I want to determine all pure-strategy Perfect Bayesian Equilibria for this task, but I cannot get very far. Side note: First number is payoff for A, second number payoff for player B.
For the last stage, I know that upper A will always choose to exit, and the bottom A will always choose to pass.
But I don't know how to proceed from there? I have ascribed beliefs for player B being in the upper and bottom node (beta and 1-beta, respectively), but am not sure what to do with it.
I tried assuming that Nature goes 'up' and in that case beta=1. Then B would choose pass because it has a greater payoff than exiting. That's where I get confused.