Information-theoretic limitations of data-based price discrimination

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. The main results of this paper are twofold. The first set of results is pricing strategy independent and reveals the fundamental information-theoretic limitation of any data-based pricing strategy in revenue generation for two cases: 3PD and uniform pricing. The second set of results proposes the $K$-markets empirical revenue maximization (ERM) strategy and shows that the $K$-markets ERM and the uniform ERM strategies achieve the optimal rate of convergence in revenue to that generated by their respective true-distribution 3PD and uniform pricing optima. Our theoretical and numerical results suggest that the uniform (i.e., $1$-market) ERM strategy generates a larger revenue than the $K$-markets ERM strategy when the sample size is small enough, and vice versa.