Approximate Revenue Maximization with Multiple Items (1204.1846v3)

Abstract: Maximizing the revenue from selling more than one good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple "one-dimensional" mechanisms, such as selling the goods separately, guarantee at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least $50\%$ when they are independent. For the case of $k>2$ independent goods, we show that selling them separately guarantees at least a $c/\log^2 k$ fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a $c/\log k$ fraction of the optimal revenue. Additional results compare the revenues from the two simple mechanisms of selling the goods separately and bundled, identify situations where bundling is optimal, and extend the analysis to multiple buyers.

• The paper demonstrates that simple selling mechanisms guarantee a significant fraction of optimal revenue, with at least 50% for two independent goods and 73% for identical distributions.
• It rigorously establishes that for k independent goods, separate selling achieves revenue at least a constant over log² k, while bundling provides a constant over log k fraction for identical items.
• The work offers practical insights for auction design and pricing strategies, and it opens new avenues in mechanism design research for multi-item sales.

Approximate Revenue Maximization with Multiple Items: An Analysis

This paper by Sergiu Hart and Noam Nisan explores the intricate domain of revenue maximization when a seller has multiple goods to sell to a single buyer. The problem, notably more complex than the single-good case, is tackled by assessing simple mechanisms, such as selling separately or as a bundle, against the maximal revenue attainable (the optimal revenue).

Key Results and Theorems

The research provides rigorous bounds for the Guaranteed Fraction of Optimal Revenue (GFOR) from simple mechanisms:

1. Two Independent Goods: Theorem A establishes that selling two goods separately guarantees at least 50% of the optimal revenue when the goods are independent. This result holds for any distribution of values.
2. Two Independent and Identically Distributed Goods: Theorem B demonstrates that selling each good separately guarantees at least 73% of the optimal revenue when the two goods are identically distributed.
3. Multiple Independent Goods: Theorem C finds that for $k > 2$ independent goods, selling them separately yields a revenue that is at least a constant over $\log^2 k$ fraction of the optimal revenue.
4. Independent and Identically Distributed Goods: Theorem D shows that bundling $k$ such goods guarantees a revenue that is at least a constant over $\log k$ of the optimal revenue.

These results underline that even simple mechanisms can secure significant portions of the potential optimal revenue under certain conditions.

Implications and Applications

The theoretical insights of the paper highlight crucial boundaries for mechanism design in economics and algorithmic game theory. The Guaranteed Fraction of Optimal Revenue (GFOR) as a benchmark serves as a tool for both economists and computer scientists working on revenue maximization problems. Practically, these findings can guide auction design and pricing strategies for multi-item sales, suggesting efficient mechanisms when full characterization of optimal solutions is infeasible.

Future Directions

The paper opens several avenues for future research. First, characterizing the distributions where separate selling or bundling is optimal remains unresolved. Moreover, establishing tighter bounds for GFOR, especially for deterministic mechanisms or when examining other forms of implementation such as Bayesian Nash, is an ongoing challenge. Another significant direction is exploring mechanisms that transcend simple selling strategies to capture even closer approximations to the optimal revenue.

In summary, this work by Hart and Nisan provides a valuable framework for understanding the efficacy of simple mechanisms in multi-item sales scenarios. It underscores areas for further exploration in both theoretical and practical aspects of economics and computer science, fostering enhanced collaboration and innovation in mechanism design.