Synthesis · The Territory

One Question, Six Institutions

Every paper here asks a single question from a different angle: given that a sender knows more than a receiver and wants something, what can the sender make the receiver believe? The answers turn out to be geometry — orders on evidence, partitions of a state space, and envelopes of a value function over beliefs. What separates the papers is the institution: whether the sender can commit, whether claims can be verified, how aligned preferences are, and how credible the channel is.

01 The one-line map

02 The spectrum of credibility

The cleanest way to hold all six in mind is a single axis: how much can the sender commit? At the left, messages are cheap and unverifiable; at the right, the sender designs and is bound by an information structure. The 2020 and 2022 papers fill in the middle and the off-axis case of verifiable evidence.

03 The conceptual chain

The arc bends toward one realization: the binding constraint on persuasion is rarely information. It is credibility — and credibility is an institution, not a fact.

1981 · Paul Milgrom

Good News and Bad News

Paul Milgrom — Representation Theorems and Applications, Bell Journal of Economics

Before we can ask what a sender will say, we must say what it means for evidence to be favorable. Milgrom makes "good news" a property of likelihoods, not payoffs.

Q What it proves

When can we say signal $x$ is unambiguously better news than signal $y$ about an ordered state $\theta$? Milgrom's answer is Bayesian and ordinal: $x$ is more favorable than $y$ exactly when the posterior after $x$ first-order stochastically dominates the posterior after $y$ — so that every decision-maker who prefers higher $\theta$ prefers the belief induced by $x$.

The representation theorem ties this posterior order to a condition on the likelihoods themselves. In the ordered-signal case, $x$ is more favorable than $y$ iff for $\bar\theta>\theta$,

This is the strict Monotone Likelihood Ratio Property (MLRP): high signals are relatively more likely under high states than under low ones.

Fig The single-crossing of likelihoods

The figure below makes MLRP concrete with two state-conditional densities. The left panel shows $f(x\mid\theta_L)$ and $f(x\mid\theta_H)$; the middle shows their likelihood ratio $f(x\mid\theta_H)/f(x\mid\theta_L)$ climbing monotonically; the right shows the consequence — the posterior $\Pr(\theta_H\mid x)$ rises smoothly with $x$. Higher signals push belief upward, for any payoff function.

1982 · Crawford & Sobel

Strategic Information Transmission

Vincent Crawford & Joel Sobel — Econometrica

Take away commitment and verification, leave a conflict of interest, and language doesn't break — it coarsens. The sender can only be trusted at the level of intervals.

Q What it proves

A better-informed sender sends a costless, non-binding, unverifiable message; the receiver then acts. With misaligned ideal actions, can anything be said credibly? The central theorem: every equilibrium is essentially a partition equilibrium. The state space splits into intervals, the message reveals only which interval holds the state, and there is a maximum number of intervals $N(b)$ set by the preference conflict $b$.

The uniform–quadratic kernel

With receiver payoff $-(y-m)^2$ and sender payoff $-(y-(m+b))^2$, the sender's ideal action sits a fixed bias $b$ above the receiver's. Each sender type wants to nudge the action up, so full revelation unravels: boundary types must be indifferent between adjacent interval messages, giving the cutoff recurrence

The largest feasible number of intervals is the biggest integer $N$ with $\,2N(N-1)\,b<1$. As $b\to 0$ communication becomes arbitrarily fine; once $b>\tfrac14$, only babbling survives.

Fig Conflict buys coarseness

The bar is the state space $[0,1]$. The figure solves the recurrence exactly for the maximal equilibrium at the chosen bias: each colored segment is an interval the receiver can distinguish, ticked at the receiver's chosen action (the interval's posterior mean) and at the sender's preferred point a distance $b$ to its right. Watch the partition collapse as conflict rises.

2011 · Kamenica & Gentzkow

Bayesian Persuasion

Emir Kamenica & Matthew Gentzkow — American Economic Review

Grant the sender commitment and strategic communication becomes information design. The whole problem collapses to one picture: the concave envelope of the sender's value over beliefs.

Q What it proves

The sender commits ex ante to a signal; the receiver sees a realization, updates by Bayes, and best-responds. The sender never picks the action — only a distribution over posterior beliefs. The single feasibility constraint is Bayes plausibility: posteriors must average back to the prior.

So the sender solves $\;\max_{\tau:\,\mathbb E_\tau[\mu]=\mu_0}\mathbb E_\tau[\hat v(\mu)]\;$ and the value is the concave envelope of $\hat v$ at the prior:

The sender cannot move the average belief — Bayes pins it. The only freedom is dispersion. Persuasion has value precisely when the sender benefits from spreading beliefs apart.

Fig The prosecutor and the judge

State $\theta\in\{\text{innocent},\text{guilty}\}$, belief $\mu=\Pr(\text{guilty})$. The judge convicts iff $\mu\ge q$. The prosecutor wants conviction, so $\hat v$ is a step: $0$ below the threshold, $1$ at or above it. A step is not concave — and that gap is the entire opportunity. The concave envelope is the line $\min(\mu/q,\,1)$; at a prior below $q$ the optimal signal splits belief into a "give-up" posterior at $0$ and a "just-convict" posterior at exactly $q$.

2020 · Lipnowski & Ravid

Cheap Talk With Transparent Motives

Elliot Lipnowski & Doron Ravid — Econometrica

Return to cheap talk, but make the sender care only about the action, not the state. One assumption swaps concavification for quasiconcavification — and names the price of having no commitment.

Q What it proves

The sender has transparent motives: utility depends on the receiver's action, not directly on $\theta$. With no commitment, every on-path message must be one the sender is willing to send after every state — and because payoffs are state-independent, that forces a flatness condition: all induced posteriors must give the sender the same payoff.

The first line is Bayes plausibility; the second is sender indifference. A payoff $s$ is securable when the prior lies in the convex hull of beliefs delivering at least $s$, and the best cheap-talk value is the quasiconcave envelope:

Fig Two envelopes, one value function

The figure plots a non-monotone sender value $v(\mu)$ together with both envelopes, computed exactly: the concave envelope (commitment) by upper convex hull, and the quasiconcave envelope (cheap talk) as $\min$ of the running maxima from the left and right. The shaded band between them is the value of commitment — what the sender forfeits by being unable to bind himself.

2022 · Fréchette · Lizzeri · Perego

Rules and Commitment in Communication

Guillaume Fréchette, Alessandro Lizzeri & Jacopo Perego — Econometrica · an experiment

A single framework nests cheap talk, disclosure, and Bayesian persuasion — then puts real subjects in it. The theory's sharpest prediction: more commitment helps and hurts, depending on whether claims can be verified.

Q What it tests

A sender first picks a committed rule $\pi_C$; after seeing the state she may revise to $\pi_R$. The message comes from the committed rule with probability $\rho$ and the revision with probability $1-\rho$ — so $\rho$ is the degree of commitment. Two regimes govern what claims are allowed:

At $\rho=0$ the unverifiable regime is cheap talk and the verifiable regime is disclosure; at $\rho=1$ both become Bayesian persuasion. Treatments use $\mu_0=\tfrac13$, threshold $q=\tfrac12$, and $\rho\in\{.2,.8,1\}$.

Fig Opposite slopes, one corner

The striking prediction: informativeness moves in opposite directions in $\rho$. Under unverifiable messages, more commitment makes the announced rule credible, so information rises. Under verifiable messages, no commitment already forces unraveling toward disclosure; more commitment lets the sender strategically pool and hide, so information falls. The two curves meet only at full commitment.

The four corners of the framework

2022 · Lipnowski · Ravid · Shishkin

Persuasion via Weak Institutions

Elliot Lipnowski, Doron Ravid & Denis Shishkin — Journal of Political Economy

Real institutions are neither perfectly credible nor pure cheap talk. Put a dial between the two and the value of persuasion doesn't slide smoothly — it can fall off a cliff.

Q What it proves

The sender commissions a study that follows its committed protocol with probability $\chi$, but with probability $1-\chi$ the sender can influence the report after seeing the state. The receiver sees the message, not its origin. The endpoints are familiar:

The construction splits messages into official-only (trusted) and influence-compatible (fabricable) classes. The official part runs concavification logic, but capped by what the sender could secure through cheap talk — else he deviates:

subject to $k\beta+(1-k)\gamma=\mu_0$ and the credibility constraint $(1-k)\gamma(\theta)\ge(1-\chi)\mu_0(\theta)$ for all $\theta$ — enough mass on the influenced component to cover the institution's possible failure in every state.

Fig The central bank and the cliff

Their central-bank example makes the discontinuity exact. The sender's value as a function of credibility $\chi$ is

Slide $\chi$ across the figure. Above $\tfrac34$ the sender already has full-commitment value. Between $\tfrac23$ and $\tfrac34$, value rises linearly with credibility. But cross below $\tfrac23$ and the value jumps down discontinuously from $\tfrac43$ to $1$: a small loss of credibility triggers a large loss of persuasive power.