Contribution
A general model of a sender who chooses what a receiver will learn, with an arbitrary finite state space, compact action space, prior and preferences. It shows that choosing a signal is the same as choosing a distribution of posteriors whose mean is the prior, and that nothing else constrains it. From that it gets a necessary and sufficient condition for the sender to gain (the concave closure of its value lies above its value at the prior) and properties of the optimal signal. The mathematics is Aumann and Maschler’s concavification, which the authors credit; what is new is the general persuasion setting and the characterisations.
Key insight
Bayes’s rule constrains only the average of the beliefs a signal produces. So a sender who controls the experiment, but not its reading, can split the prior into posteriors arranged to fall where the receiver’s action changes in its favour. The receiver who knows exactly how the experiment was designed still acts as the sender wants more often than she would without it: in the judge example, “the judge knows 70 percent of defendants are innocent, yet she convicts 60 percent of them!”
Assumptions
- Common prior μ₀ in the interior of Δ(Ω); Ω finite (compact metric in the online Appendix, unread); A compact, at least two actions, each optimal at some belief; u and v continuous.
- Commitment. Receiver observes the chosen signal π and its realization, and the realization is reported truthfully. “This makes our environment effectively nonstrategic.” Sender’s “choice of a signal establishes the meaning of messages”. The authors defend it by legal mandate (Brady v. Maryland), committed disclosure rules (grading, hospital ratings), organizational procedure (“Firms commit to the information they will seek out for performance reviews”), publicly registered studies, and direct observation (taste tests, trial software).
- No transfers, and Sender cannot affect Receiver’s payoffs.
- Sender-preferred tie-breaking: “If Receiver is indifferent between some actions at a given belief, we assume she takes an action that maximizes Sender’s expected utility.” This is what lets the optimum put Receiver exactly at indifference.
- Single receiver. Multiple receivers who care about each other’s actions and receive private signals are explicitly outside the method.
- Arbitrarily informative signals are made innocuous by redefining the state: see the garbling passage under Connections.
Key results
- Proposition 1. For a value v*, these are equivalent: some signal has value v*; some straightforward signal (realizations are recommended actions, and Receiver follows them) has value v*; some Bayes-plausible distribution of posteriors τ has E_τ v̂(μ) = v*. Construction: π(s | ω) = μ_s(ω) τ(μ_s) / μ₀(ω). Bayes plausibility is “the only restriction on the equilibrium distribution of posteriors”. The first equivalence is likened to the revelation principle.
- Corollary 2. With V the concave closure of v̂, the value of an optimal signal is V(μ₀), and Sender benefits from persuasion iff V(μ₀) > v̂(μ₀). An optimal signal exists because Sender-preferred tie-breaking makes v̂ upper semicontinuous.
- Judge example. Prior Pr(guilty) = 0.3; Receiver convicts iff the posterior is at least 0.5. The unique optimum is π(i | innocent) = 4/7, π(g | innocent) = 3/7, π(g | guilty) = 1, giving posteriors 0 (weight 0.4) and 0.5 (weight 0.6), and conviction with probability 0.6 against 0 with no information and 0.3 with full information.
- Remark 1. v̂ concave: no gain at any prior, and no disclosure is optimal. v̂ convex and not concave: gain at every prior, and full disclosure is optimal.
- Proposition 2. No “information Sender would share” (no μ with v̂(μ) > E_μ v(â(μ₀), ω)) implies no gain. Such information plus a discrete preference at the prior implies a gain. With A finite, discreteness holds at the prior generically (except on a Lebesgue-null set). The sufficient condition is useless for connected A.
- Proposition 3 (and 6 in the appendix). If v̂(μ) = ṽ(Tμ) for a linear T, Sender benefits iff ṽ lies below its concave closure at Tμ₀. For the expected-state case Ṽ is only an upper bound on the optimal value, with a counterexample (A = [0,1], Ω = {−1, 0, 1}, v = a²).
- Proposition 4. If an optimal signal induces a belief that leads to a worst action, Receiver is certain of her action there.
- Proposition 5. Under Assumption 1 (no weakly-but-not-strictly induced dominant action) and a strict gain, at any induced belief that is interior or leads to a best-attainable action, Receiver’s preference is not discrete: with finite A, she is indifferent between two actions.
- Alignment (Section IV). More aligned preferences weakly raise Sender’s optimal value, but can lower the gain, and “making preferences more aligned can reduce the extent of communication in equilibrium.”
- Lobbying. Quadratic losses; full disclosure is uniquely optimal when α > 1/2, no disclosure when α < 1/2, and the bias ω* does not matter. Observed industry-biased studies therefore suggest that “the intended constituency is not fully rational, the lobbyists cannot commit not to distort study results ex post, or the payoff functions differ”.
- Product information. A trial separates consumers into those sure the product is not for them and those just willing to buy; “Such optimal information provision allows the firm to extract all the consumer surplus.”
- Extensions. Receiver’s private information and her own later information gathering are absorbed into v̂, keeping Proposition 1 and the corollaries but losing the discreteness results.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Any Bayes-plausible distribution of posteriors is induced by some signal, so Bayes plausibility is the only constraint | strong | Proposition 1, proof in Appendix A (Carathéodory for finite support) |
| C2 | Sender’s optimal value is V(μ₀); Sender gains iff V(μ₀) > v̂(μ₀) | strong | Corollary 2 |
| C3 | With finitely many actions, information Sender would share suffices for a gain at a generic prior | strong | Proposition 2 and Lemma 1 |
| C4 | At an optimum the receiver taking the worst action is certain, and the receiver at an interior belief is indifferent | strong under Assumption 1 | Propositions 4 and 5 |
| C5 | Greater preference alignment can make the optimal signal less informative | strong as an existence claim | Section IV argument; the lobbying example for ω* |
| C6 | Commitment results bound Sender’s gain from communication in any other communication game | asserted here | proof in the online Appendix, unread |
| C7 | Systematic harm to receivers from persuasion indicates limited rationality | informal | the Conclusion’s one sentence; no theorem in the article |
Concepts
- Signal π. A finite realization space S and distributions π(· | ω). An experiment in Blackwell’s sense, though the paper does not say so.
- Distribution of posteriors τ; Bayes plausible. τ over Δ(Ω) with Σ μ τ(μ) = μ₀.
- Straightforward signal. S ⊆ A, and Receiver takes the recommended action.
- v̂(μ). Sender’s expected utility when both hold belief μ and Receiver plays â(μ).
- Concave closure V. The smallest concave function weakly above v̂ (Aumann and Maschler’s concavification).
- Default action. â(μ₀), Receiver’s action with no information.
- Persuasion and incentives. “There are two ways to induce a person to do something. One is to provide incentives, by which we mean anything which changes marginal utility—explicit payments, coercion, or supply of complementary goods. The other is to persuade, by which we mean anything which changes beliefs.”
- Discrete preference; worst action; best-attainable action; more aligned. As defined in Sections II–IV.
Connections
- Aumann and Maschler (1995) supply the formal method; the authors call their problem “mathematically analogous”. Brocas and Carrillo (2007) is the binary special case. Crawford and Sobel, Grossman, Milgrom, Kartik and Spence are the strategic-communication models from which commitment sets this one apart.
- Blackwell is not cited. The word does not occur in the published article or the NBER working paper. The garbling relation is there, unnamed: “We can define ω to be the realization of the most informative signal Sender can generate. Then, his choice of π is equivalent to a choice of how much to garble information that is potentially available.” Footnote 5 points to Ganuza and Penalva’s orders on signals by the dispersion of posteriors, and footnote 15 notes that comparing “the informativeness of sets of signals” is a complication left aside.
- Deception and bounded receivers are set aside. The NBER version names work “where Receivers are not perfect Bayesians (Mullainathan, Schwartzstein, and Shleifer 2008, Ettinger and Jehiel forthcoming)” and keeps it apart. It also says that if Sender chose his signal covertly, “the only equilibrium would be for Sender to choose the fully informative signal and the set of equilibria would be isomorphic to those of Crawford and Sobel (1982).”
Bearing on the record
The LIT’s standing named three checks. Each is answered here.
Does the paper relate persuasion to Blackwell’s order (CLAIM-050)? Not by name. It has the garbling relation, in the passage quoted above, and it uses it only to defend the assumption that Sender can generate any signal. It does not order signals by informativeness, compare the optimum with full disclosure in Blackwell’s sense, or prove anything about the order. The LIT’s phrase “preferring a garbling of full information” is therefore the record’s gloss, and as stated it is empty: under the paper’s redefinition of ω every signal is a garbling of full information, the optimum included. What the paper does show, and what bears on CLAIM-050, is that the party who chooses the experiment can strictly prefer a less informative one, and that how much it reveals is not monotone in how aligned it is with the receiver (C5). That does not conflict with Blackwell’s theorem, which ranks experiments for the person deciding; Sender’s objective is not a decision problem of Receiver’s. The bearing is the record’s: CLAIM-050 says what fidelity is for a receiver, and this paper is a reason not to assume that whoever chooses the rendering procedure wants the procedure that ranks highest. It also touches CLAIM-230: the paper fixes the state by fiat as “the realization of the most informative signal Sender can generate”, so here too the decision measure is fixed by whoever labels the state. Nothing in the paper bears on the bounded-receiver claims (CLAIM-199, CLAIM-225); its receiver is ideal.
Is it manipulation (QUESTION-070)? The paper does not say, and does not ask. It uses the word once, in the introduction, as a gloss on Sender’s aim: “If Receiver understands that Sender chose what information to convey with the intent of manipulating her action for his own benefit, can Sender still gain from persuasion?” Its own taxonomy has two kinds of influence, incentives and persuasion, with coercion among the incentives and “anything which changes beliefs” counting as persuasion. Manipulation is not a third kind. The nearest normative remark is the Conclusion’s, that systematic harm to receivers “suggests that limited rationality may be at play” (C7). Two things in the paper matter for where QUESTION-070 would draw the line, and both readings of them are the record’s:
- Harm depends on the baseline. In the judge example, the record’s arithmetic gives the judge a just verdict with probability 0.7 under the optimal signal (0.4 sure acquittals plus 0.6 × 0.5 convictions), exactly what she gets with no information and below the 1.0 of full information. In the product example the consumer, indifferent when she buys, gets her outside option either way, and the firm takes “all the consumer surplus”. So the receiver is not harmed against no information, and loses all the value the information could have had for her. An account that defines manipulation by harm has to say which baseline it uses.
- The definitions on file need intent and covertness. Susser, Roessler and Nissenbaum (LIT-1002, unread) are summarised in QUESTION-070 as requiring that influence be intentional and covert. Here the design is overt, and the paper’s aim is to show it works when overt. If those definitions are right, Bayesian persuasion is not manipulation, whatever the sender intends. The LIT’s statement that this is “often argued” was not checked against any work and still is not.
On the organized-whole part, the paper’s Sender is any optimiser of v; no intention enters the analysis beyond the maximisation. Several of its own examples are institutional procedures: grading policies, rating procedures, performance reviews, tenure rules. That a procedure adopted for any reason, or selected, would have the same effect is the record’s inference, not the paper’s.
Is the steering invisible in present structure (QUESTION-071)? In this model, no, and the LIT’s standing needs correcting. The receiver is not ignorant of who chose the information structure or why: she observes the signal, and the paper stresses that she is “fully aware that the investigation was designed to maximize the probability of conviction.” The history the LIT says would reveal the steering is common knowledge, and knowing it does not undo it. Moreover the optimum leaves a trace in present belief: by Propositions 4 and 5 the receiver is certain when she takes the sender’s worst action and indifferent when she takes another. A posterior that sits exactly on a decision boundary is the signature of sender-optimal design. So the case does not instantiate QUESTION-071’s first part (invisibility). It is a different case the question does not yet name: influence that is fully visible and still effective, because each step of the receiver’s inference is correct. Whether that helps THEORY-040 is open; the paper is about beliefs and actions, not about will or psychological structure.
The paper carries no instruction for machine-learning practice.
Limitations
- The receiver is an ideal Bayesian with a common prior. Every result depends on it; the paper treats departures as outside its scope and as evidence of limited rationality.
- Commitment is strong. The paper defends it with real settings and, in the online Appendix (unread), with an equivalence to verifiable concealment and an upper-bound result.
- One receiver. The method fails when receivers’ actions depend on others’ signals.
- No welfare or normative analysis. Receiver’s welfare is not computed anywhere in the article; the remarks on it above are the record’s.
- Not Blackwell. Informativeness is used informally, and comparing sets of optimal signals is left aside (footnote 15).
Open questions
- Whether the receiver’s indifference at the optimum is a general signature a third party could test for, not only in the finite-action case.
- Whether a definition of manipulation by function or selection history (QUESTION-070’s “what would answer it”) counts overt, committed information design, and what baseline its notion of harm uses.
- What the order on signals that persuasion induces looks like against Blackwell’s: Ganuza and Penalva’s dispersion orders, cited in footnote 5, would be the place to look.