<p>I assume you're only interested in pure strategy Nash equilibria.</p>
<p>Consider the subgame between players <span class="math-container">$1$</span> and <span class="math-container">$3$</span> after Player <span class="math-container">$1$</span> has chosen <span class="math-container">$A$</span>.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th></th>
<th><span class="math-container">$Y$</span></th>
<th><span class="math-container">$Z$</span></th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="math-container">$W$</span></td>
<td><span class="math-container">$(2,1)$</span></td>
<td><span class="math-container">$(0,0)$</span></td>
</tr>
<tr>
<td><span class="math-container">$X$</span></td>
<td><span class="math-container">$(0,0)$</span></td>
<td><span class="math-container">$(1,2)$</span></td>
</tr>
</tbody>
</table>
</div>
<p>This has two Nash equilibria (in pure strategies).</p>
<p>Similarly, you can consider the subgame between Players <span class="math-container">$1$</span> and <span class="math-container">$3$</span>  where player 1 chooses <span class="math-container">$B$</span>. Again you will find 2 Nash equilibria.</p>
<p>In order to determine what Player <span class="math-container">$1$</span> is going to do in step 1 (<span class="math-container">$A$</span> or <span class="math-container">$B$</span>) you need to see, depending on the Nash equilibria chosen in stage 2, which one she will prefer.</p>
<p>As far as I see, there are 5 equilibria.</p>

CC BY-SA 4.0
2021-05-31T15:10:03.813Z
I did the same procedure. But I made a mistake. Can you also write these 5 equilibria? I will be glad. Thank you.
studentp
CC BY-SA 4.0
2021-05-31T13:48:32.200Z
2021-05-31T13:48:32.200Z
tdm
<p><a href="https://i.sstatic.net/9TBKl.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9TBKl.jpg" alt="enter image description here" /></a></p>
<p>Find SPNE?</p>
<p>My suggestion is <span class="math-container">$\{(AW,L,Y), (BU, L,Y), (BD,R,Y)\}$</span></p>
<p>How to find :</p>
<p>firstly, I consider the first subgame where P3 and P1 play. And I choose (W,Z)</p>
<p>Secondly, I  consider the second subgame where P1 and P2 play. And I choose (U,L) and ( D,R).</p>
<p>Thirdly I consider 2 cases. In the first one Player one chooses both A and B for the 1st subgame. And player one chooses only B for the second subgame.</p>
<p>———-</p>
<p>But I am not sure about my solution. I am confused since there are 3 players</p>

<game-theory>
Find SPNE for the extensive game with imperfect information
studentp
# Find SPNE for the extensive game with imperfect information

(Asked by: username_0 on 2021-05-31)

[![enter image description here](https://i.sstatic.net/9TBKl.jpg)](https://i.sstatic.net/9TBKl.jpg)

Find SPNE?

My suggestion is $\{(AW,L,Y), (BU, L,Y), (BD,R,Y)\}$

How to find :

firstly, I consider the first subgame where P3 and P1 play. And I choose (W,Z)

Secondly, I consider the second subgame where P1 and P2 play. And I choose (U,L) and ( D,R).

Thirdly I consider 2 cases. In the first one Player one chooses both A and B for the 1st subgame. And player one chooses only B for the second subgame.

———-

But I am not sure about my solution. I am confused since there are 3 players

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2021-05-31. Score: 2 | Accepted answer

I assume you're only interested in pure strategy Nash equilibria.

Consider the subgame between players $1$ and $3$ after Player $1$ has chosen $A$.

|  | $Y$ | $Z$ |
| --- | --- | --- |
| $W$ | $(2,1)$ | $(0,0)$ |
| $X$ | $(0,0)$ | $(1,2)$ |

This has two Nash equilibria (in pure strategies).

Similarly, you can consider the subgame between Players $1$ and $3$ where player 1 chooses $B$. Again you will find 2 Nash equilibria.

In order to determine what Player $1$ is going to do in step 1 ($A$ or $B$) you need to see, depending on the Nash equilibria chosen in stage 2, which one she will prefer.

As far as I see, there are 5 equilibria.

> **Comments:**
> **username_0**: I did the same procedure. But I made a mistake. Can you also write these 5 equilibria? I will be glad. Thank you.  

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