All Substitution Is Local

2026-04-01

AI Summary

This study investigates when distinct information sources act as complements or substitutes for a Bayesian decision-maker facing a finite set of actions. By decomposing complementarity and substitutability effects and integrating Bayesian decision theory with information economics, the work introduces a “local substitutability” principle: information sources exhibit substitutability only when their combined posterior beliefs cross a decision boundary; within the region bounded by such thresholds, their interaction remains complementary regardless of statistical dependence. This insight challenges the conventional binary view of information interactions, revealing that substitutability is inherently local while complementarity is pervasive. The theoretical claims are formally verified using Lean 4, ensuring rigorous logical consistency.

Abstract

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.

Problem

• information sources
• Bayesian decision-making
• complementarity
• substitutability
• decision boundaries

Innovation

• local substitution
• Bayesian decision-making
• information complementarity
• decision boundaries
• formal verification