Continuous auction models
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.