Cap-and-Penalize: Competitive Mechanisms for Multi-Phase Regularized Online Allocation

This paper introduces a novel mechanism for online allocation with multi-phase, non-separable regularizers, termed Cap-and-Penalize (CnP), inspired by real-world applications such as cap-and-tax policies in carbon pricing. The CnP regularizer models a multi-phase cost structure, imposing a monotone convex penalty when total allocation exceeds a predefined level (soft cap) and enforcing a strict limit (hard cap) beyond which allocation is prohibited. Our contributions are twofold: (1) we propose an online mechanism for CnP-regularized allocation without per-step resource constraints, which operates as a simple and intuitive posted-price mechanism, but achieves the best-possible guarantee among all possible online algorithms; (2) we tackle the more complex setting with per-step resource constraints by decomposing the regularizer into local components, yielding a similar mechanism with time-dependent marginal pricing functions. To establish the tightness of our results in both settings, we introduce a representative function-based approach that transforms the lower-bound proof into the problem of solving an ordinary differential equation with boundary conditions. We believe that this technique has the potential to be applied to other similar online optimization problems.