# Relation between Nash equilibrium , subgame perfect equilibrium, backward induction, perfect Bayesian equilibrium

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?

• 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. May 8, 2017 at 20:11

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.)

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) 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

• 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. May 8, 2017 at 20:12
• @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 :( May 8, 2017 at 21:53