On the limitations of data-based price discrimination

Recent technological advances have enabled firms to use data to price discriminate. This paper studies third-degree price discrimination (3PD) based on a random sample of valuation and covariate data, where the covariate is continuous, and the distribution of the data is unknown to the seller. We first propose a $K$-markets empirical revenue maximization (ERM) strategy and study its rates of convergence in revenue. We then establish the fundamental information-theoretic limitation of any data-based pricing strategy and show that the $K$-markets ERM and the uniform (i.e., $1$-market) ERM strategies generate revenue converging to that of the true-distribution 3PD and uniform optima, respectively, at the optimal rate. A key takeaway from our information-theoretic limitation results is that, no sample-based 3PD strategy is able to escape from the curse of dimensionality and hence the $K$-markets ERM strategy is not an exception. This result prompts us to compare the revenues from the $K$-markets ERM and the uniform ERM in more specific cases. This comparison is ambiguous, in contrast to the classic pricing problem with a known distribution where third-degree price discrimination is at least as good as uniform pricing in generating revenue.