Revelation principle
The revelation principle is a fundamental principle in mechanism design. It states that if a social choice function can be implemented by an arbitrary mechanism (i.e. if that mechanism has an equilibrium outcome that corresponds to the outcome of the social choice function), then the same function can be implemented by an incentive-compatible-direct-mechanism (i.e. in which players truthfully report type) with the same equilibrium outcome (payoffs). In mechanism design, the revelation principle is of utmost importance in finding solutions. The researcher need only look at the set of equilibria characterized by incentive compatibility. That is , if the mechanism designer wants to implement some outcome or property, they can restrict their search to mechanisms in which agents are willing to reveal their private information to the mechanism designer that has that outcome or property. If no such direct and truthful mechanism exists, no mechanism can implement this outcome/property by contraposition. By narrowing the area needed to be searched, the problem of finding a mechanism becomes much easier. The principle comes in two variants corresponding to the two flavors of incentive-compatibility: Example. Consider the following example. There is a certain item that Alice values as  formula_1 and Bob values as formula_2. The government needs to decide who will receive that item and in what terms. Proof. Suppose we have an arbitrary mechanism Mech that implements Soc. We construct a direct mechanism Mech' that is truthful and implements Soc. Mech' simply simulates the equilibrium strategies of the players in Game(Mech). I.e: Reporting the true valuations in Mech' is like playing the equilibrium strategies in Mech. Hence, reporting the true valuations is a Nash equilibrium in Mech', as desired. Moreover, the equilibrium payoffs are the same, as desired. In correlated equilibrium. The revelation principle says that for  every arbitrary "coordinating device" a.k.a. correlating, there exists another direct device for which the state space equals the action space of each player. Then the coordination is done by directly informing each player of his action.