On this slide deck, page 12, it says "then this direct mechanism implements exactly the same allocation as the indirect mechanism we started". What is indirect mechanism in this context and in particular, what is the "indirect mechanism we started" on this page?
In direct mechanism agents directly report their preferences (preferences are observable).
In indirect mechanism agents don’t report their preferences directly. Preferences can be observed only indirectly through signals or behavior.
By Revelation Principle if some outcomes can be implemented in indirect mechanism they must be also implementable in the direct mechanism where everything is observable - that’s what the quote is trying to say.