Efficiency-Revenue Trade-offs in Auctions (1205.3077v1)

Abstract: When agents with independent priors bid for a single item, Myerson's optimal auction maximizes expected revenue, whereas Vickrey's second-price auction optimizes social welfare. We address the natural question of trade-offs between the two criteria, that is, auctions that optimize, say, revenue under the constraint that the welfare is above a given level. If one allows for randomized mechanisms, it is easy to see that there are polynomial-time mechanisms that achieve any point in the trade-off (the Pareto curve) between revenue and welfare. We investigate whether one can achieve the same guarantees using deterministic mechanisms. We provide a negative answer to this question by showing that this is a (weakly) NP-hard problem. On the positive side, we provide polynomial-time deterministic mechanisms that approximate with arbitrary precision any point of the trade-off between these two fundamental objectives for the case of two bidders, even when the valuations are correlated arbitrarily. The major problem left open by our work is whether there is such an algorithm for three or more bidders with independent valuation distributions.

• The paper demonstrates that randomized mechanisms can efficiently achieve any point on the Pareto curve of revenue and welfare trade-offs.
• The study establishes that deterministic mechanisms are (weakly) NP-hard for general cases, yet offers feasible approximations for two-bidder scenarios.
• It highlights an open problem regarding polynomial-time deterministic mechanisms for auctions with three or more bidders, motivating further research.

The paper "Efficiency-Revenue Trade-offs in Auctions" explores the nuanced relationship between two cardinal objectives in auction theory: revenue maximization and social welfare optimization. The paper situates itself within the context of agents with independent priors bidding for a single item, comparing Myerson's optimal auction, which maximizes expected revenue, to Vickrey's second-price auction, which maximizes social welfare.

The central question the paper addresses is the exploration of trade-offs between these two criteria. Specifically, it investigates auctions that aim to optimize revenue while ensuring that social welfare surpasses a predetermined threshold. The primary inquiry is focused on whether such trade-offs can be efficiently achieved using deterministic mechanisms as opposed to randomized ones.

Key findings and contributions of the paper include:

1. Polynomial-time Mechanisms with Randomization:
   • The paper asserts that if randomized mechanisms are permitted, then there exist polynomial-time algorithms capable of achieving any point on the Pareto curve, which delineates the trade-off between revenue and welfare.
2. Deterministic Mechanisms:
   • The authors explore whether similar guarantees can be made with deterministic mechanisms. They provide a negative result for this question by proving that achieving these trade-offs with deterministic mechanisms is (weakly) NP-hard.
3. Approximation for Two Bidders:
   • Despite the negative result for the general case, the paper presents a significant positive finding for the specific scenario involving two bidders. It demonstrates that there exist polynomial-time deterministic mechanisms that can approximate, with arbitrary precision, any point on the trade-off curve between revenue and welfare. This result holds even when the bidders' valuations are arbitrarily correlated.
4. Open Problem:
   • The major open problem that stems from this work is whether a similar polynomial-time deterministic algorithm exists for the case of three or more bidders, each with independent valuation distributions. This remains an unanswered question and a potential area for future research.

Overall, the paper makes substantial contributions to the understanding of how trade-offs between fundamental auction objectives can be managed and implemented, particularly highlighting the complexities and computational challenges involved in deterministic settings.