Source
Emir Kamenica and Matthew Gentzkow (2011), “Bayesian Persuasion”, American Economic Review 101(6):2590–2615, LIT-1021, read in full in NOTE-717, with Appendix A’s proofs. The online Appendix was not read.
What was actually shown
Setting. A common prior μ₀ in the interior of Δ(Ω), Ω finite; a compact action set with at least two actions; continuous payoffs. Sender chooses a signal π; Receiver observes π and its realization, which is reported truthfully. “This makes our environment effectively nonstrategic.” No transfers. When Receiver is indifferent she takes the action Sender prefers.
Proposition 1. For a value v*, these are equivalent: some signal has value v*; some straightforward signal (realizations are recommended actions, which Receiver follows) has it; some distribution of posteriors τ with mean μ₀ has E_τ v̂(μ) = v*. Bayes plausibility is “the only restriction on the equilibrium distribution of posteriors”. The proof constructs the signal: π(s | ω) = μ_s(ω) τ(μ_s) / μ₀(ω).
Corollary 2. With V the concave closure of v̂, the optimal value is V(μ₀), and Sender gains iff V(μ₀) > v̂(μ₀). An optimum exists because the tie-breaking makes v̂ upper semicontinuous. If v̂ is concave, no disclosure is optimal at every prior; if convex and not concave, full disclosure is (Remark 1).
The judge. Prior Pr(guilty) = 0.3; the judge convicts iff her posterior is at least 0.5. The unique optimum gives posteriors 0 (weight 0.4) and 0.5 (weight 0.6): conviction with probability 0.6, against 0 with no information and 0.3 with full information. The judge “is fully aware that the investigation was designed to maximize the probability of conviction”. NOTE-717 redid these numbers.
Propositions 4 and 5. If an optimal signal induces a belief leading to Sender’s worst action, Receiver is certain there. Under Assumption 1 and a strict gain, at an interior induced belief Receiver’s preference is not discrete: with finite actions she is indifferent between two.
Alignment (Section IV). More aligned preferences weakly raise Sender’s optimal value but can lower the gain, and can make the optimal signal less informative.
What could have come out otherwise. The receiver’s knowledge of the design could have undone its effect. It does not, because each posterior is correct given the signal. The NBER version adds that if Sender chose covertly, the only equilibrium would be full disclosure.
What this does not say
- That this is manipulation. The word appears once, for Sender’s aim. The paper’s taxonomy has incentives and persuasion, “anything which changes beliefs”, and no third kind. Definitions that require covertness (LIT-1002, as QUESTION-070 summarises it) would exclude this case, since the design is overt.
- That the receiver is harmed. The paper computes no welfare. On the record’s arithmetic (NOTE-717), the judge reaches a just verdict with probability 0.7 under the optimum, exactly as with no information and below the 1.0 of full information; in the product example the consumer gets her outside option either way and the firm takes “all the consumer surplus”. So whether she is harmed depends on the baseline.
- That it holds for real receivers. Every result needs an ideal Bayesian with a common prior. The paper reads systematic harm to receivers as a sign that “limited rationality may be at play”; that is one sentence of its Conclusion, not a theorem.
- That it holds without commitment or with several receivers. The sender must commit and the receiver must observe the signal. Receivers who care about each other’s actions and get private signals are outside the method.
- Anything about Blackwell’s order. Blackwell is not cited. The paper redefines the state as “the realization of the most informative signal Sender can generate”, so every signal is a garbling of full information, the optimum included. What the result does show is that the party who chooses the experiment can strictly prefer a less informative one, which Blackwell’s ranking for the decider does not forbid (NOTE-717).
- That the indifference signature is general. It is proved for finite actions under Assumption 1. Whether a third party could test for it more widely is open.
Connections
- QUESTION-071 asks whether a will can be manipulated with nothing in
its present structure showing it. This is a different case, which the
question does not yet name: influence that is fully visible and still
effective, because each step of the receiver’s inference is sound. It
also leaves a present trace, a posterior exactly on a decision boundary.
It concerns beliefs and actions, not will, so no
answersedge is filed. - QUESTION-070. The sender is any optimiser of a payoff, and several of the paper’s examples are institutional procedures (grading, ratings, performance reviews). That a selected procedure would act the same is the record’s inference (LIT-1021, Standing). The paper does not say whether this is a wrong.
- THEORY-235 says covert influence exploiting decision-making vulnerabilities harms autonomy by making the person opaque to herself. Here the influence is overt and exploits no vulnerability; the receiver is ideal.
- CLAIM-050 says what fidelity is for a receiver. This result is a reason not to assume that whoever chooses the rendering procedure wants the most informative one.
- Rules in equilibrium (THEORY-414) uses correlated equilibrium, where a public signal tells each player what to do. Here a committed signal does the same for one receiver, chosen by an interested party. The pairing is the record’s.
Note of 2026-10-21: the many-player generalization
Bergemann and Morris’s survey of information design (LIT-1313), read on 2026-10-21 in its accepted manuscript (NOTE-1034), generalizes this result to many receivers with information of their own: an omniscient designer can attain exactly the Bayes correlated equilibria, and a receiver’s own information bounds what can be induced (THEORY-844). With one uninformed receiver the obedience constraints reduce to this account’s. The account is unchanged.