For delegated choice with inspection cost and free delegation, a maximal mechanism combining a single threshold-based inspected proposal with the best no-inspection option is a tight 3-approximation.
Pandora’s problem with nonobligatory inspection: Optimal structure and a ptas
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Delegation with Costly Inspection
For delegated choice with inspection cost and free delegation, a maximal mechanism combining a single threshold-based inspected proposal with the best no-inspection option is a tight 3-approximation.