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：我们考虑一个生产者向连续统的隐私意识消费者销售产品的问题，其中生产者可以实施三级价格歧视，对不同市场细分提供不同价格。在无隐私约束的情况下，Bergemann、Brooks和Morris [2015] 刻画了所有可能的消费者-生产者效用的集合，证明其构成一个三角形。我们考虑一种通过概率性掩盖每个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-生产者效用集合形成一个凸多边形，明确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研究隐私机制对生产者和消费者效用的影响。特别地，我们证明隐私约束总是通过降低可达最大和最小效用而损害生产者。从消费者角度看，尽管隐私机制确保最小效用在非隐私情形基础上有所提升，但有趣的是，它可能会降低最大效用。最后，我们论证提高隐私水平并不必然增强这些效应。例如，生产者最大效用或消费者最小效用可能对隐私水平的提高呈现非单调行为。