<p><strong>Definition</strong>: Descendant <br>
Nodes downwards from a node X are called <em>descendants</em> of X. (Upwards are <em>ancestors</em> of X.)</p>
<p><strong>Definition</strong>: Information Set <br>
Nodes connected by a dashed line are in an <em>information set</em>. The player who chooses the action here knows that they are in a node of the set, but they do not know which node specifically. The exception is when the information set only containts one node, it is a <em>singleton</em>.</p>
<p><strong>Definition</strong>: Subgame  <br>
Take a node X. Is X's information set a singleton? If not, the game starting at X is not a <em>subgame</em>. (It will never happen during a game that a player will know for <strong>sure</strong> that they are in X, and has to reason accordingly.) <br>
Also look at X's descendants. If any of them are in an information set with nodes who are not X's descendants, then the game starting at X is not a subgame. (The player deciding in X cannot reason solely based on the descendants, because there is at least one player who will need to consider other nodes as well, since they don't know where they are.) <br>
If neither of the above apply, then the game starting at X is a subgame.</p>

CC BY-SA 4.0
2024-05-24T18:26:26.443Z
2024-05-25T17:36:25.953Z
2024-05-25T17:36:25.953Z
Giskard
<p>First lets talk about game tree 2. There are 3 subgames namely:
1). The initial root node itself
2).P2 left which is further branched out into its own game .
3). P2 right again the same.
As these two branches lead out to asymmetric games we will count them as different . To look at and find out subgame in a game tree we need to see branching out of the root node. These are the only three sub games .
But if we look at first game tree, there is no branching at P2 but there is at P3 . If we observe carefully both the branches of P3 are similar games so we can actually cancel one out. This way we are left with only 1 subgame (i.e. the game itself).</p>

CC BY-SA 4.0
2024-05-24T13:15:27.240Z
2024-05-24T13:15:27.240Z
Lksonice
<p><a href="https://i.sstatic.net/f5PF4du6.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/f5PF4du6.png" alt="enter image description here" /></a></p>
<p>This game has only 1 subgame (the game itself).</p>
<p><a href="https://i.sstatic.net/y5WEVk0w.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/y5WEVk0w.png" alt="enter image description here" /></a></p>
<p>This game has 3 subgames. Why is it different? I don't know how to judge the number of subgames.</p>

<game-theory>
Why is the number of subgames in these two games different?
PiVoyager
# Why is the number of subgames in these two games different?

(Asked by: username_0 on 2024-05-14)

[![enter image description here](https://i.sstatic.net/f5PF4du6.png)](https://i.sstatic.net/f5PF4du6.png)

This game has only 1 subgame (the game itself).

[![enter image description here](https://i.sstatic.net/y5WEVk0w.png)](https://i.sstatic.net/y5WEVk0w.png)

This game has 3 subgames. Why is it different? I don't know how to judge the number of subgames.

**Tags:** game-theory

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## 2 answers

### Answer by username_1 on 2024-05-24. Score: 1

**Definition**: Descendant   
Nodes downwards from a node X are called *descendants* of X. (Upwards are *ancestors* of X.)

**Definition**: Information Set   
Nodes connected by a dashed line are in an *information set*. The player who chooses the action here knows that they are in a node of the set, but they do not know which node specifically. The exception is when the information set only containts one node, it is a *singleton*.

**Definition**: Subgame   
Take a node X. Is X's information set a singleton? If not, the game starting at X is not a *subgame*. (It will never happen during a game that a player will know for **sure** that they are in X, and has to reason accordingly.)   
Also look at X's descendants. If any of them are in an information set with nodes who are not X's descendants, then the game starting at X is not a subgame. (The player deciding in X cannot reason solely based on the descendants, because there is at least one player who will need to consider other nodes as well, since they don't know where they are.)   
If neither of the above apply, then the game starting at X is a subgame.


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### Answer by username_2 on 2024-05-24. Score: 0

First lets talk about game tree 2. There are 3 subgames namely:
1). The initial root node itself
2).P2 left which is further branched out into its own game .
3). P2 right again the same.
As these two branches lead out to asymmetric games we will count them as different . To look at and find out subgame in a game tree we need to see branching out of the root node. These are the only three sub games .
But if we look at first game tree, there is no branching at P2 but there is at P3 . If we observe carefully both the branches of P3 are similar games so we can actually cancel one out. This way we are left with only 1 subgame (i.e. the game itself).


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