In this paper, we propose an online-matching-based model to tackle the two fundamental issues, matching and pricing, existing in a wide range of real-world gig platforms, including ride-hailing (matching riders and drivers), crowdsourcing markets (pairing workers and tasks), and online recommendations (offering items to customers). Our model assumes the arriving distributions of dynamic agents (e.g., riders, workers, and buyers) are accessible in advance, and they can change over time, which is referred to as \emph{Known Heterogeneous Distributions} (KHD). In this paper, we initiate variance analysis for online matching algorithms under KHD. Unlike the popular competitive-ratio (CR) metric, the variance of online algorithms' performance is rarely studied due to inherent technical challenges, though it is well linked to robustness. We focus on two natural parameterized sampling policies, denoted by $\mathsf{ATT}(\gamma)$ and $\mathsf{SAMP}(\gamma)$, which appear as foundational bedrock in online algorithm design. We offer rigorous competitive ratio (CR) and variance analyses for both policies. Specifically, we show that $\mathsf{ATT}(\gamma)$ with $\gamma \in [0,1/2]$ achieves a CR of $\gamma$ and a variance of $\gamma \cdot (1-\gamma) \cdot B$ on the total number of matches with $B$ being the total matching capacity. In contrast, $\mathsf{SAMP}(\gamma)$ with $\gamma \in [0,1]$ accomplishes a CR of $\gamma (1-\gamma)$ and a variance of $\bar{\gamma} (1-\bar{\gamma})\cdot B$ with $\bar{\gamma}=\min(\gamma,1/2)$. All CR and variance analyses are tight and unconditional of any benchmark. As a byproduct, we prove that $\mathsf{ATT}(\gamma=1/2)$ achieves an optimal CR of $1/2$.

翻译：本文提出了一种基于在线匹配的模型，以解决广泛存在于现实世界零工平台（包括网约车平台（匹配乘客与司机）、众包市场（配对工人与任务）以及在线推荐系统（向客户推荐物品））中的两个基本问题：匹配与定价。该模型假设动态智能体（如乘客、工人、买家）的到达分布可预先获知，且这些分布会随时间变化，我们称之为“已知异质分布”（KHD）。本文首次在KHD条件下对在线匹配算法的方差展开分析。不同于广泛使用的竞争比（CR）指标，在线算法性能的方差虽与鲁棒性紧密相关，但因固有的技术挑战而鲜有研究。我们聚焦于两种自然的参数化采样策略——$\mathsf{ATT}(\gamma)$和$\mathsf{SAMP}(\gamma)$，它们构成了在线算法设计的基础基石。我们对两种策略进行了严格的竞争比（CR）与方差分析：具体而言，当$\gamma \in [0,1/2]$时，$\mathsf{ATT}(\gamma)$在总匹配容量$B$下达到$\gamma$的竞争比和$\gamma \cdot (1-\gamma) \cdot B$的方差；相比之下，当$\gamma \in [0,1]$时，$\mathsf{SAMP}(\gamma)$达到$\gamma (1-\gamma)$的竞争比和$\bar{\gamma} (1-\bar{\gamma})\cdot B$的方差（其中$\bar{\gamma}=\min(\gamma,1/2)$）。所有竞争比与方差分析均为紧致的，且不依赖于任何基准。作为副产品，我们证明了$\mathsf{ATT}(\gamma=1/2)$能达到最优竞争比$1/2$。