AI 中文总结
本文研究离散时间连续拍卖模型，提出短视出价更新规则，推导宏观微分方程并证明出价收敛，推广至异质速度并建立流体动力学极限。
AI 中文摘要
我们研究了一个离散时间拍卖模型，其中多个卖家根据前一轮的表现更新他们的出价。竞标者旨在最大化利润，并仅根据最近的结果调整出价（短视行为）。在每一轮中，拍卖者购买竞标者提供的总数量中价格最低的 $p$ 部分。我们找到了控制宏观动力学的微分方程组，并将其作为微观模型的缩放极限推导出来。我们找到了最大价格演化 $q_t$ 的显式解，并表明从长远来看，竞标者会协调一致，即他们的出价收敛到一个仅取决于初始分布和比例 $p$ 的公共值。对于泊松分布的初始出价，我们建立了经验出价分布和最大价格轨迹的流体动力学极限，并推测了 $q_t$ 相应的高斯波动。最后，我们将模型推广到允许异质的出价更新速度：在这种情况下，最大价格速度变得与最大价格处竞标者的更新速度的调和平均值成正比。
英文摘要
We study a discrete-time auction model in which multiple sellers update their bids according to their performance in the preceding round. Bidders aim to maximize their profits and adjust their bids solely based on their most recent outcome (myopic behavior). In each round, the auctioneer purchases the lowest-priced $p$-fraction of the total quantity offered by the bidders. We find a system of differential equations governing the macroscopic dynamics and derive it as a scaling limit of the microscopic model. We find an explicit solution for the max-price evolution $q_t$ and show that, in the long run, bidders coordinate, i.e., their bids converge to a common value depending only on their initial distribution and the fraction $p$. For Poisson-distributed initial bids, we establish hydrodynamic limits for the empirical bid distribution and the max-price trajectory and conjecture the corresponding Gaussian fluctuations for $q_t$. Finally, we generalize the model to allow for heterogeneous bid-update velocities: in this case, the max-price velocity becomes proportional to the harmonic mean of the update velocities of bidders at the max-price.
Comments30 pages, 9 figures
