Distribution-Free Pricing

Abstract

Problem definition: 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. Academic/practical relevance: 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? To measure your confidence in the previous answer, what is the standard deviation of customer valuations? Methodology: We focus on the maximin profit criterion and derive distribution-free upper and lower bounds on the profit function. Results: 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 we provide easily verifiable, distribution-free, sufficient conditions that guarantee the pure bundle to be more robustly profitable than à 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 part (i.e., cluster) of a partition of all products and allows customers to choose one or multiple parts (i.e., clusters), and we provide various algorithms to compute clustered bundling heuristics. In parallel, most of our results hold for the minimax relative regret criterion as well. Managerial implications: 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 multiproduct problem.

Price and revenue bounds for bundles of information goods

“…, and i * = arg max i {𝜋 * i }, then the trivial revenue bounds are [v, v]. This compares favorably with the lower bounds given in Bakos and Brynjolfsson (1999) and Geng et al (2005) which assume small coefficients of variations for the distribution of the bundle valuation and which rapidly decay to 0 if these coefficients are moderate to large, which is a fairly realistic scenario; see also Chen et al (2019) for possible sharper bounds when the underlying variance of the product valuation is known. Therefore, any sharpening of these generic bounds without relying on additional assumptions would be of significant interest.…”

“…The extant literature has predominantly relied on certain implicit assumptions among the individual component valuations, such as independence and/or knowledge of higher moments that characterize the bundle valuation, in which case one can apply the law of large numbers, central limit theorem, or other sharper tail inequalities, like Cantelli's or Chebyshev's inequality. For example, Chen et al (2019) leverage knowledge on the mean and variance of the product valuation to derive distribution-free bounds for the single product pricing problem. Since their analysis is motivated by the sale of a single product they do not consider the possible interactions between multiple components that form a bundle.…”

Bounds and Heuristics for Multi-Product Personalized Pricing

“…They also include robust procedures to cluster customers types. Chen et al [2019] present results for single-product distribution-free pricing.…”

Convex Loss Functions for Contextual Pricing with Observational Posted-Price Data

“…Algorithms also exist under alternative knowledge about the valuation distribution. Cohen et al (2015) provide bounds when only the support of the valuation distribution is known, Azar et al (2013) and Chen et al (2019) use the the mean and variance of the valuation distribution, Elmachtoub et al (2020) uses the coefficient of deviation of the valuation distribution, while Bergemann and Schlag (2011) use a neighbourhood containing the true valuation distribution. In contrast to all this work, we have posted-price samples on whether the item sells or not at the price they were given, rather than samples or other knowledge of the valuation distribution.…”