The Competition Complexity of Auctions: A Bulow-Klemperer Result for Multi-Dimensional Bidders

TL;DR

This paper extends Bulow-Klemperer results to multi-dimensional auction environments, showing that recruiting additional bidders ensures simple mechanisms outperform optimal ones, with bounds on the number of extra bidders needed.

Contribution

It establishes the first full Bulow-Klemperer results for multi-dimensional settings, providing bounds on the number of additional bidders needed for simple mechanisms to surpass optimal revenue.

Findings

Competition complexity is at most n+2m-2 for additive valuations over m items.

Competition complexity is at least logarithmic in m.

Results extend to bidders with downward-closed constraints and matroid constraints.

Abstract

A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to $n$ bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a {\em prior-independent} mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first {\em full Bulow-Klemperer} results in {\em multi-dimensional} environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's {\em competition complexity}. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of $n$ bidders with additive valuations over $m$ independent, regular items is at most $n+2m-2$ and at least $\log(m)$. We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive $m-1$ factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.