Perception of probabilities urns task

In this task we examine whether people differ systematically in the way they process probabilities. Furthermore we want to learn whether such heterogene- ity translates into heterogeneity in both the “Will you win?” measure and risk preference measures. For example, someone who systematically perceived their chances of winning as higher than they actually are would overstate their chances of winning in the “Will you win?” task. Such a person would also overes- timate their chances of winning in a lottery and as a result demand a higher risk premium than someone who perceived probabilities realistically. Hence, the cor- relation between reported chances of winning in the “Will you win?” task and risk preference measures as we observe in the data.

The control task we report here aims to measure if and to what extent such a systematic heterogeneity in the way that subjects process signals relating to their chances of winning actually exists. In this task, subjects participate in a se- ries of lotteries, for which objective probabilities exist but are not communicated explicitly. Instead subjects are provided with a strong visual signal: the image of an urn that they see for 1.5s. The urn contains 100 balls, each of which can be either red or black (seeFigure 4.1). Subjects receive 15€ if a red ball is drawn, and nothing otherwise. Subjects’ task is to state their estimated chance of win- ning the lottery. The idea of the task is that those with a disposition towards perceiving their chances in lotteries more favorably, perceive signals of probabil- ities more positively and, hence, state seeing a higher number of red balls. If the urns task is selected for payment, either belief elicitation is paid or one of the lotteries is played out (seeSection 4.A.5in the appendix for details).

The task consists of 29 trials organized in two blocks with different urns, con- taining either a small, medium-sized, or large number of red balls, i.e., chance of winning. The sequence, in which the urns appear, as well as the position of each colored ball are pre-randomized and the same for all subjects. For an overview of the number of correct balls in each urn refer toTable 4.A.3in the appendix. To be able to control for possible distortions during the visual perception of each particular urn, there was an additional stage before the main task, in which sub- jects estimated the number of red balls in each urn without any connotation of lotteries or winning and losing. A description of how the data are aggregated as well as detailed descriptive statistics can be found in Section 4.A.6 in the appendix.

Our first observation is that subjects on average overstate the number of balls both in stage 2 when they represent probabilities and in stage 1 when they do not (seeTable 4.6). However, on average they to do less so in stage 2 (significant difference; Wilcoxon signed-rank test: p < .0001). This could either indicate that subjects estimate the number of red balls more realistically when they represent probabilities or simply reflect learning over time.

Figure 4.1. Exemplary urn. Each urn contains 100 balls and was shown for 1.5s.

Table 4.6. Urns task: Avg deviations from the correct answer

Stage 1 Stage 2 Difference

(number of red balls) (chances of winning) (Stage 2 - Stage 1)

Mean 3.432 2.680 -0.728

SD 2.775 2.518 2.466

Notes.N = 349. Stage 1 and Stage 2 averages are significantly different from 0 (t-tests: p<.0001 for both). Stage 1 and Stage 2 averages differ significantly (Wilcoxon signed-rank test: p<.0001).

If systematic misperception of probabilities manifested itself in stage 2 but not in stage 1, the difference between them would be the appropriate measure to capture that misperception. However, if it was captured by responses in both stages, an aggregate measure of both would be preferable. We remain agnos- tic about this issue and correlate both the difference and aggregate measures with risk preference measures as well as the response to “‘Will you win?” (see

Table 4.7). For any of the two, we would expect a lower degree of risk aversion (lower risk premia, higher response to the general risk question, more bombs collected in BRET) and a more optimistic response to “Will you win?”(i.e., a higher value), the more positively probabilities are perceived. Such a pattern is neither visible for the difference measure nor for the aggregate measure. Corre- lations are either not significantly different from 0 or do not go in the expected direction. Hence, the urns task does not provide evidence that the correlation

Table 4.7. Urns task: Correlations with average deviations from the true number

“Will you win?” General risk Risk Premium Risk Premium Risk Premium Risk Premium BRET question 25% of 15€ 50% of 15€ 75% of 15€ MPS

Stage 1 -0.085 0.006 0.053 0.047 0.168*** 0.068 0.092*

(number of red balls) (0.123) (0.912) (0.338) (0.396) (0.002) (0.215) (0.092)

Stage 2 -0.103* -0.033 0.082 0.057 0.113** 0.053 0.020

(chances of winning) (0.060) (0.542) (0.134) (0.300) (0.038) (0.329) (0.711) Measure 1: Differences -0.028 -0.047 0.002 -0.025 -0.093* -0.045 -0.097* (Stage 2 - Stage 1) (0.605) (0.388) (0.971) (0.654) (0.089) (0.417) (0.076) Measure 2: Aggregate -0.115** -0.027 0.057 0.039 0.150*** 0.055 0.071 (Avg (Stage 2, Stage 1)) (0.036) (0.627) (0.300) (0.477) (0.006) (0.316) (0.192) Notes.Spearman rank-order correlation coefficients. p-values in parentheses. *** p<0.01, ** p<0.05, * p<0.1. N = 335. Risk preference measures are aggregated over multiple measurements when possible, i.e., weeks 1 and 3 for choice lists, and weeks 1, 2 and 3 for the general risk question. All non-trivial trials are included in the average.

between “Will you win?” and risk preference measures is caused by systematic misperception of probabilities.