Optimal digital product auctions with unlimited supply and rebidding behavior

We consider a digital product seller who needs to determine the number of items to sell and its price over an infinite horizon. The seller indeed owns unlimited supply of the digital products like music or software. Each buyer is risk-neutral and needs one unit of the product. The number of buyers in each period and their private valuations are random. The seller conducts an auction to allocate the items in each period. The buyers who fail to gain one item in the previous periods will rebid in the subsequent auctions. Regarding the formulated dynamic program, we prove that the optimal allocation solution is a variant of the base-stock policy. Based on the known Revenue Equivalence Principle, we also prove that the generalized second-price auction and the first-price auction will result in the same expected revenue for the seller. Finally, with mild technical modifications, the results of the infinite-horizon case can be extended to the finite-horizon case even if the demand is time-varying stochastic and independent.