<p>I'm solving a 3 person game and the Subgame Perfect Nash Equilibrium I get is not a part of the Nash Equilibrium. Isn't a SPNE supposed to be a refinement over the Nash? I've seen some Nash Equilibria not being a part of the SPNE (i.e., some NE being redundant), but I wonder if a SPNE is supposed to just cut down the NE into one SPNE or if it can be something completely different.</p>
<p>The SPNE has a mixed strategy for 2 players and a pure strategy for one player.</p>
<p><a href="https://i.sstatic.net/ga5RV.jpg" rel="nofollow noreferrer">Here is the game for anyone interested.</a></p>

CC BY-SA 4.0
2021-03-05T14:53:55.207Z
SPNE is always a NE. You may want to double check your work to see if there were any mistakes.
Herr K.
CC BY-SA 4.0
2021-03-05T16:05:51.710Z
What do you mean with "part of the Nash Equilibrium"? Any SPNE of a game G is a NE of G, as the name indicates. And, by definition, it is a NE of G that induces a NE in each subgame. The induced NE of a subgame is therefore in some sense "part of" the SPNE, but the subgame need not actually be reached on the SPNE's path of play. Maybe this is your source of confusion?
VARulle
CC BY-SA 4.0
2021-03-05T17:09:31.087Z
@VARulle -- No, I meant that the SPNE I get is not a NE as. Sorry for the confusion in wording
WorldGov
CC BY-SA 4.0
2021-03-05T17:10:23.287Z
My SPNE is in mixed strategy. But the NE contains only of pure strategy. Does a SPNE always have to be a NE even if the SPNE has a mixed strategy?
WorldGov
CC BY-SA 4.0
2021-03-05T17:21:57.620Z
Have you considered the possibility that there is a NE that involves mixed strategies?
Herr K.
CC BY-SA 4.0
2021-03-06T15:51:54.890Z
I haven't. That might be the case. I'll try to find out if the Mixed Strategy Nash Equilibrium exists and if it's indeed the SPNE. Thanks!
WorldGov
<game-theory>
Should a sub-game perfect Nash Equilibrium always be a part of the Nash Equilibrium of a game?
WorldGov
# Should a sub-game perfect Nash Equilibrium always be a part of the Nash Equilibrium of a game?

(Asked by: username_0 on 2021-03-05)

I'm solving a 3 person game and the Subgame Perfect Nash Equilibrium I get is not a part of the Nash Equilibrium. Isn't a SPNE supposed to be a refinement over the Nash? I've seen some Nash Equilibria not being a part of the SPNE (i.e., some NE being redundant), but I wonder if a SPNE is supposed to just cut down the NE into one SPNE or if it can be something completely different.

The SPNE has a mixed strategy for 2 players and a pure strategy for one player.

[Here is the game for anyone interested.](https://i.sstatic.net/ga5RV.jpg)

**Tags:** game-theory
> **Comments:**
> **username_1**: SPNE is always a NE. You may want to double check your work to see if there were any mistakes.  
> **username_2**: What do you mean with "part of the Nash Equilibrium"? Any SPNE of a game G is a NE of G, as the name indicates. And, by definition, it is a NE of G that induces a NE in each subgame. The induced NE of a subgame is therefore in some sense "part of" the SPNE, but the subgame need not actually be reached on the SPNE's path of play. Maybe this is your source of confusion?  
> **username_0**: @VARulle -- No, I meant that the SPNE I get is not a NE as. Sorry for the confusion in wording  
> **username_0**: My SPNE is in mixed strategy. But the NE contains only of pure strategy. Does a SPNE always have to be a NE even if the SPNE has a mixed strategy?  
> **username_1**: Have you considered the possibility that there is a NE that involves mixed strategies?  
> **username_0**: I haven't. That might be the case. I'll try to find out if the Mixed Strategy Nash Equilibrium exists and if it's indeed the SPNE. Thanks!  

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## 0 answers

