Tight Competitive and Variance Analyses of Matching Policies in Gig Platforms

TL;DR

This work studies online matching and pricing on gig platforms under Known Heterogeneous Distributions (KHD). It introduces two LP-based sampling policies, ATT($\gamma$) with attenuations and SAMP($\gamma$) without attenuations, and derives tight competitive ratio and variance results: for ATT($\gamma$) with $\gamma\in[0,1/2]$, the CR is $\gamma$ and the variance bound is $\gamma(1-\gamma)\cdot B$, with ATT($\tfrac{1}{2}$) achieving the optimal CR of $1/2$; for SAMP($\gamma$), the CR is $\gamma(1-\gamma)$ and the variance bound is $\bar{\gamma}(1-\bar{\gamma})\cdot B$ where $\bar{\gamma}=\min(\tfrac{1}{2},\gamma)$. Both analyses are unconditional with respect to the LP benchmark and are shown to be tight via constructed instances. The results extend beyond identically distributed arrivals (KIID) to time-varying distributions, address robustness via variance, and connect to Prophet Inequality concepts, offering practical, tractable policies for matching and pricing in dynamic gig-economy platforms. The work also highlights how variance analysis complements CR in assessing risk and reliability of online policies in maximization settings.

Abstract

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 $\mathsf{ATT}(γ)$ and $\mathsf{SAMP}(γ)$, 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 $\mathsf{ATT}(γ)$ with $γ\in [0,1/2]$ achieves a CR of $γ$ and a variance of $γ\cdot (1-γ) \cdot B$ on the total number of matches with $B$ being the total matching capacity. In contrast, $\mathsf{SAMP}(γ)$ with $γ\in [0,1]$ accomplishes a CR of $γ(1-γ)$ and a variance of $\barγ (1-\barγ)\cdot B$ with $\barγ=\min(γ,1/2)$. All CR and variance analyses are tight and unconditional of any benchmark. As a byproduct, we prove that $\mathsf{ATT}(γ=1/2)$ achieves an optimal CR of $1/2$.

Figures (4)

• Figure 1: An example highlighting the tightness of competitive analysis of $\mathsf{ATT}\xspace(\gamma)$ for any $\gamma \in [0,1/2]$ and the optimality of $\mathsf{ATT}\xspace(\gamma=1/2)$, where both claims are unconditional of the benchmark LP.
• Figure 2: An example where $\mathsf{ATT}\xspace (\gamma)$ achieves the tight CR bound of $\gamma$ and the tight variance bound of $\gamma (1-\gamma) \cdot B$ simultaneously for any $\gamma \in [0,1/2]$.
• Figure 3: An example highlighting the unconditional tightness of competitive analysis of $\mathsf{SAMP}\xspace(\gamma)$ for any $\gamma \in [0,1]$.
• Figure 4: An example highlighting the tightness of variance analysis of $\mathsf{SAMP}\xspace(\gamma)$ for any $\gamma \in [0,1]$.

Theorems & Definitions (12)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2
• Example 1
• Theorem 4
• Lemma 3
• Theorem 5
• Lemma 4