No Aspiration to Win? An Experimental Test of the...

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No Aspiration to Win?

An Experimental Test of the Aspiration Level Model*

June 27, 2013

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 20% 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: C91, D81.

* 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, 77305

Fontainebleau Cedex, France. E-mail: [email protected]

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1. 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

(1952) argues that DMs will seek to minimize the probability of disaster. Indeed,

farmers minimize the probability of falling below the subsistence level (Lopes 1987),

cabdrivers aim at a daily target (Camerer et al. 1997), and investment managers try to

meet a target return (Payne, Laughhunn, and Crum 1980, 1981). 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 2008; Baucells and Villasís 2010; Camerer

1989; Etchart-Vincent and L’Haridon 2011; Fehr-Duda et al. 2010; Hershey and

Schoemaker 1980; Tversky and Kahneman 1992; Wakker 2010; Weber and Camerer

1998). Although such risk-seeking behavior seems natural, even obvious (Payne

2005), 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 2008;

Levy and Levy 2009) or take aspiration levels as the foundation of decision theory,

dispensing with utility functions (Castagnoli and LiCalzi 2006).

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 (2008) 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.

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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 1992) and as a natural aspiration level (Lopes and

Oden 1999; Pahlke, Kocher, and Trautmann 2011). The outcome zero has also been

investigated from a psychological perspective (Birnbaum et al. 1992; Mellers, Weiss,

and Birnbaum 1992; Weber, Anderson, and Birnbaum 1992), and tests of the affect

heuristic concentrate around zero (Bateman et al. 2007). 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 1999; Payne 2005; Payne,

Laughhunn, and Crum 1980), 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 20% 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 (2008) and Levy and Levy (2009) 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 xj ( j = 1, 2, . . ., n) and probabilities pj is

( ) ∑ ( ) ( ) ( ) (1)

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It is straightforward to show that this expression is equivalent to

AL ( ) ( )j jjV L p v x if we define: ( ) ( )j jv x u x for x x ; ( ) ( )j jv x u x

for x x ; and ( ) ( )j jv x u x when x coincides with the aspiration level. The result is

a utility function v that is discontinuous around the aspiration level. Figure 1

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 1 about Here ]]

2.1. Model Predictions

To derive predictions from the aspiration level model, we assume a value function as

in Figure 1: 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 1992) accommodates that risk seeking with a convex value

function in the loss domain.1 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 1 for a two-

outcome monetary lottery that yields the outcome 0 with probability p and the

outcome −y with probability 1 − p. The dashed line indicates that participating in the

lottery is preferred to receiving its expected value for sure.

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.

1 Under PT, risk attitude is described by the value function and probability weighting function. In this

paper we address only the value function.

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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 1 we consider not only the baseline lottery with outcomes −y and 0 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 risk-

averse 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

3.1. 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 ( , ; )i i i iL p x y , where x and y (x > y) are the outcomes and p is

the probability of the highest outcome x. Table 1 gives details on all the lotteries.

Those in parts 1A and 2A of the experiment involved relatively small outcomes

(ranging from −28 to +28 euros) and were incentivized; the lotteries in parts 1B and

2B involved relatively large outcomes (±500 euros) and were not incentivized.

[[ INSERT Table 1 about Here ]]

Lotteries in part 1A were of the form 12

( , ; )i i iL x y . In the base lottery we set x = 20

and y = 0. We denote this lottery by BSL , 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.

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(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 c

BSL , where { 28, 24, 20, 0.5,0, 0.5, 4, 8}c . For

instance, 0.5 12

( ,20.5;0.5)BSL . Note in particular that lotteries with c equal to −20,

−24, and −28 mirror the lotteries with c equal to 0 (the base lottery), +4, and +8 in the

sense that gains are replaced by losses. For instance, 8 12

( ,28;8)BSL whereas

28 12

( , 8; 28)BSL . Part 1B is analogous to part 1A except that all outcomes are large:

in this base lottery, we set x = 300 and y = 0. We denote the base lottery by BLL (for

“base, large”), and the shifts c in this part were from the set

{−500, −400, −300, 0, +100, +200}. Again, note that lotteries with c equal to −300,

−400, and −500 mirror the lotteries with c equal to 0, +100, and +200 (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 = 20 and y = −10. We

presented 19 mixed lotteries (p, 20; −10) with the probability p of the high outcome

varying from 0.05 to 0.95 in steps of 0.05. We will denote these lotteries as ,MS pL (for

“mixed, small”), where {0.05, ,0.95}p . In part 2B we did the same for lotteries

(p, 200; −100), which we will denote by ,ML pL (for “mixed, large”).

All choices were presented in the form of a “price list” (Andersen et al. 2006;

Binswanger 1980, 1981). 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 €1 in part 1A, €10 in part 1B, €0.5 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.

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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) > zL (resp., EV(L) < zL). If the expected value of the lottery falls within the

interval [ , ]L Hz z then we classify the choice as risk neutral.

3.2. Experimental Procedures

The experimental sessions took place in Amsterdam at the CREED laboratory in April

2009. 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: 1A, 1B, 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

1998; Starmer and Sugden 1991). Out of all questions in parts 1A 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,

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average earnings amounted to €34 and ranged from €0 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. 2006). 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 ( , ; )iz L p x y for some sure outcome z then

( , ; )iz L p x y for any z z ; likewise, if a participant indicated the preference

( , ; )iz L p x y then it was assumed that ( , ; )iz L p x y for any z z .

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.


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 ( , ; )ix L p x y and ( , ; )iy L p x y . Hence, these basic principles of

rationality are not violated.

3 For participants who made multiple switches, we take the midpoint between the first and the second

switch.

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4. Hypotheses

We formulate three hypotheses. The first two are tested in part 1 of study, and the

third one is tested in part 2.

Hypothesis 1 (H1). 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

BSL and 28

BSL , where the overall probability of reaching

the aspiration level is 0, than for the lottery 20

BSL , where the overall probability of

reaching the aspiration level is 0.5. 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 0.5

BSL will be significantly higher, and that of lottery 0.5

BSL significantly

lower, than the valuation of the lottery BSL .

Hypothesis 3 (H3). The CE of lotteries ,ML pL and ,MS pL , 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 1 of the experiment. Table 1 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.



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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 risk-

seeking 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 40%. 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 1).



We are mainly interested in how attitudes toward risk change after the shift in

payoffs. If the risk-seeking behavior of participants in 20

BSL is driven by their desire to

break even—that is, to achieve at least zero—then we should observe less risk-

seeking behavior when a constant is subtracted from all outcomes, thereby making it

impossible to achieve zero (H1). 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 20

BSL , the percentage of risk-seeking choices is significantly higher for the

lotteries 24

BSL (χ2 = 8.64, p = 0.063) and 28

BSL (χ2 = 16.06, p < 0.001). Compared with

300

BLL , the difference in percentage of risk-seeking choices is not significant for lottery

400

BLL (χ2 = 1.07, p = 0.302) but is significant for 500

BLL (χ2 = 5.06, p = 0.024). 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 BSL , is

significantly higher in lotteries 4

BSL (χ2 = 4.92, p = 0.027) and 8

BSL (χ2 = 10.56,

p < 0.001). For large outcomes, the percentage of risk-averse choices decreases. The

difference between BLL and 100

BLL is not significant (χ2 = 1.78, p < 0.182), although

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that between BLL and 200

BLL is marginally significant (χ2 = 2.77, p < 0.096). 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 BSL with lotteries 0.5

BSL and 0.5

BSL constitutes the strictest test of our

model (H2). Observe that BSL has one positive outcome and one zero outcome, 0.5

BSL

has one negative and one positive outcome, and 0.5

BSL 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 0.5

BSL (resp. 0.5

BSL )

should be substantially lower (resp. higher) than that of BSL . Yet we find no support

for H2, either. The mean CEs of the three lotteries are close to each other, ranging

only between 8.5 and 8.6. A within-subject test reveals that the shifted lotteries are

not significantly different from BSL : for 0.5

BSL , Z = 0.73 and p = 0.949; for 0.5

BSL ,

Z = 1.343 and p = 0.180 (Wilcoxon signed-rank tests,5 two-tailed).

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Hypotheses 1 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., 9.5 and 10.5) 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 0.8; two-tailed tests).

6 At the individual level we find some violations of dominance. Compared with the baseline lottery, 17

of the 48 participants report a higher CE for 0.5

BSL

; for 6 of them, this difference is strictly greater than

2. Also, 13 participants report a lower CE for 0.5

BSL

; 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. (2007) and Mellers, Weiss, and Birnbaum (1992).

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level would come from a “vertical” portion of graph as in Figure 1. 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 (1992), 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 0.1, the

CEs are significantly different at the 1% 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 (1992) 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.



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 20% 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 = 0.5 and

p = 0.6 for both the small and large outcomes; the corresponding p-values are (respectively) p = 0.033

and p = 0.096.

8 The values of the CEs are based on the parameters and functional forms described in Tversky and

Kahneman (1992). We do use a slightly lower value for the loss aversion parameter (λ is set at 1.75

instead of 2.25) to yield a better fit with the data.

mailto:.@WHY

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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 0 and 1) was at most .1

away from the expected value in at least 14 out of the 19 questions of part A (B); a

participant is risk averse if at least 14 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 .5 (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 0.15 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 B1 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 21% (27%) and 42% (25%) respectively. About 20%

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 16% (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 -0.1 and standard deviation

0.1. 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

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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 1A and 1B. 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

2009; Lopes and Oden 1999; Payne 2005) or put participants under time pressure

(Pahlke, Kocher, and Trautmann 2013). 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 2011), 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

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(aspiration level) and then analyzed data based on this information (Fellner, Güth, and

Maciejovsky 2009). Fourth, it may be that an aspiration level does not emerge until

the entire context of outcomes is known by the decision maker (Zeisberger 2012); in

part 1 of our experiment, for example, the DM does not know beforehand how much

she can win or lose. Finally, Payne (2005) 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

1992), 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 20% 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.

16

Figure 1. Example of a value function for the aspiration level (AL) model.

Figure 2. Screenshot of some questions from part 1 of the experiment.

Sheet 3 of 8 – Part 1a

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 €20 or winning €0 graphically. option A option B

1 A: 50% Chance of winning €20 and 50% Chance of receiving €0

B: Receiving €0 for sure


B: Winning €1 for sure







(deleted entries 6 to 19)





Computer Assistance:

ON

Continue »

17

Figure 3. Mean and median CEs for part 1A, as well as expected value (EV) of the

lottery and CE value predicted by prospect theory, for parameter values α = 0.88,

β = 0.88, γ = 0.61, δ = 0.69, and λ = 1.75.

18

Figure 4. Mean and median CEs for part 1B, as well as expected value (EV) of the

lottery and CE value predicted by prospect theory, for parameter values α = 0.88,

β = 0.88, γ = 0.61, δ = 0.69, and λ = 1.75.

Figure 5. Proportion of risk-seeking and risk-averse choices for lotteries in part 1A of

the experiment. Numbers on the horizontal axis are the two outcomes for each lottery.

19

Figure 6. Proportion of risk-seeking and risk-averse choices for lotteries in part 1B 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 = 0.88, β = 0.88, γ = 0.61, δ = 0.69, and λ = 1.75. Numbers on the

vertical axis are the high outcome’s probability; all values rescaled as (x + 10)/30.

0.2

.4.6

.81

pro

ba

bili

ty h

igh

outc

om

e (

p)

0 .2 .4 .6 .8 1

20

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 = 0.88, β = 0.88, γ = 0.61, δ = 0.69, and λ = 1.75. Numbers on the

vertical axis are the high outcome’s probability; all values rescaled as (x + 100)/300.

0.2

.4.6

.81

pro

ba

bili

ty h

igh

outc

om

e (

p)

0 .2 .4 .6 .8 1

21

Figure 9: Aggregate S-Shape preferences can arise from

heterogeneous aspiration levels. Panel A: a piecewise-

linear aspiration level value function with an aspiration

level at x=-0.2. 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 -0.1 and standard

deviation 0.1.

22

Table 1

Part Lottery

High

outcome

(x)

Low

outcome

(y)

Probability of

high outcome

(p)

Mean

CE

Median

CE

1A BS

L +20 0 0.5 8.5 8.5

1A 28

BSL

−8 −28 0.5 −15.5 −15.5

1A 24

BSL

−4 −24 0.5 −12.7 −13.0

1A 20

BSL

0 −20 0.5 −10.1 −10.5

1A 0.5

BSL

+19.5 −0.5 0.5 8.6 8.5

1A 0.5

BSL

+20.5 +0.5 0.5 8.6 9.5

1A 4

BSL

+24 +4 0.5 11.6 11.5

1A 8

BSL

+28 +8 0.5 15.0 14.5

1B BL

L +300 0 0.5 109.0 105.0

1B 500

BLL

−200 −500 0.5 −319.6 −310.0

1B 400

BLL

−100 −400 0.5 −232.1 −240.0

1B 300

BLL

0 −300 0.5 −134.4 −145.0

1B 100

BLL

+400 +100 0.5 221.5 220.0

1B 200

BLL

+500 +200 0.5 317.3 340.0

2A ,MS p

L +20 −10 {0.05, , 0.95}

2B ,MS pL +200 −100 {0.05, , 0.95}

23

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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 90 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

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 50% chance and €0 with 50% chance. Now suppose you think that

receiving €0, €1, 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 1, 2, and 3; and you mark

option B for questions 4, 5, and 6.

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 50% chance of

occurring. In the example to the left, we presented you a lottery that gives you €5 with 50%, chance

and €0 with 50% 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 €0 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 €20 and €0. Suppose that there is a 80% chance that

you will get €20, and 20% that you will get €0. In such cases, you can think of the lottery as an urn

with 100 balls, of which 80 are yellow, and 20 are blue. Then a ball is picked randomly, without

looking from the urn, and if it is yellow, you earn €20 while if it us blue you earn €0.

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 80% chance of winning €5, and 20% chance of

winning nothing. In that case, 80% of the surface is blue, and 20% is white.

Tip: You can make all the choices in a single table with just two clicks by turning on the computer

assistance.

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 1, 2, and 3, you can also just click once on option A at

question 3, and the computer assumes you prefer A at questions 1 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 50-

50 chance of losing €4 or losing €0.]

30

Instructions for Part 1A

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.


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 50% chance, and €0 with 50% chance. Now suppose you prefer €2 for

sure, but prefer the lottery over €1 for sure. Then you choose option A in questions 1, 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 €0, 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 1B

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

31

you subsequently. We are only interested in what you would do if you actually faced these choices, so

please think about them carefully.

Every time we ask you to indicate the option you prefer. Try to imagine that the options are offered to

you for real. Please note that there are no right or wrong answers. We are simply interested in your

preferences.

Instructions for Part 2A

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.


[Another example as above was given, with a lottery that gave then €5 with 50% chance, and €0 with

50% chance.]

Instructions for Part 2B

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

you subsequently. We are only interested in what you would do if you actually faced these choices, so

please think about them carefully.

Every time we ask you to indicate the option you prefer. Try to imagine that the options are offered to

you for real. Please note that there are no right or wrong answers. We are simply interested in your

preferences.

32

Appendix B: classification of participants

This appendix describes our procedure for classifying participants. The individual certainty equivalent

(CE) for a question is determined as described in the main text. We normalized the certainty equivalent

of a participant to (CE-y)/(x-y) and interpret the probabilities of the high outcome as utilities. The first

step is to eliminate participants that show strong violations of monotonicity. A participant is classified

as violating monotonicity if on five or more occasions the associated probability decreased by 0.15 or

more as the CE increased, or if on three or more occasions the associated probability decreased by 0.30

or more as the CE increased. The second step identified the remaining participants as risk-neutral if in

at least 14 out of the 19 questions the probability was at most 0.10 away from the expected value. The

third step identified the remaining participants as having an aspiration level if there were four (or more)

consecutive choices for which the CEs were within a bandwidth of 0.50 (Part 2A) or 5 (Part 2B). The

fourth step identified the remaining participants as risk averse if in at least 14 out of the 19 questions

the CE was below the expected value. All participants that could not be classified are, together with

those that showed violations of monotonicity, grouped as mixed. The resulting classification is shown

in the figures below.

33

Panel 1A: Risk-neutral

Panel 1B: Aspiration Level

0.5

10

.51

0.5

1

-10 0 10 20 -10 0 10 20

-10 0 10 20 -10 0 10 20

2807 2808 2810 2814

2818 2901 2906 2919

2923 2926

0.5

10

.51

0.5

1

-10 0 10 20 -10 0 10 20

-10 0 10 20 -10 0 10 20

2802 2803 2817 2822

2908 2910 2914 2917

2920 5728347

34

Panel 1C: Risk-averse

Panel 1D: Mixed

Figure B1: Individual level results in Part 2A. Certainty Equivalents are on the horizontal

axis and the probability on the high outcome on the vertical axis. The number at the top of

each figure indicates the subject number. Panel A: Risk-neutral. Panel B: Aspiration Level.

Panel C: Risk-averse. Panel D: Mixed.

0.5

10

.51

0.5

10

.51

-10 0 10 20

-10 0 10 20 -10 0 10 20 -10 0 10 20 -10 0 10 20

2804 2805 2806 2809 2811

2813 2815 2816 2819 2821

2902 2903 2907 2913 2916

2921 2922 2924 2925

0.5

10

.51

0.5

1

-10 0 10 20 -10 0 10 20 -10 0 10 20

2812 2820 2823

2904 2905 2911

2912 2915 2969

35

Panel 2A: Risk-neutral

Panel 2B: Aspiration Level

0.5

10

.51

0.5

10

.51

-100 0 100 200 -100 0 100 200 -100 0 100 200

-100 0 100 200

2805 2807 2808 2809

2810 2814 2815 2818

2901 2906 2907 2923

2926

0.5

10

.51

0.5

1

-100 0 100 200 -100 0 100 200 -100 0 100 200

2817 2902 2903

2905 2908 2912

2914 2917 2924

36

Panel 2C: Risk-averse

Panel 2D: Mixed

Figure B2: Individual level results in Part 2B. Certainty Equivalents are on the horizontal

axis and the probability on the high outcome on the vertical axis. The number at the top of

each figure indicates the subject number. Panel A: Risk-neutral. Panel B: Aspiration Level.

Panel C: Risk-averse. Panel D: Mixed.

0.5

10

.51

0.5

1

-100 0 100 200 -100 0 100 200 -100 0 100 200 -100 0 100 200

2803 2804 2806 2816

2819 2821 2823 2913

2916 2920 2921 2925

0.5

10

.51

0.5

10

.51

-100 0 100 200 -100 0 100 200

-100 0 100 200 -100 0 100 200

2802 2811 2812 2813

2820 2822 2904 2910

2911 2915 2919 2922

2969 5728347

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