Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)

The minimum detectable difference, d, that can be detected for a given sample size is given by Diggle et al. (2002), pp. 30; Liu and Wu (2005); Kupper and Hafner (1989):
d = sigma * (z_{1-alpha/2}+z_{1-beta}) * sqrt( 2 * (1+(M - 1) * rho) / (M * n))
where d = (mu0-mu1), mu0 and mu1 are the means for each treatment group, sigma is their common standard deviation, alpha=0.05 (Type I error), beta=0.20 (Type II error), therefore z_{1-alpha/2} = 1.96, z_{1-beta} = 0.84, M the number of repeated measurements (M = 2 in our case, since subjects undertake each task twice, once for self and once for others), rho is the interclass correlation coefficient between measurements, and n is the per group sample size. Assuming values for rho = {0, 0.3, 0.6, 0.9}, a target sample size of N = 100 per group can detect differences between the treatment groups in terms of switching point as follows:
For risk aversion: 0.34-0.57
For prudence: 0.43-0.73
For temperance: 0.48-0.8
Diggle, P. J., P. Heagerty, K.-Y. Liang, and S. L. Zeger (2002). Analysis of Longitudinal Data (2nd ed.). New York, USA: Oxford University Press Inc
Liu, H. and T. Wu (2005). Sample size calculation and power analysis of time-averaged difference. Journal of Modern Applied Statistical Methods 4 (2), 434–445.
Kupper, L. L. and K. B. Hafner (1989). How appropriate are popular sample size formulas? The American Statistician 43 (2), 101–105.