The Limits of Price Discrimination Under Privacy Constraints

TL;DR

This paper investigates how privacy constraints affect the limits of price discrimination by introducing a market-level privacy mechanism that hides the true market with probability $\beta$ and reveals it with $1-\beta$. Grounding the analysis in a Bayesian framework with a uniform prior over markets, the authors show that the set of feasible consumer-producer utilities forms a convex polygon, obtained as a linear image of a high-dimensional polytope, rather than the triangle found in non-private settings. Key findings include that privacy always reduces both the minimum and maximum utility achievable by the producer, while it raises the consumer's minimum utility but can lower or raise the maximum consumer utility, with nonmonotone responses to changes in $\beta$. The results extend from the two-value case ($K=2$) to general $K$-value valuations, revealing a direct effect (noise-perturbation) and an indirect effect (pricing strategy change) that reshape the feasible set; the general $K$ case yields a convex polytope as the image of a polytope under a linear map. The work provides a structural understanding of how data privacy reshapes surplus division in pricing, offering methodological tools and insights relevant for policy and platform design in privacy-aware markets.

Abstract

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.

Figures (10)

• Figure 1: The surplus triangle of bergemann2015limits
• Figure 2: Illustration of our characterization of the set of all consumer and producer utilities under a privacy mechanism (the triangle $ABC$) with a comparison to the setting without privacy (the triangle $TQR$). Both panels correspond to a setting with two consumer values: high and low. The left panel corresponds to a setting where high and low consumer values are far apart. The right panel corresponds to a setting where high and low consumer values are close, and the privacy parameter is small.
• Figure 3: Illustration of the set of possible consumer and producer utilities (the solid blue area) and its comparison with the non-private case (the triangle $TQR$) for an example with $K=5$ values.
• Figure 4: Illustration of $\mathcal{S}$ for the case $\alpha^* \geq \eta$.
• Figure 5: Illustration of $\mathcal{S}$ for the case $\alpha^* \leq \eta$.

Theorems & Definitions (36)

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