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 ΔVoI 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 ΔVoI .
s1
s2
s3
a1
12
0
3
a2
0
12
3
a3
3
3
9
Table 1: Decision problem parameters. Left : reward matrix R(a,s) ; a3 is a safe action, a1 and a2 specialize to s1 and s2 . Middle, Right : likelihood kernels for channels i and j . Channel i partially identifies s1 ; channel j partially identifies s2 . The channels are conditionally independent.
b1=(111,112,118)
b2=(41,61,127)
b3=(125,125,61)
interior of R3
near boundary
on a1/a2 boundary
VoI(i)
0
11/16
5/2
VoI(j)
3/44
1/16
5/2
VoI(j∣i)
15/176
9/64
3/32
E[Dg]
3/176
49/64
—
E[Dh]
0
11/16
—
Table 2: Summary of information values at three marked beliefs. b1 : interior complementarity (Theorem 4 ); b2 : boundary complementarity (Proposition 2 ); b3 : 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.
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 O(T3/4) 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.
Organizations often pool dispersed information into one ranking and then allow many agents to act on that shared view. In a discovery problem, this can improve beliefs while reducing coverage. We develop an exactly solvable benchmark with sixteen boxes, one target, eight searchers, and noisy private clues. Pooling raises the accuracy of the best single recommendation from 0.20 to 0.3835, but repeating that recommendation lowers group discovery from 0.8322 under decentralized clue-following to 0.3835. A coordinated eight-action portfolio using the same pooled reports reaches 0.8594, and seven coordinated actions recover the decentralized benchmark. The paradox is a protocol failure, not an information failure: a one-answer rule compresses a portfolio of available actions into one repeated choice. We then replace the planner with self-interested searchers who split a prize. The equal-split game is a potential game. Its anonymous symmetric equilibrium obeys a water-filling rule. In the canonical instance it achieves 0.5991: strictly above consensus, but below both private search and the planner. The exact mixed price of anarchy is 2 - 1/N. A sole-rescue reward, which pays only an agent who covers the target alone, makes every pure Nash equilibrium first-best. Finally, a latent common-cue model shows how correlated reports collapse effective discovery channels. The centralized planner gain rises strictly with copying, and in the canonical environment the symmetric market overtakes decentralized report-following at copying probability c = 0.788462. In a proportional large-market limit the five-protocol ordering survives exactly: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874. The contribution is a compact benchmark that separates information, allocation, incentives, and dependence into exact, reusable quantities.
Information sharing can improve a pooled estimate while eliminating independent rescue actions. This paper separates those effects in exact finite discovery models. A centralized action-budget profile shows that equal one-person accuracy can coexist with different portfolio values. Under a registered incremental-sharing protocol, a sharing step improves discovery exactly when pooled residual error contracts faster than an independent rescue attempt. Exact bounded registries exhibit compression, aggregation, neutral curves, and a bounded zero mixed class. In a two-agent Bayesian game with a hidden mixture of common and independent signal sources, the registered selected equilibrium yields a strict positive sharing interval at signal accuracy 3/5, while alternative equilibria show that the result is selection-dependent rather than universal. The models are synthetic and finite; no human or organizational data are used.