Bounds and Heuristics for Multiproduct Pricing

For a large class of demand models that allow for multiple consumer types, we present performance guarantees for simple nonpersonalized pricing heuristics relative to optimal personalized pricing. Our results demonstrate that in a general setting, the effectiveness of pricing along a positive vector depends on how the price vector aligns with optimal personalized price vectors. We propose two positive direction vectors: the "economic" and "robust" directions. The economic direction is a convex combination of the optimal personalized price vectors and aims to do well on average. The robust direction offers the best worst-case performance guarantee. By judiciously selecting pricing directions, our results also provide performance guarantees of simple pricing strategies relative to more sophisticated pricing strategies. In particular, we provide performance guarantees for nonpersonalized, optimal linear pricing relative to optimal nonlinear, personalized pricing. Our research also examines the performance of common heuristics for bundle pricing relative to optimal, personalized, bundle-size pricing. Our experiments show that performance often improves when consumer types are clustered and each cluster is offered a price direction. We compared the performance of the k-means clustering heuristic and the farthest point first clustering heuristic. Our findings indicated that k means clustering has significantly superior performance on average. This suggests that businesses could potentially benefit from implementing k-means clustering in their pricing strategies. In conclusion, our study offers valuable insights and performance guarantees for various pricing strategies and their relative effectiveness. These findings could inform pricing decisions and potentially lead to improved outcomes for firms.