Optimal strategies, Game value, Payoff matrix solution tutorial

# Optimal strategies, Game value, Payoff matrix solution tutorial

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A television campaign is conducted during the football season to promote a well-known brand X shaving cream. For each of several weeks, a survey is made, and it is found that each Week, 60% of those using brand X continue to use it and 40% switch to another brand. It is also found that of those not using brand X, 40% switch to brand X While the other 60% continue using another brand. a) Draw a transition diagram. b) Write the transition matrix.

c) If 40% of the people are using brand X at the start of the advertising campaign, what percentage will be using it 1 week later? 2 Weeks later?

The 1990 census reported that 20% of the households in Middletown were homeowners and the remainder were renters. During the next decade, 3% of the homeowners became renters and the rest continued to be homeowners. Similarly, 11% of the renters became homeowners and the rest continued to rent.

(A) Fill in the appropriate transition matrix for this process. Homeowners Renters (B) According to this transition matrix, what percentage of households were homeowners in 2000? (Round to the nearest tenth of a percent.)

(C) If the transition matrix remains the same, What percentage of the households will be homeowners in 2010?

Consumers in a certain state can choose between three long-distance telephone services: GTT, NC], and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 20% of its customers to NC] and 15% to Dash, NC] loses 20% of its customers to GTT and 15% to Dash, and Dash loses 10% of its customers to GTT and 5% to NC]. Assuming that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?

GTT's expected market share is U%. (Round to the nearest tenth as needed.) NC] 's expected market share is (Round to the nearest tenth as needed.) Dash's expected market share is U%. (Round to the nearest tenth as needed.)

An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 35% of those in the poor category are moved to satisfactory and 10% of those in the satisfactory category are moved to preferred. Also, 10% in the preferred category are moved to the satisfactory category, and 10% of those in the satisfactory category are moved to the poor category. Customers are never moved from poor to preferred, or conversely, in a single year. Assuming these percentages remain valid over a long period of time, how many customers can the company expect to have in each category in the long run? Poor = U% (Round to the nearest tenth if necessary.) Satisfactory = (Round to the nearest tenth if necessary.) Preferred = (Round to the nearest tenth if necessary.)

Once a year employees at a company are given the opportunity to join one of three pension plans, A, B, or C. Once an employee decides to join one of these plans, the employee cannot drop the plan or switch to another plan. Past records indicate that each year 10% of the employees elect to join plan A, 9% elect to join plan B, 12% elect to join plan C, and the remainder to not join any plan. (1) In the long run, What percentage of employees Will elect to join plans A, B, and C? (2) On average, how many years will it take an employee to decide to join a plan?

(1) of employees will elect to join plan A. (Type an integer or decimal. Round to the nearest tenth.)

of employees will elect to join plan B. (Type an integer or decimal. Round to the nearest tenth.)

of employees will elect to join plan C. (Type an integer or decimal. Round to the nearest tenth.) (2) It will take U years, on average, for an employee to decide to join a plan. (Round to the nearest integer.)

Two equally competitive pet shops Want to locate stores in Lake Tahoe, Where there are currently none. There are three main business centers. Tahoe City serves l9% of the population, Incline Village 44% and South Lake Tahoe 37%. If both shops locate in the same center, they split all the business equally; if they locate in different centers, they each get all the business in the center in which they locate plus half the business in the third center. Where should the pet shops locate? Set up a game matrix and solve.

Enter the values for the game matrix. The first pet shop should locate in? The second pet shop should locate in?

A town has only two banks, R and C, and both compete C about equally for the town's business. Each week each bank decides on the use of one of the following means of  promotion: TV, radio, newspaper, and mail. A marketing research firm provided the following payoff matrix, which indicates the percentage of market gain or loss for each choice of action by R and by C (we assume that any gain by R is a loss by C, and vice versa)

(A) Find the optimal strategy P * . Hint: if a row was deleted, the probability of choosing that row is zero.

Find the optimal strategy Q *

What is the value v of the game? v = (type a fully-reduced fraction or mixed number.)

(B) What is the expected value of the game for R if the bank R always chooses TV and bank C uses its optimum strategy?

(C) What is the expected value of the game for R if bank C always chooses radio and bank R uses its optimum strategy?

(D) What is the expected value of the game for R if both banks always use the paper?

You have inherited $10,000 just prior to a presidential election and you wish to invest it in solar energy and oil stocks. An investment advisor provides you with a payoff matrix that indicates your probable 4-year gains, depending on which party comes into What should you invest your money in so that you would have the largest expected gain irrespective of l10w the election turns out? The amount you should invest in solar energy is (Round to two decimal places.) The amount you should invest in oil is ESE. (Round to two decimal places.) A town has two banks, bank R and bank C, which compete about equally for the town's business. Each Week each bank decides to use only one means of promotion: TV, radio, newspaper, or mail. A research firm provided the given payoff matrix, which indicates the percentage of market gain or loss for each choice of action by R and by C (We assume that any gain by R is a loss by C, and vice versa). What are the optimum strategies for R and C? R* = C * = What is the game value? You have inherited$10,000 just prior to a presidential election and wish to invest it in solar energy and oil stocks. An investment advisor provides you with a payoff matrix that indicates your probable 4-year gains, depending on which party comes into office. How should you invest your money so that you would have the largest expected gain irrespective of how the election turns out? Invest in solar energy.

Invest in oil. (Round to the nearest dollar.)

Player R has a $2, a$5, and a $10 bill. Player C has a$1, a $10, and a$20 bill. Each player selects and shows (simultaneously) one of their three bills. If the total value of the two bills shown is even, R Wins C's bill; if the value is odd, C wins R's bill. (Which player would you rather be?) (A) Set up the payoff matrix for the game.

(B) Solve the game using the simplex method. Eliminate any recessive rows or columns.

P* =

Q* =

v =

Which player would you rather be?

A store is about to order deluxe, standard and economy grade DVD players for next year's inventory. The state of the nation's economy (fate) during the year will be a factor on sales for that year. Records over the past 5 years show that if the economy is up, the store will net 3, 2, and 1 million dollars, respectively, on sales of deluxe, standard and economy grade models; if the economy is down, the company will net 3, 2, and 4 million dollars, respectively, on sales of deluxe, standard and economy grade models. (A) Set up a payoff matrix for this problem.

(B) Find optimal strategies for both the company and fate (the economy). What is the value of the game?

Find P*, the optimal strategy for the row player, the department store.

Find the optimal strategy for the column player, the economy (fate).

Find v, the value of the game.

(C) How should the company's budget be allocated to each grade of DVD player to maximize their return irrespective of what the economy does the following year?

(D) What is the expected value of the game to the company if it orders only deluxe DVD players and fate plays the strategy "Down"? If the company plays its optimal strategy and fate plays the strategy "Down"?

The expected value of the game to the company if it orders only deluxe DVD players and fate plays the strategy "Down" is (Type an integer or a simplified fraction.)

The expected value of the game to the company if it plays its optimal strategy and fate plays the strategy "Down" is (Type an integer or a simplified fraction.)

Optimal strategies

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Excerpt from file: Math Tutorial 1. a) b) A A c) From the information given we start with the following vector matrix, To find the amount after the first week I multiplied the matrices, The percentage of people using brand X 1 week after the start of the advertising campaign is 48% . At the beginning of week 1 we

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Excerpt from file: MGT506Module5SLP Visionary Leadership, Cross-Cultural Leadership, Facilitating Change The purpose of the Session Long Project in Trident University International classes is to give you the opportunity to explore the applicability of the Module to your own life, work, and place in space and time,

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