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\title{‘Tis better to choose and lose than to never choose at all}

\author{
Nathaniel J. S. Ashby\thanks{Data
Analytics, Harrisburg University of Science and Technology, 326 Market
Street, Harrisburg, PA 17101, USA. E-mail: nathaniel.js.ashby@gmail.com)}\;\,\thanks{Technion – Israel  Institute of Technology.}
\and
Tim Rakow\thanks{King's College London, Institute of Psychiatry, Psychology, and
Neuroscience.}
\and
Eldad Yechiam\footnotemark[2]
}

\date{} % leave empty
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\newcommand{\jhead}{Judgment and Decision Making, Vol.~12, No.~6}
\newcommand{\jdate}{November 2017}
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  \jdate \hfill To choose or not to choose \qquad}
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%\begin{latexonly}
\vspace{-3mm}
\begin{onecolabstract}
%\end{latexonly}
  When decisions involve opting in or out of competition many decision
  makers will opt-in even when doing so leads to losses on average.
  In the current paper, we examine the generality of this effect in
  risky choices not involving competition. We found that re-framing a
  sure (certain) zero option as an option to observe the results of
  the other options without choosing would lead to increased
  consequential choice (i.e., increased selection of risky options
  rather than the zero option). Specifically, in two studies we
  compared the rate of consequential choice in a novel paradigm where
  decision makers decide to observe or to choose with consequence from
  a set of risky options (decisions-to-engage) to a full-feedback
  decisions-from-feedback paradigm where the choice set included a
  labeled sure zero option. Compared to decisions-from-feedback,
  participants were more likely to choose from mixed (risky) gambles
  with consequence (over a zero outcome) in decisions-to-engage. This
  occurred irrespective of whether doing so was advantageous (i.e.,
  when choice led to monetary gains on average) or disadvantageous
  (i.e., when choice led to monetary losses on average), and when
  descriptions of the options outcomes and probabilities were provided
  (Study 2). These findings provide an important boundary condition
  for the positive effects of experience on the quality of choice, and
  suggest that decision makers' preference for agency can sometimes
  induce poorer choices.

\smallskip
\noindent
Keywords: loss aversion, loss avoidance, decisions-from-experience,
decision making, decision strategy, agency.
%\begin{latexonly}
\end{onecolabstract}\bigskip
]
%\end{latexonly}

{\renewcommand{\thefootnote}{}

\footnotetext{

Supported by the Leverhulme Trust (RPG-384), the
I-CORE program (1821/12), and the Israel Science Foundation (199/12).

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}

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

Recent work suggests that increased experience with a pair of options
(i.e., making choices and receiving outcome feedback) increases the
rate of selecting options that return higher average rewards in risky
choice, that is, choices that maximize expected value (EV: Ashby \&
Gonzalez, 2017; Konstantinidis, Ashby \& Gonzalez, 2015; Yechiam \&
Hochman, 2013). This increased maximization with increasing experience
is observed both when options provide net gains or losses on average
(Ashby \& Rakow, 2016, 2017), and, under some conditions, such
experiential choice has been found to outperform choice based on a
description of probabilities and outcomes (Jessup, Bishara \&
Busemeyer, 2008).\footnote{In some contexts, such as when outcomes
  occur with low probabilities, patterns of choice consistent with EV
  maximization are greatly diminished (e.g., Barron \& Erev, 2003).}
In addition, most participants appear to employ strategies that
maximize EV, and in addition endorse strategies aimed at EV
maximization (Ashby, Konstatinous \& Yechiam, 2017), although, in some
situations, participants use strategies that maximize value in the
short run (Wulff, Hills \& Hertwig, 2015). Furthermore, experience
appears to reduce biases that can lead to divergence from EV
maximization such as loss aversion and avoidance (e.g., Erev, Ert \&
Roth, 2010; Erev, Gilat-Yihyie, Marchiori \& Sonsino, 2015; Erev, Ert
\& Yechiam, 2008; Yechiam \& Hochman, 2013; but see, Abdellaoui,
L’Haridon \& Paraschiv, 2011; Glöckner, Hilbig, Henniger \& Fiedler,
2016, for negative results).

Most studies involving decisions-from-experience have employed
paradigms like decisions-from-feedback (Barron \& Erev, 2003) or
decisions-from-samples (Hertwig, Barron, Erev \& Weber, 2004). In
decisions-from-feedback (also referred to as the clicking paradigm),
decision makers are forced to make repeated choices between options
with unknown outcomes and probabilities; thus, decision makers learn
about the outcomes and their probabilities only by making
consequential choices. In decisions-from-samples (also referred to as
the sampling paradigm), decision makers sample (simulate playing
options at no monetary cost) from available options before making one
forced consequential choice. As noted, in both tasks decision makers
are forced to make a choice between the options presented to them (as,
indeed, is the case in most laboratory experiments on decision
making).  However, many everyday choices are part of an
inter-connected chain of decisions, in which a choice between options
is predicated on a prior decision of whether or not to pick one of the
available options. For example, we must decide if we want to invest in
the stock market, play the lottery, attend an academic conference, buy
a home, or go on a vacation before we ever consider which stock,
lottery-game, talk, home, or vacation destination we prefer.\footnote{
  And indeed, the decision to choose is itself usually predicated on
  previous decisions. For example, the decision to invest follows a
  decision to save which in turn follows a decision to earn money
  which was preceded by a decision to go to college and so on. Our
  focus in this paper is a pair of decisions from such chains: the
  decision to opt in our out of a given choice, and the (possible)
  subsequent choice from a well-defined set of options.} Thus, the
studies that have found that experience frequently leads to increases
in EV maximization have not captured what may be an important nuance
of many, if not most, decisions made outside the lab (for an
exception, see Erev et al., 2010).

%Table 1. 
\begin{table*}[t!]\centering
\caption{Sets of non-zero options and their outcomes (O1 \& O2) in
points, probabilities (P1 \& P2), and expected values (EV1 \& EV2).
Points were converted at a rate of 30 points to a \$0.01. The first
column indicates which sets were used in each study.}
\begin{tabular}{llrrrrrrrrrrr}\toprule
 &  & \multicolumn{5}{c}{Option 1} & { ~ } & \multicolumn{5}{c}{Option 2} \\\cmidrule{3-7}\cmidrule{9-13}
Study & Set type & O11 & P11 & O12 & P12 & EV1 &  & O21 & P21 & O22 & P22 & EV2 \\\midrule
1 \& 2 & Gain only & 53 & 0.33 & 27 & 0.67 & 36 &  & 75 & 0.33 & 9 & 0.67 & 31 \\
1 \& 2 & Mixed +EV & 53 & 0.33 & $-$13 & 0.67 & 9 &  & 75 & 0.33 & $-$31 & 0.67 & 4 \\
1 \& 2 & Mixed --EV & 28 & 0.33 & $-$53 & 0.67 & $-$26 &  & 50 & 0.33 & $-$71 & 0.67 & $-$31 \\
2 & Gain only & 43 & 0.4 & 28 & 0.6 & 34 &  & 85 & 0.4 & 8 & 0.6 & 39 \\
2 & Mixed +EV & 43 & 0.4 & $-$12 & 0.6 & 10 &  & 85 & 0.4 & $-$32 & 0.6 & 15 \\
2 & Mixed --EV & 18 & 0.4 & $-$0.52 & 0.6 & $-$24 &  & 6 & 0.4 & $-$72 & 0.6 & $-$19 \\\bottomrule
\end{tabular}
\end{table*}

\begin{figure*}[t!]\centering
\includegraphics[clip=true,width=6in]{figA1.eps}
\caption{Screenshot of decisions-to-engage paradigm in Study 1. In the
  decisions-from-feedback condition, ``Observe'' was replaced with ``0
  pts for sure''. In Study 2 the ``Option'' labels were replaced with
  each option's outcome and its probability.}
\end{figure*}

In the current study, we examine the effect of how the decision is
presented by including or excluding the dilemma of whether to choose
or not. (For a similar design involving insurance purchases see Szrek
\& Baron, 2007.) Specifically, we contrast a condition framed as a
forced choice between options (one of which is known to always pay
zero), with a condition in which the sure zero option is framed as an
opportunity to opt out of the choice by taking a passive observational
role. In both conditions, participants learn the final outcome of all
options. One domain where such a framing difference is expected to
have an effect is in decisions where all options return net losses on
average. For example, participants in Erev et al.\ (2010) decided
whether to play a competitive game with another player. They reported
a strong tendency to play the game even if it involved large losses
and smaller gains.  One explanation for this oddity is that it
reflects a general tendency to take risks (for small amounts of money)
in repeated choices independently of the framing of the
decision. Another explanation is that participants derived some value
from competition (Franken \& Brown, 1995).

To clarify this potential impact of decision framing we examine
situations where one can “exclude” oneself from playing by deciding to
observe without consequence (e.g., in order to gather information) and
where no direct competition is involved (i.e., “games against
nature”).\footnote{However, participants may have felt as though they
  were in a competition with the researcher conducting the study
  (i.e., trying to win as much money from the researcher as
  possible).} In Study 1 we examined whether re-framing an option
paying 0 with certainty (as in a standard decision-from-feedback task)
as an “observe” option (our decisions-to-engage task) increases the
selection of (other) available options, including those producing
losses on average. Study 2 replicates and extends Study 1 by including
descriptions of the options outcomes and probabilities and including a
greater variety of choice sets. Our hypothesis is that, when given the
option of opting in or out of consequential choice, decision makers
will be more likely to choose from options with consequences rather
than receiving nothing with certainty. Thus, we test whether decision
makers are less likely to choose a sure zero option when it is framed
as not acting (i.e., observing).

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

\section{General Method}

Both studies employed two paradigms: (1) a novel task combining
the decisions-from-feedback (Barron \& Erev, 2003) and
decisions-from-samples (Hertwig et al., 2004) paradigms that we call
the decision-to-engage paradigm; and (2) a version of the
decisions-from-feedback paradigm (Barron \& Erev, 2003). In each study,
participants were randomly assigned to one of these two paradigms
(conditions).

In the decisions-to-engage condition, participants were informed that
they would face several tasks in which they would make 100 decisions
between alphabetically-labeled options (e.g., “Option A”, “Option B”,
shown in Figure 1).  Table 1 shows the sets of options in each study.
In addition, they were informed that they could instead choose to
observe by pressing an option labeled “observe”. When participants
opted to observe (i.e., clicked the observe button) they were shown
the outcomes that occurred for all available options (i.e., what they
would have won or lost had they chosen with consequence). If they
choose with consequence they saw the outcome that occurred for the
option they selected, as well as what they would have won or lost had
they instead selected the other option. Thus, full feedback was
available for all options irrespective of the choice they made. The
decisions-from-feedback condition was identical to the
decisions-to-engage condition except the observe option was labeled “0
pts.\ with certainty”. Thus, in the decisions-to-engage condition,
selecting the sure zero option was framed as taking an
inactive/passive role, while in the decisions-from-feedback it was
framed as an active choice of receiving nothing.

%Table 2. 
\begin{table*}[t]\centering
\caption{Participant details and pay.}
\begin{tabular}{@{}ccccccc@{}}\toprule
Study & N & N Decisions-from-Experience  & N Decisions-to-Engage & \% male & $M_{age}$ & Pay received\\\midrule
1 & 207 & 103 & 104 & 50\% & 34.4 & \$0.75 + choice-contingent pay ($\sim$\$1)\\
2 & 231 & 121 & 110 & 47\% & 36.2 & \$1.50 + choice-contingent pay ($\sim$\$2)\\\bottomrule
\end{tabular}
\end{table*}

% changed female to male

In both studies, irrespective of condition, outcomes remained on the
screen until a participant made the next decision; decisions for a
given choice set concluded after 100 decisions; participants
encountered all choice sets in random order; and participants were
told when they had finished their choices for each choice set and that
a new set of choices would follow. Participants were under no time
constraints when making their decisions.

As indicated in Table 1 some (non-zero) choice sets consisted of only
gains (positive outcomes), while others consisted of both gains and
losses (mixed outcomes). In half of the mixed outcome sets, all options
had positive EVs (+EV; i.e., consequential choice led to gains on
average), while, in the other half of the mixed outcome sets, all
non-zero options had negative EVs (--EV: i.e., consequential choice led
to losses on average). Study 1 used one gain only, one mixed +EV, and
one mixed --EV option-set while Study 2 included two of each. All sets
included an EV maximizing option: The safer option was always better in
Study 1 while the safer and riskier options were better with equal
frequency across sets in Study 2. Options were randomly assigned to
buttons (labels) at the start of each task.

%Figure 1 -> 2
\begin{figure}[t!]
\includegraphics[clip=true,width=\columnwidth]{fig1.eps}

\includegraphics[clip=true,width=\columnwidth]{fig2.eps}
\caption{Proportion of participants making consequential choices in
  Study 1 (top) and Study 2 (bottom): for options with positive outcomes
  only (left panel); options with mixed outcomes with a positive EV
  (+EV; middle panel); and options with mixed outcomes with a negative
  EV (--EV; right panel); by condition: decisions-from-feedback (DF)
  and decisions-to-engage (Engage). Asterisks indicate that the rate
  of consequential choice was greater than 50\%. Error bars indicate
  95\% confidence intervals.}
\end{figure}

Both studies were conducted on Amazon Mechanical Turk. Study 1 preceded
another study and lasted 20 minutes on average while Study 2 was run
alone and lasted 30 minutes on average. To ensure that participants
took their decisions seriously all consequential decisions a
participant made were realized.

\subsection{Participants}
Table 2 shows the participant characteristics across studies; we aimed
for 100 participants per condition in Study 1, while Study 2’s sample
size was based on a power analysis aiming to provide high power
($1 - \beta$ = .90) to detect the difference in choice proportions
between the decisions-from-feedback and decisions-to-engage conditions
in the mixed negative EV pair, based on the difference observed in
Study 1.

\section{Results of Study 1}

Figure 2 (top) displays the proportion of participants choosing with
consequence from the gain-only option pair (left panel) and the
mixed-outcome option sets providing positive (+EV; middle panel) and
negative EVs (--EV; right panel), plotted separately by condition
(decisions-from-feedback, DF; decisions-to-engage, Engage).

We compared the rate of consequential choice (choices in each pair
averaged by participant) in the decision-to-engage condition to the
rate of not choosing the certain zero option in the
decision-from-feedback condition using independent \textit{t}{}-tests.
We found that in the gain-only pair those in the decisions-to-engage
condition (95\%; CI\textsubscript{95\%}[.93, .98]) did not choose with
consequence significantly less than those in the
decisions-from-feedback condition (97\%; CI\textsubscript{95\%}[.95,
.99]), \textit{t}(205) = .83, \textit{p} = .41. Nevertheless, in the
mixed outcome positive-EV option pair those in the decisions-to-engage
condition (66\%; CI\textsubscript{95\%}[.59, .72]) chose with
consequence more (maximized more) so than those in the
decisions-from-feedback condition (53\%; CI\textsubscript{95\%}[.47,
.59]), \textit{t}(205) = $-$2.89, \textit{p} = .004. Similarly, in the
mixed outcome negative-EV option pair those in the decision-to-engage
(56\%; CI\textsubscript{95\%}[.49, .62]) condition chose with
consequence more than those in the decision-from-feedback condition
(45\%; CI\textsubscript{95\%}[.39, .51]), i.e., maximized EV less,
\textit{t}(205) = $-$2.49, \textit{p} = .01.

Figure 3 displays the rate of not choosing the observe or sure zero
option (i.e., the rate of consequential choice) over decisions
separately for each option-pair and condition. It appears to reflect a
decreasing in consequential choice in the decisions-to-engage
condition over time in the mixed sets.\footnote{ For additional
  analyses see the supplementary analysis.} For the final 25
decisions, the decisions-to-engage and decisions-from-feedback
conditions did not differ in the gain-only pair, \textit{t}(205) =
.94, \textit{p} = .35. However, in the mixed positive (\textit{t}(205)
= $-$2.63, \textit{p} = .009) and negative EV sets (\textit{t}(205) =
$-$3.02, \textit{p} = .003) the rate of consequential choice was
higher in the decisions-to-engage condition than in the
decisions-from-feedback condition (i.e., experience did not eliminate
the effect of frame).

% %Figure 2
% \begin{figure}[t]
% \includegraphics[clip=true,width=\columnwidth]{fig2.eps}
% \caption{Proportion of participants making consequential choices in
% Study 2: for options with positive outcomes only (left panel); options
% with mixed outcomes with a positive EV (+EV; middle panel); and options
% with mixed outcomes with a negative EV (--EV; right panel); by
% condition: decisions-from-feedback (DF) and decisions-to-engage
% (Engage). Asterisks indicate that the rate of consequential choice was
% greater than 50\%. Error bars indicate 95\% confidence intervals.}
% \end{figure}

\section{Study 2}

% Figure 3
\begin{figure}[t!]
\includegraphics[clip=true,width=\columnwidth]{figA3.eps}

\includegraphics[clip=true,width=\columnwidth]{figA4.eps}
\caption{The rate of consequential choice over decisions in Study 1
  (top) and Study 2 (bottom), plotted separately for sets containing
  gains only (Gains Only), mixed outcomes with a positive EV (Mixed
  +EV), and mixed outcomes with a negative EV (Mixed --EV), and by
  condition (black = decisions-from-feedback; red =
  decisions-to-engage).}
\end{figure}

\subsection{Changes to Method}

Study 2 was designed to address two shortcomings of Study 1 and to
examine a potential boundary for the observed framing effect. First,
due to a programming oversight, foregone outcomes were not recorded in
Study 1, precluding potentially informative model fitting. Second, in
Study 1 the EV maximizing option was always the safer option.
Therefore, in Study 2 three new sets of options were added (one gain
only, one mixed +EV, and one mixed --EV) in which the riskier option
provided the highest average payout (see Table 1). Study 2 also added
descriptive information about each option. Specifically, instead of the
options being labeled “Option A” or “Option B” they were labeled with
their possible outcomes and the probabilities of those outcomes
occurring. Thus, Study 2 allows us to examine whether the effect of
framing persists when full information is available: We might, for
instance, predict that if participants are explicitly aware that the
likelihood and size of a loss occurring is greater than that of a gain
they would show a stronger preference for the sure zero option.

\subsection{Results}

Figure 2 (bottom) displays the proportion of participants choosing
with consequence from the gain only choice sets (left panel) and the
mixed outcome choice sets providing positive (+EV; middle panel) and
negative EVs (--EV; right panel), plotted separately by condition
(decisions-from-feedback – DF; decisions-to-engage – Engage).

As in Study 1 we compared the rate of consequential choice (choices in
each pair type averaged by participant) in the decision-to-engage
condition to the rate of not choosing the sure zero option in the
standard decision-from-feedback condition. Counter to Study 1 we found
that in the gain only choice sets those in the decisions-to-engage
condition (97\%; CI\textsubscript{95\%}[.96, .99]) chose with
consequence slightly, but significantly, less than those in the
decisions-from-feedback condition (99\%; CI\textsubscript{95\%}[.99,
.99]), \textit{t}(229) = 2.88, \textit{p} = .004. Those in the
decisions-to-engage condition (77\%; CI\textsubscript{95\%}[.72, .82])
chose with consequence marginally more than those in the
decisions-from-feedback condition (71\%; CI\textsubscript{95\%}[.65,
.76]) in the mixed outcome positive EV choice sets (i.e., maximized EV
more), \textit{t}(229) = $-$1.81, \textit{p} = .07. As in Study 1, those
in the decision-to-engage condition (60\%; CI\textsubscript{95\%}[.54,
.67]) also chose with consequence more than those in the
decision-from-feedback condition (51\%; CI\textsubscript{95\%}[.45,
.56]) in the mixed outcome negative EV choice sets (i.e., maximized EV
less), \textit{t}(229) = $-$2.19, \textit{p} = .03.

%Figure 3 -> 4
\begin{figure}[t]
\includegraphics[clip=true,width=\columnwidth]{fig3.eps}
\caption{The number of participants estimated as having a given
$\lambda $ (Lambda) by condition: decisions-from-feedback (DF) and
decisions-to-engage (Engage).}
\end{figure}

Figure 3 shows the rate of not choosing the observe
or sure zero option (i.e., the rate of consequential choice) over
decisions separately for each choice set and condition and appears to
show a stronger decline in consequential choice in the mixed sets with
experience than observed in Study 1. Focusing on the final 25
decisions between the decisions-to-engage and decisions-from-feedback
conditions no differences in significance from those reported above
were found for the gain only and mixed outcome positive EV sets. The
greater rate of consequential choice in the decisions-to-engage
condition was, however, no longer significant in the mixed negative EV
choice sets, \textit{t}(229) = $-$1.50, \textit{p} = .13.
% Thus, counter to Study 1
% after a great deal of experience the gap between the two conditions
% when options provided losses on average was reduced.

% You did not test the reduction. The fact that one is significant and
% the other is not does not imply that the two are different.

Because information about the outcomes and their probabilities of
occurrence were provided, the first choice in each pair was roughly
equivalent to a decision in a descriptive choice format without
feedback. We therefore analyzed whether there was any difference in the
rate of consequential choice between the two conditions for these first
choices in each set type. No significant differences were found
(\textit{t}s {\textless} 1.48, \textit{p}s {\textgreater} .14).

While the results reported above may suggest that participants were
less sensitive to losses in the decisions-to-engage condition, another
way to measure aversion to losses is to employ computational modeling
to determine how losses are weighted in the decision process. We
accomplished this by employing a standard reinforcement learning model
including a loss aversion parameter ($\lambda $) which estimates the
weighting of losses in the decision process (Ahn, Busemeyer,
Wagenmakers \& Stout, 2008; see the supplementary analysis in the
Appendix for details regarding the model and model fitting). The model
was run for the conditions with mixed gains and losses – estimating
loss aversion in gain only problems does not make sense. In these
conditions if $\lambda $ is greater than 1 this indicates that losses
were weighted more heavily than gains --- in line with loss
aversion. However, if $\lambda $ equals 1 this indicates losses and
gains were weighted equally, while $\lambda $ {\textless} 1 indicates
that losses were weighted less than gains (i.e., the opposite of loss
aversion). Figure 4 shows a histogram of these $\lambda $ by
condition. We find that the median value of $\lambda $ in the standard
decisions from experience condition was .88 whereas in the decisions
to engage condition it reduced to .52 (Mann-Whitney \textit{Z} = 3.12,
\textit{p} = .002), suggesting that participants treated losses as if
they were less consequential in the decision-to-engage condition. It
is notable that the median participant (in either condition) appears
to weigh losses less heavily than gains; this is an unusual state of
affairs in decision research studies (i.e., losses do not loom larger
than gains in our data).

\section{Discussion}

We aimed to determine whether framing a decision as one of taking an
active or passive role would increase the rate of selecting sure zero
options, even when those options return losses on average. We find
that framing the decision as a decision of whether or not to observe
increased consequential choice: Participants chose with consequence
from mixed outcome options when they provided both positive and
negative EVs more than they did in the decisions-from-feedback
paradigm and either equal to or greater than chance. While this
increase in consequential choice decreased participants’ earnings when
options returned net losses on average (i.e., in the mixed outcome
negative EV option sets), it increased them when options returned net
gains on average (i.e., in the mixed outcome positive EV option sets).
In addition, modeling of participants’ decisions suggests that for
most participants loss aversion was reversed in both conditions,
though to a greater extent in the decision-to-engage paradigm. It
seems that when decisions are framed as a decision to opt in or out of
taking action participants are less avoidant of potential losses, even
when doing so puts them in harm’s way.

These findings add to the literature on loss aversion (the notion
that, typically, “losses loom larger than gains”; Kahneman \& Tversky,
1979; Tversky \& Kahneman, 1991). One logical consequence of loss
aversion is that a sure outcome of zero is preferred to a mixed gamble
with an EV of zero (Kahneman \& Tversky, 1979): This is implied by the
steeper relation between outcomes and subjective value for losses
compared to gains in the prospect theory value function. Indeed,
Tversky and Kahneman (1992) reported that, for the median participant,
a mixed outcome positive EV gamble (50/50 gamble between +\$202 or
–\$100) was indicated as being as attractive as a sure outcome of
\$0. Yet, in our studies, a substantial proportion of participants
chose a mixed option with a worse-than-zero EV over a sure zero option
even with substantial experience (Studies 1 \& 2) and complete
knowledge of the outcomes and probabilities (Study 2). Such choices
appear counter to loss aversion because a loss averse decision maker
would take the sure zero rather than face the risk of losing money
(nearly twice as much as they could win in the negative-EV choice sets
in Study 1 and some in Study 2)\footnote{ Note, however, that when we
  say that these choices are \textit{not} loss averse, this should
  \textit{not be misunderstood} as implying that participants are
  insensitive to the presence of losses.  Indeed, we find that the
  proportion of consequential choices is lower for those choice sets
  in which losses occur (and in which the consequential choices
  therefore have lower EVs). Thus participants are sensitive to EV and
  to the presence of losses, but are arguably insufficiently sensitive
  to these features: for +EV sets they sometimes make \textit{too few}
  consequential choices to maximize EV, while for --EV sets they often
  make too many consequential choices to maximize EV.  Thus in keeping
  with Peterson and Beach’s (1967) summary of research comparing human
  decisions against normative models, we observe another instance for
  which decision makers are “influenced by appropriate variables and
  in appropriate directions … [while] the degree of sensitivity is
  often less than optimal” (p.\ 43).}. Importantly, this adverse
tendency is greatest when the decision is framed as a
decision-to-engage; thereby adding to the growing literature which
shows that sensitivity to losses is in many ways context dependent
(e.g., see Yechiam \& Hochman, 2013; Walasek \& Stewart, 2015).

It is also possible that framing the decisions as a decision-to-engage
decreased risk avoidance. In other words, participants might not
(only) have underestimated the impact of a loss on their total
earnings, but (additionally or alternatively) might have misjudged
their relative frequency of occurrence (even when description was
provided in Study 2) which would also increase the attractiveness of
choosing from the options with consequence, or might simply found the
idea of risk taking to be fun.  However, if diminished risk aversion
of these sorts was the main contributor to the effect, we would also
expect to see the rate of playing with consequence to be greater in
the decisions-to-engage paradigm than the decisions-from-feedback
paradigm in the gain only choice sets as well. This pattern was not
observed, and was in fact reversed in Study 2 (although, in both
studies, choice with consequences was very close to ceiling).  Thus,
future studies of this decision-to-engage effect would be wise to
select sets of options designed to specifically test how changes in
risk sensitivity might contribute to the effect.

One explanation for the current results that fits irrespective of the
root cause (decreased loss aversion/avoidance and/or risk avoidance),
and can also explain why EV maximization is often observed in
decisions-from-feedback, is that in the decision-to-engage paradigm
participants developed a myopic focus on the differences in payouts
between the two options (Vlaev, Chater, Stewart \& Brown, 2011;
Stewart, Chater \& Brown 2006), rather than the “bigger picture” of
the average outcomes actually paid out (Kahneman \& Lovallo, 1993;
Hills \& Hertwig, 2010). Thus, explicitly presenting the options as a
two-stage process (a choice to engage, and – if engaging – a choice
between options) may mean that decision makers are less likely to
compare all options simultaneously. Specifically, participants may
have counted the option of not participating as disadvantageous
(Wilson et al., 2014).  Therefore, consistent with a heuristic such as
“you have to be in it to win it”, the mere possibility of gaining
something was more rewarding than doing nothing at all, while the size
of gains and losses and their frequencies of occurrence played a
lesser part in this decision (Wells \& Windschitl, 1999; Reyna \&
Brainerd, 2008). Potentially, this myopic focus emerged in the current
setting because participants perceived opting out as losing their
agency, which research suggests individuals are averse to (Paternoster
\& Pogarsky, 2009). This explanation speaks to the seductive ‘grasp’
of the casino, on-line bookmaker, or ‘betting shop’. Once the bettor
is drawn into the gambling environment (e.g., by ‘free’ bets, cheap
entertainment or cost-effective meals) the choices then encountered
there (framed as opt-in decisions) will be more attractive than the
equivalent choice encountered in a different context.

Another interesting finding observed in the present studies is that the
rate of observational choice increased when losses were possible
compared to when they were not: The EVs were also smaller in the mixed
outcome sets and outcomes varied more both of which may have played a
role. This result seems to be consistent with recent findings in the
decisions-from-samples paradigm that participants search more before
making a consequential choice if losses are possible (Lejarraga et al.,
2012; Lejarraga \& Hertwig, 2016; see also Yechiam, Zahavi \& Arditi,
2015). More generally, it is consistent with the notion of loss
attention (Yechiam \& Hochman, 2013), suggesting that losses lead to
increased vigilance and scrutiny of the environment. Importantly,
though, the current findings indicate that this effect of losses on
observational choices was not strong enough to counteract individuals’
tendency to engage in consequential choices that included more frequent
losses despite their disadvantageousness.

Our results might seem inconsistent with the “loss attention” account
(Yechiam \& Hochman, 2013) which predicts increased EV maximization in
tasks involving potential losses. Our results show a robust violation
of EV maximization both when EVs were positive (choosing without
consequence) and negative (choosing with consequence). However, our
study did not directly compare EV maximization levels in the different
domains, since problems presented in each domain were different in
terms of whether the status quo option (observe or 0 option) was
advantageous or not (in the gain domain or loss domain, respectively).

A shortcoming of the current work, and of most research exploring
decisions-from-experience in the lab, is that decisions outside of the
lab rarely happen back to back in a rapid fashion. Similarly, apart
from playing state lotteries, casino games, and some forms of
investment (and reinvestment), few decisions involve repeated choices
from options providing monetary outcomes with fixed probabilities. We
would thus urge caution in inferring the generalizability of the
current findings to real-world behaviors where decisions are more
complex. Nonetheless, our finding that framing a sure zero option as
an opt-out option can increase risk-taking (while, conversely, framing
an opt-out option as “taking zero” can decrease risk taking) is
potentially important. For example, it may speak to the seduction of
inducements to gambles that explicitly focus on what may be missed out
on by electing not to gamble, and the policy or regulatory decisions
about how casinos and bookmakers may advertise their “services”. And
there are many situations in which evaluations of products,
opportunities, people or situations are based on experiences or
observations that are in fact made in rapid succession. Thus, one may
consider multiple views of an item, several online reviews of a hotel,
or size up a new acquaintance or location based on multiple
observations – all acquired in just a few seconds. Moreover, such
evaluations are tied to decisions: once observations or experiences
have accumulated, one can walk away from the potential purchase, book
the hotel, close the door on the salesman, or flee the dark foreboding
street. What we learn from laboratory choices that require rapid
evaluation of observations may inform our understanding of such
situations (e.g., whether ``first impressions'' weigh heavily in the
evaluation and the decision that follows; Ashby \& Rakow, 2016;
Denrell, 2005).

In sum, the current results provide some insight into the role that
experience and choice architecture play in
decisions-from-experience. We find that when decisions are framed as a
decision to engage or not many decision makers choose to play with
consequence irrespective of the possibility and size of potential
losses and gains. This increase in consequential choice can be both
helpful and harmful depending upon the option payoffs, and therefore
provides an important boundary condition for when experience might aid
decision makers and when it might lead to self-harm. More generally,
our studies point to a gap in the literature. JDM researchers “know”
the typical pattern of risk preference under different conditions
(e.g., the “four-fold pattern” summarized in textbooks), because there
have been hundreds of carefully conducted studies of risky choice.
However, almost all of these studies employed forced choice paradigms.
Our studies illustrate that the patterns of preference may not look
the same when decision makers can choose not to choose and garner some
experience (e.g., the fourfold pattern has been shown to be reversed
in experiential choice; Hertwig \& Erev, 2009). Researching such
decisions will give us a fuller understanding of how preferences play
out in the decisions where “opting out” is an option.

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%\vfill
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\end{hangparas}

\bigskip

\section*{Appendix: Supplementary Analysis}

\begin{table*}[t!]\centering
\caption*{Supplementary Table 1: Median model fit (BIC) and parameter values in
the two study conditions for the problems involving mixed gains and
losses in Study 2. Standard errors appear in parenthesis.}
\begin{tabular}{@{}l@{}ccccc@{}}\toprule
Condition & Model fit  (BIC) & Diminishing sensitivity ($\gamma$) & Loss aversion ($\lambda$) & Recency  ($\phi$) & Sensitivity  ($c$)\\\midrule
Decisions-from-Feedback & 36.19 (23.0) & 0.61 (.04) & .88 (.34) & .03 (.04) & .58 (.38)\\
Decision-to-Engage & 34.01 (18.1) & 0.45 (.04) & .52 (.28) & .02 (.03) & .91 (.40)\\\bottomrule
\end{tabular}
\end{table*}

\subsubsection*{Study 1}
To explore what influenced participants to choose a higher value option
when they did choose with consequence (did not choose the sure zero
option), we performed a logistic regression predicting a choice of the
higher value option by condition, choice set type (entered linearly and
centered), and their interaction. We controlled for repeated
measurement by clustering on the level of participant (Rogers, 1991).
Only choice set type was significant with the rate of picking the
higher value option being lower when losses were large
(\textit{M}\textsubscript{Gains} = .68;
\textit{M}\textsubscript{Mixed +EV} = .53;
\textit{M}\textsubscript{Mixed --EV} = .44), Odds Ratio = .59, \textit{z}
= $-$6.68, \textit{p} {\textless} .001. Other \textit{p}s {\textgreater}
.15.

\subsubsection*{Study 2.}
To explore what influenced participants to choose a higher value
option when they did choose with consequence (did not choose the sure
zero option), we performed a logistic regression predicting a choice
of the higher value option by condition, gamble type (entered linearly
and centered), whether the riskier option was of higher value, as well
as their respective interactions. We controlled for repeated
measurement by clustering on the level of participant. We find that
the likelihood of picking the higher EV option was lower as the
overall value of the options decreased (i.e., going from the gain only
to the mixed negative EV pair), Odds Ratio = .66, \textit{z} = 5.19,
\textit{p} {\textless} .001, and increased when the riskier option was
of higher value, Odds Ratio = 1.29, \textit{z} = 2.12, \textit{p} =
.03. These effects are qualified by their interaction indicating that
participants tended to prefer the riskier options when losses were
larger (Riskier worse: \textit{M}\textsubscript{Gains} = .64;
\textit{M}\textsubscript{Mixed +EV} = .54;
\textit{M}\textsubscript{Mixed --EV} = .59; Riskier Better:
\textit{M}\textsubscript{Gains} = .61; \textit{M}\textsubscript{Mixed
  +EV} = .46; \textit{\textsubscript{M}}\textsubscript{Mixed --EV} =
.72), a preference which was rewarding when the riskier was better,
but costly when it was not, Odds Ratio = 2.55, \textit{z} = 7.23,
\textit{p} {\textless} .001.  Lastly, we find that those in
decisions-to-engage condition (\textit{M} = .63;
CI\textsubscript{95\%} [.59, .66]) made more choices for the higher
value options than those in the decisions-from-feedback (\textit{M} =
.57; CI\textsubscript{95\%} [.55, .59]) condition, Odds Ratio = 1.26,
\textit{z} = 2.01, \textit{p} = .04. None of the other interactions
reached significance, \textit{p}s {\textgreater} .07.

\subsubsection*{Ahn et al.\ (2008)’s learning model.}
This model is based on prospect theory’s value function (Kahneman \& Tversky, 1979) in that:

\noindent
$if\; x(t)>0:\; u(t) = x(t)^{\gamma} $

\noindent  
$if\; x(t)<0:\; u(t) = -\lambda | x(t) |^{\gamma}$

\noindent
where $u(t)$ is the utility for outcome $x$ in trial $t$;
$\lambda$ is the loss aversion parameter, which was constrained between 0
and 10; and $\gamma$ is the diminishing sensitivity parameter, which was
constrained between 0 and 1. The model further assumes that the
participants learn from experience using a delta learning rule, as
follows:
\[E_{j(t)} = E_{j(t-1)} + \phi [ v(t) - E_{j(t-1)} ] \]

\noindent
where $E_j$ is the expectancy (or propensity) of each option $j$, and
$\phi$ is the learning rate parameter (validation of this learning
assumption appears in Ahn et al., 2008).\footnote{ The differences in
  $\lambda$ between study conditions are replicated when using
  alternative learning assumption (e.g., a decay based learning model;
  Ahn et al., 2008).} Choices are a stochastic function of the
expectancies, as follows:\vspace{-1ex}
\[ Pr[ G_j,\, (t+1) ] = \frac{ e^{\phi E_{j(t)}} }{\sum_k e^{\phi E_{k(t)}} } \]

\vspace{-1ex}
\noindent
where $Pr[ G_j,\, (t) ]$ is the probability of selecting option
\textit{G} in trial \textit{t}, and $\theta$ determines whether one is
making more or less deterministic choices (i.e., whether the predicted
choice proportion is determined according to the expectancies). As in
Ahn et al.\ (2008) and Yechiam and Ert (2007), $\theta = 3c$, and the
range of parameter $c$ was constrained between 0 and 10 (higher values
of $c$ imply greater determinism).

The fit of the model to the current two datasets was compared to a
simple baseline model based on the optimized proportion of the choices
of the different options: The baseline model prediction is the mean
probability of selecting each alternative. A BIC test (Schwartz, 1978)
was used to compare the baseline to the learning model, as follows:
{\small
$BIC = 2[Log\,Likelihood\,Model -Log\,Likelihood\,Baseline] - k \cdot ln(t),$}

\noindent
where $k$ is the difference in the number of parameters between the
learning model and baseline model (i.e., $k = 2$). Modeling took place
as in Ahn et al.\ (2007). The results showed that model fits were
adequate (mean BIC of 99.14, median of 22.13). Because the mean of the
parameter estimates for $\lambda$ is biased for loss aversion (e.g.,
mean of $\lambda = 2$ and $\lambda = 0.5$ is 1.25 despite the fact
that the two estimates are equally loss averse and gain seeking), we
used median parameters, as shown in Supplementary Table 1.

\end{document}
