Pooling Equilibrium¶

Definition¶

In a signalling game, a pooling equilibrium is a perfect Bayesian equilibrium in which every type of the sender chooses the same message $m^*$ . On the path the receiver's belief after $m^*$ equals the common prior,

\mu(\theta\mid m^*)=p(\theta)\qquad\text{for every type } \theta,

and the receiver best responds to the prior. Every other message $m\ne m^*$ is off the equilibrium path, and the beliefs $\mu(\cdot\mid m)$ there are part of the equilibrium but not restricted by Bayes' rule.

Interpretation¶

In a pooling equilibrium the message carries no information: the receiver learns nothing it did not already know. This can happen even when types differ a lot, as long as no type gains from breaking away.

Payoffs against the off-path belief mu' after m = 1: a flat green line at the pooling payoff 0, a blue step from -1 to 1 at mu' = 1/2 for H, and a dashed step from -3 to -1 for L.

The figure checks the "pooling on the free message" example for every possible off-path belief $\mu'=\mu(1)$ $μ (1)$ . Staying at $m=0$ is rejected and pays both types $0$ (green). Deviating to $m=1$ pays $0$ if the receiver rejects, which she does when $\mu'<\tfrac{1}{2}$ $\frac{1}{2}$ , and $2$ if she accepts, which she does when $\mu'>\tfrac{1}{2}$ $\frac{1}{2}$ , minus the cost: $-1$ or $1$ for $H$ , and $-3$ or $-1$ for $L$ . The $L$ payoff never reaches $0$ , and the $H$ payoff stays below it exactly when the receiver rejects $m=1$ $1$ , so the shaded beliefs $\mu'\le\tfrac{1}{2}$ support the pooling equilibrium, while beliefs above $\tfrac{1}{2}$ make $H$ break away.

Move the off-path belief $\mu'$ : pooling on $m=0$ survives while $\mu'\le\tfrac12$ $\frac{1}{2}$ , so that the receiver rejects $m=1$ $1$ . Raise $c_H$ above $2$ and every belief supports pooling.

Whether a type gains from deviating depends on how the receiver would interpret the deviation, that is, on the off-path part of the system of beliefs. Pooling equilibria are typically supported by pessimistic off-path beliefs: any deviation is read as coming from the "bad" type, which makes deviation unattractive. Since PBE allows any off-path belief, many pooling equilibria can exist; forward induction, mentioned only briefly in the course, asks whether such beliefs are plausible.

A frequent mistake is to check only the on-path condition. A complete verification has three parts: on-path beliefs equal the prior; the receiver's actions are optimal after the pooled message and after each unused message given the stated beliefs; and no type prefers an unused message.

Key Results¶

On the path the posterior equals the prior, because all types send $m^*$ with probability one.
Fix a receiver action after $m^*$ that is a best response to the prior. Pooling on $m^*$ with that action is a PBE if and only if some off-path beliefs, with optimal receiver responses to them, make every type weakly prefer $m^*$ to every other message.
When the receiver's response is monotone in the belief, as in the example below, more pessimistic off-path beliefs make pooling easier to sustain; with sufficiently optimistic ones it can break down.
Pooling and separating equilibria can coexist and give different payoffs: in the example below type $H$ gets $0$ when pooling and $2-1=1$ in the separating equilibrium.

Examples¶

Pooling on the free message. In the signalling game example, type $H$ (prior $p=0.4$ $0.4$ ) and type $L$ send $m=0$ for free or $m=1$ at cost $c_H=1$ or $c_L=3$ $3$ . The receiver accepts ( $A$ ) or rejects ( $R$ ); accepting gives her $1$ against $H$ and $-1$ against $L$ , rejecting gives $0$ , so with belief $\mu(m)=\mathbb{P}(H\mid m)$ she accepts if $\mu(m)>\tfrac{1}{2}$ and rejects if $\mu(m)<\tfrac{1}{2}$ $\frac{1}{2}$ . The sender gets $2$ if accepted and $0$ if rejected, minus her cost.

On path: $\mu(0)=0.4<\tfrac{1}{2}$ $\frac{1}{2}$ , so the receiver rejects after $m=0$ and both types get $0$ .
Off path: let $\mu(1)=\mu'$ $μ^{'}$ . If $\mu'<\tfrac{1}{2}$ the receiver rejects after $m=1$ $1$ , and deviating gives $H$ only $-1<0$ and $L$ only $-3<0$ $0$ . If $\mu'>\tfrac{1}{2}$ she accepts, and $H$ would get $2-1=1>0$ $0$ , so $H$ deviates.

So pooling on $m=0$ $0$ , with rejection after both messages, is a PBE for every off-path belief $\mu'\in[0,\tfrac{1}{2}]$ (at $\mu'=\tfrac{1}{2}$ the receiver is indifferent and rejecting is a best response), but not for $\mu'\in(\tfrac{1}{2},1]$ $(\frac{1}{2}, 1]$ .

Pooling on the costly message. If both types chose $m=1$ $1$ , then $\mu(1)=0.4$ $0.4$ , the receiver rejects, and $L$ would get $-3$ , while $m=0$ gives at least $0$ whatever $\mu(0)$ is. Hence pooling on $m=1$ is not a PBE.

Prerequisites¶

Sources¶

Game Theory · Week 7

Aliases: pooling PBE Keywords: all types send the same message, posterior equals prior, off-path beliefs, pessimistic beliefs, uninformative signal Symbols: element of, mu, theta

Last updated: 14 October 2026 08:21