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In this slide deck, p15. it says "The revelation principle requires that the principal can fully commit to the terms of the contract. If this is not the case, an indirect mechanism, which allows for some commitment, may strictly outperform any direct revelation mechanism, which allows for no commitment."

It sounds contradictory, because the first sentence sounds like "For revelation mechanism to outperform indirect mechanism, the principal must be fully committed." But then why does direct revelation mechanism allow for no commitment in the later part of the paragraph? What exactly does "if this is not the case" mean here? What exactly does commitment mean here?

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Commitment here is with respect to the direct mechanism that implements the indirect mechanism. Let $M = \{S,(q,t)\}$ be an indirect mechanism.

By the revelation principal, there exists a direct mechanism $M' = \{\Theta,(q',t')\}$ that can implement the same outcome as the indirect mechanism.

For $M'$ to implement the same outcome, when the designer tells the agents that she is going to use mechanism $M'$, she cannot change her mind once the agents reveal her type. This requires the designer to be able to commit to this mechanism.

Suppose she could not commit. It may be the case that once the agent has revealed her type $\theta$, she may decide to change her mechanism to $M''$. Then, $M'$ cannot implement the same outcome, as she changes her mind.

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