No Aspiration to Win? An Experimental Test of the Aspiration Level Model *

Transcription

1 No Aspiration to Win? An Experimental Test of the Aspiration Level Model * June 27, 23 Enrico Diecidue, Moshe Levy, and Jeroen van de Ven Abstract In the area of decision making under risk, a growing body of literature addresses aspiration levels. Researchers often assume an aspiration level at zero because doing so helps explain several phenomena, such as risk-seeking behavior in the domain of losses. This paper describes a simple experiment designed to test this assumption. We find no support for an aspiration level at zero. For approximately 2% of the subjects we do detect non-zero aspiration levels, but the levels vary considerably across subjects. The aggregate results are consistent with prospect theory, but can also be explained by a population with heterogeneous aspiration levels. These results improve our understanding of when and why aspiration levels play a role in decision making. Keywords: Decision under risk, aspiration levels. JEL classification: C9, D8. * We thank Fabiano Prestes for programming the experimental software and Alexander Schram for research assistance. We thank Jeeva Somasundaram, Stefan Zeisberger, Stefan Trautmann, and Peter Wakker for detailed comments. INSEAD; Hebrew University of Jerusalem; University of Amsterdam. Corresponding author: Enrico Diecidue, Decision Sciences, INSEAD, Bd. de Constance, 7735 Fontainebleau Cedex, France. enrico.diecidue@insead.edu

2 . Introduction It seems plausible that, when faced with risk, decision makers (DMs) pay attention to the probability of reaching a target or aspiration level. In an early contribution, Roy (952) argues that DMs will seek to minimize the probability of disaster. Indeed, farmers minimize the probability of falling below the subsistence level (Lopes 987), cabdrivers aim at a daily target (Camerer et al. 997), and investment managers try to meet a target return (Payne, Laughhunn, and Crum 98, 98). It is entirely conceivable that the probability of nonnegative returns from the risky option can explain the many experimental results related to risk seeking in the domain of losses (Abdellaoui, Bleichrodt, and L Haridon 28; Baucells and Villasís 2; Camerer 989; Etchart-Vincent and L Haridon 2; Fehr-Duda et al. 2; Hershey and Schoemaker 98; Tversky and Kahneman 992; Wakker 2; Weber and Camerer 998). Although such risk-seeking behavior seems natural, even obvious (Payne 25), few studies explicitly test for the existence of an aspiration level. Among the few exceptions is Payne, who concludes that the overall probability of winning or losing should be part of any descriptive theory of choice. Decision theory models integrate these probabilities with aspiration levels (Diecidue and van de Ven 28; Levy and Levy 29) or take aspiration levels as the foundation of decision theory, dispensing with utility functions (Castagnoli and LiCalzi 26). This paper provides evidence from a laboratory experiment designed to investigate aspiration levels. We focus on two-outcome lotteries and propose a novel test to determine whether DMs consider the overall probability of a strictly positive or negative outcome. One advantage of our test is that it isolates an aspiration level with respect to a reference point in a simple setting. The distinguishing feature of this approach is that aspiration levels are formulated in terms of probabilities namely, the overall probability of reaching (or not) the aspiration level. Diecidue and van de Ven (28) show that the combination of expected utility and aspiration level can be reformulated as expected utility with a discontinuous utility function. Toward the end of discerning such a discontinuity, we propose lottery choices to our participants, the outcomes of which we vary to manipulate the probability of achieving the aspiration level. 2

3 Our data can reveal the aspiration level s exact location, and our main focus is on the zero level of aspiration. Many experiments have shown that this zero level plays a special role in decisions under risk. It is taken as the reference point in prospect theory (PT; Tversky and Kahneman 992) and as a natural aspiration level (Lopes and Oden 999; Pahlke, Kocher, and Trautmann 2). The outcome zero has also been investigated from a psychological perspective (Birnbaum et al. 992; Mellers, Weiss, and Birnbaum 992; Weber, Anderson, and Birnbaum 992), and tests of the affect heuristic concentrate around zero (Bateman et al. 27). Most theories that incorporate a reference point or aspiration level fail to specify its location; however, in many experiments it is often taken to be the outcome zero. Our study, as most of the experimental literature in decision under risk, relies on simple two-outcome lotteries: Although there is consensus that aspiration levels matter for multi-outcome lotteries (Lopes and Oden 999; Payne 25; Payne, Laughhunn, and Crum 98), little is known about aspiration levels for two-outcome lotteries. Our framework allows examining different possible aspiration levels. While we find evidence for aspiration levels for 2% of the subjects, the levels are heterogeneous, and we do not find any special importance for the level of zero. At the aggregate level, the results are consistent with prospect theory, but they can also be explained by a population of subjects with heterogeneous aspiration levels. 2. The Aspiration Level Model Diecidue and van de Ven (28) and Levy and Levy (29) introduce models that build on two intuitions. First, decision makers are concerned with aspiration levels and, in particular, with the overall probability of meeting an aspiration level; second, DMs will likely exhibit some sensitivity to the level and likelihood of all other outcomes. In these models, DM preferences are expressed as a combination of expected utility and the aspiration level. We denote by P(x + ) (resp. P(x )) the overall probability of reaching an outcome strictly above (resp. below) the aspiration level. The valuation of a lottery L with outcomes x j ( j =, 2,..., n) and probabilities p j is ( ) ( ) ( ) ( ) () 3

4 It is straightforward to show that this expression is equivalent to V AL ( L) p v( x ) if we define: v( x ) u( x ) for x x ; v( x ) u( x ) j j j j for x x ; and v( x ) u( x ) when x coincides with the aspiration level. The result is j j a utility function v that is discontinuous around the aspiration level. Figure illustrates such a utility function v (the solid line), when the aspiration level is set at zero. As the graph shows, the utility function v jumps at the aspiration level. [[ INSERT Figure about Here ]] 2.. Model Predictions To derive predictions from the aspiration level model, we assume a value function as in Figure : The key element is the jump at the aspiration level. The theory does not impose any restrictions on the concavity/convexity of the smooth parts of the function. A value function as in the figure is consistent with two major findings from laboratory studies namely, people exhibit risk-averse behavior in the domain of gains but risk-seeking behavior in the domain of losses. Prospect theory (PT) (Tversky and Kahneman 992) accommodates that risk seeking with a convex value function in the loss domain. However, the aspiration level model offers a different explanation for risk-seeking behavior in that domain. Faced with the choice between a sure negative outcome and a lottery that features some outcomes at or above the aspiration level, a DM may prefer the lottery simply because it offers the only chance of reaching the aspiration level. We illustrate this dynamic in Figure for a twooutcome monetary lottery that yields the outcome with probability p and the outcome y with probability p. The dashed line indicates that participating in the lottery is preferred to receiving its expected value for sure. j j j 2.2. Testing for an Aspiration Level Our experimental design is based on two-outcome monetary lotteries. We elicit the certainty equivalent (CE) for each lottery in other words, the sure amount of money that renders the DM indifferent to participating in the lottery. We deal with two simple types of manipulations, as described next. Under PT, risk attitude is described by the value function and probability weighting function. In this paper we address only the value function. 4

5 First, we elicit CEs of different two-outcome lotteries constructed from a baseline lottery by adding or subtracting a constant amount to all outcomes. For instance, in Figure we consider not only the baseline lottery with outcomes y and but also the so-called shifted lottery with outcomes y z and z. The aspiration level model predicts that small changes in outcomes that are at or near the aspiration level can have a substantial effect on valuations of the lottery and on attitudes toward risk. In our example, participants exhibit risk-seeking behavior in the baseline lottery but riskaverse behavior in the shifted lottery. Second, we present participants a mixed lottery with one positive and one negative outcome and then systematically vary the probability of obtaining the positive outcome. For every probability, we elicit the CE for that mixed lottery. 2 If there is a discontinuity in evaluation then we should observe a vertical segment when we plot the probability of the high outcome as a function of the CE; i.e., the CE will be constant for a range of values of the probability of the high outcome. 3. Experimental Design and Procedures 3.. Experimental Method and Task The experiment consisted of four different parts in which participants made choices between a sure amount of money and a two-outcome monetary lottery. We will denote a lottery i by ( p, x ; y ), where x and y (x > y) are the outcomes and p is i i i i the probability of the highest outcome x. Table gives details on all the lotteries. Those in parts A and 2A of the experiment involved relatively small outcomes (ranging from 28 to +28 euros) and were incentivized; the lotteries in parts B and 2B involved relatively large outcomes (±5 euros) and were not incentivized. [[ INSERT Table about Here ]] Lotteries in part A were of the form (, x ; y ). In the base lottery we set x = 2 i 2 i i and y =. We denote this lottery by L, where the subscript B and S denote 2 Because of the utility function s discontinuity, certainty equivalence in the aspiration level model is not well defined everywhere. The CE in our context is defined as the minimum certain amount of money that is (weakly) preferred to the lottery. 5

6 (respectively) base and small. The other lotteries in this part originate from the base lottery by imposing a shift that is, by adding a constant c to all outcomes. We denote the shifted lotteries by instance, 2 c L, where c { 28, 24, 2,,,, 4, 8}. For (,2;). Note in particular that lotteries with c equal to 2, 24, and 28 mirror the lotteries with c equal to (the base lottery), +4, and +8 in the sense that gains are replaced by losses. For instance, 28 2 (,28;8) whereas 8 2 (, 8; 28). Part B is analogous to part A except that all outcomes are large: in this base lottery, we set x = 3 and y =. We denote the base lottery by L BL (for base, large ), and the shifts c in this part were from the set { 5, 4, 3,, +, +2}. Again, note that lotteries with c equal to 3, 4, and 5 mirror the lotteries with c equal to, +, and +2 (respectively) but with gains replaced by losses. We did not incorporate very small shifts in this part. In part 2A of the experiment, lottery outcomes were fixed at x = 2 and y =. We presented 9 mixed lotteries (p, 2; ) with the probability p of the high outcome varying from.5 to.95 in steps of.5. We will denote these lotteries as L MS, p (for mixed, small ), where p {.5,,.95}. In part 2B we did the same for lotteries (p, 2; ), which we will denote by L ML, p (for mixed, large ). All choices were presented in the form of a price list (Andersen et al. 26; Binswanger 98, 98). Each price list consisted of a number of binary choices between the lottery and increasing amounts of sure money (ranging from the lowest to the highest outcome of the lottery). The incremental steps by which the sure amount varied were relatively small compared to the difference in possible outcomes of the lotteries; this allows us to infer a relatively precise certainty equivalent (step sizes were in part A, in part B, in part 2A, and 5 in part 2B). Price lists are built such that a participant always prefers the lottery in the first decision but always prefers the sure amount of money in the last decision. The CE of a lottery is determined by the average of the two points at which the participant switches from the lottery to the sure amount. 6

7 We classify a choice as risk averse (resp., risk seeking) if the expected value EV(L) of the lottery is larger (resp., smaller) than the upper (resp., lower) bound that is, if EV(L) > z L (resp., EV(L) < z L ). If the expected value of the lottery falls within the interval [ z, z ] then we classify the choice as risk neutral. L H 3.2. Experimental Procedures The experimental sessions took place in Amsterdam at the CREED laboratory in April 29. Altogether, 48 students from the University of Amsterdam participated in the study. Of these, 52% were female; the mean age was 22. Participants were recruited from the CREED database. Participants were welcomed to the lab, instructed about the general procedure of the experiment, and assigned to a computer. They first received some general instructions that explained the experimental setup. Participants were told that they would each receive an endowment of 28 and that it would be possible to earn a considerable amount of additional money but also that it was possible to incur some losses that would be subtracted from their endowment (though participants were also informed that they could never lose more than their endowment). At this stage, everyone had the option to opt out of the experiment, but no one did. Next, we handed out the endowments in (unsealed) envelopes and provided the participants with detailed written instructions (see Appendix). The motive for giving the endowments up front was to create a feeling among participants that they owned the money, so that any subtraction would feel as a genuine loss. Before participants started with the actual experimental questions, they received some test questions to verify that they understood the task. Then they proceeded with the questions of the different parts, which were always in the same order: A, B, 2A, 2B. Within each part, however, the order of questions was randomized. To determine final payment for participants, we employed the random incentive system (Cubitt, Starmer, and Sugden 998; Starmer and Sugden 99). Out of all questions in parts A and 2A, one was randomly selected at the end of the experiment for each participant and then paid out according to the decision made. Depending on their stated preference, participants either received the sure amount of money or played the chosen lottery. Overall, 7

8 average earnings amounted to 34 and ranged from to 56. Each session lasted for about 45 minutes. The experiment was computerized using a web-based design. Figure 2 shows a screenshot of one group of lottery questions. To help visualize the lottery, we always showed a probability pie chart that was divided into two parts and colored to reflect the likelihood of each possible outcome. Because the price list for each question was lengthy, participants had the option of using computer auto-complete assistance (for discussion of a similar procedure, see Andersen et al. 26). The assistance made it possible for a participant to fill in all entries of a given list with just two mouse clicks. When computer assistance was enabled, the software would autocomplete choices for entries by assuming monotonicity of preferences. For instance, if at any point a participant prefers the sure amount of money to some lottery, then it is reasonable to assume that the same participant will also prefer any larger sure amount to that same lottery. Formally, if z ( p, x; y) for some sure outcome z then i z ( p, x; y) for any z z; likewise, if a participant indicated the preference i z ( p, x; y) then it was assumed that z ( p, x; y) for any z z. i However, participants always had the option to change any entry in their list of responses before proceeding, and they could disable computer assistance at any time. [[ INSERT Figure 2 about Here ]] It is worthwhile to note that, for any given question, a participant rarely switched more than once between the options. Thus, a participant hardly ever preferred the sure outcome to the lottery for some value yet not for a higher value of the sure outcome. 3 One reason is that nearly all participants used computer assistance to autocomplete their replies. Moreover, no choices were made that violated the conditions x ( p, x; y) and y ( p, x; y). Hence, these basic principles of rationality are not violated. i i i 3 For participants who made multiple switches, we take the midpoint between the first and the second switch. 8

9 4. Hypotheses We formulate three hypotheses. The first two are tested in part of study, and the third one is tested in part 2. Hypothesis (H). Risk-seeking behavior in the loss domain is explained by the existence of an aspiration level at zero. Thus, the proportion of risk-seeking choices will be lower for the lotteries 24 and the aspiration level is, than for the lottery 28, where the overall probability of reaching 2, where the overall probability of reaching the aspiration level is. The same statement applies to the corresponding lotteries with large outcomes. Hypothesis 2 (H2). Adding or subtracting a small amount from the outcome at the aspiration level leads to a significantly different valuation of the lottery. Thus, the CE of lottery will be significantly higher, and that of lottery lower, than the valuation of the lottery L. significantly Hypothesis 3 (H3). The CE of lotteries L ML, p and L, MS p, as a function of the probability of the high outcome, is constant around the aspiration level, indicating a jump in the value function. 5. Results We first focus on part of the experiment. Table reports the mean and median certainty equivalents, which are also plotted in Figure 3 and Figure 4 for small and large outcomes, respectively. The CEs satisfy (weak) dominance except in one case (as detailed in what follows). Certainty equivalent values reflect risk aversion for gains and risk seeking for losses: CEs tend to exceed the lottery s expected value when losses are involved, and they are usually below the expected value when gains are involved. [[ INSERT Figure 3 about Here ]] [[ INSERT Figure 4 about Here ]] 9

10 The bar graphs in Figures 5 and 6 show the percentage of risk-seeking and risk-averse choices for each lottery. Focusing first on the base lotteries (i.e., those involving the zero outcome), we find that for lotteries with a positive outcome most choices are risk averse (52% for small outcomes, 65% for large outcomes) and there are few riskseeking choices (6% for both small and large outcomes). In contrast, for the base lotteries involving a negative outcome, the percentage of risk-averse choices is lower in both cases: 25% each of risk-averse and risk-seeking choices for small outcomes; for large outcomes, the respective percentages are 8% and 4%. Our finding that choices tend to be risk averse for gains and risk seeking for losses is consistent with results reported in the extant literature (see the works cited in Section ). [[ INSERT Figure 5 about Here ]] [[ INSERT Figure 6 about Here ]] We are mainly interested in how attitudes toward risk change after the shift in payoffs. If the risk-seeking behavior of participants in 2 is driven by their desire to break even that is, to achieve at least zero then we should observe less riskseeking behavior when a constant is subtracted from all outcomes, thereby making it impossible to achieve zero (H). However, we find no support for this hypothesis; to the contrary, we observe a higher percentage of risk-seeking choices in such shifted lotteries. We use the nonparametric McNemar test for related samples in order to check for differences (two-tailed test with a correction for continuity). Compared with lottery 2, the percentage of risk-seeking choices is significantly higher for the lotteries 24 (χ 2 = 8.64, p =.63) and 28 (χ 2 = 6.6, p <.). Compared with 3 BL, the difference in percentage of risk-seeking choices is not significant for lottery 4 BL (χ 2 =.7, p =.32) but is significant for 5 BL (χ 2 = 5.6, p =.24). The percentage of risk-averse choices is stable with respect to shifted outcomes. Turning now to the domain of gains, we find that for small outcomes the percentage of risk-averse choices increases after a shift and, compared with L, is significantly higher in lotteries 4 (χ 2 = 4.92, p =.27) and 8 (χ 2 = 6, p <.). For large outcomes, the percentage of risk-averse choices decreases. The difference between L BL and L BL is not significant (χ 2 =.78, p <.82), although

11 that between L BL and 2 L BL is marginally significant (χ 2 = 2.77, p <.96). As is apparent in Figures 5 and 6, the proportion of risk-seeking choices is both small and stable in the gains domain. Comparing lottery L with lotteries and constitutes the strictest test of our model (H2). Observe that has one negative and one positive outcome, and L has one positive outcome and one zero outcome, has only strictly positive outcomes. Our design does not allow a direct comparison (in terms of behavior toward risk) between these lotteries, 4 so instead we look for differences in the aggregate CE. If there exists an aspiration level at zero that plays a significant role then despite the minuscule change in outcomes the CE of should be substantially lower (resp. higher) than that of L (resp. ). Yet we find no support for H2, either. The mean CEs of the three lotteries are close to each other, ranging only between 8 and 8.6. A within-subject test reveals that the shifted lotteries are not significantly different from L : for, Z =.73 and p =.949; for Z =.343 and p =.8 (Wilcoxon signed-rank tests, 5 two-tailed). 6, Hypotheses and 2 are based on the assumption of an aspiration level at zero. Of course, it is possible that participants have a different aspiration level. The choices they make in parts 2A and 2B of our experiment can be used to detect nonzero aspiration levels. We have mentioned that support for the existence of an aspiration 4 As with all the questions, we gave participants a list of sure integer amounts of money. This means that the expected value (i.e., 9 and ) of these two shifted lotteries had to fall somewhere between two sure amounts. In that case, then, participants are less likely to be classified as risk neutral and thus more likely to be classified either as risk seeking or risk averse than in other cases. 5 Results are similar if instead we use t-tests. There are no significant differences between any pair of the lotteries (with p-values always exceeding.8; two-tailed tests). 6 At the individual level we find some violations of dominance. Compared with the baseline lottery, 7 of the 48 participants report a higher CE for 2. Also, 3 participants report a lower CE for ; for 6 of them, this difference is strictly greater than ; for 4 of these participants, the difference is strictly greater than 2. Such violations of dominance for small changes in outcomes have been found elsewhere; see, for example, Bateman et al. (27) and Mellers, Weiss, and Birnbaum (992).

12 level would come from a vertical portion of graph as in Figure. So in Figures 7 and 8 we plot the probability p against the mean and median certainty equivalent. These CEs are transformed as in Tversky and Kahneman (992), so that any point on the diagonal represents a risk-neutral choice. Thus, these figures depict the value function aggregated across all subjects. The plots in these two figures do not exhibit any vertical parts. The CEs are significantly different for almost all adjacent probabilities and, for all pairs of lotteries whose probabilities differ by at least., the CEs are significantly different at the % level with only two exceptions (Wilcoxon signed-rank tests, two-tailed). 7 We conclude that there is no support for H3 at the aggregate level. In short, we find no support for any of the three hypotheses. We report that the mean and median CEs are remarkably close to the PT predictions of Tversky and Kahneman (992) at the aggregate level. Figures 3, 4, 7, and 8 plot the certainty equivalents for prospect theory and show that they are always near the CEs of participants in this experiment. 8 Our findings challenge the assumption of a strictly zero aspiration level and indicate the need for further research investigating alternative aspiration levels. [[ INSERT Figure 7 about Here ]] [[ INSERT Figure 8 about Here ]] How can these results be reconciled with the previous findings in the literature supporting the importance of aspiration levels? One possible explanation is that individuals do have aspiration levels, but these levels are not necessarily at zero, and are heterogeneous across subjects. While the individual level data is rather noisy, it does lend some support to this explanation. Eyeballing the data reveals that, at the individual level, the choices of about 2% of subjects do exhibit a vertical segment when plotted in this domain; this evidence suggests there is heterogeneity in 7 The two exceptions occur when we compare the probabilities of a high outcome with p = and p =.6 for both the small and large outcomes; the corresponding p-values are (respectively) p =.33 and p = The values of the CEs are based on the parameters and functional forms described in Tversky and Kahneman (992). We do use a slightly lower value for the loss aversion parameter (λ is set at.75 instead of 2.25) to yield a better fit with the data. 2

13 aspiration levels. It is therefore possible that nonzero aspiration levels play an important role at the individual level. The individual-level value function plots are shown in Appendix B. We classified participants based on the value of the CEs: a participant is risk neutral if the CE (normalized to be between and ) was at most. away from the expected value in at least 4 out of the 9 questions of part A (B); a participant is risk averse if at least 4 of the choices had a CE below the expected value. A participant has an aspiration level if there are four or more consecutive choices for which the CE is within a bandwidth of (in part 2A) or 5 (in part 2B). In addition there are participants with a large number of violations of monotonicity (five or more CE increased by at least when the probability decreased) or that are not captured by the above classification. We labeled these participants as mixed (a more detailed descripted of the classification procedures can be found in Appendix B). Our classification is, of course, to some extent arbitrary. Eyeballing the individual utility functions within each class suggests, however, that the procedure gives reasonable results (see Figures B and B2 in Appendix B). Based on this classification we find that in part A (B) the large majority of subjects are risk neutral and risk averse 2% (27%) and 42% (25%) respectively. About 2% of the subjects reveal preferences consistent with the aspiration level model, with a jump in their value function. However, the aspiration levels are typically nonzero, and they are heterogeneous across subjects. The remaining 6% (29%) of participants are mixed, with no clear pattern emerging from their choices. It is surprising that such a diversity at the individual level leads to CEs that at the aggregate level are very close to PT (Figure 7 and 8). We suspect that the aggregate results are generated by individuals with heterogeneous aspiration levels. Figure 9 provides a flavor of how this may come about. Consider a population of individuals with a simplified piece-wise linear aspiration level value function as in panel A. All individuals have the same type of value function, but each with a different aspiration level. Panel B shows an example of the aggregate value function when the aspiration level is normally distributed in the population with mean -. and standard deviation.. The aggregate value function conforms to the prospect theory S-shape value function, even though none of the individuals is represented by such a S-shape. This 3

14 is, of course, a very simplified picture made to convey the basic idea. It is clear that in practice preferences are not all of the same type, which further complicates matters. Figures 5 and 6 lend some indirect support for the heterogeneous aspiration level explanation. The figures show that the proportion of risk-seeking choices increases the more negative the shift in parts A and B. If individuals have heterogeneous aspiration levels, the more negative the shift the more individuals will have the lower outcome below their aspiration level, which will make them act ask risk-seekers. 6. Conclusion Aspiration levels are receiving increased attention in the theoretical literature on decision making under risk. The role of aspiration levels in decision making is a natural and psychologically intuitive one. Models incorporating aspiration levels have been proposed, as an alternative to expected utility and prospect theory, to explain some frequently observed behavior in particular, risk seeking in the loss domain with a minimum number of assumptions. In order to test the main implications of models that assume zero aspiration levels, we designed a simple experiment based on the eliciting CEs for two-outcome lotteries. We do not find any support for an aspiration level near zero. It is remarkable that our study, which aimed to assess the effects of an aspiration level at zero, found no evidence for such an intuitive idea and instead actually yielded strong evidence in favor of prospect theory at the aggregate level. Several different factors may explain why our results differ from those in the existing literature. First, our design is based on lotteries with only two possible outcomes; other studies use more complex lotteries with multiple outcomes (Levy and Levy 29; Lopes and Oden 999; Payne 25) or put participants under time pressure (Pahlke, Kocher, and Trautmann 23). Second, the evidence from existing studies may be driven by a reference point and loss aversion instead of by the overall probabilities resulting from an aspiration level (Ert and Erev 2), and disentangling the two effects is complex. Third, studies reporting evidence of aspiration levels in financial decision making have first collected the individual value of a target return 4

15 (aspiration level) and then analyzed data based on this information (Fellner, Güth, and Maciejovsky 29). Fourth, it may be that an aspiration level does not emerge until the entire context of outcomes is known by the decision maker (Zeisberger 22); in part of our experiment, for example, the DM does not know beforehand how much she can win or lose. Finally, Payne (25) provides the cleanest evidence in favor of an aspiration level at zero; although his results (for lotteries with multiple outcomes) are not compatible with the standard parameterization of PT (Tversky and Kahneman 992), those results are consistent with alternative parameterizations of that theory. Payne argues that it is only for complex tasks that a DM uses aspiration levels as a heuristic. It is certainly conceivable that nonzero aspiration levels play an important role in decision making. We have evidence of a nonzero aspiration level for 2% of the participants. We suggested that heterogeneous levels can be consistent with both aspiration levels at the individual level, and aggregate results conforming to PT. The evidence presented here challenges the notion of simply assuming an aspiration level at zero, and it opens the way for additional research dedicated to examining other aspiration levels. There is more to be discovered about the circumstances such as the decision problem s complexity under which aspiration levels play a role. These explorations require that we modify theoretical models of aspiration levels (which typically do not integrate complexity into the decision process) and will thereby help to shape further theoretical developments. 5

16 Figure. Example of a value function for the aspiration level (AL) model. Sheet 3 of 8 Part a For each of the decisions below, please indicate whether you prefer Option A or Option B. The chart to the right represents the probabilities of winning 2 or winning graphically. option A option B A: 5% Chance of winning 2 and 5% Chance of receiving B: Receiving for sure 2 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning for sure 3 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning 2 for sure 4 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning 3 for sure 5 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning 4 for sure (deleted entries 6 to 9) 2 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning 9 for sure 2 A: 5% Chance of winning 2 and 5% Chance of receiving B: Winning 2 for sure Computer Assistance: ON Continue» Figure 2. Screenshot of some questions from part of the experiment. 6

17 Figure 3. Mean and median CEs for part A, as well as expected value (EV) of the lottery and CE value predicted by prospect theory, for parameter values α =.88, β =.88, γ =.6, δ =.69, and λ =.75. 7

18 Figure 4. Mean and median CEs for part B, as well as expected value (EV) of the lottery and CE value predicted by prospect theory, for parameter values α =.88, β =.88, γ =.6, δ =.69, and λ =.75. Figure 5. Proportion of risk-seeking and risk-averse choices for lotteries in part A of the experiment. Numbers on the horizontal axis are the two outcomes for each lottery. 8

19 Figure 6. Proportion of risk-seeking and risk-averse choices for lotteries in part B of the experiment. Numbers on the horizontal axis are the two outcomes for each lottery Figure 7. On the horizontal axis are mean CEs (solid dots) and median CEs (open dots) for part 2A as well as CE value predicted by prospect theory (solid line) for parameter values α =.88, β =.88, γ =.6, δ =.69, and λ =.75. Numbers on the vertical axis are the high outcome s probability; all values rescaled as (x + )/3. 9

20 probability high outcome (p) Figure 8. On the horizontal axis are mean CEs (solid dots) and median CEs (open dots) for part 2B as well as CE value predicted by prospect theory (solid line) for parameter values α =.88, β =.88, γ =.6, δ =.69, and λ =.75. Numbers on the vertical axis are the high outcome s probability; all values rescaled as (x + )/3. 2

21 Figure 9: Aggregate S-Shape preferences can arise from heterogeneous aspiration levels. Panel A: a piecewiselinear aspiration level value function with an aspiration level at x=-.2. Panel B: The aggregate preferences of a heterogeneous population on individuals with value functions as in A, but with the aspiration level distributed normally with mean -. and standard deviation.. 2

22 Table High Low Probability of outcome outcome high outcome Mean Median Part Lottery (x) (y) (p) CE CE A A A A A A A A B B B B B B L L BL BL BL BL BL BL 2A L MS, p +2 {.5,,.95} 2B L MS, p +2 {.5,,.95} 22

23 References Abdellaoui, Mohammed, Han Bleichrodt, and Olivier L Haridon. 28. A tractable method to measure utility and loss aversion under prospect theory. Journal of Risk and Uncertainty 36(3): (August 5, 22). Andersen, Steffen, Glenn W. Harrison, Morten Igel Lau, and E. Elisabet Rutström. 26. Elicitation using multiple price list formats. Experimental Economics 9(4): (August 9, 22). Bateman, Ian, Sam Dent, Ellen Peters, Paul Slovic, and Chris Starmer. 27. The affect heuristic and the attractiveness of simple gambles. Journal of Behavioral Decision Making 2(4): (August, 22). Baucells, Manel, and Antonio Villasís. 2. Stability of risk preferences and the reflection effect of prospect theory. Theory and Decision 68(-2): (July 24, 22). Binswanger, Hans P. 98. Attitudes toward Risk: Experimental measurement in rural India. American Journal of Agricultural Economics (3): Attitudes towards Risk: Theoretical implications of an experiment in rural India. Economic Journal (364): Birnbaum, Michael H., Gregory Coffey, Barbara A. Mellers, Robin Weiss Utility Measurement: Configural-Weight Theory and the Judge s Point of View. Journal of Experimental Psychology: Human Perception and Performance 8: Camerer, Colin F An experimental test of several generalized utility theories. Journal of Risk and Uncertainty 2(): (August 5, 22). Camerer, Colin F, L. Babcock, G. Loewenstein, and R. Thaler Labor Supply of New York City Cabdrivers: One Day at a Time. The Quarterly Journal of Economics 2(2): (August 6, 22). Castagnoli, E., and M. LiCalzi. 26. Expected Utility Without Utility. Theory and Decision 4: Cubitt, Robin P., Chris Starmer, and Robert Sugden On the validity of the random lottery incentive system. Experimental Economics (2): (August 9, 22). 23

24 Diecidue, Enrico, and Jeroen van de Ven. 28. Aspiration Level, Probability of Success and Failure, and Expected Utility. International Economic Review 49(2): (August 7, 22). Ert, Eyal, and Ido Erev. 2. On the Descriptive Value of Loss Aversion in Decisions under Risk. SSRN Electronic Journal. (December 3, 22). Etchart-Vincent, Nathalie, and Olivier L Haridon. 2. Monetary incentives in the loss domain and behavior toward risk: An experimental comparison of three reward schemes including real losses. Journal of Risk and Uncertainty 42(): (August 6, 22). Fehr-Duda, Helga, Adrian Bruhin, Thomas Epper, and Renate Schubert. 2. Rationality on the rise: Why relative risk aversion increases with stake size. Journal of Risk and Uncertainty 4(2): (August 6, 22). Fellner, Gerlinde, Werner Güth, and Boris Maciejovsky. 29. Satisficing in financial decision making a theoretical and experimental approach to bounded rationality. Journal of Mathematical Psychology 53(): (August, 22). Hershey, John C., and Paul J.H. Schoemaker. 98. Prospect theory s reflection hypothesis: A critical examination. Organizational Behavior and Human Performance 25(3): (August 6, 22). Levy, H., and M. Levy. 29. The safety first expected utility model: Experimental evidence and economic implications. Journal of Banking & Finance 33(8): (August 7, 22). Lopes, L Between Hope and Fear: The Psychology of Risk. Advances in Experimental Social Psychology 2: Lopes, L., and GC Oden The Role of Aspiration Level in Risky Choice: A Comparison of Cumulative Prospect Theory and SP/A Theory. Journal of Mathematical Psychology 43(2): (August 6, 22). Mellers, Barbara, Robin Weiss, and Michael Birnbaum Violations of dominance in pricing judgments. Journal of Risk and Uncertainty 5(). Pahlke, Julius, Martin G. Kocher, and Stefan Trautmann. 23. Tempus Fugit: Time Pressure in Risky Decisions. SSRN Electronic Journal. (August 8, 22), Forthcoming in Management Science. 24

25 Payne, John W. 25. It is Whether You Win or Lose: The Importance of the Overall Probabilities of Winning or Losing in Risky Choice. Journal of Risk and Uncertainty 3(): (August 7, 22). Payne, John W, Dan J Laughhunn, and Roy Crum. 98. Translation of Gambles and Aspiration Level Effects in Risky Choice Behavior. Management Science 26(): Further Tests of Aspiration Level Effects in Risky Choice Behavior. Management Science 27(8): Roy, A D Safety First and the Holding of Assets. Econometrica 2(3): Starmer, C., and R. Sugden. 99. Does the random-lottery incentive system elicit true preferences? American Economic Review (4): 9. Tversky, Amos, and Daniel Kahneman Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty 5(4): (July 2, 22). Wakker, Peter P. 2. Prospect Theory for Risk and Ambiguity. Cambridge: Cambridge University Press. Weber, Elke U, Carolyn J Anderson, and Michael H Birnbaum A theory of perceived risk and attractiveness. Organizational Behavior and Human Decision Processes 52(3): (August 6, 22). Weber, Martin, and Colin F. Camerer The disposition effect in securities trading: an experimental analysis. Journal of Economic Behavior & Organization 33(2): (August 6, 22). Zeisberger, Stefan. 22. The Importance of the Overall Probability of a Loss in Repeated Investment Tasks. SSRN Electronic Journal. (December 3, 22). 25

26 Appendix A: Instructions for Participants Welcome Thank you very much for participating. Today you will take part in an experiment in which you will be asked to make choices between lotteries and sure amounts of money. The experiment takes up to 9 minutes to complete (but we expect, on average, a shorter time). As compensation you will receive 28 as a show up fee. In addition, as we will explain later in more detail, you will have the chance to play for real one of your choices in which you can win or lose money. The amount of money you can win or lose is very substantial, so we strongly advise you to take every question seriously. How is the experiment going to work? The experiment consists of 4 different parts in which you will make individual decisions. You will get specific instructions for each part. How do you earn money? Your earnings are determined as follows. You will see a number of questions. Every time, you indicate which of the options you prefer (a sure amount of money, or playing a lottery). At the end of the experiment, after you have completed all questions, the computer will randomly select one question. Your answer to that question determines your earnings. If you have chosen the sure amount of money, you receive that specific amount of money from that question. If you have chosen to play the lottery, the lottery will be played and you can win or lose the amounts of money from that question. Remember that you can also lose money. This will be subtracted from your show up fee. In any case you cannot lose more than your show up fee. Please note that only one question will be selected, but neither you (nor we) know which question will be selected in advance. Because you don't know which question will be selected, it is in your best interest to answer every question seriously and truthfully. Note also that there is no right or wrong answer, it all depends on your own preferences. Important: please note that we have a strict no deception policy. Whatever is written in the instructions is true. At any point during the experiment, you can decide to stop. However, if you decide to stop you will lose all of your earnings, including the show up fee, and we ask you to remain seated until the end of the experiment. Please keep this sheet with you throughout the entire experiment 26

27 If you have any questions at any time, please raise your hand and wait until somebody comes to you. * * * General Instructions Please read through the instructions carefully. If you have questions at any time, please feel free to ask one of the experimenters. Please note that on the sheets next to your computer you can find the very same instructions which are displayed on the screen. In case you forget parts of the instructions after you have already started the survey, you can use these sheets for reference. Now please start reading the instructions below. The task In the experiment, we present a series of choices. Every time, you have the choice between a lottery and a sure amount of money. We ask you to indicate which option you prefer. The way it works is as follows. You will see a table with a list of questions. For every question, you have the option to choose between a lottery, and an amount of money you can get for sure. For each table, the lottery is the same for every question. However, the amount of money you can get for sure increases with every question. Relative to the lottery, it starts with a low amount of money and increases up to a high amount of money. You are asked to indicate at which point you think the option to receive a sure amount of money is sufficiently attractive so that you prefer it to the option with the lottery. For instance, in the example below, you are asked to choose between a sure amount of money and a lottery that gives you 5 with 5% chance and with 5% chance. Now suppose you think that receiving,, or 2 for sure is worse than playing the lottery, but that receiving 3, 4, or 5 for sure is better than playing the lottery. Then you mark option A for questions, 2, and 3; and you mark option B for questions 4, 5, and 6. 27

28 Every question presents you with a choice between a lottery, and a sure amount of money. Some lotteries have two amounts of money that are equally likely, i.e., they both have a 5% chance of occurring. In the example to the left, we presented you a lottery that gives you 5 with 5%, chance and with 5% chance. In this case, you can also think of the lottery as flipping a fair coin, where you get 5 if heads comes up, and if tails comes up. Often, we present you with a lottery with two amounts of money that are not equally likely. For instance, suppose the two amounts of money are 2 and. Suppose that there is a 8% chance that you will get 2, and 2% that you will get. In such cases, you can think of the lottery as an urn with balls, of which 8 are yellow, and 2 are blue. Then a ball is picked randomly, without looking from the urn, and if it is yellow, you earn 2 while if it us blue you earn. To help you visualize the lottery, you will see a diagram right next to the table. These diagrams look as in the example below. In the diagram, you can see the relative chances of amounts of money you can get by playing the lottery. Every amount of money is visualized by a different color. The surface of each color, represents how likely it is that you win that amount of money. For instance, in the left diagram below, you are equally likely to receive 5 (blue) or nothing (white), so the surfaces are equally sized. In the right diagram, you have an 8% chance of winning 5, and 2% chance of winning nothing. In that case, 8% of the surface is blue, and 2% is white. Tip: You can make all the choices in a single table with just two clicks by turning on the computer assistance. 28

29 You will see several tables, and each table consists of a list of questions. For your convenience, we give you the option to make use of the "computer assistance". The computer assistance makes it possible for you to fill in a complete table, with just two mouse clicks. If you click on alternative B at one choice, the computer assumes that you also prefer option B for all following choices, since option B becomes more attractive. Similarly, if you click on alternative A at one choice, for all decisions above, the computer assumes that you prefer option A as well. Thus, instead of clicking option A for questions, 2, and 3, you can also just click once on option A at question 3, and the computer assumes you prefer A at questions and 2 as well. Similarly, instead of clicking option B for questions 4, 5, and 6, you only need to click on option B at question 4, and the computer assumes you prefer B at questions 5 and 6 as well. If you prefer to answer your choices manually for all questions, simply turn the computer assistance off by clicking the drop-down at the bottom of the table and selecting "Off". By default, the computer assistance is turned on. Please note that there are no right or wrong answers. We are just interested in your preferences. [A second practice question was given that presented participants with a lottery in which they had a 5-5 chance of losing 4 or losing.] 29

30 Instructions for Part A In this part, we present you again with some choices between a predetermined amount of money and a lottery. This time, the choices you make are for REAL MONEY. We will not pay you for every question. Only one of all questions will be selected at the end of the experiment, and your choice at that question will be implemented for real money. Neither you, nor we, know which question will be selected. Every question is equally likely to be selected. Every decision you make counts as a question. To emphasize: only one question out of all tables will be selected for payment, and not one question out of every table. The question is chosen by the computer after you make your choices. If for that decision you have chosen alternative B you will simply win or lose the amount of money specified. If you have chosen alternative A the gamble will be played out. Since you do not know which of your decisions will be selected, it is important that you think carefully about each decision and consider whether you prefer to get the money for sure or to play the gamble. Please note that there are no right or wrong answers. We are just interested in your preferences. For instance, in the example below, you are asked to choose between a sure amount of money, and a lottery that gives you 5 with 5% chance, and with 5% chance. Now suppose you prefer 2 for sure, but prefer the lottery over for sure. Then you choose option A in questions, and 2, and option B for questions 3, 4, 5, and 6. Now if the computer draws question 2 at the end of the experiment, you will play the lottery and receive either 5 or, depending on the outcome of the lottery. If, on the other hand, the computer draws question 5, you receive 4 for sure. Instructions for Part B Next we present you some hypothetical choices. There are always two options. With Option A, the amount of money you win or lose depends on the outcome of a lottery. With Option B you can win or lose a predetermined amount of money for certain. In this part, all decisions you make are hypothetical and will not be paid to you for real. Thus, your choices in this part have no effect on your earnings and have no effect on what choices will be given to 3