# Optimal (and Benchmark-Optimal) Competition Complexity for Additive Buyers over Independent Items

## TL;DR

This paper establishes tight bounds on the number of additional bidders needed for simple auction mechanisms to outperform optimal auctions in revenue, specifically for additive bidders over independent items, with results matching known benchmarks.

## Contribution

It provides the first tight bounds on competition complexity for additive bidders over independent items, matching the Eden et al. benchmark up to constants.

## Key findings

- Competition complexity is at most $n(\ln(1+m/n)+2)$ and $9\sqrt{nm}$.
- Bounds are tight up to constants in different regimes.
- Competition complexity is $\omega(1)$ even for two items.

## Abstract

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 $9\sqrt{nm}$. 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.