The Revelation Principle feels like one of those results that flip flops between trivially obvious and absurdly impossible… I’m currently in an “absurdly powerful” frame of mind.
I guess the principle is mostly useful for impossibility results? Given an arbitrary mechanism, will you usually be able to decompose it to find the associated incentive compatible mechanism?
I’m on board with “absurdly powerful”. It underlies the bulk of mechanism design, to the point my advisor complains we’ve confused it with the entirety of mechanism design.
The principle gives us the entire set of possible outcomes for some solution concept like dominant-strategy equilibrium or Bayes-Nash equilibrium. It works for any search over the set of outcomes, whether that leads to an impossibility result or a constructive result like identifying the revenue-optimal auction.
Given an arbitrary mechanism, it’s easy (in principle) to find the associated IC direct mechanism(s). The mechanism defines a game, so we solve the game and find the equilibrium outcomes for each type profile. Once we’ve found that, the IC direct mechanism just assigns the equilibrium outcome directly. For instance, if everyone’s equilibrium strategy in a pay-your-bid/first-price auction was to bid 90% of their value, the direct mechanism assigns the item to the person with the highest value and charges them 90% of their value. Since a game can have multiple equilibria, we have one IC mechanism per outcome. The revelation principle can’t answer questions like “Is there a mechanism where every equilibrium (as opposed to some equilibrium) gives a particular outcome?”