Think More or Read More? A Token-Budget Theory of Reasoning with Shared Context

Reasoning systems increasingly improve answers by incorporating additional context into the model's input, from multi-agent debate, where agents read their peers' responses, to self-refine, where a model revisits its own drafts. In these systems, communication consumes context as one agent's output becomes another's input. We study the regime where reading and generation draw on one fixed token window, so every token spent reading is a token that cannot be spent reasoning. We develop a **token-budget theory** that formalizes this trade-off, modeling accuracy as a function of the reasoning budget remaining after reading and aggregating votes with a Condorcet jury model that permits correlated errors. For multi-agent debate, the theory gives a closed-form prediction for the optimal number of agents. This quantity is estimated from single-agent measurements, allowing the optimal group size to be determined without running debate, and the same accounting extends to a heterogeneous set of different models. We empirically verify our theory and show that, at the predicted group size, our method outperforms strong baselines under the tightest calibration spend, where debate-calibrated rules can afford only one or two questions. Self-refine obeys the same accounting, with its own draft in place of a peer message. The theory further predicts that communication pays only when the gain in the accuracy ceiling exceeds the reasoning room spent on reading, with this condition tightening as the window shrinks. Across five open models and four reasoning tasks, self-consistency matches or beats deployed debate at 93% of 980 matched window-and-group-size settings, and a paired bootstrap on the same cells finds self-consistency significantly ahead at 57% and debate at 0.5%. The gap is largest under tight windows and nearly disappears at a loose one, consistent with the theory. Under a shared window, the value of added context depends less on what a model reads than on how much room remains to think.