Dynamic Auctions with Demand Learning

Posted: 3 Sep 2014

See all articles by Archis Ghate

Archis Ghate

University of Washington; Independent

Date Written: September 1, 2014


The minimum bid is a key design variable in retail auctions as it affects the number of posted bids and the clearing price. Retailers often conduct sequential, single-unit auctions online as a revenue generation and inventory clearing tool, and need to dynamically decide the minimum bid in each auction depending on the inventory level at that time. To reduce inventory costs, it may in fact be optimal to scrap some of the inventory rather than holding it until it is auctioned off. In auctions of fashion goods, seasonal products, new products, and where the seller is new to the market, the seller may be uncertain about the market response. Such a seller may want to dynamically learn the demand by observing the number of posted bids. This introduces an additional, exploration versus exploitation trade-off. We formulate a Markov decision process (MDP) to study this dynamic auction-design problem. We consider a Vickrey mechanism, and a random number of bidders with private, independent, and identically distributed valuations.

We first develop a clairvoyant model where the seller knows the demand distribution. We prove that it is optimal to scrap all inventory above a certain threshold and then auction the remaining units. We derive a first order necessary condition, which states that, in any inventory level, the bidders' virtual value (as defined by Myerson) at an optimal minimum bid should equal the seller's marginal profit. This is an intuitive generalization of Riley and Samuelson's well-known result from the one single-unit auction case. Under the standard regularity condition that the virtual value is strictly increasing, our first order necessary condition is also sufficient and leads to a structured value iteration algorithm for efficiently computing optimal policies. In that case, we prove that optimal minimum bids (i) decrease with increasing inventory levels, and (ii) are higher than optimal minimum bids in one single-unit auction. We also prove that optimal scrapping thresholds and optimal minimum bids both decrease with increasing inventory costs.

We illustrate our results and perform sensitivity analyses using examples with a Poisson-distributed number of bidders and Beta-distributed private valuations.

In the second half of the paper, we assume that the number of bidders is Poisson distributed but the seller does not know its mean. The seller uses a mixture-of-Gamma prior on this mean and updates this belief over the sequence of auctions while simultaneously deciding minimum bids and scrapping quantities. This results in a high-dimensional Bayesian MDP whose exact solution is intractable. We therefore implement a certainty equivalent control (CEC) heuristic instead. We extend our clairvoyant structural result to CEC and numerically investigate the effect of initial inventory and mean demand on CEC performance.

Keywords: sequential auctions, optimal learning, bayesian statistics

JEL Classification: D44, C61

Suggested Citation

Ghate, Archis and Ghate, Archis, Dynamic Auctions with Demand Learning (September 1, 2014). Available at SSRN: https://ssrn.com/abstract=2490280 or http://dx.doi.org/10.2139/ssrn.2490280

University of Washington ( email )

Seattle, WA
United States

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