I am a TA in a basic class on game theory in the economics department. I am ashamed to admit, but I do not entirely understand how all the different kinds of equilibria connected to each other. The class is taught in german, so I hope I am using the correct terms. I will now write down how I understand everything, and would kindly ask someone who knows more about the topic to correct me at appropriate points.
Side note: When I say "more general", I mean that it is applicable to a wider range of problem, while when I say "stricter", I mean that it is applicable to the same kinds of problems, but puts more restrictions in place (, and eliminates some equilibria).
The games:
There are games with perfect and complete information. There are also games with imperfect information, where one player is unsure about the decisions of other players at some point. There are also games with incomplete information, where some players are nor only in the dark about some other players action, but also about the types of other players.
Games can also be split in simultaneous and sequential games. The simultaneous games can be understood as a special simple case of the sequential games with imperfect information. (A special case of the simultaneous games are signalling games, but I don't think it is very relevant for my question.)
For simplicity sake, I limit myself to one-shot games, but I am pretty sure that all this works equally well for (in)finitely repeated games. But this is really just an educated guess, math is weird, so there might be a top of exceptions.
Now, how do we go about solving them:
The simplest option is the Nash-Equilibrium (NE) in pure strategies. "Each player picks an action." This one works really well for simultaneous games, but doesn't find all equilibria. This is mostly applied to simple simultaneous games, or simple sequential games.
An extension of the NE in pure strategies, is the NE in mixed strategies. "Each player picks a probability distribution over all his actions." (With probabilities of 1 and 0 we get the former case.) This method finds ALL equilibria, even this implausible ones. This is applied in pretty much the same cases as the former.
A stricter option, which only becomes relevant in sequential games with several subgames (if there is only one (sub)game this yields the same equilibria) is the subgame perfect equilibrium. It can be found through backwards induction. In comparison to the NE (in mixed strategies, because pure strategies are just a special case) it yields less or the same number of equilibria, eliminating the ones which are not subgame perfect. Those can be equilibria with the same equilibrium path, but irrational actions beside the path (and therefore irrelevant because of it) or equilibria with a different path, which are supported only through an incredible threat.
Now there are only two concepts known to me left, which need to be somehow tied into the picture. The bayesian NE (BNE) and the perfect bayesian equilibrium (PBE). I am not entirely sure about the next parts, so please look closely at them.
The BNE is a more general version of the NE. It is required when we are dealing with games of incomplete information.
Here the players maximize their expected utility like in the NE with mixed strategies (and like they actually ALWAYS do), but they do it now not only with regard to the other players decision, but also with regard to natures "decisions"/other players types.
Thats about it. It is not really conceptually different, it is only that the expected utility has to be calculated with different typen in mind using the bayes rule for probabilities of actions. The distribution of types is common knowledge. The relationship between the NE and the BNE is as similar as the one between the pure and mixed strategy NE. A NE is a special case of the BNE where there is no "nature-move", because there are no different types of players. Therefore technically the BNE could be applied to all the simpler games, but we would not gain anything, as we ended up in the normal NE anyway.
The last concept, the PBE, is actually a stricter concept, than the BNE. It not only requires that players consider the probability of other players having different types, but also that the players update their beliefs wherever possible, using the bayes rule.
Here is a little picture, of how I tried to put everything together.
