The revelation principle fails when the format of each agent's strategy is an action

In mechanism design theory, a designer would like to implement a social choice function which specifies her favorite outcome for each possible profile of agents' private types. The revelation principle asserts that if a social choice function can be implemented by a mechanism in equilibrium, then there exists a direct mechanism that can truthfully implement it. This paper aims to propose a failure of the revelation principle. At first we point out that in any game the format of each agent's strategy is either an abstract message or a real action. For any given social choice function, if the mechanism which implements it in Bayesian Nash equilibrium has action-format strategies, then ``honest and obedient'' will not be an equilibrium strategy in the corresponding direct mechanism. Consequently, the revelation principle fails.