I've come up with a rather complex game, illustrated below, and need some help to get started on solving for the (weak) perfect Bayesian equilibria.
I've written the beliefs of player 1 as variables next to nodes. $P_1$ chooses $A$ or $B$ then $P_2$ must simultaneously pick $a$ or $b$ ($P_2$ acts simultaneously if $P_1$ chooses $A$ or $B$, similarly for $A'$ or $B'$).
A first step I have seen in the literature for computing PBE is to write out all the choices of both players and nature, then enumerate through all the separating, pooling, and semi-separating cases.
- $P_1$ has three information sets
- Info set 1: $P_1$ can choose between $A$, $B$, and $C$
- Info set 2: $P_1$ can choose between $A'$ and $B'$
- Info set 3: $P_1$ can choose between $A'$ and $B'$
- $P_2$ has four information sets
- Info set 1: $P_2$ can choose between $a$ and $b$
- Info set 2: $P_2$ can choose between $a$ and $b$
- Info set 3: $P_2$ can choose between $a'$ and $b'$
- Info set 4: $P_2$ can choose between $a'$ and $b'$
- Nature makes two choices:
- the original state of nature
- the outcome of player 1's move $C$
Are the above cases correct? Do I need to check separating, pooling, semi-pooling for each combination? Is there a more efficient way to do this?
Note: There are two moves by Nature (N). $P_1$ doesn't see the outcome of nature's first move, $P_2$ doesn't see the outcome of nature's second move. As far as I understand, one can translate this game into one where nature settles all of the randomness at the root node, but I think this makes the tree (possibly much) bigger. I'm hoping it can be analyzed in its current form.
Issue: The third and fourth information states for player 2 don't seem to be correct as written. Player $2$ should have a belief over the outcome of nature's second move, but I'm not sure how to denote that.
Any guidance would be greatly appreciated!