Combinatorial Auctions via Posted Prices (1411.4916v1)

Abstract: We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity.

• The paper introduces anonymous posted price mechanisms that are simple, transparent, and DSIC for combinatorial auctions.
• It shows that for fractionally subadditive valuations, posted prices secure at least half of the expected optimal social welfare.
• The work extends to valuations with complements using the MPH-k hierarchy, offering practical market design implications.

Combinatorial Auctions via Posted Prices

The paper "Combinatorial Auctions via Posted Prices" by Michal Feldman, Nick Gravin, and Brendan Lucier explores the application of anonymous posted price mechanisms within the context of combinatorial auctions in a Bayesian setting. The research addresses how these mechanisms can be used to achieve a desirable trade-off between computational efficiency and economic outcomes, specifically when agent preferences are fractionally subadditive.

Key Contributions

1. Anonymous Posted Price Mechanisms: The paper proposes anonymous posted price mechanisms as a simple, transparent, and incentive-compatible approach to combinatorial auctions. In these mechanisms, item prices are posted beforehand, and agents sequentially choose their preferred bundles from available and unsold items.
2. Approximation Guarantees: It is demonstrated that for fractionally subadditive valuations (which encompass submodular functions), posted prices can ensure at least half of the expected optimal social welfare. This result is achieved by leveraging sample access to valuation distributions and algorithmic queries, allowing prices to be computed in polynomial time while ensuring dominant strategy incentive compatibility (DSIC).
3. Valuations with Complements: The authors extend their findings to valuations with complements, showing that the approximation factor deteriorates linearly with the level of complementarity. Specifically, this is handled through the Maximum over Positive Hypergraph-k (MPH-k) hierarchy, which provides a structured approach to account for various degrees of complementarity within valuation functions.
4. Practical Implications: The research implies the first polynomial-time constant-factor DSIC mechanism for Bayesian settings with submodular combinatorial auctions, given access to demand query oracles. This offers a significant advance over previous methods that either resulted in polylogarithmic approximations or depended on exponentially large supports of valuation distributions.

Theoretical and Practical Implications

From a theoretical perspective, the paper advances the design of auction mechanisms by offering a middle ground between overly complex auction designs and simplistic but potentially inefficient posted price mechanisms. Despite the computational hardness typically associated with these auctions, especially for submodular bidders, the paper shows that substantial welfare can still be achieved in a tractable manner.

Practically, the findings have implications for large-scale market design, such as cloud computing resource allocation and spectrum auctions. The simplicity and transparency of posted price mechanisms, combined with the welfare guarantees provided, render them appealing even in settings with incomplete information and unpredictable buyer valuations.

Future Directions

The paper paves the way for further exploration in several domains:

• Extension to More General Valuation Classes: While the paper focuses on subadditive and MPH-k functions, extending these results to broader valuation classes remains an open question.
• Order Selection Optimization: Investigating whether randomized or strategically chosen consumer arrival orders could enhance outcomes might provide more robust insights.
• Applications to Repeated Auctions: Exploring the dynamics of posted price mechanisms in repeated auction settings or multi-stage markets may also yield significant insights into their long-term efficacy.

In conclusion, this paper contributes to the interface between algorithmic game theory and market design, proposing feasible solutions for highly complex allocation problems in large markets using posted price mechanisms. The approximation guarantees achieved bode well for further analyses that could push the boundaries of efficiency and applicability in combinatorial auctions.