# Find SPNE for the extensive game with imperfect information

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

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.

• I did the same procedure. But I made a mistake. Can you also write these 5 equilibria? I will be glad. Thank you. Commented May 31, 2021 at 15:10