a mixed outcome experiment with 
only two possible outcomes with equal probabilities ($
x_{1},\frac{1}{2};-x_{2},\frac{1}{2}$) and observed $V=0$. Given 
symmetric distribution, $\theta =0$, giving, $V=0= \mu - \beta 
\sigma $, or, $\beta = \mu /\sigma $. Now, $ \mu 
=\frac{1}{2}x_{1}-\frac{1}{2}x_{2}$. Also, $\sigma 
^{2}=\frac{1}{2}(x_{1}-\mu )^{2}+ \frac{1}{2} (-x_{2}-\mu 
)^{2}=\frac{1}{2}(x_{1}-\frac{1}{2}x_{1}+\frac{1}{2}x_{2})^{2}+ 
\frac{1}{2} (-x_{2}-\frac{1}{2}x_{1}+\frac{1}{2}x_{2})^{2}$. That 
yields, $\sigma =\frac{1}{2}x_{1}+\frac{1}{2}x_{2}$. Thus, $ \beta = 
\mu /\sigma = \frac{x_{1}-x_{2}}{x_{1}+x_{2}}$. Tversky \& Kahneman 
(1992) reported four problems of this kind that yield $\beta =0.42, 
0.34, 0.34, 0.30$. An average is taken and converted to an equivalent 
fraction for convenience, giving $\beta =\frac{1}{3}$, which is in 
the range established above. 

\renewcommand{\baselinestretch}{1.00} \large\normalsize

Now, with an estimate of $\beta $ at hand, $\alpha $ can be estimated 
as follows. Assume that there exists a point where, $\frac{V}{x}=p$. 
Then, from Equation 4, $0= \alpha (1-2p)-\beta \lbrace p(1-p)\rbrace 
^{1/2}$ yielding, $\alpha =\frac{\beta \lbrace p(1-p)\rbrace 
^{1/2}}{1-2p}$. Tversky \& Kahneman (1992) considered $p\le 0.1$ as 
low. Following that, $p=0.1$ is taken as the point of transition from 
low probability to moderate probability. Data from the same study shows 
that $\frac{V}{x}\sim p$ at $=0.1$ . At that point, with $\beta
=\frac{1}{3}$ , $\alpha =\frac{1}{8}$ is obtained, which is in the 
range established above.

It must be recognized that $\alpha $ and $\beta $ represent a 
psychological weighting process and as such are likely to vary with 
individual differences and experimental factors. The rest of this paper uses 
$\alpha =\frac{1}{8}$ and $\beta =\frac{1}{3}$ as ``prior'' 
parameters for purpose of calculations to illustrate how the SSOT model 
functions. These rough parameters are not intended to be used for 
comparison of accuracy or predictive power of various models.

\section{Results}

It has been well argued in Birnbaum (2008) as to how the so-called 
``new paradoxes'' refute Cumulative Prospect Theory, Rank-Dependent 
Utility, and Rank-and Sign-Dependent Utility Theories in favor of a 
class of models that includes the transfer of attention exchange (TAX) 
model. Here, it is examined if SSOT is also capable of explaining key 
drivers of these new paradoxes, viz., event-splitting, stochastic 
dominance and restricted branch independence. It will be shown first 
that SSOT can reproduce some basic behavioral observations in 
decision-making under risk, including the ``fourfold pattern''.

%Table 2
\begin{table*}[t]\centering
 \caption{Event-splitting problems 1.1 (row 1) and 1.2 (row 2) 
from Birnbaum (2008). Prior CPT does not, but prior SSOT and prior TAX 
predict preference reversal due to event-splitting of gambles in 1.2.}
 \begin{tabular}{c@{\quad}cccccccc}\toprule
 & & & \mc{6}{c}{Calculated cash equivalents} \\\cmidrule{4-9}
 \mc{2}{c}{Choice} & \% Choosing & \mc{2}{c}{Prior SSOT} & \mc{2}{c}{Prior TAX} & \mc{2}{c}{Prior CPT} \\\cmidrule(lr){1-2}
 First gamble & Second gamble & second gamble & First & Second & First & Second & First & Second \\\midrule
 \begin{tabular}[c]{@{}r@{}}A: 85 to win \$100\\ 10 to win \$50\\ 05 to win \$50\end{tabular} & \begin{tabular}[c]{@{}r@{}}B: 85 to win \$100\\ 10 to win \$100\\ 05 to win \$7\end{tabular} & 62 & 81.7 & 83.7 & 68.4 & 69.7 & 82.2 & 79.0 \\\midrule
 \begin{tabular}[c]{@{}r@{}}A': 85 to win \$100\\ 15 to win \$50\end{tabular} & \begin{tabular}[c]{@{}r@{}}B': 95 to win \$100\\ 05 to win \$7\end{tabular} & 26 & 82.2 & 78.1 & 75.7 & 62.0 & 82.2 & 79.0 \\\bottomrule
 \end{tabular}
\end{table*}

\subsection{Fourfold pattern}

Tversky \& Kahneman (1992) described a ``fourfold pattern'': risk 
aversion for gains and risk seeking for losses of high probability; risk 
seeking for gains and risk aversion for losses of low probability. Table 
1 shows selected results illustrating this fourfold pattern and shows 
how SSOT can account for them. 

Consider $(100,5\%;0,95\%)$. (i) The \textit{reference} $\mu 
=100\ast5\%=5$. (ii) $\theta =(1-2\ast5\%)\ast100=90$. It is multiplied with 
coefficient $\alpha =\frac{1}{8}$. Thus, the impact $\alpha \theta 
=\frac{1}{8}\ast90\approx 11$. This is positive and is taken as \textit{
opportunity}. (iii) The standard deviation, $\sigma =\lbrace 
5\%\ast95\%\rbrace ^{\frac{1}{2}}100=22$. Since the gamble is in gains 
domain, $b=-1$. Thus, $b\sigma =-22$. With coefficient $\beta 
=\frac{1}{3}$ we have $\beta b\sigma =\frac{1}{3}(-22)\approx -7$. 
This is negative and is taken as \textit{threat}. The final \textit{
value}, $V=5+11-7=9>5$(expected value). This is in line with the 
reported relationship. 

As $p$ increases, $\theta $ decreases, crossing 0 when $
p=\frac{1}{2}$, and turning negative after that. In the case of $
(100,95\%;0,5\%)$, $\mu =95$. $\theta =(1-2\ast95\%)\ast100=-90$, 
leading to negative impact of $\alpha \theta =\frac{1}{8}\ast(-90)\approx -11
$. Standard deviation and $b$ do not change so $
V=95-11-7=77<95$(expected value). 

\renewcommand{\baselinestretch}{1.1} \large\normalsize

For $(-100,5\%;0,95\%)$, $\mu =-5$. $\theta =(1-2\ast5\%)(-100)=-90
$, leading to negative impact of $\alpha \theta =\frac{1}{8}\ast(-90)\approx 
-11$\textit{. } Due to domain change, $b= +1$, hence, $\beta 
b\sigma =\frac{1}{3}\ast(+1)\ast22\approx 7$,. Therefore, $
V=-5-11+7=-9<-5$(expected value). 

\pagebreak

For $(-100,95\%;0,5\%)$, we have, $\mu =-95$, $\theta 
=(1-2\ast95\%)(-100)=90$, $\alpha \theta =\frac{1}{8}\ast90\approx 11$
and $b= +1$, hence, $\beta b\sigma 
=\frac{1}{3}\ast(+1)\ast22\approx 7$. Therefore, $V=-95+11+7=-77>-95$(expected 
value). Thus, the model reproduces the fourfold pattern of Table 1.

\renewcommand{\baselinestretch}{1.00} \large\normalsize

More generally, consider gambles of type $(x,p;0,1-p)$. In the gains 
domain, spread is perceived as \textit{threat}. $ \theta $, which 
equals $(1-2p)x$, is perceived as an \textit{opportunity} for $p 
\le \frac{1}{2}$. For any given $x$, this factor becomes stronger 
as $p$ reduces. At low $p$, it overrides the \textit{threat} 
factor when $\alpha (1-2p)>\beta \lbrace p(1-p)\rbrace ^{1/2}$. In 
the losses domain, spread is perceived as \textit{opportunity}. $
\theta $, which equals $(1-2p)(-x)$, is perceived as \textit{threat
} for $p \le \frac{1}{2}$. Thus, for low $ p$, it is \textit{
threat} aversion and otherwise it is \textit{opportunity} seeking. In 
this model, decision-makers can be simultaneously \textit{opportunity} 
seeking and \textit{threat }averse.

%Table 3 
\begin{table*}[t] \centering
 \caption{Violation of stochastic dominance in Birnbaum (2008) 
problem 3.1.Values $\alpha (\theta _{+}- \theta )\approx -\alpha 
\frac{(1-p)x}{2}=-0.6$ and $\alpha (\theta -\theta _{-})\approx 
-\alpha \frac{px}{2}=-5.4$ predict slight reduction in value moving 
from $G_{0} $ to $G_{+} $ and relatively higher increase in value 
moving from $G_{0} $ to $G_{-}$. Prior CPT is not, but prior SSOT 
and prior TAX are consistent with the observed data.}
 \begin{tabular}{c@{\quad}cccccccc}\toprule
 & & & \mc{6}{c}{Calculated cash equivalents} \\\cmidrule{4-9}
 \mc{2}{c}{Choice} & \% Choosing & \mc{2}{c}{Prior SSOT} & \mc{2}{c}{Prior TAX} & \mc{2}{c}{Prior CPT} \\\cmidrule(lr){1-2}
 First gamble & Second gamble & second gamble & First & Second & First & Second & First & Second \\\midrule
 \begin{tabular}[c]{@{}r@{}}$G_{+} $: 90 to win \$96 \\ 05 to win \$14\\ 05 to win \$12\end{tabular} & \begin{tabular}[c]{@{}r@{}}$G_{-}$: 85 to win \$96\\ 05 to win \$90\\ 10 to win \$12\end{tabular} & 73 & 70.6 & 74.9 & 45.8 & 63.1 & 70.3 & 69.7 \\\midrule
 \begin{tabular}[c]{@{}r@{}}$G_{0} $ : 90 to win \$96\\ 10 to win \$12\end{tabular} & $-$ & $-$ & 70.8 & $-$ & 58.1 & $-$ & 70.1 & $-$ \\\bottomrule
 \end{tabular}
\end{table*}

\subsection{Event-splitting effect}

\renewcommand{\baselinestretch}{1.1} \large\normalsize

A simple case of event splitting: $A (x,p;0,1-p)$ split to
$B (x,p-r;x,r;0,1-p)$. Obviously, there is no difference in $\mu$,
since, $ (p-r)x+rx=px$ . There is no difference in $\sigma $ either as
$p(x-px)^{2}+(1-p)(0-px)^{2}=(p-r)(x-px)^{2}+r(x-px)^{2}+(1-p)(0-px)^{2}
$.\footnote{It is important to note that none of traditional measures
  of higher moments will change either, since for a moment of order
  $m$,
  $p(x-px)^{m}+(1-p)(0-px)^{m}=(p-r)(x-px)^{m}+r(x-px)^{m}+(1-p)(0-px)^{m}$. Thus,
  no moments-only model (for example, mean-variance or
  mean-variance-skewness models) will be able to explain change in
  value due to event-splitting.} However, there is change in
$\theta $. $\theta _{A}=\frac{\frac{x}{2}-px}{\frac{2-1}{2}}=(1-2p)x$
and $\theta _{B}=\frac{\frac{2x}{3}-px}{\frac{3-1}{3}}=(1-1.5p)x$.

Take any gamble $G_{base} (x_{1},p_{1};x_{2},p_{2}\ldots 
x_{i},p_{i}\ldots x_{n},p_{n})$ with $x_{i}>0$, pick its element $k
$ , $(x_{k},p_{k})$ and split it to generate elements $
(x_{k},p_{k}-r)$ and $(x_{k},r)$ for another gamble $G_{split}$. 
Then, from Equation 2.3, $\theta 
_{base}=\frac{n}{n-1}(\sum{\frac{x_{i}}{n}- \mu )}$ and $\theta 
_{split}=\frac{n+1}{n}(\sum{\frac{x_{i}}{n+1}+\frac{x_{k}}{n+1}- \mu )}
$. Thus, $\theta _{split}-\theta _{base}= 
\frac{x_{k}}{n}-\frac{\sum{x_{i}}- \mu }{n(n-1)}$. Therefore, $
V_{split}>V_{base}$ if,$ x_{k}>\frac{\sum{x_{i}}- \mu }{(n-1)}$. 

For a binary gamble $(x,p;0,1-p)$, $x>0$, splitting of higher 
branch satisfies this condition ($x>\frac{x- px}{(2-1)}$) and leads 
to increase in value. Splitting of lower branch ($0<\frac{x- 
px}{(2-1)}$) leads to decrease in value. Because splitting in this 
model can either increase or decrease the value of a gamble, this model 
violates the property of branch-splitting independence identified and 
tested by Birnbaum (2007). 

An example of event-splitting non-independence from Birnbaum (2008) is
illustrated in Table 2. 1.1A is produced by splitting lower branch of
1.1A' (causing small decrease in value) and 1.1B is produced by
splitting higher branch of 1.1B'(causing significant increase in
value). The majority prefers A to B, and a majority prefers B' to A'.

\subsection{Violation of stochastic dominance}

That SSOT predicts violation of stochastic dominance is demonstrated in 
this section through a recipe simplified from Birnbaum (2008). Take a 
base gamble $G_{0}(x,p;0,1-p)$, with $x>0$. Now, modify it to 
generate a stochastically dominating gamble $G_{+}(x,p;y,q;0,1-p-q)$ 
where $y$ is a small positive quantity and $q$ is a relatively small 
probability. Next, generate a stochastically dominated gamble $
G_{-}(x,p-s;z,s;0,1-p)$ where $z$ is a positive quantity slightly 
lower in value to $x$ and $s$ is a relatively small probability. 

\renewcommand{\baselinestretch}{1.2} \large\normalsize

From Equations 2, for $G_{0}(x,p;0,1-p)$:\\
$\mu _{0}=px$, $\theta 
_{0}=\frac{x/2-px}{(2-1)/2}=x-2px=(1-2p)x$, and\\
$\sigma _{0}=p(x-px)^{2}+(1-p)(0-px)^{2}=p(1-p)x^{2}$. 

For $G_{+}(x,p;y,q;0,1-p-q)$:\\
$\mu _{+}=px+qy$, $\theta 
_{+}=\frac{\frac{x+y}{3}-(px+qy)}{\frac{3-1}{3}}=\frac{x+y}{2}-(\frac{3}{2})(px+qy)=\frac{(1-3p)}{2}x+\frac{(1-3q)}{2}y
$, and\\
$\sigma _{+}=p(x-\mu _{+})^{2}+q(y-\mu _{+})^{2}+(1-p-q)(0-\mu 
_{+})^{2}=px^{2}+qy^{2}-\mu _{+}^{2}=p(1-p)x^{2}+q(1-q)y^{2}-2pqxy$. 

%\newpage

For $G_{-}(x,p-s;z,s;0,1-p)$, substituting $r=p-s$ (replaced back 
in the last step):\\
$\mu _{-}=rx+sz$, $\theta 
_{-}=\frac{\frac{x+z}{3}-(rx+sz)}{\frac{3-1}{3}}=\frac{x+z}{2}-(\frac{3}{2})(rx+sz)=\frac{(1-3r)}{2}x+\frac{(1-3s)}{2}z$ and $\sigma _{-}=r(x-\mu _{-})^{2}+s(z-\mu _{-})^{2}+(1-p)(0-\mu 
_{-})^{2}=rx^{2}+sz^{2}-\mu 
_{-}^{2}=r(1-r)x^{2}+s(1-s)z^{2}-2rsxz=p(1-p)x^{2}+(2sp-s-s^{2})x^{2}+ 
s(1-s)z^{2}-2(p-s)sxz$.

The impact on mean is straightforward: $\mu _{+}- \mu =qy\approx 0 $
and, $\mu -\mu _{-}=s(x-z)\approx 0$. The difference in $\sigma $ is 
as follows: $\sigma _{+}^{2}- \sigma ^{2}= q(1-q)y^{2}-2pqxy= qy 
((1-q)y-2px)\approx 0, $since, $qy\approx 0$ and $\sigma ^{2}-\sigma 
_{-}^{2}=-(2sp-s-s^{2})x^{2}- s(1-s)z^{2}+2(p-s)sxz 
=s(-2px^{2}+x^{2}+sx^{2}- 
(1-s)z^{2}+2(p-s)xz)=s(x^{2}-z^{2}+sx^{2}+sz^{2}-2sxz-2px^{2} +2pxz)= 
s(x-z)(x+z+s(x-z)-2px) \approx 0$, since, $s(x-z)\approx 0. $

The differences in $\theta $, which are decisive, can be calculated: $
\theta _{+}- \theta =\frac{(1-3p)}{2}x+\frac{(1-3q)}{2}y-(1-2p)x= 
\frac{-1+p}{2}x+\frac{(1-3q)}{2}y\approx -\frac{(1-p)x}{2}$ and $
\theta -\theta _{-}=(1-2p)x-\lbrace 
\frac{(1-3r)}{2}x+\frac{(1-3s)}{2}z\rbrace 
=\frac{(-p)}{2}x+\frac{(1-3s)}{2}(x-z)\approx -\frac{px}{2}$. Negative 
signs in both these expressions imply violation of stochastic dominance. 
In summary, similar to event-splitting effect, violation of stochastic 
dominance in cases such as the one described here is driven by $\theta 
$. $G_{0} $ to $G_{+}$ is splitting of the lower branch leading to 
reduction in value; while $G_{0}$ to $G_{-}$ is splitting of the 
higher branch leading to increase in value.

For illustration, calculations for one experimental set from Birnbaum
(2008) are shown in Table 3. It is reported that a majority (73\%) of 
subjects preferred G$_{-}$ over G$_{+}$ despite expected values 
being 87.3 and 87.7 respectively. SSOT values these prospects at 74.9 
and 70.6, consistent with the majority preference, driven by change in $
\theta $ while deltas in $\mu $ and $\sigma $ are very small.

%Table 4
\begin{table*}[t]\centering
 \caption{Violation of restricted branch independence in 
Birnbaum (2008) problems 13.1 (row 1) and 13.2 (row 2). While, $R$ is 
preferred over $S$ in 13.1, the preference switches in 13.2. Prior CPT 
is not, but prior SSOT and prior TAX are consistent with the observed 
data.}
 \begin{tabular}{c@{\quad}cccccccc}\toprule
 & & & \mc{6}{c}{Calculated cash equivalents} \\\cmidrule{4-9}
 \mc{2}{c}{Choice} & \% Choosing & \mc{2}{c}{Prior SSOT} & \mc{2}{c}{Prior TAX} & \mc{2}{c}{Prior CPT} \\\cmidrule(lr){1-2}
 First gamble & Second gamble & second gamble & First & Second & First & Second & First & Second \\\midrule
 \begin{tabular}[c]{@{}r@{}}S: 25 to win \$44\\ 25 to win \$40\\ 50 to win \$5\end{tabular} & \begin{tabular}[c]{@{}r@{}}R: 25 to win \$98\\ 25 to win \$10\\ 50 to win \$5\end{tabular} & 40 & 18.5 & 17.8 & 20.0 & 19.2 & 19.8 & 29.8 \\\midrule
 \begin{tabular}[c]{@{}r@{}}S': 50 to win \$111\\ 25 to win \$44\\ 25 to win \$40\end{tabular} & \begin{tabular}[c]{@{}r@{}}R': 50 to win \$111\\ 25 to win \$98\\ 25 to win \$10\end{tabular} & 62 & 62.8 & 66.7 & 57.2 & 60.7 & 69.4 & 62.3 \\\bottomrule
 \end{tabular}
\end{table*}

\renewcommand{\baselinestretch}{1.05} \large\normalsize

\subsection{Violation of restricted branch independence }

Consider two gambles with the same number of branches and the same 
probability distribution, $S=(x_{1},p_{1};x_{2},p_{2}\ldots 
x_{i},p_{i}\ldots x_{n},p_{n})$ and $R=(y_{1},p_{1};y_{2},p_{2}\ldots 
y_{i},p_{i}\ldots y_{n},p_{n})$ having a common branch such that $
x_{n}=y_{n}=z$. Restricted branch independence assumes that a change 
in $z$ will not change the preference relationship between $S$ and $
R$. Suppose $S$ is preferred over $R$, then, under SSOT, $
V_{S}>V_{R}$. Then, if $\frac{\partial V_{s}}{\partial z}\ge 
\frac{\partial V_{R}}{\partial z}$, $V_{S}>V_{R}$ for all $z$. 
Otherwise, with increase in $z$, the gap in values will close and 
preference may get switched. A standard case is analyzed to understand 
how this derivative function works. For convenient tracking, label $
p_{n}=r$. Assume, $x_{i}\ge 0$, for all $i$, such that $\mu \ge 
0$ and $b= -1$. Also, introduce an additional constraint $0<\beta 
<1$ to model a typical spread-averse agent.

\renewcommand{\baselinestretch}{1.1} \large\normalsize

Differentiating Equation 3 w.r.t. $z$,
$\frac{\partial V}{\partial z}= \frac{\partial \mu }{\partial
  z}+\alpha \frac{\partial \theta }{\partial z}-\beta \frac{\partial
  \sigma }{\partial z}$. Now, $ \frac{\partial \mu }{\partial z}=r$ ,
$\frac{\partial \theta }{\partial z}=\frac{n}{n-1} (\frac{1}{n}-r)$
and
$ \frac{\partial \sigma }{\partial z}=\frac{1}{2\sigma }\frac{\partial
  \sigma ^{2}}{\partial z}=\frac{r(z-\mu )}{\sigma }$. Therefore,
$ \frac{\partial V}{\partial z}=r+\alpha \frac{n}{n-1}
(\frac{1}{n}-r)-\beta \frac{r(z-\mu )}{\sigma }$. Thus, if,
$ \frac{\partial V_{s}}{\partial z}\ge \frac{\partial V_{R}}{\partial
  z}$ , then
$-\beta \frac{r(z-\mu _{S})}{\sigma _{S}}\ge -\beta \frac{r(z-\mu
  _{R})}{\sigma _{R}}$. That is,
$\frac{(z-\mu _{S})}{\sigma _{S}}\le \frac{(z-\mu _{R})}{\sigma _{R}}$
or
$z\le \frac{\mu _{S}\sigma _{R}-\mu _{R}\sigma _{S}}{\sigma
  _{R}-\sigma _{S}} $. This is not guaranteed and will depend on all
the constituents of $S$ and $R$. Assume that for $z^\prime$ this condition
is not satisfied.  Then, the corresponding second gamble ($R^{\prime}$) will
be preferred over the corresponding first gamble ($S^{\prime}$), that is, a
violation of type $SR^{\prime}$ will be observed. Note that this condition
($z\le \frac{\mu _{S}\sigma _{R}-\mu _{R}\sigma _{S}}{\sigma
  _{R}-\sigma _{S}}$) is not dependent on $\theta $, $\alpha $ or
$\beta $ . Thus, the same pattern applies to the gambles with uniform
probabilities as well and also for any value of the coefficients and
the only type of violation predicted by SSOT is $SR^{\prime}$. In summary,
violation of restricted branch independence is driven by the change in
$\sigma $.

As an example from Birnbaum (2008, Table 4), for Problem 13.1, $R$ is 
preferred over $S$ and $(z=5)\le (\frac{\mu _{S}\sigma _{R}-\mu 
_{R}\sigma _{S}}{\sigma _{R}-\sigma _{S}}=18)$. Thus, this preference 
holds as long as $z\le \frac{\mu _{S}\sigma _{R}-\mu _{R}\sigma 
_{S}}{\sigma _{R}-\sigma _{S}}$ holds. However, for Problem 13.2, $
(z=111)>(\frac{\mu _{S}\sigma _{R}-\mu _{R}\sigma _{S}}{\sigma 
_{R}-\sigma _{S}}=49)$, and the preference reversal is predicted. 

\renewcommand{\baselinestretch}{1.00} \large\normalsize

\section{Discussion}

As the previous section shows, SSOT is capable of explaining several 
empirical phenomena explained by CPT and TAX both (fourfold pattern) or 
by TAX only (event-splitting effect, violations of stochastic dominance 
and violation of restricted branch independence). 

Event-splitting effect and violations of stochastic dominance in the 
examples described above are related and are explained by the $\theta $
component under SSOT. 

Violations of restricted branch independence are explained by the $
\sigma $ component. 

\subsection{The Opportunity-Threat Theory}

Let $S$ be a set of future states ($s\in S$) of the world of which
exactly one state will obtain. Assume that it is unknown to the
decision-maker as to which state will obtain. Further, assume that $S$
can be mapped to a consequence set $X$ through some function
$\phi (s)=(x,p)$ where the first term ($x$) denotes the objective
magnitude of the consequence and the second term ($p$) denotes the
objective probability of that consequence. These objective
probabilities sum to 1. In addition, assume that the outcome set $X$
can be mapped by the decision-maker to a mental set $M$ through some
functions $u(x)=u$ and $\pi (p)=\pi $ corresponding to subjective or
implied utilities and subjective or implied probabilities,
respectively.  These subjective or implied probabilities also sum to
1. Also, assume that the decision-maker is able to map this mental set
$M$ to a decision-making single number $V$ (called \textit{value})
such that $V(M)=V$.

Two key assumptions underlie this ($M$ to $V$) mapping process:
\begin{itemize}
\item First, the decision-maker has a referencing algorithm that
  generates \textit{reference} with which $M$ can be divided into two
  mutually exclusive and collectively exhaustive subsets $M_{up}$ (the
  \textit{upside} set, containing elements considered more or equally
  preferred compared to \textit{reference}) and $M_{dn}$ (the
  \textit{downside} set, containing elements considered less preferred
  compared to \textit{reference}).
\item Second, that there is available a netting algorithm that maps
  $ M_{up}$ and $M_{dn}$ into $Y_{up}$, the \textit{relative upside }
  set, containing elements of $M_{up}$ net of \textit{reference} and
  into $Y_{dn}$, the \textit{relative downside} set, containing
  elements of $M_{dn}$ net of \textit{reference}), respectively.
\end{itemize}
With these assumptions, the following definitions are stated:
\newcounter{numberedCntD}
\begin{enumerate}
\item $Y_{agg}=agg(Y_{up},Y_{dn})$, where $agg$ is a function that
  measures direction and impact of aggregation of $Y_{up}$ and
  $Y_{dn} $.
\item $Y_{dis}=dis(Y_{up},Y_{dn})$, where $dis$ is a function that
  measures direction and impact of distance between $Y_{up}$ and
  $ Y_{dn}. $
\item $opportunity=Y_{agg}>0$, $Y_{dis}>0$.
\item $threat=Y_{agg}<0$, $Y_{dis}<0$. 
\setcounter{numberedCntD}{\theenumi}
\end{enumerate}
Then, the Opportunity-Threat Theory (OTT) asserts that
\[V=f(reference, opportunity, threat) \tag{5} \] and that the
decision-maker will prefer or be indifferent to $M_{1}$ compared to
$M_{2}$ iff $V(M_{1})\ge V(M_{2})$, except by way of error.

To illustrate, assume a roll of dice that pays \$1, if rolled 1, \$2,
if rolled 2 etc. Thus, $M$ = (1,1/6; 2,1/6; 3,1/6; 4,1/6; 5,1/6;
6,1/6). Further, assume an obvious reference point that is the average
(3.5). Outcomes higher or equal to average are mapped to $M_{up}$ =
(4,1/6; 5,1/6; 6,1/6) and rest are mapped to $M_{dn}$ = (1,1/6; 2,1/6;
3,1/6). Now, assume a simple netting algorithm that is to subtract the
average from each of the outcomes. Thus, the relative sets are
constructed as $Y_{up}$ = (0.5,1/6; 1.5,1/6, 2.5,1/6) and $Y_{dn}$ =
($-$2.5,1/6, $-$1.5,1/6, $-$0.5,1/6). To complete the illustration,
assume $agg(.) = EV(Y_{dn})+EV(Y_{up})$ and
$dis(.) = EV(Y_{dn})-EV(Y_{up})$ (so that it accounts for spread
aversion). Then, these numbers are respectively, $Y_{agg} = 0$ (note that
it was a symmetric gamble) and $Y_{dis} = -1.5$. Thus, opportunity = 0 and
threat = $-$1.5. Assuming V(.) = reference + opportunity + threat, $V$ =
3.5 + 0 $-$ 1.5 = 2. Thus, this gamble will be valued at
\$2.00 instead of expected value of \$3.50.

In general, OTT applies to cases involving monetary outcomes or 
otherwise. It allows incorporation of different measures of \textit{
opportunity} and \textit{threat}. It also permits different 
attitudes to opportunity and threat. It does not require symmetry in 
attitudes in gains and losses domains. In moral situations, \textit{
opportunity} may lie in domain of morally correct and \textit{threat} 
may lie in domain of morally incorrect. In such cases attitudes may be 
modelled with binary parameters. 

Several parameters that can serve as \textit{reference} have been
discussed in the literature, for example, status quo (Thaler, 1980;
Samuelson \& Zeckhauser, 1988), omission (Baron \& Ritov, 1994) or
aspiration (van de Ven \& Diecidue, 2008). OTT allows incorporation of
different points of reference and even multiple points of
reference. In a special case, the point of reference happens to be
Expected Utility. In that case, if either the agent is neutral to both
\textit{opportunity} and \textit{threat} or net impact of
\textit{opportunity} and \textit{threat} cancel out, the model reduces
to Expected Utility.

\subsection{A structural comparison of SSOT, CPT and TAX}

To structurally compare SSOT with CPT and TAX, a working model of OTT
was developed. Consider,
$X= (x_{1},p_{1};x_{2},p_{2}\ldots x_{n},p_{n})$, with $n$ denoting
the number of exhaustive and mutually exclusive future states and
$\sum{p_{i}}=1$. Assume
$ X_{up}= (x_{1},p_{1};x_{2},p_{2}\ldots x_{i},p_{i}\ldots
x_{k},p_{k})$ and
$X_{dn}= (x_{k+1},p_{k+1};x_{k+2},p_{k+2}\ldots x_{j},p_{j}\ldots
x_{n},p_{n})$, such that the upside set has $k$ elements and the
downside set has $n-k$ elements. Define,
$V(X_{up})= \sum{(x}_{i}- \mu )$, $i=1 \ldots k$ and
$V(X_{dn})= \sum{(x}_{j}- \mu ), j=k+1 \ldots n $. Further, define
$V(O)=V(X_{up})+V(X_{dn}) $ analogous to $\theta $ and
$V(T)=\vert V(X_{up})-V(X_{dn})\vert $ analogous to $\sigma $ (using
range as a measure of spread instead of standard deviation).  Then,
analogous to Equation 2,
\[V= \mu + \alpha^{\prime} V(O)+ \beta^{\prime} b V(T) \tag{6.1} \]
where $\mu = \sum_{i=1}^{n}{p_{i}x_{i}}$ and $\alpha^{\prime}$ and $
\beta^{\prime}$ are coefficients for respective terms.

Now, assuming non-negative domain ($b=-1$) and for simplicity $
V(X_{O})-V(X_{T})>0$, expansion of all the terms in Equation 5 yields,
\[V=\sum_{i=1}^{k}{w_{i}x_{i}} +\sum_{j=k+1}^{n}{w_{j}x_{j}}
  \tag{6.2} \] where
$w_{i}=(\alpha^{\prime}-\beta^{\prime})+\lbrace
1-(\alpha^{\prime}-\beta^{\prime})k-(\alpha^{\prime}+\beta^{\prime})(n-k)\rbrace
p_{i}$ and
$w_{j}=(\alpha ^{\prime}+\beta^{\prime})+\lbrace
1-(\alpha^{\prime}-\beta ^{\prime})k-(\alpha
^{\prime}+\beta^{\prime})(n-k)\rbrace p_{j}$.

Equation 6.2 reveals that there is an implied rank-order. However,
there are only two ranks --- greater than expected value and lower
than expected value. Also, the weights carry a component that is not a
product of $p $. Thus, the SSOT model is significantly different from
CPT. Now, Equation 6.2 is also analogous to a two-branch
gamble. Clearly, there is transfer of weight from the upper branch to
the lower branch. SSOT thus has similarities with TAX. However,
in this formulation, all gambles reduce to a two-branch
gamble. Therefore, potentially, there could be differences from TAX
that will need to be explored.

SSOT, in this paper is shown to explain certain key phenomena
underlying the new paradoxes. TAX has been very extensively studied
and explains a wide range of empirical findings. SSOT may or may not
be able to explain all those findings. Also, TAX and SSOT treat
probabilities in a fundamentally different manner (non-linear vs.\
linear, respectively). I thus speculate that TAX cannot be a special
case of SSOT, although under limiting conditions (for example, binary
gambles with equal probabilities) SSOT and special TAX appear
identical.

\subsection{Areas of future research and conclusion}

Further, areas of future research should include exploration of nature
and stability of the coefficients $\alpha $ and $\beta $ used in SOT.
In SSOT, a single value of $\alpha $ and a single value of $\beta $
was assumed for convenience. One question is whether these
coefficients should have different values in different domains
(positive, negative and mixed). Moreover, SOT should be extended to a
stochastic model and to probability distributions that may not be
discrete. The nature and impact of error should also be
explored. Another important area will be to understand how this model
can be applied in ambiguous (Ellsberg, 1961; Camerer \& Weber, 1992)
situations.  Implications for% decision analysis, in general, and
investment decisions (Markowitz, 1952), in particular, also remain to
be explored.

In conclusion, a new theory of decision-making under risk is proposed. 
This Opportunity-Threat theory relies on analyzing risk into its 
components. A simplified special model (SSOT) of this theory is able to 
explain a range of empirical phenomena that are explained by TAX but not 
by CPT. 

\begin{figure*}[t!]\centering
\includegraphics[width=6in]{fig1.eps}\medskip
\caption{Consider a simple binary gamble with utility, 
probability pairs ($u_{1}, p_{1};u_{2}, p_{2}$), where $
u_{2}>u_{1}>0$ and $p_{1}+p_{2}=1$. Panel \textbf{a} shows the 
expected value $\mu $ that is analogous to the center of gravity 
balancing the weights of the shaded areas. Imagine that the 
decision-maker wants to find a value equivalent $\mu ^{\prime}$ that is 
certain ($p=1$). This then becomes the decision-enabling center of 
gravity ($\mu ^{\prime},1$) shown in panel \textbf{b}.}
\end{figure*}

\section*{References}
\begin{hangparas}{1em}{1}

Allais, M. (1953). L'extension des théories de l'équilibre économique 
général et du rendement social au cas du risque. Econometrica, \textit{
Journal of the Econometric Society}, 269-290. 

Baron, J., \& Ritov, I. (1994). Reference points and omission bias.
\textit{Organizational Behavior and Human Decision Processes}, 
\textit{59}, 475-498.

Birnbaum, M. H. (1974). The nonadditivity of personality impressions.
\textit{Journal of Experimental Psychology Monograph}, \textit{102},
543-561.

Birnbaum, M. H. (2007). Tests of branch splitting and branch-splitting 
independence in Allais paradoxes with positive and mixed consequences. 
\textit{Organizational Behavior and Human Decision Processes}, \textit{102}, 
154-173.

Birnbaum, M. H. (2008). New paradoxes of risky decision making. \textit{
Psychological Review}, \textit{115}(2), 463-501.

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

\vfill
\break

Birnbaum, M. H., \& Stegner, S. E. (1979). Source credibility in social 
judgment: Bias, expertise, and the judge's point of view.\textit{
Journal of Personality and Social Psychology}, \textit{37}(1), 
48-74.

Camerer, C., \& Weber, M. (1992). Recent developments in modeling 
preferences: Uncertainty and ambiguity. \textit{Journal of Risk and 
Uncertainty}, \textit{5}, 325-370.

Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. \textit{
The Quarterly Journal of Economics}, 643-669.

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

Markowitz, H. (1952). Portfolio selection. \textit{The Journal of 
Finance}, \textit{7}(1), 77-91.

Samuelson, W., \& Zeckhauser, R. (1988). Status quo bias in decision 
making. \textit{Journal of Risk and Uncertainty}, \textit{1}(1), 
7-59.

Thaler, R. (1980). Toward a positive theory of consumer choice. \textit{
Journal of Economic Behavior \& Organization}, \textit{1}(1), 
39-60.

Tversky, A., \& Kahneman, D. (1992). Advances in prospect theory: 
Cumulative representation of uncertainty. \textit{Journal of Risk and 
Uncertainty}, \textit{5}(4), 297-323.

van de Ven, J., \& Diecidue, E. (2008). Aspiration level, probability of 
success and failure, and expected utility. \textit{International 
Economic Review}, \textit{49}, 683-700.

Von Neumann, J., \& Morgenstern, O. (1944). \textit{Game theory and 
economic behavior}. Princeton, Princeton University.\textbf{ }

\end{hangparas}

\newpage

\renewcommand{\baselinestretch}{1.1} \large\normalsize

\subsection*{Appendix: Derivation of $\theta $ and $\psi $ parameters}

Special Opportunity-Threat (SOT) model introduces parameters $\theta $
and $\psi $, which are derived as shown in Figure 1: 

Following Figure 1, characterize the linear shift (on x-axis) as an
aggregation parameter, net of upside and downside,
$\theta = \mu^{\prime} - \mu$. Now, if $\mu ^{\prime} $plays the role
of center of gravity, $ \sum_{i}^{n}{(1-p_{i})(u_{i}-\mu^{\prime})}=0$. Simple expansion leads to
$ \sum_{i}^{n}{(u_{i}-\mu^{\prime})}-\sum_{i}^{n}{p_{i}(u_{i}-\mu
  ^{\prime})}=n\bar{u}-n\mu ^{\prime}-\mu +\mu ^{\prime}=0$, where
$\bar{u}= \frac{\sum_{i=1}^{n}{u_{i}}}{n}$. That gives,
$\mu ^{\prime}=\frac{n\bar{u}-\mu }{n-1}$. Thus,
$\theta =\mu ^{\prime}- \mu =\frac{n\bar{u}-\mu }{n-1}-\mu
=\frac{n(\bar{u}-\mu )}{(n-1)}$.  Alternatively, $\theta $ can be
viewed as sum of $(u_{i}-\mu )$ residuals divided by the degree of
freedom $(n-1)$.

Now, only a fraction of this factor is actually incorporated in the
decision-making process depending on psychological weighting. Thus,
the center of gravity ends up at
$\mu^{\prime\prime}=\mu + \alpha \theta $ ($\mu^{\prime\prime}$ is not
shown in the figure). Finally, distance parameter is calculated as
standard deviation of outcome values around this new point.
$\psi ^{2}= \sum_{i=1}^{n}{p_{i}(u_{i}-\mu^{\prime\prime})^{2}}$. Now,
$
\sum_{i=1}^{n}{p_{i}(u_{i}-\mu^{\prime\prime})^{2}}=\sum_{i=1}^{n}{p_{i}(u_{i}-\mu
  -\alpha .\theta )^{2}}=\sum_{i=1}^{n}{p_{i}(u_{i}-\mu )^{2}+}
\sum_{i=1}^{n}{p_{i}(\alpha .\theta )^{2}}$. Thus,
$\psi = \lbrace \sigma ^{2}+ (\alpha \theta )^{2}\rbrace ^{1/2}$.

%\textbf{Appendix B: Data and calculations}

%Provided in a supplementary spreadsheet

\end{document}
