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：本文根据随机抽样的估值和共变数据研究了三度价格歧视(3PD),其中共变数据是连续的,数据的分配是卖方所不知道的。本文的主要结果有两个方面。第一组结果是价格战略独立,揭示了基于数据的价格战略在创造收入方面对以下两种情况的基本信息理论限制:3PD和统一定价。第二组结果提出了以K为单位的市场实证收入最大化战略,并表明以K为单位的机构风险管理和统一的机构风险管理战略实现了收入与其各自真实分配3PD和统一定价选择所产生的收入的最佳趋同率。我们的理论和数字结果表明,统一(即以美元为单位的)机构风险管理战略在抽样规模足够小的情况下产生比以K美元为单位的机构风险管理战略更大的收入,反之亦然。