When does consulting one information source raise the value of another, and when does it diminish it? We study this question for Bayesian decision-makers facing finite actions. The interaction decomposes into two opposing forces: a complement force, measuring how one source moves beliefs to where the other becomes more useful, and a substitute force, measuring how much the current decision is resolved. Their balance obeys a localization principle: substitution requires an observation to cross a decision boundary, though crossing alone does not guarantee it. Whenever posteriors remain inside the current decision region, the substitute force vanishes, and sources are guaranteed to complement each other, even when one source cannot, on its own, change the decision. The results hold for arbitrarily correlated sources and are formalized in Lean 4. Substitution is confined to the thin boundaries where decisions change. Everywhere else, information cooperates. Code and proofs: https://github.com/nidhishs/all-substitution-is-local.

翻译：当咨询一个信息源会提升还是降低另一个信息源的价值？我们针对面临有限行动的贝叶斯决策者研究了这一问题。这种相互作用分解为两种对立的力量：互补力，衡量一个信息源如何将信念移动到另一个信息源变得更有用的位置；以及替代力，衡量当前决策被解决的程度。它们的平衡遵循局部化原理：替代需要观测跨越决策边界，但仅跨越本身并不能保证替代。只要后验概率仍停留在当前决策区域内，替代力就会消失，且各信息源保证相互互补，即使某个信息源本身无法单独改变决策。该结论对任意相关的信息源均成立，并在 Lean 4 中形式化。替代仅限于决策发生变化的薄边界。在其他所有地方，信息相互合作。代码与证明：https://github.com/nidhishs/all-substitution-is-local