A Costly Bayesian Implementable Social Choice Function May Not be Truthfully Implementable

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The revelation principle is a fundamental theorem in many economics fields. In this paper, we construct a simple labor model to show that a social choice function which can be implemented costly in Bayesian Nash equilibrium may not be truthfully implementable. The key point is the strategy cost property given in Section 4: In the direct mechanism, each agent only reports a type and will not pay the strategy cost which would be paid by himself when playing strategies in the original indirect mechanism. As a result, the revelation principle may not hold when agents' strategies are costly in the indirect mechanism.