All Substitution Is Local

Nidhish Shah; Shaurjya Mandal; Asfandyar Azhar

econ.THApr 1, 2026

All Substitution Is Local

Authors: Nidhish Shah, Shaurjya Mandal, Asfandyar Azhar

Organizations: Harvard University · University of Oxford

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.

Figures & tables

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 $\Delta\operatorname{VoI}$ .

	$s_{1}$	$s_{2}$	$s_{3}$
$a_{1}$	12	0	3
$a_{2}$	0	12	3
$a_{3}$	3	3	9

Table 1: Decision problem parameters. Left : reward matrix

R(a,s)

;

a_{3}

is a safe action,

a_{1}

and

a_{2}

specialize to

s_{1}

and

s_{2}

. Middle, Right : likelihood kernels for channels

i

and

j

. Channel

i

partially identifies

s_{1}

; channel

j

partially identifies

s_{2}

. The channels are conditionally independent.

	$b_{1}=(\tfrac{1}{11},\,\tfrac{2}{11},\,\tfrac{8}{11})$	$b_{2}=(\tfrac{1}{4},\,\tfrac{1}{6},\,\tfrac{7}{12})$	$b_{3}=(\tfrac{5}{12},\,\tfrac{5}{12},\,\tfrac{1}{6})$
	interior of $\mathcal{R}_{3}$	near boundary	on $a_{1}/a_{2}$ boundary
$\operatorname{VoI}(i)$	$0$	$11/16$	$5/2$
$\operatorname{VoI}(j)$	$3/44$	$1/16$	$5/2$
$\operatorname{VoI}(j\mid i)$	$15/176$	$9/64$	$3/32$
$\mathbb{E}[D_{g}]$	$3/176$	$49/64$	—
$\mathbb{E}[D_{h}]$	$0$	$11/16$	—

Table 2: Summary of information values at three marked beliefs.

b_{1}

: interior complementarity (Theorem 4 );

b_{2}

: boundary complementarity (Proposition 2 );

b_{3}

: substitution (Theorem 6 ). Entries marked — indicate that the individual forces are not separately defined at a boundary belief where

V

has a kink.

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.

Value of Information

Value of information studies whether acquiring information can improve decisions enough to justify its cost.

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Bayesian persuasion studies how an informed sender can influence the behavior of a receiver through strategic information disclosure. Standard models assume the sender is the receiver's only source of information, yet in many applications receivers also consult external sources the sender can neither observe nor control. We study an online Bayesian persuasion problem in which a binary-action receiver has access to a fixed signaling scheme that is unknown to the sender. Over

T

rounds, the sender commits to a signaling scheme and sends a signal; the receiver combines it with its private signal and acts, while the sender observes only the action. We design a learning algorithm that achieves regret

\widetilde{O}(T^{3/4})

relative to the optimal scheme of a sender who knows the private signaling scheme of the receiver, with polynomial dependence on the sizes of the state space and the receiver's signal alphabet. Our key insight is reducing the problem of learning the exponentially large belief-space partitioning induced by the private scheme to a one-dimensional change-point detection problem.

Abstract

Value of Information

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