Risk and Rationality: Uncovering Heterogeneity in Probability Distortion

Transcription

1 Risk and Rationality: Uncovering Heterogeneity in Probability Distortion Adrian Bruhin Helga Fehr-Duda Thomas Eer February 17, 2010 Abstract It has long been recognized that there is considerable heterogeneity in individual risk taking behavior but little is known about the distribution of risk taking tyes. We resent a arsimonious characterization of risk taking behavior by estimating a finite mixture model for three different exerimental data sets, two Swiss and one Chinese, over a large number of real gains and losses. We find two major tyes of individuals: In all three data sets, the choices of roughly 80% of the subjects exhibit significant deviations from linear robability weighting of varying strength, consistent with rosect theory. 20% of the subjects weight robabilities near linearly and behave essentially as exected value maximizers. Moreover, individuals are cleanly assigned to one tye with robabilities close to unity. The reliability and robustness of our classification suggest using a mix of reference theories in alied economic modeling. KEYWORDS: Individual Risk Taking Behavior, Latent Heterogeneity, Finite Mixture Models, Prosect Theory JEL CLASSIFICATION: D81, C49 AUTHORS AFFILIATIONS: Adrian Bruhin, University of Zurich, Switzerland, bruhin@iew.uzh.ch Helga Fehr-Duda, ETH Zurich, Switzerland, fehr@econ.gess.ethz.ch Thomas Eer, ETH Zurich, Switzerland, eer@econ.gess.ethz.ch

2 1 Introduction Risk is a ubiquitous feature of social and economic life. Many of our everyday choices, and often the most imortant ones, such as what trade to learn and where to live, involve risky consequences. While it has long been recognized that individuals differ in their risk taking attitudes, comaratively little is known about the distribution of risk references in the oulation. 1 Since references are one of the ultimate drivers of behavior, knowledge of the comosition of risk attitudes is aramount to redicting economic behavior. Economic models often allow for heterogeneity, but this heterogeneity is usually defined by the boundaries of the standard model of references, exected utility theory (EUT ). The emirical evidence, however, reveals that heterogeneity in risk taking behavior is of a substantive kind, i.e. some eole evaluate risky rosects consistently with EUT, whereas other eole deart substantially from exected utility maximization (Hey and Orme, 1994). Moreover, it seems to be the case that rational decision makers, revealing EUT references, constitute only a minority of the oulation (Lattimore, Baker, and Witte, 1992). To imrove descritive erformance a lethora of alternative theories have been develoed. Unfortunately, no single best fitting model has been identified so far (Harless and Camerer, 1994; Starmer, 2000) and, deending on the individual, one or the other model fits better. This finding oses a serious roblem for alied economics. What the modeler needs is a arsimonious reresentation of risk references that is emirically well grounded and robust, and not a host of different functionals. Providing such a arsimonious characterization of heterogeneity in risk taking behavior is the objective of this aer. Our method is based on a literature on classifying individuals which has been 1 Excetions include Dohmen, Falk, Huffman, Sunde, Schu, and Wagner (2005); Eckel, Johnson, and Montmarquette (2005); Harrison, Lau, Rutström, and Sullivan (2005); Harrison, Lau, and Rutström (2007). 1

3 recently adoted by the social sciences. On the basis of statistical classification rocedures, such as finite mixture models, investigators have tried to discover which decision rules eole actually aly when laying games or dealing with comlex decision situations (El-Gamal and Grether, 1995; Stahl and Wilson, 1995; Houser, Keane, and McCabe, 2004; Houser and Winter, 2004). The finite mixture aroach does not require fitting a model for each individual, which is - given the usual quality of choice data - frequently imossible and often not desirable in the first lace. Instead, our method reveals latent heterogeneity by estimating the roortions of distinct behavioral tyes in the oulation and assigning each individual to one endogenously defined behavioral tye, characterized by a unique set of arameter values. We aly such a finite mixture model to choice data from three different exeriments, two of which were conducted in Zurich, Switzerland. The third exeriment took lace in Beijing, Peole s Reublic of China. We analyze 448 subjects decisions over real monetary gains and losses, which comrise a total of nearly 18,000 observations. All three exeriments were designed in a similar manner and served to elicit certainty equivalents for binary lotteries. Using a flexible sign-deendent functional as basic behavioral model, we show the following main results. First, the estimation rocedure renders a robust classification of risk taking behavior across all three data sets. Moreover, the roortions of these distinct tyes in their resective oulations are very similar. Second, almost all the exerimental subjects are unambiguously assigned to one distinct tye. Measuring the quality of classification by the Normalized Entroy Criterion (Celeux and Soromenho, 1996), ambiguity of assignments turns out to be extremely low. Thus, we observe hardly any mixed tyes, i.e. individuals with a high robability (of say 0.4) of being one tye and a high robability (of say 0.6) of being another tye. This clean segregation suggests that the classification rocedure is able to cature the distinctive characteristics 2

4 of each behavioral tye. Third, without restricting arameter values a riori, we find that, in all three data sets the minority tye, who constitutes about 20% of the oulation, weights robabilities and values monetary outcomes near linearly. Consequently, this grou of individuals can essentially be characterized as exected value maximizers. This result is articularly interesting in the light of Rabin s calibration theorem (Rabin, 2000), which shows that exected utility maximizers should be aroximately risk neutral for small stakes, tyically encountered in laboratory exeriments, if behavior under high stakes is to remain within a lausible range of risk aversion. Therefore, we label subjects belonging to this grou of nearly risk neutral eole as EUT tyes. Moreover, the EUT grou remains robust to increasing the number of tyes in the mixture. Fourth, the majority of individuals, labeled CPT tyes, are characterized by significant deartures from linear robability weighting, consistent with rosect theory. As three-grou classifications show, this grou s behavior can be characterized as a mixture of two different tyes: In all three data sets a roortion of aroximately 30% of the subjects dislay ronounced deartures from linear robability weighting, whereas the relative majority of 50% differ less radically from linear robability weighting. Finally, within the class of CPT tyes, we find major differences between Swiss and Chinese behavior. Sensitivity to changes in robabilities is generally lower for the Chinese subjects than for the Swiss. While the majority CPT grous robability weighting curves do not differ dramatically between countries, the minority grous dislay diametrically oosed atterns of robability weighting. In articular, the minority Chinese CPT grou weight robabilities extremely favorably, rendering them risk seeking over a considerable range of robabilities. The minority Swiss CPT grou, however, is characterized by the oosite behavior. Thus, our analysis rovides a deeer understanding for the finding that, on average, the Chinese tend to be more risk seeking than 3

5 Westerners (Kachelmeier and Shehata, 1992). Our results show that the classification rocedure successfully uncovers latent heterogeneity in the oulation. If there is heterogeneity of a substantive kind, as the data suggest, basing redictions on a single reference theory is inaroriate and may lead to biased results (Wilcox, 2006). EUT references should be taken account of alongside rosect theory references, even if rational EUT individuals constitute only a minority in the oulation. As the literature on the role of bounded rationality under strategic comlementarity and substitutability shows, the mix of rational and irrational actors may be decisive for aggregate outcomes (Haltiwanger and Waldman, 1985, 1989; Fehr and Tyran, 2005; Camerer and Fehr, 2006; Fehr and Tyran, 2008). Deending on the nature of strategic interdeendence the behavior of even a minority of layers may drive the aggregate outcome. Therefore, the mix of tyes in the oulation is a crucial variable in redicting market outcomes. Since the finite mixture model rovides a robust and reliable classification of individuals, the resulting estimates of grou sizes and grou-secific arameters may serve as valuable inuts for alied economics. The finite mixture method has been used by others in the context of modeling risk taking. However, to the best of our knowledge, there is no revious study showing a nearly identical classification of risk reference tyes for three indeendent data sets. Additionally, our analysis breaks new ground by showing that EUT tyes emerge endogenously and by extending classification to three grous. Related work by Harrison and colleagues (Andersen, Harrison, and Rutström, 2006; Harrison, Humhrey, and Verschoor, 2009; Harrison and Rutström, 2009) alies finite mixture models as well, but differs from our aroach. Their estimation rocedure is based on the a riori assumtion that choices, irresective by whom they were taken, are either EUT consistent or CPT consistent, i.e. it sorts choices by re-defined decision model. In contrast, we aim at classifying individuals by endogenously defined tye. Therefore, if 4

6 there is a grou of eole whose behavior can best be described by EUT they should get identified by the classification rocedure. Furthermore, in certain decision situations choices of EUT individuals and CPT individuals do not differ substantially from one other and, therefore, both decision models fit equally well there. Consequently, deending on the data available, classification by EUT - and CPT-consistent decisions may differ markedly from classification by decision makers tyes. A recent study by Conte, Hey, and Moffat (2009) is also dedicated to finite mixture modeling of risk taking behavior. Their results for British subjects corroborate our conclusions: Even though their work differs from ours in set of lotteries, elicitation method and estimation rocedure and restricts one behavioral tye to be EUT a riori they also find that, in the domain of gains, 80% of the individuals exhibit nonlinear robability weighting whereas 20% are assigned to EUT. The aer is structured as follows. Section 2 describes the exerimental design and rocedures of the three exeriments. The functional secification of the behavioral model and the finite mixture model are discussed in Section 3. Section 4 resents descritive statistics of the data and the results of the classification rocedure. Section 5 concludes. 2 Exerimental Design In the following section we describe the exerimental setu and rocedures. The exeriments took lace in Zurich in 2003 and 2006 as well as in Beijing in In Zurich, all subjects were recruited from the subject ool of the Institute for Emirical Research in Economics, which consists of students of all fields of the University of Zurich and the Swiss Federal Institute of Technology Zurich. In Beijing, subjects were recruited by flier distributed at the camuses of Peking University and Tsinhua University. Since all three exeriments are based on the 5

7 same design rinciles, we will resent the rototye exeriment Zurich 2003 in detail and describe to what extent the other two exeriments deviate from the rototye. The main distinguishing features of the different exeriments are summarized in Table 1. Table 1: Differences in Exerimental Design Number of: Zurich 03 Zurich 06 Beijing 05 Subjects Lotteries Observations 8,906 4,669 4,225 Procedure comuterized comuterized aer and encil Framing abstract and contextual abstract and contextual contextual We elicited certainty equivalents for a large number of two-outcome lotteries. One half of the lotteries were framed as choices between risky and certain gains ( gain domain ), the other half were resented as choices between risky and certain losses ( loss domain ). 2 For each decision in the loss domain, subjects were endowed with a secific monetary amount, which served to cover otential losses and equalized exected ayoffs of corresonding gain and loss lotteries. In the Zurich 2003 and the Beijing exeriments, 50% of the subjects were confronted with decisions framed in the standard gamble format. The other 50% of the subjects had to make choices framed in contextual terms, i.e. gains were reresented as risky or sure investment gains, losses as reair costs and insurance remiums, resectively. The Zurich 2006 exeriment was based on contextually framed lotteries only. In Zurich, outcomes x 1 and x 2 ranged 2 There were no mixed lotteries involving both gains and losses. 6

8 from zero Swiss Francs to 150 Swiss Francs 3. The ayoffs in the Beijing 2005 exeriment were commensurate with the comensation in Zurich and varied between 4 and 55 Chinese Yuan 4. Exected ayoffs er subject amounted to aroximately 31 Swiss Francs and 20 Chinese Yuan, resectively, which was considerably more than a local student assistant s hourly comensation, lus a show u fee of 10 Swiss Francs and 20 Chinese Yuan, thus generating salient incentives. Probabilities of the lotteries higher gain or loss x 1 varied from 5% to 95%. The gain lotteries for Zurich 2003 are resented in Table 2. The other two exeriments essentially included a subset of these. The lotteries aeared in random order on a comuter screen 5, in Beijing on aer. Table 2: Gain Lotteries (x 1, ; x 2 ), Zurich 2003 x 1 x 2 x 1 x 2 x 1 x Outcomes x 1 and x 2 are denominated in Swiss Francs (CHF). 3 At the time of the exeriments, one Swiss Franc equaled about 0.76 and 0.84 U.S. Dollars, resectively. 4 At the time of the exeriment, one Chinese Yuan equaled about 0.12 U.S. Dollars. 5 The exeriment was rogrammed and conducted with the software z-tree (Fischbacher, 2007). 7

9 Figure I: Design of the Decision Sheet Decision situation: 22 Otion A Your Choice: Otion B Guaranteed ayoff amounting to: 1 A o B 20 2 A o B 19 3 A o B 18 4 A o B 17 5 A o B 16 6 A o B 15 7 A rofit of CHF 20 with A o B 14 8 A o B 13 9 robability 75% A o B A o B and a rofit of CHF 0 with A o B A o B 9 13 robability 25% A o B 8 14 A o B 7 15 A o B 6 16 A o B 5 17 A o B 4 18 A o B 3 19 A o B 2 20 A o B 1 OK In the comuterized exeriments, the screen dislayed a decision sheet containing the secifics of the lottery under consideration and a list of 20 equally saced certain outcomes, ranging from the lottery s maximum ayoff to the lottery s minimum ayoff, as shown in Figure I. 6 The subjects had to indicate in each row of the decision sheet whether they referred the lottery or the certain ayoff. The lottery s certainty equivalent was calculated as the arithmetic mean of the smallest certain amount the subject referred to the lottery and the subsequent certain amount on the list, when the subject had, for the first time, reorted reference for the lottery. For examle, if the subject had decided as indicated by the small circles in Figure I, her certainty equivalent would amount to 13.5 Swiss Francs. Before subjects were ermitted to start working on the real decisions, they 6 The format of the decision sheet for the Beijing exeriment was identical to the Zurich one. 28 8

10 had to correctly calculate the ayoffs for two hyothetical choices. In the comuterized exeriments, there were two trial rounds to familiarize the subjects with the rocedure. At the end of the exeriment, one row number of one decision sheet was randomly selected for each subject, and the subject s choice in that row determined her ayment. Subjects were aid in rivate afterward. The subjects could work at their own seed, the vast majority of them needed less than an hour to comlete the exerimental tasks as well as a socio-economic questionnaire. 3 Econometric Model This section discusses the secification of the finite mixture model, which allows controlling for latent heterogeneity in risk taking behavior in a arsimonious way. For the urose of classifying subjects according to risk taking tye, we need to secify three ingredients of the mixture model: the basic theory of decision under risk, the functional form of the decision model, and the secification of the error term. The underlying theory of decision under risk should be able to accommodate a wide range of different behaviors. Sign- and rank-deendent models cature reference deendence and nonlinear robability weighting. Therefore, a flexible aroach in the sirit of cumulative rosect theory (CPT ) lends itself to describing risk taking behavior. Moreover, CPT nests EUT as secial case. 7 If there is a grou of eole, whose behavior can best be described by EUT, these individuals should be identified by the finite mixture estimation as a unique grou exhibiting the redicted behavior. Suose that there are C different tyes of individuals in the oulation. 7 The bulk of revious research has been conducted under the tacit assumtion that utility is defined over lottery outcomes rather than lottery outcomes integrated with total wealth. We extend the model to accommodate the ossibility of integration in Section

11 According to CPT, an individual belonging to a certain grou c {1,..., C} values any binary lottery G g = (x 1g, g ; x 2g ), g {1,..., G}, where x 1g > x 2g, by v (G g ) = v(x 1g )w( g ) + v(x 2g )(1 w( g )). The function v(x) describes how monetary outcomes x are valued, whereas the function w() assigns a subjective weight to every outcome robability. The lottery s certainty equivalent ĉe g can then be written as ĉe g = v 1 [v(x 1g )w( g ) + v(x 2g )(1 w( g ))]. In order to make CPT oerational, we have to assume secific functional forms for the value function v(x) and the robability weighting function w(). A natural candidate for v(x) is a sign-deendent ower function x α if x 0 v(x) = ( x) β otherwise, which can be conveniently interreted and has turned out to be the best comromise between arsimony and goodness of fit in the context of rosect theory (Stott, 2006). Our secification of the value function seems to lack a rominent feature of rosect theory, loss aversion, caturing that [...]most eole find symmetric bets of the form (x, 0.5; x, 0.5) distinctly unattractive (Kahneman and Tversky (1979),. 279). In this interretation, loss aversion measures a decision maker s attitude towards mixed lotteries, encomassing both gains and losses. 8 Our lottery design does not contain any mixed lotteries, however. When there are only single-domain lotteries and loss aversion is introduced into our model in the conventional way, i.e. by assuming 8 Köbberling and Wakker (2005) view loss aversion as comonent of risk attitudes which is logically indeendent from basic utility: Prosects [...] will exhibit considerably less risk aversion if [...] they are nonmixed than if [...] they are mixed. [...] [T]he difference in risk aversion between them is due to loss aversion (. 125). 10

12 v(x) = λ( x) β for x < 0 and λ > 0, (Tversky and Kahneman, 1992), the arameter of loss aversion λ is not identifiable: λ cancels out in the definition of the certainty equivalent ce of a loss lottery (x 1, ; x 2 ) with x 1 < x 2 0, as λ( ce) β = λ( x 1 ) β w() + λ( x 2 ) β (1 w()) holds for any value of λ. Consequently, when there are no mixed lotteries available, estimating such a arameter is neither feasible nor meaningful in the first lace. Obviously, this argument rests on the assumtion that subjects reference oint with resect to which gains and losses are defined is equal to zero. However, subjects might encode ositive ayments as gains only if they exceed a certain threshold, which would turn some of the objectively given gain lotteries into mixed ones, containing both subjective gains and losses. While in rincile ossible, estimating this reference oint is questionable when there are no mixed lotteries from the onset, which would rovide valuable additional information for locating the reference oint reliably. To comlicate matters, near linear value functions, as is redominantly the case for our data, ose severe identification roblems. 9 assume a zero reference oint. For these reasons, we stick to common ractice and Turning to the second comonent of the model, a variety of functional forms for modeling robability weights w() have been roosed in the literature (Quiggin, 1982; Tversky and Kahneman, 1992; Prelec, 1998). We use the two-arameter secification suggested by Goldstein and Einhorn (1987) and Lattimore, Baker, and Witte (1992): w() = δ γ, δ 0, γ 0. δ γ + (1 ) γ We favor this secification because it has roven to account well for individual heterogeneity (Wu, Zhang, and Gonzalez, 2004) and the arameters are nicely 9 Previous attemts at estimating model arameters simultaneously with the reference oint are extremely scarce and suggest that the reference oint is of negligible magnitude (Harrison, List, and Towe, 2007). Their exerimental design included mixed lotteries, however. 11

13 interretable. The arameter γ < 1 largely governs the sloe of the curve and measures sensitivity towards changes in robability. The smaller the value of γ, the more strongly the robability weighting function dearts from linear weighting. 10 The arameter δ largely governs curve elevation and measures the relative degree of otimism. The larger the value of δ for gains, the more elevated is the curve, the higher is the weight laced on every robability and, consequently, the more otimistically the rosect is valued, ceteris aribus. For losses, the oosite holds. Linear weighting is characterized by γ = δ = 1. In a sign-deendent model, the arameters may take on different values for gains and for losses. We now turn to the third ste of model secification. In the course of the exeriments, we measured risk taking behavior of individual i {1,..., N} by her certainty equivalents ce ig for a set of different lotteries. Since CPT exlains deterministic choice, we have to add an error term ɛ ig in order to estimate the arameters of the model based on the elicited certainty equivalents. The observed certainty equivalent ce ig can then be written as ce ig = ĉe g + ɛ ig. There may be different sources of error, such as carelessness, hurry or inattentiveness, resulting in accidentally wrong answers (Hey and Orme, 1994). The Central Limit Theorem suorts our assumtion that the errors are normally distributed and simly add white noise. Furthermore, we allow for three different sources of heteroskedasticity in the error variance. First, for each lottery, subjects had to consider 20 certain outcomes, which are equally saced throughout the lottery s range x 1g x 2g. Since the observed certainty equivalent ce ig is calculated as the arithmetic mean of the smallest certain amount referred to the lottery and the subsequent amount on the list, the error is roortional to the lottery range. 11 Second, as the subjects 10 If linear robability weighting is acceted as standard of rationality γ < 1 can be interreted as index of dearture from rationality (Tversky and Wakker, 1995). 11 See Wilcox (2009) for a similar aroach in the context of discrete choice under risk. 12

14 may be heterogeneous with resect to their revious knowledge, their attention san as well as their ability of finding the correct certainty equivalent, we exect the error variance to differ by individual. Third, lotteries in the gain domain may be evaluated differently from the ones in the loss domain. Therefore, we allow for domain-secific variance in the error term. This yields the form σ ig = ξ i x 1g x 2g for the standard deviation of the error distribution, where ξ i denotes an individual domain-secific arameter. Note that the model allows to test for both individual-secific and domain-secific heteroskedasticity by either imosing the restriction ξ i = ξ, or by forcing all the ξ i to be equal in both decision domains. Both tyes of restrictions are rejected by their corresonding likelihood ratio tests in all three samles with -values close to zero. Therefore, we control for all three tyes of heteroskedasticity in the estimation rocedure. Having discussed all the necessary ingredients, we now turn to the secification of the finite mixture model. The basic idea of the mixture model is assigning an individual s risk-taking choices to one of C tyes of behavior, each characterized by a distinct vector of arameters θ c = (α c, β c, γ c, δ c) 12. When estimating the model arameters the number of tyes C is held fixed. otimum number of classes is determined by estimating mixture models with varying C and alying some suitable test to decide among them (see Section 4.2). We denote the roortions of the C different tyes in the oulation by π c. Given our assumtions on the distribution of the error term, the density of tye c for the i-th individual can be exressed as f (ce i, G; θ c, ξ i ) = G g=1 1 σ ig φ ( ceig ĉe g σ ig where φ denotes the density of the standard normal distribution. Since we do not know a riori to which grou a certain individual belongs to, the roortions π c are interreted as robabilities of grou membershi. Therefore, each individual 12 The vectors γ c and δ c contain the domain-secific arameters for the sloe and the elevation of the robability weighting functions. ), The 13

15 density of tye c has to be weighted by its resective mixing roortion π c, which, of course, is unknown and has to be estimated as well. Summing over all C comonents yields the individual s contribution to the model s likelihood L. The log likelihood of the finite mixture model is then given by ln L (Ψ; ce, G) = N C ln π c f (ce i, G; θ c, ξ i ), i=1 c=1 where the vector Ψ = (θ 1,..., θ C, π 1,..., π C 1, ξ 1,..., ξ N ) summarizes all the arameters of the model. The arameters are estimated by the iterative Exectation Maximization (EM) algorithm (Demster, Laird, and Rubin, 1977) 13, which rovides an additional feature: In each iteration, the algorithm calculates by Bayesian udating an individual s osterior robability τ ic of belonging to grou c. The final osterior robabilities reresent a articularly valuable result of the estimation rocedure. Not only do we obtain the robabilities of individual grou membershi, but we also have a method of judging the quality of classification at our disosal. If all the τ ic are either close to zero or one, all the individuals are unambiguously assigned to one secific grou. The τ ic can be used to calculate a suitable measure of entroy, such as the Normalized Entroy Criterion (Celeux and Soromenho, 1996), in order to gage the extent of ambiguous assignments. If classification has been successful, i.e. if genuinely distinct tyes have been identified, we should observe a low measure of entroy. 13 Various roblems may be encountered when maximizing the likelihood function of a finite mixture model and, therefore, a customized estimation rocedure was used that can adequately deal with these roblems. Details of the estimation rocedure, written in the R environment (R Develoment Core Team, 2006), are discussed in the sulementary materials available on-line. 14

16 4 Results In the following section we resent descritive statistics of the raw data and the results of the finite mixture estimations. 4.1 Descritive Statistics At the level of observed data, risk taking behavior can be conveniently summarized by relative risk remia RRP = (ev ce)/ ev, where ev denotes the exected value of a lottery s ayoff and ce stands for its certainty equivalent. RRP > 0 indicates risk aversion, RRP < 0 risk seeking, and RRP = 0 risk neutrality. In the context of EUT, risk references are catured solely by the curvature of the utility function, which in turn determines the sign of relative risk remia. Hence, the sign of RRP should be indeendent of, the robability of the more extreme lottery outcome. In Figures II through IV, median risk remia sorted by show a systematic relationshi between RRP and, however: In all three data sets subjects choices dislay a fourfold attern, i.e. they are risk averse for low-robability losses and high-robability gains, and they are risk seeking for low-robability gains and high-robability losses. Therefore, at a first glance, average behavior is adequately described by a model such as CPT rather than EUT. As the following sections show, the median RRP s gloss over an imortant feature of the data as there is substantial latent heterogeneity in risk taking behavior. 15

17 Figure II: Median Relative Risk Premia, Zurich 2003 Gain Domain Loss Domain RRP RRP Figure III: Median Relative Risk Premia, Zurich 2006 Gain Domain Loss Domain RRP RRP

18 Figure IV: Median Relative Risk Premia, Beijing 2005 Gain Domain Loss Domain RRP RRP Model Selection So far we have not addressed the issue whether a finite mixture model is actually to be referred over a single-comonent model in the first lace, and what the number of grous C in the mixture model, often termed model size, should be. In order to deal with these questions the researcher needs a criterion for assessing the correct number of mixture comonents. The literature on model selection in the context of mixture models is quite controversial, however, and there is no best solution. 14 For this reason, rather than relying on a single measure, we examine several criteria with differing characteristics to get a handle on the roblem of model selection. Obviously, the classical information criteria, the Akaike Information Criterion AIC and the Bayesian Information Criterion BIC are a natural starting oint for our analysis. Unfortunately, the AIC criterion is order-inconsistent, 14 The roblem of identifying the number of classes is one of the issues in mixture modeling with the least satisfactory treatment (Wedel (2002),. 364). For examle, a standard likelihood ratio test is not aroriate here (Cameron and Trivedi (2005),. 624). 17

19 i.e. the robability that it is minimized at the true model size does not aroach unity with increasing samle size, and it tends to overfit models (Atkinson, 1981; Geweke and Meese, 1981; Celeux and Soromenho, 1996). The BIC, on the other hand, has been roved to be consistent under suitable regularity conditions but may suffer from over- or underestimating the number of mixture comonents (Biernacki, Celeux, and Govaert, 2000). Aside from these roblems, both classical criteria share the rincile of trading off model arsimony against goodness of fit, but do not directly measure the ability of the mixture to rovide well searated and nonoverlaing comonents, which, ultimately, is the objective of estimating mixture models. Therefore, Celeux and Soromenho (1996) roose the Normalized Entroy Criterion NEC, which is based on the osterior robability of grou membershi τ ic. Biernacki, Celeux, and Govaert (1999) argue that the NEC criterion aears to be less sensitive than AIC and BIC. However, the NEC focuses solely on the quality of classification and does not take model fit into account. Ideally, what the researcher would like to have at her disosal is a criterion that delivers both an assessment of model fit, making allowance for arsimony, and the quality of classification. Biernacki, Celeux, and Govaert (2000) therefore suggest to modify the BIC criterion by factoring in a enalty for mean entroy. When the mixture comonents are well searated, mean entroy is close to zero and its weight in their roosed Integrated Comleted Likelihood Criterion ICL is negligible. In the one-comonent case there is no entroy by definition, and therefore ICL coincides with BIC. While there is no theoretical justification for this aroach, simulations seem to show a suerior erformance comared to other heuristic criteria, such as NEC (Biernacki, Celeux, and Govaert, 2000), as well as comared to AIC and BIC (McLachlan and Peel, 2000). As different criteria may come u with conflicting results concerning the correct number of mixture comonents, model selection is a difficult roblem. One way of dealing with this issue is to use one s central research question as 18

20 guide line. 15 Our concern here is twofold: First, given the vast heterogeneity in individual risk taking behavior, it is doubtful whether a single-comonent model is adequate. Therefore, the crucial question is whether C > 1 should be referred to C = Second, considering the heated disute about the right model of choice under risk, another objective of our study is to identify relative grou sizes of EUT and Non-EUT tyes. Bearing these objectives in mind, we calculated values for four different criteria, AIC, BIC, NEC as well as ICL, and three different model sizes, C {1, 2, 3}, resented in Table 3. According to these criteria, the model size which minimizes the resective criterion value should be referred. Table 3: Model Selection Criteria Zurich 03 AIC BIC NEC ICL C = 1-38,398-35,815 n.a. -35,815 C = 2-39,629-36, ,991 C = 3-40,504-37, ,807 Zurich 06 AIC BIC NEC ICL C = 1-20,858-19,297 n.a. -19,297 C = 2-22,173-20, ,566 C = 3-22,622-20, ,968 Beijing 05 AIC BIC NEC ICL C = 1-18,485-16,529 n.a. -16,529 C = 2-19,585-17, ,582 C = 3-19,965-17, , Cameron and Trivedi (2005) argue in this context: Therefore, it is very helful in emirical alication if the comonents have a natural interretation (. 622). 16 Parameter estimates for C = 1 are resented in the sulementary materials available on-line. 19

21 As AIC, BIC, and therefore also ICL, are highest at C = 1 for all three data sets, C > 1 is clearly favored over C = 1. As the NEC criterion is not defined for C = 1, Biernacki, Celeux, and Govaert (1999) argue in favor of a multi-comonent model if there is a C > 1 with NEC(C) 1, which is the case here. We therefore conclude that a finite mixture model is suerior to a single-comonent model, given the unanimous recommendation by all four criteria. With regard to the choice between C = 2 and C = 3, the three-grou classifications seem to be favored by all criteria but NEC. Given the minimum level of NEC at C = 2, a two-grou classification is referable if the central issue is a arsimonious reresentation of risk taking tyes rather than best model fit. As entroy is generally extremely low for both the two-grou and threegrou classifications, both model sizes seem quite sensible, however. Before we infer from these results that we should choose C = 3, we take a closer look at the difference between the two-grou and three-grou classifications. 17 What is of secial interest here is whether one grou remains fairly stable and the other grou gets subdivided into two new ones when model size is increased, 17 Since there is quite some heterogeneity within the majority grou, it is to be exected that even finer segmentations imrove model fit. However, when we extend the number of grous beyond three, multimodality of the log likelihood function becomes a severe roblem as, deending on the randomly drawn start values, even a stochastic extension of the EM algorithm tends to converge to local maxima. For oorly drawn start values the estimation algorithm diverges with one grou getting smaller in each iteration, which might indicate that the likelihood is unbounded (McLachlan and Peel (2000),. 54). Therefore, estimating larger models may ask too much of our data. See also the discussion of overarametrization in Cameron and Trivedi (2005) (. 625). Nevertheless, in the case of Zurich 06 we were able to estimate four and five-grou models: In both cases the relative size of the minority grou declines only slightly. This finding suorts our conjecture that heterogeneity is articularly ronounced within the majority grou, whereas the minority grou is fairly homogeneous and robust to model size. Since we are not able to resent results for all three data sets, we do not discuss these findings here. 20

22 or whether the individuals get reshuffled to three new tyes. If the latter were the case, a two-grou secification would clearly be misleading. In order to answer this question we examine relative grou sizes and transition atterns of individuals tye assignment. Table 4: Relative Grou Sizes Zurich 03 Tye I Tye II / IIa Tye IIb C = % 82.9 % C = % 27.3 % 56.0 % Zurich 06 Tye I Tye II / IIa Tye IIb C = % 77.7 % C = % 29.8 % 48.2 % Beijing 05 Tye I Tye II / IIa Tye IIb C = % 79.7 % C = % 29.3 % 50.8 % Table 4 dislays the estimated relative grou sizes of the behavioral tyes for model sizes C = 2 and C = 3. As the ercentages reveal, all the Tye I grous remain stable with resect to relative grou size. Moreover, with a few excetions, Tye I individuals remain Tye I when model size is increased: Only a total of 2% of the individuals move into or out of Tye I when an additional comonent is introduced into the finite mixture model. 18 Increasing model size results in a decomosition of the original Tye II grous into two subtyes, Tye IIa Tye IIb, as there is still considerable heterogeneity within these grous. Thus, from the oint of view of identifying Tye I individuals, the two-grou classifications are informative by themselves and rovide the most arsimonious 18 Across all three data sets, only 2 individuals are newly assigned to Tye I and 7 individuals leave Tye I, when C is increased from 2 to 3. 21

23 classification whereas three grous render a more detailed descrition of Tye II individuals. To kee interretation of grahs manageable we will resent results for C = 2 when contrasting Tye I with Tye II, and for C = 3 when discussing subtyes of Tye II behavior Clean and Robust Segregation of Behavioral Tyes In order to be of value to alied economics, a classification of risk taking behavior should meet two conditions: First, it should be clean, i.e. all the individuals should be clearly associated with one secific risk taking tye. Second, the classification should be robust across different exeriments based on the same design rinciles. Regarding the first condition, entroy criteria, based on the osterior robabilities of grou membershi, can be used to evaluate the quality of classification. One such measure is the Normalized Entroy Criterion NEC, introduced in the revious section. If all the individuals can be clearly assigned to one of the different behavioral grous, the osterior robabilities of grou membershi τ ic are close to zero or one, and NEC 0. A τ ic distinctly different from zero or one indicates that the individual is classified as a mixed tye belonging to grou c with robability τ ic and to the other grou(s) with robability 1 τ ic. As Table 3 shows, NEC always lies in the vicinity of zero, irresective of model size C being 1, 2 or 3, i.e. there are hardly any mixed tyes with ambiguous grou affiliation. The high quality of classification can also be inferred directly from the distributions of the individuals osterior robabilities of grou membershi. In Figure V, based on C = 2, τ EUT denotes the osterior robability of belonging to the first tye, which can indeed be characterized, as we will demonstrate below, as exected utility maximizers. 20 As the distributions of τ EUT show, 19 The interested reader is referred to Bruhin, Fehr-Duda, and Eer (2007) for an extensive discussion of C = As grou membershi is stable, histograms of τ EUT for C = 3 are qualitatively the same. 22

24 the individuals osterior robabilities of behaving consistently with EUT are either close to one or close to zero for ractically all the individuals in all three data sets, indicating an extremely clean segregation of subjects to tyes. Our result is quite remarkable as it substantiates that there are distinct tyes in the oulation, be they Swiss or Chinese. And it also shows that the underlying behavioral model rovides a sound basis of discriminating between them. With resect to the second criterion, robustness of classification, Figure V illustrates the robably most striking result of our study, namely similar distributions of tyes across all three data sets. In all three histograms of Figure V, there are about four times as many individuals with τ EUT close to zero, comared to individuals with τ EUT close to one. This finding is mirrored by the estimates of the relative grou sizes, dislayed in Table 4, which shows a stable roortion of Tye I of about 20%, irresective of model size C. Moreover, it can be shown that the hyothesis of the same distribution of tyes revailing in all three data sets cannot be rejected. Similarly, when model size is increased to C = 3, relative grou sizes turn out to be of equal magnitudes in all three data sets and are statistically indistinguishable from one another. Therefore, classification is not only unambiguous, but also results in roughly equal mixing roortions, demonstrating that classification is robust across exeriments. This finding leads us to the next question. Do the resective tyes identified in each data set also exhibit similar atterns of behavior? This question will be addressed in the following sections, dedicated to the characterization of the endogenously defined tyes of behavior. 23

25 Figure V: Distribution of Posterior Probability of Assignment to EUT, τ EUT (C = 2) Zurich 2003 Zurich 2006 Beijing 2005 Relative Frequencies Relative Frequencies Relative Frequencies ! EUT! EUT! EUT Figure VI: Tye-Secific Probability Weighting Functions, Zurich 2003 Probability Weights: Gain Domain Probability Weights: Loss Domain w() EUT Tyes CPT Tyes w() EUT Tyes CPT Tyes 24

26 Table 5: Classification of Behavior (C = 2) EUT Tyes CPT Tyes Parameters ZH 03 ZH 06 BJ 05 Pooled ZH 03 ZH 06 BJ 05 Pooled π Gains (0.026) (0.025) (0.020) (0.013) α (0.014) (0.018) (0.102) (0.011) (0.021) (0.026) (0.107) (0.013) γ (0.022) (0.020) (0.033) (0.021) (0.015) (0.015) (0.014) (0.009) δ Losses (0.015) (0.019) (0.052) (0.012) (0.022) (0.028) (0.074) (0.013) β (0.018) (0.023) (0.084) (0.013) (0.028) (0.047) (0.107) (0.019) γ (0.043) (0.020) (0.040) (0.023) (0.017) (0.014) (0.013) (0.009) δ (0.062) (0.033) (0.065) (0.026) (0.028) (0.044) (0.053) (0.016) ln L 20,185 11,336 10,108 41,385 Parameters Individuals Observations 8,906 4,669 4,225 17,800 Standard errors (in arentheses) are based on the bootstra method with 4,000 relications. Parameters include additional estimates for ξ i for domain- and individual-secific error variances. ZH stands for Zurich, BJ for Beijing. 25

27 Figure VII: Tye-Secific Probability Weighting Functions, Zurich 2006 Probability Weights: Gain Domain Probability Weights: Loss Domain w() EUT Tyes CPT Tyes w() EUT Tyes CPT Tyes Figure VIII: Tye-Secific Probability Weighting Functions, Beijing 2005 Probability Weights: Gain Domain Probability Weights: Loss Domain w() EUT Tyes CPT Tyes w() EUT Tyes CPT Tyes 4.4 Characterization of the Minority Tye Irresective of model size, the first tye of individuals encomasses about 20% of the subjects in all three data sets, thus constituting the minority tye. In 26

28 order to characterize risk taking behavior we examine the arameter estimates of the value functions and robability weighting functions. Table 5 dislays, for C = 2, the tye-secific arameter estimates of the finite mixture model and their standard errors, obtained by the bootstra method with 4, 000 relications (Efron and Tibshirani, 1993). 21 When model size is increased to three grous, arameter estimates, resented in Tables 7 to 10 in Aendix A, remain unchanged for the minority tye, as grou membershi does not change substantially. Therefore, from the oint of view of identifying this tye of individuals, model size is not a crucial issue and the two-grou classifications nicely contrast the distinctive characteristics of the minority tye with the majority one s. Concerning robability weighting, Table 5 dislays almost identical arameter estimates across all three data sets as well as the ooled data. Without having imosed any restrictions on the arameters, we find that the minority tyes robability weighting functions are roughly linear, as the arameter estimates for both γ and δ are close to one. Since the robability weights are a nonlinear combination of these arameters, inference needs to be based on γ and δ jointly. Therefore, we constructed the 95%-confidence bands for the robability weighting curves by the bootstra method. Figures VI, VII, and VIII contain the grahs of the tye-secific robability weighting functions for each decision domain. The gray solid lines corresond to the estimated curves for the first tye, referred to as EUT tye, and the gray dashed lines delimit their resective confidence bands. For both gains and losses, the confidence bands for the first tye in fact include the diagonal over a wide range of robabilities, demonstrating high congruence with linear robability weighting. Where the confidence bands do not include the diagonal, the curves still lie extremely close to linear weighting. In sum, in all three data sets, we find the first behavioral 21 [U]nless the samle size is very large, the standard errors found by an information-based aroach may be too unstable to be recommended (McLachlan and Peel (2000),. 68). 27

29 tye to exhibit near linear robability weighting. With resect to the valuation of monetary outcomes, the second element of the decision model, the estimated arameters α and β also dislay a high degree of conformity. As can be inferred from the standard errors in Table 5, the 95%-confidence intervals of each single curvature estimate contains unity, imlying that the hyothesis of linear value functions cannot be rejected. Together with near linear robability weighting, this result justifies regarding the first tye of individuals as largely consistent with exected value maximization, and therefore EUT. 4.5 Characterization of the Majority Tyes As the discussion on model selection revealed, model size makes a difference when characterizing the majority tyes. Due to the stability of the minority EUT grous in all three data sets the behavior of the majority grous can be described by a mixture of two different subtyes. As the majority grous exhibit inverted S-shaed robability weighting curves, aarent in Figures VI, VII, and VIII, we label them CPT tyes and their corresonding subtyes as CPT-I and CPT-II. CPT-I and CPT-II grous are characterized by secific varieties of nonlinear robability weighting as Figures IX to XI show. The difference between CPT-I and CPT-II tyes manifests itself redominantly in relative strength of otimism: the elevation of the robability weighting curves, measured by δ, 22 differs substantially between CPT-I and CPT-II. CPT-II individuals, who constitute the relative majority of aroximately 50% in all three data sets, exhibit moderately S-shaed robability weighting curves with δ in the vicinity of one. The remaining 30% of the individuals, however, are characterized by differing atterns of behavior. Swiss CPT-I individuals are systematically 22 Parameter estimates are resented in Tables 7 to 10 in Aendix A. 28

30 less otimistic than Swiss CPT-II tyes, whereas the Chinese CPT-I grou encomasses highly otimistic individuals, overweighting gain robabilities and underweighting loss robabilities over a wide range of robabilities. This secific feature of Chinese CPT-I tyes might exlain the revalence of high risk tolerance in the Chinese oulation, documented by revious research (Kachelmeier and Shehata, 1992). The three-grou classifications constitute a valuable iece of information when more disaggregate estimates of risk taking behavior are called for. When the focus of research lies on a arsimonious characterization of risk taking tyes, juxtaosing rational decision makers, not rone to robability distortions, with non-rational ones, two-grou classifications are sufficiently informative due to the stability of EUT grou membershi. Figure IX: Probability Weights CPT-I vs. CPT-II, Zurich 2003 Probability Weights: Gain Domain Probability Weights: Loss Domain w() CPT I Tyes CPT II Tyes w() CPT I Tyes CPT II Tyes 29

31 Figure X: Probability Weights CPT-I vs. CPT-II, Zurich 2006 Probability Weights: Gain Domain Probability Weights: Loss Domain w() CPT I Tyes CPT II Tyes w() CPT I Tyes CPT II Tyes Figure XI: Probability Weights CPT-I vs. CPT-II, Beijing 2005 Probability Weights: Gain Domain Probability Weights: Loss Domain w() CPT I Tyes CPT II Tyes w() CPT I Tyes CPT II Tyes 4.6 Observed Behavior by Tye So far we have characterized the different behavioral tyes by their estimated arameter values. The obvious question arises whether the discriminatory ower 30

32 of the classification rocedure can also be traced at the behavioral level. After assigning the subjects to one of the tyes, EUT, CPT-I and CPT-II, based on their osterior robability of grou membershi τ ic, the observed relative risk remia can be broken down by tye as deicted in Figure XII, aggregated for the ooled data set. As can be seen, median RRP of the EUT tyes are close to zero, reflecting near risk neutral behavior in accordance with exected value maximization. When tracing behavior of the CPT-I and CPT-II tyes at the level of observed RRP in Figure XII, we find a fourfold attern of risk attitudes, with distinctive deartures from risk neutrality. Consistent with the characterizations before, CPT-I tyes exhibit more ronounced deviations. These findings document that individuals tye assignment is highly congruent with observed behavioral differences. Obviously, the tye-secific median relative risk remia do not differ greatly at = 0.5. In decision situations when the more extreme reward materializes with a 50% chance, the tyical CPT individual will not over- or underweight its robability significantly, and therefore her behavior will often not be distinguishable from a tyical EUT tye s. This consideration can be illustrated by means of Figure XIII, which dislays the deartures of average CPT behavior, aggregated over both subtyes CPT-I and CPT-II, from EUT, measured by the tye-secific differences in median normalized certainty equivalents. Each circle in Figure XIII corresonds to one secific lottery layed in any of the three exeriments, encomassing a total of 59 gain and 59 loss lotteries, ordered by the robability of the more extreme lottery outcome. At a gain robability of 25%, for instance, CPT lottery evaluations, on average, exceed EUT ones by u to 30% of their corresonding exected values. The dashed lines in the grahs reresent the case when median CPT behavior does not differ from median EUT behavior. Positive values in the grahs indicate that, on average, CPT tyes are relatively more risk seeking than EUT tyes. The oosite holds for negative 31