We consider a producer's problem of selling a product to a continuum of privacy-conscious consumers, where the producer can implement third-degree price discrimination, offering different prices to different market segments. In the absence of privacy constraints, Bergemann, Brooks, and Morris [2015] characterize the set of all possible consumer-producer utilities, showing that it is a triangle. We consider a privacy mechanism that provides a degree of protection by probabilistically masking each market segment, and we establish that the resultant set of all consumer-producer utilities forms a convex polygon, characterized explicitly as a linear mapping of a certain high-dimensional convex polytope into $\mathbb{R}^2$. This characterization enables us to investigate the impact of the privacy mechanism on both producer and consumer utilities. In particular, we establish that the privacy constraint always hurts the producer by reducing both the maximum and minimum utility achievable. From the consumer's perspective, although the privacy mechanism ensures an increase in the minimum utility compared to the non-private scenario, interestingly, it may reduce the maximum utility. Finally, we demonstrate that increasing the privacy level does not necessarily intensify these effects. For instance, the maximum utility for the producer or the minimum utility for the consumer may exhibit nonmonotonic behavior in response to an increase of the privacy level.

翻译：我们考虑一位生产商向连续统中具有隐私意识的消费者销售产品的问题，该生产商可实施三级价格歧视，即对不同市场细分群体提供差异化定价。在无隐私约束的情形下，Bergemann、Brooks与Morris[2015]刻画了所有可能的消费者-生产商效用集合，证明该集合构成一个三角形。我们考虑一种通过概率化遮蔽各市场细分来提供保护程度的隐私机制，并证明由此产生的消费者-生产商效用集合构成一个凸多边形，该多边形可被显式刻画为某个高维凸多面体到$\mathbb{R}^2$的线性映射。这一刻画使我们能够探究隐私机制对生产商与消费者效用的影响。具体而言，我们证明隐私约束始终会损害生产商利益，其可实现的最大效用与最小效用均会降低。从消费者视角看，尽管隐私机制保障了最小效用相较非隐私场景有所提升，但有趣的是，它可能降低最大效用。最后，我们证明提高隐私保护水平并不必然加剧上述效应。例如，生产商的最大效用或消费者的最小效用可能随隐私保护水平提升呈现非单调变化。