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 spent in a mechanism have not been fully discussed. In this paper, we investigate the correctness of the revelation principle when strategies of agents are costly. We point out two key results: (1) The strategy of each agent in the direct mechanism is just to report a type. Each agent does not need to take any other action to prove himself that his reported type is truthful, and should not play the strategy as specified in the indirect mechanism. Hence each agent should not spend the strategic costs in the direct mechanism (see Proposition 1); (2) When strategic costs cannot be neglected in the indirect mechanism, the proof of revelation principle given in Proposition 23.D.1 of the book ``A. Mas-Colell, M.D. Whinston and J.R. Green, Microeconomic Theory, Oxford University Press, 1995'' 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.