  We present tight bounds and heuristics for personalized, multi-product pricing problems. Under mild conditions we show that the best price in the direction of a positive vector results in profits that are guaranteed to be at least as large as a fraction of the profits from optimal personalized pricing. For unconstrained problems, the fraction depends on the factor and on optimal price vectors for the different customer types. For constrained problems the factor depends on the factor and a ratio of the constraints. 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 optimal 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 optimal personalized mixed bundle pricing. 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 that illustrate the tradeoffs between using the economic factor and the robust factor for each cluster, as well as the tradeoffs between using a clustering heuristic with a worst case performance of two and a machine learning clustering algorithm. In our experiments economically motivated factors coupled with machine learning clustering heuristics performed best. 
Bounds and Heuristics for Multi-Product Personalized Pricing