Deliberation with Overlapping Evidence

How should AI agents deliberate? Multi-agent AI systems increasingly improve their answers through deliberation, yet their agents often hold overlapping knowledge from searching the same corpora, calling the same tools, or sharing training data. This paper studies how AI agents should deliberate to reach the omniscient forecast: the Bayesian forecast given everyone's pooled private data. We first prove that exchanging posteriors (the belief protocol) cannot reach the omniscient forecast. We then propose the anecdote protocol, where agents share not only their beliefs but also the clues behind them, and prove that it reaches the omniscient forecast. Finally, we compare the two protocols with Bayesian and LLM agents and find that the anecdote protocol outperforms the belief protocol for both types of agents.