Optimal minimum bids and inventory scrapping in sequential, single-unit, Vickrey auctions with demand learning

Retailers often conduct sequential, single-unit auctions and need to decide the minimum bidin each auction. To reduce inventory costs, it may be optimal to scrap some of the inventoryrather than holding it until it is auctioned off. In some auctions, the seller may be uncertainabout the market response and hence may want to dynamically learn the demand by observingthe number of posted bids. We formulate a Markov decision process (MDP) to study thisdynamic auction-design problem under the Vickrey mechanism.We first develop a clairvoyant model where the seller knows the demand distribution. Weprove that it is optimal to scrap all inventory above a certain threshold and then auction theremaining units. We derive a first order necessary condition whereby the bidders’ virtual valueat an optimal minimum bid equals the seller’s marginal profit. This is a generalization of Rileyand Samuelson’s result from the one, single-unit auction case. When the virtual value is strictlyincreasing, this necessary condition is also sufficient and leads to a structured value iterationalgorithm.We then assume that the number of bidders is Poisson distributed but the seller does notknow its mean. The seller uses a mixture-of-Gamma prior on this mean and updates this beliefover several auctions. This results in a high-dimensional Bayesian MDP whose exact solutionis intractable. We therefore propose and compare two approximation methods called certaintyequivalent control (CEC) and Q-function approximation. Numerical experiments suggest thatQ-function approximation can attain higher revenues than CEC. ∗University of Washington, Seattle, USA; [email protected]