<p>1- Backward induction solution is Nash equilibrium solution. <strong>TRUE</strong></p>

<p>2- Not all Nash equilibria are sequentially rational. <strong>TRUE</strong>; <em>NE allows for sequentially irrational behavior off equilibrium path.</em></p>

<p>3- All Backward induction solutions are sequentially rational. <strong>TRUE</strong></p>

<p>4- SPNE solutions are Nash equilibria. <strong>TRUE</strong></p>

<p>5- SPNE solutions are sequentially rational if game has at least one proper sub game. <strong>FALSE</strong> <em>(Counterexample below: the game has one proper subgame; $(XC,L)$ is an SPNE but involves a sequentially irrational strategy $L$ by player 2).</em> If only has improper sub game then it may not be sequentially rational. <strong>TRUE</strong> <em>(Treat the subgame after $X$ as a game of its own.)</em></p>

<p><a href="https://i.sstatic.net/hNpdo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/hNpdo.png" alt="enter image description here"></a></p>

<p>6- PBE solutions are Nash equilibrium and SPNE. <strong>UNCERTAIN / TRUE (in most cases)</strong>; <em>it actually depends on how exactly you define PBE (there are many definitions). See <a href="http://econweb.ucsd.edu/~jwatson/PAPERS/WatsonPBE.pdf" rel="nofollow noreferrer">Watson (2017)</a> for more details.</em></p>

<p>7- PBE solutions are sequentially rational no matter what. <strong>TRUE</strong></p>

<p>8- WPBE solutions are Nash equilibria. And sequentially rational no matter what. <strong>TRUE</strong></p>

CC BY-SA 3.0
2017-05-08T20:12:33.277Z
7,8 also seems to depend on the definition. The definition for PBE I knew stated that the strategies were sequentially rational and the beliefs consistent.
Giskard
CC BY-SA 3.0
2017-05-08T21:53:15.030Z
@denesp: Thanks. I misread 5, and have fixed my answer. I agree with you on 7 and 8 after thinking about it a bit harder. Those statements are true by definition. I must have had the wrong thing for breakfast today `:(`
Herr K.
CC BY-SA 3.0
2017-05-08T19:32:41.333Z
2017-05-08T21:48:53.947Z
2017-05-08T21:48:53.947Z
Herr K.
<p>My claims are</p>

<p>1- Backward induction solution is Nash equilibrium solution.</p>

<p>2- Not all Nash equilibria are sequentially rational</p>

<p>3- All Backward induction solutions are sequentially rational</p>

<p>4- SPNE solutions are Nash equilibria </p>

<p>5- SPNE solutions are sequentially rational if game has at least one 
proper sub game. If only has improper sub game then it may not be sequentially rational.</p>

<p>6- PBE solutions are Nash equilibrium and SPNE</p>

<p>7- PBE solutions are sequentially rational no matter what.</p>

<p>8- WPBE solutions are Nash equilibria. And sequentially rational no matter what.</p>

<p>Are these true? What do you think? Anything want to add?</p>

CC BY-SA 3.0
2017-05-08T20:11:02.433Z
As @HerrK. points out in his answer whether these statements are true or false depend on the exact definitions you are using. Please include them in your question.
Giskard
<game-theory>
Relation between Nash equilibrium , subgame perfect equilibrium, backward induction, perfect Bayesian equilibrium
Kuantew
# Relation between Nash equilibrium , subgame perfect equilibrium, backward induction, perfect Bayesian equilibrium

(Asked by: username_0 on 2017-05-08)

My claims are

1- Backward induction solution is Nash equilibrium solution.

2- Not all Nash equilibria are sequentially rational

3- All Backward induction solutions are sequentially rational

4- SPNE solutions are Nash equilibria

5- SPNE solutions are sequentially rational if game has at least one
proper sub game. If only has improper sub game then it may not be sequentially rational.

6- PBE solutions are Nash equilibrium and SPNE

7- PBE solutions are sequentially rational no matter what.

8- WPBE solutions are Nash equilibria. And sequentially rational no matter what.

Are these true? What do you think? Anything want to add?

**Tags:** game-theory
> **Comments:**
> **username_1**: As @HerrK. points out in his answer whether these statements are true or false depend on the exact definitions you are using. Please include them in your question.  

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## 1 answers

### Answer by username_2 on 2017-05-08. Score: 3 | Accepted answer

1- Backward induction solution is Nash equilibrium solution. **TRUE**

2- Not all Nash equilibria are sequentially rational. **TRUE**; *NE allows for sequentially irrational behavior off equilibrium path.*

3- All Backward induction solutions are sequentially rational. **TRUE**

4- SPNE solutions are Nash equilibria. **TRUE**

5- SPNE solutions are sequentially rational if game has at least one proper sub game. **FALSE** *(Counterexample below: the game has one proper subgame; $(XC,L)$ is an SPNE but involves a sequentially irrational strategy $L$ by player 2).* If only has improper sub game then it may not be sequentially rational. **TRUE** *(Treat the subgame after $X$ as a game of its own.)*

[![enter image description here](https://i.sstatic.net/hNpdo.png)](https://i.sstatic.net/hNpdo.png)

6- PBE solutions are Nash equilibrium and SPNE. **UNCERTAIN / TRUE (in most cases)**; *it actually depends on how exactly you define PBE (there are many definitions). See [Watson (2017)](http://econweb.ucsd.edu/~jwatson/PAPERS/WatsonPBE.pdf) for more details.*

7- PBE solutions are sequentially rational no matter what. **TRUE**

8- WPBE solutions are Nash equilibria. And sequentially rational no matter what. **TRUE**

> **Comments:**
> **username_1**: 7,8 also seems to depend on the definition. The definition for PBE I knew stated that the strategies were sequentially rational and the beliefs consistent.  
> **username_2**: @denesp: Thanks. I misread 5, and have fixed my answer. I agree with you on 7 and 8 after thinking about it a bit harder. Those statements are true by definition. I must have had the wrong thing for breakfast today `:(`  

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