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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.

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

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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.

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  • $\begingroup$ I did the same procedure. But I made a mistake. Can you also write these 5 equilibria? I will be glad. Thank you. $\endgroup$
    – 1190
    May 31 at 15:10

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