What is the difference between a Rational Expectations Equilibrium (REE) with asymmetric information and Bayesian Nash Equilibrium (BNE)? Since agents in both cases play some game or have a strategic behavior what is the difference? I would like an extended answer with some examples to understand it because I have not found any specific details until now (maybe I am searching in the wrong direction). Thank you in advance!
The idea of a rational expectations equilibrium is more general than BNE. It simply means that the belief system of agents is consistent with the model and incorporates all available information. This abstract idea can be applied for games, markets or other types of interactions. BNE is a solution concept for non-cooperative games. The expectations being rational also poses no restriction on how agents in a model update their beliefs, while BNE assumes beliefs are updated according to Bayes rule.
Let me be more concrete. If you are modeling a competitive market, the REE is not a Nash equilibrium. Players are simply maximizing giving prices and information (instead of given the strategies of other players). In contrast, if you are modeling a market where the buyers or sellers are strategic (for example a Cournot model with incomplete information), the typical solution concept is BNE and it will also be an REE.
To sum up, any BNE is an REE, but the converse is not true.