Bounds and Heuristics for Multi-Product Personalized Pricing

We present tight bounds and heuristics for personalized, multi-product pricing problems. Under mild conditions we show that offering a non-personalize price in the direction of a positive vector (factor) has a tight profit guarantee relative to optimal personalized pricing. An optimal non-personalized price is the choice factor, when known. Using a factor vector with equal components results in uniform pricing and has exceedingly mild sufficient conditions for the bound to hold. A robust factor is presented that achieves the best possible performance guarantee. As an application, our model yields a tight lower-bound on the performance of linear pricing relative to personalized non-linear pricing, and suggests effective non-linear price heuristics relative to personalized solutions. Additionally, our model provides guarantees for simple strategies such as bundle-size pricing and component-pricing with respect to personalized mixed bundling policies. Heuristics to cluster customer types are also developed with the goal of improving performance by allowing each cluster to price along its own factor. Numerical results are presented for a variety of demand models, factors and clustering heuristics. In our experiments, economically motivated factors coupled with machine learning clustering heuristics performed best.