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.

翻译：本文研究基于估值和协变量数据的第三度价格歧视（3PD），其中协变量是连续的，并且数据分布未知于卖家。本文的主要结果有两个。第一组结果与定价策略无关，并揭示了任何基于数据的定价策略在两种情况下，即3PD和统一定价下的收入生成中的基本信息论限制。第二组结果提出了$K$-市场经验收入最大化（ERM）策略，并表明$K$-市场ERM和统一ERM策略实现了收入收敛的最优率，分别由其相应的真分布3PD和统一定价的最优值生成。我们的理论和数值结果表明，当样本大小足够小时，统一（即$1$-市场）ERM策略生成的收入比$K$-市场ERM策略更高，反之亦然。