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 $9sqrt{nm}$. When $n leq m$, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When $n geq m$, the second bound is optimal for the benchmark introduced in [EFFTW17a] 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 $n geq m$ regime, the competition complexity of $n$ bidders with additive valuations over even $2$ i.i.d. regular items is indeed $omega(1)$. Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables.
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