# Is there a systematic approach to find equilibria in sequential games?

I know that one can use backward induction to find one particular subgame perfect NE. And I know that wherever possible one can represent the game in normalform and then find all NE.

But is there a systematic way of finding all extensive form specific subtypes of NE, like sequential or weak perfect Bayesian equilibria? What about the rest of the subgame perfect equilibria that can't be found by backward induction?

Do I always have to guess and proof that it really is the kind of equilibrium I think it is?

• 1. You seem to imply that backward induction sometimes finds equilibria that are not subgame perfect. Could you provide an example? 2. You also seem to imply that in some games you cannot check which equilibria are subgame perfect with backward induction. Could you provide an example? Commented Feb 13, 2017 at 10:47
• I guess I was trying to be overly cautious in formulating my first sentence. In fact I can't possibly imagine how backward induction can result in a non SPE. Concerning your second point: I was not trying to imply that. I think you can always check if a given NE is a SPE using backward induction. But it was not so clear to me that you can find all SPE's that way. Commented Feb 13, 2017 at 12:31
• It cannot. You could consider editing your question to make it clearer. I am still unsure about what it is you are trying to accomplish. Commented Feb 13, 2017 at 12:33
• I am just trying to figure out how to start finding let's say all PBE of a game without having to look closely and more or less guessing the equilibrium. @Bayesian's answer is a good step forward for me. Commented Feb 13, 2017 at 12:39
• Then you can upvote and/or accept it. Commented Feb 13, 2017 at 12:40

You can first find all NE. Then you check which ones are subgame-perfect. Then you proceed and check for which of the NE you can find beliefs that are consistent with the definition of PBE. You can go on and refine the set of equilibria further by kicking out all equilibria that do not satisfy the additional requirements of your stricter equilibrium concept.

If by "backward induction" you mean solving the game backwards subgame-by-subgame, then by definition you find all the SPNE (= all NE in which the equilibrium strategy profile also constitutes NE in all subgames).

• Okay that makes sense. So is solving the corresponding Normalform game the only way to find all NE? Commented Feb 13, 2017 at 12:37
• That is usually the easiest way. You can also intersect best-response correspondences. Commented Feb 13, 2017 at 12:56