The competition complexity of an auction setting is the number of additional bidders needed such that the simple mechanism of selling items separately (with additional bidders) achieves greater revenue than the optimal but complex (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. Our main result settles the competition complexity of $n$ bidders with additive values over $m < n$ independent items at $\Theta(\sqrt{nm})$. The $O(\sqrt{nm})$ upper bound is due to [BW19], and our main result improves the prior lower bound of $\Omega(\ln n)$ to $\Omega(\sqrt{nm})$. Our main result follows from an explicit construction of a Bayesian IC auction for $n$ bidders with additive values over $m<n$ independent items drawn from the Equal Revenue curve truncated at $\sqrt{nm}$ ($\mathcal{ER}_{\le \sqrt{nm}}$), which achieves revenue that exceeds $\text{SRev}_{n+\sqrt{nm}}(\mathcal{ER}_{\le \sqrt{nm}}^m)$. Along the way, we show that the competition complexity of $n$ bidders with additive values over $m$ independent items is exactly equal to the minimum $c$ such that $\text{SRev}_{n+c}(\mathcal{ER}_{\le p}^m) \geq \text{Rev}_n(\mathcal{ER}_{\le p}^m)$ for all $p$ (that is, some truncated Equal Revenue witnesses the worst-case competition complexity). Interestingly, we also show that the untruncated Equal Revenue curve does not witness the worst-case competition complexity when $n > m$: $\text{SRev}_n(\mathcal{ER}^m) = nm+O_m(\ln (n)) \leq \text{SRev}_{n+O_m(\ln (n))}(\mathcal{ER}^m)$, and therefore our result can only follow by considering all possible truncations.

翻译：拍卖设定的竞争复杂性是指所需额外竞拍者的数量，使得简单机制（即单独出售物品，并引入额外竞拍者）所获得的收益，能够超过未引入额外竞拍者时最优但复杂（随机化、依赖先验、贝叶斯真实）机制的收益。我们的主要结果解决了$n$个加性估值买家在$m < n$个独立物品上的竞争复杂性，其值为$\Theta(\sqrt{nm})$。上界$O(\sqrt{nm})$来自[BW19]，而我们的主要结果将先前的下界$\Omega(\ln n)$提升至$\Omega(\sqrt{nm})$。这一结果源于我们显式构造了一个贝叶斯激励相容（IC）拍卖，针对$n$个加性估值买家在$m<n$个独立物品上的情况，物品分布来自截断于$\sqrt{nm}$的等收益曲线（$\mathcal{ER}_{\le \sqrt{nm}}$），其收益超过$\text{SRev}_{n+\sqrt{nm}}(\mathcal{ER}_{\le \sqrt{nm}}^m)$。在此过程中，我们证明$n$个加性估值买家在$m$个独立物品上的竞争复杂性，恰好等于满足对所有$p$有$\text{SRev}_{n+c}(\mathcal{ER}_{\le p}^m) \geq \text{Rev}_n(\mathcal{ER}_{\le p}^m)$的最小$c$（即某些截断等收益曲线体现了最坏情况下的竞争复杂性）。有趣的是，我们还表明当$n > m$时，未截断的等收益曲线并不体现最坏情况下的竞争复杂性：$\text{SRev}_n(\mathcal{ER}^m) = nm+O_m(\ln (n)) \leq \text{SRev}_{n+O_m(\ln (n))}(\mathcal{ER}^m)$，因此我们的结果只能通过考虑所有可能的截断来得出。