Robust One-Shot Price Experimentation

Abstract

We study the benefits of one-shot price experimentation for a seller in setting a price with limited information about the customer valuation of the product. More specifically, the seller only knows the exact purchase probability associated with a single historical price and aims to maximize the worst-case revenue ratio compared to an oracle with complete knowledge of the value distribution. We assume that the customer value distribution satisfies that the virtual value function is non-decreasing and then assess the benefits of one-time price experimentation for the seller, who can first choose an experiment price and then set a final price based on the historical and experimental purchase probabilities. We analytically characterize the optimal distributionally robust experiment price point and obtain its performance guarantee for any historical purchase probability. We find that, perhaps surprisingly, when the purchase probability of the historical price is above approximately 29.5%, the firm should experiment with a markup price that is higher than the historical price; otherwise, with a markdown price. The intuition behind this is that with the purchase probability of the historical price at or above 29.5% if the markup experiment price does not improve profit beyond the historical price, the firm can fall back to choose a final price lower than the historical price. Moreover, we show the optimal experiment-then-price policy follows a keep-or-adjust structure: the seller keeps using a predetermined price if the experiment price does not yield higher profits than the historical price, and otherwise, adjusts to a final price based on the experiment's outcome. Finally, we show that as the purchase probability of the historical price increases from 0 to 1, the value of experimentation, measured as the incremental competitive ratio, first increases, then decreases, then increases, and finally decreases.

Keywords: experimentation, robust pricing, distribution free, limited information, competitive ratio, markup, markdown