Distribution-Free Pricing

Abstract

We study a monopolistic robust pricing problem in which the seller does not know the customers' valuation distribution for a product but knows its mean and variance. This minimal requirement for information means that the pricing managers only need to be able to answer two questions: How much will your targeted customers pay on average? And, to measure your confidence in the previous answer, what is the standard deviation of customer valuations. We first focus on the maximin profit criterion and derive distribution-free upper and lower bounds on the profit function. By maximizing the tight profit lower bound, we obtain the optimal robust price in closed form as well as its distribution-free, worst-case performance bound. We then extend the single-product result to study the robust pure bundle pricing problem where the seller only knows the mean and variance of each product, and provide easily verifiable, distribution-free, sufficient conditions that guarantee the pure bundle to be more robustly profitable than a la carte (i.e., separate) sales. We further derive a distribution-free, worst-case performance guarantee for a heuristic scheme in which customers choose between buying either a single product or a pure bundle. Moreover, we generalize separate sales and pure bundling to a scheme called clustered bundling that imposes a price for each mutually exclusive subset of products and allows customers to choose one or multiple subsets, and provide various algorithms to compute clustered bundling heuristics. In parallel, most of our results hold for the minimax relative regret criterion as well. The robust price for a single product is in closed form under the maximin profit or minimax relative regret criterion and hence is easily computable. Its interpretation can be easily explained to pricing managers. We also provide efficient algorithms to compute various mixed bundling heuristics for the multi-product problem.