Question

Consider an extensive-form game in which player 1 is one of two types: A and B. Suppose that types A and B have exactly the same preferences; the difference between these types has something to do with the payoff of another player. Is it possible for such a game to have a separating PBE, where A and B behave differently?

Answer 1

No there will only be one PBE in this case. The case can be better potrayed by the gift game example on the Wikipedia page. Here the payoff for the player depends on the payoff of the player 2 only. If 2 accepts the payoff of 2 is 1, which makes the payoff of A type 1 and B type 0. Similarly if payoff of 2 is 0, the payoff of A is also 0 and that of B is 1. This is a hypothetical payoff matrix mentioned above but the gist of the argument is the same. The PBE will be the same as the gift example of the wiki page.