<p>I have not previously come across three player games and I am trying to solve for the subgame perfect NE/equilibria for the following game, where player 3's decision node is linked to the decisions of both player 1 and player 2. In two player games, I would transform this to a normal -form game with a 2 player matrix but not sure if that's also the way to go for this three player game.</p>
<p>I only find 2 subgames in the game, not sure if there are other subgames I may be overlooking/missing.<a href="https://i.sstatic.net/YWPP9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YWPP9.png" alt="the image below is the game for which i am trying to solve" /></a></p>

CC BY-SA 4.0
2023-03-20T10:12:45.047Z
There are no proper subgames in this game, just the trivial one, i.e. the game itself. So all NE are subgame perfect.
VARulle
CC BY-SA 4.0
2023-03-20T10:24:11.933Z
Is this because player 3's decision nodes are linked to both P1 and P2. In the event that their decision nodes were only linked, for example to P1, then we would have 2 proper subgames, including the game itself. Is that understanding correct?
Kay101
CC BY-SA 4.0
2023-03-21T11:52:17.713Z
Partly yes. The two decision nodes of P3 are included in an information set (denoted by the set with the dashed borderline). Subgames must not cut through information sets. If there were no such (non-singleton) information set, then there would be 3 proper subgames: One starting at P2's node and 2 starting at P3 left and right nodes, respectively.
VARulle
<microeconomics><game-theory>
Subgames in imperfect information games
Kay101
# Subgames in imperfect information games

(Asked by: username_0 on 2023-03-20)

I have not previously come across three player games and I am trying to solve for the subgame perfect NE/equilibria for the following game, where player 3's decision node is linked to the decisions of both player 1 and player 2. In two player games, I would transform this to a normal -form game with a 2 player matrix but not sure if that's also the way to go for this three player game.

I only find 2 subgames in the game, not sure if there are other subgames I may be overlooking/missing.[![the image below is the game for which i am trying to solve](https://i.sstatic.net/YWPP9.png)](https://i.sstatic.net/YWPP9.png)

**Tags:** microeconomics, game-theory
> **Comments:**
> **username_1**: There are no proper subgames in this game, just the trivial one, i.e. the game itself. So all NE are subgame perfect.  
> **username_0**: Is this because player 3's decision nodes are linked to both P1 and P2. In the event that their decision nodes were only linked, for example to P1, then we would have 2 proper subgames, including the game itself. Is that understanding correct?  
> **username_1**: Partly yes. The two decision nodes of P3 are included in an information set (denoted by the set with the dashed borderline). Subgames must not cut through information sets. If there were no such (non-singleton) information set, then there would be 3 proper subgames: One starting at P2's node and 2 starting at P3 left and right nodes, respectively.  

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