We study the problem of delegated choice with inspection cost (DCIC), which is a variant of the delegated choice problem by Kleinberg and Kleinberg (EC'18) as well as an extension of the Pandora's box problem with nonobligatory inspection (PNOI) by Doval (JET'18). In our model, an agent may strategically misreport the proposed element's utility, unlike the standard delegated choice problem which assumes that the agent truthfully reports the utility for the proposed alternative. Thus, the principal needs to inspect the proposed element possibly along with other alternatives to maximize its own utility, given an exogenous cost of inspecting each element. Further, the delegation itself incurs a fixed cost, thus the principal can decide whether to delegate or not and inspect by herself. We show that DCIC indeed is a generalization of PNOI where the side information from a strategic agent is available at certain cost, implying its NP-hardness by Fu, Li, and Liu (STOC'23). We first consider a costless delegation setting in which the cost of delegation is free. We prove that the maximal mechanism over the pure delegation with a single inspection and an PNOI policy without delegation achieves a $3$-approximation for DCIC with costless delegation, which is further proven to be tight. These results hold even when the cost comes from an arbitrary monotone set function, and can be improved to a $2$-approximation if the cost of inspection is the same for every element. We extend these techniques by presenting a constant factor approximate mechanism for the general setting for rich class of instances.

翻译：我们研究了带检查成本的委托选择问题（DCIC），这是Kleinberg和Kleinberg（EC'18）提出的委托选择问题的一个变体，同时也是Doval（JET'18）提出的非强制检查潘多拉盒问题（PNOI）的扩展。在我们的模型中，代理人可能策略性地误报所提议元素的效用，这与标准委托选择问题中假设代理人如实报告所提议方案效用的设定不同。因此，在给定每个元素外生检查成本的情况下，委托方可能需要检查所提议元素及其他备选方案，以最大化自身效用。此外，委托行为本身会产生固定成本，因此委托方可自行决定是否进行委托或亲自检查。我们证明DCIC确实是PNOI的泛化形式，其中策略性代理人的侧信息需以特定成本获取，这暗示了其NP难性（由Fu、Li和Liu在STOC'23中证明）。我们首先考虑无成本委托场景，即委托行为本身无额外成本。我们证明：在无成本委托的DCIC问题中，纯委托结合单次检查的机制与无委托的PNOI策略的并集机制能达到3倍近似比，且该界限被证明是紧的。这些结论在检查成本来自任意单调集函数时依然成立；若每个元素的检查成本相同，近似比可提升至2倍。我们进一步扩展这些技术，为广泛实例类别的一般性场景提出了常数倍近似机制。