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\title{Is broad bracketing always better? How broad decision framing leads to more optimal preferences over repeated gambles}

\author{
Elizabeth C. Webb\thanks{Columbia
University, Department of Marketing, 3022 Broadway, Uris Hall 511, New
York, NY, 10027. Email: ecw2144@gsb.columbia.edu.}
\and
Suzanne B. Shu\thanks{UCLA Anderson School of Management,
Department of Marketing.}
}

\date{} % leave empty
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\newcommand{\jref}{http://journal.sjdm.org/vol12.4.html}
\newcommand{\jhead}{Judgment and Decision Making, Vol.~12, No.~4}
\newcommand{\jdate}{July 2017}
\pagestyle{myheadings} \markright{\protect\small \href{\jref}{\jhead}, \jdate \hfill Broad bracketing in repeated gambles \qquad}
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\href{\jref}{\jhead}, \jdate, pp.\ 382-395
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{\small \href{\jref}{\jhead}, \jdate, pp.\ 382--395}
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%\begin{latexonly}
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\begin{onecolabstract}
%\end{latexonly}
  The effect of choice bracketing — the consideration of repeated
  decisions as a set versus in isolation — has important implications
  for products that are inherently time-sensitive and entail varying
  levels of risk, including retirement accounts, insurance purchases,
  and lottery preferences. We show that broader choice brackets lead
  to more optimal risk preferences across all risk types, including
  negative expected value and pure-loss gambles, suggesting that broad
  decision framing can help individuals make better choices over risks
  more generally. We also examine the mechanism behind these
  bracketing effects. We find that bracketing effects work by
  attenuating (magnifying) the weight placed on potential losses for
  positive EV (non-positive EV) gambles and by providing aggregated
  outcomes that might not otherwise be calculated. Thus, decision
  frames that provide probability distributions or aggregated outcomes
  can help individuals maximize expected value across different types
  of risky prospects.

% removed part about "largely unexplored" - I'm sure someone will
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\smallskip
\noindent
Keywords: choice bracketing, myopic loss aversion, risk-taking, risk perception, repeated gambles
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]
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{\renewcommand{\thefootnote}{}

\footnotetext{

  This work is part of the first author’s dissertation. The authors
  would like to thank John Payne, Stephen Spiller, Eric Johnson, Scott
  Shriver, Sanjay Sood, and Aimee Drolet for their invaluable feedback
  on this paper in its various stages. The authors would also like to
  thank the many seminar participants at UCLA, Columbia University,
  and The University of Pennsylvania who provided feedback on this
  project along the way.

Copyright: \copyright\ 2017.
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}
Individuals often face financial decisions in which outcomes and
feedback accrue over time or over several repeated transactions. Many
financial risks with positive expected value (EV) can thus be thought
of as either segregated transactions (disconnected single trials of a
repeated gamble) or as an integrated set (a probability distribution
over many trials of a repeated gamble). For example, in the context of
portfolio allocation, an investor can evaluate returns on a short-term
basis (e.g., once a day, once a week, or once a quarter) or on a
long-term basis (e.g., once a year or once every couple of years).
Assuming that the underlying risk associated with the portfolio does
not change over time, the information available to the investor is
identical under either evaluation strategy. However, short-term
evaluations will usually lead to more experienced losses, as even
positive EV assets typically entail some chance of loss. In contrast,
long-term evaluations of these same multiple positive EV risks will
lead to higher cumulative gains, less experienced losses, and a better
sense of the underlying probability distribution. If investors are loss
averse, information that helps an investor visualize or understand how
positive EV outcomes are aggregated over time can lead to investment
decisions with higher value outcomes. 

What happens if these repeated financial risks have a negative EV rather
than a positive EV, such as lotteries, or even entail only losses, such
as insurance? How would the presentation of information about a series
of primarily negative outcomes (monthly premiums, daily lottery
tickets) affect the decision to take such risks? Arguably, an
individual would again make different choices depending on whether the
risks are represented as occurring over only one or a few trials or
aggregated over a larger set of trials. In contrast to positive EV
risks, repeated non-positive EV risks will result in larger cumulative
losses relative to a narrow evaluation strategy. Thus, individuals may
shift their preferences depending on the decision frame (broad or
narrow). More generally, we evaluate whether information about
aggregated outcomes (broader bracketing) leads to decisions with higher
payoffs for non-positive EV gambles as well.

We demonstrate that how a repeated financial risk is represented —
either broadly or narrowly — affects preferences for that
risk. Specifically, for positive EV gambles, individuals prefer a smaller certain gain in isolation (narrow bracketing), but prefer the gamble, with its higher EV, over the certain amount when the probability distribution is provided (broad bracketing). For negative EV gambles and pure-loss gambles,
individuals prefer the gamble (with its larger potential loss) in
isolation (narrow bracketing), but prefer a smaller certain loss with
its higher EV when they see the probability distribution over many
trials (broad bracketing). Our results thus imply that broad
bracketing prevents preference reversals for repeated financial risks
and leads to more optimal preferences (i.e., choices that maximize the
expected value of returns), whether those risks are predominantly
positive or negative.

\section{Choice Bracketing }
Choice bracketing describes how choices are grouped together (Read et al., 1999). When encountering repeated related choices, individuals can consider each of those choices separately in
isolation, or they can consider those choices as a set with
cumulative, aggregated outcomes. The former is considered narrow
bracketing — only one or a small number of choices are considered
together — while the latter is considered broad bracketing — each choice is
considered in the context of many other choices in a set. Thus, narrow
bracketing is focused on local outcomes, while broad bracketing is
focused on more global outcomes. For example, consider the choice of
whether or not to eat dessert at the end of a meal. Bracketing this
choice narrowly means deciding whether the pleasure of eating the
dessert outweighs the costs (e.g., caloric intake, breaking a diet).
Bracketing this choice broadly means considering all times (before and
after) that one may eat dessert and considering the cumulative effects
of those desserts combined. Thus, an individual may choose to
impulsively eat each dessert when considered alone, but if presented
with the cumulative effect of eating dessert every meal, may reject
that set outright (Ainslie, 1975). This means the individual’s choices are
different depending on the bracketing involved — narrow or broad. This is
called a bracketing effect and occurs whenever choices differ depending
on the bracket through which they are evaluated.

Bracketing effects subsume many concepts. Specifically, in their
paper defining choice bracketing, Read et al.\ (1999) consider
sequential and simultaneous choice, narrow and broad decision frames,
isolated and distributed choice, acts and patterns, and local and
overall value functions, all as forms of narrow and broad choice
bracketing. For this reason, we use the terminology of “choice
bracketing,” though many of our effects could also be defined in other
ways (e.g., narrow or broad framing). Choice bracketing can also be
applied to many different choices — not just risky choice. For example,
research on choice bracketing and mental budgeting has found that which
product categories are combined together in a budget category can have
a direct effect on how budgets are formed and followed (Read et al.,
1999; Thaler, 1999). However, we focus specifically on the effects of
choice bracketing as they relate to decisions over repeated financial
risks.

\renewcommand\arraystretch{1.0}
\begin{figure*}[t!]\centering
\caption*{Figure 1: Example of Within-Subject Bracketing Manipulation
by Gamble Type, Study 1.}
{\sf
\begin{tabular}{llll}\toprule
Narrow Bracket Version & \mc{3}{l}{Broad Bracket Version}\\\midrule
\mc{4}{c}{Positive Expected Value}\\\midrule
The gamble:		&	\mc{3}{l}{The gamble:}\\
10\% *\ \ \ \ \ \ \ \ \ \ \ \ Win \$0.75         & 1\% & \# & Win \$12\\
90\% ********* Lose \$0.01         & 1\% & \# & Win \$11\\
The gamble is played 90 times.		         & 4\% & \#\#\#\# & Win \$10\\
		         & 7\% & \#\#\#\#\#\#\# & Win \$9\\
The certain amount: \$3		         & 12\% & \#\#\#\#\#\#\#\#\#\#\#\# & Win \$8\\
		         & 16\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Win \$7\\
		         & 18\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Win \$6\\
		         & 16\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Win \$5\\
		         & 12\% & \#\#\#\#\#\#\#\#\#\#\#\# & Win \$4\\
		         & 7\% & \#\#\#\#\#\#\# & Win \$3\\
		         & 4\% & \#\#\#\# & Win \$2\\
		         & 1\% & \# & Win \$1\\
		         & 1\% & \# & Win \$0\\[1ex]
			 & \mc{3}{l}{The certain amount: \$3}\\\midrule
\mc{4}{c}{Negative Expected Value}\\\midrule
The gamble: & \mc{3}{l}{The gamble:}\\
10\% *\ \ \ \ \ \ \ \ \ \ \ \ Win \$0.25   & 2\% & \#\# & Lose \$8\\
90\% ********* Lose \$0.10         & 5\% & \#\#\#\#\# & Lose \$7\\
The gamble is played 60 times.		         & 13\% & \#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$6\\
		         & 22\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$5\\
The certain amount: --\$2 (Lose \$2)		         & 26\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$4\\
		         & 20\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$3\\
		         & 9\% & \#\#\#\#\#\#\#\#\# & Lose \$2\\
		         & 2\% & \#\# & Lose \$1\\[1ex]
		 & \mc{3}{l}{The certain amount: --\$2 (Lose \$2)}\\\midrule
\mc{4}{c}{Pure-Loss}\\\midrule
The gamble:		& \mc{3}{l}{The gamble:}\\
90\% ********* Lose \$0.10         & 1\% & \# & Lose \$11\\
10\% *\ \ \ \ \ \ \ \ \ \ \ \ Lose \$0.50         & 7\% & \#\#\#\#\#\#\# & Lose \$10\\
The gamble is played 50 times.		         & 7\% & \#\#\#\#\#\#\# & Lose \$9\\
		         & 10\% & \#\#\#\#\#\#\#\#\#\# & Lose \$8\\
The certain amount: --\$4 (Lose \$4)		         & 28\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$7\\
		         & 15\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$6\\
		         & 12\% & \#\#\#\#\#\#\#\#\#\#\#\# & Lose \$5\\
		         & 14\% & \#\#\#\#\#\#\#\#\#\#\#\#\#\# & Lose \$4\\
		         & 2\% & \#\# & Lose \$3\\
		         & 1\% & \# & Lose \$2\\[1ex]
		 & \mc{3}{l}{The certain amount: --\$4 (Lose \$4)}\\\bottomrule
\end{tabular}}
\end{figure*}
\renewcommand\arraystretch{1.2}

To manipulate bracketing, we use an exact replication of a technique
used in previous research (Benartzi \& Thaler, 1999). Specifically, in
their Study 2, Benartzi and Thaler (1999) showed subjects repeated
positive EV mixed gambles described either in words (i.e., “\textit{N}
plays of gamble \textit{1}, with probability \textit{p }of winning
\textit{x}, probability \textit{1-p }of losing \textit{y}”) (narrow bracketing)
or as a visual display of the distribution of possible outcomes for the
aggregated set of gambles (broad bracketing). An example of this
manipulation is shown in Figure 1.\footnote{ The example for the
Positive EV gamble is taken directly from Benartzi and Thaler’s (1999)
materials. We requested these materials from the authors and used the
same format in our studies. For the non-positive EV gambles, which were
not used by the original authors, we created new gambles that conformed
to the same general principles used in the construction of the positive
EV gambles.} The difference between the narrow and broad bracket is
only in how the information is displayed; in the narrow format the
number of trials is described in words, while in the broad format the
number of trials is only implicit in the distribution.

\section[Repeated Gambles \& Myopic Loss Aversion ]{Repeated Gambles \&
Myopic Loss Aversion }
The question of how individuals approach repeated plays of a gamble, and
its implications for investment behavior, has been well explored in the
literature (see, e.g., Keren, 1991; Klos et al., 2005;
Redelmeier \& Tversky, 1992; Thaler \& Johnson, 1990; Wedell \&
Böckenholt, 1994). This work has generally found that individuals are
more willing to accept multiple plays of a gamble than any single trial
or play of that gamble. Thus, broad bracketing (combining repeated
plays of a gamble into one choice set) reduces risk aversion and leads
to expected value maximization.\footnote{Note that broad bracketing’s
effects on expected value maximization may also lead to overall utility
maximization if outcomes are not monetary, yet utilities are additive
over time. With monetary gambles over short periods, it is safe to
assume that the utilities are a function of total monetary gain or
loss.}

Perhaps one of the most well-known demonstrations of how choice
bracketing aligns with repeated gambles is the work on myopic loss
aversion (Benartzi \& Thaler, 1995; Benartzi \& Thaler, 1999; Gneezy \& Potters, 1997; Langer \&
M. Weber, 2001; Thaler et al., 1997). In Benartzi
and Thaler’s (1999) work on the equity premium puzzle, the authors were
able to change risk preferences through choice bracketing such that
individuals’ choices over repeated positive EV gambles were less
risk-averse when the gambles were broadly bracketed versus narrowly
bracketed. The authors describe these findings as supportive of the
concept of myopic loss aversion: individuals are loss averse for mixed
gambles, but also myopic, since they appear to consider the gambles in
isolation (narrowly) rather than thinking about each one as a piece of
a larger set of gambles with an overall outcome distribution that
favors gains (broadly). Benartzi and Thaler conclude that broader framing
attenuates the effect of loss aversion, and changes preferences towards
what is predicted by expected value calculations. 

\pagebreak

Benartzi and Thaler's (1999) framing work and related empirical tests
of myopic loss aversion  have clearly demonstrated that broad
bracketing leads to more normative and financially optimal choices in a
world of positive EV risks (Benartzi \& Thaler, 1995; Gneezy \&
Potters, 1997; Haigh \& List, 2005; Thaler et al., 1997). However, less
work has been done to understand how bracketing and loss aversion
combine when the expected value of the risks is not positive. The
original context for myopic loss aversion was the U.S. stock market,
which has a positive EV over time (Benartzi \& Thaler, 1995). However,
there are also environments where individuals face choices with
negative outcomes or negative expected values over time, such as
insurance policies or state lotteries. In this paper, we investigate
whether broader choice bracketing can lead to better choices for all
gamble types. More specifically, we investigate whether bracketing
changes risk preferences for negative EV gambles and gambles over pure
losses. We do this by assessing preferences for a gamble versus a sure
loss (negative certainty equivalent) where the sure loss is a smaller
absolute value loss than the expected value of the gamble. Our findings
imply that broad bracketing can be used to help individuals make more
optimal choices over risky prospects that have a negative EV or solely
entail losses, not just positive EV ones. 

\section[Mechanisms Underlying Bracketing Effects for Repeated
Gambles]{Mechanisms Underlying Bracketing Effects for Repeated
Gambles}

\renewcommand{\baselinestretch}{1.02} \large\normalsize

Beyond extending the usefulness of broad bracketing to other types of
gambles, our work also explores more deeply what mechanisms underlie
the documented bracketing effects. Prior studies on myopic loss
aversion and choice bracketing (Benartzi \& Thaler, 1999) have not
clearly identified the reasons for the effects, beyond the finding that
broad brackets overcome loss aversion by effectively showing that
losses are almost entirely compensated for by gains (for positive EV
gambles). Other studies have proposed that broadly bracketed gambles
reduce risk perception or the level of risk associated with the gambles
(Coombs \& Meyer, 1969; Read et al., 1999). Thus, the bracketing effect
could be due to outcome aggregation, which changes the expected
benefits and costs of the prospect (under the assumption that
individuals cannot construct the probability distribution when a gamble
is described narrowly), or due to a change in the perceived level of
risk which makes the gamble more (or less) acceptable in accordance
with the individual’s risk attitude.

In addition to testing the main effect of bracketing type on risk
preferences, in Study 1, we test whether bracketing effects occur
because of changes in perceived risk, such that for positive EV
(negative EV, pure-loss) gambles, broad brackets reduce (increase)
perceived risk. Risk perception is a subjective judgment that
represents the beliefs or feelings that individuals have about the
level of risk inherent in the prospect under consideration (Blais \&
E. U. Weber, 2006; Holtgrave \& E. U. Weber, 1993; Klos et al., 2005; E. U. Weber \&
Hsee, 1998; E. U. Weber et al., 2002).  In psychological models
of risk-return, risk perception has been shown to account for much of
the variance in risk preferences (E. U. Weber \& Hsee, 1998; E. U. Weber \&
Milliman, 1997). Previous work on perceived risk has also shown that
framing effects can result from differences in risk perception across
the frames (Mellers et al., 1997; Schwartz \& Hasnain,
2002). Given the work on risk perception and framing, it is possible
that bracketing effects are explained entirely by changes in risk
perception. To this end, we ask each subject to rate perceived risk
after they make a choice for each gamble (measured via a 7-point Likert
scale, see E. U. Weber et al., 2006). 

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig0.eps}
\caption*{Figure 2: Schematic Illustration of Broad-Trial
  Condition. Red dots represent the probability of losing; black, winning.}
\end{figure}

In Study 2, we directly test whether the effect of broad bracketing
comes from adding up the cumulative effects of repeated trials. We
hypothesize that there are two possible components of these adding up
effects. First, adding together the outcomes of multiple gambles can
affect loss aversion by changing the perceived size and probability of
an overall loss. For example, the aggregation of multiple positive EV
gambles will result in a distribution of outcomes where cumulative
losses are very unlikely, and the majority of possible outcomes are
positive. In contrast, for non-positive EV gambles, aggregation will
result in a probability distribution with a larger number of more
probable cumulative losses. Thus, broad brackets can either increase or
decrease the salience of the losses (situational weight on losses),
depending on the type of gamble. 

%ecw0716: added a paragraph break here - don't feel strongly about this though

The second possible component of
adding up effects results from individuals not making sufficient
adjustment from the outcomes of a single gamble to the effects of many
such gambles. Decision makers are likely to evaluate the outcomes of a
gamble, especially the higher probability outcome, within a single
trial and insufficiently adjust for the multiplicative effect of many
trials. For example, with a positive EV gamble, individuals may focus
on the higher probability gain from a single trial of the gamble and not multiply it
appropriately for repeated plays, thus underappreciating the impact of
the large number of trials. 

To test the separate effects of these two explanations, we manipulate
broad bracketing in three ways: (1) narrow, (2) broad, and (3)
broad-trial. The broad manipulation provides the full distribution of
aggregated outcomes. The broad-trial manipulation presents each trial
as a block with ten colored dots. Within the block, dots are colored
either red (to represent the probability of losing) or black (to
represent the probability of winning). For example, a positive EV
gamble that has a 10\% chance of winning \$0.75 and a 90\% chance of
losing \$0.01, played 90 times, would have 90 blocks. Within each
block, there would be one black dot and nine red dots.  Figure 2
provides an example, using a different gamble. By including the
broad-trial format, we can assess whether simply making the
repeated-trial nature of the gamble more salient changes risk
preferences or if explicit outcome aggregation is needed for a
bracketing effect to occur. We also directly measure situational
weight on losses and the importance of the number of trials to see
which hypothesized mechanism is reported by subjects as having a
larger influence on their decisions.

\section{Overview and Contribution}
To summarize our approach and preview the contributions of this paper,
we extend the work on bracketing effects by examining how broad
bracketing influences not only positive EV mixed gambles, but also
negative EV (mixed) and pure-loss gambles. In two studies, we show
bracketing effects (different preferences under broad brackets versus
narrow brackets) across all gamble types. However, we find that the
direction of this effect is different for positive EV gambles versus
non-positive EV gambles (negative EV and pure-loss). Specifically, for
positive EV gambles, subjects prefer the certain (smaller) gain
when evaluating a choice in a narrow bracket, but prefer the gamble when
evaluating the same choice in a broad bracket. For negative EV and pure-loss
gambles, subjects prefer the gamble when evaluating a choice in a narrow
bracket, but prefer the certain (smaller) loss when evaluating the same choice
in a broad bracket. Across gamble types, our empirical findings imply
that broader choice brackets lead to more optimal choice strategies, as
defined by maximizing expected value.

We also evaluate possible mechanisms for our documented bracketing
effects. We specifically focus on whether bracketing effects are the
result of outcome aggregation that affects the weight placed on losses, or of trial
aggregation, under which the higher salience of repeated gambles
overcomes insufficient adjustment from the single gamble. Ultimately,
we find that the bracketing effect for repeated-play gambles is
primarily due to changing the weight placed on losses via outcome
aggregation rather than an effect of highlighting the repeated trials
inherent in the risky prospect. These findings suggest that targeted
interventions that help individuals calculate aggregated outcomes and
change the salience of losses can improve decision-making for all risk
types.

\renewcommand{\baselinestretch}{1.03} \large\normalsize

\section{Study 1: Broad Bracketing Produces More Optimal Preferences for
all Gamble Types}
In Study 1, we begin by replicating Benartzi and Thaler’s (1999)
Study 2, in which we ask subjects to state their preferences across
several gambles. Using the original study design, subjects see the same
gambles in both a narrow bracket (a text description of the gamble with
the number of trials specified) and in a broad bracket (a probability
distribution of the outcomes across all trials). We then extend the
authors’ work by including negative EV and pure-loss gambles under the
same set-up. We replicate the authors’ findings for positive EV gambles
by showing that individuals switch from predominantly choosing the
offered certainty equivalent (which is less than the EV of the gamble)
under the narrow bracket to predominantly choosing the gamble under the
broad bracket. For negative EV and pure-loss gambles, the opposite
occurs: individuals switch from predominantly choosing the gamble under
the narrow bracket to predominantly choosing the offered (negative)
certainty equivalent (which is a smaller loss than the EV of the gamble) under the broad bracket. This pattern of results suggests that broad bracketing can be used to help individuals make
more normatively optimal decisions over risk. 

\begin{table*}[t]\centering
\caption*{Table 1: List of Gambles Used in Study 1.}
\begin{tabular}{lllccl}\toprule
  & Type & Bracket & EV & \mc{1}{C{.5in}}{Certain\newline Amount} & Description\\
  Gamble 1 & Positive EV & Narrow & \$6  & \$3  & 10\% win \$0.75, 90\%
                                                  lose \$0.01, 90 trials\\
  Gamble 2 & Positive EV & Narrow & \$6  & \$3  & 90\% win \$0.10, 10\%
                                                  lose \$0.50, 150 trials\\
  Gamble 3 & Positive EV & Narrow & \$6  & \$3  & 50\% win \$0.25, 50\%
                                                  lose \$0.15, 120 trials\\
  Gamble 4 & Positive EV & Broad & \$6  & \$3  & Truncated
                                                 distribution\\
  Gamble 5 & Positive EV & Broad & \$6  & \$3  & Full distribution\\
  Gamble 6 & Positive EV & Narrow & \$60  & \$30  & High-stakes: 50\% win
                                                    \$2.50, 50\% lose \$1.50, 120 trials\\
  Gamble 7 & Positive EV & Broad & \$60  & \$30  & High-stakes truncated
                                                   distribution\\
  Gamble 8 & Pure-Loss & Narrow & {}-\$7 & {}-\$4 & 90\% lose \$0.10, 10\%
                                                    lose \$0.50, 50 trials\\
  Gamble 9 & Pure-Loss & Narrow & {}-\$7 & {}-\$4 & 50\% lose \$0.15, 50\%
                                                    lose \$0.25, 35 trials\\
  Gamble 10 & Pure-Loss & Narrow & {}-\$7 & {}-\$4 & 10\% lose \$0.75,
                                                     90\% lose \$0.01, 80 trials\\
  Gamble 11 & Pure-Loss & Broad & {}-\$7 & {}-\$4 & Truncated
                                                    distribution\\
  Gamble 12 & Pure-Loss & Narrow & {}-\$70 & {}-\$40 & High-stakes: 50\%
                                                       lose \$2.50, 50\% lose \$1.50, 35 trials\\
  Gamble 13 & Pure-Loss & Broad & {}-\$70 & {}-\$40 & High-stakes
                                                      truncated distribution\\
  Gamble 14 & Negative EV & Narrow & {}-\$4 & {}-\$2 & 50\% win \$0.15,
                                                       50\% lose \$0.25, 80 trials\\
  Gamble 15 & Negative EV & Narrow & {}-\$4 & {}-\$2 & 10\% win \$0.25,
                                                       90\% lose \$0.10, 60 trials\\
  Gamble 16 & Negative EV & Narrow & {}-\$4 & {}-\$2 & 90\% win \$0.01,
                                                       10\% lose \$0.50, 100 trials\\
  Gamble 17 & Negative EV & Broad & {}-\$4 & {}-\$2 & Truncated
                                                      distribution\\
  Gamble 18 & Negative EV & Narrow & {}-\$40 & {}-\$20 & High-stakes: 50\%
                                                         win \$1.50, 50\% lose \$2.50, 80 trials\\
  Gamble 19 & Negative EV & Broad & {}-\$40 & {}-\$20 & High-stakes
                                                        truncated distribution\\\bottomrule
  \mc{6}{p{6in}}{Notes: (1) The gambles are ordered sequentially for ease of
  exposition — in the actual study, all subjects saw the
  Positive EV gambles first, and then the non-positive EV
  gambles with Pure-Loss and Negative EV types counterbalanced), and the order of the
  gambles within each of the three types was randomized for each subject.  (2)
  The truncated distributions show all outcomes that occur with a
  probability of 1\% or greater. (3) High-stakes gambles are a version of
  one of the narrowly bracketed gambles with the outcomes multiplied by
  10.}
\end{tabular}
\end{table*}

\subsection[Method]{Method}
Study 1 was conducted online with 144 subjects\footnote{ We recruited
  150 subjects. One-hundred-forty-nine subjects successfully completed
  the survey. Of those, five subjects were dropped because they did
  not complete a second part of the survey (the loss aversion section
  described later).} (39.6\% female, M\textsubscript{age} = 35.7
years) recruited through Amazon’s Mechanical Turk (mTurk). This
study was a replication and extension of Benartzi and Thaler’s (1999)
Study 2. We ran the exact same study but with two additional gamble
types: negative EV and pure-loss, since the authors of the original
study asked only about positive EV mixed gambles. In their study,
Benartzi and Thaler (1999) asked subjects to consider $N$ independent
trials of a gamble with a probability $p$ of winning an amount $x$,
and a probability $1-p$ of losing an amount $y$. The bracketing
manipulation involved either describing the gambles as “being played
$N$ times” (narrow bracket) or providing the full probability
distribution of outcomes from the repeated plays (broad bracket). We
used this same bracketing manipulation across all three gamble
types. For the pure-loss gambles, subjects considered $N$ independent
plays with a probability $p$ of losing an amount $x$, and a
probability $1-p$ of losing an amount $y$. The negative EV gambles
appeared the same as the positive EV gambles except that they had an
expected value less than zero, while the positive EV gambles had an
expected value greater than zero. A list of all of the gambles used in
Study 1 is shown in Table 1, with additional explanation provided
below. We also provide the gambles (as displayed to subjects) and the
complete Study 1 materials in \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 2}.

We measured risk preference as the choice between the gamble and a
certainty equivalent (CE). The certain amount that was offered to subjects
varied by gamble type. For the positive EV gambles, the certain amount
was \$3, which is less than the EV of the gamble.\footnote{ It is
  important to note that the EV is equal for all of the gambles within
  a gamble type (positive EV, negative EV, pure-loss), with the
  exception of the high-stakes gambles, where the EV (and individual
  outcomes) are multiplied by ten.} For the negative EV gambles, the
certain amount was --\$2 (lose \$2). This amount is greater (a smaller
loss) than the EV of the negative EV gambles. Similarly, for the
pure-loss gambles, the certain amount was --\$4 (lose \$4). This
certain amount is greater (a smaller loss) than the EV for the
pure-loss gambles. To ensure that choices were incentive compatible,
ten subjects were randomly chosen to have one of their positive EV
choices played for real (i.e., if they chose the gamble, the gamble
would be played out for them, if they chose the certain amount they
would be given the certain amount for that question). All subjects
were told about this possibility before beginning the survey.

To summarize, Study 1 has a 3 (Gamble Type: Positive EV, Negative EV,
Pure-Loss) x 2 (Bracket: Broad, Narrow) completely within-subjects
design. Each subject was asked to evaluate seven positive EV gambles,
six negative EV gambles, and six pure-loss gambles. Since only the
Positive EV gambles were used for the incentive compatible lottery
task, all subjects saw the positive EV gambles first. The order of the
Positive EV gambles was randomized once and then counterbalanced across
subjects. After the positive EV gambles, subjects saw the non-positive
EV (negative EV and pure-loss) gambles. The order of the non-positive
EV gambles was randomized for each subject and the two conditions were
counterbalanced across subjects.

Following Benartzi and Thaler (1999), four of the gambles from each
type were presented in a narrow bracket and two were presented in a
broad bracket. The EV for all of these gambles was the same (the EV
for all narrowly bracketed gambles and the two broadly bracketed
gambles within a gamble type was the same).  For all gamble types, one of the narrow and
one of the broad bracket gambles was a high-stakes version that had
outcomes multiplied by ten. These two gambles had the same EV. This
version was included to ensure that the same pattern of preferences
holds over larger possible outcomes.  For most of the broadly
bracketed gambles, we truncated the distribution to exclude any
outcome with less than a one percent chance of occurring (mimicking
Benartzi and Thaler’s (1999) approach). However, to verify that the
bracketing effect still occurs when small probability losses are
included in the distribution, we also added one positive EV gamble
with a non-truncated probability distribution (the seventh positive EV
gamble).\footnote{ We do not include a full distribution version of
  the gamble for the non-positive EV gambles since we do not expect
  loss aversion for these gamble types.  While it’s possible that
  small probability gains could change preferences, the gambles we
  used for the non-positive EV gamble types do not have even small
  probability gains when outcomes are aggregated across all trials.}
The gambles we used for each gamble type were constructed to have
approximately the same aggregated payoff distribution, but different
characteristics. In this sense, the bracketing manipulation is a
framing effect since the information in both versions of the problem
(narrow versus broad) is identical, and only the presentation of that
information changes. An example of the bracketing manipulation for
each gamble type is shown in Figure 1.

To see whether differences in risk perception explain the bracketing
effects, as found in prior literature (Mellers et al., 1997; Schwartz
\& Hasnain, 2002), we asked subjects to provide risk perception
ratings for each of the gambles. This allows us to compare perceived
risk levels for the same gamble in a narrow bracket and a broad
bracket, thus testing whether the bracketing effect we observe is
simply due to changes in perceived risk. Risk perception was collected
for each gamble. After making a choice for the gamble, risk
perception was measured on a scale from one (“Not at all Risky”) to
seven (“Extremely Risky”) (Blais \& E. U. Weber, 2006).\footnote{ At the end
  of the survey, we also measured loss aversion at the
  individual-level using the Dynamic Experiments for Estimating
  Preferences or DEEP method (Toubia et al., 2013). This measure was included to test whether individual-level (but not gamble-specific) loss aversion predicts risk
  preferences. Finally, after completing the DEEP questions, we asked
  subjects to self-report gender and age.}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig2.eps} 
\caption*{Figure 3: Percent of Subjects
Choosing the Gamble by Gamble \& Bracket Type, Study 1.\\
\small{Notes: (1) Broad Bracket and Narrow Bracket collapse across all gamble
choices within that bracket for a given gamble type (e.g., the number
displayed for the Broad Bracket Positive EV gambles is the average
choice share across the broadly bracketed truncated gamble,
non-truncated gamble, and high-stakes version of the gamble). (2) The
differences between the Narrow Bracket and Broad Bracket manipulations
are significant at the \textit {p} {\textless} 0.001 level for all gamble types.}}
\end{figure}

\renewcommand{\baselinestretch}{1.03} \large\normalsize

\subsection{Results }
We first compare risk preferences (as measured by subjects’ choice of
the gamble\footnote{ Since the DV asked for a choice of the gamble,
  the certain amount, or indifference between the two, we focus on the
  choice of the gamble. Thus, subjects who chose indifference are
  counted as preferring the certain amount. While this is imprecise in
  terms of capturing true indifference, it provides a conservative
  measure for risk-taking preferences, which we believed was a better
  approach for the analyses we report. Treating indifference as
  choosing the gamble does not significantly affect the pattern of
  results as only a small proportion of subjects chose indifference
  for each gamble (of total choices within a gamble type, 3.1\%,
  8.0\%, and 6.0\% of subjects chose indifference for the Positive EV,
  Pure-Loss, and Negative EV gambles, respectively).}) by bracket
(broad versus narrow) for each gamble type (positive EV, negative EV,
pure-loss). These results by gamble type and bracket are summarized in
Figure 3.

For the positive EV gambles, we see a replication of Benartzi and
Thaler’s (1999) findings: subjects are relatively risk averse when
considering the gambles in the narrow bracket, but relatively
risk-seeking when considering them in the broad bracket. If we average
the gambling choice shares across all of the narrowly bracketed gambles
(Gambles 1--3, 6) and compare this to the average for the broadly
bracketed gambles (Gambles 4, 5, 7), the difference in choice shares is
highly significant (M\textsubscript{Narrow} = 33.7\% vs.
M\textsubscript{Broad} = 72.2\%, \textit{t}(283.55) = --9.19, \textit{p}
{\textless} 0.001). While we present this information collapsed across
all of the individual gambles, the pattern of results holds for each
narrowly bracketed version (excluding the high-stakes version) of the
gamble when compared to the broadly bracketed gamble (both truncated
and full versions, but not high-stakes version). For example, if we
compare the choice shares for Gamble 1 (narrow) to Gamble 4 (broad,
truncated distribution) the choice shares are significantly different
(M\textsubscript{Gamble1} = 36.1\% vs.\ M\textsubscript{Gamble4} =
77.1\%, \textit{t}(281.05) = --7.68, \textit{p} {\textless} 0.001). For a summary of
choices by each individual gamble, see Figure A1 in \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 1}. These
results demonstrate a bracketing effect for positive EV gambles:
displaying the same financial risk in different bracketing formats
leads to a preference reversal. Specifically, subjects are
relatively risk averse over the gamble when presented in a narrow
bracket, but become relatively risk-seeking when that same gamble is
presented in a broad bracket.

The bracketing effect is significant for the high-stakes version of
the positive EV gamble as well (when the outcomes are multiplied by
ten) (M\textsubscript{NarrowHigh-Stakes} = 35.4\% vs.
M\textsubscript{BroadHigh-Stakes }= 75.0\%, \textit{t}(283.22) =
--7.34, \textit{p} {\textless} 0.001). Further, the gamble with the
non-truncated probability distribution still garnered significantly
higher choice shares for the gamble compared to the narrowly bracketed
(non-high-stakes) gambles (M\textsubscript{Narrow} = 33.1\% vs.
M\textsubscript{BroadNon-Truncated }= 64.6\%, \textit{t}(259.56) =
--6.39, \textit{p} {\textless} 0.001; \textit{ps }{\textless} 0.001 for
all pairwise comparisons between the narrowly bracketed gambles and
the non-truncated broad gamble).\footnote{ It is interesting to note
  that there is a significant difference in gamble choice shares
  between the truncated distribution broad gamble (gamble 4) and the
  full distribution (non-truncated) broad gamble (gamble 5), such that
  subjects are significantly less likely to choose the gamble with the
  full distribution than with the truncated distribution
  (M\textsubscript{BroadTruncated} = 77.1\% vs.
  M\textsubscript{BroadFull} = 64.6\%, \textit{t}(281.36) = 2.35,
  \textit{p }= 0.02). We attribute this attenuation in gamble choice
  shares to loss aversion since the full distribution shows small
  probability, larger losses compared to the truncated distribution.
  (Note that the full distribution also shows small probability,
  larger gains but given loss aversion, we expect the effect of the
  losses to have a larger impact on choice shares, which is also
  supported by an attenuation, rather than increase, in gamble shares
  for the full distribution gamble.) Even though the difference between
  these two gambles is significant, the difference between the full
  distribution broad gamble and the narrowly bracketing gambles is
  large and significant. Thus, a bracketing effect still occurs.}
Thus, we have replicated the bracketing effect found by Benartzi and
Thaler (1999).

Next, we turn to the gamble choice shares for the two gamble types not
included in Benartzi and Thaler’s (1999) original study: negative EV
and pure-loss. As Figure 3 shows, we see the opposite pattern of
results for these gamble types compared to the positive EV gambles.
Subjects are significantly more likely to accept the gamble when
displayed in a narrow bracket than when displayed in a broad
bracket\footnote{ For this comparison we calculate the average gamble
  choice share across all narrowly bracketed (non-high-stakes) gambles
  and compare it to the average gamble choice share for the
  non-high-stakes broadly bracketed gamble (e.g., for Negative EV
  gambles, it compares the average across gambles 14-16 to gamble
  17).}  (for negative EV gambles: M\textsubscript{Narrow} = 41.9\%
vs.  M\textsubscript{Broad} = 8.3\%, \textit{t}(254.15) = 8.25,
\textit{p }{\textless} 0.001; for pure-loss gambles:
M\textsubscript{Narrow} = 42.8\% vs.\ M\textsubscript{Broad} = 11.1\%,
\textit{t}(273.04) = 7.55, \textit{p} {\textless} 0.001). As with the
positive EV gambles, this pattern of results holds both when combined
across bracket type and when individual gambles are analyzed
separately (as shown in Figure A1, \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 1}). The results
also hold for the high-stakes version of the gambles
(M\textsubscript{NarrowHigh-StakesNegEV} = 50.0\%,
M\textsubscript{BroadHigh-StakesNegEV} = 11.8\%, \textit{t}(244.51) =
7.68, \textit{p} {\textless} 0.001;
M\textsubscript{NarrowHigh-StakesPure-Loss} = 42.4\%,
M\textsubscript{BroadHigh-StakesPure-Loss} = 8.3\%, \textit{t}(224.50)
= 7.19, \textit{p} {\textless} 0.001). Thus, the results for the
non-positive EV gambles also show a significant bracketing effect:
risk preferences reverse across bracket types, such that subjects are
relatively more risk-seeking when evaluating narrowly bracketed
gambles and relatively more risk averse when evaluating those same
gambles in a broad bracket.  These results extend the previous research by
showing that broad bracketing can lead to more optimal choices across
many risk types, not just risks with positive expected values. It
appears that broad bracketing (via outcome aggregation) consistently
helps individuals adopt more normative choice strategies over
sequential risks.

\paragraph{Risk perception.} To test whether changes in risk
perception may explain bracketing effects, we measured risk perception
for each gamble. If perceived risk varies by the type of bracket used
(and behavior follows beliefs), then we would expect to see higher
perceived risk for the narrowly bracketed positive EV gambles compared
to the broadly bracketed positive EV gambles. For the non-positive EV
gambles, we would expect to see lower perceived risk for the narrowly
bracketed gambles compared to the broadly bracketed gambles. We do see
this expected pattern, with perceived risk higher (lower) for the
narrowly bracketed positive EV (non-positive EV) gambles compared to
the broadly bracketed versions of the gambles (positive EV:
M\textsubscript{Narrow} = 3.66 vs.\ M\textsubscript{Broad} = 2.86,
\textit{t}(286) = 5.15, \textit{p }{\textless} 0.001; negative EV:
M\textsubscript{Narrow} = 4.44 vs . M\textsubscript{Broad} = 5.10,
\textit{t}(286) = --3.67, \textit{p }{\textless} 0.001; pure-loss:
M\textsubscript{Narrow} = 4.19 vs.\ M\textsubscript{Broad} = 5.07,
\textit{t}(286) = --4.48, \textit{p }{\textless} 0.001) (risk
perception ratings for each gamble are shown in Figure A2, Appendix
1).\footnote{Within-subject regressions of choice on bracketing, EV
  (positive vs.\ non-positive), and their interaction, with and
  without the inclusion of risk perception as a covariate, indicate
  that inclusion of risk perception in the models generally reduced
  the interaction term. Because risk perception may not be perfectly
  measured, we cannot say whether risk perception can fully account
  for the interaction.} This suggests that bracketing effects work, at
least in part, by changing the perceived risk associated with the
gamble.

Overall, Study 1 confirms our main predictions: broad bracketing leads
to more optimal risk preferences compared to narrow bracketing for all
types of risks (positive EV, negative EV, and pure-loss).

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

\section{Study 2: Possible Mechanisms Behind Bracketing Effects}
The findings from Study 1 suggest that broad bracketing helps
individuals make more optimal choices over many different types of
financial risk. In Benartzi and Thaler’s (1999) original study, the
authors suggest that broad bracketing attenuates myopic loss aversion.
While the authors did not directly test this mechanism, the implication
is that broad bracketing works both by overcoming loss aversion (the
weight put on losses relative to gains) and by overcoming insufficient
adjustment for the larger number of trials. In Study 2, we introduce a
measure that we refer to as “situational weight on losses” to directly
test how the weight put on possible losses within the task shifts with
the bracketing manipulation.\footnote{ In Study 1, we found that loss
  aversion as an individual trait variable does not attenuate the
  documented bracketing effects; however, it does not rule out the
  possibility that broad brackets change the weight put on losses
  within the context of the gamble.} Since we use non-positive EV
gambles, a conventional measure of loss aversion is not applicable
across all gamble types. To maintain consistency across gamble types,
we focus only on the weight placed on losses (or the importance of
potential losses in the decision calculus) without making a comparison
to the weight placed on gains. We expect the situational weight placed
on losses to be higher (lower) for the narrowly bracketed positive EV
(non-positive EV) gambles compared to the broadly bracketed gambles
for each type. This is because losses will be more salient in these
bracketing formats (i.e., for broadly bracketed positive EV gambles,
the distribution shows the small probability of incurring a loss
across all trials; for broadly bracketed non-positive EV gambles, the
distribution shows the large probability of incurring a loss even
greater than the sure loss (CE) across all trials).

Study 1 is also limited in its ability to test whether broadly bracketed
presentations help individuals recognize the impact of a large number
of trials on cumulative outcomes, thus overcoming insufficient
adjustment from a single gamble to many repeated ones. We directly test
this by using an additional bracketing manipulation that makes salient
the number of repeated trials without directly aggregating the outcomes
across trials (i.e., without providing the probability distribution).
We call this manipulation Broad-Trial because we emphasize the number
of trials (see Figure 2 for an illustration of this manipulation). By including this version of the bracketing manipulation, we are also able to test whether bracketing effects are attributable to a
lack of consideration of the aggregated outcomes. If the Broad-Trial
manipulation is statistically different from the Narrow bracketing
manipulation, we know that bracketing works, in part, by facilitating
mental adjustment for more trials. If only the Broad manipulation
(probability distribution format) is statistically different from the
Narrow manipulation (and the Narrow and Broad-Trial conditions are
statistically equivalent), this would suggest that broad bracketing
primarily works because of its impact on the situational weighting of
losses (via outcome aggregation). We also add a measure rating the
importance of the number of trials to try to address the mental
adjustment individuals make for the repeated trials inherent in the
gamble, and whether the Broad and Broad-Trial conditions facilitate
this adjustment.  Thus, in Study 2 we are focused on further
disentangling the process behind the bracketing effects documented in
Study 1. 

\begin{table*}[t]\centering
\caption*{Table 2: List of Gambles Used in Study 2.}
\begin{tabular}{llccl}\toprule
 & Type & EV & \mc{1}{C{.5in}}{Certain\newline amount} & Description\\\midrule
Gamble 1 & Positive EV & \$6  & \$3  & 50\% win \$0.25, 50\% lose
\$0.15, 120 trials\\
Gamble 2 & Positive EV & \$60  & \$30  & High-stakes: 50\% win \$2.50,
50\% lose \$1.50, 120 trials\\
Gamble 3 & Pure-Loss & --\$7 & --\$4 & 50\% lose \$0.15, 50\% lose
\$0.25, 35 trials\\
Gamble 4 & Pure-Loss & --\$70 & --\$40 & High-stakes: 50\% lose
\$2.50, 50\% lose \$1.50, 35 trials\\
Gamble 5 & Negative EV & --\$4 & --\$2 & 50\% win \$0.15, 50\% lose
\$0.25, 80 trials\\
Gamble 6 & Negative EV & --\$40 & --\$20 & High-stakes: 50\% win
\$1.50, 50\% lose \$2.50, 80 trials\\\bottomrule
\mc{5}{L{5.7in}}{Note: The gambles are ordered sequentially for ease of
exposition  —  in the actual study, the order of the gambles was randomized
across subjects within each condition (Narrow, Broad,
Broad-Trial).}\\
\end{tabular}
\end{table*}

\renewcommand{\baselinestretch}{1.02} \large\normalsize

\subsection[Method]{Method}
Study 2 was conducted online through mTurk with 291
subjects\footnote{ We recruited 300 subjects. Nine subjects
were dropped either because they had the same IP address as another
subject or because they did not complete the second part of the
survey (the DEEP loss aversion measure).} (37.5\% female,
\textit{M}\textit{\textsubscript{age}} = 33.2 years), and was
structured similarly to Study 1. A main difference in Study 2 is that
the bracketing manipulation is now a between-subjects factor, so
subjects saw all gambles in one presentation format only. Thus,
Study 2 is 3 (Bracket Type: Broad, Broad-Trial, Narrow) x 3 (Gamble
Type: Positive EV, Negative EV, Pure-Loss) mixed design. Bracket Type
is a between-subjects factor and Gamble Type is within-subjects. In
each Bracket Type, subjects evaluated six gambles total: two
positive EV, two negative EV, and two pure-loss. One of the gambles in
each gamble type was a high-stakes version, as in Study 1. The specific
gambles are listed in Table 2. Subjects saw all gambles in the
bracket type they were assigned to, and the gambles were the same
across conditions, so that only the format they were displayed in
differed. An example of all of the gambles and other study materials
can be found in \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 2}.

Subjects in the Broad condition saw all gambles in the broad bracket
format used in Study 1 (i.e., they saw the truncated probability
distribution for the full set of gambles). Subjects in the Narrow
condition saw all gambles in the narrow bracket format used in Study 1
(i.e., they saw the gambles described in text only). Finally, subjects
in the Broad-Trial condition saw the static text information (as in
the Narrow presentation) but also saw colored dots representing the
full set of gambles and their outcomes, with red dots for losses and
black dots for gains.\footnote{ For the pure-loss gambles, the dots
  were two different shades of red, since all outcomes were losses, to
  distinguish the two different outcomes and their associated
  probabilities from each other.} For example, one of the positive EV
gambles used was a 50\% chance to win \$0.25 and a 50\% chance to lose
\$0.15, played 120 times. In the Broad-Trial condition, this gamble
was described as in the Narrow condition (120 plays of a gamble with
these outcomes) but below the text description there was an
illustration of 120 blocks (representing each trial) with five red
dots (representing loss probabilities) and five black dots
(representing gain probabilities) per block (see Figure 2). Again,
broad bracketing could work by helping individuals properly adjust the
EV for the multiple repeated trials (rather than just focusing on a
single trial of the gamble) or by aggregating outcomes and making
losses more or less salient as a result. To this end, the Broad-Trial
condition illustrates the number of choices without explicitly
aggregating outcomes so we can distinguish whether the bracketing
effect from Study 1 is driven by a focus on the larger number of
trials (addressing insufficient adjustment) or by providing
information about the probability and size of cumulative losses
(changing the situational weight placed on losses).

The order of the gambles was randomized and counterbalanced across
subjects within a condition. For each gamble, subjects were asked to
choose between (1) taking the gamble, (2) taking the certain amount,
and (3) expressing indifference between the gamble and certain
amount. After making their choice, subjects were asked to rate the
perceived risk for each gamble (as in Study 1). In addition to risk
perception, we also asked all subjects to rate the importance of
losses (situational weight on losses) and the importance of the number
of trials (a measure of adjustment for the repeated trials) for each
gamble. For the situational weight on losses measure we asked
subjects, “how important was the chance of losing money in your
decision of whether or not to take the gamble?” For the adjustment
measure, we asked subjects, “how important was the number of trials in
your decision of whether or not to take the gamble?” Subjects
responded to both measures on a seven-point scale ranging from one
(“Not at all Important”) to seven (“Extremely Important”). The
situational weight on losses measure is especially interesting in that
it allows us to test whether the weight placed on potential losses is
affected by the bracketing manipulations.\footnote{ At the end of the
  survey, after all of the gambling questions, we asked subjects to
  respond to questions measuring individual-level loss aversion
  (lambda) via the DEEP method, and to self-report gender and age, as we did in Study
  1.}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig3.eps} 
\caption*{Figure 4: Choice Shares for the Gamble Across Bracketing
Conditions and Gamble Type, Study 2.\newline
{\small
Notes: (1) Broad, Broad-Trial, and Narrow collapse
across the two gambles within each bracket for a given gamble type
(e.g., the number displayed for the Broad Bracket Positive EV gambles
is the average choice share across the two broadly bracketed gambles
each subject saw). (2) The differences between the Narrow and Broad
conditions are significant at the \textit{p} {\textless} 0.001 level for all
gamble types. The differences between the Broad and Broad-Trial
conditions are significant at the \textit{p} {\textless} 0.001 level  for all
gamble types. The differences between the Broad-Trial and Narrow
conditions are not significant (\textit{p} {\textgreater} 0.10) for all gamble
types. (3) Error bars are for standard errors.}}
\end{figure}

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

\subsection{Results}
Our analysis proceeds as follows: first we address how the
bracketing manipulations affect overall risk preferences (as measured
through choice shares); next we turn to process variables — situational
weight on losses and importance of the number of trials; we then
briefly address perceived risk.

\paragraph{Risk Preferences.} We first evaluated how risk preferences
(choice shares) varied across bracketing conditions for each gamble
type. These results are displayed in Figure 4 (further details are
shown in Figure A3, \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 1}). As Figure 4 shows, we replicate our
bracketing effect from Study 1 across gamble types: subjects in the
Broad bracketing condition are significantly more likely to take the
positive EV gambles than subjects in the Narrow condition
(M\textsubscript{Broad }= 77.5\% vs.\ M\textsubscript{Narrow} =
63.0\%, \textit{t}(188.16) = 2.58, \textit{p }= 0.005, one tailed);
significantly less likely to take the negative EV gambles than
subjects in the Narrow condition (M\textsubscript{Broad }= 12.0\% vs.\
M\textsubscript{Narrow} = 41.7\%, \textit{t}(145.86) = --5.91,
\textit{p }{\textless} 0.001); and significantly less likely to take
the pure-loss gambles than subjects in the Narrow condition
(M\textsubscript{Broad }= 9.5\% vs.\ M\textsubscript{Narrow} = 36.5\%,
\textit{t}(144.40) = --5.41, \textit{p }{\textless} 0.001).

The same pattern of results emerges when comparing the Broad condition
to the Broad-Trial condition, suggesting that the Broad condition also
leads to a bracketing effect relative to the Broad-Trial condition.
Specifically, subjects in the Broad condition were more likely to take
the positive EV gambles than in the Broad-Trial condition
(M\textsubscript{Broad }= 77.5\% vs.\ M\textsubscript{Broad-Trial} =
53.0\%, \textit{t}(190.81) = --4.31, \textit{p }{\textless} 0.001);
less likely to take the negative EV gambles than in the Broad-Trial
condition (M\textsubscript{Broad }= 12.0\% vs.\
M\textsubscript{Broad-Trial} = 44.9\%, \textit{t}(155.82) = 6.89,
\textit{p }{\textless} 0.001); and less likely to take the pure-loss
gambles than in the Broad-Trial condition (M\textsubscript{Broad }=
9.5\% vs.\ M\textsubscript{Broad-Trial} = 41.4\%, \textit{t}(151.94) =
6.59, \textit{p }{\textless} 0.001). It appears that a manipulation
that illustrates the number of trials does not have the same effect on
risk preferences as providing the probability distribution/outcome
aggregation. This suggests that the bracketing effects from Study 1
did not occur because the broader bracket helped subjects sufficiently
adjust the EV for the gamble across all trials (i.e., broad bracketing
does not work by correcting for insufficient adjustment).

\begin{figure}[t]
 \includegraphics[width=\columnwidth]{fig4.eps} 
\caption*{Figure 5: Average Importance of the Number of Trials, Study
2.\newline
{\small
Notes: (1) The Narrow condition combines the Narrow and Broad-Trial
conditions. (2) Broad and Narrow collapse across all gamble choices
within the specified condition for a given gamble type (e.g., the
number displayed for the Broad, Positive EV gambles is the average
importance rating across the two broadly bracketed gambles each
subject saw). (3) The differences between the Narrow and Broad
conditions are significant at the \textit{p} {\textless} 0.001 level for all
gamble types. (4) Error bars are for standard errors.}}
\end{figure}

Finally, if we compare the Broad-Trial and Narrow conditions, we see no
statistically significant differences between the two in terms of
choosing the gamble (for positive EV gambles:
M\textsubscript{Broad-Trial }= 53.0\% vs.\ M\textsubscript{Narrow} =
63.0\%, \textit{t}(192.00) = --1.64, \textit{p }= 0.10; for negative EV
gambles: M\textsubscript{Broad-Trial }= 44.9\% vs.\ 
M\textsubscript{Narrow} = 41.7\%, \textit{t}(191.84) = 0.54, \textit{p
}= 0.59; and for pure-loss gambles: M\textsubscript{Broad-Trial }=
41.4\% vs.\ M\textsubscript{Narrow} = 36.5\%, \textit{t}(192.47) =
0.81, \textit{p }= 0.42). Thus, the Broad-Trial bracketing format is
statistically equivalent to the Narrow bracketing format. This result
was unexpected, as we predicted that the Broad-Trial manipulation
would produce a bracketing effect relative to the Narrow condition
through more sufficient adjustment for more trials, however, this was
not empirically confirmed. Since there was no difference in risk
preferences between the Narrow and Broad-Trial conditions, for the
remainder of our analyses we combine the Broad-Trial and Narrow
conditions (and refer to the combined data as the Narrow condition)
unless otherwise noted.\footnote{ For all of the analyses described
  below, specific details are provided by condition (Narrow, Broad,
  Broad-Trial) in \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 1}.}

% is this true about A1?

\begin{figure}[t]
 \includegraphics[width=\columnwidth]{fig5.eps} 
\caption*{Figure 6: Average Situational Weight on Losses, Study 2.\newline
{\small
Notes: (1) Narrow combines the Narrow and Broad-Trial
conditions. (2) Broad and Narrow collapse across all gamble choices
within the specified condition for a given gamble type (e.g., the
number displayed for the Broad, Positive EV gambles is the average
situational weight on losses across the two broadly bracketed gambles
each subject saw). (3) The differences between the Narrow and Broad
conditions are significant at the \textit{p} {\textless} 0.001 level for all
gamble types. (4) Error bars are for standard errors.}}
\end{figure}

\paragraph{Other Measures. }In previous work on myopic loss aversion, the
bracketing effect between broad and narrow framing has been attributed
to both an attenuation of loss aversion and the salience of multiple
trials. We measured both the importance of potential losses and the
importance of the number of trials directly in Study 2 in order to
distinguish between these two proposed mechanisms. First, we look at
the importance of the number of trials (a proxy for adjustment to the
single gamble for the multiple trials). We would expect this variable
to be of greater importance in the Broad condition compared to the
Narrow condition since the number of trials is more explicitly
illustrated in this condition. 

This expectation, however, was not confirmed in the data, as
summarized in Figure 5 (more details about this variable are provided
in Figure A5, \href{http://journal.sjdm.org/17/17317/Appdendices.pdf}{Appendix 1}). As Figure 5 shows, the importance of
the number of trials in the decision to gamble is significantly
\textit{lower} in the Broad condition than in the Narrow condition
across all gamble types (positive EV: M\textsubscript{Broad }= 4.32
vs.  M\textsubscript{Narrow} = 5.17, \textit{t}(207.35) = 4.83,
\textit{p }{\textless} 0.001; negative EV: M\textsubscript{Broad }=
4.31 vs.\ M\textsubscript{Narrow} = 5.13, \textit{t}(186.39) = 4.63,
\textit{p }{\textless} 0.001; pure-loss: M\textsubscript{Broad }= 4.36
vs.  M\textsubscript{Narrow} = 5.04, \textit{t}(193.93) = 3.50,
\textit{p }{\textless} 0.001). As with the risk preference results,
this suggests that the bracketing effect in the Broad condition is not
attributable to greater adjustment to the single gamble in order to
account for the multiple trials. We believe that this measure is
significantly lower (less important) in the Broad condition relative
to the Narrow condition because the number of trials is implicit in
the distribution (and subjects may not realize that for there to be a
probability distribution, there must be multiple trials underlying the
gamble).\footnote{ It should also be noted that the ratings for the
  importance of number of trials for the Broad-Trial condition is not
  significantly different from the ratings for the Narrow
  condition. This suggests that making the number of trials visually
  more salient does not increase adjustment more than simply providing
  information for the number of trials in text format.}

Next, we consider the effects of bracketing on the importance of losses
(our situational weight on losses measure). We predicted that the
situational weight on losses measure would be significantly lower in
the Broad bracketing condition compared to the Narrow condition for
positive EV gambles, since the Broad bracketing condition would show
the low probability of losses across all trials. In contrast, losses
should be more important and the measure should be significantly higher
in the Broad condition for negative EV and pure-loss gambles, since the
Broad condition for these gamble types shows the higher probability of
incurring a loss even larger than the sure loss (CE). Thus, the weight
placed on the possible loss in the gamble should be greater in the
Broad condition relative to the Narrow condition (where the cumulative
losses are not as clear to subjects).

The results for situational loss aversion are summarized in Figure 6
(for more details on this measure, see Figure A6 in Appendix
1). Again, our hypotheses were not completely confirmed. For the
positive EV gambles, the results are as expected: the situational
weight on losses is significantly lower in the Broad bracketing
condition compared to the Narrow condition (M\textsubscript{Broad }=
3.68 vs.  M\textsubscript{Narrow }= 4.93, \textit{t}(163.36) = 6.01,
\textit{p }{\textless} 0.001). This suggests that the bracketing
effect between the Broad and Narrow conditions is attributable to
changes in the weight placed on losses for positive EV gambles. For
non-positive EV gambles, the evidence for a bracketing effect on the
weight placed on losses appears somewhat weaker, although the ratings
are near the ceiling, with little room to show effects, very likely
because of the large number of losses involved. For negative EV
gambles this effect is not quite significant (M\textsubscript{Broad }=
5.82 vs.\ M\textsubscript{Narrow }= 5.65, \textit{t}(216.19) = --1.44,
\textit{p }= 0.08, one tailed). For pure-loss gambles, the situational
weight on losses is rated as significantly higher in the Broad
condition than in the Narrow condition (M\textsubscript{Broad }= 6.05
vs.\ M\textsubscript{Narrow}\textit{ }= 5.74, \textit{t}(249.56) =
--2.38, \textit{p }= 0.01, one tailed).

The results suggest that the bracketing effect is being driven, in
large part, by reducing (positive EV gambles) or increasing
(non-positive EV gambles) the situational weight placed on losses.
Accordingly, for positive EV gambles, the broad bracket allows
subjects to realize the overwhelming likelihood of a gain relative to
a loss and, as a result, to weight losses as less important; for the
negative EV gambles, the broad bracket allows subjects to realize that
the chance of a gain is unlikely, and thus they weight the cumulative
losses as more important; for the pure-loss gambles, the broad bracket
allows subjects to realize that cumulative losses from the gamble are
significantly greater than the sure loss, and thus they weight these
higher losses as more important.

\pagebreak

Overall, Study 2 has replicated the bracketing effects from Study 1 for
all gamble types, and has provided further process evidence for these
effects. Across two studies we now know that broad bracketing (via
provision of the probability distribution) leads to more optimal risk
preferences for gambles of all types. Our results related to the
importance of the number of trials suggests that this significant
bracketing effect is not due to adjustments made to the single gamble
for the multiple trials. Further, our results for this measure suggest
that individuals do not consider the multiple trial factor when
evaluating a probability distribution (most likely, subjects in
Study 1 did not realize that the narrow and broad brackets provided
different representations of the same gamble). Our findings for the
importance of losses provide evidence that bracketing does affect the
decisional weights placed on losses and this is partly responsible for
changing preferences between brackets. For positive EV gambles, broad
bracketing reduces the weight placed on losses and increases the
attractiveness of the gamble, while for non-positive EV gambles, broad
bracketing increases the weight placed on losses and decreases the
attractiveness of the gamble.

While the situational weight placed on losses is significantly
affected by bracketing manipulations, it does not account for the
entire effect of bracketing on risk preferences.  In combination with
the findings regarding the importance of the number of trials, the
documented bracketing effects appear to be driven in part by changing
the weights placed on losses, but also continue to reflect cognitive
capacity constraints that prevent individuals from aggregating outcomes
and probabilities across multiple trials. Importantly, this implies
that interventions targeted at changing risk preferences over repeated
gambles should focus on removing these cognitive constraints by
explicitly providing probability distributions or aggregated outcomes.

\section{Summary and Discussion}
In two studies, we replicate the findings of previous research showing
that broad bracketing (via the provision of a probability distribution)
leads to relative risk-seeking for positive EV gambles. We were able to
extend these findings by confirming that this result occurs because of
a decrease in the weight placed on losses that occurs when there is an
attenuation of cognitive constraints related to probability
distribution construction. We further extend the findings related to
choice bracketing and decision framing by demonstrating that broad
bracketing can also lead to more optimal risk preferences (reduced
risk-seeking) for gambles over losses (i.e., negative EV and pure-loss
gambles). This bracketing effect is driven by changes in the
situational weight placed on losses (increasing the weight placed on
losses) and by alleviating cognitive constraints. Thus, the same
mechanism lies behind the bracketing effects for all gamble types, but
the direction of the effect varies depending on the expected value
domain. 

We find that a bracketing effect is only produced via provision of a
probability distribution (outcome aggregation) — a similar effect is
not found when using a visual aid to illustrate the number of
trials. This suggests that bracketing manipulations must explicitly
address insufficient adjustment (failing to properly adjust the single
gamble for the multiple trials).\footnote{ This is consistent with
  previous work that tried to disentangle the effects of aggregating
  choices (making multiple choices at once) and aggregating outcomes
  (across choices) (Moher \& Koehler, 2010). The authors of that work
  similarly found that outcome aggregation was necessary to produce a
  bracketing effect.} While providing a probability distribution does
this and increasing the visual salience of the number of trials does
not, this does not rule out the possibility that another form of broad
bracketing could produce a bracketing effect without the explicit
provision of the distribution. Identifying other possible broad
bracketing manipulations is a potentially interesting avenue for
future research.

Our results suggest that in order to help individuals make better
decisions over risk, they should be provided with a probability
distribution over returns (or aggregated outcomes), rather than just a
text description or a visual aid depicting the number of “trials”
inherent to the decision. Doing so will lead to relative risk-seeking
over positive EV investments, and relative risk aversion over negative
EV or pure-loss investments (versus risk aversion and risk-seeking,
respectively). Our findings also suggest that, more generally,
bracketing or framing manipulations that decrease (increase) the weight
placed on losses will lead to more optimal risk preferences for
positive EV (non-positive EV) risks.

Ultimately, a large part of the bracketing effect for all gamble types
is attributable to cognitive capacity constraints. Individuals change
the weight placed on the expected costs/benefits when provided with the
probability distribution, and they are not able to replicate or
construct the distribution on their own even when the number of trials
and cumulative nature of the risk is made more salient. When presented
with a description of the risk, individuals rely on heuristics or
simplified calculations to determine their preferences (i.e.,
calculating the EV of one trial and insufficiently adjusting). The
provision of the probability distribution does not engage more
deliberate or careful thought; rather it provides information that can
be used quickly and efficiently to determine preferences that are
ultimately more optimal. This further implies that it is unlikely that
many subjects even realized that the probability distributions from
Study 1 were representative of the other gambles they had encountered.
Thus, the simple provision of a probability distribution can improve
decision-making over risk, and does so without the need for behavioral
change or effortful thought processes.

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\end{hangparas}

\end{document}
