99% Revenue with Constant Enhanced Competition

The enhanced competition paradigm is an attempt at bridging the gap between simple and optimal auctions. In this line of work, given an auction setting with $m$ items and $n$ bidders, the goal is to find the smallest $n' \geq n$ such that selling the items to $n'$ bidders through a simple auction generates (almost) the same revenue as the optimal auction. Recently, Feldman, Friedler, and Rubinstein [EC, 2018] showed that an arbitrarily large constant fraction of the optimal revenue from selling $m$ items to a single bidder can be obtained via simple auctions with a constant number of bidders. However, their techniques break down even for two bidders, and can only show a bound of $n' = n \cdot O(\log \frac{m}{n})$. Our main result is that $n' = O(n)$ bidders suffice for all values of $m$ and $n$. That is, we show that, for all $m$ and $n$, an arbitrarily large constant fraction of the optimal revenue from selling $m$ items to $n$ bidders can be obtained via simple auctions with $O(n)$ bidders. Moreover, when the items are regular, we can achieve the same result through auctions that are prior-independent, {\em i.e.}, they do not depend on the distribution from which the bidders' valuations are sampled.