Estimating Correlated Valuations for Data-Driven Bundle Pricing with Copula Inference

Bundling is pervasive in many markets. Optimal bundle pricing requires learning the joint distribution of consumer valuations for items in the bundle, that is, consumers' willingness to pay for the items. However, retailers typically do not observe this quantity. In this work, we develop a statistically consistent and efficient inference procedure for fitting a copula model over the latent consumer valuations, using only transaction data. We show that this method is computationally tractable by transforming the seemingly intractable likelihood function (i.e., integrals over the latent valuations) into distribution function evaluations, which can be estimated efficiently. The inference procedure which utilizes copula estimates correlated valuations over the bundle items and allows arbitrary marginal distributions. It offers significant flexibilities compared to existing methods in the bundling literature which typically limit to certain distribution families (e.g., uniform or normal) or restrictive dependence structure (e.g., independence or perfect correlation). We also extend the method to settings that exhibit intra-category substitution effect, where each bundle component could face competition from similar products in the same category. We show that the joint valuation estimation can be integrated into a choice modeling framework which is popular in many industries. Lastly, we show how the method can have a real impact in practice with a case study in the telecommunications industry that investigated phone-plan bundles. Our results highlight the importance of modeling the joint dependence across product valuations and show that there is room for a significant increase in profits over current practice