Pooling Subjects

Table 3.1 outlines our results if we pool all subjects. The second column of Table 3.1 states in how many decisionsBn was chosen overAn for eachn= 2,3,4. In this column

no distinction between the decisions were made other than whether they were designed to elicit risk aversion, prudence, or temperance. The hypothesis that the probability of making a risk-averse choice in a second-order decision, a prudent choice in a third-order decision, or a temperate choice in a fourth-order decision is equal or lower than 1/2 is rejected by binomial tests at the 1%-level for all three tests (p= 0.0000;N = 1876).

Table 3.1: Descriptive Results when Pooling Subjects

Choices Across Within Within In Mixed Eliciting Domains Gains Losses Gambles Risk Aversion 56 % 55 % 57 % 57 % (N) (1876) (871) (469) (536) Prudence 56 % 60 % 55 % 52 % (N) (1876) (670) (469) (737) Temperance 56 % 58 % 54 % 56 % (N) (1876) (670) (469) (737)

In decisions eliciting third-order risk preferences, 56% of the choices are in favor of the prudent choice, that is B3 was preferred over A3. Deck and Schlesinger (2010), Ebert and Wiesen (2011), and Noussair, Trautmann, and van de Kuilen (2011) also observed a majority of prudent choices. At first glance their results indicate a higher consistency with the concept of prudence since they found 61% (in Deck and Schlesinger, 2010), 65% (in Ebert and Wiesen, 2011), and 69% (in Noussair, Trautmann, and van

de Kuilen, 2011) of choices in favor of the prudent alternative. Note, however, that in their experiments subjects could only make real monetary gains. The result which is best comparable to their analysis is therefore reported in the third column of Table 3.1. Here, we find that 60% of choices in gains are in favor of the prudent choice, which is closer to the results of the other experiments.

In decisions eliciting fourth-order risk attitudes, 56% of the choices are in favor of the temperate choice. This stands in contrast to the results of Deck and Schlesinger (2010). They report that only 38% of the choices favor the temperate choice and conclude that individuals are on average intemperate. In their experiment, only four decisions for each subject elicited temperance. Noussair, Trautmann, and van de Kuilen (2011) used five decisions for each subject and found that 60% of the choices were temperate. Our result on temperance, based on 28 decisions for each subject, adds confidence that the prevalent choice pattern indeed exhibits temperance. There are two reasons why we expect this result to be robust. First, our design was in final outcomes only and thus could be very easily understood by subjects and second, our analysis of temperance rests on 1876 observations while Deck and Schlesinger (2010) only used 396 observations to elicit temperance. This result could have strong consequences since the results of Deck and Schlesinger (2010) on intemperance led them to conclude that CPT better explains data on higher-order risk attitudes than EUT with CRRA or CARA functions. When choosing only temperate choices in gains as the appropriate comparison to the results of Deck and Schlesinger (2010) the gap between our and their result even broadens by two percentage points.

A striking feature of the results in Table 3.1 is the fact that the percentage of risk- averse over risk-seeking, of prudent over imprudent, and of temperate over intemperate choices are so close that they all are rounded to 56% (the more exact percentages are 56.24% risk-averse, 55.70% prudent, and 56.02% temperate choices).6 _{Later, we will} explore whether this stems from the fact that the same individuals who are risk averse, are also prudent and temperate. But it already indicates that third- and fourth-order risk preferences are equally present as second-order risk preferences among subjects.

Based on these first results we can summarize the following. If we would accept the assumptions that individuals are homogeneous and observations are independent, we could state with a high degree of confidence that the behavior of the representative sub- ject exhibits stochastic risk aversion, stochastic prudence, and stochastic temperance. 6_{If we compute the 90% confidence intervals of the underlying population percentages we receive}

54.14%-58.14%, 53.78%-57.61%, and 54.11%-57.93%, respectively. The fact that these intervals widely overlap again highlights the possibility that the underlying percentage of choosing the ‘averse’ option may well be equal or very close for different orders.

This first picture, however, is based on risk preferences across domains. Columns 3 to 5 of Table 3.1 state for each domain separately in how many decisionsBn was chosen over

An (for each n = 2,3,4). Choices eliciting risk aversion are very homogeneous across

domains. Also, choices eliciting temperance do not show much variation. Only choices eliciting prudence seem to slightly depend on the domain they lie in. We will make more precise statements in the following section which explores the relations of higher- order risk preferences within different domains in more detail. For each of the nine cells distinguishing domains in Table 3.1, we conduct a binomial test to determine whether our observed pattern across domains can also be observed within different domains. For all these nine cells, i.e. for all domains and orders, we can reject the hypothesis that the underlying probability of choosing Bn is equal or below 1/2 (with one marginally

insignificant exception for prudence in mixed domains).7 Based on all twelve binomial tests, we can now formulate our first result.

Result 3.1 Aggregated choices over all subjects represent stochastic risk aversion, pru- dence, and temperance not only across domains but also within the gain, loss, and mixed domain.

While Result 3.1 is based on pooling subjects, the remaining results are based on a within-subject analysis.