We have this signaling game :
- sender type $t$ is uniformly distributed among $[0, 1]$. She takes action A if $t < \phi$, B if $t > \phi$
- if receiver takes action A' when seeing A and B' when seeing B, it is optimal for sender not to deviate. If the receiver react in any other way, it is optimal to deviate. A' is optimal for receiver if $t < \phi$ and B' if $t > \phi$
So this is an equilibrium.
What happens if $\phi = 1$ ?
If $\phi = 1$, and sender always send A, and receiver A', everything stays OK.
Sender will not send B if she is sure receiver respond B'.
But is responding B' a credible threat as $P(t > 1) = 0$ ?