Cheap Talk Model: Crawford‑Sobel Equilibria and Bias Limits

Cheap talk describes communication that costs nothing and does not directly affect payoffs. Unlike signaling games, where costly actions such as education or pricing convey information, cheap‑talk messages are purely verbal. Credible information can still be transmitted even when the sender’s and receiver’s interests diverge, provided the bias between them is not extreme.

The Crawford‑Sobel Model

Crawford and Sobel (1982) formalize cheap talk with a sender (advisor) who observes the true state (t) in the interval ([0,1]). The receiver (decision‑maker) observes only the sender’s message (m) and then chooses a decision (y). Both players share a quadratic loss structure. The receiver’s loss is ((y-t)^2), so the optimal decision matches the state. The sender’s ideal decision is shifted by a bias (\beta>0); their loss is ((y-(t+\beta))^2). Both agree on the direction in which the state influences the ideal decision.

Equilibrium Types

Babbling Equilibrium

In the babbling equilibrium the sender always sends the same message, regardless of the state. The receiver ignores the message and selects the prior mean, (y=1/2). No information is transmitted.

Perfect Communication

A fully revealing equilibrium cannot exist because the sender always prefers to report a higher state to move the receiver’s decision toward (t+\beta). Any attempt at perfect communication gives the sender an incentive to misreport, breaking incentive compatibility.

Partitional Equilibrium

The sender can sustain communication by partitioning the state space into intervals and reporting only which interval the state falls into. The receiver updates beliefs to a uniform distribution over the reported interval and chooses its midpoint as the optimal decision.

Partitional Dynamics

In a two‑cell equilibrium the sender reports whether the state lies below or above a threshold (t_1). The receiver’s optimal decision for the lower interval is its midpoint, and similarly for the upper interval. The sender is indifferent at the threshold, which determines (t_1). Solving the indifference condition yields

[ t_1 = \tfrac12 - 2\beta . ]

The equilibrium exists only if (\beta < \tfrac14); otherwise the threshold would fall outside ([0,1]) and the partition collapses to babbling.

Generalizing to a (k)-cell partition produces (k-1) thresholds. The existence condition becomes

[ \beta < \frac{1}{2k(k-1)} . ]

As the number of cells increases, the intervals become narrower and the communication more precise.

Implications of Bias

Bias and informativeness move in opposite directions. A larger bias forces the sender to rely on coarser partitions, reducing the amount of information the receiver can extract. When (\beta) exceeds (\tfrac14), only the babbling equilibrium survives, and the receiver always chooses the prior mean.

When bias is small enough to allow multiple cells, both parties benefit. The receiver obtains a decision closer to the true state, while the sender still secures a decision shifted by (\beta) but gains from the higher precision of the receiver’s action. The highest feasible number of cells (k) maximizes welfare for both sender and receiver, subject to the bias constraint.