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和协变量数据的第三级价格歧视（3PD），其中协变量是连续的，且数据分布对卖方未知。本文的主要结果有两方面。第一组结果独立于定价策略，揭示了在3PD和统一定价两种情形下，任何基于数据的定价策略在收入生成方面的基本信息论局限。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策略的收入收敛速度达到其各自真实分布下的3PD和统一定价最优值。我们的理论和数值结果表明，当样本量足够小时，统一（即1-市场）ERM策略产生的收入大于K-市场ERM策略，反之亦然。