Scrip: When Does a Shared Budget Help Collective Reasoning?

How should a collective solver allocate a limited budget between acquiring evidence and weighting predictions? We study this question through Scrip, where frozen language agents purchase documents, tools, or peer rationales from the same account that determines their predictive influence. Verified outcomes update account balances during training, and evaluation uses a frozen trained state. Across controlled multi-hop and forecasting tasks, selective acquisition and outcome-based weighting explain most of the improvement over a judge. Coupling acquisition and influence adds a smaller gain at the original budget and a larger gain under scarcity. The advantage is also present under non-oracle retrieval, where retrieval noise narrows it. We characterize the mechanism with two analyses. Matched-report replay identifies sealed Scrip exactly with a cost-adjusted online pool, while a convex-hull bound locates the limit of probability aggregation when reports lack complementary information. These results identify when a shared budget helps collective reasoning and when the task calls for richer communication.