The revelation principle does not always hold when strategies of agents are costly

The revelation principle asserts that for any indirect mechanism and equilibrium, there is a corresponding direct mechanism with truth as an equilibrium. Although the revelation principle has been a fundamental theorem in the theory of mechanism design for a long time, so far the costs related to strategic actions of agents have not been fully discussed. In this paper, we propose the notion of profit function, and claim that the definition of Bayesian Nash equilibrium of mechanism should be based on the profit function instead of the utility function when strategies of agents are costly (see Definition 23.D.1'). After then, we derive two key results: (1) The strategic action of each agent in a direct mechanism is just to report a type, and each agent does not need to spend any strategic cost occurred in any indirect mechanism (see Proposition 1); (2) When strategies of agents are costly, the proof of revelation principle is wrong (see Proposition 2). We construct a simple labor model to show that a Bayesian implementable social choice function is not truthfully implementable (see Proposition 4), which contradicts the revelation principle.