Closing a Mental Account: The Realization Effect for Gains and Losses

How do risk attitudes change after experiencing gains or losses? For the case of losses, Imas (Am Econ Rev 106:2086–2109, 2016) shows that subsequent risk-taking behavior depends on whether these losses have been realized or not. After a realized loss, individuals’ risk-taking decreases, whereas it increases after an unrealized (paper) loss. He refers to this asymmetry as the realization effect. In this study, we derive theoretical predictions for risk-taking after paper and realized gains, and for investment opportunities with different skewness. We experimentally test these predictions and, at the same time, replicate Imas’ original study. Independent of a prior gain or loss, we show that subsequent risk-taking is higher when outcomes remain unrealized. However, we find no evidence of a realization effect for non-positively skewed lotteries. While the first result suggests that the effect is more general, the second result reveals its boundary conditions.


Introduction
Many risky endeavors, be it a night at the casino or an investment in a stock, involve instances in which individuals must decide whether to continue, to abandon, or to double down on a previous decision. They often view such episodes in isolation, even though normative theory suggests integrating them into a broader perspective of total wealth. They instead engage in mental accounting (Thaler, 1985(Thaler, , 1999, which refers to a cognitive process to categorize outcomes by their source or purpose. Prior outcomes within a mental account, perceived as a gain or a loss, obtain special relevance for this account and affect subsequent risk taking (Thaler and Johnson, 1990).
Existing theory can account for these different reactions by a variety of models or arguments. On the one hand, risk-seeking behavior after a prior loss and risk-averse behavior after a prior gain are often explained by prospect theory (Kahneman and Tversky, 1979).
After a loss, the relevant part of the prospect theory value function to evaluate further outcomes is convex, which implies risk-seeking behavior. In contrast, a prior gain will situate a person in the gain domain for which the value function is concave, which implies risk-averse behavior.
On the other hand, more risk-seeking behavior after gains and more risk-averse behavior after losses can be motivated by the house money effect (Thaler and Johnson, 1990) and the hedonic editing hypothesis (Thaler, 1985). The house money effect describes a situation in which prior gains can be used to wager in subsequent gambles. People find it easier to part with money not coming from their own pocket. In addition, hedonic editing allows them to offset future losses against earlier gains. For losses, it is argued that they become more painful when they follow on the heels of prior losses (Barberis et al., 2001).
A unifying framework to resolve the conflicting evidence has been recently proposed by Imas (2016). It builds on the distinction between realized and unrealized outcomes, whereby a realization is defined "as an event in which money or another medium of value is transferred between accounts" (Imas, 2016(Imas, , p. 2091. He argues that individuals behave differently depending on whether a loss is realized or whether it is still unrealized (a paper loss). Experimentally, he replicates prior findings that participants become more risk-averse after a realized loss, while they become more risk-seeking after a paper loss. He labels the difference in risk taking between paper and realized losses the "realization effect" and explains its occurrence with cumulative prospect theory (Tversky and Kahneman, 1992) and choice bracketing (Read et al., 1999;Rabin and Weizsäcker, 2009), an idea directly related to mental accounting.
The proposed framework sheds light on why both, risk-averse as well as risk-seeking behavior, can be observed after the same prior outcome. However, drawing general conclusions from realization for subsequent risk taking requires some caution. First, Imas's (2016) theoretical and experimental elaboration focuses exclusively on losses, and second, it tests the realization effect for an investment opportunity with a positively skewed distribution of outcomes. We argue that the literature is still in need of empirical and theoretical clarification about how prior outcomes -losses as well as gains -affect subsequent risk taking, and in particular, under which conditions a distinction between paper and realized outcomes leads to differential risk-taking behavior. In this study, we contribute to this goal by examining two major research questions: (1) Does the realization effect exist for gains as well? (2) Does the realization effect depend on the skewness of the underlying investment opportunity?
To this end, we derive theoretical predictions for risk-taking behavior after gains and for investment opportunities with positive skewness, no skewness, and negative skewness. We model loss averse investors, who open a mental account at the beginning of an investment episode and close it upon realization. Paper gains and losses alter the balance of the mental account and can thereby affect risk taking. Paper gains act as a cushion against future losses and thus invite higher risk taking, which is absent after gains are realized. We thus predict a realization effect for gains. Skewness comes into play mainly via the size of potential gains and losses relative to the account balance. With non-positive skewness, losses become less probable but larger. They threaten to exceed the paper gain cushion, attenuating the realization effect after gains. Likewise after paper losses, more probable but smaller gains take away the potential to break even, which is a major motivation for higher risk taking after losses. We thus predict a smaller or absent realization effect for non-positively skewed lotteries.
We conduct three well-powered experiments to test these predictions. In the first experiment, we replicate the main experiment by Imas (2016) using an identical design, which examines a series of positively skewed investment opportunities. The importance of replication for scientific progress in economics has been highlighted recently (Maniadis et al., 2014;Camerer et al., 2016;Christensen and Miguel, 2018). At the same time, the experiment allows to address our first research question about a realization effect for gains. Not only is risk taking after gains arguably as important as after losses, but it shares a similar conflict in previous empirical results and theory. If there is evidence for a realization effect in the gain domain as we predict, the proposed framework would have broader implications than those already suggested for the loss domain.
To answer the second question, we analyze in two further experiments boundary conditions for the realization effect. In particular, we depart from positively skewed lotteries used so far and examine how symmetric or negatively skewed lotteries affect risk taking behavior after paper and realized outcomes. Not only does positive skewness encourage risk-seeking behavior as it is often associated with gambling (e.g., lotteries or casinos), but the underlying distributions of most financial investment opportunities (e.g., stocks or funds investments) are less or not at all positively skewed. In order to establish the validity of the realization effect for these settings, it is essential to confirm whether the effect is indeed reduced as theory predicts.
The first experiment, which replicates study one by Imas (2016), involves a sequence of four positively skewed lotteries, each of which represents the throw of a die. One lucky number (out of six) wins seven times the stake invested into the lottery, while the stake is lost for all other outcomes. Up to EUR 2.00 can be invested in each lottery. After the third lottery, previous earnings are either paid out to participants or remain unrealized, which defines the two treatments in the experiment (realization treatment and paper treatment). The relevant comparison then is what participants do in the fourth and final lottery depending on realization. We use a larger sample size (N =203) than the original study to ensure sufficient statistical power and to be able to examine outcome histories that occur less frequently.
We first confirm that participants take less risk after a realized loss compared to a paper loss. However, the difference of 16 cents in average invested amounts between treatments is smaller than in the original experiment (38 cents), and the realization effect is not statistically significant. While we confirm a decrease in risk taking in the realization treatment, we cannot corroborate an increase in risk taking in the paper treatment. Standard replication measures show that the replication is at least partially successful.
Exploiting observations in which participants have obtained a gain at the time of realization, we find a similar investment pattern as for losses. Participants take significantly less risk after a realized gain than after a paper gain. The realization effect is larger for gains than for losses with a difference of 22 cents in average investment between treatments. In the paper treatment, participants seem to gamble with the house's money, while in the realization treatment they have closed the mental account and regard gains from the lottery as their personal money. Given the consistent direction of the realization effect for gains and losses, we test for the realization effect unconditional of a particular outcome history. The results show a positive and strongly significant realization effect (p < .01) in the full sample.
In addition to own experimental data, we analyze data from the original study by Imas (2016) with respect to gains. 1 Although limited in the number of observations, the realization effect for gains is strong and consistent with our results. Thus, we find evidence for a realization effect for gains in two independent samples. Moreover, pooling the data from both studies, we find a positive and strongly significant realization effect (p < 0.001) for gains and losses. To test for the theoretical relation between the realization effect after gains and the house money effect, we examine the invested amounts after a paper gain. In almost all cases, participants do not invest more than what they have gained in the lotteries. This implies that they gamble with the house's money, but do not touch their initial experimental endowment.
In experiments two and three, we examine how other distributions of outcomes affect risk-taking behavior after paper and realized gains and losses. We keep the basic experimental setup, but change the probability of gains. Instead of a positively skewed lottery, participants invest in a symmetric or negatively skewed lottery, respectively. By construction this also increases the heterogeneity of outcome histories prior to realization. We find neither in the symmetric lottery nor in the negatively skewed lottery a statistically significant realization effect for gains or losses (total sample size N =304). In contrast to the positively skewed environment in the first study, participants tend to invest similarly after a paper outcome and a realized outcome. This finding is in line with theoretical work by Barberis (2012) and 1 The data is publicly available via the AER website. Imas (2016) restricts his analysis to participants, who have lost in all lotteries up to round three (when realization takes place).

Imas (2016) in which individuals form contingent plans over a sequence of lotteries.
The results across all experiments suggest boundary conditions for the realization effect. Figure 1 depicts the magnitude of the realization effect we find, conditional on the outcome history as well as the skewness of the investment opportunity. Increased risk taking after paper gains and losses requires positive skewness, while decreased risk taking after realized gains and losses does not. The absence of the realization effect for non-positively skewed lotteries is thus primarily driven by an absence of increased risk taking after paper outcomes.
This includes the absence of loss chasing which seems to be limited to positive skewness environments.
The remainder of the paper is organized as follows. In section 2, we derive theoretical predictions for the experiments, in particular for risk-taking behavior after gains and for lotteries with different skewness and review the prior literature. Section 3 presents the experimental design and the main results. A final section concludes.

Theory and Literature
To understand the behavior of participants in the experiments, we build on the model by Barberis et al. (2001). In addition to standard consumption-based utility, they consider utility derived directly from the fluctuations of financial wealth. In particular, agents react to gains and losses from their risky assets, which makes the model suitable for the analysis of behavior after gains and losses. Prior theory used to motivate the realization effect does not generate clear predictions for risk-taking behavior after gains. We introduce two departures from the main model in Barberis et al. (2001), which are the distinction between paper outcomes and realized outcomes, and a different value function after losses. The first is a natural extension to accommodate a treatment of paper and realized outcomes, the second takes into account the empirically observed behavior in the loss domain.

Basic framework
The full utility specification in Barberis et al. (2001) includes utility from consumption u(C t ) and utility derived from fluctuation of financial wealth v(X t , B t , Z t ). We concentrate on the latter as it represents the important part for evaluating risk-taking behavior after gains and losses. X t is the gain or loss a participant experiences in lottery t. 2 B t is the bet size a participant selects for lottery t. And Z t is a mental account, which reflects whether a participant perceives himself up or down in the game. Mental accounting describes the cognitive processes people use to organize and evaluate their financial activities (Thaler, 1985(Thaler, , 1999. A key implication is that people do not consider money across different mental accounts as perfect substitutes, but rather categorize money based on its origin or purpose and assign it to separate accounts. Outcomes within a mental account are evaluated jointly whereas outcomes in different mental accounts are evaluated separately (Thaler, 1999).
The three variables X t , B t and Z t jointly determine the utility derived from fluctuations of financial wealth. A difference to the more general model arises from the fact that only part of a participant's endowment is invested into the risky lottery. Still B t can be interpreted as a participant's risky asset holdings. The outcome of lottery t is X t = R t B t − B t with gross return R t . We abstract from a risk-free rate, as no return is paid on money not invested into the lottery. If a participant loses in the lottery, then X t = −B t . If a participant wins, then X t = (x − 1)B t with x > 1 as the multiple that is applied to a winning bet. The lottery will thus either generate a loss or a gain. Besides these potential outcomes, participants take their prior gains and losses into account. Z t is the mental account, which reflects prior outcomes: While Barberis et al. (2001) leave open what exactly this mental account (or "historical benchmark") is, in our context we will assume that it is the sum of prior gains and losses.
A participant can thus be in the gain domain (Z t > 0), in the loss domain (Z t < 0) or at break-even (Z t = 0). In particular, Z 1 = 0 as no lottery has yet been played. In this situation, utility from changes in financial wealth is described by: The parameter λ > 1 captures loss aversion. We further assume that realizing a gain or a loss resets the benchmark to zero as the mental account is closed. The intuition is that when a stock is sold, the proceeds are mentally transferred from the account investment to consumption. Paper losses may consequently not be regarded as final and possess the potential to rebound (Shefrin and Statman, 1985). The idea that realization affects decision making has been tested in an experimental asset market (Weber and Camerer, 1998). When stocks are automatically sold after each period, the disposition effect is significantly reduced. The automatic selling procedure closes existing mental accounts, and stocks are no longer charged by prior experiences of gains or losses. 3 This means that after realizing lottery outcomes, a participant is effectively in the same decision situation as before entering the first lottery: H1. After a gain or a loss is realized, risk-taking behavior will be similar as in a decision without prior history. Barberis and Xiong (2009) study the implications of realized and paper outcomes as well.
In two alternative models, they define prospect theory preferences either over total gains and losses or realized gains and losses. They discover that the model based on realized outcomes predicts the disposition effect more reliably.

Behavior after gains
One main idea of the model is that prior gains serve as a cushion against losses which are felt less severely as long as they do not exceed prior gains. This is consistent with the "house money effect" predicting that people take more risk in presence of a prior gain (Thaler and Johnson, 1990). When offered a risky lottery, individuals evaluate prior paper gains (house money) and the risky prospect jointly within the same mental account. Since the house money is integrated with future outcomes, losses can be offset and are perceived as less painful than usual. 4 Formally, losses up to the level of prior gains are not subject to loss aversion: 3 Barberis et al. (2001) consider this plausible although they exclude this possibility for their analysis: "However, larger deviations-a complete exit from the stock market, for example-might plausibly affect the way [Z t ] evolves. In supposing that they do not, we make a strong assumption, but one that is very helpful in keeping our analysis tractable (p.13)." We assume that realizing all gains or losses is perceived similar to an exit from the market. 4 The idea is consistent with Arkes et al. (1994) who argue that windfall gains are spend more readily than other types of assets and Peng et al. (2013) who argue that the psychological value of losing parts of a prior gains is relatively low.
This means that losses up to Z t are evaluated at the more gentler rate of 1 instead of λ. Accordingly, a paper gain reduces loss aversion when compared to a realized gain. This is particularly true for small bet sizes B t < Z t , which do not jeopardize the whole gain cushion. Realization closes the respective mental account for prior gains and triggers the internalization of house money. Prior gains are no longer available to offset potential losses.
Without integration, individuals evaluate a risky lottery separately from the previous gain and do not use the more gentler rate of 1 instead of λ anymore. This reasoning is also graphically illustrated in Panel A of Figure 2. We hypothesize: H2. After a paper gain people are more prone to take risks than after a realized gain.
H2a. They avoid bet sizes that run the risk to lose more than the sum of prior gains.
Hypothesis 2 may shed light on seemingly contradictory results in the empirical literature: Less risk taking after a prior gain versus more risk taking after a prior gain. While the house money effect predicts a higher propensity to gamble after a prior gain than before (or after a loss), the disposition effect describes the opposite behavior. Investors show a tendency to sell winning stocks too early and to keep losing stocks too long (Shefrin and Statman, 1985;Odean, 1998;Weber and Camerer, 1998). Intuitively, the trading behavior behind the disposition effect is in line with prospect theory (Kahneman and Tversky, 1979). A winning stock moves an investor into the gain domain of the prospect theory value function. As the value function is assumed to be concave for gains, it implies risk-averse behavior and a higher likelihood to sell the stock.
Further tests are similarly inconclusive for risk taking after gains. Weber and Zuchel (2005) show in lottery experiments that participants become more risk-seeking after a gain, while Franken et al. (2006) find in a gambling task that previous gains lead to less risk taking. Clark (2002) does not find evidence in either direction following gains in a public goods experiment. However, bettors on the horse track take more risk after a previous gain

Behavior after losses
When a mental account is in the red, i.e., a participant has experienced an overall loss, then the outcomes of a lottery are evaluated in the following way: The expression represents the mirror image of the situation after gains and again reflects the idea of an open mental account in which a loss is not final. Gains that make up for prior losses are particularly attractive and are valued at a rate of λ. Barberis et al. (2001) assume that losses on the heels of prior losses are more painful than usual and let loss aversion rise in Z t . However, the results by Imas (2016) for paper losses question this idea, as people take more risk after a series of losses. The traditional view inspired by prospect theory also favors higher risk taking after losses (Kahneman and Tversky, 1979). While the channel in prospect theory is higher risk tolerance, in the piecewise linear (risk-neutral) utility function used here, it could manifest in a decreasing loss aversion parameter (consistent with a learning effect documented by Merkle (2019)). We thus depart from the assumption of higher loss aversion after a prior loss and instead propose a constant loss aversion parameter. The extent of loss chasing will depend on how people's preferences react to prior losses.
When offered a risky lottery, individuals evaluate prior paper losses and the risky lottery jointly within the same mental account. They thus evaluate further losses at the same rate as gains reducing these losses. By contrast, realization closes the respective mental account, internalizes the prior losses and resets the reference point to Z t = 0 (see also Panel B of Figure 2). Note that equations 3 and 4 simplify to equation 2 in this case. We thus expect participants to take more risk when confronted with a paper loss (mental account still open) than with a realized loss (mental account closed): H3. After a paper loss people are more prone to take risks than after a realized loss.
H3a. They favor bet sizes that give them the opportunity to break even.
For risk taking after losses similarly inconclusive empirical evidence as for gains has been found. There is strong empirical support for an increase in risk taking after experiencing a loss, which has been demonstrated in the lab (Gneezy and Potters, 1997;Weber and Zuchel, 2005;Langer and Weber, 2008;Andrade and Iyer, 2009) as well as in the field (Coval and Shumway, 2005;Meier et al., 2020). Such loss chasing has been identified as a source for gambling problems (Zhang and Clark, 2020), and might be driven by impulsive action (Verbruggen et al., 2017). On the other hand, several studies report a decrease in risk taking after losses (Massa and Simonov, 2005;Shiv et al., 2005;Frino et al., 2008). Imas (2016) points out how the different results can be reconciled by distinguishing paper losses and realized losses (in line with H3). The presented findings almost exclusively rely on positively skewed gambles, for other skewness environments there is hardly any evidence (see also Nielsen, 2019).
Hypothesis 3a does not follow directly from the introduced theory, as gains are treated equally up to the point where they exceed prior losses (X t > −Z t ). However, already Thaler and Johnson (1990) report such a break-even effect. Moreover, there is evidence that finally realizing an outcome is associated with an immediate burst of utility (Barberis and Xiong, 2012;Frydman et al., 2014). Such realization utility implies that agents also care about the level of Z t , in particular when they anticipate that the respective mental account will be closed.
In the experiment, the final lottery represents the last opportunity to influence cumulative outcomes Z T which are automatically realized at the end of the experiment. Lotteries that allow to change the sign of Z T should be especially attractive. A sufficiently large multiplier x as found in positively skewed lotteries usually allows to break even. Depending on accumulated losses, it might not even be necessary to increase risk.

The realization effect and skewness
In our model, a positively skewed lottery is prone to the realization effect as it offers a high potential gain and limited loss. In the gain domain, the cushion provided by Z t will be able to absorb most of a possible loss and induce risk taking unless the mental account is closed. In the loss domain, the lottery almost always offers the chance to break even, as the multiplier x applied on the bet B t is sufficiently high. This also induces risk taking, which is why a strong realization effect can be expected for positively skewed lotteries independent of the prior outcome.
In contrast, symmetric and negatively skewed lotteries are characterized by a lower but more probable gain, and a higher but less probable loss. A reasonable assumption is that probabilities and payoffs of the lotteries are altered simultaneously so that their expected payoff remains (about) constant. 5 It is then more likely that previous gains cannot completely cushion a potential loss, which might deter people from risk taking. Figure 2 illustrates this by the size of the mental account balance Z 1 in period one relative to the bet size B 2 in period two. The smaller account balance Z 1 after an initial gain only allows for smaller bets if people do not want to risk their endowment. We predict no reaction to skewness for risk-taking behavior after realized gains, as it is independent of prior history (see H1). Consequently, the realization effect should be reduced.
In the loss domain, symmetric or negatively skewed lotteries offer less potential to break even. Initial losses (−Z 1 ) are larger relative to potential gains xB 2 . However, it is still possible to recoup prior losses at least partly, making the lottery somewhat more attractive than after losses are realized and mental accounts closed.
H4. The realization effect is reduced or absent for symmetric and negatively skewed lotteries.
Previous empirical studies have shown in various domains that skewness influences risk taking and that positively skewed lotteries tempt individuals to engage in more risk taking.
For example, individual investors have a preference for lottery-type stocks, characterized by low prices, high volatility and large positive skewness (Kumar, 2009). Further evidence for positive skewness-loving investment behavior comes from horse race betting and state lotteries (Golec and Tamarkin, 1998;Garrett and Sobel, 1999). This is in line with Grossman and Eckel (2015), who find increased risk taking in an experimental study with positively skewed lotteries. While most of the literature on dynamic risk taking concentrates on positively skewed lotteries, there are many situations in every-day decision making in which outcome distributions are less or not at all positively skewed. For example, investors in the stock market or corporate managers usually face less lottery-like investment opportunities. Given this gap in the literature on risk taking for non-positively skewed lotteries, the second objective of this study is to investigate whether the realization effect can be generalized to symmetric and negatively skewed lotteries.
Our model is broadly consistent with the theory provided by Imas (2016). The common prediction is that risk taking after a paper loss is higher than a) before a paper loss and b) after a realized loss. However, we explicitly model a mental account (represented by Z t ), while Imas (2016) invokes a mere shift in the reference point. This difference becomes apparent when deriving predictions for the gain domain. An agent with a paper gain might take less risk in his model compared to an agent with a realized loss or no history. 6 As this defies, for example, the presence of a house money effect, we find this approach not appealing for understanding behavior after gains.
In the main model by Imas (2016), the proof for the general existence of a realization effect after losses relies on features of a positively skewed lottery. The effect is not necessarily absent for symmetric or negatively skewed lotteries, but in these cases depends on preferences (e.g., the degree of loss aversion). Similar to our model, a reduced aggregate realization effect can be expected in a population with heterogeneous preferences. Both models rely on myopic decision makers, who take only the next round of a lottery into account. An alternative is allowing for people to make contingent plans on their investments after gains and losses (e.g., Barberis, 2012). Contingent plans may alter the existing skewness of asset returns, for example make them more positively skewed by planning to cut losses. In Online Appendix A, we discuss such models in more detail.

Experimental Design and Results
The design of the experiments is based on Imas (2016), who studies a version of the investment lottery by Gneezy and Potters (1997). Participants receive a total endowment which can be invested over several rounds in the same lottery. In each round, participants can invest a maximum amount E in the lottery, which is a constant fraction of the total endowment.
They thus decide on their lottery investment (B t ) and how much they want to invest risk-free (E − B t ). For simplicity, the risk-free investment provides no interest. With probability p, the lottery returns the invested amount times a multiple x, with probability 1 − p the investment is lost. A participant can thus either make a gain of (x − 1)B t or a loss of −B t . The expected payoff in each round is: Lotteries are structured in such a way that px > 1, which means that the lottery has a positive expected payoff, and the expected payoff increases in the bet size B t . Otherwise the lottery would be unattractive to risk-averse participants. After the investment decision is made, the outcome of the lottery is determined and revealed to participants. In the following round, the same lottery is played again. Importantly, investment possibilities in later rounds do not depend on prior payoffs as the maximum investment E is a constant fraction of the total endowment.
The total number of lottery rounds in all experiments is four. In the realization treatment, participants invest over three rounds, and outcomes are realized at the end of the third round. After this an additional lottery takes place. In the paper treatment, all four rounds are played consecutively and there is no special significance of the turn between the third and final round. However, to keep information between treatments constant, participants in both treatments are informed about their earnings at the end of the third round on the screen. The main analysis thus relies on the risk taking behavior in the final round, as the first three rounds are identical between treatments.

Design and participants
In the first experiment, we replicate the original design by Imas (2016). In each round, participants decided how much to invest in a positively skewed lottery. The lottery succeeded with a probability of 1/6 and paid seven times the invested amount or it failed with a probability of 5/6 and the invested amount was lost. Considering this experimental design, the conditions under which the realization effect occurs turn out to be arguably restrictive.
Imas (2016) focuses his attention to sequences of prior losses, excluding all histories involving a gain. 7 In addition, the nature of the lotteries is such that participants bet on the throw of a six-sided die and win (seven-fold) if their predetermined "lucky number" comes up. This results in a positively skewed lottery. In the first experiment we extend the analysis to the gain domain, while in experiments two and three we introduce different types of skewness.
Participants were randomly assigned to either a realization treatment or a paper treatment as described above. After entering the laboratory, each participant received an envelope which contained the endowment of EUR 8.00. The instructions asked participants to count the money (see Online Appendix B for the experimental instructions). The lotteries were framed 7 In expectation, only (5/6) 3 = 58% of observations enter the analysis.
in terms of the throw of a six-sided die and always proceeded in the same way. First, each participant was randomly assigned a success number between 1 and 6, which was displayed on the computer screen. Then participants decided how much to invest in the lottery up to a maximum of EUR 2.00. As soon as all participants had entered the amount, the experimenter rolled a large die in front of the room. All participants received the opportunity to check whether the die was fair. If the success number matched the rolled number, the participant won the lottery and obtained seven times the invested amount (plus the amount invested risk-free). If the success number did not match the rolled number, the participant lost the invested amount and kept the amount not invested. For the next round a new success number was assigned. As in the original experiment, all results of the die roll were written on a board in front of the room.
In the realization treatment, outcomes were realized at the end of the third round.
Participants who lost money by that time, took the lost amount out of the envelope and handed it back to the experimenter. Participants who won received additional money from the experimenter. After this, participants made one last investment decision in a final round and were paid accordingly. In the paper treatment, outcomes were not realized at the end of the third round. Outcomes were merely communicated on the screen as in the realization treatment, but no physical transfer of money took place. 8 At the end of round four, all outcomes were realized for both groups. As in the original experiment, the time between rounds was normalized across treatments. Consistent with hypotheses H2 and H3, we predict that participants in the paper treatment (after gains and losses) will invest more in the final lottery than participants in the realization treatment.
Experiment one was programmed in z-Tree (Fischbacher, 2007) and conducted in the Mannheim Experimental Laboratory (mLab). We selected a sample size of N > 200 participants to obtain statistical power of at least 90% to detect an effect of the size of the original realization effect at the 5% significance level (Camerer et al., 2016). We recruited 203 people via ORSEE (Greiner, 2015) from a university-wide subject pool to participate in a study on decision making. Participants were on average 23 years old and the number of female (n = 108) and male (n = 95) participants was relatively similar (see Table 1).

Replication results
We first examine the replication of the realization effect for losses. The analysis centers on the change of investment between round three and four, as realization takes place before round four. To test for the realization effect, we are mainly interested in three comparisons: The difference in the change of investment between the paper and realization treatment (between-treatment comparison) and the change of investment for each treatment separately (within-treatment comparisons). Panel A of Table 2 shows the amounts invested in the lottery for participants who have a total loss by the end of round three which means that they lost in each of the first three rounds. 9 Investments do not differ significantly across treatments over the first three rounds. In the final round, participants in the paper treatment invest slightly more, while participants in the realization treatment invest less. This pattern is consistent with a realization effect as stated in hypothesis H3, which predicts a positive difference in differences (DiD = 0.16, t(113) = 1.58, p = 0.12).
However, compared to results of study one by Imas (2016)  Interestingly, when focusing on the investment behavior within treatment, the realization effect we find is primarily driven by a decrease in risk taking in the realization treatment (−0.12, t(57) = 1.64, p = 0.11), while the effect in the original data is primarily driven by an increase in risk taking in the paper treatment. We can confirm that participants tend to take less risk after a realized loss, but we cannot replicate that participants increase risk taking after a paper loss (0.04, t(56) = 0.57, p = 0.57). In other words, we do not find loss chasing in the paper treatment which ultimately explains the overall less pronounced realization effect for losses as compared to Imas (2016).
One reason for the non-robust results after paper losses might be that the positively skewed lottery offers participants the chance to break even without necessarily having to increase risk taking. Whether or not some participants still increase their risk taking will depend on their prospect theory preference parameters. 10 When comparing the invested amount in round four to the invested amount in round one, we find that participants are more risk averse after a realized loss than without any prior outcome (−0.22, t(57) = 2.49, p = 0.02). This is inconsistent with hypothesis H1, but in line with the idea of Barberis et al. (2001) who argue that individuals become more sensitive to future losses after a previous loss. In general, the changes in risk taking between round three and four are not particularly large when compared to the changes observed for earlier rounds (see Online Appendix D). We find some significant results for earlier rounds across all three experiments, but we cannot identify a systematic pattern behind these changes. Significance occurs mostly between round one and round two, which suggest that participants try out the lottery first before making considerable adjustments to their bet size. Importantly, by round three risk-taking behavior is very similar between treatments.
As a further test for replication, we pool our data with the original data by Imas (2016).
Hereby, we are able to obtain a meta-analytic estimate of the effect (Camerer et al., 2016).
In the pooled data we obtain a strongly significant realization effect after losses (DiD = 0.24, t(165) = 3.10, p < 0.01). We conclude that the evidence on the outcome of the replication is mixed. We find a weaker but directionally consistent realization effect after losses.

Results for gains
We next examine participants with a gain at the end of the third round. Given the considerable upside potential of the lottery, most participants who succeeded in at least one lottery faced positive net earnings at the end of the third round. The overall sample of 203 participants splits into 115 participants with a loss by the end of round 3 analyzed above, 71 participants with a gain by the end of round three, and 17 participants who have zero net earnings by the end of round three (due to not investing in the lottery at all). Of the 71 participants with a gain, 65 won the lottery once and 6 won twice. 11 Panel B of Table 2 shows the invested amounts for these participants. In most cases, changes in investment in rounds one to three do not differ significantly across treatments. 12 Consistent with Hypothesis H2, the change in risk taking between round three and four is significantly different between the paper and the realization treatment (DiD = 0.22, t(69) = 2.16, p = 0.03). This realization effect for gains is somewhat larger than the replicated effect for losses. Within treatment, participants in the paper treatment take significantly more risk ( To back-up this finding, we turn again to the original data by Imas (2016), which have not been analyzed with regard to risk taking after gains. As before, we only use observations of participants with a gain at the end of round three.Despite the relatively small sample size (N=24), we nevertheless find evidence for a realization effect after gains in his data. As shown in Table 3, participants take more risk in the paper treatment than the realization treatment considering changes between round three and four. Consistent with the results from our experiment, the realization effect is positive and statistically significant (DiD = 0.55, t(22) = 2.29, p = 0.03). Within treatment, participants take more risk after a paper gain (0.47, t(8) = 1.99, p = 0.08) and tend to take less risk after a realized gain (−0.08, t(14) = 0.67, When we pool the data from both studies, we find a strong realization effect for gains (DiD = 0.29, t(93) = 2.96, p < 0.01). We thus find experimental evidence for a realization effect for gains in two independent samples. The studies were conducted with student populations from different universities, in different countries, and at different points in time.
While the p-value in both samples is similar (p = 0.03), the combined evidence provides far stronger support to hypothesis H2.
Irrespective of whether the prior outcome is a gain or loss, risk taking is thus higher when outcomes remain unrealized. This finding allows us to analyze the existence and strength of 11 Table D.4 in Online Appendix D provides more details about participants' average earnings after round three conditional on the outcomes in each round. 12 We also do not find significant changes in investment before and after the round in which a participant wins across treatments; see Online Appendix D, Table D.5.
the effect independent of the sign of the prior outcome. Therefore, we run OLS regressions for the entire sample with the change in invested amount between round three and four as the dependent variable. We include a treatment indicator taking a value of one for the realization treatment. Table 4 shows in column (1) the results of the baseline regression. We observe a strong realization effect, with those in the realization treatment taking significantly less risk.
Unsurprisingly, the economic magnitude is in between those estimated for gains and losses separately. The positive constant provides evidence for an increase in risk taking in the paper treatment. Controlling for gains and losses after round three by a gain indicator (gain=1) does not affect the main result (Column 2). Interacting the treatment and gain variables allows us to test whether the realization effect is stronger after previous gains or losses. The negative but insignificant coefficient of the interaction term hints at a stronger realization effect after gains.
We run the same regressions on the data from Imas's (2016) study 1. Columns (4) to (6) in Table 4 display the results. A strong realization effect also exists in his data independent of prior gains and losses. The effect in his data is even more pronounced in economic magnitude than in our data. The combined effect independent of the prior outcome in the pooled data is (DiD = 0.25, t(283) = 4.38, p < 0.001).
A relevant assumption about the realization effect is that people are less loss averse for money they keep in the mental account for house money (paper gains) than for their own money that they keep in a different mental account (realized gains). This assumption has testable implications for the amount people are willing to bet (hypothesis H2a). We predict that participants avoid bet sizes that run the risk to lose more than the sum of prior gains.
Since participants can invest up to EUR 2.00 in each round and lose at a maximum their invested amount, the subsample of interest are participants who have earnings between EUR 8.00 and EUR 10.00 after round three (i.e., gains between EUR 0 and EUR 2). If mental accounting is important, participants are expected not to invest more than their current paper gains (house money) in round four. Figure 3 plots the earnings after round three against the invested amount in round four. The maximum invested amount of participants in this subsample was EUR 1.00. All dots above the line represent participants who invest less than their house money in round four, which restricts their potential losses to less than their previous gains. Dots below the line represent participants who risk to lose more than their prior gains. Consistent with hypothesis H2a, 11 out of 12 participants invest less or exactly as much money as they previously gained.
Essential for the realization effect is that the used realization mechanism is effective in closing a mental account. We tested an alternative realization mechanism in two versions of an online experiment, one of which is an identical replication of the online study in Imas (2016). As a physical transfer of money is not feasible online, participants in the realization treatment initiate a transfer of money between accounts by typing the command "closed".
We successfully replicate the realization effect using this alternative realization mechanism in the original design by Imas (2016), but discover that the effect is rather fragile when modestly changing the design. We find that the framing of how the last round is related to the preceding three matters for whether risk taking increases or decreases in the realization treatment of the online experiment. 13 We conclude that in an online environment proper realization is more difficult to achieve and mental accounts may remain open using the described procedure.
Complete results are reported in Online Appendix E.

Experiment 2 and 3 3.2.1 Design and participants
Experiment two and three address the question whether the realization effect depends on the skewness of the underlying investment opportunity. We take the same experimental design as in experiment one except for the investment opportunity which we change to either a symmetric (experiment two) or a negatively skewed lottery (experiment three). In line with hypothesis H4, we predict a reduced or absent realization effect in these settings.
Participants were again endowed with EUR 8.00 at the beginning of the experiment and could invest up to EUR 2.00 in each of four subsequent lottery rounds. In experiment two (symmetric lottery) participants could invest in a lottery which succeeded with a probability of 1/2 and payed 2.33 times the invested amount. With a probability of 1/2 the lottery failed and the invested amount was lost. Instead of one success number for the role of the die, participants received three success numbers. In experiment three (negative skewness) participants could invest in a lottery which succeeded with a probability of 5/6 and payed 1.4 times the invested amount or failed with a probability of 1/6. Instead of a success number they received one failure number.
The multiplier for the gain case was adjusted to keep the expected payoff of each lottery equal to the expected payoff of the lottery in experiment one. While the objective of experiment three was to create a mirror image of the original positively skewed lottery, a complete reversal of gains and losses was infeasible as losses cannot exceed the endowment (by laboratory rules).
Instead of a seven-fold loss, we thus have to restrict the loss to the invested amount. Still, participants are expected to experience many small gains and occasionally (relatively) large losses.
As before, participants were randomly assigned to either a realization treatment, in which outcomes were realized by the end of the third round, or a paper treatment. The procedure in the two treatments was the same as in experiment one. Both experiments were conducted in the Mannheim Experimental Laboratory (mLab) and the AWI Experimental Laboratory at the University of Heidelberg. 14 We recruited 304 participants in total, 95 of them were assigned to experiment two and 209 to experiment three. A smaller sample size was required in experiment two as a symmetric lottery generates sufficient observations for gains and losses more easily. The demographics of participants in experiments two and three are similar to those in experiment one (see Table 1).

Results of experiment 2 (symmetric lottery)
We first analyze the investment behavior of participants who accumulate a loss by the end of round three. Panel A of Table 5 presents the invested amounts for those participants.
Investments do not differ significantly across treatments in the first three rounds. Comparing the changes in investment between round three and four across treatments, the realization effect points in the expected direction (DiD = 0.08, t(35) = 0.54, p = 0.59), but is small and statistically insignificant. When analyzing the invested amounts within each treatment, we find that participants who have a paper loss by the end of round three do not increase their investment (0.00), and participants who have a realized loss tend to slightly decrease their investment (−0.08, t(14) = 0.73, p = 0.48). Participants thus seem not to invest differently after a paper or a realized loss. In particular, we do not observe more risk taking after paper losses.
Panel B of Table 5 shows the invested amounts of participants with an accumulated gain by the end of round three. Similar to losses, the realization effect cannot be observed in the symmetric lottery setting (DiD = −0.03, t(55) = 0.18, p = 0.86) for gains. The change in investment between round three and four in the paper treatment (−0.11, t(26) = 0.69, p = 0.50) and the realization treatment (−0.08, t(29) = 0.89, p = 0.38) points in the same direction. After a paper as well as a realized gain, participants tend to invest similarly.
Consistent with hypothesis H4, we find no evidence for a realization effect after gains or losses when the investment opportunity is symmetric.
Looking at investments on participant level, we find that 53% of the participants do not change their invested amount between round three and four (fairly independent of treatment).
Any overall effect would thus have to rely on a subset of participants to make strong changes in their investments. We also find that the absence of the realization effect does not depend on the round(s) in which participants win in the lottery.
Finally, we test whether participants in the paper treatment do not increase their investment after a loss because their losses are too high to break even in the final lottery. In contrast, the positively skewed lottery always allowed to break even. We split the sample of participants with accumulated losses into those who have earnings by the end of round 3 which are smaller than EUR 5.34 and those who have earnings between EUR 5.34 and EUR 8.00 (the highest possible gain in the final lottery is 2.33 * 2 − 2 = EUR 2.66). Despite the resulting small sample size, we find that participants with paper losses tend to invest differently depending on whether break-even is possible or not. Those who cannot break even, tend to decrease the invested amount in round four by on average EUR 0.38, whereas participants who can break even tend to increase the invested amount by EUR 0.11. That people favor bet sizes that allow them to break even is consistent with hypothesis H3a.
However, given the small sample size of participants with a paper loss (N =22), the effect remains insignificant and has to be interpreted with caution.

Results of experiment 3 (negatively skewed lottery)
We again start by examining the investment behavior of participants who accumulated a loss by the end of round three. Most of these participants lost only once, but remained in the loss domain. Panel A of Table 6 shows the investments in all rounds for these participants by treatment. Levels and changes in investment between rounds do not differ significantly across treatments. Considering the difference of the changes in investment from round three to round four across treatments, the realization effect points in the expected direction (DiD = 0.05, t(68) = 0.36, p = 0.72), but is small and statistically insignificant. Participants in both treatments react similarly to a loss by slightly reducing their investments (−0.04, t(31) = 0.45, p = 0.66 and −0.09, t(37) = 1.17, p = 0.25).
The investments for participants with gains by the end of round three are displayed in Panel B of Table 6. As for losses, we do not find a significant realization effect for gains in this setting. Participants do not invest differently after a paper and a realized gain (0.00 and 0.00). In fact the investments on average do not change at all between round three and round four. Results change very little if we restrict the sample to those participants who experience three successes in a row (N =121). In line with hypothesis H4, we do not find evidence for a realization effect when participants invest in a negatively skewed lottery. This supports theoretical predictions that the realization effect depends on the positive skewness of the lottery.

Conclusion
In this paper, we examine whether and under which conditions a distinction between realized and unrealized prior outcomes leads to differential subsequent risk taking. We formalize our thoughts in a model of mental accounts that people use to keep track of their paper gains and losses. A mental account is closed when an investment episode ends and outcomes are realized. For losses, recent experimental evidence finds that individuals take less risk after a realized loss and more risk after a paper loss, which is referred to as the realization effect.
It is tempting to conclude from this result that realization per se has a strong effect on subsequent behavior. We first ask, whether-as our theory predicts-the finding generalizes to the gain domain, i.e., whether a realization effect can also be observed after gains. Second, we identify positive skewness as a necessary condition to observe the realization effect. As such, our results show that conclusions about the universality of the realization effect have to be drawn with some caution.
The main objectives and findings from our study can be summarized as follows: We replicate the result by Imas (2016) for losses, extend the analysis to gains and test the boundary conditions of the effect with respect to the skewness of the investment opportunity.
Using the same experimental setting and a larger sample size than the original study, we show that the realization effect exists also for gains. We thus show that the framework of realization is independent of the sign of prior outcomes as it holds not only for losses, but also for gains. However, at the same time, the effect turns out to be sensitive to changes in the skewness of the underlying investment opportunity. We do not find differential risk taking after paper and realized outcomes for non-positively skewed lotteries. This finding documents the importance of learning more about the conditions under which the effect arises and informs judgments about its external validity.
The results confirm theoretical predictions that a realization effect mostly occurs in positively skewed lotteries. The analysis of risk taking in non-positively skewed lotteries, in particular, in negatively skewed lotteries has received less attention in the literature. One recent exception is contemporaneous work by Nielsen (2019), who examines risk taking under negatively skewed outcome distributions for realized and unrealized losses. Using a different realization mechanism and a different investment task in which individuals can choose the skewness of their preferred option, she finds no realization effect for negatively skewed outcomes. Her finding is in line with our results and further supports the conclusion that the realization of outcomes does not always induce differences in risk taking compared to settings in which outcomes remain unrealized.   For illustrative purposes only two rounds of a lottery are displayed and outcomes are either on paper (left diagrams) or realized after the first round (right diagrams). Each diagram plots the round of the lottery on the x-axis and the earnings on the y-axis. Endowments are the same in t=0, which then adjust depending on the outcome of the first lottery in t=1. In round 2, the chosen investment B 2 determines the potential earnings indicated by the horizontal bars. Color coding shows whether outcomes are evaluated as gains (green) or losses (red).
Whether an outcome is evaluated as a gain or loss depends on the mental account and its reference point. For example, in the left diagram of Panel A, the paper gain from the first lottery enters a newly opened mental account shown in yellow. Outcomes in round 2 are evaluated against this previous gain which offsets potential losses. The right diagram of Panel A shows the same situation when instead the gain is realized. The respective mental account is closed, the previous gain is internalized, and the reference point shifts to the new wealth level. In round 2, there is no cushion against a potential loss which is indicated in red. Figure 3: Testing the mental accounting assumption. The figure plots the earnings by the end of round three against the investment in round four for each participants who has earnings between EUR 8.00 and EUR 10.00 by the end of round three. Participants with earnings below EUR 8.00 are excluded as they made a loss and participants with earnings above EUR 10.00 are excluded as they cannot lose more than what they previously gained (given than the investment per round cannot be more than EUR 2.00 which also presents the highest possible loss per round). All dots above the diagonal line present participants who invest less than what they previously gained and all dots below the diagonal line exhibit all participants who invest more than what they previously gained.

Online Appendix A. The Realization Effect and Skewness
In what follows, we use a non-myopic framework to explain more precisely how the realization effect and the skewness of the investment opportunity are related. To do so, we build on previous work by Barberis (2012) and Imas (2016) [in an alternative to his main model]. Barberis (2012) shows that cumulative prospect theory can explain sequential risk-taking behavior and demonstrates how the skewness of a lottery affects people's propensity to take risk. Therefore, his model is relevant to analyzing the relationship between realization and skewness. To explain differential risk-taking behavior after paper and realized losses in a non-myopic case, Imas (2016) uses the findings from Barberis (2012) framework. In the following, we will therefore also refer to Imas (2016).
A key ingredient of cumulative prospect theory is probability weighting (Tversky and Kahneman, 1992). People tend to overweight small probabilities while underweighting large probabilities. As a result of probability weighting, Barberis (2012) shows that (1) people are willing to invest in a symmetric lottery with negative expected payoff and (2) prospect theory predicts an inconsistency in subsequent investment behavior. To understand these predictions, we briefly review his model. In the model, cumulative prospect theory generates inconsistency in sequential risk-taking environments, which is captured by the difference between people's ex-ante plans and actual behavior. People initially optimize over a set of potential gambling plans. For a wide range of parameters people prefer the "loss-exit" plan, where they plan to continue gambling if they win and stop gambling if they start accumulating losses. However, after they begin gambling, people actually deviate from this "loss-exit" plan: They continue gambling when they lose and stop gambling when they have a significantly large gain.
The reasoning is as follows: The "loss-exit" plan makes accepting risk initially attractive. A key characteristic of this plan is that its perceived distribution of outcomes over all rounds is positively skewed. Since small probabilities of winning in this plan are overweighted, gambling becomes highly attractive. In other words, following this plan limits the downside (they stop gambling after losing) while it retains the potential upside (they continue gambling after winning), making the overall lottery distribution much more positively skewed than the one of a single lottery. However, over the course of rounds, the probabilities of the prospective outcome distribution change, becoming less positively skewed. The difference in skewness over final outcomes before and after the individual starts gambling is what generates inconsistent behavior. Barberis (2012) shows that for a wide range of preference parameter values, the described probability weighting effect outweighs the loss aversion effect, and thus people are willing to begin gambling in the first place with the "loss-exit" plan in mind. The trade-off between probability weighting and loss aversion will be important to I our line of argument.
When do people deviate from the plan? People only show inconsistent behavior over the course of lotteries if the plan makes it attractive enough to accept risk in the first place. This means that the overall lottery distribution needs to be sufficiently positively skewed such that small probabilities within this plan are initially overweighted, but over the course of rounds become less overweighted.
More precisely, the probability weighting effect needs to dominate the loss aversion effect. Otherwise, the person would stick to his plan and not deviate from it. Now, what does realization do to gambling behavior? Imas (2016) has theoretically shown for losses that realization brings people closer to their initial plan if they suffer from inconsistent behavior. The argument is the following: Realization closes the respective mental account and internalizes the paper outcome. For a loss, this means that there is no option to break-even anymore and therefore people stop chasing losses. In addition to this, investing in the lottery after a realization becomes less attractive as the overall distribution becomes less positively skewed the more rounds have already been played. How realization influences risk taking after gains requires a little more explanation: For a gain, realization removes the possibility to offset future losses by previous gains. Therefore, losses are more painful after realized gains than after paper gains, which decreases people's willingness to take risk after a realized gain compared to a paper gain. In addition to this, the progress in rounds decreases the attractiveness of investing in the lottery because probability weighting changes over time. Once gains which were initially unlikely and therefore overweighted occurred, they are not perceived as unlikely anymore and consequently less overweighted. The lower attractiveness of the lottery combined with the larger sensitivity to future losses, makes investing in the lottery after a realized gain less attractive than after a paper gain. Realization decreases people's propensity to gamble after a gain -they deviate from the ex-ante plan.
Realization of a loss brings people closer to their initial plan (Imas, 2016), whereas realization of a gain does not. How can this prediction be used to explain the relationship between the realization effect and the skewness of lotteries? First, we consider a symmetric lottery and assume that it is played over four rounds (as in experiment 2). We assume that people form an optimal "loss-exit" plan as described above. However, this plan has an overall lottery distribution which is not very positively skewed compared to, for example, the one that would emerge from a positively skewed lottery. This has a key implication: People are less likely to deviate from the optimal plan for losses and more likely to deviate from it for gains. They are predicted to act inconsistently after gains while they act consistently after losses. This implication follows directly from Barberis (2012) who finds that at least 26 rounds are necessary to observe inconsistency with symmetric lotteries for the usually assumed preference parameter values of Tversky and Kahneman (1992). This can easily be seen in our setting of either a symmetric lottery with p=1/2 versus a positively skewed lottery with p=1/6 played over four rounds. The most favorable outcome (4 successes) occurs with a probability of 1/16 for the symmetric lottery compared to 1/1296 for the positively skewed lottery. Therefore, the more positively skewed the lottery is, the less rounds are needed to observe inconsistency.
This implication is essential to understand why the realization effect is less likely to be found for symmetric lotteries. A necessary condition to find differential behavior between paper and realized outcomes is that people deviate from their optimal plan after a loss and stick to it after a gain. As explained, this occurs if probability weighting is very pronounced due to the degree of skewness of the lottery distribution. The skewness can be affected in two ways: Either it can be increased by providing a positively skewed lottery from the beginning or by extending the number of rounds people can invest in the lottery. Since the number of rounds is fixed over all our experiments, it is the skewness of the symmetric lottery that makes deviations from the ex-ante plan after a loss unlikely and after a gain likely. After a paper loss, people are actually willing to gamble, but do not do so with a symmetric lottery over a few rounds because the probability weighting effect does not outweigh the loss aversion effect.
There is a second argument adding to this reasoning: when abstracting from the role of exante plans, a symmetric lottery provides little opportunity to recover from a paper loss because the potential upside is relatively small compared to the downside. Consistent with the effect of probability weighting, the decreased chance to break-even makes deviations from the ex-ante plan after a paper loss less likely. As explained by Imas (2016), realization of a loss brings people closer to the initial plan and circumvents inconsistent behavior. However, if there is no inconsistent behavior, realization should have little effect: people will adhere to their ex-ante plan after a loss and behave similarly after a paper and realized loss.
A similar reasoning works for gains. After a gain, the loss aversion effect outweighs the probability weighting effect. Although, previous paper gains cushion future losses which decreases loss aversion, the progress in rounds in a symmetric lottery is accompanied by a strong reduction in probability overweighting. Ultimately, the unattractiveness of the lottery dominates and the person is less likely to continue gambling after a paper gain. This means that the person shows inconsistency in his investment decisions. There is another point adding to the unattractiveness of a symmetric lottery: The relatively large downside can potentially wipe out people's previous gains as well as parts of their own money if they gamble again. Consistent with our previous line of argument, the risk to be wiped off by large losses makes deviations from the ex-ante plan after a paper gain more likely. As explained above, realization presents another way to prevent people from gambling after a gain.
However, if there is inconsistent behavior after paper gains as well as after realized gains, realization III has little effect in preventing people who else gamble from gambling: people will not adhere to their ex-ante plan after a gain and behave similarly after a paper and realized gain.
The same reasoning as above applies to negatively skewed lotteries. As the probability weighting works in the opposite direction (losses are overweighted and gains underweighted) and the potential upside relative to the downside becomes even less attractive compared to the symmetric lottery, realization will have little effect because people are already predicted to stick to their ex-ante plan after paper losses and deviate from it after paper gains.

Online Appendix B. Experiment Instructions
This appendix demonstrates the instructions of experiment 1. Moreover, this appendix presents exemplary screenshots of experiment 1.
B.1 Instructions in the paper treatment of experiment 1 Welcome to our experimental study on decision making. The experiment will take about 30 minutes.
All the money you earn is yours to keep. You receive 8.00 Euro in an envelope. This is your money which you can use to participate in the experiment. Please check that the envelope contains 8.00 Euro. The experiment consists of 4 successive rounds of investment decisions. You will have 8.00 Euro in total to invest. Each round you must decide how much of 2.00 Euro you would like to invest in a lottery: With a probability of 1/6 (16%) the lottery will "succeed" and you will make 7 times the amount you invested. With a probability of 5/6 (84%) the lottery will "fail" and you will lose the amount you invested. The procedure in each round is the same.
First, you are assigned one success number between 1 and 6. It is displayed on the computer screen. Second, you enter the amount you would like to invest in the lottery. The amount can be up to 2.00 Euro. When everyone is ready, the experimenter will roll a six-sided die in front of the class.
If the rolled number is your success number, you will win the round and you will earn 7 times the amount invested. If the rolled number is not your success number, you will lose the invested amount.
The outcome of the lottery is reported each round. Afterwards, you get a new success number and make the same decision in the next round.
At the end of the four rounds, your game payment will be the 8.00 Euro you started with plus your net earnings from the investments. Note that net earnings can be positive or negative.

B.2 Instructions in the realization treatment of experiment 1
Welcome to our experimental study on decision making. The experiment will take about 30 minutes.
All the money you earn is yours to keep. You receive 8.00 Euro in an envelope. This is your money which you can use to participate in the experiment. Please check that the envelope contains 8.00 Euro. The experiment consists of 3 successive rounds of investment decisions. You will have 6.00 Euro in total to invest. Each round you must decide how much of 2.00 Euro you would like to invest in a lottery: With a probability of 1/6 (16%) the lottery will "succeed" and you will make 7 times the amount you invested. With a probability of 5/6 (84%) the lottery will "fail" and you will lose the amount you invested. The procedure in each round is the same.
V First, you are assigned one success number between 1 and 6. It is displayed on the computer screen. Second, you enter the amount you would like to invest in the lottery. The amount can be up to 2.00 Euro. When everyone is ready, the experimenter will roll a six-sided die in front of the class.
If the rolled number is your success number, you will win the round and you will earn 7 times the amount invested. If the rolled number is not your success number, you will lose the invested amount.
The outcome of the lottery is reported each round. Afterwards, you get a new success number and make the same decision in the next round.
At the end of the three rounds, your game payment will be the 8.00 Euro you started with plus your net earnings from the investments. Note that net earnings can be positive or negative. After the three rounds, you begin with the next part of the experiment. In the next part, you make one more decision.

Online Appendix D. Additional Results from the Experiment
This appendix presents additional results from the experiments. In Table D.1, table D.2 and table   D.3, we investigate dynamic risk taking more generally and report the changes in risk taking over all rounds prior to the final round for the positively skewed, symmetric and negatively skewed lottery, respectively. In Table D.4, we provide an overview of the outcomes after round 3 for the participants in the positively skewed lottery.      The table shows how many participants have negative, positive or zero net earnings after  round 3, the average earnings after round 3, the standard deviation of the earnings after round 3 and how the participants in the respective outcome category (loss, gain and no gain/loss) randomly split between the paper and the realization treatment. For those participants who have a gain after round 3, we report in which round they have a success. A comparable sample split for participants who have a loss after round 3 provides no additional information, because participants only end up in the domain of losses after round 3 if they do not gain in any of the rounds. This goes back to the large upside potential of the positively skewed lottery.

Online Appendix E. Online Experiment
This appendix presents results of an experiment which was conducted online. The main difference to experiments 1-3 is that we used another realization mechanism and a different framing of the final investment round in the online experiment.
In his study, Imas (2016) tests the boundary conditions of the realization effect with respect to the realization mechanism.While in the original experiment money is physically transferred to participants by the experimenter, a robustness experiment considers an electronic transfer of money between different accounts. The process of sending money is initiated by participants. They have to type "closed" to transfer and realize outcomes from the first three lottery rounds. Based on the results, this realization mechanism is sufficient to produce a similarly strong realization effect. Imas (2016) concludes that a physical transfer of money is not necessary to show the realization effect.
We challenge this conclusion for two reasons. First, it is instrumental for the realization effect that the investor recognizes the difference between paper and realized outcomes and the point in time of realization. Not only is this distinction presumably less salient for a virtual transfer of money, but it remains open whether the two separate electronic accounts also constitute separate mental accounts. Secondly, literature in psychology shows that people perceive a physical transfer of money differently than an electronic transfer of money. For example, paying with a credit card is perceived less painful than paying with cash explained by the transparency of the payment outflow (Raghubir and Srivastava, 2008). Likewise realization utility might be felt less intensely when relying on this realization mechanism. We therefore replicate the two different mechanisms in our experiment.
Furthermore, we examine how framing affects the realization effect. We initially did not intend to investigate this question as part of the online experiment. However, after we ran the online experiment, we noticed that we deviated from the original design in how we labeled the final round in the realization treatment. This change in framing turns out to make a difference. We will therefore also report the results of an additional treatment in which we replicate the framing of the final round exactly as in Imas (2016).

E.1 Design and participants
We discuss that the realization mechanism might be a major determinant for differential risk taking after realized and unrealized outcomes. The less strong realization effect identified in the prior experiments casts some doubt on whether a weaker realization mechanism is sufficient to generate the effect. We thus in experiment 4 replicate study 2 by Imas (2016), which uses an online experiment without physical transfer of money.

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The structure of the online experiment was similar to the laboratory experiments with the main differences that the stakes were smaller and the realization mechanism was modified. Participants were paid a fixed amount of $0.30 plus their earnings at the end of the experiment. They received an endowment of $1.00 in the beginning of the experiment to be used in four subsequent investment decisions. In each round, participants were randomly assigned a success number and decided how much of $0.25 to invest in the same lottery as in experiment 1. Afterwards, they rolled a virtual six-sided die and the rolled number (randomly generated) was presented on the computer screen. If the success number matched the rolled number, the invested amount was multiplied by seven; if not, the invested amount was lost. Participants learned the outcome and continued with the next round with a new success number.
Participants were randomly assigned to either the paper or the realization treatment. In the realization treatment, earnings were reported at the end of the third round and participants were asked to type "closed" in a dedicated window to realize their position. It was explained that any money they lost up to this round would be withdrawn from their account and transferred to the experimenter and any money they won would be credited to their account. Afterwards, the same lottery was offered for an additional investment decision. Participants in the paper treatment view their earnings after round 3. However, they did not initialize any transfer of money before continuing with the final investment decision. The design matches the original design by Imas (2016) as close as possible using the same stakes and most importantly the same realization mechanism.
As mentioned in the previous section, we unintentionally deviated from the original design with respect to the framing of the final round in the realization treatment. Instead of "Round 1" we labeled the final round "Additional Round". 15 We initially did not expect this change in framing to affect the outcome of the experiment since we assume a proper realization mechanism to be robust against relatively minor framing effects. Nevertheless, we run a second realization treatment (Realization Round 1) on Amazon mechanical Turk in which we revert the framing in the final round from "Additional Round" to "Round 1".
Experiment 4, including all three treatments, was programmed in SoSciSurvey, a platform to create academic survey studies and conducted online using the labor market of Amazon mechanical Turk which allowed us to get access to a more representative sample of the population. 16 We recruited 471 individuals for the experiment, again guided by a power analysis. Participants were on average 36 years old, 42% of participants were male, 39% stated that they attended a statistics class, and the level of cognitive reflection was lower than in the student sample (1.84 correct answers out of 4). 15 We noticed this deviation from the original design after we ran the online experiment. 16 https://www.soscisurvey.de/index.php?id=index&lang=en.

E.2 Results
We first examine the investment decisions of participants who lost in each round prior to round 4.
Panel A of Table E.1 shows investments in all rounds by treatment for this group. Investments are not statistically different in round 1 to 3 across treatments. Focusing on the changes in investment between round 3 and 4 across treatments, we cannot find evidence for a realization effect. The difference of the changes in investment between treatments is insignificant and points in the wrong direction (DiD = −1.48, t(197) = 1.37, p = 0.17). To understand this result better, we consider the investment by treatment group. In line with a realization effect, participants in the paper treatment take more risk after a paper loss, but only marginally and not statistically significant (0.44, t(103) = 0.64, p = 0.52). Unlike in experiment 1, participants in the realization treatment take significantly more risk after losses (1.92, t(94) = 2.29, p = 0.02). This result is contradictory to the realization effect which predicts the opposite.
As an explanation to these findings, it appears natural to question the modified realization mechanism. In this experiment, an electronic transfer of money between accounts instead of a physical transfer of money is supposed to induce a mental realization of earnings. We argue that by merely typing "closed", participants do not perceive the difference between realized and unrealized outcomes and do not derive disutility from the realization of a loss (Barberis and Xiong, 2012).
Since the realization is unrecognized, the mental account remains open and the opportunity to break even by taking more risk persists. The investment pattern of participants in the realization treatment is consistent with this reasoning. The mechanism might rather emphasize the existence of a loss and stimulate more risk taking.
Further evidence for the assumption that participants did not perceive the electronic transfer of money between accounts as a realization might come from the gain domain. Panel B of Table   E.1 summarizes the investments for participants who have a gain by the end of round 3. Indeed, we neither find a realization effect for gains. Participants do not take less risk after a realized gain compared to a paper gain, but rather the opposite (DiD = −1.67, t(152) = 1.38, p = 0.17). As for losses, participants tend to take more risk after a realized gain than before a realized gain (1.50, t(64) = 1.81, p = 0.07). Again this is inconsistent with a realization effect, but consistent with the assumption that participants in the realization treatment did not part with the money they gained.
In the remaining part, we show the results of the "Realization (Round 1 framing)" treatment.
To remind the reader, we only change the label of the final round in the realization treatment from "Additional Round" to "Round 1". The change in framing is implemented to exactly align this version with the original design of the online study by Imas (2016). Consistent with the realization effect, participants now tend to take less risk after a gain and a loss in the realization treatment XXV when using the proposed change in framing (see Table E.1). In the slightly altered version, we thus successfully replicate the online study by Imas (2016). One possible reason is that the altered framing of the final round strengthens the triggered process of closing the respective mental account.
The "Round 1" frame makes the beginning of a new investment episode clearer to participants, while the "Additional Round" frame might rather give participants the impression that they still have the chance to recover from the previous loss as they are offered an additional lottery. This would imply a continuation of the investment episode rather than and end.
We conclude that the realization effect is sensitive to the realization mechanism if money is not physically transferred. Presumably, participants in the realization treatment of the online experiment do not part with the money in the same way as those participants in the experiments with physical transfer do. Therefore, the realization effect becomes vulnerable to circumstantial effects such as framing. Future research could follow up on potential other effects which may be interrelated with the realization effect.  ). Panel A is restricted to participants who have a loss by the end of round three, Panel B shows averages for all participants who have a gain by the end of round three. Both panels show results by treatment (paper, realization and realization round 1 framing) and differences between treatments. In this experiment the realization is a non-physical transfer of money. Participants in the realization treatment had to type the word "closed" in a respective window after round 3 to realize their earnings. In the realization round 1 framing treatment, the final round was named "Round 1" instead of "Additional Round". Change is the difference between the investment in the final round and round three. N provides the number of participants for each treatment-outcome combination. T-values of a two-sided t-test are shown in parentheses.