This problem has 3 parts:
Compute all pure strategy Nash equilibria.
Is this NE (a part of) weak perfect Bayesian equilibria? If yes, what are the beliefs, if not, which strategy or strategies are not sequentially rational?
Compute all (pure and mixed) weak perfect Bayesian equilibria.
(see the picture for a game tree).
Now, this is a way I would have normally approach a problem like this:
i. Since there are no sub-games (except for the whole game itself), I would redraw the whole tree in normal form.
ii. find Nash equilibria by the iterative elimination of strictly dominated strategies and by underlying the best responses.
iii. find mixed strategies based on previous steps and by assuming that players mix to make each other indifferent.
iv. then I would check for sequential rationality by backward induction. Afterwards, I would get stuck.
However, in this case, I struggle even with the application of basic NE and redrawing of the tree because I have never seen information nodes crisscrossing each other like this.
