<p>Is every subgame perfect Nash equilibrium a Nash equilibrium? In perfect information games, every SPNE is the NE of its corresponding subgame, and that makes it an NE. But I think someone asked a similar question on this site and <a href="https://economics.stackexchange.com/users/1601/giskard">@Giskard</a> and <a href="https://economics.stackexchange.com/users/25512/varulle">@VARulle</a> replied with a counter-example. It was most likely the entry-deterrence model.</p>
<p>If anyone can help me with what might have been the original question and its possible counter-example, I would be able to read that up. It seemed interesting.</p>

CC BY-SA 4.0
2022-10-26T07:40:01.217Z
Is every game a subgame of itself?
Michael Greinecker
CC BY-SA 4.0
2022-10-26T08:00:44.263Z
@MichaelGreinecker Yes.
Rick_Morty
CC BY-SA 4.0
2022-10-26T09:06:07.297Z
Actually, with the other question we pointed out that that information is available in most textbooks. Could you let us know what you have read so far?
Giskard
CC BY-SA 4.0
2022-10-26T16:04:49.030Z
@Citrus There is your answer.
Michael Greinecker
CC BY-SA 4.0
2022-10-26T19:41:39.583Z
Hint: The "NE" in "SPNE" stands for "NE".
VARulle
CC BY-SA 4.0
2022-10-27T00:41:57.513Z
@VARulle I was waiting for you. I get the hint, but you made a comment that was *probably* saying that all SPNEs are not NEs or something like that. I am curious what it was as I can't track that exact post.
Rick_Morty
CC BY-SA 4.0
2022-10-27T00:58:14.670Z
@Giskard Was waiting for you as well! I can't track the exact comment. Can you tell me which "information [available in most textbooks]" are you talking about? I have studied a few cases of normal form and extensive form games (finite and infinite) given perfect information, and also NE, SPNE, games involving markets (if you consider that) and a bit of evolutionary games.
Rick_Morty
CC BY-SA 4.0
2022-10-27T03:53:43.933Z
Which textbook though? What definition of SPNE are you using? By information, I mean the veracity of the statement in your title.
Giskard
CC BY-SA 4.0
2022-10-27T07:47:27.023Z
SPNE is not defined on page 162 of this book, only subgames are. There is no need to link to a digital copy of the book, just provide the title and please copy the definition of SPNE here.
Giskard
CC BY-SA 4.0
2022-10-27T07:59:56.010Z
@Giskard Sorry, it's on p. 163. Def.: *A subgame perfect equilibrium is a strategy profile $s^*$ with the property that in no subgame can any player $i$ do better by choosing a strategy different from $s_i^*$, given that every other player $j$ adheres to $s_j^{*}$.*
Rick_Morty
CC BY-SA 4.0
2022-10-27T08:03:16.370Z
@Giskard On the very next pg., there's another definition of SPNE which is too long for a comment. [Here's an image](https://i.imgur.com/eshUdXc.png) of the same. On that very pg., there's a small explanation why every SPNE is an NE.
Rick_Morty
CC BY-SA 4.0
2022-10-27T08:10:16.390Z
So starting from the empty history, taking the game as its own subgame, this gives you $u_i(s^*) \geq u_i(r_i,s^*_{-i})$ for all $r_i$, right?
Giskard
CC BY-SA 4.0
2022-10-27T08:15:54.647Z
@Giskard Yes, by definition
Rick_Morty
CC BY-SA 4.0
2022-10-27T08:47:36.887Z
You have probably seen a comment of me to a question where someone said that he knows that not every NE is an SPNE, but he can't find an example for this. That's also odd, but somewhat different. An SPNE is by the most common definition a NE satisfying an additional condition, so your question is somewhat trivial. (The definition you cite doesn't state it that way, but together with the fact that a game is a subgame of itself it also establishes that an SPNE is a NE.)
VARulle
<microeconomics><game-theory>
Is every SPNE an NE?
Rick_Morty
# Is every SPNE an NE?

(Asked by: username_0 on 2022-10-26)

Is every subgame perfect Nash equilibrium a Nash equilibrium? In perfect information games, every SPNE is the NE of its corresponding subgame, and that makes it an NE. But I think someone asked a similar question on this site and [@Giskard](https://economics.stackexchange.com/users/1601/giskard) and [@VARulle](https://economics.stackexchange.com/users/25512/varulle) replied with a counter-example. It was most likely the entry-deterrence model.

If anyone can help me with what might have been the original question and its possible counter-example, I would be able to read that up. It seemed interesting.

**Tags:** microeconomics, game-theory
> **Comments:**
> **username_1**: Is every game a subgame of itself?  
> **username_0**: @MichaelGreinecker Yes.  
> **username_2**: Actually, with the other question we pointed out that that information is available in most textbooks. Could you let us know what you have read so far?  
> **username_1**: @Citrus There is your answer.  
> **username_3**: Hint: The "NE" in "SPNE" stands for "NE".  
> **username_0**: @VARulle I was waiting for you. I get the hint, but you made a comment that was *probably* saying that all SPNEs are not NEs or something like that. I am curious what it was as I can't track that exact post.  
> **username_0**: @Giskard Was waiting for you as well! I can't track the exact comment. Can you tell me which "information [available in most textbooks]" are you talking about? I have studied a few cases of normal form and extensive form games (finite and infinite) given perfect information, and also NE, SPNE, games involving markets (if you consider that) and a bit of evolutionary games.  
> **username_2**: Which textbook though? What definition of SPNE are you using? By information, I mean the veracity of the statement in your title.  
> **username_2**: SPNE is not defined on page 162 of this book, only subgames are. There is no need to link to a digital copy of the book, just provide the title and please copy the definition of SPNE here.  
> **username_0**: @Giskard Sorry, it's on p. 163. Def.: *A subgame perfect equilibrium is a strategy profile $s^*$ with the property that in no subgame can any player $i$ do better by choosing a strategy different from $s_i^*$, given that every other player $j$ adheres to $s_j^{*}$.*  
> **username_0**: @Giskard On the very next pg., there's another definition of SPNE which is too long for a comment. [Here's an image](https://i.imgur.com/eshUdXc.png) of the same. On that very pg., there's a small explanation why every SPNE is an NE.  
> **username_2**: So starting from the empty history, taking the game as its own subgame, this gives you $u_i(s^*) \geq u_i(r_i,s^*_{-i})$ for all $r_i$, right?  
> **username_0**: @Giskard Yes, by definition  
> **username_3**: You have probably seen a comment of me to a question where someone said that he knows that not every NE is an SPNE, but he can't find an example for this. That's also odd, but somewhat different. An SPNE is by the most common definition a NE satisfying an additional condition, so your question is somewhat trivial. (The definition you cite doesn't state it that way, but together with the fact that a game is a subgame of itself it also establishes that an SPNE is a NE.)  

---

## 0 answers

