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All Substitution Is Local

Nidhish Shah, Shaurjya Mandal, Asfandyar Azhar

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.

All Substitution Is Local

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.

Paper Structure

This paper contains 28 sections, 5 theorems, 13 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decision problem.
  4. Channels.
  5. Auxiliary functions.
  6. Bregman divergence.
  7. Martingale identity.
  8. The Localization Principle
  9. Two Opposing Forces.
  10. When Posteriors Stay Inside The Decision Region.
  11. When Posteriors Cross A Decision Boundary.
  12. Illustration
  13. Interior complementarity.
  14. Boundary complementarity.
  15. Substitution.
  16. ...and 13 more sections

Key Result

Proposition 2

where $\hat{b} = \hat{b}(i, o)$, the functions $g$ and $h$ are defined in eq:g--eq:h, and both expected Bregman divergences are non-negative. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Belief-simplex geometry of the interaction. Left: the sign of $\Delta\operatorname{VoI}$ partitions the simplex into complement and substitute regions. Middle, Right: the two constituent forces; the complement force is diffuse across the interior while the substitute force concentrates sharply along decision boundaries. Their difference determines the sign of $\Delta\operatorname{VoI}$.
  • Figure 2: Phase transition along $b(t) = (1/4+t,\, 1/6,\, 7/12-t)$. Complement force (red) and substitute force (blue). Shading indicates the complement (red) and substitute (blue) regimes; the interaction boundary (where the forces cross) precedes the decision boundary.

Theorems & Definitions (13)

  • Definition 1: Second-order interaction
  • Proposition 2: Bregman decomposition
  • proof
  • Corollary 3: Independence from channel ordering and costs
  • proof
  • Theorem 4: Interior complementarity
  • proof
  • Remark 5
  • Theorem 6: Converse: substitution requires boundary-crossing
  • proof
  • ...and 3 more