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\title{Reversing the endowment effect}

\author{
Campbell Pryor\thanks{School of Psychological Sciences, University of Melbourne, VIC, Australia}
\and 
  Amy Perfors\footnotemark[1]% indicates same as 1st thanks
\and
  Piers D. L. Howe\footnotemark[1] % indicates same as 1st thanks
}


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\newcommand{\jref}{http://journal.sjdm.org/vol13.3.html}
\newcommand{\jhead}{Judgment and Decision Making, Vol.~13, No.~3}
\newcommand{\jdate}{May 2018}
\pagestyle{myheadings} \markright{\protect\small \href{\jref}{\jhead}, \jdate \hfill Reversing the Endowment Effect \qquad}
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\href{\jref}{\jhead}, \jdate, pp.\ 275-286
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\begin{onecolabstract}
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  When given a desirable item, people have a tendency to value this
  owned item more than an equally-desirable, unowned item. Conversely,
  when the endowed item is undesirable, in some circumstances people
  have a tendency to swap it for an equally undesirable item, a
  phenomenon known as the reversed endowment effect. The fact that the
  endowment effect can reverse for undesirable items has been taken as
  evidence against loss aversion being the underlying cause of the
  endowment effect. This study represents the first time that the
  reversed endowment effect has been observed for choices with real
  consequences. However, we find that the reversed endowment effect
  occurs only when participants' ability to compare the available
  choice options is limited. We further show that these endowment
  reversals can also be induced for choices between desirable options
  and removed for choices between undesirable options by manipulating
  the expectations participants have when making a choice. Finally, we
  show that our data, including endowment reversals, can in principle
  be explained by loss aversion.

\smallskip
\noindent
Keywords: endowment effect, endowment reversal, prospect theory, loss aversion, undesirable, bads, gambles, prior expectations, comparisons, time pressure, reference points
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{\renewcommand{\thefootnote}{}
\footnotetext{ % note blank lines above and below acknowledgment
This research is supported by an Australian Government Research Training Program (RTP) Scholarship

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

\saythanks

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\section{Introduction}

The endowment effect, the finding that mere ownership of an item tends
to increase the value assigned to it, has maintained consistent
research interest for decades (Thaler, 1980). Valuation paradigms have
found that people often require more money to give up ownership of an
object than they would be willing to pay to acquire that same object
(Kahneman, Knetsch \& Thaler, 1990). Furthermore, preferential choice
paradigms have found that preference for an item tends to be greater
when it is owned than when an alternative item is
endowed. Interestingly, Brenner, Rottenstreich, Sood and Bilgin
(2007) found that this preference for keeping endowed items reverses
when the items are undesirable. For example, they found that when
choosing between a speeding fine and attending traffic school,
participants preferred to switch to whichever option they were not
endowed with.

Brenner et al.'s (2007) results are particularly interesting from a
theoretical perspective because they argue against the most prominent
explanation of the endowment effect, prospect theory. Prospect theory,
and related theories, propose that people assign greater value to
items they own because they evaluate other options relative to their
endowed state (Kahneman \& Tversky, 1979). Thus, an
equivalently-valuable, non-endowed alternative will have some
advantages (gains) and some disadvantages (losses) relative to the
endowed option. Importantly, prospect theory assumes that people are
loss averse such that the relative losses of the non-endowed
alternative loom larger than that alternative's relative gains. Thus,
people perceive that switching to the non-endowed alternative has a
net negative value and consequently choose to keep the endowed
option. As Brenner et al.\ point out, this explanation focuses only on
the gains and losses of an alternative relative to the endowed option,
meaning whether the options are both desirable or both undesirable
should be irrelevant. As a result, under these assumptions, prospect
theory cannot explain why the endowment effect reverses for
undesirable options.

An explanation of the reversed endowment effect, proposed by Brenner
et al.\ (2007), revolves around the idea that \textit{all} attitudes
towards the endowed option are amplified. When both options are
desirable, the endowed option becomes more desirable and thus
favoured; when both options are undesirable, the endowed option
becomes less desirable and less favoured. Brenner et al.\ (2007)
suggested that this amplification occurs because losing the endowed
option itself is weighted more than gaining an alternative item. Thus,
losing a desirable option that you own is particularly negative
(explaining the endowment effect) whilst losing an undesirable option
that you own is particularly positive (explaining the endowment
reversal). Attentional theories such as salience theory have a similar
interpretation, suggesting that we amplify the valence of the endowed
option because we process it first, making the desirable or
undesirable features of the endowed option stand out more (Bordalo,
Gennaioli \& Shleifer, 2012, 2013). Amplification effects may contribute towards a number of other judgement biases such as the omission bias (Baron \& Ritov, 1994) and serial presentation effects (Houston, Sherman \& Baker, 1989).

Whilst Brenner et al.'s (2007) observed endowment reversals have been
taken as support for this amplification effect, recent research
suggests the story is not so simple. Dertwinkel-Kalt and Köhler (2016)
looked at whether the reversed endowment effect, which was previously
observed for hypothetical choices, generalised to incentivised
choices. Participants chose between tedious, undesirable tasks that
they subsequently either had to complete (incentivised condition) or
did not have to complete (hypothetical condition). Under the
hypothetical condition, they found data consistent with Brenner et
al.'s endowment reversal. However, when the choices were incentivised
they found that participants returned to preferring the endowed
task. While amplification effects can explain an endowment reversal in
the hypothetical condition, they cannot explain the preference for
undesirable endowments in the incentivised condition.

Dertwinkel-Kalt and Köhler (2016) explained their observed difference
between incentivised and hypothetical choices with a dual-process
model. They suggested that, in line with salience theory, the endowed
option is processed first. For hypothetical choices, people are not
motivated to engage in any further processing. This leads to favouring
desirable endowments or avoiding undesirable endowments in the manner
predicted by salience theory. However, when participants are
sufficiently incentivised, they will engage in the more effortful
process of comparing the options. These direct comparisons engage loss
aversion, which then shifts preference towards the endowed option
regardless of desirability.

Dertwinkel-Kalt and Köhler's (2016) explanation of their results
suggests that endowment reversals should tend to occur when one's
ability to process and compare the choice stimuli is limited. The
first aim of this paper is to explicitly test this prediction by
limiting comparisons between the choice options during incentivised
choices. We do this in two ways: by hiding information about the
non-endowed alternative to explicitly replicate the lack of processing
of the non-endowed options hypothesized by Dertwinkel-Kalt and Köhler,
and by enforcing response deadlines in order to limit the time
available to compare the choice options. If limited comparisons
between the options are important to the occurrence of endowment
reversals, then the endowment effect should reverse between desirable
and undesirable choice settings under these limited-comparison
conditions while no such reversal should occur when comparisons are
freely allowed. We found that endowment reversals occurred only when
comparisons were limited.

The second aim of this paper is to test an alternative explanation of
the reversed endowment effect. If the reversed endowment effect is
dependent on comparisons with the non-endowed alternative being
limited, an interesting implication is that these reversals could be
due to people evaluating the endowed option relative to their
expectations rather than due to amplification of the endowed item. For
example, people have a tendency to not update their beliefs and
expectations entirely, when presented with new information (Anderson,
1983; Nissani \& Hoefler-Nissani, 1992). Thus, people may not
perfectly update their reference point when endowed with an item and
instead, continue to be influenced by the relatively neutral reference
state they had prior to being presented with the choice scenario. When
comparisons with the non-endowed alternative are limited, desirable
endowments would exceed these more neutral expectations and therefore
be preferred (the endowment effect) while undesirable endowments would
be inferior to these more neutral expectations and therefore tend to
be avoided (the reversed endowment effect).

This explanation of the reversed endowment effect based on imperfect
updating of the reference point makes an interesting
prediction. Previous theories based on amplification of the endowed
option have suggested that reversals of the endowment effect occur
specifically when the choice options are undesirable. However, the
theory we present here suggests that endowment reversals are not based
on the absolute desirability of the options but instead on their value
relative to one's previous reference point. This leads to the
prediction that it will be possible to prevent the endowment effect
reversing for undesirable options or even induce a reversed endowment
effect for desirable options by shifting people's previous reference
points in certain ways. Specifically, if someone had a previous
reference point that was superior to the subsequently endowed option
(regardless of whether that endowed option is desirable or
undesirable) then this will tend to bias their reference point at the
time of making a decision such that they expect an outcome that is
superior to the endowed option if they switch away from the endowed
option. This leads to the prediction that endowment reversals will
occur both for desirable and undesirable choices if participants have
a previous reference state that is superior to the endowed
option. Conversely, it is predicted that the endowed option will be
preferred, regardless of whether it is desirable or undesirable, if
someone had a previous reference point that was inferior to the
subsequently endowed option.

In Experiment 3 we tested this prediction using a practice gamble to
induce a superior or inferior reference point prior to the choice of
interest. We found that regardless of whether the endowed option was
desirable or undesirable, the endowment effect would reverse if the
reference point was superior to the endowed option but would not
reverse if it were inferior to the endowed option. We also found that
a model based on prospect theory that incorporates these assumptions
could, in principle, explain our data. This shows that endowment
reversals cannot, in principle, be taken as evidence against prospect
theory, as has previously been claimed. Of course, this does not mean
that prospect theory \textit{is }necessarily the best explanation of
our data and to emphasise this point we concluded by considering a
number of alternative explanations for our data.

\section{Experiment 1A}
This experiment directly tested the hypothesis that whether the
endowment effect reverses or not depends on the extent to which the
two options are compared (Dertwinkel-Kalt \& Köhler, 2016). It did
this by sometimes showing the non-endowed alternative (thereby
allowing it to be compared to the endowed option) but other times
hiding it so as prevent comparisons. This design thus forced
participants to process the stimuli in the manner that the
dual-process model of Dertwinkel-Kalt and Köhler (2016) assumed occurs
for hypothetical vs incentivised choices.

We used risky gambles, traditionally referred to as ``prospects'', as
the choice stimuli. This was done for two reasons. First, it allowed
us to present incentivised choices by informing participants that the
outcome of the gambles could influence their payment. Second, with
gambles it is easy to create directly equivalent desirable and
undesirable choices. Previous studies have used undesirable scenarios
that do not have a clear equivalent desirable scenario. For example,
the undesirable tasks used by Dertwinkel-Kalt and Köhler (2016)
consisted of sorting confetti or filling a grid with `1's and
`0's. For neither of these tasks is there an obvious equivalent
desirable task to compare to. Gambles allow desirable options (a
probability of winning a certain amount) to be precisely matched to
equivalent undesirable options (the same probability of losing the
same amount). Finally, it is worth noting that we presented choices
between two gambles rather than one gamble and a non-risky (certain)
alternative simply to reduce the biases people have been found to show
towards increasingly certain outcomes (Kahneman \& Tversky, 1979),
which might otherwise have overpowered any effect of the endowment
manipulation.

Given that the choices we presented were incentivised, the
dual-process model proposed by Dertwinkel-Kalt and Köhler (2016)
predicted that the endowed option would be preferred for both
desirable and undesirable gambles when comparisons were allowed. When
comparisons were prevented by hiding information about the non-endowed
gamble, it was predicted that a regular endowment effect would be
observed for desirable gambles, but a reversed endowment effect would
be observed for undesirable gambles.

\subsection{Method}

200 participants (mean age = 32, 51\% female) were recruited online
through Microworkers.com and completed the experiment in a web
browser. All participants in all experiments presented here were from
English-speaking countries and provided informed consent before
participating. The task was advertised as paying US\$0.20 plus a
potential additional bonus. However, upon completion, all participants
were paid \$0.80, corresponding to the best possible outcome that
could be expected based on bonuses offered during the experiment. The
experiment took around 2 minutes to complete.

Upon beginning the experiment, participants were informed that they
were currently earning \$0.50 for participation in the experiment but
had to choose a gamble which could affect their payout. The gambles
could win them up to \$0.30 or lose them up to \$0.30 (reducing their
payout to as low as \$0.20). Participants were presented with a screen
that was completely empty except for two boxes labelled ``Gamble A''
and ``Gamble B'' respectively. They were asked to choose one of the
boxes before any information about the gambles was
presented. Whichever box they chose was designated as the endowed
gamble.

\begin{figure*}[t!]\centering
\includegraphics[width=6in,clip=true]{model_fit_all.eps}
\caption{Bar chart of observed \textbf{endowment preference} (error
  bar's represent 95\% CIs) across the different conditions of the
  experiments reported here. The crosses (Xs) represent the predicted
  \textbf{endowment preference} when the model of loss aversion
  described later in this paper was simultaneously fit to our
  different experimental conditions.}
\label{fig:figure1}
\end{figure*} 

After choosing a box, participants were then shown a gamble and
endowed with this gamble. To be clear, their choice of box was, in
reality, irrelevant because whichever box they chose was then assigned
the values associated with the endowed gamble allocated to them. The
endowed gamble was chosen from the following two gambles with
equivalent expected value: 40\% chance of \$0.30 or 80\% chance of
\$0.15, such that half of participants were endowed with the former
gamble and half were endowed with the latter. In the
\textsc{desirable} condition, participants were told they had a chance
of winning the stated amount. In the \textsc{undesirable} condition,
they were told they had a chance of losing the stated
amount. Participants then clicked a button to continue.

If the participant was in an \textsc{allowed} comparison condition,
then they were now shown the non-endowed gamble alongside the endowed
gamble. If the endowed gamble was a 40\% chance of \$0.30 then the
alternative gamble was an 80\% chance of \$0.15, and vice
versa. Alternatively, if the participant was instead in the
\textsc{hidden} condition then the alternative gamble was not shown
and instead was represented by a question mark, making them unable to
compare the endowed gamble to the non-endowed alternative. The
dependent variable of interest was whether participants then chose to
keep the endowed gamble or swap to the non-endowed gamble.

These conditions amounted to a 2x2x2 between-subjects design crossing
\textbf{desirability} (\textsc{desirable} or \textsc{undesirable}) x
\textbf{comparisons} (\textsc{allowed} or \textsc{hidden}) x
\textbf{endowed} \textbf{gamble} (40\% of \$0.30 or 80\% of
\$0.15). The latter variable was included to control for possible
biases in the sample such as risk seeking or risk aversion, and we
therefore collapsed across it.\footnote{A chi-square test looking at
  endowment preference based on which option was endowed found no
  significant difference in overall endowment preference between when
  the 40\% chance of \$0.30 gamble or the 80\% chance of \$0.15 gamble
  was endowed
  ($\chi^2(1,200) = 0.33, p = .565, OR = 1.18, 95\%CI[0.67, 2.07]$).}
Through controlling for such biases, we could expect that if
participants were unaffected by the endowing of an item then
\textbf{endowment preference }(the percentage of participants choosing
the endowed item) should be approximately 50\%. Endowment share
greater than 50\% would reflect preference for the endowed gamble over
the non-endowed alternative.

\subsection{Results}
The average \textbf{endowment preference} and 95\%CI for each
condition is shown in Figure \ref{fig:figure1}. How were people's
choices affected by the desirability of the options and whether
comparisons were possible? We addressed this question with a 2x2
logistic regression with \textbf{desirability} (\textsc{undesirable} =
0, \textsc{desirable} = 1) and \textbf{comparisons} (\textsc{hidden} =
0, \textsc{allowed} = 1) as predictors and \textbf{endowment
  preference} (non-endowed alternative chosen = 0, endowment chosen =
1) as the outcome variable. There were significant main effects of
both \textbf{desirability} ($OR=2.93,\ p=.013,\ 95\%CI [1.31, 6.77]$)
and \textbf{comparisons} ($OR=6.73,\ p<.001,\ 95\%CI[2.96,
16.78]$). Importantly, there was also a significant interaction
($OR=0.25,\ p=.025,\ 95\%CI[0.07, 0.83]$) with the endowed gamble
being significantly more likely to be chosen in the \textsc{desirable}
condition compared to the \textsc{undesirable} condition when the
non-endowed alternative was \textsc{hidden}
($\chi^2(1,100)=6.83,\ p=.009,\ OR=2.93,\ 95\%CI[1.30, 6.65]$). On the
other hand, in the \textsc{allowed }condition, no significant
difference was found between the \textsc{desirable} and
\textsc{undesirable} conditions
($\chi^2(1,100)=0.46,\ p=.50,\ OR=0.74,\ 95\%CI[0.30,
1.79]$). Consistent with the proposition of Dertwinkel-Kalt and Köhler
(2016), the desirability of the gamble affected choices only when the
comparisons were hindered. Experiment 1B looks at whether these
results generalise to non-risky choice stimuli.

\section{Experiment 1B}
\subsection{Method}
200 English-speaking participants (mean age = 35, 53\% female) were recruited online through Amazon's Mechanical Turk. The design of Experiment 1B was exactly the same as Experiment 1A except for the changes outlined below. 

Participants were informed that they would have to complete a button-pressing task during this experiment. Participants in the \textsc{desirable }condition were initially informed that they would ``be paid \$0.40 for completing the task on the next page, which takes about 1.5 minutes and requires clicking on very small buttons''. However, they were then asked to choose one of two boxes which might change the task involved or the monetary payout. For half of the participants in the \textsc{desirable} condition, their chosen box endowed them with a ``20 second task instead, which requires clicking on large buttons''. The other half of participants in the \textsc{desirable} condition were endowed with an additional \$0.10 for completing the experiment. We collapsed across these endowed gamble conditions when analysing the data to control for possible biases towards a shorter task or higher payout affecting \textbf{endowment preference}.\footnote{Whether the shorter task or higher payout was endowed did not significantly affect \textbf{endowment preference} ($\chi^2(1,200)=0.32,\ p=.57,\ OR=1.17,\ 95\%CI[0.67,2.05]$).} 

Participants in the \textsc{allowed} comparison condition were shown the other possible option (e.g. participants endowed with an additional \$0.10 were shown that the other box offered the 20 second task). As in Experiment 1A, all participants were then informed that they had the option to switch to the other box if they wished and asked to choose which they would prefer. 

The procedure for participants in the \textsc{undesirable} condition was exactly the same and had the same potential end-points as those in the \textsc{desirable} condition, but the available options were phrased as being undesirable. Specifically, participants were initially informed that they would be ``paid \$0.50 for completing the task on the next page, which takes about 20 seconds and involves clicking on large buttons''. The two boxes either endowed participants with completing ``a tricky 1.5 minute task instead, which requires clicking on very small buttons'' or a reduction in their payout of \$0.10. 

Regardless of their allocated condition or decision, upon deciding which option to keep, participants were presented with a debriefing statement and informed that the choice was actually only hypothetical and that they did not have to complete any additional task. They were paid \$0.50 for participation, which took around 1 to 2 minutes.

\subsection{Results}
Figure \ref{fig:figure1} shows the average endowment preference and 95\% CI for each experimental condition in Experiment 1B. 

A 2x2 logistic regression found a significant main effect of \textbf{comparisons} ($OR=4.03,\ p=.001,\ 95\%CI[1.75, 9.76]$) and of \textbf{desirability} ($OR=8.14,\ p<.001,\ 95\%CI[3.42,20.69]$) on \textbf{endowment preference}. Consistent with the finding from Experiment 1A, a significant interaction between \textbf{desirability} and \textbf{comparisons} was observed ($OR=0.14,\ p=.002,\  95\%CI[0.04, 0.47]$). Specifically, the endowed option was chosen significantly more in the \textsc{desirable} than the \textsc{undesirable} condition for the \textsc{hidden} comparison condition, ($\chi^{2}(1,100)=23.08,\ p<.001,\ OR=8.14,\  95\%CI[3.32,19.94]$), but not for the \textsc{allowed} comparison condition ($\chi^{2}(1,100)=0.16,\ p=.685,\ OR=1.18,\ 95\%CI[0.53,2.61]$).

\section{Experiment 2}
Experiments 1A and 1B limited comparisons by hiding the non-endowed
alternative. This manipulation represents the most extreme extent to
which participants could fail to compare options when making a choice,
and, indeed, responses could be explained in terms of expectations
about what the alternative might be. Experiment 3 looks at this latter concern. Experiment 2 looks at whether the
findings from the previous experiments generalise to a setting in
which both options are presented, and comparisons are instead limited
simply by enforcing a response deadline.

\subsection{Method}
100 English-speaking participants (mean age = 36, 50\% female) were
recruited online through Amazon's Mechanical Turk. Experiment 2 used
the same design as the \textsc{desirable} and \textsc{undesirable
  allowed} comparison conditions of Experiment 1A except participants
were now instructed at the start of the experiment that a countdown
timer might appear during the experiment.\footnote{As in the earlier
  experiments, we collapsed across which option was endowed when
  testing our key research questions. An exploratory analysis did not
  find a significant difference in \textbf{endowment preference} based
  on whether the 40\% chance of \$0.30 or 80\% chance of \$0.15 gamble
  was endowed in the \textsc{time-pressure} conditions of Experiment 2
  ($\chi^2(1,100)=0.50,\ p=.478,\ OR=0.75,\ 95\%CI[0.33,1.68]$).} To
encourage rapid responding, they were instructed that failure to make
a choice before the timer reached zero might reduce their potential
bonus payout. When participants were offered the option to swap their
endowed gamble with the non-endowed alternative, a timer always
appeared on the screen counting down from ten seconds.

Five participants (three from the \textsc{undesirable} condition and
two from the \textsc{desirable} condition) were excluded from the
analysis due to failing to respond within the ten second
deadline. These participants were still paid the full bonus for
participating. The conclusions drawn from this experiment remained the
same when these participants were included in the following analyses.

\subsection{Results}
Figure \ref{fig:figure1} shows the mean \textbf{endowment preference}
share and 95\%CI for the \textsc{desirable} and \textsc{undesirable}
conditions.

To test the prediction that there should be a greater difference
between desirable and undesirable choices in terms of endowment
preference when decisions are made under time pressure, we compared
the \textsc{time-pressure} condition of Experiment 2 to the
\textsc{allowed} comparison condition of Experiment 1A. A 2x2 logistic
regression predicted \textbf{endowment preference} based on
\textbf{desirability} and \textbf{comparisons }(\textsc{time-pressure
  =} 0, \textsc{allowed} = 1). A significant main effect of
\textbf{comparisons} was found\textbf{
}($OR=5.1,\ p<.001,\ 95\%CI[2.17,12.63]$), though not of
\textbf{desirability} ($OR=0.74,\ p=.500,\ 95\%CI[0.30,1.79]$). Of key
interest, there was a significant interaction between
\textbf{desirability} and \textbf{comparisons}
($OR=0.25,\ p=.026,\ 95\%CI[0.07,0.84]$) with the endowed gamble being
significantly more likely to be chosen in the \textsc{desirable}
condition compared to the \textsc{undesirable} condition when the
choice was made under \textsc{time-pressure}
($\chi^{2}(1,100)=6.57,\ p=.010,\ OR=0.34,\ 95\%CI[0.15,0.78]$). Taken
together, the interaction between desirability and whether comparisons
are encouraged or limited observed across Experiments 1A, 1B and 2
provides support for the hypothesis that endowment reversals are more
likely to occur when comparisons between the options are limited.

\section{Experiment 3}
Given the important role of comparison processes implied by our
previous experiments, Experiment 3 tested our hypothesis that the
expectations participants hold prior to a choice influence the
occurrence of endowment reversals.

\subsection{Method}
200 participants (50 per condition, mean age = 32, 102 females) were
recruited via Microworkers. The task was advertised as paying US\$0.25
however, all participants were subsequently paid \$0.75 for
participation. Experiment 3 was a 2x2 between-subjects design crossing
\textbf{desirability} (\textsc{desirable} or \textsc{undesirable}) x
\textbf{prior reference} \textbf{point} (\textsc{superior} or
\textsc{inferior} reference point relative to the endowed
gamble). Experiment 3 followed essentially the same design as the
\textsc{hidden} condition of Experiment 1A except for a few key
changes. Firstly, the values of gambles shifted slightly. The endowed
gamble was now a 50\% chance of winning (losing) US\$0.20 for all
participants in the \textsc{desirable} (\textsc{undesirable})
condition. The base rate that participants were told they were earning
before partaking in the gamble was also shifted up slightly to
\$0.55. These changes were made to avoid the practice gamble
approaching ceiling or floor levels in terms of percentage and payout
amounts while remaining within certain payout limitations implemented
by Microworkers.

Secondly, and of key importance, Experiment 3 included an initial
'practice' trial that participants took part in before they were then
offered a choice of gambles. As in our previous experiments,
participants were presented with two empty boxes, chose one and were
presented with the supposed gamble corresponding to that box. However,
participants did not choose whether they would want to keep that
gamble. Instead, participants were now reminded that this choice was
purely meant as practice; it was irrelevant to the subsequent choice
and would not affect their payout in any way. The practice gamble was
not resolved, meaning it had no effect on participants' payments and
participants were not shown any outcome of the practice gamble. This
practice trial was actually used to induce a higher or lower reference
point. Specifically, for half of the participants, this practice trial
presented a gamble that was \textsc{superior} to the endowed gamble in
the following trial. In the \textsc{desirable} condition, this
superior gamble was a 75\% chance of winning \$0.30. In the
\textsc{undesirable} condition, this superior gamble was a 25\% chance
of losing \$0.10. For the other half of participants, this practice
trial was \textsc{inferior} to the subsequently endowed gamble. For
\textsc{desirable} choices, the inferior gamble was a 25\% chance of
winning \$0.10 and for \textsc{undesirable} choices the inferior
gamble was a 75\% chance of losing \$0.30.

To summarise, participants were first presented with a practice trial
and were instructed that it was just an example and would not
influence their payout. In this practice trial, participants were
presented with a screen that was completely empty except for two boxes
labelled ``Gamble A'' and ``Gamble B'' respectively. They were asked
to choose one of the boxes before any information about the gambles
was presented. Regardless of which box they chose, they were then
presented with the practice gamble. They were not told the gamble
corresponding to the other box. The practice trial then ended, without
resolving the practice gamble, and the participants were invited to do
the main experiment. As before, participants were presented with two
boxes labelled ``Gamble A'' and ``Gamble B'' respectively and were
invited to choose one of them. Regardless of which one they chose,
they would be shown a gamble of 50\% chance of winning (losing)
\$0.20. They were not shown the gamble corresponding to the other
box. Based on this information alone, they were then asked which
gamble (i.e. A or B) they wished to take.

\subsection{Results}
Figure \ref{fig:figure1} shows the observed \textbf{endowment
  preference} share and 95\% confidence intervals for each of the
conditions in Experiment 3. Were preferences towards the endowment
influenced by the desirability of the options and the relative value
of the practice gamble? A logistic regression was run predicting
\textbf{endowment preference} (non-endowed gamble chosen = 0, endowed
gamble chosen = 1) based on \textbf{desirability}
(\textsc{undesirable} = 0, \textsc{desirable} = 1) and \textbf{prior
  reference} \textbf{point} (\textsc{inferior} = 0, \textsc{superior}
= 1). As predicted, a significant main effect of the \textbf{prior
  reference point} was found, with the odds of choosing the endowed
gamble being significantly higher when the practice gamble was
\textsc{inferior} than when it was \textsc{superior} to the endowment
($OR=5.25,\ p<.001,\ 95\%CI[2.39,13.46]$). We found no significant
main effect of \textbf{desirability}
($OR=1.34,\ p=.509,\ 95\%CI[0.56,3.23]$) and no significant
interaction between \textbf{desirability} and \textbf{prior reference
  point} ($OR=0.58,\ p=.365,\ 95\%CI[0.17,1.89]$). The data are
consistent with our prediction that the participants' choices would be
determined by the value of the gamble in the practice trial relative
to the endowed gamble.

\section{Modelling the data}\label{section:_Hlk505436285}
We present here a computational model based on prospect theory
(Kahneman \& Tversky, 1979). The goal is not to show that such a model
offers the best explanation of our results, especially given we do not
have sufficient data to verify a number of the model's more flexible
assumptions. Instead, we wish to assess the extent to which, under
reasonable assumptions, a model based on loss aversion can, in
principle, quantitatively explain the existence of both the endowment
effect and its reversal. This is especially important given previous
claims that endowment reversals are inconsistent with loss aversion
(Brenner et al., 2007). The key addition to prospect theory in the
model presented here is that we assume people's reference points are
influenced not only by their current endowed state but also their
previous reference state.

Prospect theory assumes that each option has a value associated with
it representing the subjective evaluation of any potential outcomes
weighted as a function of the probability of each outcome occurring
(Kahneman \& Tversky, 1979). Preference for one option over another is
based on the difference in weighted value between these two
options,where an option with a higher value will be preferred to one
with a lower value (Kahneman \& Tversky, 1979). To arrive at
quantitative predictions of choice proportions we include noise,
$\sigma$, as a free parameter in the model such that the final
difference is logistically distributed with a mean equal to the
difference in value between options and a standard deviation equal to
the noise parameter (Wang \& Fischbeck, 2004). Following Wang and
Fischbeck (2004), Equation \ref{eq1} shows the probability of
preferring an endowed option, $\epsilon$, over an alternative option,
$a$, using a logistic function of the difference of the value of the
two options:
\begin{equation} \label{eq1}
	P(\varepsilon \vert a)=\frac{1}{ 1+e^{-\frac{V(\varepsilon )-V(a)}{\sigma
}}}
\end{equation}

Here, $V$ is the overall subjective value of an option across its
different possible outcomes. We define any option, $x$, as a set of
outcomes, $o^{x}$, with a corresponding set of probabilities,
$p^{x}$. $V$ is defined as the sum of the subjective value $v$ of each
of the possible $n$ outcomes of an option, weighted as a function
$\pi$ of the probability of each outcome occurring, as shown in
Equation \ref{eq2} (Kahneman \& Tversky, 1979).
\begin{equation} \label{eq2}
	V(x)=\sum_{i=1}^{n}{\pi (p_{i}^{x})v(o_{i}^{x}\vert r)}
\end{equation}

Equation \ref{eq2} acknowledges that the outcomes of $x$ are evaluated relative to a reference point, $r$. The function $\pi (p_{i}^{x})$ was proposed in prospect theory to weight the different possible outcomes of $x$ by a nonlinear function of the probability of the outcome occurring (Kahneman \& Tversky, 1979). For simplicity, we follow the lead of Kõszegi and Rabin (2006) and assume that people's evaluations of gambles is such that $\pi (p_{i}^{x})=p_{i}^{x}$. This assumption is likely to breakdown for very small or very large probabilities, but should be reasonable for all other values (Kahneman \& Tversky, 1979). In our gamble experiments, we avoided extremely large or extremely small probabilities for this reason. The function $v(o_{i}^{x} \vert r)$ calculates the subjective value of an outcome relative to the reference point, as outlined in Equation \ref{eq3}.
\begin{equation} \label{eq3}
	v(o_{i}^{x}|r)=\mu (o_{i}^{x}-r)
\end{equation}

Equation \ref{eq4} specifies the value function, $\mu$. Specifically, as seen in Equation \ref{eq4}, we utilise the piecewise power function proposed by Tversky and Kahneman (1992), along with the median values they estimated for both the loss aversion parameter (2.25) and the exponent of the power function (0.88). These parameter estimates are used given their status as reasonable defaults to allow for model fitting in the absence of information to the contrary (e.g. Barberis \& Xiong, 2009; Hens \& Vlcek, 2011; Koop \& Johnson, 2012). 

This function satisfies the criteria for a value function set out by Kahneman and Tversky (1979) and Tversky and Kahneman (1991). In particular, it is a negatively accelerating function (due to the exponent of the power function being less than 1) for both gains and losses, but steeper in the losses domain than in the gains domain (due to the loss aversion parameter being greater than 1) to reflect loss aversion.
\begin{equation} \label{eq4}
	\mu (z) = \begin{cases}
	z^{0.88},& \text{if } z\geq 0 \\
    -2.25(-z)^{0.88}, & \text{if } z<0
    \end{cases}   
\end{equation}

To ensure our model was not reliant on the exact specification of the value function, we also fit the model with the value function used by Usher and McClelland (2004) in their Leaky Competing Accumulator model. Despite differences between the two value functions, they each provided essentially equivalent fits to our data (see Appendix A).

Equation \ref{eq3} assumes that the reference point is certain but the option being considered can be risky. In the case that the reference point also has uncertainty, we must consider each of the possible outcomes of the choice option as well as each of the possible outcomes of the reference point in order to determine the value of the choice option relative to the reference point. Kõszegi and Rabin (2006, p. 1137) provide a general equation for such a case, which allows for integration across continuous probability distributions of different outcomes. Given we are only dealing with discrete probabilities in this study, and we have assumed that $\pi (p_{i}^{x})=p_{i}^{x}$, we can simplify the equation used by Kõszegi and Rabin (2006) to represent the value of an option, $x$, with $n$ possible outcomes relative to a risky reference point ($r$) with $m$ possible outcomes to that shown in Equation \ref{eq5}.
\begin{equation} \label{eq5}
	V(x \vert r)= \sum_{i=1}^{n}{\sum_{j=1}^{m}{p_{i}^{x}p_{j}^{r}v(o_{i}^{x}\vert o_{j}^{r})}}  
\end{equation}

In prospect theory, the value of an option is determined relative to the reference point. A common assumption for choice tasks where one option is endowed is to assume that the reference point is equivalent to the endowed option, $\varepsilon$ (Brenner et al., 2007; Koop \& Johnson, 2012). In this paper we argue that the reference point may not be determined entirely by the current endowed state. Instead, it may also be influenced by prior expectations. 

In terms of how multiple pieces of reference information can be incorporated into the current reference state, Ordónez, Connolly and Coughlan (2000) found data consistent with the idea that people treat separate pieces of reference information as independent rather than averaging that information into a single reference point. Thus, a simple way in which the current reference state could incorporate information from both the previous reference state and the current endowed state is if people compare options to their current endowed state some of the time and compare options to the previous reference state the rest of the time. We can represent this with a discrete probability distribution where, at any point in time, the reference state $r$ is set equal to the endowed option $\varepsilon$ with probability $\gamma$, and is set equal to the previous reference state $s$ otherwise. This is shown in Equation \ref{eq6}, where $\gamma$ is a free parameter and $0<\gamma<1$.
\begin{equation} \label{eq6}
\begin{split}
	P(r=\varepsilon ) &=\gamma \\ 
	P(r=s) &= 1-\gamma
\end{split}
\end{equation}

Though Ordónez et al. (2000) favoured the approach of having independent reference points, Appendix B shows that, in our case, essentially equivalent fits can be obtained if we incorporate the previous reference state and current endowed state into a single reference point. Specifically, the model in Appendix B assumes that the current reference state is represented as a probability distribution across possible outcomes, where beliefs about the probabilities of outcomes are updated based on previous expectations and the current endowment. Our goal is not to distinguish between such approaches but simply to show that loss aversion can explain the endowment reversals under various reasonable assumptions.

In Experiments 1A, 1B and 2 participants did not have any basis for forming strong prior expectations of attaining non-zero outcomes. Thus, for these experiments, we set the previous reference state $s$ as a 100\% chance of winning \$0.00 (i.e. not expecting to win or lose any money) and, in the case of Experiment 1B, expecting no change in the time required to complete the task either. In Experiment 3 participants viewed a practice gamble before completing the experimental choice in order to influence their previous reference state, $s$. Consequently, for Experiment 3, $s$ in Equation \ref{eq6} is set equal to the relevant practice gamble. For example, for the \textsc{superior desirable} \textsc{condition} of Experiment 3, $s$ is set as a 75\% chance of winning \$0.30 and a 25\% chance of winning nothing.

We assume that participants treated the non-endowed alternative as equal to the reference point whenever the non-endowed option was hidden, or choices were made under time pressure. We made this assumption because the reference point represents the participant's expectations. If the non-endowed alternative was not processed, the participant had to decide whether or not to accept the endowed option based purely on whether it exceeded what they expected to get from switching.

To accommodate the fact that Experiment 1B used non-risky choice
options, we firstly set $p_{i}^{x} = 1$ for these non-risky options
given their outcomes were certain. Given the options in Experiment 1B
could vary along two attributes (changes in the time commitment of the
task and changes to monetary payout) we also included an additional
free parameter, $\delta$, to approximately equate these two
attributes. Specifically, if an outcome $i$ related to changes in the
time commitment, the value presented to participants,
$o_{i}^{x_{\mathit{presented}}}$, was multiplied by $\delta$, as shown in Equation 7.
\begin{equation*} \label{eq7}
	o_{i}^{x} = \begin{cases}
	\delta o_{i}^{x_{\mathit{presented}}}, \text{{\hspace{1.9in}} (7)}\\ \text{{\ } when } o_{i}^{x_{\mathit{presented}}} \text{relates to time commitment} \\
    o_{i}^{x_{\mathit{presented}}},\\ \text{{\ }when } o_{i}^{x_{\mathit{presented}}} \text{relates to monetary payout} \\
    \end{cases}   
\end{equation*}

\subsection{Model fitting}
When fitting the model to our data, we set $\gamma$, $\sigma$ and
$\delta$ as free parameters and simultaneously fit the model to the
endowment preference share across our experimental conditions. Values
of the free parameters were found that maximised the likelihood of the
data using the Nelder-Mead algorithm implemented by the ``optim''
function in R (R Core Team, 2016). The specific parameter values
obtained were: $\gamma=0.664$, $\sigma=0.510$ and $\delta=0.136$. The
endowment preference share predicted by the model for each of the
experimental conditions is presented in Figure \ref{fig:figure1}. A
chi-square test failed to find a significant difference between the
predicted endowment preference count based on the model and the
observed endowment preference count,
$\chi^{2}(9,\ N=495)=7.58,\ p=.870$. Our model can therefore
adequately describe the data.

\section{Discussion}
In previously published papers, the reversal of the endowment effect
for undesirable options had been observed only for hypothetical
choices. To our knowledge, this paper represents the first time that
such reversals have been observed for incentivised choices that have
real consequences. However, we found that such reversals occurred only
when comparisons between the endowed and non-endowed alternative were
limited, whether that occurred by reducing the time available to
compare the options or by hiding information about the non-endowed
alternative. On the other hand, when comparisons were allowed and
encouraged, participants generally preferred the endowed option
regardless of desirability. We replicated these findings for non-risky
(more money vs.\ easier task) choices. We also showed that whether or
not the endowment effect reversed depended on the participant's prior
experience. In particular, when comparisons were limited we showed
that the endowment effect reversed for both desirable and undesirable
gambles if the participant had previously been shown a more desirable
practice gamble. Conversely, when shown a less desirable practice
gamble, participants preferred the endowed gamble regardless of
whether it was desirable or undesirable.

\subsection{Theories of the endowment effect and endowment reversal}
One of the theoretical points we wished to address in this paper was
whether, as has been previously proposed, endowment reversals are
inconsistent with prospect theory. We found that we were able to model
all of the above findings using a straight-forward extension of
prospect theory. While we cannot show whether the assumptions of this
model are true, we were able to demonstrate that endowment reversals
are, at least in principle, consistent with prospect theory.

In applying prospect theory to endowment reversals, we made the
assumption that people have expectations that are influenced both by
their current endowed state and their previous reference
state. However, this explanation is not restricted to prospect theory
and loss aversion. This assumption could also be incorporated into
other theories of the endowment effect. For example, Gal (2006)
suggests that people show inertia, in that they will not change their
current state of affairs without sufficient motivation to do so. Gal
proposes that it is this inertia, and the general difficulty in
determining which option is best in order to motivate change, that
leads to the endowment effect. If participants are comparing the
endowed option to their more neutral expectations, it may be that an
undesirable endowed option is sufficiently inferior to these more
neutral expectations to motivate switching. While we have focused
primarily on loss aversion in order to address previous claims that
the reversed endowment effect is evidence against this explanation of
the endowment effect, it is worth emphasising that such other
mechanisms also provide viable explanations for our results.

We assumed above that participants' expectations (or reference points)
were a function of their current endowed state and their previous
reference state. However, it is worth considering the other reference
points people could have held. A large number of these possible
reference points, such as the expected value of the gambles, the best
possible outcome or expecting an outcome of zero, are independent of
the endowment and therefore do not help us explain the effect of
endowing an option. Research into ambiguity aversion, the finding that
people tend to favour known risks over unknown risks, might suggest
that people are generally pessimistic, expecting outcomes that are
worse than their known endowed state (Becker \& Brownson, 1964;
Camerer \& Weber, 1992; Keren \& Gerritsen, 1999). Counter to our
results though, this would mean that participants will tend to prefer
the known endowed option, regardless of desirability. Thus, our
results cannot simply be explained by ambiguity aversion.

One alternative assumption about participants' expectations that could
explain endowment reversals is that participants may perceive the
non-endowed alternative to be riskier. For example, during the
\textsc{hidden} condition of Experiment 1A and the
\textsc{time-pressure} condition of Experiment 2, participants may
have expected the non-endowed gamble to have approximately the same
expected value as the endowed gamble but expected that the non-endowed
gamble could just as likely have small potential outcomes as extreme
potential outcomes. Importantly, there is evidence that people are
risk-seeking for losses and risk-averse for gains (Kahneman \&
Tversky, 1979). This means that risk seeking during undesirable
scenarios could explain preference for the non-endowed gamble while
aversion to this risk might explain the observed preference for the
endowment in the \textsc{desirable} condition. Thus, it is possible
that perception of the non-endowed option as riskier could explain, or
have contributed to, the endowment reversals we observed in
Experiments 1 and 2. It should be noted that additional mechanisms
would be needed to explain why the endowment effect occurs when
comparisons are encouraged and why a previous gamble can affect
whether or not the endowment effect reverses, as observed in
Experiment 3.

\subsection{Implications}
The occurrence of endowment reversals shows that endowing an item can
have completely opposite effects on preference for that item;
potentially either increasing or decreasing its perceived
value. Understanding when these endowment reversals are likely to
occur is particularly important whenever trying to use ownership as a
way of manipulating behaviour, either in laboratory studies or as part
of real-world nudging strategies. We have shown that these endowment
effect reversals can occur for choices with real consequences and that
they can occur for both desirable and undesirable options depending on
people's expectations. Our observation that endowment reversals occur
when comparisons are limited also has interesting implications for
when we might expect the endowment effect to reverse. For example, the
mode in which choice information is presented could be expected to
influence the extent to which options are compared and the subsequent
likelihood of endowment reversals occurring. Choice information
presented in tables tends to elicit detailed comparisons (Noguchi \&
Stewart, 2014) while information presented in blocks of text is
difficult to extract and compare (Kelly, 1993; Smerecnik et al.,
2010). Thus, for example, presenting choices in text may increase the
potential for endowment reversals to occur compared to when options
are presented in a table.

Individuals may also systematically differ in their tendency to show
endowment reversals. People with high need for cognition, for example,
tend to engage in deeper and broader comparisons in decision
situations (Levin, Huneke \& Jasper, 2000). Thus, low need for
cognition may be associated with a greater tendency to show endowment
reversals due to a tendency not to compare the options. Individual
differences can be crucial to our understanding of how manipulations
are affecting choice (e.g. Liew, Howe \& Little, 2016). Need for
cognition offers a potentially interesting individual-level variable
that could explain variation in how endowments affect choice.

\subsection{Conclusion}
In this study, we set out to provide a greater understanding of when
the endowment effect is likely to reverse and, thus, the theoretical
implications of these reversals. Two factors were found to influence
whether endowments increased or decreased preference: how the choice
options were being compared and the likely expectations held when
going into the choice. We emphasise the importance of considering
these two factors both when studying the endowment effect and if
attempting to implement endowment manipulations in real-world nudging
strategies. Simple changes in the extent to which options are compared
and the expectations people carry into a decision situation can
determine whether ownership increases or decreases preference for an
option.

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

Anderson, C. A. (1983). Abstract and concrete data in the perseverance of social theories: When weak data lead to unshakeable beliefs. \textit{Journal of Experimental Social Psychology, 19}(2), 93--108. 

Barberis, N., \& Xiong, W. (2009). What drives the disposition effect? An analysis of a long-standing preference-based explanation. \textit{The Journal of Finance, 64}(2), 751--784. 

Baron, J., \& Ritov, I. (1994). Reference Points and Omission Bias. \textit{Organizational Behavior and Human Decision Processes, 59}(3), 475-498.

Becker, S. W., \& Brownson, F. O. (1964). What Price Ambiguity? Or the Role of Ambiguity in Decision-Making. \textit{Journal of Political Economy, 72}(1), 62--73.

Bordalo, P., Gennaioli, N., \& Shleifer, A. (2012). Salience in experimental tests of the endowment effect. \textit{The American Economic Review, 102}(3), 47--52. 

Bordalo, P., Gennaioli, N., \& Shleifer, A. (2013). Salience and consumer choice. \textit{Journal of Political Economy, 121}(5), 803--843. 

Brenner, L., Rottenstreich, Y., Sood, S., \& Bilgin, B. (2007). On the psychology of loss aversion: Possession, valence, and reversals of the endowment effect. \textit{Journal of Consumer Research, 34}(3), 369--376. 

Camerer, C., \& Weber, M. (1992). Recent developments in modeling preferences: Uncertainty and ambiguity. \textit{Journal of Risk and Uncertainty, 5}(4), 325--370. 

Dertwinkel-Kalt, M., \& Köhler, K. (2016). Exchange asymmetries for bads? Experimental evidence. \textit{European Economic Review, 82}, 231--241. 

Gal, D. (2006). A psychological law of inertia and the illusion of loss aversion. \textit{Judgment and Decision Making, 1}(1), 23--32. 

Hens, T., \& Vlcek, M. (2011). Does prospect theory explain the disposition effect? \textit{Journal of Behavioral Finance, 12}(3), 141--157. 

Houston, D. A., Sherman, S. J., \& Baker, S. M. (1989). The influence of unique features and direction of comparison of preferences. \textit{Journal of Experimental Social Psychology, 25}(2), 121-141

Kahneman, D., Knetsch, J. L., \& Thaler, R. H. (1990). Experimental tests of the endowment effect and the Coase theorem. \textit{Journal of Political Economy, 98}(6), 1325--1348. 

Kahneman, D., \& Tversky, A. (1979). Prospect theory: An analysis of decision under risk. \textit{Econometrica, 47}(2), 263--292. 

Kelly, J. D. (1993). The effects of display format and data density on time spent reading statistics in text, tables and graphs. \textit{Journalism \& Mass Communication Quarterly, 70}(1), 140--149. 

Keren, G., \& Gerritsen, L. E. M. (1999). On the robustness and possible accounts of ambiguity aversion. \textit{Acta Psychologica, 103}(1), 149--172. 

Koop, G. J., \& Johnson, J. G. (2012). The use of multiple reference points in risky decision making. \textit{Journal of Behavioral Decision Making, 25}(1), 49--62. 

Kõszegi, B., \& Rabin, M. (2006). A Model of reference-dependent preferences. \textit{The Quarterly Journal of Economics, 121}(4), 1133--1165. 

Levin, I. P., Huneke, M. E., \& Jasper, J. D. (2000). Information processing at successive stages of decision making: Need for cognition and inclusion-exclusion effects. \textit{Organizational Behavior and Human Decision Processes, 82}(2), 171--193. 

Liew, S. X., Howe, P. D. L., \& Little, D. R. (2016). The appropriacy of averaging in the study of context effects. \textit{Psychonomic Bulletin \& Review, 23}(5), 1639--1646. 

Nissani, M., \& Hoefler-Nissani, D. M. (1992). Experimental studies of belief dependence of observations and of resistance to conceptual change. \textit{Cognition and Instruction, 9}(2), 97--111. 

Noguchi, T., \& Stewart, N. (2014). In the attraction, compromise, and similarity effects, alternatives are repeatedly compared in pairs on single dimensions. \textit{Cognition, 132}(1), 44--56. 

Ordónez, L. D., Connolly, T., \& Coughlan, R. (2000). Multiple reference points in satisfaction and fairness assessment. \textit{Journal of Behavioral Decision Making, 13}(3), 329--344. 

R Core Team. (2016). R: A language and environment for statitical computing. Vienna, Austria: R foundation for statistical computing. Retrieved from https://www.R-project.org/

Smerecnik, C. M., Mesters, I., Kessels, L. T., Ruiter, R. A., De Vries, N. K., \& De Vries, H. (2010). Understanding the positive effects of graphical risk information on comprehension: Measuring attention directed to written, tabular, and graphical risk information. \textit{Risk Analysis, 30}(9), 1387--1398. 

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

Tversky, A., \& Kahneman, D. (1991). Loss aversion in riskless choice: A reference-dependent model. \textit{The Quarterly Journal of Economics, 106}(4), 1039--1061. 

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

Usher, M., \& McClelland, J. L. (2004). Loss aversion and inhibition in dynamical models of multialternative choice. \textit{Psychological Review, 111}(3), 757--769. 

Wang, M., \& Fischbeck, P. S. (2004). Incorporating framing into prospect theory modeling: A mixture-model approach. \textit{Journal of Risk and Uncertainty, 29}(2), 181--197. 
\end{hangparas}

\bigskip
\section*{Appendix A} %The asterisk means that these divisions are not numbered.
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In order to assess the extent to which the predictions of the model outlined in this paper were dependent on the specific value function used, we refit the model using the value function employed by Usher and McClelland (2004) in their Leaky Competing Accumulator (LCA) model, as shown in Equation \ref{eqA1}. As can be seen in Figure \ref{fig:figureA1}, the predictions of the model with the LCA value function were essentially equivalent to the predictions of the model presented in this paper. The parameter estimates obtained in this case were $\gamma=0.637$, $\sigma=0.792$ and $\delta=0.172$.
\begin{equation} \label{eqA1}
\mu(z)= \begin{cases}
log(1+z),& \text{if } z\geq 0 \\
    -\{(log(1+\lvert z\rvert) + [log(1+\lvert z \rvert)]^2\}, & \text{if } z<0
    \end{cases}
\end{equation}

\begin{figure*}[b!]\centering
\includegraphics[width=6in,clip=true]{model_fit_all_LCA_and_new_model.eps}
\caption{Observed endowment preference share (error bars represent 95\%CIs) for Experiments 1A, 1B, 2 and 3. The crosses represent the predicted endowment preference share of the model using the value function from Equation \ref{eq4}. The dots represent the predicted endowment preference share of the model using the LCA value function from Equation \ref{eqA1}.}
\label{fig:figureA1}
\end{figure*} 

% check figure - name has changed

\newpage

\section*{Appendix B} %The asterisk means that these divisions are not numbered.
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\begin{figure*}[t!]\centering
\includegraphics[width=6in, clip=true]{model_fit_all_mixture_v_prob.eps}
\caption{Observed endowment preference share (error bars represent 95\% CIs) across Experiments 1A, 1B, 2 and 3. The crosses represent the predicted endowment preference share of the model described in this paper. The dots represent the predicted endowment preference share of the model presented in this appendix, which combines the previous reference state and the current endowed state according to Equation \ref{eqB1}}
\label{fig:figureB1}
\end{figure*} 
In this paper we presented a model that set the current reference point to either the current endowed option or the previous reference state with a certain probability. We present here an alternative method of combining the previous reference state and current endowed state to assess whether the predictions of our model are consistent across assumptions about how information is integrated. 

Here we treat both the previous reference state and the current endowed state as probability distributions of different outcomes. We then set the current reference state equal to a weighted average of the probabilities in each of these probability distributions. Thus, the current reference point, $r$, is equal to a probability distribution across all of the possible outcomes in both the current endowed state, $\varepsilon$, and the previous reference state, $s$, where the probability of each possible outcome is simply equal to the weighted average of the probability of that outcome in the previous reference state and the current reference state. This is shown in Equation \ref{eqB1}, where $p_{i}^{x}$ represents the probability of outcome $i$ in probability distribution $x$ and $\gamma$ represents the impact that the current endowment has on expectations relative to the impact that the previous referent point has, with the restriction that $0<\gamma<1$. The parameter estimates obtained when this model was fit to our data were $\gamma=0.899$, $\sigma=1.921$ and $\delta=0.198$.
\begin{equation} \label{eqB1}
	p_{i}^{r}=\gamma p_{i}^{\varepsilon}+(1-\gamma ) p_{i}^{s}
\end{equation}
As an example, assume the current endowment is a 50\% chance of winning \$0.20 and a 50\% chance of winning nothing, whilst the previous reference state was a 25\% chance of winning \$0.10 and a 75\% chance of winning nothing. If $\gamma=0.5$, then the current reference state would be a 25\% chance of winning \$0.20, a 12.5\% chance of winning \$0.10 and a 62.5\% chance of winning nothing. As can be seen from Figure \ref{fig:figureB1}, this model fits our data approximately as well as the previous model, which provides further evidence that our results can be explained in terms of loss aversion across a number of reasonable model assumptions.

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