
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Question
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_thumbnail.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.
Expert Solution

arrow_forward
Step 1
Given,
We have to calculate the expected payoff of the game with the given above payoff matrix using the mixed strategies supplied.
Step by stepSolved in 2 steps

Knowledge Booster
Similar questions
- helparrow_forwardScenario. You, as a property investor, are interested in understanding which factor(or factors) drives the prices of investment properties. A dataset is collected whichcontains the prices (in thousand dollars, as denoted by apart price) for 50 one-bedroom apartments in city X, their corresponding rents per week (in dollars, asdenoted by rent) and the costs to hold each of these properties per week (in dollars,as denoted by cost of property). Following the procedures below to analyse thedataset ’assign2 data.csv’ by using Rstudio. Please only include relevant outputsfrom Rstudio in your solution and attach the R codes as appendice ( forattach R codes). (a). Import the data into Rstudio, draw two scatter plots: apart priceversus rent and apart price versus cost. (b). Fit the following two linear models:Model 1: apart price = b0 + b1 × rentModel 2: apart price = c0 + c1 × costWrite down the equations of the two models with correct coefficients. (c). Written down the p-values from the…arrow_forward(PUBLIC HEALTH) What is an example of a deterministic model of causality and probabilistic (stochastic) causality?arrow_forward
- For the payoff matrix shown , find the optimal strategy for the row and column player and then find expected payoff P = 4 -3 -2 2arrow_forwardASAP PLEASE If your trading systerm has a losing percentage of 45%, how likely it is to have a series of four losses in a row? 1% 2.25% 45% 4.10%arrow_forwardDevarrow_forward
- How do I solve the following: You are an investment manager for Simple Asset Management, a company that specializes in developing simple investment portfolios consisting of no more than three assets such as stocks, bonds, etc., for investors who like to keep things simple. One of your more popular investments is called the All World Fund and is composed of global stocks with good dividend yields. A client is interested in constructing a portfolio that consists of the All World Fund and the Treasury Index Fund, which consists of U.S. Treasury securities (government bonds). You calculate a 7.8% expected return on the All World Fund with a return standard deviation (a measure of risk) of 18.90%. The expected return of the Treasury Index Fund is 5.50% with a return standard deviation of 4.6%. To analyze the relationship between the two investments, you also calculate the covariance between the two of –12.4. Which graph below best represents the expected returns for the…arrow_forwardA politician can emphasize jobs or the environment in her election campaign. The voters can be concerned about jobs or the environment. A payoff matrix showing the utility of each possible outcome is shown. The political analysts feel there is a 0.51 chance that the voters will emphasize jobs. Voters Jobs Environment +25 Jobs - 10 Environment - 15 +30 Which strategy should the candidate adopt? What is the expected utility? Which should the candidate emphasize to gain the highest utility? Jobs Environment What is the expected utility of the strategy chosen in the previous step? 7.05 (Type an integer or a decimal.)arrow_forwardplease answer in text form and in proper format answer with must explanation , calculation for each part and steps clearlyarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Mathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,

Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education

Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,

