Question 1198162: State University is about to play Ivy College for the state tennis championship. The State team has two players (A and B), and the Ivy team has three players (X, Y, and Z). The following facts are known about the players’ relative abilities: X will always beat B; Y will always beat A; A will always beat Z. In any other match, each player has a ½ chance of winning. Before State plays Ivy, the State coach must determine who will play first singles and who will play second singles. The Ivy coach (after choosing which two players will play singles) must also determine who will play first singles and second singles. Assume that each coach wants to maximize the expected number of singles matches won by the team. Use game theory to determine optimal strategies for each coach and the value of the game to each team
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! Certainly, let's analyze this tennis championship scenario using game theory.
**1. Define the Game**
* **Players:** State Coach and Ivy Coach
* **Strategies:**
* **State Coach:**
* **Strategy 1:** Player A plays first singles, Player B plays second singles.
* **Strategy 2:** Player B plays first singles, Player A plays second singles.
* **Ivy Coach:**
* **Strategy 1:** Player X plays first singles, Player Y plays second singles.
* **Strategy 2:** Player X plays first singles, Player Z plays second singles.
* **Strategy 3:** Player Y plays first singles, Player X plays second singles.
* **Payoffs:**
* The payoff to the State Coach is the expected number of singles matches won by the State team.
* The payoff to the Ivy Coach is the expected number of singles matches won by the Ivy team.
**2. Construct the Payoff Matrix**
| State Coach \ Ivy Coach | Strategy 1 (X, Y) | Strategy 2 (X, Z) | Strategy 3 (Y, X) |
|---|---|---|---|
| **Strategy 1 (A, B)** | 0.5 | 1 | 0 |
| **Strategy 2 (B, A)** | 1 | 0.5 | 1 |
* **Explanation:**
* **Row 1, Column 1 (A, B) vs. (X, Y):**
* A loses to Y (0 points for State).
* B loses to X (0 points for State).
* **Total: 0 points**
* **Row 1, Column 2 (A, B) vs. (X, Z):**
* A loses to Y (0 points for State).
* B beats Z (1 point for State).
* **Total: 1 point**
* **Row 1, Column 3 (A, B) vs. (Y, X):**
* A loses to Y (0 points for State).
* B loses to X (0 points for State).
* **Total: 0 points**
* **Row 2, Column 1 (B, A) vs. (X, Y):**
* B loses to X (0 points for State).
* A beats Z (1 point for State).
* **Total: 1 point**
* **Row 2, Column 2 (B, A) vs. (X, Z):**
* B loses to X (0 points for State).
* A loses to Y (0 points for State).
* **Total: 0 points**
* **Row 2, Column 3 (B, A) vs. (Y, X):**
* B loses to X (0 points for State).
* A beats Z (1 point for State).
* **Total: 1 point**
**3. Determine Optimal Strategies**
* **State Coach:**
* Strategy 2 (B, A) dominates Strategy 1, as it yields higher or equal payoffs in all cases.
* **Optimal Strategy:** Play B first and A second.
* **Ivy Coach:**
* Strategy 1 (X, Y) dominates Strategy 2 and Strategy 3, as it yields higher or equal payoffs in all cases.
* **Optimal Strategy:** Play X first and Y second.
**4. Determine the Value of the Game**
* With these optimal strategies:
* State Coach expects to win 1 match out of 2.
* Ivy Coach expects to win 1 match out of 2.
* **Value of the Game:** 0.5
**Conclusion:**
* The optimal strategy for the State Coach is to play Player B first and Player A second.
* The optimal strategy for the Ivy Coach is to play Player X first and Player Y second.
* The value of the game is 0.5, indicating that both teams can expect to win one of the two singles matches on average.
This analysis demonstrates how game theory can be used to determine the best course of action for each coach in this competitive scenario.
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