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The game you have provided is a *sequential game* represented as an extensive form (tree). To convert this into *normal form* (which is a matrix representation of the strategies and payoffs), we need to follow these steps:

1. **Identify the players**: From the diagram, we have two players:
   - Player 1 chooses between \( U \) (up) and \( D \) (down).
   - Player 2 chooses between \( A \) and \( B \) after Player 1's choice.

2. **Determine the strategies**:
   - Player 1 has two strategies: \( U \) or \( D \).
   - Player 2's strategy depends on the information they have. Since Player 2 might not know which node they are at (this is called imperfect information), Player 2 will have to choose \( A \) or \( B \) at both nodes (after Player 1's move). Thus, Player 2 has four strategies:
     - \( AA \): Play \( A \) after \( U \) and \( A \) after \( D \).
     - \( AB \): Play \( A \) after \( U \) and \( B \) after \( D \).
     - \( BA \): Play \( B \) after \( U \) and \( A \) after \( D \).
     - \( BB \): Play \( B \) after \( U \) and \( B \) after \( D \).

3. **Build the payoff matrix**: The matrix will show the payoffs for all combinations of Player 1 and Player 2 strategies. We can read the payoffs from the tree.

|               | \( AA \) (A,A) | \( AB \) (A,B) | \( BA \) (B,A) | \( BB \) (B,B) |
|---------------|----------------|----------------|----------------|----------------|
| **\( U \)**   | (3,3)           | (3,3)          | (5,1)          | (3,6)          |
| **\( D \)**   | (4,2)           | (9,0)          | (4,2)          | (2,2)          |

- If Player 1 plays \( U \) and Player 2 plays \( A \) at both nodes, the outcome is (3,3).
- If Player 1 plays \( D \) and Player 2 plays \( B \) after both moves, the outcome is (2,2).

This matrix is now in *normal form*, with Player 1's strategies listed in rows and Player 2's strategies in columns.

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Would you like more details or have any questions about this game representation? Here are 5 related questions you might consider:

1. What is the Nash equilibrium for this game in normal form?
2. How does the concept of subgame perfect equilibrium apply to this sequential game?
3. Can we solve this game using backward induction, and what would be the solution?
4. How would the outcome change if Player 2 had perfect information?
5. What if Player 1 had more than two strategies to choose from?

**Tip**: In extensive form games, it's helpful to first look for Nash equilibria using backward induction before converting to normal form!

How do I represent this game tree in normal form?

Learn how to convert a sequential game, represented in extensive form (game tree), into normal form. This tutorial walks you through identifying players, their strategies, and building the payoff matrix, along with concepts such as Nash equilibrium and backward induction.

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