Saddle Point

Chapter 4 · begins on page 141

Signals, Disclosure, and Persuasion

Megaphone back to it on the cover
Try it yourself Persuasion by design What can a committed test-designer make a sceptical audience believe? Open the engine →

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 S1 · Chapter 4 · page 168

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

A test designed by the sender; Model S1


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: 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

The 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
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 168Run this model →

Model S2 · p. 170

How much information can be transmitted when interests differ and talk is free?

Model S2 · Chapter 4 · page 170

How much information can be transmitted when interests differ and talk is free?

Cheap talk with a bias; Model S2


1Decision-makers
sender (informed) and receiver (acts)
2Time
state, then message, then action
3Information
the sender knows ; the receiver knows only the message
4Commitment
none (contrast: the sender commits to a policy in Proposition 4.7 (p. 171))
5Payoffs
quadratic losses; the sender is biased upwards by
6Strategies
message rules ; action rules

The analysis contract

(i) concept
perfect Bayesian equilibrium
(ii) deviations
every message for every type; every action
(iii) beliefs
Bayes' rule on the path; off-path messages can be mapped to on-path ones
(iv) selection
all partition equilibria are listed; the most informative one is singled out
(v) attainment
attained; exact boundaries; indifference and best replies checked exactly
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 170

Model S3 · p. 172

When does a costly signal separate the types, and what does separation cost?

Model S3 · Chapter 4 · page 172

When does a costly signal separate the types, and what does separation cost?

Two types and a costly signal; Model S3


1Decision-makers
sender of two types; a competitive receiver (reward = expected type)
2Time
type, then signal, then reward
3Information
the type is private; the signal is observed
4Commitment
none
5Payoffs
reward minus signalling cost
6Strategies
signal levels for each type

The analysis contract

(i) concept
perfect Bayesian equilibrium in pure strategies
(ii) deviations
every signal level for each type
(iii) beliefs
off the path, beliefs put all mass on the low type (the most pessimistic)
(iv) selection
the separating set is reported; the least-cost separating equilibrium is singled out
(v) attainment
attained; closed form
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 172

Model S4 · p. 174

Can information hurt the player who receives it, and what can that player do about it?

Model S4 · Chapter 4 · page 174

Can information hurt the player who receives it, and what can that player do about it?

A producer who may know more; Model S4


1Decision-makers
producer and buyer
2Time
one simultaneous move; in (E) the experiment is chosen first
3Information
(U), (K), (Q) or (E); the buyer knows which, but never the state
4Commitment
none, except the commitment to an experiment in (E)
5Payoffs
as in Definition 4.11 (p. 174); not zero-sum (and zero-sum in the counterpart)
6Strategies
producer: action for each observation; buyer: one of three moves

The analysis contract

(i) concept
Bayesian Nash equilibrium (all equilibria, mixed included); in (E), the producer's best experiment
(ii) deviations
every strategy of each producer type and of the buyer
(iii) beliefs
common prior; the buyer updates on the information structure, not on a result it cannot see
(iv) selection
all equilibria are listed, including the continuum of buyer mixtures at the boundary ; ties are reported both ways
(v) attainment
attained, except the value in (E), which is attained only under the producer-favourable pure tie and is otherwise a supremum
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 174

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 S5 · Chapter 4 · page 177

What does a published test buy the regulator, at whose expense, and how much of it survives if results can be revised?

A pass/fail test with revisable results; Model S5


1Decision-makers
regulator (designs and reports the test) and creditors (roll over or run), treated as one receiver
2Time
test published before the state is known; result reported; then creditors act
3Information
endogenous: the test and, with probability , the regulator's own report after seeing the state
4Commitment
the published test binds with probability ; is Model S1's commitment, is cheap talk
5Payoffs
creditors: or from rolling over, from running; regulator: , or
6Strategies
binary tests; the regulator's report in the revisable case; the creditors' threshold rule

The analysis contract

(i) concept
for , the regulator-optimal experiment; for , the regulator's best perfect Bayesian equilibrium given the published test
(ii) deviations
all experiments (by Theorem 4.2 (p. 168)); every report of the revising regulator in each state
(iii) beliefs
Bayes' rule from the common prior and ; a message never sent is read as bad news
(iv) selection
the indifferent creditor rolls over (regulator-favourable)
(v) attainment
attained; exact; checked against enumeration of two-posterior splits and of every binary test on a grid
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 177

Model S6 · p. 180

Which kind of cost lets a threat separate the resolved sender from the irresolute one?

Model S6 · Chapter 4 · page 180

Which kind of cost lets a threat separate the resolved sender from the irresolute one?

A public threat with a cost of backing down; Model S6


1Decision-makers
sender of two types; receiver of two types, one of which always resists
2Time
types, then message, then concession or resistance, then fight or back down
3Information
both types are private; the message is observed
4Commitment
none beyond the message's cost: is paid only on backing down after , on sending
5Payoffs
as in the definition; the responsive receiver strictly prefers resisting below belief that the sender is resolved, conceding above it, and is indifferent at , where it may mix
6Strategies
a probability of threatening for each sender type; a probability of conceding to each message; fight or back down

The analysis contract

(i) concept
perfect Bayesian equilibrium, mixed strategies allowed
(ii) deviations
both messages for each type; both responses to each message; fight or back down
(iii) beliefs
off the path, all mass on the irresolute type (the most pessimistic for the sender, as in Model S3)
(iv) selection
the separating equilibrium where it exists, otherwise the equilibrium in which the irresolute type mixes; pooling on silence reported in Remark 4.19 (p. 183)
(v) attainment
attained; closed form
In the book: Chapter 4, Signals, Disclosure, and Persuasion, page 180

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 4.4 · Chapter 4 · page 169

Model S1

In words 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.


Here if and otherwise, so . The optimal test passes every good case and a share of the bad cases; a pass leaves the posterior at exactly , a fail at . The receiver acts with probability , against with no test and with a perfectly informative one. The receiver's expected payoff under the optimal test is — the same as with no information — and under full disclosure.

The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 169

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 4.6 · Chapter 4 · page 170

Partition equilibria; uniform–quadratic case

In words 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.


Every equilibrium is outcome-equivalent (almost surely the same induced actions and payoffs; not necessarily the same message vocabulary) to a partition of into intervals, the sender revealing only which interval contains and the receiver choosing its midpoint. The boundaries are , , and such an equilibrium exists if and only if . The receiver's expected loss is and the sender's is that plus . At at most intervals can be sustained, with boundaries ; the receiver's loss is , and for .

The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 170

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 4.7 · Chapter 4 · page 171

With commitment, the biased sender tells everything

In words 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.


If the sender can commit to an information policy before learning , and the receiver responds with , the sender's expected loss is . It is minimised exactly when almost surely — full disclosure, up to informational equivalence of the message alphabet — which gives the sender a loss of and the receiver a loss of , both better than in any cheap-talk equilibrium. The identity uses all three stated ingredients: the constant additive bias, the common quadratic metric, and the posterior-mean response; it does not extend to arbitrary quadratic losses.

The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 171

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 4.9 · Chapter 4 · page 173

Separation and its cost

In words 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.


A separating equilibrium with the low type at and the high type at exists if and only if (weak incentive constraints; Spence's original statement has the strict interval). In the least-cost one () the high type receives and the low type ; in the zero-cost pooling benchmark — pooling at — both receive . Separation burns an expected in signalling costs. The high type weakly prefers least-cost separation to zero-cost pooling exactly when its share is at most (indifferent at ); above it, separation leaves both types worse off than that benchmark.

The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 173

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 4.15 · Chapter 4 · page 178

Model S5

In words 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.


Let and let , so that without information creditors run.

  • Commitment (). If , the optimal test passes every sound bank and a share of weak ones; a pass leaves creditors exactly at and a fail at , and the regulator's value is , above the value of full disclosure and of no disclosure. If , full disclosure is the unique optimum; if it is one of many; in both cases . (For creditors roll over without information. If no test is needed, and ; if full disclosure remains optimal, with value ; if the value is , attained both by no disclosure and by full disclosure.)
  • Creditors. Under the optimal test with creditors' expected payoff is , their payoff without information; under full disclosure it is .
  • Discretion (). If , creditors run with probability one in every equilibrium, and the regulator receives . If , truthful reporting is an equilibrium and gives : commitment is then worth nothing.
  • Partial credibility (). The regulator's best equilibrium value is if and if . Above the threshold it is attained by a published test that passes every sound bank and a share of weak ones — a share that falls as falls and is at — the revising regulator reporting a pass in both states.
  • At the base values: , a pass occurs with probability , against under full disclosure and with no disclosure; creditors receive , against under full disclosure; a weak bank is funded with probability ; and .
The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 178

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 .

Proposition 4.18 · Chapter 4 · page 181

Model S6

In words 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 .


  • Cheap talk. In every equilibrium the responsive receiver resists after every message sent with positive probability. The resolved sender receives , the irresolute , the responsive receiver ; a fight occurs with probability .
  • Sunk cost. No supports a separating equilibrium: deterring the irresolute type needs , while the resolved type pays at most .
  • Audience cost. The separating equilibrium — the resolved type threatens, the irresolute does not, the responsive receiver concedes to a threat and resists silence — exists if and only if . In it the resolved sender receives , the irresolute , the responsive receiver ; a fight occurs with probability , and the cost is never paid.
  • For there is an equilibrium in which the resolved type threatens, the irresolute type threatens with probability , and the responsive receiver resists silence and concedes to a threat with probability ; a threat is conceded overall with probability . At : , the resolved sender receives , the irresolute , the responsive receiver ; the cost is paid with probability and a fight occurs with probability .
The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 181

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.

Theorem 4.2 · Chapter 4 · page 168

Concavification; binary state

In words 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.


Let be the sender's expected payoff when the receiver holds posterior , and let be bounded and upper semicontinuous. A distribution of posteriors can be induced by some experiment if and only if its mean is the prior. The sender's best value is , where is the smallest concave function above , and under these assumptions it is attained by an experiment with at most two results. Without upper semicontinuity, is in general only a supremum.

The proof is in the book: Chapter 4, Signals, Disclosure, and Persuasion, page 168

The worked case

Parallaxes in history

1875 · Chapter 4 · p. 156

Before a shared number, a shared standard

Then

Metre Convention, 1875: seventeen states fund a bureau that ties measurements to prototypes

Now

Carbon border adjustment, 2026: comparable emissions figures need a common measurement rule (Section 4.6 (p. 153))

1875 · Chapter 4 · page 156

Before a shared number, a shared standard

Then

Metre Convention, 1875: seventeen states fund a bureau that ties measurements to prototypes

Now

Carbon border adjustment, 2026: comparable emissions figures need a common measurement rule (Section 4.6 (p. 153))


What happened

On 20 May 1875 seventeen states signed at Paris the Metre Convention D [Bureau International des Poids et Mesures 1875; the count of signatories from BIPM's page on the Convention]. Article 1 created a permanent scientific body, the International Bureau of Weights and Measures, at Paris, at the common expense of the parties; Article 3 placed it under the exclusive direction and supervision of an International Committee, itself answerable to a General Conference of delegates of all the contracting governments. Article 6 fixed the Bureau's work: comparisons and verifications of the new prototypes of the metre and the kilogram; custody of the international prototypes; periodic comparisons of national standards with them; comparisons of geodetic, precision and non-metric standards. Article 9 financed it by contributions on a scale based on population. The edition dates each article: Articles 7 and 8 carry the date of the amending convention of 1921, the rest 1875 D [Bureau International des Poids et Mesures 1875, Articles 1, 3, 6, 9].

The model

Field 3 R. In Model S1 and in the worked case below the experiment is a kernel — the probability of a pass in each state — that the receiver knows. The treaty is the institution that makes a kernel common knowledge for physical quantities: a measurement made in one country means the same as one made in another because both are tied, by published comparisons, to one prototype. The two situations above have the same structure: a field called “emissions” is a signal whose kernel is set by a boundary, a method and a verifier, not by the database that stores it.

What could be known

In 1875 a national standard was compared by bringing it to the prototype: comparability moved at the speed of a physical object and was renewed at the intervals Article 6 calls periodic. A product passport moves a figure in a fraction of a second to anyone with the right to read it. What transmission does not do is what Article 6 did: establish, by procedure, that two figures measure the same thing. The remedy of 1875 was institutional (a bureau, a committee, a conference, a budget rule), not a faster channel, and the accreditation framework and registry above are institutions of the same kind. Computation widens what can be combined and checked; it also lengthens the chain between the reader of a dashboard and the measurement, and the signal is specified only as well as that chain.

Where the parallel stops

A metrology convention is not a strategic-disclosure problem: no party in 1875 chose which measurements to reveal, and the Bureau's duties were comparison, not persuasion. Nor should the present definitions of units be projected back: the convention governs prototypes and their comparison, nothing more.

In the book: Chapter 4, page 156All sixteen parallaxes on the timeline →

1895–1899 · Chapter 4 · p. 157

A statement in the House

Then

Westminster, 1895: a minister tells the Commons how an expedition's advance would be viewed

Now

Model S6: a public statement is the move to which a cost of backing down can attach (Proposition 4.18 (p. 181))

1895–1899 · Chapter 4 · page 157

A statement in the House

Then

Westminster, 1895: a minister tells the Commons how an expedition's advance would be viewed

Now

Model S6: a public statement is the move to which a cost of backing down can attach (Proposition 4.18 (p. 181))


What happened

On 28 March 1895, in a debate in the House of Commons on England and France in Africa, raised on the vote for the Foreign Office, the Under-Secretary for Foreign Affairs, Sir Edward Grey, said that “the advance of a French Expedition under secret instructions right from the other side of Africa, into a territory over which our claims have been known for so long, would be not merely an inconsistent and unexpected act, but it must be perfectly well known to the French Government that it would be an unfriendly act, and would be so viewed by England” D [House of Commons 1895, cols. 405–406]. Three and a half years later a second Blue Book on Fashoda, reported by The Spectator of 29 October 1898 as published that week, included a despatch covering a letter from Major Marchand, who wrote of Fashoda, “which I entered on July 10th last”. On 9 February 1899 Mr Brodrick, answering a question put to the Under-Secretary for Foreign Affairs, told the Commons that discussions were taking place with a view to settling a line of delimitation between the French and Anglo-Egyptian spheres; on 24 March he said that the accounts which had appeared of the agreement between Great Britain and France as to the delimitation of territory in Africa were in the main correct, that he hoped to lay it on the Table in a few days, and that it stipulated for equal treatment of British and French trade between the Nile and Lake Chad D [House of Commons 1899, col. 331; House of Commons 1899, col. 345].

The model

Fields 3 and 4 of Model S6 R. A statement in the Commons is public and recorded, and the same audience can later compare it with what is done; a Blue Book published while a dispute is open puts correspondence on the same footing. In the model these are the moves to which a cost of backing down can attach; a message to which no cost attaches is free talk. The model needs more than the channel: a cost large enough, and a sender uncertain whether it will be resisted.

What could be known

The documents of 1895–99 record the channel and the dates, and nothing about : whether any audience would have penalised a retreat, and by how much. A statement made today is recorded and searchable at once and for good, which makes a retreat cheaper to observe. In the model, observation is a precondition of the cost, not the cost: the cost is what an audience does with what it observes, the quantity that survey experiments measure (Tomz 2007).

Where the parallel stops

The box reads documents for their channel. It does not analyse the crisis of 1898, say why any statement was made or published, or claim that the statement of 1895 shaped what followed; no number of the model comes from the record. A historical study that counts the Fashoda crisis among a dozen great-power crises argues that audience costs were not a major factor in any of them (Trachtenberg 2012, pp. 3, 13–17, 32).

In the book: Chapter 4, page 157All sixteen parallaxes on the timeline →

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.

The life, in moments →

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.

The life, in moments →

Main works

  • The Strategy of Conflict (1960)
  • Arms and Influence (1966)
  • Micromotives and Macrobehavior (1978)