Stalking information: Bayesian inventory management with unobserved lost sales

Retailers are frequently uncertain about the underlying demand distribution of a new product. When taking the empirical Bayesian approach of Scarf (19 59), they simultaneously stock the product over time and learn about the di stribution. Assuming that unmet demand is lost and unobserved, this learnin g must be based on observing sales rather than demand, which differs from s ales in the event of a stockout. Using the framework and results of Braden and Freimer (1991), the cumulative learning about the underlying demand dis tribution is captured by two parameters, a scale parameter that reflects th e predicted size of the underlying market, and a shape parameter that indic ates both the size of the market and the precision with which the underlyin g distribution is known. An important simplification result of Scarf (1960) and Azoury (1985), which allows the scale parameter to be removed from the optimization, is shown to extend to this setting. We present examples that reveal two interesting phenomena: (1) A retailer may hope that, compared t o stocking out, realized demand will be strictly less than the stock level, even though stocking out would signal a stochastically larger demand distr ibution, and (2) it can be optimal to drop a product after a period of succ essful sales. We also present specific conditions under which the following results hold: (1) Investment in excess stocks to enhance learning will occ ur in every dynamic problem, and (2) a product is never dropped after a per iod of poor sales. The model is extended to multiple independent markets wh ose distributions depend proportionately on a single unknown parameter. We argue that smaller markets should be given better service as an effective m eans of acquiring information.