Optimal Auction Design with Deferred Inspection and Reward

Consider a mechanism run by an auctioneer who can use both payment and inspection instruments to incentivize agents. The timeline of the events is as follows. Based on a pre-specified allocation rule and the reported values of agents, the auctioneer allocates the item and secures the reported values as deposits. The auctioneer then inspects the values of agents and, using a pre-specified reward rule, rewards the ones that have reported truthfully. Using techniques from convex analysis and calculus of variation, for any distribution of values, we fully characterize the optimal mechanism for a single agent. Using Border's theorem and duality, we find conditions under which our characterization extends to multiple agents. Interestingly, the optimal allocation function, unlike the classic settings without inspection, is not a thresholding strategy and instead is an increasing and continuous function of the types. We also present an implementation of our optimal auction and show that it achieves a higher revenue than auctions in classic settings without inspection. This is because the inspection enables the auctioneer to charge payments closer to agent's true value without creating incentives for them to deviate to lower types