Analogous interpretations between SPNE & NE vs. PBE & BNE

Question

In a (finite horizon) sequential game of complete information, any subgame perfect Nash equilibrium (SPNE) necessarily must specify the equilibrium of the last stage game as a Nash equilibrium.

I am wondering if an analogous result holds for sequential games of incomplete information, that is, in a perfect Bayesian equilibrium (PBE) does the last stage game form a Bayesian Nash equilibrium (BNE)?

Answer 1

In PBE, the information sets are not subgames, and a subgame begins at a singleton information set, where the current state is common knowledge. For example, the standard signaling game with two types, two messages for the sender, and two actions for the receiver has only one subgame, the whole game. So the "last subgame" you mention in your question is not necessarily the same thing as the terminal decision nodes or terminal simultaneous move subgame in a complete information dynamic game.

But if you solve for a PBE and then wipe away the beliefs, the players are still mutually best-responding, so it constitutes a Bayesian Nash Eqm of the game. The problem is that, like with complete information, there might be many patterns of best responses which cannot make sense when compared across subgames. But when you solve for a PBE, usually you sketch out a set of mutual best responses and then check what the beliefs are that support those choices (or show there aren't any such beliefs). That's already a BNE for the game if there are no profitable deviations, it just might not survive the stronger criterion of Bayesian beliefs, because there is dynamically sub-optimal behavior somewhere, just as in the comparison with NE and SPNE.

Answer 2

In a perfect Bayesian equilibrium (PBE) does the last stage game form a Bayesian Nash equilibrium (BNE)?

The short answer is yes, but it can be misleading.

First, any proper subgame of an incomplete information game will have complete information (among the players that are still making choices in the subgame). This is because the definition of a subgame does not allow you to break information sets. Therefore, in any proper subgame, BNE and NE are the same thing.

Second, you cannot simply define a PBE as a strategy profile that induces a BNE in every subgame. The reason is that players make choices at each information set and many information sets will not start a new subgame (as Renar pointed out in their answer). PBE must induce rational choices at all information sets, even those that do not start a new subgame.