When to Match: A Cost-Balancing Principle for Dynamic Markets

Abstract: Matching platforms, from ridesharing to food delivery to competitive gaming, face a fundamental operational dilemma: match agents immediately to minimize waiting costs, or delay to exploit the efficiency gains of thicker markets. Yet computing optimal policies is generally intractable, sophisticated algorithms often rely on restrictive distributional assumptions, and common heuristics lack worst-case performance guarantees. We formulate a versatile framework for multi-sided matching with general state-dependent cost structures and non-stationary arrival dynamics. Central to our approach is a cost-balancing principle: match when accumulated waiting cost reaches a calibrated proportion of instantaneous matching cost. This equilibrium condition emerges from fluid-limit analysis and motivates a simple, adaptive Cost-Balancing (CB) algorithm requiring no distributional assumptions. We prove that CB achieves a competitive ratio of $(1+\sqrtΓ)$ under adversarial arrivals, where $Γ$ quantifies economies of scale, guaranteeing cost within a constant factor of the offline optimum. In contrast, standard greedy and threshold policies can incur unbounded costs in adversarial scenarios. We further establish a universal lower bound of $(\sqrt{5}+1)/2$ (the golden ratio), quantifying the fundamental price of uncertainty in online matching. Experiments on game matchmaking and real-world food delivery data demonstrate practical effectiveness, with CB consistently outperforming industry-standard heuristics.