Chapter 4 · begins on page 141
Signals, Disclosure, and Persuasion
The models
Each model in the book is stated on one card: who decides, when, with what information and commitment, what they get, what they may do, and the question it answers.
Model S1 · p. 168
How much action can a designed test produce, and at whose expense?
Model S2 · p. 170
How much information can be transmitted when interests differ and talk is free?
Model S3 · p. 172
When does a costly signal separate the types, and what does separation cost?
Model S4 · p. 174
Can information hurt the player who receives it, and what can that player do about it?
Model S5 · p. 177
What does a published test buy the regulator, at whose expense, and how much of it survives if results can be revised?
Model S6 · p. 180
Which kind of cost lets a threat separate the resolved sender from the irresolute one?
What is proved, in words
The book follows each result with one sentence that says what it means without symbols. The formal statement is a click away; the proofs are in the book.
Proposition 4.4 · Model S1 · p. 169
The optimal test passes every good case and just enough bad ones to keep a pass persuasive; the action rate is twice that of full disclosure, against , while the receiver ends no better off than with no test at all.
Proposition 4.6 · Partition equilibria; uniform–quadratic case · p. 170
In the uniform–quadratic model with a constant positive bias, a sender without commitment can convey only which of finitely many intervals the state lies in; as the bias grows, the largest number of intervals falls in steps, and in every such equilibrium both sides do worse than under commitment to disclose.
Proposition 4.7 · With commitment, the biased sender tells everything · p. 171
If the biased sender can commit to its disclosure policy in advance, telling everything is best for both sides — under the stated constant bias, quadratic losses and posterior-mean response.
Proposition 4.9 · Separation and its cost · p. 173
A costly signal can separate types, but its cost can be pure waste: separation burns , and the high type strictly prefers it to free pooling only when it is rare enough, and is indifferent at the threshold.
Proposition 4.15 · Model S5 · p. 178
When creditors would run without information (), a regulator that would rather keep a weak bank funded than see it run designs a test that passes just enough weak banks to hold creditors at their threshold, and creditors gain nothing from it; if its results can be revised, nothing is lost until the chance that they stand falls below a threshold, and then everything is — and the less credible the regulator, the harsher the binding part of its test must be.
Proposition 4.18 · Model S6 · p. 181
Free talk cannot carry a threat that every type wants to make; a cost paid up front deters the irresolute sender only at a price the resolved sender will not pay, because a concession is worth more to the irresolute; a cost paid only on backing down falls on the type that would back down, and separates the types once it reaches .
Theorem 4.2 · Concavification; binary state · p. 168
A sender who designs the test before knowing the state can produce any spread of posteriors that averages to the prior, and its best value is read off the smallest concave function above its payoff.
The worked case
From page 158 · sources checked
Bank stress tests as a committed information policy
What does a published test buy the supervisor, at whose expense, and how much of it depends on the test's binding?
The mapping, the network and what the mathematics licenses →Parallaxes in history
1875 · Chapter 4 · p. 156
Before a shared number, a shared standard
Metre Convention, 1875: seventeen states fund a bureau that ties measurements to prototypes
Carbon border adjustment, 2026: comparable emissions figures need a common measurement rule (Section 4.6 (p. 153))
1895–1899 · Chapter 4 · p. 157
A statement in the House
Westminster, 1895: a minister tells the Commons how an expedition's advance would be viewed
Model S6: a public statement is the move to which a cost of backing down can attach (Proposition 4.18 (p. 181))
The people
Chapter 4 · Signals, Disclosure, and Persuasion
A. Michael Spence
born 1943
Economist who studied at Princeton, Oxford and Harvard, where he received his doctorate. He developed the theory of market signalling — how an informed party can credibly convey its type by an action that is costlier for other types — and taught at Harvard and Stanford, where he was dean of the Graduate School of Business. He shared the 2001 Nobel Memorial Prize in Economic Sciences with George Akerlof and Joseph Stiglitz for analyses of markets with asymmetric information.
Main works
- “Job market signaling” (Quarterly Journal of Economics 87, 1973)
- Market Signaling: Informational Transfer in Hiring and Related Screening Processes (1974)
Chapter 4 · Signals, Disclosure, and Persuasion
Thomas C. Schelling
1921 – 2016
Economist who received his doctorate from Harvard and taught at Yale, at Harvard and at the University of Maryland; he spent a year at the RAND Corporation. He studied bargaining, commitment and credibility — how a player can gain by limiting its own options or information — and later how individual choices aggregate into collective patterns. He shared the 2005 Nobel Memorial Prize in Economic Sciences with Robert Aumann for having enhanced the understanding of conflict and cooperation through game-theory analysis.
Main works
- The Strategy of Conflict (1960)
- Arms and Influence (1966)
- Micromotives and Macrobehavior (1978)