Comparing Subjects

While Table 3.1 already showed how aggregated choices represent risk aversion, prudence, and temperance, Figures 3.1, 3.2, and 3.3 rather present these results on the 8_{If we examine the second-order decisions over the mixed domain, we find some indications that}

loss aversion is present. Subjects chooseB2 noticeably more often (65 % versus 50%) if this choice

provides a sure gain (decisions 21 and 26) than if this choice leads to a sure loss (decisions 2 and 22). However, the number of decisions is too small to allow for a detailed formal analysis.

level of the individual subject. Based on Result 3.2, we do not distinguish between the gain, loss, and mixed domain but only between orders of risk preferences. We count how often each individual chooses Bn over An for each n and show the relative frequencies

of subjects with a certain number of risk-averse, prudent, and temperate choices. The dark distributions in Figures 3.1, 3.2, and 3.3 show the empirical frequency distributions of the respective number of risk-averse, prudent, and temperate choices made by the subjects. The light distributions show the frequency distributions that would have occurred if every subject had chosen randomly between Bn and An. This would also

have been the distribution if every subject had been stochastically risk neutral, since this is equivalent to the behavior of choosing Bn with a probability of 1/2.

Figure 3.1: Characterizing Subjects: Risk Aversion

Concerning second-order risk preferences, the empirical distribution in Figure 3.1 does clearly not follow random behavior.9 The median of the empirical distribution (17 risk-averse choices) is above the median of the random distribution (14 risk-averse choices). More subjects are located in the right tail than in the left tail of the empirical distribution. This indicates risk aversion. In order to see how many subjects are in fact risk-averse, we can classify them using binomial tests. For each subject, it is tested whether her probability of choosing B2 is equal or lower than 1/2. Likewise, in order to classify risk-seeking subjects, it is tested whether the probability of choosing A2 is equal or lower than 1/2. Then, subjects choosing B2 in 0-10 out of 28 decisions are risk-seeking and those choosing B2 in 18-28 out of 28 decisions are risk-averse. The 9_{We can reject that the two distributions are equal with a two-sided Kolmogorov-Smirnov test}

(p= 0.017; D = 0.3793; N = 58). We can further reject that the empirical distribution follows a normal distribution with a two-sided Shapiro-Wilk test (p= 0.0617;W = 0.9319;N = 29).

remaining subjects, those who choose B2 in 11-17 out of 28 decisions, stay unclassified and cannot be distinguished from choosing randomly.10 We find that 42% of the subjects can be classified as risk-averse and only 24% as risk-seeking. The remaining 34% cannot be distinguished from behaving randomly. Therefore, the number of subjects who are risk-averse is almost twice as high as the number of risk-seeking subjects.

Figure 3.2: Characterizing Subjects: Prudence

Figure 3.2 shows our results for third-order risk preferences. As before, the two distributions are distinct.11 _{The median of the empirical distribution is again higher} in the actual distribution (15 prudent choices) and the right tail is fatter than the left tail. This means subjects tend to be prudent. When performing the same classification as for second-order risk preferences, we find that 33% of all subjects are prudent, 15% are imprudent, and 52% cannot be distinguished from random behavior. Similar as with risk aversion, more than twice as many subjects can be classified as being prudent rather than imprudent. Ebert and Wiesen (2011) use a comparable classification and find that 47% are prudent, 8% are imprudent, and 45% remain unclassified for the 16 decisions subjects made in their experiment. Deck and Schlesinger (2010) analyze only six decisions and a comparable classification would yield 14% prudent, 2% imprudent, and 84% unclassified subjects. Moreover, a comparable classification based on five prudent decisions in Noussair, Trautmann, and van de Kuilen (2011) would show that approximately 45% are prudent, 13% are imprudent, and 42% cannot be distinguished

10_{This classification implements a one-sided binomial test at the 10%-significance level.}

11_{We can reject the null hypothesis that subjects were choosing randomly in favor of the one-}

sided alternative that subjects are prudent with a Kolmogorov-Smirnov test (p= 0.077;D= 0.2759;

N = 58). Also, we can again reject that the empirical distribution follows a normal distribution with a two-sided Shapiro-Wilk test (p= 0.0091;W = 0.8987;N = 29).

from being neither prudent nor imprudent.12 So, our within-subject result on prudence seems to be in line with existing evidence.13

Figure 3.3: Characterizing Subjects: Temperance

Finally, Figure 3.3 shows the results of temperate choices against the distribution that would occur under random behavior. It shows that individuals tend to be temper- ate, the median individual chooses 15 times the temperate option. The left tail has only slightly more mass than what would be predicted under random behavior, indicating that the share of subjects who are clearly intemperate is very low. In contrast, the right tail of the distribution has significantly more mass than under the random distribution.14 A classification of subjects shows that 36% are temperate, 15% are intemperate, and 49% remain unclassified. More than twice as many subjects are therefore temperate rather than intemperate. A comparable classification of fourth-order risk preferences with the data of Deck and Schlesinger (2010) yields the opposite pattern. Here, only 6% are temperate while 21% of the subjects are intemperate. The remaining 73% can-

12_{Noussair, Trautmann, and van de Kuilen (2011) do not provide exact data on this and these}

approximations are therefore inferred from the graphs displaying the empirical distribution over the five choices subjects made.

13_{When comparing our empirical third-order distribution with the ones of Deck and Schlesinger}

(2010), Ebert and Wiesen (2011), and Noussair, Trautmann, and van de Kuilen (2011), it can be observed that they share central features. The empirical medians are above the medians of the random distributions (4 vs. 3 prudent choices in Deck and Schlesinger, 2010; 11 vs. 8 prudent choices in Ebert and Wiesen, 2011; and 4 vs. 2 or 3 prudent choices in Noussair, Trautmann, and van de Kuilen, 2011) and more mass is in the right tail of their empirical distributions than in a random counterpart.

14_{A two-sided Kolmogorov-Smirnov test for equality of the two distributions yields a clear rejection}

(p= 0.038;D = 0.3448; N = 58). Furthermore, the empirical distribution does not follow a normal distribution when tested with a two-sided Shapiro-Wilk test (p= 0.0028;W = 0.9185;N= 29).

not be distinguished from random behavior in a classification based on four temperate choices subjects made in their experiment.

While the empirical distribution in Figure 3.3 might not look surprising when com- pared with the empirical distributions of risk aversion (Figure 3.1) and prudence (Figure 3.2), it is very different from the results of Deck and Schlesinger (2010). The empirical distribution of Deck and Schlesinger (2010) is skewed to the right, while ours is skewed to the left. Deck and Schlesinger (2010) conclude that subjects must be intemperate.15 They deduce that conventional functional forms of EUT are not reconcilable with indi- viduals being intemperate. Based on our data on temperance, we come to a different conclusion. Since we elicited 28 decisions for each subject whereas Deck and Schlesinger (2010) elicited only four, we are confident that our result on temperance is robust.

Although being also based on only five temperate choices, the results of Nous- sair, Trautmann, and van de Kuilen (2011) provide further confidence for our result on fourth-order risk preferences. A comparable classification with their data would show that approximately 34% are temperate, 15% are intemperate, and 51% cannot be dis- tinguished from being neither temperate nor intemperate.16 _{Also, the median of their} empirical distribution is not lower than under random behavior (3 vs. 2 or 3 temperate choices) and it has more mass in the right tail.

Result 3.3 Of those subjects that can be distinguished from random behavior, approxi- mately twice as many are stochastically risk-averse, prudent, and temperate rather than stochastically risk-seeking, imprudent, and intemperate.

Result 3.3 complements Result 3.1 and highlights that the prevalent pattern of risk preferences satisfies risk aversion, prudence, and temperance also on a within-subject level.