Question 1198577: 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 CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Certainly, let's analyze this tennis match 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 each team is the expected number of singles matches won.
* We'll represent the payoffs in a matrix, with State Coach's strategies as rows and Ivy Coach's strategies as columns.
**2. Construct the Payoff Matrix**
| | Ivy Coach: Strategy 1 (X, Y) | Ivy Coach: Strategy 2 (X, Z) | Ivy Coach: Strategy 3 (Y, X) |
|-----------|--------------------------|--------------------------|--------------------------|
| State: Strategy 1 (A, B) | 0.5 | 1 | 0 |
| State: Strategy 2 (B, A) | 0 | 0.5 | 1 |
* **Explanation of Payoffs (Example):**
* **Row 1, Column 1 (State: A, B vs. Ivy: X, Y):**
* A loses to Y (0 points for State)
* B loses to X (0 points for State)
* Total points for State: 0
**3. Find the Optimal Strategies**
* **Identify Dominant Strategies:**
* A dominant strategy is one that always yields a better payoff for a player, regardless of what the opponent does.
* **State Coach:**
* If Ivy plays Strategy 1 (X, Y), State's best response is Strategy 2 (B, A) to win the second singles match.
* If Ivy plays Strategy 2 (X, Z), State's best response is Strategy 1 (A, B) to win the second singles match.
* If Ivy plays Strategy 3 (Y, X), State's best response is Strategy 2 (B, A) to win both singles matches.
* **No dominant strategy for the State Coach.**
* **Ivy Coach:**
* If State plays Strategy 1 (A, B), Ivy's best response is Strategy 3 (Y, X) to win both singles matches.
* If State plays Strategy 2 (B, A), Ivy's best response is Strategy 1 (X, Y) to win both singles matches.
* **No dominant strategy for the Ivy Coach.**
* **Find the Nash Equilibrium:**
* A Nash Equilibrium is a set of strategies where no player can unilaterally improve their payoff by changing their strategy, given the other player's strategy.
* In this game, there is **no pure-strategy Nash Equilibrium**.
* **Mixed Strategies:**
* Since there is no pure-strategy Nash Equilibrium, the coaches might consider mixed strategies (randomizing their choices).
**4. Value of the Game**
* Since there is no clear dominant strategy or pure-strategy Nash Equilibrium, the exact value of the game is difficult to determine without further analysis.
**Conclusion**
This tennis match scenario presents a complex game with no obvious dominant strategies or pure-strategy Nash equilibria. The coaches might need to employ mixed strategies (randomizing their player orders) and potentially use game theory concepts like minimax strategies to determine the optimal playing order and the expected value of the game.
**Note:**
* This analysis assumes that the players perform consistently according to their relative strengths.
* Psychological factors and unexpected performances can also influence the outcome of the matches.
I hope this analysis provides a good understanding of the game theory concepts involved in this tennis match scenario!
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