Revenue Maximization for Selling Multiple Correlated Items

TL;DR

This paper investigates simple auction mechanisms for selling multiple correlated items to a single buyer, demonstrating that such mechanisms can achieve a constant fraction of the optimal revenue even under weak correlations.

Contribution

It extends prior work by adapting core decomposition techniques to correlated settings, providing a simple mechanism with guaranteed revenue approximation.

Findings

A modified core decomposition lemma applicable to correlated valuations.

A simple mechanism achieves a constant fraction of optimal revenue under weak correlation.

The approach does not fully extend to strong correlations.

Abstract

We study the problem of selling $n$ items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e., desirabilities of the buyer for the items are not drawn independently. Ideally, the goal is to design a mechanism to maximize the revenue. However, it has been shown that a revenue optimal mechanism might be very complicated and as a result inapplicable to real-world auctions. Therefore, our focus is on designing a simple mechanism that achieves a constant fraction of the optimal revenue. Babaioff et al. propose a simple mechanism that achieves a constant fraction of the optimal revenue for independent setting with a single additive buyer. However, they leave the following problem as an open question: "Is there a simple, approximately optimal mechanism for a single additive buyer whose value for $n$ items is sampled from a common base-value distribution?" Babaioff et al. show a constant approximation factor of the optimal revenue can be achieved by either selling the items separately or as a whole bundle in the independent setting. We show a similar result for the correlated setting when the desirabilities of the buyer are drawn from a common base-value distribution. It is worth mentioning that the core decomposition lemma which is mainly the heart of the proofs for efficiency of the mechanisms does not hold for correlated settings. Therefore we propose a modified version of this lemma which is applicable to the correlated settings as well. Although we apply this technique to show the proposed mechanism can guarantee a constant fraction of the optimal revenue in a very weak correlation, this method alone can not directly show the efficiency of the mechanism in stronger correlations.