Dynamic Offer Selection and Buy-Now Decisions on Resale Platforms: A Finite-Horizon Dynamic Programming Model

Digital resale platforms allow buyers to either purchase at a posted price or submit a lower offer subject to uncertain seller acceptance. This study develops a stylized finite-horizon dynamic program in which the buyer jointly chooses whether to bid and how much to offer while retaining a buy-now fallback. A savings transformation yields an exact participation boundary and a one-dimensional characterization of the optimal offer. The optimal discount increases with continuation saving; hence offers are less aggressive when more opportunities remain and rise after successive rejections. Each opportunity is modeled as a conditional exposure event to an exogenous seller pool; platform-specific visibility, matching, competing-buyer preemption, and strategic seller response are outside the baseline model. Numerical illustrations use assumed parameters and are internal model calculations, not statistical estimates or evidence of platform performance. Under the reference specification, the adaptive policy yields an expected payment of 283.34, only 0.13 units (4.42 basis points of buy-now) below the best fixed rule optimized for the same ten-opportunity horizon. A normalized sensitivity map shows that the model-implied gain increases with a wider feasible gap and lower opportunity cost but remains below 1% of buy-now in the stationary region examined. The contribution is a theoretical policy characterization, not a deployable buyer-facing tool; deployment would require estimating seller tolerance from accepted, rejected, and expired offer logs and testing the policy out of sample or in the field.