A Bayesian implementable social choice function may not be truthfully implementable

The revelation principle is a fundamental theorem in many economics fields such as game theory, mechanism design and auction theory etc. In this paper, I construct an example to show that a social choice function which can be implemented in Bayesian Nash equilibrium is not truthfully implementable. The key point is that agents pay cost in the indirect mechanism, but pay nothing in the direct mechanism. As a result, the revelation principle may not hold when agent's cost cannot be neglected in the indirect mechanism.