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A prosecutor wants convictions; the judge wants accurate decisions. Only 30% of defendants are guilty. By committing in advance to a noisy evidence rule, the prosecutor can induce conviction after 60% of signals. Some of those signals are false positives, and the judge accounts for that.

Information design and Bayesian persuasion

Cheap talk failed because the speaker could choose a message after learning the state. Flip that: let the sender commit to an information policy in advance, before the state is known. By choosing a possibly noisy mapping from states to signals, a sender can shift a rational receiver’s beliefs without later changing the rule.

From earlier lessons

Two quick ones before we start.

In cheap talk, a biased expert can convey only...

A rational receiver updates beliefs from a signal using...

Predict, then play

Only 30% of defendants are guilty, and the judge convicts only when the posterior probability of guilt reaches 50%. The prosecutor commits to a signal: it always sends favourable-to-guilt evidence when the defendant is guilty, and sends the same signal for some fraction of innocent defendants too. Tune that false-positive rate.

Why did that happen?

Take a guess, then reveal the reason

The trick is committing before the fact. The signal is informative but noisy: it raises the probability of guilt from 30% to 50%, even though some innocent defendants receive it. Increase the false-positive rate too far and the judge stops convicting; reduce it and the prosecutor wastes persuasive capacity. The optimum sits at the judge’s threshold.

The math

The prior is $\Pr(\text{guilty}) = 0.3$ , and the judge convicts iff the posterior is at least $\tfrac12$ . The signal reports “guilty” always when guilty, and with probability $x$ when innocent. By Bayes, a “guilty” report yields posterior

\Pr(\text{guilty}\mid\text{"guilty"}) = \frac{0.3}{0.3 + 0.7\,x}.

Set this to the threshold $\tfrac12$ : $0.3 = 0.5(0.3 + 0.7x)$ gives $x = \tfrac{3}{7}$ . Then the chance of a “guilty” signal is $0.3 + 0.7\cdot\tfrac37 = 0.6$ , and every such signal induces conviction. The label is shorthand for evidence favourable to guilt, not a literal report that the state is guilty.

Definition, Bayesian persuasion

The sender commits to a signal structure (a mapping from states to message probabilities) before the state is realized, and the receiver best-responds to the posterior each message induces. Any signal structure is feasible as long as the message-posteriors average back to the prior (Bayes plausibility). The sender’s problem is to pick the best such distribution of posteriors, which is read off the concave closure of the sender’s payoff. Commitment, absent in cheap talk, is what gives the messages credibility.

The two extremes lose. Full disclosure convicts only the truly guilty, $30\%$ . Secrecy leaves the posterior at the prior $0.3 < 0.5$ , convicting no one. A committed noisy signal beats both for the prosecutor.

Work it out

Push the posterior to the threshold

Let the signal flag an innocent as “guilty” with probability $x$ . A “guilty” report then has posterior $0.3 / (0.3 + 0.7x)$ . For the judge to convict, this must reach $\tfrac12$ , and to convict as often as possible the prosecutor wants $x$ as large as it can be while still hitting $\tfrac12$ exactly.

Solve $0.3/(0.3 + 0.7x) = 0.5$ : multiply out to $0.3 = 0.15 + 0.35x$ , so $x = 3/7 \approx 0.43$ . The probability of a “guilty” report is $0.3 + 0.7 \cdot \tfrac37 = 0.6$ . Every such report convicts, so the conviction rate is $60\%$ . Raising $x$ past $3/7$ drops the posterior below $\tfrac12$ and the judge stops convicting; lowering it wastes convincing power. The optimum sits exactly at the threshold.

Key idea

Commit to an honest information policy in advance and you can move a rational receiver far beyond the prior. The optimal signal is just convincing enough, pushing beliefs exactly to the receiver’s threshold and no further.

Check yourself

Check yourselfapply

The optimal signal pushes the posterior after a 'guilty' report to exactly 0.5, not higher. Why not aim higher to be safe?

Check yourselfanalyse

Does Bayesian persuasion require the sender to lie?

Check yourselfbrings back: cheap talk

Why can persuasion move beliefs so much more than cheap talk?

Sources

Kamenica, E., and Gentzkow, M. (2011). Bayesian persuasion. American Economic Review, 101(6), 2590-2615.