Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
Sample size: 500 individuals.
We use a within-subjects design, which significantly reduces the number of participants. We consider testing Corollary 1 (our "experiment 1") as one experiment with a within-subjects design, and the cost-bounding exercise (our experiment two block 1 and and experiment two block 2, the only difference between which is the level of the "high" reward ) as another experiment with a within-subjects design.
For each of these two experiments, for a small effect size (Cohen's d_z - not: NOT Cohen's d) of 0.2, Type I error (alpha) of 0.05, a power level of 0.8, and a two-sided ("difference is greater than 0" for experiment 2, or, for testing corollary 1, that the difference in revelation earlier is greater than the difference for revelation later) test, the R package "pwr" yields that 199 participants are necessary, which we round up to 200. Thus, 200 individuals are necessary for each of two experiments. We allow participants to take part in both experiments; given the randomness in the experiment (which implies that not all of the 200 participants will have all of the necessary data), and assuming (conservatively) that about 200 participants (across both experiments together) will not have enough data, we arrive at the necessary number: 200 (for the first experiment) + 200 (for the second experiment) + 100 (to account for missing data issues)=500 participants.
Not that the effect size used for this computation is small (Cohen (1998) defines "small" to be 0.2 or less), and thus, this setup can detect quite small differences in behavior.