Algorithmic Pricing via Virtual Valuations (0808.1671v1)

Abstract: Algorithmic pricing is the computational problem that sellers (e.g., in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami et al. (2005) propose this problem and give logarithmic approximations (in the number of consumers) when each consumer's values for bundles are known precisely. Subsequently several versions of the problem have been shown to have poly-logarithmic inapproximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multi-parameter settings; however, logarithmic approximations are inadequate for this purpose. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unit-demand variant of this problem. Here a consumer has a valuation for each different item and their value for a set of items is simply the maximum value they have for any item in the set. We assume that the preferences of the consumers are drawn from a distribution, the standard assumption in economics; furthermore, the setting of a specific set of customers with known preferences, which is employed in all prior work in algorithmic pricing, is a special case of this general problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Our work complements these existing works by considering the case where the consumer's valuations for the different items are independent random variables. Our main result is a constant approximation that makes use of an interesting connection between this problem and the concept of virtual valuations from the single-parameter Bayesian optimal mechanism design literature.

• The paper extends Myerson’s auction theory by applying virtual valuations to achieve a constant-factor approximation in unit-demand pricing.
• It shows that a single virtual price can yield a 3-approximation for general distributions and a refined 2.17-approximation when distributions are identical.
• It demonstrates that near-optimal pricing is computable in polynomial time under the monotone hazard rate condition, enabling practical algorithmic pricing.

Insightful Overview of "Algorithmic Pricing via Virtual Valuations"

The paper, "Algorithmic Pricing via Virtual Valuations," explores the challenges in algorithmic pricing and mechanism design, particularly focusing on the unit-demand setting where consumer valuations for different items are drawn independently from known distributions. This work extends the principles of Bayesian optimal mechanism design, well-understood in single-parameter settings, to multi-item scenarios where approximations are necessary to achieve practical solutions in the absence of exact characterizations.

The paper diverges from prior research that operate under precisely known consumer preferences by incorporating distributions to model consumer valuations. This aligns with standard economic assumptions and allows the authors to aim for constant factors in approximation rather than the logarithmic factors in the number of consumers that previous methods achieved. The authors leverage techniques from Myerson’s optimal auction theory, applying the concept of virtual valuations to extend those insights into pricing contexts outside traditional auction theory.

Core Contributions and Results

The paper introduces the Bayesian Unit-demand Pricing Problem (BUPP) and contrasts it with the classic Bayesian Single-item Auction Problem (BSAP). For product distributions, the authors demonstrate:

• The optimal revenue from a single-item auction forms an upper bound for revenue from optimal unit-demand pricing.
• By setting a single virtual price across items, they obtain a constant-factor approximation of the optimal pricing revenue. Specifically, they achieve a 3-approximation for general distributions and a refined 2.17-approximation when distributions are identically distributed.
• Importantly, they show that computing a near-optimal virtual price is feasible in polynomial time for distributions satisfying the monotone hazard rate condition.

Methodological Approach

The authors develop a formal equivalence between the BUPP pricing problem and Myerson's auction design through the concept of virtual valuations, mediated by a single virtual price that offers a tractable solution. The extension of Myerson's work entails setting reserve prices analogously to auctions but in pricing settings without direct competition among consumers. The paper discusses techniques to handle non-regular distributions using an ironing procedure on the virtual valuations, which preserves the essential properties needed for approximation while admitting solutions in polynomial time.

Implications and Future Directions

Practically, the results imply that algorithmic pricing solutions grounded in auction theory can be broadly applied to market scenarios with independent consumer preference distributions. Theoretically, the paper bridges a substantial gap in multi-parameter mechanism design by extending known single-parameter insights to more complex domains.

Future developments could explore NP-hardness in precise optimal pricing solutions within BUPP and extend the algorithmic frameworks here to combinatorial or correlated settings where current models fall short. Further, exploring pricing mechanisms with lottery elements could expand applicability where consumers are presented with probabilistic item distributions at fixed prices.

Ultimately, the paper contributes essential frameworks and computational approaches for pricing strategies that approximate optimal solutions in complex economic settings with independent item demands. This work not only underscores the versatility of virtual valuation methodologies but also sets a foundational platform for subsequent research in economic mechanism design and algorithmic pricing strategies.