Nash Credit Routing for Optimizing Responses Beyond GRPO

A GRPO update can improve its sampled objective while reducing the likelihood of an individual successful response. Repeated, directionally similar successes can also dominate the positive update through multiplicity alone. We introduce Nash Credit Routing (NCR), a constrained correction to GRPO that reallocates a fixed positive-credit budget across successful responses and selectively refunds penalties on failed-response segments. The correction remains near GRPO, preserves at least its first-order sampled-objective improvement, and causes no additional first-order likelihood harm to any sampled success. NCR assigns positive bargaining powers by minimizing directional collision under a multiplicity-calibrated ratio constraint, then maximizes weighted Nash welfare over normalized likelihood gains. Across the evaluated models and mathematics benchmarks, NCR achieves higher macro-average pass@1 and pass@128 than the compared baselines.