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An analyst who wants you to invest tells you the outlook is bright. The words cost nothing, and you know the analyst is biased. Should you ignore the message entirely? Not quite. Even a slanted source can tell you something, up to a point.

Cheap talk

Signaling worked because the signal was costly. Here messages are free: an expert can say anything. You might think free words carry no information, but they can, partially. How much gets through depends on how far the expert’s interests diverge from yours.

From earlier lessons

Two quick ones before we start.

A signal is credible in Spence's model because it is...

A biased expert and a decision-maker who disagree about the ideal action will find communication...

Predict, then play

An expert privately knows a state between 0 and 1 and sends a message; you then pick an action. You want to match the state; the expert wants you to pick a bit higher, by a bias $b$ . Move the bias and watch how finely the expert can credibly speak.

Why did that happen?

Take a guess, then reveal the reason

The expert cannot lie freely and still be believed: you know the bias and discount accordingly. But if the expert only has to place the state in a broad bracket, it has no reason to misrepresent which bracket, because within a bracket its preferred action and yours are close enough. So credible communication is coarse: the expert reveals a region, not a point. The wider the bias, the wider the brackets must be to keep the expert honest, until only one bracket is left and the message means nothing.

The math

The state $\theta$ is uniform on $[0,1]$ . You want to choose an action equal to $\theta$ ; the expert wants $\theta + b$ . Messages are costless, so the only equilibria are partition equilibria: the expert reveals which interval of a partition contains $\theta$ , and you take the interval’s midpoint as your action.

Definition, cheap talk (Crawford-Sobel)

With costless messages and a conflict of interest $b$ , informative equilibria partition the state space into finitely many intervals; the expert reports the interval, and the receiver best-responds with its mean. The maximum number of intervals falls as $b$ rises, and for $b \ge 1/4$ (in the uniform-quadratic case) only the one-interval babbling equilibrium survives.

Consider the two-interval equilibrium with boundary type $a_1$ . The intervals $[0,a_1]$ and $[a_1,1]$ induce actions $y_1 = a_1/2$ and $y_2 = (a_1+1)/2$ . The boundary type must be indifferent between the two induced actions, given its own target $a_1 + b$ :

2(a_1 + b) = y_1 + y_2 = a_1 + \tfrac12 \quad\Rightarrow\quad a_1 = \tfrac12 - 2b.

At $b = \tfrac18$ this gives $a_1 = \tfrac14$ : the expert can credibly say only whether the state is below or above $\tfrac14$ . A two-interval equilibrium needs $a_1 > 0$ , so it exists only when $b < \tfrac14$ .

Work it out

The most informative equilibrium at b = 1/8

Try two intervals split at $a_1$ . The low interval $[0, a_1]$ induces the action $a_1/2$ ; the high interval $[a_1, 1]$ induces $(a_1 + 1)/2$ . For the split to hold, the type exactly at $a_1$ must not prefer to be taken for the other interval. Its ideal action is $a_1 + b$ , so indifference requires it to sit halfway between the two induced actions: $2(a_1 + b) = a_1/2 + (a_1+1)/2$ .

Solving, $a_1 = \tfrac12 - 2b$ . At $b = \tfrac18$ , $a_1 = \tfrac14$ , and the receiver’s two actions are $0.125$ and $0.625$ . Push the bias to $\tfrac14$ and $a_1$ hits $0$ : the low interval vanishes and only babbling remains. Costless words inform, but a larger bias buys coarser talk, not the same message shifted.

Key idea

Costless messages can still transmit information, but only coarsely, and the amount shrinks as the speaker’s bias grows. Past a threshold, a biased expert can convey nothing credible at all.

Check yourself

Check yourselfapply

With bias b = 1/8, what can the expert credibly communicate?

Check yourselfanalyse

Someone claims a bigger bias just shifts the expert's message upward without losing information. Why is that wrong?

Check yourselfbrings back: signaling

How does cheap talk differ from Spence-style signaling?

Sources

Crawford, V. P., and Sobel, J. (1982). Strategic information transmission. Econometrica, 50(6), 1431-1451.