A Note on 'The Limits of Price Discrimination' by Bergemann, Brooks, and Morris

Abstract: This note revisits the analysis of third-degree price discrimination developed by Bergemann et al. (2015), which characterizes the set of consumer-producer surplus pairs that can be achieved through market segmentation. This was proved by means of market segmentation with random prices, but it was claimed that any segmentation with possibly random pricing has a corresponding direct segmentation, where a deterministic price is charged in each market segment. However, the latter claim is not correct under the definition of market segmentation given in the paper, and we provide counterexamples. We then propose an alternative definition to resolve this issue and examine the implications of the difference between the two definitions in terms of the main result of their paper.

• The paper demonstrates that surplus outcomes attainable via stochastic pricing cannot always be replicated by deterministic direct segmentation.
• It provides explicit counterexamples highlighting the failure of equivalence between general segmented pricing and direct segmentation strategies.
• The study introduces a revised segmentation definition that restores equivalence in surplus attainability, emphasizing implications for both theoretical analysis and practical mechanism design.

Revisiting Direct Segmentation in Third-Degree Price Discrimination

Summary and Context

This note provides a critical reassessment of a central result in "The Limits of Price Discrimination" by Bergemann, Brooks, and Morris (2015), which examines third-degree price discrimination by monopolists segmenting markets based on consumer characteristics. The original paper establishes a complete characterization of attainable consumer-producer surplus pairs via stochastic segmentation and claims that any outcome implementable with stochastic pricing is also implementable through "direct segmentation"—deterministic pricing within each segment.

The present note identifies falsity in this claim: the asserted equivalence between general segmented pricing and direct segmentation does not universally hold under the original definitions. By constructing explicit counterexamples, the author demonstrates the existence of surplus outcomes attainable via stochastic pricing that cannot be equivalently replicated by any direct segmentation. The paper also proposes rigorous revisions to the definitions of segmentation to restore the equivalence and reassesses the surplus attainment results under both the original and revised frameworks.

Main Contributions

Counterexamples to Equivalence of Stochastic and Direct Segmentation

The analysis delivers concrete counterexamples showing that for some markets and segmentation structures, the joint distribution of consumer types and prices achievable by allowing stochastic pricing cannot be replicated by any direct segmentation (i.e., partitioning the market into at most $K$ submarkets where each receives a deterministic price $v_k$). These counterexamples hold whether the original stochastic pricing rule is itself randomized or deterministic, demonstrating that the failure of equivalence is rooted in the definition of segmentation rather than the presence of randomization.

The core insight is that the mapping from general pricing strategies (possibly mixed) to deterministic pricing at the segment level is not always surjective on the set of induced joint distributions of valuations and prices. Thus, restricting analysis to direct segmentation is not without loss of generality.

Alternative Definition Restoring Equivalence

To remedy this, the author proposes an amended segmentation definition in which segmentation is represented as a distribution over type-price pairs $(x, v_k)$. Here, the monopolist can index segments not just by their conditional valuation distributions, but by both the distribution and its assigned price. Under this definition, any feasible outcome under general segmentation is also feasible under direct segmentation, restoring the desired equivalence in surplus attainability.

Impact on Surplus Triangle Characterization

A critical result from Bergemann et al. is that all surplus tuples $(u, \pi)$ with $u \ge 0$, $\pi \ge \pi^*$, and $u+\pi \le w^*$ (where $u$ is consumer surplus, $\pi$ is producer surplus, $w^*$ is total surplus, and $\pi^*$ is surplus under uniform monopoly pricing) are attainable through segmentation. This work re-examines the surplus triangle under the original definition, considering the true set of attainable surplus pairs through direct segmentations. The author establishes that, except in specific cases where the aggregate market coincides with a "characteristic market" (e.g., constant elasticity equal to $-1$), the set of surplus pairs achievable through direct segmentation still coincides with the triangle described above. The characterization is exact except for special degenerate cases, such as unit-elastic demand, where not all points in the triangle are attainable.

Technical Analysis

The core technical apparatus includes:

• Rigorous definition of market segmentations as probability distributions over consumer types, and precise distinctions between stochastic and direct (deterministic) segmentation.
• Algebraic constructions demonstrating the inability to match certain distributions over valuations and prices through direct segmentation, by carefully comparing marginal and conditional distributions.
• Convex analytic arguments to establish the set of surplus pairs attainable under alternative segmentation schemes and when exact equivalence with the Bergemann et al. surplus triangle obtains.
• Formalization of the revised segmentation definition, mapping any (possibly stochastic) pricing rule to an outcome-equivalent direct segmentation under the new framework.

Implications and Future Directions

The paper corrects an important point of methodology for the analysis of price discrimination in theoretical IO. The results clarify that the set of feasible surplus outcomes can depend nontrivially on whether the monopolist’s segmentation procedure can separate "segments" with identical valuation distributions but different assigned prices—an important distinction for practical mechanism design and for empirical identification strategies.

The findings advise caution for subsequent works leveraging Proposition 2 from Bergemann et al., making it clear that under the original framework, surplus triangle characterizations need to be interpreted with these subtleties in mind. The precise technical conditions under which direct segmentation is without loss are now more accurately circumscribed.

The revised definitions present an avenue for unifying and simplifying welfare analyses in the presence of price discrimination, and may inform new mechanism design models where observable segment distinctions and posted pricing protocols are constrained by information-theoretic or regulatory considerations.

Future theoretical directions could include:

• Full characterization of surplus attainability for alternative forms of demand and under observable and unobservable consumer heterogeneity.
• Algorithmic implementations of segmentation schemes under practical constraints where the revised definition may only be approximated.
• Analysis of the robustness of surplus attainment to additional institutional constraints, such as restrictions on segment granularity, menu complexity, or commitment.

Conclusion

This note corrects the previous equivalence assertion on the attainability of surplus pairs under direct segmentation in third-degree price discrimination. Through formal counterexamples and updated definitions, it precisely delineates the mapping between general stochastic segmentation and direct deterministic segmentation, providing both theoretical clarification and updated surplus attainability results. The work refines the foundations of modern price discrimination theory and offers formal tools for subsequent research on welfare and implementation in segmented monopoly markets.