<p>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.</p>

<p>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).</p>

CC BY-SA 3.0
2017-02-13T12:37:00.587Z
Okay that makes sense. So is solving the corresponding Normalform game the only way to find all NE?
FloodLuszt
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2017-02-13T12:56:24.517Z
That is usually the easiest way. You can also intersect best-response correspondences.
Bayesian
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2017-02-13T11:15:27.650Z
2017-02-13T11:15:27.650Z
Bayesian
<p>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.</p>

<p>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?</p>

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

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2017-02-13T10:47:25.890Z
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?
Giskard
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2017-02-13T12:31:46.113Z
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.
FloodLuszt
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2017-02-13T12:33:14.143Z
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.
Giskard
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2017-02-13T12:39:46.447Z
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.
FloodLuszt
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2017-02-13T12:40:53.783Z
Then you can upvote and/or accept it.
Giskard
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2017-02-13T12:47:17.933Z
I tried to upvote but I guess I need more reputation for it to be visible. So is accepting good practice even if you ask follow up questions?
FloodLuszt
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2017-02-13T14:14:24.357Z
@FloodLuszt if you don't feel that an answer fully addresses your question then you can ask in a comment for clarification. But if you have a completely new follow-up question then it is good practice to post it as a separate, new question (otherwise we run the risk of answers and questions being out of sync with each other).
Ubiquitous
<game-theory><nash-equilibrium><algorithms>
Is there a systematic approach to find equilibria in sequential games?
FloodLuszt
# Is there a systematic approach to find equilibria in sequential games?

(Asked by: username_0 on 2017-02-13)

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 username_3 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?

**Tags:** game-theory, nash-equilibrium, algorithms

> **Comments:**
> **username_1**: 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?  
> **username_0**: 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.  
> **username_1**: 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.  
> **username_0**: 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. @username_3's answer is a good step forward for me.  
> **username_1**: Then you can upvote and/or accept it.  
> **username_0**: I tried to upvote but I guess I need more reputation for it to be visible. So is accepting good practice even if you ask follow up questions?  
> **username_2**: @username_0 if you don't feel that an answer fully addresses your question then you can ask in a comment for clarification. But if you have a completely new follow-up question then it is good practice to post it as a separate, new question (otherwise we run the risk of answers and questions being out of sync with each other).  

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## 1 answers

### Answer by username_3 on 2017-02-13. Score: 1 | Accepted answer

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).

> **Comments:**
> **username_0**: Okay that makes sense. So is solving the corresponding Normalform game the only way to find all NE?  
> **username_3**: That is usually the easiest way. You can also intersect best-response correspondences.  

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