The Encyclopedia of Economic Principals

Chapter 23

Information Acquisition, Disclosure, Persuasion, and Attention

What people learn depends on who designs, sends and pays for the information.

Four of the chapter's worked examples, made interactive: a committed disclosure rule, cheap talk from a biased staff, voluntary disclosure that unravels, and the limit on how much prices can reveal.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

How much bad news can a good signal hide?

How often can a committed regulator say "retain" and still be believed?

A committed rule can pool some bad states with the good one, as long as the favorable message stays just credible enough. The average posterior must still equal the prior.

Equation, written in LaTeX: \Pr(G\mid retain)=\frac{0.30}{0.30+0.70x}.

Equation, written in LaTeX: \frac{0.30}{0.30+0.70x}=0.50 \Rightarrow x=\frac{3}{7}.

Equation, written in LaTeX: 0.30+0.70(\frac{3}{7})=0.60.

Equation, written in LaTeX: 0.60(0.50)+0.40(0)=0.30.

Scroll sideways for the whole equation

G is a sound bank, with prior probability set below; depositors retain only if Pr(G | message) is at least the threshold. The rule sends "retain" always when sound and with probability x when fragile.

Predict first. If depositors demand 0.75 instead of 0.50, does retention fall by more or less than half?

Your prediction

Choose an example

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Figure: How much bad news can a good signal hide? Bars of retention probability: full revelation 0.30, no disclosure 0, optimal rule 0.60.
Depositor retention threshold: 0.5, Prior probability bank is sound: 0.3
Constructed example: the chapter's hypothetical bank (prior 0.30, thresholds 0.50 and 0.75); thresholds of 0.40 and 0.60 and priors of 0.20 and 0.40 are added.

Calculated values

Pooling probability x
0.4286
Retention, optimal rule
0.6000
Retention, full revelation
0.30
Retention, no disclosure
0

Under no disclosure the posterior stays at 0.30, so everyone withdraws. Setting 0.30 / (0.30 + 0.70x) = 0.50 gives x = 0.30(0.50) / (0.50 x 0.70) = 0.4286. The retain message is sent with probability 0.30 + 0.70(0.42857) = 0.6000, against 0.30 under full revelation. Bayes plausibility: 0.6000(0.50) + 0.4000(0) = 0.3000.

Worked steps

  1. x = 0.30(0.50) / (0.50 x 0.70) = 0.4286
  2. Retention = 0.30 + 0.70(0.42857) = 0.6000
  3. Check: 0.6000(0.50) + 0.4000(0) = 0.3000

Use the idea

When reading a disclosure policy, ask how many bad cases it lumps into the favorable label; a stricter audience forces a cleaner label.

Where the conclusion applies

Commitment to the rule, Bayesian depositors who know it, and a single threshold. Retaining funds in a fragile bank is still a loss to depositors.

Check your understanding: At prior 0.40 and threshold 0.40, why does the regulator need no garbling at all?
The prior 0.40 already meets 0.40, so x = 1 and retention is 1.0.

Chapter 23 source: section "Bayesian persuasion".

Demonstration 2 of 4

Cheap talk with a biased staff

How much bias can a two-message report survive?

The staff member at the cutoff must be indifferent between the two actions. The larger the bias, the lower that cutoff, until no informative split is left.

Equation, written in LaTeX: a_L=\frac{t}{2}, a_H=\frac{t+1}{2}.

Equation, written in LaTeX: t+0.10=\frac{a_L+a_H}{2}=\frac{2t+1}{4}.

Equation, written in LaTeX: 4t+0.40=2t+1 \Rightarrow t=0.30.

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theta is uniform on [0, 1]. The committee wants a = theta, the staff wants theta + b. Staff with theta below t say "low"; a_L and a_H are the committee's actions after each message.

Predict first. At what bias does the two-message equilibrium first disappear?

Your prediction

Choose an example

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Figure: Cheap talk with a biased staff. Unit interval of inflation pressure with bias 0.10: cutoff 0.30, actions 0.150 and 0.650.
Staff bias b: 0.1
Constructed example: the chapter's hypothetical policy staff (bias 0.10 and 0.30); biases of 0.05 and 0.20 are added.

Calculated values

Cutoff t
0.30
Action after low
0.15
Action after high
0.65
Two messages informative
yes

With b = 0.10, the boundary condition t + 0.10 = (2t + 1) / 4 gives t = (1 - 4 x 0.10) / 2 = 0.30. The committee takes a_L = 0.30 / 2 = 0.150 after "low" and a_H = (0.30 + 1) / 2 = 0.650 after "high".

Worked steps

  1. t = (1 - 4 x 0.10) / 2 = 0.30
  2. a_L = 0.30 / 2 = 0.150
  3. a_H = (0.30 + 1) / 2 = 0.650

Use the idea

Coarse reports from an advisor with different aims are not a failure of language; the conflict of interest limits how fine the categories can be.

Where the conclusion applies

A uniform state, known bias, quadratic style preferences and the two-message equilibrium; babbling always exists too.

Check your understanding: With b = 0.20, what actions does the committee take after each message?
t = (1 - 0.80) / 2 = 0.10, so a_L = 0.05 and a_H = 0.55.

Chapter 23 source: section "Cheap-talk partition equilibrium".

Demonstration 3 of 4

Unraveling from the top, stopped by a fee

Who reveals quality when silence is priced at the average of the silent sellers?

Silence is priced at the average of those who stay silent, so the best silent seller always wants out. A disclosure fee stops the cascade once quality minus fee falls below that price.

Equation, written in LaTeX: \frac{40+60+80+100}{4}=70.

Equation, written in LaTeX: \frac{40+60+80}{3}=60.

Equation, written in LaTeX: \frac{40+60}{2}=50.

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Four equally likely sellers with qualities 40, 60, 80 and 100. Buyers pay expected quality; a certificate reveals quality truthfully for the fee set below.

Predict first. Does a fee of 25 stop unraveling earlier or later than 15?

Your prediction

Choose an example

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Figure: Unraveling from the top, stopped by a fee. Four seller bars with qualities 40 to 100; disclosing types 100 and 80; final silent price 50.
Disclosure fee: 15
Constructed example: the chapter's hypothetical equipment sellers (qualities 40 to 100, fees 0 and 15); fees of 5 and 25 are added.

Calculated values

Final silent price
50
Disclosing types
100 and 80
Silent types
40 and 60

With a fee of 15: Silent price 70; 100 - 15 = 85 > 70, so 100 discloses; Silent price 60; 80 - 15 = 65 > 60, so 80 discloses; Silent price 50; 60 - 15 = 45 < 50, so 60 stays silent. The process stops with 40 and 60 silent at a price of 50.

Worked steps

  1. Silent price 70; 100 - 15 = 85 > 70, so 100 discloses
  2. Silent price 60; 80 - 15 = 65 > 60, so 80 discloses
  3. Silent price 50; 60 - 15 = 45 < 50, so 60 stays silent

Use the idea

If certificates are cheap, read silence as bad news; a costly certificate leaves room for decent sellers to stay quiet.

Where the conclusion applies

Sellers know their quality, disclosure is verifiable, buyers are aware of the option and price at expected quality. A seller exactly indifferent is shown as staying silent.

Check your understanding: With fee 25, which sellers disclose and what is the silent price?
100 - 25 = 75 > 70 so 100 discloses; silent price 60; 80 - 25 = 55 < 60, so only 100 discloses.

Chapter 23 source: section "Disclosure unraveling".

Demonstration 4 of 4

Why prices cannot be fully revealing

As prices get more accurate, does anyone still pay to research?

Informed traders earn only from the gap between price and value. The more prices reveal, the smaller that gap, until the reward no longer covers the research that made prices informative.

Equation, written in LaTeX: 0.75(\$10)+0.25(\$30)=\$15.

Equation, written in LaTeX: 0.975(\$1)+0.025(\$39)=\$1.95<\$2.

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A stock pays $120 or $80 with equal prior. Price matches the state with probability pi and equals expected value given the price. Research costs $2.

Predict first. As prices get more accurate, does the analyst's profit rise or fall?

Your prediction

Choose an example

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Figure: Why prices cannot be fully revealing. Bars for gross profit 15.00 and net profit 13.00 against a research cost of 2.
Probability price matches state: 0.75
Constructed example: the chapter's hypothetical stock (payoffs 120 and 80, cost 2, accuracy 0.75 and 0.975); an accuracy of 0.90 is added.

Calculated values

Prices
110 and 90
Gross profit
$15.00
Net profit
$13.00
Research worthwhile
yes

Prices are 0.750(120) + 0.250(80) = 110 and 0.750(80) + 0.250(120) = 90. In the high state the analyst buys and earns 120 - 110 = 10 with probability 0.750 or 120 - 90 = 30 with probability 0.250: 0.750(10) + 0.250(30) = 15.00. Net of the 2 cost: 13.00. Research is worthwhile.

Worked steps

  1. Prices: 110 and 90
  2. Gross = 0.750(10) + 0.250(30) = 15.00
  3. Net = 15.00 - 2 = 13.00

Use the idea

Some mispricing is the fee markets pay for research; perfectly efficient prices would leave no one paid to make them so.

Where the conclusion applies

Two payoffs, one share traded, competitive rational prices and a fixed research cost.

Check your understanding: At pi = 0.90, is research still worthwhile?
Prices 116 and 84; gross 0.90(4) + 0.10(36) = 7.20; net 5.20 > 0, yes.

Chapter 23 source: section "Grossman-Stiglitz impossibility".