Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)
For each primary pairwise comparison, we have 500 spectators per treatment (e.g., No Response x Full Internalization), each making five decisions, yielding 2,500 punishment decisions per treatment. Using a two-sided two-sample t-test with a significance level of 0.05, 80% power, a baseline punishment share of 0.5, and a standard deviation of 0.5, the minimum detectable effect is approximately 4.0 percentage points, treating decisions as independent for this calculation.