https://file.aiquickdraw.com/mathbot/file/172693463181379knz6d5.jpg

Let's analyze the questions and tables in the images you uploaded.

### Question 2
The table provided lists utility values for options across three categories (X, Y, Z). We are told that a person prefers:
- X to Y
- Y to Z
- Hence, X to Z

This means that the utility for X should be higher than the utility for Y, and the utility for Y should be higher than the utility for Z.

**Step-by-step analysis of options:**
- **Option 1:** \( X = 1 \), \( Y = 2 \), \( Z = 3 \)
  - \( Z > Y > X \) (opposite of what we want)
  
- **Option 2:** \( X = 8 \), \( Y = 9 \), \( Z = 10 \)
  - \( Z > Y > X \) (again, opposite of the desired preference)
  
- **Option 3:** \( X = -10 \), \( Y = -20 \), \( Z = -90 \)
  - \( X > Y > Z \) (correct, as it matches the given preference)
  
- **Option 4:** \( X = -1 \), \( Y = 0 \), \( Z = 1 \)
  - \( Z > Y > X \) (again, this doesn't match the desired preference)

**Conclusion for Question 2:** The correct answer is **Option 3**.

### Question 5
The question asks to classify the type of game described in Question 2. Let's break down the options:

- **a. A normal form game:** A strategic game represented in a matrix form where players' strategies and payoffs are laid out, which seems to match the format of the table.
  
- **b. An extensive form game:** This would be a game represented using a decision tree, which doesn't apply to the format seen here.

- **c. A zero-sum game:** A game in which one player's gain is another player's loss. However, there is no indication that this is a zero-sum game, as no player-to-player interaction is shown in the matrix.

- **d. Both answers a. and c. are correct:** Since c. (zero-sum game) doesn't apply, this option is incorrect.

**Conclusion for Question 5:** The correct answer is **a. A normal form game**.

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Do you need further details on these questions, or would you like clarification? Here are 5 additional related questions for further exploration:

1. How can you identify a zero-sum game in a matrix format?
2. What are the key differences between normal form games and extensive form games?
3. How does utility theory apply to decision-making in economics?
4. How would the solution change if the preferences were reversed (i.e., Z to Y to X)?
5. What strategies can players use in normal form games to maximize their payoffs?

**Tip:** In game theory, always start by identifying the players, strategies, and payoffs when classifying the type of game you're dealing with!

The game presented in Question 2 would be best described as: a) A normal form game, b) An extensive form game, c) A zero-sum game, d) Both answers a. and c. are correct.

Explore utility preferences among options X, Y, and Z to determine a person’s utility function, and classify the game type in normal form and zero-sum terms. Perfect for advanced high school and college-level students studying game theory.

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