<p>As afreelunch only answered in the comment section, I'll answer it here.</p>

<p>The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call <span class="math-container">$S$</span> the set of all SPNE and <span class="math-container">$N$</span> the set of all NE in a game. Then:</p>

<p><span class="math-container">$$S \subseteq N.$$</span></p>

<p>If the sequential game has only one subgame (see the subgame definition <a href="https://en.wikipedia.org/wiki/Subgame" rel="nofollow noreferrer">here</a>), the game itself, then we will have. </p>

<p><span class="math-container">$$S = N.$$</span></p>

<p>And my reasoning was right. </p>

CC BY-SA 4.0
2019-04-14T22:54:13.217Z
2019-04-14T22:54:13.217Z
Marcelo Gelati
<p>I want to know if my thinking is correct. 
Look at the following game.</p>

<p><a href="https://i.sstatic.net/rAgl6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/rAgl6.png" alt="Simultaneous game"></a></p>

<p>As the game has only one subgame (i.e., the game itself) then the Nash Equilibria will coincide with the subgame perfect equilibria.</p>

<p>In this case, we have two Nash equilibria: {U, u} and {D, d}. By my statement before, the subgame perfect equilibria will be {U, u} and {D, d} too. </p>

<p>Am I right?</p>

CC BY-SA 4.0
2018-11-06T20:22:54.657Z
The presentation suggests that it's a sequential game. If so, then "u" is not a strategy, so {U,u} is not a Nash equilibrium.
Acccumulation
<microeconomics><game-theory>
Find a subgame perfect equilibrium and a Nash equilibrium
Marcelo Gelati
# Find a subgame perfect equilibrium and a Nash equilibrium

(Asked by: username_0 on 2018-11-06)

I want to know if my thinking is correct.
Look at the following game.

[![Simultaneous game](https://i.sstatic.net/rAgl6.png)](https://i.sstatic.net/rAgl6.png)

As the game has only one subgame (i.e., the game itself) then the Nash Equilibria will coincide with the subgame perfect equilibria.

In this case, we have two Nash equilibria: {U, u} and {D, d}. By my statement before, the subgame perfect equilibria will be {U, u} and {D, d} too.

Am I right?

**Tags:** microeconomics, game-theory

> **Comments:**
> **username_1**: The presentation suggests that it's a sequential game. If so, then "u" is not a strategy, so {U,u} is not a Nash equilibrium.  

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## 1 answers

### Answer by username_0 on 2019-04-14. Score: 0 | Accepted answer

As afreelunch only answered in the comment section, I'll answer it here.

The SPNE (Subgame Perfect Nash Equilibrium) is a refinement of the NE (Nash Equilibrium). So, let's call $S$ the set of all SPNE and $N$ the set of all NE in a game. Then:

$$S \subseteq N.$$

If the sequential game has only one subgame (see the subgame definition [here](https://en.wikipedia.org/wiki/Subgame)), the game itself, then we will have.

$$S = N.$$

And my reasoning was right.


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