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：本文研究了基于随机样本的估值与协变量数据的三级价格歧视（3PD），其中协变量为连续型，且数据的分布对卖方未知。本文的主要结果分为两部分。第一部分结果独立于定价策略，揭示了在两种情形下（3PD与统一定价）任何基于数据的定价策略在收入生成中的根本性信息论局限。第二部分结果提出了$K$市场经验收益最大化（ERM）策略，并证明$K$市场ERM与统一ERM策略在收入收敛速率上可分别达到其对应真实分布3PD与统一定价最优解的最优速率。我们的理论与数值结果表明，当样本量足够小时，统一（即1市场）ERM策略产生的收入高于$K$市场ERM策略，反之亦然。