Effect on treatment group = Effect of the drug + effect of belief in being treated
Effect on placebo group = Effect of belief in being treated
I think that the mathematical formulation of Mark's concern about using charter school lotteries as follows:
Effect in charter school group = Effect of the charter school intervention + effect of belief in being treated + baseline
Effect on students in standard schools = baseline - (any discouragement effect from losing the lottery)
So if the effect in the charter school students is greater than that in students who lose the lottery and are placed in standard schools, that skill doesn't separate the two sources of variation (placebo versus innate effect of the intervention).
The real test would be to introduce some sort of placebo intervention (a charter school that used standard educational approaches??). But that is not an easy thing to accomplish, for obvious reasons. I suspect that this is why psychologists end up having to use deception as part of their studies, despite the ethical quagmires this produces.
UPDATE: missing link fixed