Competition complexity formalizes a compelling intuition: rather than refining the mechanism, how much additional competition is sufficient for a simple mechanism to compete with an optimal one? We begin the study of this question in multi-unit pricing for welfare maximization using prophet inequalities. An online decision-maker observes $m \geq k$ nonnegative values drawn independently from a known distribution, may select up to $k$ of them, and aims to maximize the expected sum of selected values. The benchmark is a prophet who observes a sequence of length $n \geq k$ and selects the $k$ largest values. We focus on the widely adopted class of single-threshold algorithms and fully characterize their $(1-\varepsilon)$-competition complexity. Notably, our results reveal a sharp competition-induced phase transition: in the absence of competition, single-threshold algorithms are fundamentally limited to a $1-1/\sqrt{2kπ}$ fraction of the prophet value, whereas even a $1\%$ multiplicative increase beyond $n$ observations suffices to achieve a $1-\exp(-Θ(k))$ fraction. Another notable result happens when $k=1$: we show that the $(1-\varepsilon)$-competition complexity is exactly $\ln(1/\varepsilon)$, fully resolving an open question by Brustle et al. [Math. Oper. Res. 2024]. Our analysis is based on infinite-dimensional linear programming and duality arguments.

翻译：竞争复杂度形式化了一个引人深思的直觉：与其改进机制，需要增加多少额外竞争才能使一个简单机制与最优机制相竞争？我们通过先知不等式，从多单位定价中的福利最大化问题出发，开始研究这一问题。在线决策者观察到从已知分布中独立抽取的 $m \geq k$ 个非负数值，最多可选择其中 $k$ 个，目标是最大化所选数值的期望和。基准是一个先知，他观察长度为 $n \geq k$ 的序列并选择其中最大的 $k$ 个值。我们聚焦于广泛采用的一类单阈值算法，并完整刻画了其 $(1-\varepsilon)$-竞争复杂度。值得注意的是，我们的结果揭示了一个由竞争引发的急剧相变：在没有竞争的情况下，单阈值算法本质上只能达到先知值的 $1-1/\sqrt{2kπ}$ 比例，而即使观测数量仅需超过 $n$ 的 $1\%$ 乘性增量，就足以达到 $1-\exp(-Θ(k))$ 的比例。另一个值得注意的结果出现在 $k=1$ 时：我们证明了 $(1-\varepsilon)$-竞争复杂度恰好为 $\ln(1/\varepsilon)$，这完全解决了 Brustle 等人 [Math. Oper. Res. 2024] 提出的一个开放性问题。我们的分析基于无限维线性规划与对偶论证。