How much persuasion survives when information comes in separate parts?

Persuasion without a complete picture

Classical Bayesian persuasion studies how a designer can shape decisions by choosing what information to disclose. The receiver understands the signaling rules and updates beliefs accordingly. Persuasion works through “pooling”: grouping different underlying situations into a signal whose average payoff makes action attractive.

A designer who sees everything can combine strengths and weaknesses across an entire situation. But a supplier may test only its own component, or a regional branch may assess only its own market. The preprint asks what remains possible when each source reports on just one part.

These are mathematical results within a theoretical model, not measurements of actual buyers, investors or corporate approvals.

What the model assumes

The receiver’s payoff from acting is the sum of independent, potentially positive or negative contributions, each with a finite mean. One designer knows their probability distributions and commits in advance to all the reporting rules.

Each source observes only its own contribution and uses independent private randomness. The receiver sees the signals, calculates the expected total payoff and acts with a probability that cannot decrease as that expectation rises. A simple example is a threshold rule: act only when the expected payoff reaches a specified cutoff.

The sources follow the designer’s rules; the model does not give them competing objectives or strategic incentives. Its restriction concerns access to information, not a limit on message length.

A guarantee that does not deteriorate with the number of sources

Let q denote the best action probability under a joint experiment that can condition its signal on every contribution, formally a supremum. When the designer knows the receiver’s decision rule, the authors prove that separate-source experiments can achieve at least q²/144. For a known threshold rule, the stronger bound is q²/36.

The constants do not depend on the number of sources or their distributions. For a known threshold rule, a straightforward alternative—having each source pool its top-q probability mass—gives a baseline guarantee of q raised to the number of sources. Here q is the optimal joint action probability for that threshold specifically. This baseline can shrink exponentially as more sources are added.

The new bounds are worst-case guarantees, not predictions of typical performance. Nor do they establish that decentralized information preserves a fixed percentage of centralized success: squaring q can entail a substantial loss when q is small.

An example makes the distinction concrete. With two independent contributions, each equally likely to be 1 or −2, and an action threshold of zero, the optimal joint experiment achieves an action probability of 75%. The optimal separate-source experiment achieves 60%. Its rules are asymmetric: one source reveals nothing, while the other selectively pools outcomes.

Why selective pooling works

The main construction uses “sparse pooling.” Rather than relying on many sources to send pooled signals simultaneously, it keeps their total pooling probability low. Successful events can then involve just one pooling source while all the others disclose their actual contributions.

For each source, the construction calculates how much favorable probability mass it can group into a signal strong enough to help the total reach the receiver’s threshold. If enough such capacity exists, pooling probabilities are scaled down. If too little exists, the authors prove that revealing all contributions already supplies the required guarantee.

For explicitly specified finite distributions with rational values and probabilities, the appendix also provides deterministic polynomial-time constructions. These use approximations to tail probabilities rather than enumerating the potentially enormous distribution of the total.

When the receiver’s rule is unknown

A second result removes knowledge of the decision rule. The authors construct one experiment from the prior distributions alone that works simultaneously for every allowed rule.

For every positive ε, its action probability is at least cε times q raised to 2 + ε, where cε depends only on ε. The paper gives a sharper expression for the loss beyond q² and shows that its order is unavoidable up to a constant in the exponent. A pure constant-times-q² universal guarantee is impossible in general, even when the experiment needs to cover only two thresholds.

Where the guarantees stop

The quadratic exponent itself is unavoidable even with two sources and a known threshold. Independence and addition are also essential to the general guarantee: the authors construct correlated examples where separate-source success becomes arbitrarily small despite joint success of one-half. For a broader, nonadditive payoff class, they show an exponentially small separate-source success probability with the same joint benchmark.

Another counterexample shows that joint success arbitrarily close to certainty need not imply decentralized success approaching certainty uniformly across source counts: separate-source success can remain at most 1 − 1/e.

The authors disclose extensive AI assistance in developing proofs, searching literature and drafting the paper, and state that they verified and take responsibility for its contents. Open questions include sharper constants, sequential reporting and sources with their own incentives.