Cap-and-Penalize: Competitive Mechanisms for Multi-Phase Regularized Online Allocation
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.016 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".