# All Substitution Is Local

arXiv: https://arxiv.org/abs/2604.01443 · HTML: https://mlsanity.com/paper/2604.01443/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Informational Substitutes](https://mlsanity.com/paper/1703.08636/) (2017)
- [Dynamically Aggregating Diverse Information](https://mlsanity.com/paper/1910.07015/) (2019)
- [Optimal Dynamic Allocation of Attention](https://mlsanity.com/paper/1812.06967/) (2018)
- [Payoffs-Beliefs Duality and the Value of Information](https://mlsanity.com/paper/1908.01633/) (2019)

## Abstract (arXiv)

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.