Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Calculate the expected payoff of the game with the given payoff matrix using the mixed strategies supplied.
![**Title:** Calculating the Expected Payoff using a Payoff Matrix and Mixed Strategies
**Objective:** Learn how to calculate the expected payoff of a game using a given payoff matrix alongside mixed strategies.
**Payoff Matrix (P):**
\[
P = \begin{bmatrix}
2 & 0 & -1 & 2 \\
-1 & 0 & 0 & -2 \\
-2 & 0 & 0 & 1 \\
3 & 1 & -1 & 1
\end{bmatrix}
\]
**Mixed Strategies Provided:**
- Row player's strategy: \( R = [0, 0.5, 0, 0.5] \)
- Column player's strategy: \( C = \begin{bmatrix} 0.5 \\ 0.5 \\ 0 \\ 0 \end{bmatrix} \)
**Instructions:**
1. **Understanding the Matrix:** Each element in the payoff matrix represents the payoff from the row player's perspective for each combination of strategies by the row and column players.
2. **Calculate the Expected Payoff:** Use the formula for the expected payoff involving matrix multiplication. Multiply the row player's strategy vector \( R \) by the payoff matrix \( P \), and then by the column player's strategy vector \( C \).
3. **Application:** This calculation helps in determining the average outcome both players can expect if they adhere to their provided mixed strategies.
**Details:**
The expression to compute the expected payoff \( E \) is:
\[
E = R \times P \times C
\]
Perform matrix multiplication to determine \( E \).
**Conclusion:** By following these steps, you can determine the expected payoff when two players engage in a strategic game using mixed strategies. This understanding is crucial for strategic decision-making in competitive environments.](https://content.bartleby.com/qna-images/question/9a670ca7-8bea-4835-b14f-d42c1bdbe3d1/870d4a2c-787a-43a7-b1d4-f13f69593773/mdnzq3p_processed.png)
Transcribed Image Text:**Title:** Calculating the Expected Payoff using a Payoff Matrix and Mixed Strategies
**Objective:** Learn how to calculate the expected payoff of a game using a given payoff matrix alongside mixed strategies.
**Payoff Matrix (P):**
\[
P = \begin{bmatrix}
2 & 0 & -1 & 2 \\
-1 & 0 & 0 & -2 \\
-2 & 0 & 0 & 1 \\
3 & 1 & -1 & 1
\end{bmatrix}
\]
**Mixed Strategies Provided:**
- Row player's strategy: \( R = [0, 0.5, 0, 0.5] \)
- Column player's strategy: \( C = \begin{bmatrix} 0.5 \\ 0.5 \\ 0 \\ 0 \end{bmatrix} \)
**Instructions:**
1. **Understanding the Matrix:** Each element in the payoff matrix represents the payoff from the row player's perspective for each combination of strategies by the row and column players.
2. **Calculate the Expected Payoff:** Use the formula for the expected payoff involving matrix multiplication. Multiply the row player's strategy vector \( R \) by the payoff matrix \( P \), and then by the column player's strategy vector \( C \).
3. **Application:** This calculation helps in determining the average outcome both players can expect if they adhere to their provided mixed strategies.
**Details:**
The expression to compute the expected payoff \( E \) is:
\[
E = R \times P \times C
\]
Perform matrix multiplication to determine \( E \).
**Conclusion:** By following these steps, you can determine the expected payoff when two players engage in a strategic game using mixed strategies. This understanding is crucial for strategic decision-making in competitive environments.
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