Ironing in Dynamic Revenue Management: Posted Prices & Biased Auctions

We consider the design of the revenue maximizing mechanism for a seller with a fixed capacity of C units selling over T periods to buyers who arrive over time. The buyers have single unit demand and multi-dimensional private information-both their value for the object and the deadline by which they must make a purchase are unknown to the seller. This contrasts with previous work where buyers have single dimensional private information-deadlines are publicly observed and only values are private. Here, the optimal mechanism can be computed by running a dynamic stochastic knapsack algorithm. However, these mechanisms are only optimal with private deadlines when the calculated allocation rule is monotone-buyers with higher values and later deadlines should be allocated with higher probability. Such monotonicity only arises in very special cases. By contrast, in the classic static environment of Myerson [7] monotonicity is only violated for 'irregular' value distributions. Myerson characterizes the optimal mechanism by a procedure he calls 'ironing.' We characterize the optimal mechanism in our general dynamic environment by providing the dynamic counterpart of ironing. We show that only a subset of the monotonicity constraints can bind in a solution of the seller's dynamic programming problem. The optimal mechanism can be characterized by 'relaxing' these constraints with their appropriate dual multiplier. Further, the optimal mechanism can be implemented by a series of posted prices followed by a 'biased' auction in the final period where buyers have the auction biased in their favor depending on their arrival time. Our theoretical characterization complements the existing computational approaches for ironing in these settings (e.g. Parkes et al. [10]).