Optimal Multi-Dimensional Mechanism Design: Reducing Revenue to Welfare Maximization (1207.5518v1)

Abstract: We provide a reduction from revenue maximization to welfare maximization in multi-dimensional Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson's result to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder's type t_i is transformed into a virtual type f_i(t_i), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. We obtain our reduction from revenue to welfare optimization via two algorithmic results on reduced forms in settings with arbitrary feasibility and demand constraints. First, we provide a separation oracle for determining feasibility of a reduced form. Second, we provide a geometric algorithm to decompose any feasible reduced form into a distribution over virtual VCG allocation rules. In addition, we show how to execute both algorithms given only black box access to an implementation of the VCG allocation rule. Our results are computationally efficient for all multi-dimensional settings where the bidders are additive. In this case, our mechanisms run in time polynomial in the total number of bidder types, but not type profiles. For generic correlated distributions, this is the natural description complexity of the problem. The runtime can be further improved to poly(#items, #bidders) in item-symmetric settings by making use of recent techniques.

• The paper introduces a reduction method from revenue to welfare maximization, linking multi-dimensional auctions with virtual VCG rules.
• It details a separation oracle that verifies the feasibility of reduced form auctions under complex supply and demand constraints.
• A geometric decomposition algorithm efficiently constructs revenue-optimal mechanisms for settings with additive and correlated bidder valuations.

Overview of "Optimal Multi-Dimensional Mechanism Design: Reducing Revenue to Welfare Maximization"

The paper by Cai, Daskalakis, and Weinberg explores the complex issue of multi-dimensional Bayesian mechanism design, extending Myerson's foundational single-item auction framework to more intricate scenarios involving multiple items and complex feasibility constraints. The authors propose a significant methodological advancement: a reduction from revenue maximization to welfare maximization. This approach is operationalized through the development of virtual VCG (Vickrey-Clarke-Groves) allocation rules, facilitating the derivation of revenue-optimal mechanisms in settings with additive and correlated bidders.

Key Contributions

The authors introduce a novel reduction mechanism from revenue to welfare optimization, thereby bridging a crucial gap in multi-dimensional auction theory. This is accomplished through two primary algorithmic contributions:

1. Separation Oracle for Reduced Form Auctions: The paper provides a separation oracle that determines the feasibility of reduced form auctions. This oracle identifies if a proposed auction mechanism aligns with complex feasibility constraints adapted from both supply and demand side requirements.
2. Geometric Decomposition Algorithm: The authors employ a geometric algorithm to decompose feasible reduced forms into distributions over virtual VCG allocation rules, enabling the construction of revenue-optimal mechanisms.

Moreover, their work is computationally efficient, particularly for settings where bidders have additive valuations, making it feasible to run these mechanisms in polynomial time relative to the number of items and bidders. This also extends to settings with correlated distributions, though at a computational complexity dependent on the distribution's descriptive scale.

Theoretical Impacts and Computational Efficiency

A notable theoretical insight offered by this research is the characterization of all feasible Bayesian mechanisms via virtual VCG allocation rules. This framework elegantly encapsulates the auction design problem, allowing the transformation of bidder types into virtual types. Such a transformation is fundamental to maximizing virtual welfare and thereby approaching optimal revenue.

The methodology developed is efficient for additive bidder settings, leveraging characteristics such as item-symmetry to further reduce computational complexity. The proposed framework is not restricted by additivity assumptions, showing broad applicability across various auction configurations.

Practical Implications and Extensions

In practical terms, this reduction technique can substantially simplify the design of auction mechanisms in marketplaces with multiple heterogeneous items. The approach is robust enough to handle arbitrary feasibility constraints, potentially encompassing a wide variety of real-world applications such as spectrum auctions, online advertising, and complex procurements.

Significantly, the authors extend their framework to incorporate budget constraints, maintaining computational tractability while accommodating critical real-world financial constraints faced by bidders.

Future Directions

This paper sets the stage for further exploration into more generalized settings in auction theory. Key areas for future work include the extension to non-additive bidder valuations and incorporating additional market complexities such as dynamic and repeated auctions. Furthermore, investigating improved approximation algorithms for revenue optimization in correlated bidder settings would be a logical progression, leveraging the foundational reduction strategy laid out by this research.

In conclusion, the authors make an important stride in multi-dimensional mechanism design, offering tools that provide both theoretical insight and practical utility in complex auction environments. This work stimulates further research into expansive auction frameworks, encouraging enhanced interdisciplinary collaborations between economics and computer science.