npbuilds Bayesian Persuasion — The Information Architect
The modern theory of strategic information design. Classical information economics asks "how do informed parties reveal or conceal what they know?" Bayesian persuasion flips this: "how should a sender design the information structure that a receiver will observe?" This reframing, formalized by Kamenica & Gentzkow (2011), has transformed how we think about media, regulation, recommendation systems, and platform design. Grounded primarily in Kamenica & Gentzkow (2011), Bergemann & Morris (2016), and Crawford & Sobel (1982).
The Bayesian Persuasion Framework
The Setup (Kamenica & Gentzkow 2011)
- A state of the world ω is drawn from a known prior distribution
- A sender designs a signal (experiment/test/information structure) before observing ω
- The signal generates a realization that the receiver observes
- The receiver updates beliefs (Bayes' rule) and takes an action that affects both players
- The sender commits to the signal structure before the state is realized
The key insight: The sender chooses a distribution over the receiver's posterior beliefs, subject to the constraint that posteriors must be a mean-preserving spread of the prior (Bayes plausibility). The optimal signal is found by concavification — taking the concave closure of the sender's value function over the receiver's belief space.
Concavification
The sender's problem reduces to geometry:
- For each possible posterior belief μ, compute the sender's expected payoff V(μ) given the receiver's optimal action at belief μ
- Take the concave closure (concavification) of V — the smallest concave function ≥ V
- The sender's optimal payoff equals the concavified value at the prior belief
- The optimal signal is any signal that induces posteriors on the concave closure, with the prior as their average
Geometric intuition: The sender can achieve any payoff that lies on or below the concave closure of V. The concavification represents the best the sender can do by "mixing" different information revelations.
Example: The Prosecutor
A prosecutor wants a judge to convict. The judge convicts iff P(guilty) ≥ 0.5. The prior is P(guilty) = 0.3.
Without persuasion: judge acquits (0.3 < 0.5). The prosecutor's payoff is 0.
Optimal signal: design an "investigation" that sometimes produces "evidence of guilt" and sometimes "evidence of innocence." The optimal investigation:
- When guilty → always produce "evidence of guilt"
- When innocent → produce "evidence of guilt" with probability 3/7, "evidence of innocence" with probability 4/7
Result: when "evidence of guilt" is produced, P(guilty | evidence) = exactly 0.5 → judge convicts. This happens with probability 0.3(1) + 0.7(3/7) = 0.6. The prosecutor gets conviction 60% of the time, up from 0%.
Cheap Talk — Crawford & Sobel (1982)
Strategic communication without commitment. The sender observes the state and sends a costless message. The receiver takes an action.
Key difference from Bayesian persuasion: In cheap talk, the sender talks after observing the state. There's no commitment — the sender can say whatever they want. Credibility comes only from the equilibrium structure.
The Crawford-Sobel Model
- State ω ~ Uniform[0,1]
- Sender prefers the receiver's action to be biased upward by b > 0 relative to what the receiver wants
- Equilibrium: the state space is partitioned into intervals. The sender reports which interval contains the state, but not the exact state.
Key results:
- Only partition equilibria exist — the sender reports "the state is in interval [aₖ, aₖ₊₁]"
- More intervals (more informative communication) when sender-receiver interests are more aligned (b is small)
- As b → 0, communication approaches full revelation
- As b → ∞, no information is transmitted (babbling equilibrium only)
- Multiple equilibria exist; the most informative equilibrium has the most intervals
Applications: Expert advice to decision-makers, political communication, analyst recommendations, lobbying.
Verifiable Disclosure
An intermediate model: the sender can choose whether to disclose verifiable information (but cannot lie).
Unraveling Theorem (Grossman 1981, Milgrom 1981)
If information is verifiable and all types can disclose:
- The best type discloses (to separate from the rest)
- The second-best type, now worst among non-disclosers, also discloses
- Cascade: Full unraveling → everyone discloses
When unraveling fails:
- Disclosure costs (small costs can sustain non-disclosure)
- Uncertainty about whether the sender has information
- Multi-dimensional information (disclosing one dimension obscures another)
- Naive receivers who don't penalize non-disclosure
Information Design with Multiple Receivers
Bergemann & Morris (2016, 2019) generalize persuasion to multiple agents who interact strategically.
The information designer chooses a Bayes correlated equilibrium (BCE) — an information structure plus a strategy profile where each agent best-responds to their signal given the correlation structure. The set of BCE outcomes characterizes everything achievable through information design.
Key insight: Information design in games is equivalent to choosing a Bayes correlated equilibrium. This connects persuasion to correlated equilibrium and unlocks results from the equilibrium theory.
Applications:
- Platform design: What should Uber show drivers about ride requests? What should Amazon show sellers about competitor prices?
- Financial regulation: How much should rating agencies reveal about asset quality?
- Media and polarization: How do media outlets' information design choices affect political polarization?
- Stress tests: Bank stress tests as Bayesian persuasion — regulators design the information revealed to markets
Modern Extensions
Dynamic Bayesian persuasion: Sender reveals information over time. Timing matters — early revelation forecloses future options. Applications to clinical trials, news cycles, investment.
Competition in persuasion: Multiple senders compete to influence a receiver. The receiver benefits from competition (more information). Results connect to media economics and political advertising.
Robust persuasion: Sender designs signals when the prior is uncertain or when the receiver may have private information. Connects to robust mechanism design.
Attention and persuasion: The receiver has limited attention. The sender must design information that both informs and attracts attention. Models of clickbait, sensationalism, and media strategy.
Sources
Read references/sources.md for the full bibliography — founding papers (Kamenica & Gentzkow, Crawford & Sobel, Bergemann & Morris), surveys, and modern extensions.
When This Applies
- Designing what information to show users (recommendation systems, search results, ratings)
- Analyzing media strategy, political communication, or advertising
- Evaluating disclosure policies for regulators (stress tests, mandatory reporting)
- Platform information design (what drivers/sellers/buyers see)
- Any setting where a party chooses what information others observe