Matching Chairs II

Uncertainity about the numbers of available chairs

In this variant of the two-move version, Ann and Beth both know that the shop owner either has 2 L-chairs and 4 O-chairs, or he has 3 L-chairs and 4-O-chairs, or he has 3 L-chairs and 2 O-chairs. Both also know that each of these three cases is equally likely. How would this information, or rather disinformation, influence the way they play?

Mention Harsanyi's trick of transforming incomplete information into imperfect information here.

The extensive form as a DAG is shown below:

Modify this extensive form into one where the random moves are done at the very end, or at least as late as possible.

Both players have different knowledge

We have seen above that if both players are unsure about the same things, the game's character is closer to that of sequential games with perfect information that of imperfect information games. This changes however if both players have different, let's say complementing, information.

Again Ann and Beth know that the shop owner either has 2 and 4, or 3 and 4, or 3 and 2 L-chairs and 4 O-chairs, and that all three cases are equally likely. When Ann arrives in the morning, the shop owner tells her at least the number of L-chairs. When Beth arrives and Ann brags, Beth complains and the shop owener has to tell her something as well, but he tells her the number of O-chairs. How do both play?

Here is the extensive form:

Different to the previous case, the moves of the random player Nature can not be put towards the end of the game. The game also has no subgames, therefore the only way to attack this game is to translate into normal form and solve.