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Provably Near-Optimal Simple Markdown Readjustment Pricing Policies

We consider a canonical revenue management problem wherein a monopolist seller seeks to maximize expected total revenues from selling a fixed inventory of a product to customers who arrive sequentially over time, and the seller is restricted to implement a pricing policy that is non-increasing over time (i.e., markdown pricing). Gallego and van Ryzin (1994) show that the simplest non-increasing price policy, the fixed price policy, is asymptotically optimal in the high-volume regime where both the seller's initial inventory and the length of the selling horizon are proportionally scaled. Specifically, the revenue loss of the fixed price policy is O( k^{1/2}), where k is the system's scaling parameter. In the present paper, we present a novel real-time pricing policy. This policy repeatedly updates the fixed price policy in Gallego and van Ryzin (1994) by taking into account the volatility of the historic sales and forcing the price process to be non-increasing over time. We show that if the seller updates the price for only a single time, then the revenue loss of our policy can be arbitrarily close to O( k^{1/3} \ln k). If the seller updates the prices with a frequency O( \ln k / \ln \ln k ), then the revenue loss of our policy can be arbitrarily close to O( ( \ln k )^3 ). These results are novel and show the power of dynamic pricing in the presence of the markdown restriction.