I am finding the revelation principle very confusing. So I was watching the Game Theory online course by Stanford and UBC. In their video about revelation principle, here are the original words from the slides about revelation principle:
It turns out that any social choice function that can be implemented by any mechanism can be implemented by a truthful, direct mechanism.
Is this a correct statement? I feel like it is too strong. As far as I can understand from [Myerson 1981]'s result, it seems the revelation principle there only hold for the indirect Bayesian mechanism, not "any" indirect mechanism.
Also, another paper by [Maskin et al. 1979] has the following statements from their paper:
[From Paragraph after Proof of Theorem 7.1.1] Yet there is an even more compelling reason for turning our attention to indirect mechanisms, which is that there are some important Social Choice Rules which cannot be implemented in Nash strategies by direct mechanisms but which can be implemented by appealing to indirect mechanisms.
Does this paragraph from Maskin et al's paper mean there exist some social choice functions that cannot be implemented by a direct mechanism but can be implemented by an indirect mechanism? Isn't this contradicting with the online course? I am wondering which of them is wrong, or did I misunderstand something?