Preference for Skew in Lotteries: Laboratory Evidence and Applications

Transcription

1 Preference for Skew in Lotteries: Laboratory Evidence and Applications Thomas Astebro a, José Mata b, Luís Santos-Pinto c, a Haute École Commerciale, Paris b Universidade Nova de Lisboa, Faculdade de Economia c University of Lausanne, Faculty of Business and Economics This version: March 25, 2009 Abstract This paper uses a laboratory experiment to investigate how the skew of a payoff distribution influences choices under risk. We find a statistically significant preference for positive skew under low and high stakes. We find that the preference for skew is driven by subjective distortion of probabilities and the shape of the utility function. We find that 54% of subjects make choices consistent with Expected Utility Theory (EUT) whereas 46% do not. Among the EUT individuals 47.5% love skew, 49% are indifferent to it, and 3.5% are skew averse. Among the non-eut individuals 58% like skew and 42% are skew averse. We discuss the economic implications of our findings for gambling in state lotteries and inventive activity. We show that a RDU model calibrated with our estimated parameters is able to explain why individuals only buy from 1 to 10 lottery tickets and why some individuals become inventors. JEL Codes: D81; C91. Keywords: Risk; Skew; Gambling; Inventive Activity; Lab Experiment. We thank Simon Parker for providing data and Xavier Gabaix, Botond Koszegi, Glenn Harrison, and Joel Sobel for helpful comments and suggestions. We also thank seminar participants at HEC Paris, Universidade Católica Portuguesa, and University of Lausanne. Gordon Adomdza and Won Kno provided research assistance. Corresponding author. Luís Santos-Pinto. University of Lausanne, Faculty of Business and Economics, Extranef 211, CH-1015, Lausanne, Switzerland. Ph: Fax: address: LuisPedro.SantosPinto@unil.ch. 1

2 1 Introduction People may have a preference for positive skew in risky decisions. For example, people tend to overbet on the long-shot horse with low probability of winning large returns rather than the favorite with the greatest expected return (Joseph Golec and Maurry Tamarkin, 1998). And when people buy lottery tickets, Forrest, Simmons and Chesters (2000) and Garrett and Sobel (1999) show that people are more concerned with the size of the top prize than the expected value of the lottery. In addition, positive skew may effect significant life choices, not just gambling. Three-quarters of all people that enter self-employment face higher variance but lower expected return than in employment (Barton Hamilton, 2000.) Further, 97% of inventors will not break even on their investments but face a very skew distribution of returns conditional on succeeding (Thomas Astebro, 2003.) 1 The two latter results could be explained by risk loving, but Charles Holt and Susan Laury (2002) [HL] show that in choices over real-money lotteries people generally are risk averse, trading off returns for less risk, and that risk aversion increases markedly as stakes go up. Additional experiments using their method confirm low to modest risk aversion also among entrepreneurs (Julie Ann Elston, Glenn Harrison and Elisabet Rutstrom, 2005.) Gambling in state lotteries may also be related to optimism, pleasure from gambling, and/or anticipatory feelings. On the other hand, the decision to become an entrepreneur or an inventor may be a result of preferences for being one s own boss or of overly confident perceptions of skill. 2 However, a preference for positive skew remains as a factor that might play a role in these decisions. This paper uses a laboratory experiment to test how positive skew influences risky choices. We ask subjects to make several sequences of 10 pairwise choices between a safe and a risky lottery and we count the number of safe 1 Evidence from financial markets is consistent with investors having a preference for positive skew. For example, securities that make the market portfolio more negatively skewed earn positive abnormal average returns. The estimates for the annualized premium range from 2.5% Kraus and Litzenberger (1976) to 3.6% Harvey and Siddique (2000). 2 See Hey (1984), Conlisk (1993), and Caplin and Leahy (2001). While some of these hypotheses are empirically supported, all of them are simply qualitative in nature and explain gambling in state lotteries without quantifying it. 2

3 lottery choices. The design is based on Holt and Laury (2002) with two important modifications. First, we consider lotteries that have more than two outcomes so that we can use the third moment around the mean to manipulate skew and variance separately. Second, we use a graphical display to represent lottery payoffs. We consider three different skew treatments. In the first treatment the safe and risky lotteries have symmetric prize distributions. We call this treatment the zero skew condition. In the second treatment moderate skew the safe lotteries are symmetric and the risky lotteries all have skew equal to 2. In the third treatment maximum skew the safe lotteries are symmetric and the risky lotteries all have skew equal to 3.2 (the maximum possible skew in our framework). We also consider two monetary stakes conditions: low and high stakes (20 times low stakes). We find that the mean number of safe choice in the zero skew treatment is 3.86 and 4.62, in the low and high stakes conditions, respectively. Introducing positive skew significantly reduces the number of safe choices. In the moderate skew treatment, the mean number of safe choices is 3.59 and 4.11, in the low and high stakes conditions, respectively. The mean number of safe choices in the maximum skew treatment is 3.25 and 3.79, in the low and high stakes conditions, respectively. Thus, the number of safe choices decreases monotonically with an increase in skew under low and high stakes and increasing the stakes increases the average number of safe choices for all skew treatments. All these effects are statistically significant. To explain these choices we resort to the structural estimation. The results obtained in the zero skew treatment are consistent with EUT and with findings in Holt and Laury (2002, 2005) and Harrison et al. (2005, 2007). The only difference is that we find slightly less risk aversion than they do. The large number of risk choices observed in the maximum skew condition could be explained by EUT if most individuals were risk seekers. But that would contradict the safe choices observed in for the zero skew condition. Thus, the choices made in the zero skew and in the maximum skew conditions are at odds with the expected utility model. Since EUT is not able to reconcile subjects choices between the zero skew and the maximum skew treatments we use consider rank-dependent utility (RDU) as an alternative model of risky choices. 3 3 It is well known that RDU models are able to accommodate probability-dependent risk attitude; that is, the preferences are consistent with both risk-averse and risk-seeking 3

4 To perform the estimation we model RDU using Prelec s (1998) probability weighting function and consider three different specifications for the utility function: power, hyperbolic absolute risk aversion (HARA), and expopower. Power utility is the benchmark approach of most empirical work on risk aversion. The HARA utility nests the power utility and provides a simple and flexible characterization of the trade-off between risk aversion and preference for skew. The expo-power utility offers a more flexible representation of risk preferences across different stakes than the power or HARA utility. We find that RDU explains subjects lottery choices better than EUT across the three utility functions. Using the HARA-RDU specification we find an estimate of.82 for the probability weighting parameter. In the power-rdu specification the estimate is.93 and in the expo-power RDU specification is.97. Thus, we find evidence for an inverted s-shaped probability weighting function across the three specifications. We show that the HARA-RDU model explains behavior better than the power-rdu or expo-power RDU models. These results are obtained for the average individual. Taking advantage of our design we divide subjects into groups. We find that 80 subjects (54%) make choices consistent with EUT whereas 78 (46%) do not. Among the subjects whose choices are consistent with EUT, 38 (47.5%) like skew, 39 (49%) are indifferent to it, and 3 (3.4%) are skew averse. Among the subjects whose choices are consistent with RDU, we find that 45 (58%) like skew and 23 (42%) are skew averse. Our findings have implications on a number of economic settings. Consider gambling in state lotteries. Empirical evidence shows that (i) despite a return of only $.53 on the dollar, public lotteries are extremely popular, (ii) most players buy from 1 to 10 tickets, (iii) ticket sales are an increasing function of the mean, a decreasing function of the variance, and an increasing function of the skew of the prize distribution, and (iv) low-income individuals spend a higher percentage of their income on lottery tickets than do wealthier individuals. It is well known that a risk neutral or a risk averse EUT individual will never play a state lottery since such gambles offer a negative expected return. It is also well known that buying a lottery ticket can be perceived as an behavior depending on the probability distribution of outcomes. If individuals overweight small probabilities and underweight medium to large probabilities in evaluating risky gambles, then they will make risky choices in gambles involving small probabilities of large gains. 4

5 attractive gamble for a RDU individual since a lottery ticket offers a very small probability of a very large gain and a RDU individual overestimates small probabilities. However, the decision to buy lottery tickets has not been studied in detail. We formalize an individual s decision to buy lottery tickets and calibrate it using the estimated parameters. We find that a RDU individual with HARA utility buys from 0 to 7 lottery tickets for probability weighting levels between.75 and 1. Additionally, we find that, for our estimated parameters, the HARA-RDU specification matches real world data on lottery ticket purchases better than the power-rdu or expo-power RDU specifications. We also show that, for most probability weighting levels, Prospect Theory (PT) predicts that if a person wants to buy one lottery ticket, then she will want to buy at least twenty tickets which is clearly counterfactual. Finally, we show that a RDU model calibrate with our estimated parameters can explain the decision to become an inventor. To make this point we use data on inventive activity taken from Astebro (2003). The data shows that the distribution of returns to inventive activity has a negative mean, high variance, and a positive skew. Thus, a risk neutral or a risk averse EUT agent would not be an inventor. The data also shows that inventors face non-negligible probabilities of making very large gains or suffering substantial losses. Since PT typically assumes that losses hurt twice as much as equally sized gains, this makes it hard for it to explain this type of behavior. In fact, we show that a PT agent only becomes an inventor for unreasonably high levels of probability distortion. The reminder of the paper is organized as follows. Section 2 explains the experimental design. Section 3 presents the findings. Section 4 provides a brief overview of the rank-dependent utility and shows that RDU and HARA can explain the experimental results better than EUT and better than RDU with power or expo-power utility. Section 5 discusses the economic significance of the findings. The Appendix contains experimental design details, the data and summary statistics. 5

6 2 Experimental Design 2.1 Holt and Laury (2002) HL study the trade-off between risk and return using lotteries with two outcomes. Subjects are given the choice between the pairs of lotteries in Table 1. insert Table 1 here It is expected that subjects start by choosing the safe lottery (S) in the top row as it has both higher expected value and lower variance. As one proceeds down the table, the expected values of both lotteries increases, but the expected value of the risky (R) lottery increases more. When the probability of the high-payoff ticket in the R choice increases enough (moving down the table), a person should cross over to choose R. Different persons will switch at different points, the switching point being determined by the degree of risk aversion. For example, a risk-neutral person would choose S four times before switching to R. Even the most risk-averse person should switch over by decision 10 in the bottom row, since R then yields a sure payoff of $3.85. Indeed, HL find that 26% switch from S to R on the fourth decision, another 26% switch on the fifth row, and 13% switch on the sixth row. Using a standard constant relative risk aversion utility function, the distribution of choices implies risk neutrality for the first group, with slight relative risk aversion for the second and risk aversion for the third group. HL also find that when stakes are increased 20 times or more, the number of safe choices increases significantly. 19% delay to choose R until the fifth row, 23% select on the sixth row and 22% on the seventh row, and the whole distribution is shifted to the right. 2.2 Introducing Skew To find out how positive skew influences choices under risk we make two changes in Holt and Laury s design. First, we consider lotteries that can have from two to ten different prizes. Increasing the number of prizes allows us to control the mean, variance, and skew of the sequences of lotteries. Second, lotteries are presented in a graphical display rather than in a table 6

7 as in HL. Researchers have found graphical display of data sometimes to be superior to tabular display. The advantage seems to depend on data structure and task complexity, where for either simple or very complex tasks or unclear data structures there seems to be little advantage of graphical display. 4 Figure 1a shows our graphical representation for the first choice between S and R in HL lotteries. Each of the lines in HL table is depicted in one graph of the kind in Figure 1a. People are presented with the 10 choices in sequence and are provided with a review section where any decision may be revisited. Figure 1b shows one of the skew treatments. Figure 1b clearly displays a pattern, and since the various lotteries have different patterns due to skew we believed subjects more easily can interpret the graphical than the tabular display. insert Figures 1a here insert Figure 1b here The first comparison we will make is between the results obtained when the experiment is conducted offering the same choice options with the tabular display of HL and with our graphical display. (The full set of lotteries are available in an Appendix.) We then introduce skew systematically. We modify the lotteries to have the same mean and variance as in HL, but while the safe option has no skew, the risky option has skew equal to 0, 2 and 3.16 in the different treatments. We focus on positive skew as this seems to be the most interesting case in a number of circumstances, but our exercise could be extended to lotteries with negative skew. If the expected payoff were to be maintained the higher levels of skew would lead to some negative payoffs. We therefore add $1 to HLs payoffs. We test whether there is any effect of adding $1. We also introduce lotteries with 20 times the payoffs to study the impact of skew under large stakes. 4 See, e.g. Cheri Speier, 2006; Meyer J, Shamo MK, Gopher D 1999; MIH Hwang and BJP Wu,

8 Compared to HL, we randomize subjects allocation to treatments and the order of treatments, we do not offer hypothetical gambles, and we do not run treatments with very high stakes (50 and 90 ). 2.3 Treatments and Subjects Instructions were kept identical to those in HL, with some necessary revisions to account for the difference in presentation formats and the fact that the task was conducted online rather than with paper and pencil. Subjects also took a knowledge test on their understanding of the instructions and had to pass to move forward to the experiment. (Example instructions are in Appendix.) Table 2 presents the different treatments. There are three basic treatment sequences: a) subjects perform 1 set of HL prize distribution choices (T1-T4) followed by 3 sets of low stakes choices with 3 skews and then draw a prize from one of the 4 sets; b) Subjects perform 3 sets of high stakes choices with 3 different skews and then draw a prize from one of the 3 sets; c) Subjects perform 3 sets of low stakes choices with 3 different skews. Subjects are then given the option between (A) drawing a final prize from the three low stakes distributions and finish the experiment or (B) not drawing a prize and moving on to the high stakes choices. If subjects choose option (B) they will do the three sets of high stakes choices with 3 skews and draw a prize from those. All subjects presented with this option chose B. Using posters we recruited 148 students at the University of Waterloo during the Spring These were randomly allocated across 10 treatments in six pre-determined treatment sequences as outlined in Table 2. Each subject performed three or more treatments in the sequences displayed in Table 3. Each of the sequences in the table was performed by at least 24 subjects. At the end of the experiment one prize is chosen randomly and subjects were awarded their prize plus a $5 participation fee. insert Table 2 here insert Table 3 here 8

9 Subjects in our sample are briefly described in Table 4. Some of the subjects did not respond to all of the demographics questions, the number of observations available for each item being presented along with the corresponding summary statistics. insert Table 4 here 3 Findings 3.1 Skew and Incentive Effects To test the hypothesis that individuals display a preference for positive skew we considered three different treatments. In the first treatment the safe and risky lotteries have symmetric prize distributions. We call this treatment the zero skew condition. In the second treatment moderate skew the safe lotteries are symmetric and the risky lotteries all have skew equal to 2. In the third treatment maximum skew the safe lotteries are symmetric and the risky lotteries all have skew equal to 3.2 (the maximum possible skew in a lottery with ten prizes). Table 5 shows the number of safe choices under the different conditions. Individuals display a monotonic preference for skew in both the low stakes and high stakes condition. We also find that increasing the stakes increases the average number of safe choices for all skew conditions. That relative risk aversion increases with the scale of cash payoffs is consistent with Holt and Laury (2002). insert Table 5 here We performed OLS regressions with the number of safe choices as the dependent variable and included dummies for moderate and maximum skew, and for high stakes (see Table 6). We find that the number of safe choices decreases by 0.34 from zero skew to moderate skew and by 0.67 from zero skew to maximum skew. When confronted with high stakes lotteries subjects make significantly less risky choices. The least squares regression provides easy interpretation of estimated coefficients. The coefficient for the 9

10 constant in column (i) implies that under low stakes and no skew subjects make on average 3.9 safe choices - which is risk neutral behavior. 5 insert Table 6 here Our main results are robust to the inclusion the subjects socio-demographic characteristics described in Table 4. The high stakes coefficient decreases somewhat from 0.60 to 0.46, but it remains significant. By design, all subjects perform all skew conditions and skew conditions are, therefore, orthogonal to subjects. The minor differences that we observe are therefore due to the fact that observations in the two tables are not exactly the same as we do not observe all characteristics for all subjects. This estimation reveals that males, non-white individuals and part-time students make riskier decisions. All other characteristics, except those related to income, are non-significant once controlling for everything else in the model. Since all subjects make repeated decisions under different conditions we can also estimate individual random and fixed effects models. These estimations deliver the same coefficients for skew conditions, but the precision is significantly increased, because the panel-data models correct for heterogeneity in risk preferences, which appear to be large. Increased precision means that the effect of moderate skew is now significant at the 1% level, rather than at the 5% level. The high stakes coefficient is approximately 0.5, with p-values below 1% (see Table 6). Socio-demographic characteristics does not seem to be important. Only gender and race have some effect on choices; males and non-whites make riskier choices than females and whites, although statistical significance is reduced when accounting for individual effects. 6 5 As a robustness check, we also ran an ordered-probit where the dependent variable is the number of times the safe choice was made. The results from such a model are more difficult to interpret, since the marginal impact of the coefficients depend on the level of the covariates and the specific choice being considered. The qualitative results did not present changes from those in Table 6. 6 A reason for including the individual characteristics explicitly is that the response of individuals to skew might depend on demographics. To test this we included interactions between the skew variables and each of the demographic characteristics that are captured by a dummy (gender, decision maker in the house, race, graduate student status, major, full time student status). In all cases we cannot reject the null that the effects are identical. 10

11 3.2 Robustness Our subjects perform their choices in different sequences (recall table 3). Moreover, some of them went through the initial round which tested framing effects. We would therefore like to know whether choices are affected by order and by the initial treatment. The inclusion of such controls in the regression does not change the results regarding skew. Moreover, the controls are not jointly significant. The set of four dummies indicating which initial treatment (if any) the subject had performed has a p-value of 0.113, while the p-value for the five order dummies is Thus, we conclude that there is no meaningful order effect. We have noted earlier that previous research has found graphical display of data to be superior to tabular display, in particular when the tasks are neither very simple nor very complex. Subjects making fully consistent choices would never switch back and forth between the safe and risky options. Some of them, however, did. We find that subjects make fewer inconsistent choices in our graphical display as compared to the tabular display. 32% of the subjects switched back and forth when data was presented in the tabular display, and only 6% did so in the graphical display, which seems to indicate that the graphical display is cognitively superior to the tabular display for this task. 7 There seems to be some indication in our data that the tabular display leads subjects to make a somewhat larger number of safe choices. This may be due to a lesser ability to fully understand the data in the tabular than in the graphical display, as indicated by the higher proportion of switching back and forth for the tabular display. The observed average number of safe choices is 5.3, 3.2, 4.6 and 3.6 for treatments T1 to T4, respectively. We reject that hypothesis that the mean number of safe choices in T1 is equal to that in T2, but not that T3 is equal to T4. For all samples, except for T1, we cannot reject the hypothesis of risk neutral behavior. Finally, adding one dollar to the prizes has a small and not significant effect. However, as we have seen, multiplying the stakes by a factor of twenty has a significant effect. 7 We checked the robustness of our regression results by defining two additional variables. One is defined as the decision before the one in which subjects make their first risky decision. The other is their last safe decision. The three variables are identical for subjects that never go back and forth. Results did not show any remarkable changes, except that the effect of being in the moderate skew condition is somewhat more imprecisely estimated, in particular when the variable is their last safe decision. 11

12 4 Estimation of Utility Functions In this section we use structural estimation of utility functions to compare alternative explanations for the experimental results. To model subjects choices we focus on Expected Utility Theory (EUT) and Rank-Dependent Utility (RDU). Together with Prospect Theory (PT) these are the most influential models of decision making under risk and uncertainty. Since our experiment does not involve losses PT reduces to RDU if there is no asset integration Rank Dependent Utility RDU assumes that individuals subjectively distort probabilities via a probability weighting function. 9 Consider a lottery (p 1, x 1 ;...; p n, x n ), yielding outcome x j with probability p j, j = 1,.., n. The probabilities p 1,..., p n are nonnegative and sum to one. According to RDU the utility of this lottery is evaluated as n j=1 π ju(x j ),where u is the utility function and the π j s are called decision weights. The decision weights are nonnegative and sum to one. Expected utility is the special case of the general weighting model where π j = p j for all j. If x 1 > x 2 >... > x n, then where, for each j, RDU(p 1, x 1 ;...; p n, x n ) = n j=1 π ju(x j ), π j = w(p p j ) w(p p j 1 ), with w(0) = 0 and w(1) = 1. The function w(p) is the decision weight generated by the probability p when associated with its best outcome. Thus, the decision weight π j of outcome j depends only on its probability p j and its ranking position. 8 Prospect Theory has four key features. The first is reference dependence, that is, individuals care about gains and losses relative to a status quo and not about final wealth levels. The second is loss aversion, that is, that losses loom larger than equally sized gains. The third is probability weighting. The fourth is that the value function is concave for gains but convex for losses (diminishing sensitivity in the domain of gains and losses). 9 Rank-dependent models were introduced by Quiggin (1982) for decision under risk (known probabilities) and by Schmeidler (1989) for decision under uncertainty (unknown probabilities). 12

13 Many observed weighting functions are not completely convex or concave but exhibit a mixed pattern. They are concave for small probabilities and convex for moderate and high probabilities. This pattern is called inverted s-shaped probability weighting. The pattern implies that subjects pay much attention to best and worst outcomes, and little attention to intermediate outcomes. 10 To model RDU we use Prelec s (1998) probability weighting function: w(p) = exp( ( ln(p)) η ), for 0 < p 1 and 0 < η, (1) where the parameter η determines the degree of distortion of probabilities. When η = 1 there is no distortion of probabilities and we are back to EUT. If η (0, 1), then the function captures the inverted s-shaped pattern and the further away η is from 1 the higher the degree of probability weighting. If η > 1, then we have a s-shaped pattern where small probabilities are underestimated and large probabilities are overestimated The Risk-Skew Trade-off To model the utility function we focus on hyperbolic absolute risk aversion, power, and expo-power utilities. These three utility functions represent different types of risk preferences and risk-skew trade-offs. Consider the utility function: u(x) = (α + x) 1 β, (2) where x is wealth, β < 1, and α + x > An individual with this utility function is risk seeking if β < 0, risk neutral if β = 0, and risk averse if 0 < β < 1. This utility function belongs to the family of hyperbolic absolute risk aversion utility functions, also known as HARA. 13 The coefficient of 10 Inverted s-shaped probability weighting predicts that people make risky choices in gambles that yield gains with small probabilities such as in public lotteries. It also predicts that people make safe choices in gambles that yield losses with small probabilities, a phenomenon relevant for insurance. See Quiggin (1982). 11 An alternative would be to use Kahneman and Tversky s (1992) probability weighting function. However, Ingersoll (2008) shows that Kahneman and Tversky s function is nonmonotonic. That is not the case with Prelec s probabilit weighting function. 12 Note that nothing prevents α from being negative. 13 The HARA family of utility functions was introduced in Merton (1971). The HARA family has an Arrow-Pratt s coefficient of absolute risk-aversion that is an hyperbolic function of wealth since u (x)/u (x) = β/(α + x). 13

14 relative risk aversion of the HARA is u (x)x u (x) = β 1 + α/x. (3) From (3) we see that when α > 0 the coefficient of relative risk aversion is increasing with wealth and is bounded above by β. We know from Tsiang (1972) that a risk averse individual has preference for positive skew if the third derivative of his utility of wealth is positive. As a manifestation of such preference a risk averse individual will be prepared to accept a lower expected return (but not a negative one) or a higher level of overall riskiness if the distribution of payoffs is more skewed to the right. The HARA and many other utility functions satisfy this property. 14 The trade-off between risk aversion and skewness preference is given a choice-theoretical characterization by Chiu (2005). He shows that if two lotteries are strongly skewness comparable, then the function u (x)/u (x) has the interpretation of measuring the strength of an individual s preference for skew against his risk aversion. 15 The higher the ratio u (x)/u (x) the more important is the preference for skew and the less important is risk aversion. From (2) we have that u (x) u (x) = 1 + β, for 0 < β < 1. (4) α + x The power utility function is the benchmark model of most empirical work on risk aversion. 16 A power utility with coefficient of relative risk aversion equal to β is given by u(x) = x 1 β, (5) 14 Tsiang (1972) shows that decreasing absolute risk aversion implies a preference for skew but the reverse is not true. To see this consider the coefficient of absolute risk aversion ARA = u (x)/u (x). The derivative of ARA with respect to wealth is u (x)u (x) [u (x)] 2 [u (x)]. We see that for an individual to display decreasing absolute risk 2 aversion it must be that u (x) > [u (x)] 2 > 0. In contrast, for an individual to display u (x) increasing absolute risk aversion it must be that u (x) < [u (x)] 2 u (x). Thus, individuals with decreasing ARA have a preference for skew and those with increasing ARA might prefer skew or be averse to it. 15 This ratio also determines precautionary savings. 16 See Harrison et al. (2005, 2007) and Harrison and Rutström (2008). 14

15 where β < 1. We see from (5) that the HARA nests the power utility since setting α = 0 in (2) gives us a power utility. So, the power utility, like the HARA, also displays a preference for skew. From (5) we have that u (x) u (x) = 1 + β, for 0 < β < 1. (6) x Comparing (4) to (6) we see that if α > 0 then 1+β for any given x. This means that, for any given wealth level, the power utility function displays a stronger preference for skew relative to aversion to risk than the HARA. The expo-power was first proposed by Saha (1993) and is defined as u(x) = 1 φ [ 1 exp ( φx 1 θ )], > 1+β x α+x with φ > 0 and θ < The expo-power exhibits decreasing absolute risk aversion and increasing relative risk aversion when 0 < θ < 1 and φ > 0. It exhibits increasing absolute and relative risk aversion when θ < 0 and φ > 0. Thus, the expo-power is more flexible than the HARA or the power utility to capture different types of absolute risk aversion. The main disadvantage of the expo-power is that its function u (x)/u (x) does not provide a simple characterization of the risk-skew trade-off. 4.3 Estimation Results The rank dependent utility of lottery i is RDU i = n j=1 π ju(x j ), and the expected utility of lottery i is obtained by setting π j = p j for all prizes j. To model subjects choices we follow the approach of Holt and Laury (2002) and Harrison and Rutström (2008) and use maximum likelihood to estimate the relevant parameters. First we calculate the index RDU = RDU S RDU R, where RDU S is the rank dependent utility of the safe (left) lottery and RDU R is the rank dependent utility of the risky (right) lottery. This latent index, 17 The expo-power converges to the power utility when φ converges to 0. 15

16 based on latent preferences, is then linked to the observed choices using a logistic cumulative distribution function Φ( RDU i ). This probit function takes any argument between ± and transforms it into a number between 0 and 1. Thus, we have the logit link function, Pr(Choice = S) = Λ( RDU). insert Table 7 here The estimation results reported in Table 7 indicate that the HARA utility with RDU is the specification that performs best. Since the HARA utility nests the Power utility, the comparison between the two amounts to a test on the significance of the α coefficient. This parameter is indeed significant and we conclude that the HARA generalization is indeed useful. The Expo- Power is non-nested with the HARA but since the number of parameters is the same in the two models the comparison between the two models are based on the value of the likelihood function. The HARA utility performs better than the Expo-Power, both in the EUT and RDU cases. Still, the RDU is preferred to the EUT in all models, since the η coefficient is highly significant in all the RDU estimations. In summary, the HARA with RDU is the alternative that does a better job in explaining our data. In a further extension to the regression equation in column (iv) of Table 6, we estimate subject specific responses to variation in skew. In order to perform this exercise, we start from the observation that the effect of going from the no skew to the high skew treatment is almost exactly twice as much as going from no skew to medium skew. In fact, defining a variable that takes the value 0 in the no skew case, 1 in the medium skew case, and 2 in the high skew case, and using this variable in the regression returns an estimated skew coefficient of and no detectable change in the High Stakes coefficient. We used this variable to estimate subject specific responses to skew by estimating interaction terms between this variable and the individual fixed effects. The estimated coefficients were then used to classify subjects into groups according to their preference for skew. Group specific utility functions were then estimated. Table 8 reports the results Estimation of the parameter α was difficult in columns (iv) and (v). A grid search over the parameter space revealed that the best value for α was close to 1 in both cases, but also the likelihood function did not improve much over fixing the parameter in zero. 16

17 insert Table 8 here The groups are formed according to whether choices are consistent with EUT or with RDU and according to preference for skew. We classify all subjects in columns (ii), (iii), and (iv) of Table 8 as making choices consistent with EUT since their choices are not too extreme in the direction of skew seeking or aversion. 19 Thus, we have 80 EUT subjects and 78 RDU subjects. Among the EUT individuals we classify the 39 subjects in column (iii) as indifferent to skew since they either did not change their choice across the three skew treatments or if they did they went back and forth without displaying a particular pattern. We classify the 38 subjects in column (ii) as being EUT skew seekers since they make systematically more risky choice as the skew increases. We classify the 3 subjects in column (iv) as being EUT skew averse since they make systematically less risky choices as the skew increases. 20 Among the RDU individuals we classify the 45 subjects in column (i) as skew seekers since they display an extremely strong preference for skew which is confirmed by an estimate for the average probability weighting parameter of.57. We classify the 23 subjects in column (v) as skew averse since they exhibit an very strong aversion to skew which is confirmed by an estimate for the average probability weighting parameter of Economic Significance In this section we show that a rank-dependent expected utility model calibrated with our estimates is able to explain several stylized facts of gambling in public lotteries and the decision to become an inventor. We also show that PT is not able to explain these behavior as well as RDU. 5.1 Gambling in Public Lotteries Research on gambling in public lotteries shows that: 19 The probability weighting parameter η is not different from 1 in column (iii). In the remaining columns, the coefficient is statistically different from unity. 20 There are only three subjects classified in the group in column (iv). The η parameter is very imprecisely estimated and it is not statistically different from 0 or 1. 17

18 1. Despite a return of only $.53 on the dollar, public lotteries are extremely popular. In a 2007 Gallup poll, 46 percent of Americans reported participation in state lottery gambling Most players only buy a small number of tickets. For example, a lottery ticket in the lottery Euromillions is the selection and purchase of a combination of 5 digits (from 1 to 50) and 2 stars (from 1 to 9). Each ticket costs e2.00. The gross expected value of buying one ticket is approximately e.80. Most players buy from 1 to 5 tickets Ticket sales are an increasing function of the mean of the prize distribution (better bets are more attractive ones), a decreasing function of the variance in the prize distribution (riskier bets are less attractive), and an increasing function of the skew of the prize distribution Low-income individuals spend a higher percentage of their income on lottery tickets than do wealthier individuals. 24 In this section we show that RDU can explain these stylized facts but PT cannot. To do that we start by formalizing the lottery ticket purchase problem. Let each lottery ticket win a unique prize of value G with probability p and be sold at price c > pg > Since buying a single lottery has negative expected value (c > pg) a risk neutral or a risk averse EUT agent will never buy a lottery ticket. The rank-dependent utility of buying n lottery tickets is RDU(n) = [1 w(np)] u(z nc) + w(np)u(z + G nc), A ticket in the US lottery Powerball is the selection and purchase of a combination of five white balls out of 59 balls and one red ball out of 39 red balls. Each ticket costs $1.00. The gross expected value of buying one ticket is approximately $ See Walker and Young (2001). 24 See Haisley et al. (2008) and the references cited in the paper. 25 This is a simplified lottery. Real world public lotteries usally have multiple prizes (second prize, third prize, and others). 18

19 where z is initial wealth and w( ) is the probability weighting function, given by (1). The optimal n for a RDU agent is the solution to max [1 w(np)] u(z nc) + w(np)u(z + G nc) n s.t. w(p) = exp( ( ln(p)) η ) nc z u(z) RDU(n) To calibrate the lottery parameters we set G = $10 6, p = 10 6, and c = $2. Thus, a person that buys a single state lottery ticket faces a lottery with mean $1, median $2, standard deviation $1, 000, and skew For the power utility we consider four values for β: 0,.25,.50, and We calibrate the HARA and expo-power utilities with the estimates obtained with the RDU model: α = 60.7, β =.73, φ =.024, and θ =.502. We solve this problem for three levels of initial wealth $10, $100 and $1, 000 and sixteen levels of the probability weighting parameter η. 27 Table 9 summarizes the results for initial wealth of $10, Table 10 for initial wealth of $100, and Table 11 for initial wealth $1, 000. Each cell in the tables gives us the optimal number of lottery tickets that a RDU individual buys for each utility function and probability weighting level. insert Table 9 here insert Table 10 here 26 Some recent estimates for the parameter β are: β = 0.67, β = 0.52, β = 0.48, for private-value auctions, Cox and Oaxaca (1996), Goeree et al. (1999), and Chen and Plott (1995), respectivaly, β = 0.44 for several asymmetric matching pennies games, Goeree et al. (2002), and β = 0.45 for 27 one-shot matrix games, Goeree and Holt (2000). Campo et al. (2000) estimate β = 0.56 for field data from timber auctions. Holt and Laury (2002) find β to be between 0.3 and 0.5. Harrison et al. (2005) find β = 0.37 for low stakes and β = 0.57 for high stakes. 27 The literature has found a variety of estimates for the parameter η in the domain of gains. Tversky and Kahneman (1992) find η = 0.61, Camerer and Ho (1994) find η = 0.56, Wu and Gonzales (1996) find η = 0.71, Abdellaoui et al. (2000) find η = 0.60, and Harrison (2008) finds η = 0.92 and η =

20 insert Table 11 here The first column of Tables 9, 10, and 11 displays the optimal lottery ticket purchases of a RDU individual with linear utility. Such a person buys from 0 to 500 lottery tickets depending on his initial wealth and probability weighting level. This happens because he overestimates the probability of winning the lottery but suffers no disutility from facing a risky gamble so he will buy as many tickets as his wealth permits. Clearly, the predicted lottery ticket purchases of an individual with a linear utility are inconsistent with real world data on lottery ticket purchases. The second, third and fourth columns of Tables 9, 10, and 11 display the optimal lottery ticket purchases of a RDU individual with power utility with exponent.75,.50 and.25, respectively. We see that an increase in the concavity of the utility function reduces lottery ticket purchases across all probability weighting levels. An individual with power utility with exponent.75 buys from 0 to 482 tickets while an individual with exponent.50 buys from 0 to 167 tickets and an individual with exponent.25 only buys from 0 to 38 tickets. The fourth column displays the optimal lottery ticket purchases of a RDU individual with a power utility calibrated by the β estimate obtained from the power-rdu model (the estimate is.755 which is equivalent to an exponent of ). We see that such an individual buys public lottery tickets when η.65 (initial wealth $10), η.775 (initial wealth $100) or η.825 (initial wealth $1, 000). However, these thresholds for lottery ticket purchases are far from the estimate obtained for η in the power-rdu model:.928. The fifth column displays the optimal number of lottery ticket purchases by a RDU individual with a HARA utility calibrated by the α and β estimates obtained from the HARA-RDU model. Such an individual buys from 0 to 44 lottery tickets. Moreover, we see that for η between.90 and 0.75, an interval that contains the estimate obtained for η in the HARA-RDU model.815, he only buys from 0 to 7 tickets. This prediction matches closely real world data on lottery ticket purchases. The sixth column displays the optimal number of lottery ticket purchases by an individual with an expo-power utility calibrated by the φ and θ estimates obtained from the expo-power RDU model. We see that such an individual only plays a public lottery if η.60 (initial wealth $10), η.675 (initial wealth $100) or η.8 (initial wealth $1, 000). These thresholds 20

21 for lottery ticket purchases are far from the estimate obtained for η in the expo-power RDU model:.974. Thus, for our estimated parameters, the predicted behavior of a RDU individual with a HARA utility matches real world data on lottery ticket purchases better than the predicted behavior of a RDU individual with a power or an expo-power utility. Tables 9, 10, and 11 also show that increases in initial wealth lead to less than proportional increases in lottery ticket purchases for all RDU specifications. This is consistent with the fact that low-income individuals spend a higher percentage of their income on lottery tickets than high income individuals. Let us now see what are the predictions of Prospect-Theory for gambling in state lotteries. The PT value of buying n lottery tickets is P T (n) = [1 w(np)] v( nc) + w(np)v(g nc), where v(x) is the gain-loss value function. The optimal n for a PT agent is the solution to max [1 w(np)] v( nc) + w(np)v(g nc) n s.t. w(p) = exp( ( ln(p)) η ) 0 P T (n) A standard assumption in PT papers is that v(x) is a power function given by { x v(x) = 1 r, x 0 λ( x) 1 r, x < 0, where λ is the loss aversion coefficient. 28 We set λ = 2 another standard assumption of PT papers and consider six values for r: 0,.10,.25,.35,.50 and Table 12 summarizes the results for PT According to Wakker and Zank (2001) the PT model with power utility is generally assumed in parametric tests and is currently the most used nonexpected utility form. 29 Tversky and Kahneman (1992) consider v(x) = x α for x 0 and v(x) = λ( x) β for x < 0. They estimate the parameters and find that the median estimate for λ is 2.25 and the median estimate for the exponent of the value function is.88 for both gains and losses. 30 A PT individual with an expo-power value function calibrated with the estimated parameters does not buy lottery tickets for.60 η 1. 21

22 insert Table 12 here The first column in Table 12 displays the optimal lottery ticket purchases of a PT with a linear gain-loss value function. Such an individual buys from 0 to 5,511 lottery tickets. The remaining five columns in Table 12 display the optimal lottery ticket purchases of a PT individual with a power value function with exponent.90,.75,.65,.50 and.25, respectively. An increase in the concavity of the value function reduces lottery ticket purchases across all levels of probability weighting levels. An individual with power utility with exponent.90 buys from 0 to 2,901 tickets, one with exponent.75 buys from 0 to 667 tickets, one with exponent.65 buys from 0 to 140 tickets, one with exponent.50 buys from 0 to 2 tickets, and one with exponent.25 always buys 0 tickets. Comparing the predictions of PT and RDU across power-value and powerutility functions that have the same exponent we find that: (1) a PT individual switches from not playing the lottery to playing it at higher levels of probability distortion than a RDU individual, (2) a PT individual with a power value function with exponent less than or equal to.50 hardly buys any lottery tickets whereas a RDU individual with a power utility function with exponent less than or equal to.50 buys some tickets, and (3) if probability weighting is sufficiently high, an increment in probability distortion leads to larger incremental ticket purchases for a PT individual than for a RDU individual. These results are driven by PT s assumptions of loss aversion and convexity in the domain of losses. Switching from not playing the lottery to playing it gives a very small probability of a large gain but a very large probability of a small loss. Since losses are worth twice as much as equally sized gains, a PT individual must be more optimistic to participate in a lottery than a RDU individual. The interaction between loss aversion and concavity/convexity of the value function makes it very unattractive for a PT individual to purchase lottery tickets for sufficiently concave/convex value functions. The assumption of convexity in the domain of losses implies diminishing sensitivity to increasing losses. Thus, for sufficiently high probability weighting levels, buying an additional lottery ticket is more attractive for a PT individual than to a RDU individual. In short, both RDU and PT predict no purchases of lottery tickets for low levels of probability weighting and some purchases of lottery tickets for suf- 22

23 ficiently high levels of probability distortion. However, RDU predicts much less purchases of lottery tickets than PT for most probability weighting level. Thus, RDU provides a better description of gambling in state lotteries than PT. 5.2 Inventors Astebro (2003), using a sample of 1,091 Canadian inventors, finds that the distribution of returns to inventions has negative mean, large standard deviation, and positive skew. Since the mean return is negative a risk neutral or a risk averse EUT agent prefers not to become an inventor. We will now show that RDU can explain why individuals become inventors. To perform the analysis we use Astebro s data on the costs and returns to invention. We take the point of view of a representative inventor that faces a set of projects from which she will pick a single project. We assume that the probability the representative investor selects any of these projects is uniform. Summary statistics for the distribution of the net present value of a typical inventive project from this sample are: mean $36, median $7, 298, standard deviation $178, 978, skew 20.64, minimum $1, 497, 456, and maximum $4, 835, The largest loss suffered was close to 1.5 million Canadian dollars and the largest gain was close to 5 million Canadian dollars. In an inventive activity the payoff from being very successful is substantially higher than the payoff from being very unsuccessful due to the positive skew in returns. An RDU individual will be attracted to inventive activity since he overestimates the probability of being very successful and of being very unsuccessful (the low probability events) but underestimates the probability of being moderately successful or unsuccessful (the high probability events). To study the behavior of an RDU individual who faces this type of risk we consider the same utility functions as in the previous Section. Initial wealth is set to 1.5 million Canadian dollars so that the worst project can be undertaken. Table 13 reports the findings. Cells are defined as in the prior Tables. 31 The net present values are the discounted sum of development cost, opportunity cost of labor, and revenues. Most inventions did not reach the market. In those cases the revenue is assumed to be zero. 23