# 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 '17 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 '17 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 '17 at 21:53