Decision making and age: Factors influencing decision making under uncertainty

Accepted Manuscript

Decision making and age: Factors influencing decision making under uncertainty Alec N. Sproten , Carsten Diener , Christian J. Fiebach , Christiane Schwieren PII: DOI: Reference:

S2214-8043(18)30315-X 10.1016/j.socec.2018.07.002 JBEE 371

To appear in:

Journal of Behavioral and Experimental Economics

Received date: Revised date: Accepted date:

7 November 2016 7 July 2018 9 July 2018

Please cite this article as: Alec N. Sproten , Carsten Diener , Christian J. Fiebach , Christiane Schwieren , Decision making and age: Factors influencing decision making under uncertainty, Journal of Behavioral and Experimental Economics (2018), doi: 10.1016/j.socec.2018.07.002

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Two experiments were run to investigate age effects on uncertain decisions Uncertainty condition, feedback, and learning requirements were manipulated No age differences were found in risky conditions with a priori probabilities But well under ambiguity with feedback and under risk with statistical probabilities Several explanations for these effects are discussed

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Decision making and age: Factors influencing decision making under uncertainty Authors - affiliations:

Alec N. Sproten - University of Erlangen-Nuremberg, Lehrstuhl für Volkswirtschaftslehre, insb. Wirtschaftstheorie, Lange Gasse 20, 90403 Nürnberg, Germany. Correspondence to: [email protected]

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Carsten Diener - Department of Applied Psychology, Division for Clinical and Biologic Psychology, SRH University of Applied Sciences Heidelberg, Maria-Probst-Str. 3, 69123 Heidelberg, Germany. Christian J. Fiebach - Department of Psychology, Goethe University Frankfurt, Mertonstr. 17, 60325 Frankfurt am Main, Germany.

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Christiane Schwieren - Alfred Weber Institut für Wirtschaftswissenschaften, Heidelberg University, Bergheimer Str. 58, 69115 Heidelberg, Germany.

Abstract:

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In the present study, we investigate how decision making under uncertainty is affected by age. We ran two experiments with young and older adults, systematically manipulating (1) uncertainty conditions

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(risk and ambiguity), (2) feedback on decisions and (3) requirements of the task regarding executive functions. Experiment 1 aims at investigating risk with a priori probabilities and ambiguity and the effects of feedback in a card game (N = 200; older adults: 97). The results reveal no age differences in

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choice behaviour under risk with a priori probabilities. If feedback is provided, we find that older adults are less ambiguity averse than young adults, whereas there is no significant age difference if no feedback is provided. Moreover, the presence of feedback had a positive effect on the propensity to gamble in uncertain conditions by influencing subjective probabilities for both age groups. Experiment 2 uses the Balloon Analogue Risk Task (BART) to investigate decision making with statistical probabilities (N = 100; older adults: 50). Here we report older adults being more risk averse, an effect that can be explained with age differences in sensitivity to prior choices. We support the results by comparing them to survey data and conclude that differences in uncertainty-processing exist between young and older adults. Possible explanations of these differences are discussed.

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Keywords: age differences, experiment, risk, ambiguity, feedback

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JEL classification: J14, C91

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1. Introduction: “The aging workforce” currently is under much scrutiny in both popular media and scientific research. A large number of children born after 2000 in western countries could live to 100 years of age and older – with increasing shares ageing in good cognitive health (Vaupel 2010). Therefore, many economic decisions will be taken by individuals at higher age, and it becomes more and more important to understand the decision making of cognitively healthy older adults. While older adults generally avoid physical risks, face changes in medical risk taking (Hanoch, Rolison & Freund, 2018)

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or perceive themselves as less risk seeking while using the internet (White, Gummerum, Wood & Hanoch 2017) assuming the same being true for financial risk taking may be a foregone conclusion (Samanez-Larkin & Knutson, 2015). Although much research on individual decision making relies on student populations (Henrich et al. 2010), the number of studies focusing on age differences and lifespan changes in decision making has seen strong increases over the last years, with the current article

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contributing to this research.

By now, a number of (economic) experiments investigates age effects on decision making by comparing age groups (see e.g. Best & Charness 2015 for an overview and meta-analysis of studies on age effects in risky decision making). Older adults are commonly assumed to face changes in decisionmaking abilities (Peters et al. 2000) and age differences in decision making under uncertainty are a

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central finding in the literature (Mata et al. 2011). Yet, the literature is not always consistent: Whereas some studies report increasing risk aversion with age, other studies find no age differences (e.g.

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Carstensen & Hartel 2006; Mata et al. 2011). Many explanations have been drawn to explain these effects and we will refer to the most widespread explanations below.

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The goal of the present paper is to add further evidence to the picture on age effects on decision making under uncertainty by studying two age groups (younger and older adults) deciding on two experimental tasks with complementary aspects of uncertainty. We focus on uncertain conditions with

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varying information on the probabilities of realizing a positive outcome. Further, we investigate the impact of feedback on outcomes on behaviour in subsequent decisions, as (a) there is agreement in the

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literature that aging implies a decrease in the ability to learn from feedback (e.g. Mathewson et al., 2008) and (b) feedback is a central tool in information design when it comes to developing decision aids for older adults. We investigate decision making of elderly participants in both decision making under risk and decision making under ambiguity. Within both domains, we compare a situation without feedback and with feedback, where feedback does not allow for learning (as the task is designed in a way that feedback does not provide any information which can be used to increase outcomes in subsequent decisions), hence only affecting emotions and subjective uncertainty via a pathway where feedback may lead to regret for not having made the outcome-maximizing decision or joy for having just done so. We then conduct a game that adds learning to uncertain conditions,

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ACCEPTED MANUSCRIPT combining all three factors (uncertainty, feedback and learning), where participants have to recur to feedback to learn about probabilities during the task. Uncertainty describes a situation in which the outcome of a given condition is bound to some (known or unknown) probabilities. As early as in the beginning of the 20th century, uncertain conditions were classified in different subcategories, defined by their “probability situations” (Knight 1921): estimates, statistical probabilities (sometimes also called decisions from experience), and a priori probabilities (sometimes also called decisions from description). In current research, it is more common to refer to

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‘ambiguity’ instead of estimates and to ‘risk’ instead of statistical and a priori probabilities. Ambiguity (i.e., estimates) is defined as uncertain conditions in which probabilities cannot be computed empirically because of conflicting or absent information. Statistical probabilities are defined as the option to learn the risk of a choice empirically by referring to prior choices with similar outcomes. This constitutes a rather “psychological” definition of risk, as, from an economic perspective, they are

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at the border between ambiguity and risk with a priori probabilities. Nevertheless, the concept of statistical probabilities is well defined and choice models can (and will) be estimated (cf. infra). A priori probabilities define the level of risk by assignment of explicit (or easily computable) probabilities. We structure the following literature review based on the aspects that interest us: risk vs. uncertainty, and feedback.

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Age and risk

Several research groups have shown that risk taking decreases with age (Chaubey 1974; Deakin et al.

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2004; Dohmen et al. 2012; Hallahan et al. 2004), whereas other studies have shown that risk preferences do not change (Ashman et al. 2003; Dror et al. 2000; Henninger et al. 2010; Zamarian et al. 2008). A number of explanations for this, at the first glance, lack of consistency has been

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developed over the years: most centrally the observed effects might reflect differences in the demand of the tasks used with respect to executive functions. A further explanation might be additional age-

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related changes in value assessments (Samanez-Larkin & Knutson 2015) or an inverse-U shaped relationship between risk preferences and age (see Josef et al. 2016 for self-assessed risk preferences);

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other reasons, such as group sampling, may as well play a role, yet to a lesser degree and are much more difficult to control for between studies. For example, Mata et al. (2011) and Best and Charness (2015) show in their meta-analyses that older adults are more risk avoidant than young adults when confronted with statistical probabilities but not when confronted with a priori probabilities, the former drawing more heavily on executive functions than the latter: when confronted with statistical probabilities, the ability to learn from prior experiences is crucial (Hertwig & Erev 2009; Hau et al. 2008), while for a priori probabilities, mathematical abilities (i.e. understanding of probabilities) play an important role. One must however be aware of an additional distinction between such tasks: conditions with statistical probabilities can be interpreted as conditions that start out with ambiguity

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ACCEPTED MANUSCRIPT and, while exploring, turn more towards risk (while in conditions with a priori probabilities, ambiguity should be absent from the beginning; Rolison et al. 2012).

Ambiguity While the effects of age on risky decision making received considerable attention in the literature, evidence on ambiguity is sparser. Some authors showed that older adults behave similar to young

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adults under ambiguity (Tymula et al., 2013), others provide evidence that older adults are more ambiguity averse (Zamarian et al. 2008) or behave less consistently with prior indications of choice likelihood ratings under ambiguity (Rolison & Parchur 2017). These three studies further stand out from the literature by assessing decision making in risky and ambiguous conditions in a within subject design. This is important as older adults may exhibit larger heterogeneity in choice behaviour (due to e.g. differences in preservation of executive functions or simply life-long experiences) and within-

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subject comparisons are a way to avoid finding false positives (or negatives) in the corresponding analyses. Zamarian and colleagues find that normal aging does not affect decision making under risk, but does affect decision making under ambiguity: Older adults demonstrate less payoff maximizing behaviour than young adults. One might note that this study clearly reveals differences between uncertain choices on two tasks, but, from an economic perspective, the measure used to assess

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ambiguity, the Iowa Gambling Task (IGT; Bechara et al. 2005), has been under scrutiny for lacking a concise definition of optimal decision making. In addition, the IGT may assess other factors than

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ambiguity attitude alone: Learning in a situation of risk with statistical probabilities (as operationalized by Henninger, Madden & Huettel 2010), differences between “hot” and “cold” decision making (Buelow & Suhr 2009), or reactions to negative and positive feedback. Tymula et al.

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(2013), on the other hand, show that, in a task encompassing risk taking with a priori probabilities and ambiguity, older adults are more risk averse than young adults, but behave similarly to young adults

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under ambiguity.

The role of feedback Cognitive demands of behavioural tasks can in fact be a driving force behind age differences in decision making. We therefore try to influence the cognitive demands of the task by studying the effects of feedback on decisions. Feedback can have two effects in a situation of risk: (1) influencing emotions and reinforcing behaviour in general, but it also provides (2) an opportunity to learn. Thus, in contrast to feedback that

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ACCEPTED MANUSCRIPT cannot be used to learn, such as in risk conditions with a priori probabilities, in situations with statistical probabilities differences in learning abilities between old and young participants can determine differences in behaviour. In ambiguous situations, feedback reduces perceived uncertainty (Hogg & Mullin 1999; Kramer 1994), even if it is not informative for future decisions. It also influences emotions and decision making in general (Baumeister et al. 2007) and has been shown to affect behaviour more strongly in young than in older adults (Bellucci & Hoyer 1975; Tripp & Alsop 1999). To be more precise, when

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investigating dual-process models of decision making, tasks providing immediate feedback on decisions are assumed to assess “hot” decision making (hence with a strong emotional component; Loewenstein 2000), whereas tasks with delayed feedback are classified as “cold”, inducing little emotions. To add to the complexity of the problem, older adults seem to use different strategies than younger individuals in decision making, using more affective strategies where young adults rely more

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on deliberative strategies (Wood et al. 2005). Or as Huang, Wood, Berger and Hanoch (2015) put it: declines in deliberative abilities lead to declines in description-based decision making, whereas the age-related preservation of affective abilities is linked to no age effects in experience-based decisions. This is particularly important in the light of the “risk as feelings” hypothesis (Loewenstein et al. 2001), meaning that (a) feedback, by reducing perceived uncertainty, influences the emotional state of decision makers and (b) older adults´ decision making is more dependent on this emotional state.

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According to Huang, Wood, Berger and Hanoch 2013, older adults will rely less than younger adults on “mathematical” decision strategies in tasks carrying some affective component by providing

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feedback, such as the task applied in this paper.

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The goal of our study is to advance the field towards a better understanding of the relationship between aging and decision making under financial uncertainty by systematically looking at the

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factors (a) risk versus ambiguity, (b) “uninformative” feedback (which cannot be used to learn about the structure of the task1 and thus only has an “emotional” component) and (c) “informative” feedback

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(allowing for learning and being cognitively more demanding). The current study follows closely the design by Mamerow, Frey and Mata (2016), spanning the afore-listed task differences: self-assessed risk preferences, risk with statistical and risk with a priori probabilities. The ambiguity conditions of the task in the current paper mirror more closely Tymula et al.’s task, as the learning requirements are kept to a minimum. . Note that here, feedback does not carry the opportunity to learn, as this leads to different results – older adults reacting more stongly to negative feedback (Eppinger & Kray 2011).

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There is still an opportunity to learn: if, for example, a participant chose to ignore the probabilities of the task or did not understand the set-up, feedback could be used to learn. For the remainder of the paper, we however work with the hypothesis that all, or at least the majority, of participants could not learn from feedback.

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ACCEPTED MANUSCRIPT We report results from two experiments in which we observe behavioural differences in uncertain decisions between young and older adults (Table 1a summarizes the experimental conditions and Table 1b provides an overview of the hypotheses which will be described below). In the following, we posit hypotheses about behaviour. We describe our experiments and analyse the results. Finally, we discuss the findings in the light of the current interest in age differences in decision making.

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2. The Experiments 2.1. Basic Design and Hypotheses

In the first experiment, we investigate age differences in behaviour under risk (with a priori probabilities) and under ambiguity either with or without uninformative feedback on decision

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outcomes.

As most of the literature suggests differences between young and old adults both in risky and ambiguous decisions, we hypothesize that older adults have a different propensity to gamble than young adults in (H1) risky and (H2) ambiguous conditions. The task used to assess these hypotheses is designed such that feedback can only influence emotions and the perception of probabilities, but does not provide additional relevant information. Therefore, age differences in learning should not play a

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role. As feedback reduces subjective uncertainty (Trope, 1979), even if not informing about objective probabilities, providing feedback on choices with a priori probabilities should result in behaviour

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indicative of a smaller distance between subjective and objective probabilities in both age groups (H3). Assuming that most participants are a priori uncertainty averse, we therefore expect gambling

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behaviour to be more frequent, and an estimated probability weighting function to be closer to the actual probabilities when feedback is provided. Given results that uninformative feedback influences behaviour more strongly in young than in older adults, and that older adults employ more emotional

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strategies for decision making (while assuming that feedback adds an emotional component to tasks investigating behaviour under uncertainty), two possible outcomes arise: feedback will c.p. increase

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older participants´ probability to gamble (H3a) or older adults will demonstrate a smaller difference in behaviour between feedback and non-feedback treatments (H3b). Based on the literature reviewed and the methodological reflections given so far, we hypothesize that whether probabilities are given or have to be learned influences the difference between the age groups. In the second experiment, we therefore vary this factor: we administer a risk task with statistical probabilities to our participants (statistical probabilities by definition include feedback on decision outcomes). Feedback now allows for learning. Since older participants have been shown to learn slower (Kausler, 1994), we expect changes in behaviour based on feedback rather for young than for old participants. Learning in this task should in general lead to more risk taking over time (i.e., over

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ACCEPTED MANUSCRIPT the repetition of trials (Mata et al., 2011)). Thus, we expect older participants to take less risk than young participants (H4a). Alternatively, one could describe a task where probabilities in the beginning are unknown and can be learned over time as a situation where ambiguity prevails at the onset of the task, with ambiguity being gradually transformed to risk by gaining experience. Ideally, if such a task is played for a sufficient number of trials, participants will know the probability structure and end up in a situation of risk. For age effects this implies that we should expect an increase in the difference in risk taking between young and older participants over the repetition of trials (H4b). Comparing both

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kinds of risk tasks, we would expect that age has a larger effect on behaviour on a task where learning is required. Thus, we expect a difference in risk taking between the two risk tasks within the group of elderly but not for the young participants (H5). Finally, in order to get an impression of the representativeness of our results compared to the general population, we also include a widely used self-assessment question for risk preferences, and compare answers on this question to data from a

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representative sample of the German population (Josef et al. 2016).

Table 1a : Overview of the Experimental Design and Conditions )Condition Exp. Risk Ambig. Risk Feedback Feedback (a_priori) (stat.) (no learning, (no learning, uninformative) informative) 1 1 Yes Yes 2 1 Yes Yes 3 1 Yes Yes 4 1 Yes Yes 5 2 Yes

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Feedback (learning)

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Table 1b: Overview of the Expected Effects Expected Effect Age differences in uncertainty preferences

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Feedback reduces subjective uncertainty Older adults are less uncertainty averse than young adults Older adults are more uncertainty averse than young adults Statistical probabilities allow for learning through feedback Older adults are more risk averse than young adults Distance between young and older adults increases over time Difference in risk taking between risk with a priori probabilities and with statistical probabilities is more pronounced in older adults

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Hyp. H1, H2, H4 H3 H3a H3b

Exp. 1, 2 1

2 H4a H4b H5

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3. Sample, Procedures and Results 3.1. Experiment 1 3.1.1. Participants

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ACCEPTED MANUSCRIPT A total of 200 adults (103 young adults, 97 older adults; cf. Table 2) participated in the experiment. All of the young adults were students at the Universities of Mannheim or Heidelberg and were on average 23.30 years old (SD = 2.82, min: 19, max: 32). The older adults were healthy with an average age of 67.77 years (SD = 5.98, minimum age: 58 years, maximum: 88 years). The majority of the older adults held a college or university degree and were retired. We recruited them by word of mouth advertisement at an adult education centre in Heidelberg and by means of an article appearing in a local newspaper, explaining the need for participants with some computer literacy. Thereby we

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generated a group of older adults not representative of the population, but well-matched to the student sample with respect to education and cognitively active lifestyle. Participants’ health was assessed using an extensive questionnaire on physical and mental disorders, but no cognitive test battery was applied. We are confident that, if cognitive impairments existed in our participants, they only could have been mild: all older participants were able to understand the experimental setup, were able to use

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a computer, and many were recruited in classes at the adult education centre.

Table 2: Participants Young Older Number 103 97 Male/Female 44/59 35/62 Mean Age (SD) 23.30 (2.82) 67.77 (5.98) Years of education* (SD) 12.70 (1.09) 11.99 (2.41)

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3.1.2. Procedures

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*until graduation from school

In accordance with our hypotheses 1-3, we applied a three-factorial design with two between subject

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factors (age group and presence/absence of feedback) and one within-subject factor (uncertainty type: risk vs. ambiguity) to investigate decision making under uncertainty.

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We used a computerized decision making tasks that will be described in detail in the following: The risk and ambiguity task (RAT) is a computerized card game introduced by Hsu and colleagues

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that measures risk behaviour with a priori probabilities and ambiguity behaviour (Hsu et al. 2005). Participants see card decks on a computer screen. These decks always contain some number of red and of blue cards. For some decisions, the exact amount of red and blue cards in the deck is revealed to the participants (see Figure 1), whereas for other decisions participants only know the total number of cards in the deck. Further, both the uncertain and the sure payoffs are displayed. Participants always have the choice to either place a bet that a chosen colour would be drawn, or to receive a sure amount of money. Participants make 48 of these choices between a gamble and a sure amount of money. In half of the trials, participants are faced with a priori risky decisions (i.e. the probabilities of winning are given), and in half of the trials they are faced with ambiguous decisions (i.e. the probabilities of

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ACCEPTED MANUSCRIPT winning the gamble are unknown and participants do not receive any information on the underlying distribution of cards). Risky and ambiguous gambles alternate. Responses are made by selecting an option in the card game displayed on the screen. When betting, participants receive the indicated amount of ECU if the colour they chose was drawn (i.e. if a randomly drawn number matched the probability of the game); otherwise the payoff was zero. When participants choose the certain amount, they receive the amount automatically, without any influence of the drawn card colour on their gain. Over all choices card distribution (respectively the total number of cards) and outcome vary.

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Probabilities of winning range from 0.1 to 0.9, and payoffs of the gambles vary between € 2.40 and € 4.00, certain payoffs range from € 1.00 to € 3.00. Whereas participants receive full feedback on their performance in the feedback treatments, no feedback is provided in the other treatments. Specifically, in the feedback treatments, the screen displayed after each choice which colour had been selected by the participant, if any, which colour was drawn and which amount the participant had won. In the no-

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feedback treatments, only the information about participants’ choice, but no information about the drawn colour or the outcome of the gamble was shown. To avoid risk hedging, only two of the gambles were selected randomly at the end of the game to determine participants’ payoffs. Participants were allowed as much time as they needed to make their choices. The experiment took place at the former Collaborative Research Centre 504 Lab (SFB-504) of the

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University of Mannheim (young participants) and at the Alfred Weber Institute Lab (AWI-Lab) of Heidelberg University (young and older participants). Participants first had to fill in a general

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demographic questionnaire. Subsequently, the experimental tasks by which we examined behaviour under uncertainty started. Participants received a show-up fee of € 3. Depending on their choices participants earned on average € 4.80 (S.D. 0.50) in addition to the show-up fee. Although this might

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seem little as compared to the possible wealth older adults might dispose of, various authors (e.g. Camerer and Hogarth, 1999) show that within most economic games, the relative wealth of

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participants does not influence choice behaviour. Also, in accordance with Hogarth (2005), we want to argue that, although in the study of decision making there is a bias towards studying “important”

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decisions, the aggregate effects of many “small” decisions might be more important for the quality of our lives. We therefore are confident that a laboratory experiment with small amounts at stake is a valid approach to study age differences in decision making.

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Figure 1: Upper part: Timeline of the game with full feedback. Lower part: Screens presented to the participants. Left screen: choice between an ambiguous gamble and a sure amount of money. Right screen: choice between a risky gamble and a sure amount of money. In the original, colors were red and blue.

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3.1.3. Statistical methods and results

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In a first parametric analysis propensity to take risky gambles was measured by the number of times participants chose a gamble ranging from 0 (no gamble chosen at all) to 24 (always chosen the gamble instead of the sure payoff) instead of a sure amount of money in risk trials. Propensity to take

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ambiguous gambles was measured, mutatis mutandis, the same way in ambiguous trials. Descriptive statistics on the acceptance of gambles can be found in Figure 2 and Table 3a.

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Feedback

Table 3a: Age Effects Split by Feedback Decision Mean(S.E.) Mean(S.E.) Young Older Risk 15.706(0.785) 15.673(0.829) Ambiguity 14.196(0.918) 15.673(1.004) Risk 13.019(0.760) 13.688(1.031) Ambiguity 8.635(0.841) 14.333(1.128)

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Yes Yes No No

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Figure 2: Box-plots of risky and ambiguous gambles split by age group and feedback

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Hypotheses 1, 2 and 3:

A poisson regression was run to estimate the number of accepted gambles in the RAT based on the age

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group, uncertainty condition, and feedback condition and their respective interactions. Ambiguous conditions and older age interacting with the presence of feedback influence the propensity to gamble negatively (see Table 3b).

Feedback and older age interacting with the presence of ambiguity

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influence the propensity to gamble positively. Older age alone has only a marginal (yet positive) impact on the propensity to gamble and feedback does not interact with the uncertainty condition.

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Table 3b: Regression Results for the RAT Parameter Coefficient S.E. p Par. estimate 95% C.I. Ambiguity -.301 .090 < .01 .740 .620 - .883 Older age .140 .082 .089 1.150 .979 – 1.351 Feedback .270 .072 < .001 1.309 1.136 – 1.509 Older age * ambiguity .258 .094 < .01 1.294 1.076 – 1.557 Older age * feedback -.218 .095 < .05 .804 .667 - .969 Ambiguity * feedback .121 .097 .211 1.129 .934 – 1.364 Intercept 2.521 .058 < .001 12.444 11.105 – 13.943 N=200; Poisson regression with robust standard errors.

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ACCEPTED MANUSCRIPT In confirmation of the results of the Poisson regression, a nonlinear stochastic choice model (see Appendix 2 for a detailed description of the model) provides evidence that older age does not significantly influence the curvature of an assumed utility function, whereas the model shows that the utility function of participants becomes less concave in the presence of feedback (i.e. the willingness to take risks increases). Ambiguity preferences are influenced by age and the presence of feedback. It appears that feedback changes the weighting of probabilities, and that younger adults overestimate the probabilities in ambiguous decisions more strongly than older adults.

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To summarize, it appears that older age overall positively influences the propensity to accept gambles, that ambiguous gambles are chosen less often than risky gambles and that feedback increases the propensity to gamble. In addition, older adults are more prone to gamble under ambiguity, while younger adults react more strongly to feedback (which increases the propensity to gamble).

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3.2. Experiment 2

Experiment 2 is a follow-up study to experiment 1 and aims at testing hypothesis 4a and 4b. To do this, 100 of the participants of experiment 1 played an additional risk task based on statistical probabilities (thus feedback now allows participants to learn about probabilities of winning which are

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a priori not given). In addition, these participants answered a risk question which we could compare to data from a representative sample of the German population to test external validity of our results

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(Wagner, Frick, & Schupp, 2007), i.e. to test how closely the answers on this question in our sample match the answers given by a similar sample in the general population.

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3.2.1. Procedures

To measure decisions with statistical probabilities, participants made sequential choices on the

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Balloon Analogue Risk Task (description below). In order to assess self-assessment of risk preferences, participants also responded to the following item (same question as in the SOEP questionnaire (Wagner, Frick, & Schupp, 2007)): “On a scale from 0 to 10, how would you assess

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your willingness to take risks?” A value of 0 corresponded to not willing to take risks at all, and a value of 10 to high willingness to take risks. Depending on their choices on the experimental task, participants on average earned an additional € 4.75 (S.D. 0.46). The questionnaire question was not incentivized.

3.2.1.1. The balloon analogue risk task (BART): In the BART (Lejuez et al., 2002), participants see a series of 20 virtual balloons, one after another, and can earn money by pumping up the balloon. The money is stored on a temporary account until participants decide to collect the money (transfer it to a permanent account) and to proceed to the next

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ACCEPTED MANUSCRIPT balloon, or until the balloon pops and the money is lost. Participants do not receive detailed information on the probabilities that a balloon pops. Participants know that at some point, every balloon pops, and that the explosion could happen at any moment. It could pop at the first pump or when the balloon fills the entire screen (we implicitly assume that common-sense would tell participants that the probability that a balloon pops after the first pump is quite low, and is increasing the larger the balloon gets). When a balloon is filled beyond its individual exploding point, it pops on the screen. Whenever a balloon pops, the money of the temporary account is lost. Then, the next

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empty balloon appears on the screen. At every moment during a trial, the participant can interrupt the pumping and press a “collect money” button. The pressing of this button transfers the money from the temporary account to a permanent account, where it cannot be lost.

After each money transfer or explosion, the trial ends, followed by a new balloon that appears on the screen until the participant has seen a total of 20 balloons. The chance that a balloon pops is randomly

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generated, with a starting probability of P = 1/64 (0.0156). If the balloon does not pop at the first pump, the probability that it pops at the second pump is P = 1/63, P = 1/62 at the third pump, and so on until the 64th pump, where the probability will be P = 1/1. Every pump increases the money on the temporary account that can be lost by popping, and diminishes the relative additional earnings by pumping. After the first pump, participants risk only 5 cents on the temporary account by an additional pump but can increase their potential earnings by 100% by executing this pump. In contrast, after the

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30th pump, participants risk the 1.5 ECU on the temporary account for an additional increase in

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earnings of only 3.3%.

Note that optimal behaviour that maximizes payoffs in expected terms is to pump 32 times. In general, the earlier a participant stops, the more risk averse she or he is.

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3.2.2. Statistical methods and results

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To investigate experience and age effects on the BART, we applied a fully identified repeatedmeasures general linear model, with number of pumps as dependent variable, trial number as withinsubjects factor, and age as between-subjects factor. Analysing the within-subject factor trial number, a

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significant increase in participants’ readiness to take risks over the course of the experiment can be observed (F(13.239) = 5.575; p = .001), independently of age group2. Young adults are found to be more risk-seeking (F(1) = 7.346; p = .008), but the change in risk taking over the course of the experiment is not influenced by age group (trial*age group: F(13.239) = 0.867; p = .590), thus disconfirming H4b. To 2

For the repeated-measures general linear model, the Mauchly test for sphericity revealed a violation of the sphericity-assumption (Mauchly’s W = .015; df = 189; approximate χ² = 385.685; p < .001). Therefore, Greenhouse-Geisser-corrected results for the repeated-measures analysis will be reported (the corrected results, however, do not reveal any fundamental differences compared to conventional, uncorrected results). Although results are visualized with logarithmic fitting functions, the statistics reported are on a linear model as a logarithmic transformation of the data does not reveal qualitative differences in the analyses.

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ACCEPTED MANUSCRIPT visualize our results, we applied a logarithmic fitting function to the behavioural measures of each age

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group (Figure 4).

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Figure 4: Average behaviour of young and older adults on the BART. Error bars represent 95% confidence intervals. Thin lines are observed behaviour, thick lines correspond to logarithmic fitting functions (young = 14.451 + 1.858 × ln(x); older = 13.491 + 0.649 × ln(x)). When taking a modelling approach to estimate each individuals’ choice function, in which a participants’ sensitivity to gains and her initial estimate of the probability of winning are central to decision making (see Appendix 3 for a detailed description of the model), it appears that there are no

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age differences in the sensitivity to gains, in the initial estimate of the probability of winning and in the variance of this estimate (or participants’ “optimism”). Yet older participants exhibit a larger

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response sensitivity. This means that the probability of pumping after each additional pump is higher in young adults and hence that young adults take more risk on average. When combining these results

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with the outcomes of the RAT, we can show that the initial estimate of the probability of winning can be predicted by behaviour on the RAT, both under risk and under ambiguity (with risk and ambiguity behaviour correlating in opposite direction) and that the measure of sensitivity to gains is predicted by

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risk behaviour alone (Table 7 in Appendix 3). Unlike with the RAT, but as hypothesized based on the literature, young adults are more risk-seeking

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compared to older adults (H4a). Although we confirm the underlying assumption of H4b that the BART measures both components of risk and of ambiguity, when controlling for behaviour on the RAT, no significant age differences remain. Hypotheses H4b (increase of distance between young and older adults over trials) and H5 (stronger effect for older adults when comparing statistical to a priori probabilities) thus need to be rejected.

3.2.3. Representativeness of the sample of participants To get an idea about the external validity of our results and to help us understanding if (but not how so) the selection of a highly educated sample of participants can be a driving force behind the observed

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ACCEPTED MANUSCRIPT effects, we compare the results of the (non-incentivized) measure of risk preferences observed in our sample to a representative sample of the German population (from 2009 - we are aware that selfreported scales of risk preferences only correlate weakly with behaviour on tasks assessing risk (Frey et al., 2017), but the intention of this section is rather to provide a better understanding of the participant sample). The German Socio-Economic Panel (GSOEP; Wagner, Frick, & Schupp, 2007) contains different questions related to risk attitudes. The main measure assesses “willingness to take risks, in general” (called henceforth general risk preference). Roughly 13,000 persons answered the

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item relevant to our analysis. We use a sample that is spread over a wide age range (18-88 years of age), with a mean age of 53.57 years (SD 16.05) and compare it to the part of our sample that answered the same non-incentivized self-report measure of risk preferences. In our sample, despite significant differences in risk behaviour between young and older adults, no significant difference was found in the self-assessment of risk attitudes. Young adults on average reported a value of 4.98 (SD: 0.239) and older adults on average were at 5.49 (SD: 0.225), hence also not differing significantly (t(98)

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= 1.534, p = .128).

Given that our sample consisted of students and highly educated older adults, we intended to provide a rough estimate of how much of our results are driven by education. To control for the effects of education on the relationship between self-reported risk preference and age, we provide evidence from the SOEP by bootstrapping 1000 random samples of young and older adults (each N = 200; 100 older

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adults) with a minimum of 13 years of education. Young adults were younger than 30 years (mean: 25.60, SD: 1.365), older adults were between 58 and 88 years of age (mean: 67.89, SD: 4.957). It

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appears that, as in our experimental group, young and older adults do not differ significantly on the self-assessment of their own general risk attitude (t(198) = 1.296, p = .196). On a scale from 0 to 10,

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young adults reported an average of 4.45 (SD: 2.045) for the willingness to take risks, and older adults reported a mean value of 4.09 (SD: 2.088). Thus, while older and younger adults on average do not differ between our study and the general population (all p > .1) it is noteworthy that standard

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deviations are an order of magnitude higher in the sample of the GSOEP, i.e. preferences in our study were much more homogenous than in a highly educated random sample in the general population. In a

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second step, we used a linear regression model with age and gender as predictors of risk preferences. We also introduced years of education as additional predictor into a second regression model. Examination of the results reveals that both age (β = -.172, S.E. = .001, p < .001) and being female (β = -.212, S.E. = .038, p < .001) are negatively related to the willingness to take risks in the sample from the SOEP. After introducing educational level into the regression, the effects of age (β = -.170, S.E. = .001, p < .001) and gender (β = -.208, S.E. = .038, p < .001) still hold. Education appears to also have a positive effect on risk taking: the higher the level of education, the more likely participants consider themselves to take risks (β = .059, S.E. = .005, p < .001). The non-significant age-difference in self-

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ACCEPTED MANUSCRIPT assessed risk preference on the SOEP question in our experiment might be (at least partly) explained by the high educational background of our participants.

4. Discussion The aim of this study was to gain a better understanding of older adults’ decision making in situations of financial uncertainty and to investigate age differences in the propensity to gamble in uncertain

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situations. Based on findings in the literature, we hypothesized that older adults do have a different propensity to gamble than young adults in risky (H1) and ambiguous (H2) conditions. We also expected that providing feedback on choices with a priori probabilities should result in behaviour indicative of a smaller distance between subjective and objective probabilities in both age groups. We thus expected gambling behaviour to be more frequent and an estimated probability weighting

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function to be closer to the actual probabilities when feedback is provided (H3). We also hypothesized that feedback will c.p. increase older participants´ probability to gamble (H3a) under risk and under ambiguity. Concurrently, we hypothesized that older participants c.p. do react less to feedback than young participants, thus demonstrating a smaller difference in behaviour between feedback and nonfeedback treatments (H3b). We finally hypothesized that whether probabilities are given or have to be learned has an influence on age differences. We expected older participants to take less risk than

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young participants on a task with statistical probabilities (H4a) and an increase in the difference in risk taking between young and older participants over time (H4b). Comparing both kinds of risk tasks, we

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expected that age has a larger effect on behaviour on a task where learning is required. Within our group of elderly we hypothesized a difference in risk taking between the two risk tasks which should

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be less pronounced for the young participants (H5). We provide support for most, but not all of our hypotheses. Older adults appear to be equally willing

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to take risks as young adults when considering a priori probabilities (i.e., as tested in the RAT task in experiment 1, disconfirming H1). Ambiguity behaviour differs with age (confirming H2). Feedback increases the propensity to gamble, both in decisions under risk and in decisions under ambiguity,

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however, the effect is strongest in young adults (confirming H3b). When making decisions based on statistical probabilities in the BART task, older adults gamble less than young adults (confirming H4a). If learning is required, age does not appear to have a larger effect on behaviour than in a task where learning is not required (rejecting H5). Regarding H1, we did not find any significant age differences in the average number of risk-taking responses on the RAT. When estimating a hypothesized utility function, it also appeared that age did not influence the curvature of the utility function. This indicates that, when dealing with decisions under risk where probabilities are explicitly known, older adults behave similarly to young adults

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ACCEPTED MANUSCRIPT (when decisions are made in the gain domain; Cooper, Blanco & Maddox 2016). Results are different when it comes to ambiguous choices (H2): our findings in decision-making behaviour under conditions of ambiguity showed that older adults gamble more than young adults in ambiguous conditions, without feedback, and that older adults tend to estimate the probability of gaining more realistically than young adults. Studies that have assessed cognitive processes and strategies in the elderly might provide explanations for the observed behaviour. The most likely factor that could have influenced the results is experience. Older adults have had a lifetime to decide and develop strategies

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for decisions under ambiguity. Hence, their schemas or memory traces are more developed than those of young adults. One survey of bank managers, for example, revealed that older managers’ business decisions were more aggressive than the decisions of younger managers (Brouthers et al. 2000), and various studies found that uncertain investments increased until a particular age (Riley Jr & Chow 1992; Schooley & Worden 1999; Jianakoplos & Bernasek 1998). An alternative explanation for the age difference in the propensity to gamble under ambiguity is given by Mata et al. (2007). In their

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study, they found a difference in strategies used by young and older adults to make a decision: older adults looked up less information and took more time to process it. If we apply this to the fact that ambiguity is a condition with less information available than risk, we can hypothesize that ambiguous gambles are easier to process for older adults.

A novelty of this study was the investigation of the effects of feedback on uncertain choices (H3).

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Feedback had a salient effect on the propensity to gamble, both by decreasing the level of risk aversion (i.e. increasing the coefficient of curvature of the utility function) and by diminishing the weighting of

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ambiguous probabilities. Young adults gamble significantly more in the presence of feedback than without it. The main reason for this effect lies in the fact that decisions under uncertainty strongly rely

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on subjective probabilities. Feedback is an important factor in reducing subjective uncertainty (Hogg & Mullin 1999), which in our experiment led participants to gamble more than in the absence of feedback. By design, in our RAT experiment, feedback could not have been used to learn about the

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probabilities of winning (all trials had to be considered separately because each trial is an independent event). Two hypotheses arise therefore. First, the greater experience of older participants might let

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them estimate probabilities more accurately than young adults, which would lead to a lower reactivity to feedback in terms of subjective probabilities. Second, older adults might simply not (or to a lower extent) update their beliefs about subjective probabilities. The results of the first experiment provide support for the first hypothesis, but not for the second. Several factors might play a role explaining the age differences found in statistical but not in a priori probabilities (H4). The most salient factor certainly is the different learning requirements of the tasks. Learning requirements are absent for the RAT. In contrast, performance on the BART relies on the participant’s ability to learn the probabilities of outcomes from experience. When learning from experience is required to approach optimal (i.e. payoff-maximizing, risk neutral) behaviour, older

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ACCEPTED MANUSCRIPT adults appear to be more risk averse than young adults in a limited set of trials. One reason for the observed age difference could be that learning rates of young adults are generally steeper than those of older adults (Merriam et al. 2007), and thus young adults approach optimal behaviour faster than older adults. The results from our sample did not support this reasoning. A non-significant interaction effect between age group and the propensity to gamble over trials indicated that the rate of learning did not significantly differ between young and older adults. We observed that in both age groups the propensity to gamble increases over the trials with a similar pace, and that the response sensitivity is

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the factor that differentiates young from older adults: young adults are more willing to perform an additional pump. At the same time, both age groups were risk averse. Risk-taking behaviour, on average, stays below the risk neutral behaviour of 32 pumps per balloon over 20 trials. The difference between young and older adults mainly lies in the starting values. Older adults begin the game with a lower average number of pumps than their younger counterparts, and then subsequently increase the number of pumps at a pace that is not significantly different from young adults. These findings are

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consistent with evidence that older adults are more cautious than young adults (Rolison, Hanoch & Wood 2012; Rolison, Wood & Hanoch 2017; Starns & Ratcliff 2010) and that older adults only display differences in learning performance when learning demands are high, which arguably is not the case in the BART (Rolison, Hanoch & Wood 2012). It also is of importance to note that we observe correlations in behaviour between the RAT and BART. Although recent evidence suggests

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that the link between behavioural tasks is rather weak (Pedroni et al., 2017), these correlations provide us with some confidence that the same concepts of risk and ambiguity were captured across both tasks.

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The current study also has its limitations. We selected the older participants in a way that eased comparability with our young participants: All older participants were healthy, highly educated and

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practised a cognitively active lifestyle. Overall, older adults are very heterogeneous in their cognitive abilities, and activity might preserve cognitive ability with aging. The analysis of the response to the self-assessment question has shown that, even when selecting only highly educated participants, the

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responses on the question were more homogenous in our sample than in the general population. We have shown that whereas age has a negative impact on the willingness to take risks in the population,

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the level of education of participants has a positive impact. Although it is difficult to conclude that the non-representativeness of our sample has an effect on the results, we still think that there is a possibility that we reduced some of the effects claimed in other studies to be attributable to age by selecting highly educated participants. This does not mean that we are not dealing with large selection effects in the current study, but we refer the reader for a more detailed analysis of the effects of education, health, and other factors influencing the willingness to take risks, to other authors (Dohmen et al. 2011), as this was not the aim of our study. Nevertheless, future experiments should focus on age differences in more heterogeneous participant groups on both sides of the age range (Henrich et al. 2010).

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ACCEPTED MANUSCRIPT In this experiment, we wilfully did not try to rule out cohort effects. All western populations are rapidly growing older, so that there is an immediate need to describe older adults’ behaviour. Certainly it also is of great importance to understand whether and how cohorts differ (see e.g. Malmendier & Nagel, 2009), but our first aim was to describe the differences between young and older adults, paving the way for future work where one should try to minimize the impact of cohort effects. In the same line, how behaviour develops over the life-course is an interesting topic to extend the current work. With evidence that risky decision making follows an inverse-U shaped function over the life course,

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studies spanning more age segments can provide additional insights missed by the current paper.

5. Conclusion

In conclusion, we observed that older adults’ decision-making behaviour differs from that of younger adults. In risky conditions, older adults behave like young adults if a priori probabilities apply, but in risky conditions were statistical probabilities are part of the decision and in ambiguous conditions, age

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differences appear. Young adults are more ambiguity averse if no feedback is provided, and older adults exhibit a higher risk aversion when it comes to statistical probabilities. Despite several possible explanations for our results, more work will be needed to fully understand the causes of our findings. On a more practical level, our work contributes to growing evidence that older adults’ decision making differs from that of younger adults. In our societies, older adults represent a growing part of the

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population, and a part of the population that will work until a higher age, thus also making financial decisions at a higher age. Understanding how decision making is affected by age becomes crucial for

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numerous situations of everyday life, and further research is needed to understand how older adults make their decisions, to help employers, policy makers, financial institutions, but also older adults

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themselves, to cope with the effects of the demographic change. 6. Acknowledgements

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This research has been made possible by a FRONTIER-Fonds grant from the University of Heidelberg to the authors. The authors are much obliged to Daniela Jopp and Rui Mata for their helpful

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comments.

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ACCEPTED MANUSCRIPT Appendix 1: Instructions of the RAT and list of choices Appendix 1: Instuctions: In the current game, you are requested to resolve decision problems. You are getting paid for two of

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the decisions chosen at random. Your possible gain is presented to you on the screen in experimental currency units (ECU). 10 ECU correspond to 1€ - at the end of the experiment, the ECU will be converted into euro. If you have won for example 70 ECU, your payment would be 7€.

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In the current game, card decks will be presented to you. These card decks always contain some

amount of red and of blue cards. In some situations, the exact amount of red and blue cards in the deck is revealed to you, in other situations, you only will know the total amount of cards in the deck.

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In the game, you have always the choice either to bet that a certain colour will be drawn out of the deck, or to receive a sure amount of money. If you bet, you will receive the indicated amount of ECU

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if the colour you chose is drawn; otherwise you don't receive anything. If you choose the certain amount, you will get the amount automatically, without any influence of the drawn card colour on

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your gain.

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Prior to the game, you will have the possibility to get familiar with the task on two examples.

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The amount which you will be paid within each trial will be presented to you on the computer screen during the experiment.

If you have any questions, please raise your hand. One of the experimenters will come to your table and answer your question privately. Choices on the RAT

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ACCEPTED MANUSCRIPT Nr. Red Blue Total ECU bet ECU certain 1 2 18 20 12 8 2 18 12 30 16 12 3 9 1 10 19 12 4 3 9 12 16 13 5 3 12 15 20 10 6 21 9 30 20 15 7 8 32 40 16 10 8 5 10 15 12 8 9 6 14 20 18 12 10 16 24 40 16 8 11 4 36 40 20 14 12 18 9 27 15 10 13 26 13 39 12 8 14 10 20 30 12 10 15 3 2 5 18 12 16 27 3 30 14 11 17 12 28 40 16 14 18 4 1 5 14 8 19 8 12 20 18 9 20 7 21 28 12 8 21 24 8 32 19 13 22 8 2 10 16 10 23 7 3 10 13 10 24 15 5 20 12 6 Ambiguity 1 20 20 10 2 10 16 7 3 40 20 12 4 30 13 7 5 40 17 7 6 30 18 6 7 20 20 11 8 30 12 7 9 15 20 12 10 30 20 12 11 12 19 6 12 15 16 7 13 40 14 8 14 28 18 8 15 40 20 11 16 10 16 9 17 32 19 9 18 39 16 6 19 10 20 11 20 27 20 8 21 5 17 8 22 20 12 5 23 5 18 11 24 20 16 6 Red = number of red cards in the deck, Blue = number of blue cards in the deck, Total = total amount of cards in the deck, ECU bet = amount to win when betting, ECU certain = amount to earn with the sure option

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Condition Risk

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Appendix 2 – Instructions of the BART (text translated from the instruction n screens) BART: text on the instructions screen Now you're going to see 20 balloons, one after another, on the screen. For each balloon, you will use the mouse to click on the button that will pump up the balloon. Each click on the mouse pumps the

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balloon up a little more. BUT remember, balloons pop if you pump them up too much. It is up to you to decide how much to pump up each balloon. Some of these balloons might pop after just one pump. Others might not pop until they fill the whole screen.

You get MONEY for every pump. Each pump earns 0.05 ECU. But if a balloon pops, you lose the and click on the button labeled "Collect €€€".

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money you earned on that balloon. To keep the money from a balloon, stop pumping before it pops

After each time you collect €€€ or pop a balloon, a new balloon will appear. At the end of the experiment, you will be paid the amount earned on the game.

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Click the left mouse button to see the summary.

BART: text on the summary screen

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You make 0.05 ECU for each pump.

You save the money from a balloon when you click "Collect €€€".

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You lose money from a balloon when it pops.

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There are 20 balloons.

You will be paid the exact amount you earned on the game. Now, do you have any questions? Click the left mouse button to begin.

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ACCEPTED MANUSCRIPT Appendix 2: Modelling Approach for the RAT In accordance with Hsu et al. (2005), a parametric analysis to estimate coefficients of risk and ambiguity preferences on the RAT was conducted via a nonlinear stochastic choice model, combining data from all treatments. Participants’ utility functions for money are assumed to follow a power function u(x,α) = xα which is conveniently characterized by one parameter and widely used in empirical estimations of this sort. Participants are assumed to weight probabilities according to pγ 1

γ [pγ +(1-p) ]γ

(Kahneman and Tversky 1979).

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Prospect-theory’s probability weighting function w(p,γ)=

The α parameter of the utility function is interpreted as the risk aversion coefficient, i.e., the curvature of the utility function. The γ parameter is interpreted as the ambiguity aversion coefficient, i.e. how much do people underestimate ambiguous probabilities (Hsu et al., 2005). We assume that people combine these weighted probabilities and utilities linearly, so that their weighted subjective expected utility is U(p,x,α,γ) = w(p,γ)u(x,α). The tasks are binary choices in which participants either choose a

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gamble to win x (with probability p) or 0, or a certain payoff c. For the risky decks, the probabilities are the ratios of the cards. For the ambiguous decks, we assume p = 0.5. Based on Hsu et al. (2005), we constrain γ = 1 in all risky conditions and estimate γ from behavioural data in the ambiguity conditions. A small γ (< 1) indicates that small probabilities are overestimated whereas large probabilities are underestimated, and vice-versa for a large (> 1) value of γ. Including both factors α

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and γ in the model for ambiguity reflects the fact that choices between ambiguous gambles and sure outcomes are always influenced both by attitude towards ambiguity and by risk attitude. The

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probability that the participant chooses the gamble rather than the sure amount of money c is given by 1

𝑈(𝑝,𝑥,𝛼,𝛾)𝜇

the formula 𝑃(𝑝, 𝑥, 𝑐, 𝛼, 𝛾) =

1

1

, where µ is the amount of stochastic errors, or the

𝑈(𝑝,𝑥,𝛼,𝛾)𝜇 +𝑢(𝑐,𝛼)𝜇

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amount of “randomness”, in participants’ choices. Besides this baseline model, we introduce in a second model age group and the presence or absence of feedback as predictor variables for α and γ,

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and age group as predictor for the occurrence of stochastic errors. In the model without controlling for age and feedback variables, it appears that our participants tend to

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exhibit a concave utility function with α = .738 (S.E.: .044) and to underestimate ambiguous probabilities with γ = .734 (S.E.: .040). In the second model, when controlling for age groups and the presence or absence of feedback (cf. Table 4), it appears that feedback has a significant effect on α, increasing the willingness to take risks (i.e. rendering u(x,α) less concave). Older age, on the other hand, does not significantly influence the curvature of the utility function. Ambiguity preferences are influenced by age and the presence of feedback. It appears that feedback changes the weighting of probabilities, and that younger adults overestimate the probabilities in ambiguous decisions more

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ACCEPTED MANUSCRIPT strongly than older adults. It also appears that older adults make more stochastic errors than young adults3.

p

Constant Old no feedback

1.217 -0.194 -0.586

0.100 0.238 0.121

< .001 .414 < .001

Constant Old no feedback

0.446 0.267 1.601

0.022 0.146 0.198

< .001 .068 < .001

γ

μ

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Constant 0.054 0.010 <.001 Old 0.179 0.073 .014 Nbr. of observations: 9600; S.E. adjusted for 200 participants

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Parameter α

Table 4: Utility coefficients Predictor Coeff. S.E.

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When excluding the 57 participants (26 young adults and 31 older adults) that chose inconsistently at least once, the results remain qualitatively similar, yet the significant age effect on µ disappears (p = .351).

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ACCEPTED MANUSCRIPT Appendix 3: Modelling Approach for the BART Wallsten, Pleskac and Lejuez (2005) have evaluated different decision making models applied to the BART. According to these authors, in the best-fitting model, prior to beginning with each balloon, participants engage in a prospect-theory-like evaluation of the expected outcomes to determine an optimal number of pumps. Their probability of pumping then decreases within each balloon such that it equals 0.5 at the optimal stopping point. Participants also incorrectly treat the conditional probability of explosion as constant rather than increasing over pumps. This leads to the assumption that

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participants treat an event as binomial distribution with ph being the subjective probability that the balloon will not explode at pump number h. The prior probability 𝑝 ̂0 can then be represented as a beta distribution (with parameters a0 – estimated size of p prior to observing any data – and m0 – certainty about the value of p – with m0 > a0 > 0) where E(𝑝 ̂) 0 =

𝑎0 , 𝑚0

the initial estimate of the probability may be

interpreted as a measure of optimism that the balloon will not explode and where var(𝑝 ̂) 0 refers to the

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variance in the participants’ initial estimate. High variance subjects will be more sensitive to new data from the balloon, while low variance participants will be less sensitive. It is assumed that a decision maker, instead of sequentially evaluating each pump, may determine an optimal number of pumps prior to each balloon (optimal in the sense that the number of pumps optimizes the perceived prospecttheoretic gains). The reference point is set to the holdings of the temporary account (0) and therefore

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all evaluations are made in terms of gains. The expected gain for i pumps on balloon h is E h(i) = πh,i(ix)γ, with πh,i being the probability of pumping balloon h i times successively without exploding (x is the amount gained by increasing the balloon; in our case 5₵). The number of pumps that maximize

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this equation depends on how πh,i is expressed. Assuming that participants use Bayes’ rule to update beliefs about probabilities with experience, we can write the participants’ estimate of p following

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experience with h balloons as 𝑝ℎ =

𝑎0 + ∑ℎ−1 (𝑚ℎ′ −𝑑ℎ′ ) ℎ′ =1 𝑚0 +∑ℎ−1 𝑚 ℎ′ =1 ℎ′

(with dh’ taking the value of 1 if the balloon

exploded, and 0 else). Therefore, we can assume, given that the probability of the balloons not

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exploding is assumed to be stationary over pumps with an estimated value of ph for all i, that πh,i = 𝑝ℎ𝑖 . When denoting the optimizing number of pumps as gh, this results in an optimization of Eh(i) when −𝛾 . ln(𝑝ℎ )

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The response model uses a logistic function constructed in a way that the probability of a

participant taking the ith pump on the hth balloon, rh,i strictly decreases with each pump (it equals 0.5 when i = gh): 𝑟ℎ,𝑖 =

1 1+𝑒

𝛽𝛿ℎ,𝑖

with 𝛿ℎ,𝑖 = 𝑖 − 𝑔ℎ . β measures the sensitivity of a participant to her prior

evaluation and remains constant with experience. The functions (more specifically the γ, β, E(𝑝 ̂) 0 and 𝑣𝑎𝑟(𝑝 ̂) 0 parameters) are estimated individually for each participant using the maximum likelihood method and compared on the group level to make inferences about age differences. A more comprehensive description of the model can be found in Wallsten et al. (2005). Table 6 displays the results:

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ACCEPTED MANUSCRIPT Table 6: Model estimates of the BART Mean (S.E.) young Mean (S.E.) older t(df) p β .355 (.035) .521 (.067) -2.180(71.462) .033 γ .806 (.053) .677 (.062) 1.599(98) .113 .967 (.004) .964 (.005) .469(98) .640 E(𝑝 ̂) 0 3 .411 (.125) .400 (.122) .063 .950 𝑣𝑎𝑟(𝑝 ̂) ∗ 10 (98) 0 t-tests for independent samples, N=100. Degrees of freedom (df) adjusted for unequal variances in β.

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It appears that, although there are no age differences in the γ parameter or in the initial estimate of the probability and in the variance of this estimate (or participants’ “optimism”), older participants exhibit a higher value of the β parameter, thus a larger response sensitivity. This means that the probability of pumping after each additional pump is higher in young adults and hence that young adults take more risk on average. A multivariate regression analysis further reveals that β can be predicted by behaviour on the RAT, both under risk and under ambiguity (with risk and ambiguity behaviour correlating in

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opposite direction) and that γ, the measure of diminishing sensitivity to gains, is predicted by risk behaviour alone (Table 7). E(𝑝 ̂) ̂) 0 and 𝑣𝑎𝑟(𝑝 0 on the other hand do not correlate with behaviour on the RAT.

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Table 7: Regression results β γ Model 1 Model 2 Model 1 Model 2 Risk -.013 (.007)* -.020(.009)* .019(.007)** .023(.010)* Ambiguity .013 (.006)* .018(.008)* -.007(.006) -.006(.009) Age group .103 (.080) .184(.186) -.101(.086) -.246(.202) Age * risk -.014(.013) .009(.014) Age * ambiguity .009(.012) .003(.013) Constant .513 (.106)*** .535(.125)*** .517(.114)*** .461(.136)*** R² .102 .114 .096 .102 Multivariate linear regression. N = 100. Standard errors in parentheses. Regression coefficients for E(𝑝 ̂) ̂) 0 and 𝑣𝑎𝑟(𝑝 0 are not reported to increase legibility. * p < .05, ** p < .01, *** p < .001

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