The revelation principle fails when the format of each agent's strategy is an action
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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. We point out that in any game the format of each agent's strategy is either an informational message or a realistic action, and the action format is very common in many practical cases. The main result is that: For any given social choice function, if the mechanism which implements it has action-format strategies, then "honest and obedient" will no longer be the Bayesian Nash equilibrium of the direct mechanism, actually the social choice function can only be implemented "dishonestly and disobediently" in Bayesian Nash equilibrium by the direct mechanism. Consequently, the revelation principle fails when the format of each agent's strategy is an action.
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