We study 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. We consider a privacy mechanism that provides a degree of protection by probabilistically masking each market segment. 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：本文研究生产者向连续统的隐私敏感消费者销售产品的问题，其中生产者可实施三级价格歧视，向不同市场细分提供差异化定价。我们考虑一种隐私保护机制，该机制通过概率性掩蔽各市场细分来提供一定程度的隐私保护。我们证明，由此产生的所有消费者-生产者效用集合构成一个凸多边形，并明确表征为某个高维凸多胞形到$\mathbb{R}^2$的线性映射。该表征使我们能够探究隐私机制对生产者效用和消费者效用的影响。具体而言，我们证明隐私约束总会损害生产者利益，同时降低其可达到的最大效用和最小效用。从消费者视角看，尽管隐私机制确保其最小效用相较于非隐私场景有所提升，但有趣的是，该机制可能降低其最大效用。最后，我们证明提高隐私水平并不必然强化这些效应。例如，生产者的最大效用或消费者的最小效用可能随隐私水平提升呈现非单调变化。