Engine · Chapter 4, Signals, Disclosure, and Persuasion

Persuasion by design

What can a committed test-designer make a sceptical audience believe?

The situation

Prior μ₀ that the state is good % 30

Receiver's threshold θ % 50 acts when its posterior reaches θ

The test

the sender's optimal test design your own

The book's scenario

How often the receiver acts

\frac{3}{5}

≈ 0.6000

against $0$ with no test and $\frac{3}{10}$ under full disclosure. The sender's ceiling, the concave closure at the prior, is $\frac{3}{5}$ .

What the test does

The test	pass $1$ of good cases, $\frac{3}{7}$ of bad ones
A pass	probability $\frac{3}{5}$ , posterior $\frac{1}{2}$ , receiver acts
A fail	probability $\frac{2}{5}$ , posterior $0$ , receiver does not act
Receiver's expected payoff	$0$ (no test: $0$ ; full disclosure: $\frac{3}{10}$ )

D The book certifies: pass $\frac{3}{7}$ of bad cases, action rate $\frac{3}{5}$ against $\frac{3}{10}$ under full disclosure, receiver's payoff $0$ against $\frac{3}{10}$ . Agreement, exactly.

the sender's payoff at each posterior: 1 once the receiver's belief reaches θ = 1/2, 0 below · its concave closure, the most a test can deliver at each prior · the prior, split by the test into two posteriors; the sender's value is the height of the chord above the prior. The optimal test puts one posterior exactly at θ and the other at 0.

The arithmetic

$P (pass) = μ_{0} P (pass ∣ good) + (1 - μ_{0}) P (pass ∣ bad) = \frac{3}{5}$

$μ_{pass} = \frac{μ _{0} P ( pass ∣ good )}{P ( pass )} = \frac{1}{2} \geq θ$

The receiver's payoff from acting at belief $μ$ is $μ - (1 - μ) θ / (1 - θ)$ , zero exactly at $θ$ ; the optimal test leaves it with nothing because every pass lands on that line. Optimal share of bad cases passed: $μ_{0} (1 - θ) / (θ (1 - μ_{0})) = \frac{3}{7}$ .

The model it runs

Model S1 · p. 168

How much action can a designed test produce, and at whose expense?

1Decision-makers: sender (designs the test) and receiver (acts)
2Time: the test is chosen before the state is realised; then the result is seen and the receiver acts
3Information: endogenous: the sender's choice of experiment (field 6) determines what the receiver observes
4Commitment: the sender commits to the test and cannot alter or hide its result
5Payoffs: sender: $1$ if the receiver acts; receiver: as in Definition 4.1 (p. 168)
6Strategies: binary tests (sufficient here); the receiver's rule is fixed by its threshold

Analysis contract

(i) concept: sender-optimal experiment against a best-responding receiver (commitment to an information policy)
(ii) deviations: all experiments: by Theorem 4.2 (p. 168) it suffices to consider distributions of posteriors
(iii) beliefs: Bayes' rule from the common prior
(iv) selection: the indifferent receiver acts (sender-favourable)
(v) attainment: attained; exact; checked against enumeration of two-posterior splits