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This problem describes a first-price auction between Sultan Hassanal Bolkiah and Sheikh Zayed Bin Sultan Al Nahyan, where both participants bid in increments of hundreds of millions of pounds. The Sultan's bids are restricted to odd numbers (1, 3, 5, 7, or 9), while the Sheikh's bids must be even numbers (2, 4, 6, 8, or 10). The highest bidder wins and pays their own bid price, and the payoff is the difference between their valuation and the amount paid.

### Part (a): Strategic Form Matrix
To write down the strategic form, we can create a payoff matrix with the Sultan's bids on one axis and the Sheikh's on the other axis. For each combination of bids, we calculate the payoffs based on the following rules:
- If the Sultan's bid is higher, he wins and pays his bid, so his payoff is \( V_S - \text{bid}_S \), where \( V_S = 8 \) (Sultan's valuation).
- If the Sheikh's bid is higher, he wins and pays his bid, so his payoff is \( V_{Sh} - \text{bid}_{Sh} \), where \( V_{Sh} = 7 \) (Sheikh's valuation).
- If they bid the same amount, neither wins, and both get a payoff of 0.

Let's set up the matrix where rows represent the Sultan's bids (1, 3, 5, 7, 9) and columns represent the Sheikh's bids (2, 4, 6, 8, 10):

| Sultan/Sheikh | 2           | 4           | 6           | 8           | 10          |
|---------------|-------------|-------------|-------------|-------------|-------------|
| **1**         | (7, 5)      | (7, 3)      | (7, 1)      | (7, -1)     | (7, -3)     |
| **3**         | (5, 5)      | (5, 3)      | (5, 1)      | (5, -1)     | (5, -3)     |
| **5**         | (3, 5)      | (3, 3)      | (3, 1)      | (3, -1)     | (3, -3)     |
| **7**         | (1, 5)      | (1, 3)      | (1, 1)      | (1, -1)     | (1, -3)     |
| **9**         | (-1, 5)     | (-1, 3)     | (-1, 1)     | (-1, -1)    | (-1, -3)    |

Each cell contains two numbers: the first is the Sultan’s payoff, and the second is the Sheikh’s payoff.

### Part (b): Nash Equilibria
To derive the Nash equilibria, we need to analyze the strategic choices where no player has an incentive to deviate from their chosen bid, given the other player's strategy.

1. **Sultan's Best Responses**: The Sultan will prefer bidding 1, 3, or 5 when the Sheikh bids any number greater than 6. However, bidding 7 or 9 gives him negative payoffs when the Sheikh bids high, so the Sultan avoids these unless he knows the Sheikh will bid very low.
2. **Sheikh's Best Responses**: The Sheikh should bid low (e.g., 2 or 4) when the Sultan bids 1 or 3, to maximize his payoff, but he should avoid high bids when the Sultan is also bidding high.

To formally identify Nash equilibria, we look for cells where both players’ strategies are mutual best responses. For example, if the Sheikh bids 2, the Sultan’s best response is 1, since this maximizes his payoff. Likewise, if the Sultan bids 1, the Sheikh has no better option than bidding 2.

One Nash equilibrium is:
- **(Sultan: 1, Sheikh: 2)** with payoffs of (7, 5).

Would you like a further explanation or more examples of Nash equilibria? Feel free to ask for clarifications!

### 5 Related Questions:
1. What would happen if the Sultan and Sheikh both had odd or even bid restrictions?
2. How would the outcome change if both bidders' valuations were identical?
3. Can there be multiple Nash equilibria in this type of auction? Why or why not?
4. How would this auction change if the bidders had more than five bidding options?
5. How would introducing a second-price auction affect the strategic form?

### Tip:
In first-price auctions, bidding slightly below your valuation is often optimal since overbidding can lead to negative payoffs if you win.

Queen Elizabeth has decided to auction off the crown jewels. Sultan Hassanal Bolkiah and Sheikh Zayed Bin Sultan Al Nahyan are the two bidders. The Sultan’s bid must be an odd number between 1 and 9, while the Sheikh’s bid must be an even number between 2 and 10. The highest bidder wins the jewels and pays their own bid price. The Sultan values the jewels at 8 (hundreds of millions of pounds), while the Sheikh values them at 7. What is the strategic form of this game, and what are the Nash equilibria?

Explore the strategic form of a first-price auction between the Sultan and the Sheikh, where the Sultan bids odd numbers and the Sheikh bids even numbers. Learn how to calculate the Nash equilibria and understand the bidding strategies in this game theory problem.

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