Competition Complexity in Multi-Item Auctions: Beyond VCG and Regularity

TL;DR

This paper analyzes how many extra bidders are needed in simple multi-item auctions to match the revenue of optimal mechanisms, showing that under certain conditions, fewer bidders are required than previously thought, especially with strong regularity assumptions.

Contribution

It introduces tighter bounds on competition complexity in multi-item auctions under $ ext{α}$-strong regularity, improving upon prior regularity-based results and highlighting the reduced need for additional bidders.

Findings

Competition complexity is $ heta(n/α)$ under strong regularity.

Grand bundle auctions have constant competition complexity in certain cases.

Results also apply when competing against the first-best social welfare benchmark.

Abstract

We quantify the value of the monopoly's bargaining power in terms of competition complexity--that is, the number of additional bidders the monopoly must attract in simple auctions to match the expected revenue of the optimal mechanisms (c.f., Bulow and Klemperer, 1996, Eden et al., 2017)--within the setting of multi-item auctions. We show that for simple auctions that sell items separately, the competition complexity is $\Theta(\frac{n}{\alpha})$ in an environment with $n$ original bidders under the slightly stronger assumption of $\alpha$-strong regularity, in contrast to the standard regularity assumption in the literature, which requires $\Omega(n \cdot \ln \frac{m}{n})$ additional bidders (Feldman et al., 2018). This significantly reduces the value of learning the distribution to design the optimal mechanisms, especially in large markets with many items for sale. For simple auctions that sell items as a grand bundle, we establish a constant competition complexity bound in a single-bidder environment when the number of items is small or when the value distribution has a monotone hazard rate. Some of our competition complexity results also hold when we compete against the first best benchmark (i.e., optimal social welfare).