§3.2 — the credible updating rule forces the prior, $1$ or $0$, and puts no restriction when both types strictly prefer to stay

Consider the two-type signaling game with prior $0<\delta<1$ and an arbitrary payoff function $\pi(t,s,d)$. Fix a pure profile with beliefs, a signal $s$ off its equilibrium path, the equilibrium payoffs $u^*_t$, and a candidate posterior $\gamma$ after $s$.

1. If $\gamma$ is credible (Definition 2) and both types strictly prefer to deviate to $s$ under $\gamma$, then $\gamma=\delta$.
2. If $\gamma$ is credible, the high-quality type weakly prefers to deviate ($\pi(H,s,\gamma)\ge u^*_H$) and the low-quality type strictly prefers to stay ($\pi(L,s,\gamma)<u^*_L$), then $\gamma=1$.
3. If $\gamma$ is credible, the low-quality type weakly prefers to deviate and the high-quality type strictly prefers to stay, then $\gamma=0$.
4. If both types strictly prefer to stay under $\gamma$ (so $\mathbb T'\cup\mathbb T''=\emptyset$), then $\gamma$ is not credible: the credible updating rule puts no restriction on the off-path belief.

This is the form in which the credible updating rule is applied in the equilibrium analysis: it pins the off-path posterior to the prior, to $1$ or to $0$.

Differences from the source:

• generalization: "in our queueing setting" (payoffs λ^O, λ^U, the same for both types) → any payoff function π(t, s, d); reason: the argument uses only Definition 2; the queueing payoffs are an instance
• reading: Definition 2(b): T′ and T″ "remain unchanged under the above posterior belief" → T′ and T″ computed at the posterior γ itself; reason: the paper calls the procedure "essentially a fixed-point argument" (p.547)
• encoding: "the credible updating rule puts no restrictions on the off-equilibrium-path belief" when T′ ∪ T″ is empty → a γ under which both types strictly prefer to stay is not credible; reason: a perfect sequential equilibrium is tested only against credible beliefs, so such γ impose nothing