AI 中文总结
本文针对两物品拍卖问题，在买家估值分布密度非递减等条件下，证明至少一个最小估值足够高时最优机制为确定性机制，还推测该结论可扩展至三物品场景。
AI 中文摘要
考虑向单个买家出售两个异质物品时，设计收益最优拍卖机制的问题，假设买家对物品的估值分布在正半轴支撑集$[c_i,c_i+b_i]$上具有正、非递减且连续可微的密度函数。本文证明，当至少一个最小估值（即$c_1$或$c_2$）足够高时，最优机制是确定性的；提供了计算$(c_1,c_2)$阈值的方法，超过该阈值后最优机制为确定性机制。同时给出了个体销售机制为最优的买家估值分布的充分条件。研究表明，当$c_1$低而$c_2$高时，卖家最优策略是以最小估值$c_2$出售物品2，从而将问题简化为仅针对物品1的一维场景最优机制求解。基于有前景的初步结果，本文推测该结论可扩展至三物品场景：当$c_1$和$c_2$低但$c_3$高时，卖家最优策略是以最小估值$c_3$出售物品3，进而将问题简化为针对物品1和2的二维场景最优机制求解。
英文摘要
Consider the problem of designing a revenue-optimal auction mechanism when two heterogeneous items are sold to a single buyer having independent valuations over the items. The distributions of the buyer's valuation for the items are assumed to have densities that are positive, nondecreasing, and continuously differentiable on their support sets $[c_i,c_i+b_i]$ in the positive axis. I prove that the optimal mechanism is deterministic if at least one of the minimum valuations (i.e., either $c_1$ or $c_2$) is sufficiently high. I provide a method to calculate the threshold of $(c_1,c_2)$ beyond which the optimal mechanism is deterministic. I also provide a sufficient condition on the distributions of buyer's valuations for which the individual sale mechanism is optimal. I show that when $c_1$ is low and $c_2$ is high, it is optimal for the seller to sell item $2$ at the minimum valuation $c_2$, thus effectively reducing the problem to finding the optimal mechanism in the one-dimensional setting only for item $1$. I conjecture with promising preliminary results that this result can be extended to the three-item setting. Specifically, I conjecture that when $c_1$ and $c_2$ are low but $c_3$ is high, it is optimal for the seller to sell item $3$ at the minimum valuation $c_3$, thus effectively reducing the problem to finding the optimal mechanism in the two-dimensional setting for items $1$ and $2$.
