Optimal (and benchmark-optimal) competition complexity for additive buyers over independent items

The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. We prove that the competition complexity of n bidders with additive valuations over m independent items is at most n(ln(1+m/n)+2), and also at most 9√nm. When n ≤ m, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When n ≥ m, the second bound is optimal for the benchmark introduced by Eden et al. up to constant factors, even when the items are i.i.d. and regular. We further show that, while the Eden et al. benchmark is not necessarily tight in the n ≥ m regime, the competition complexity of n bidders with additive valuations over even 2 i.i.d. regular items is indeed ω(1). Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables.