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Ray Atkinson, Graduate Student

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3). Consumers in a certain state can choose between three long-distance

Resolved Question:

3). Consumers in a certain state can choose between three long-distance telephone services: GTT, NCJ, and Dash. Aggressive marketing by all three companies results in continual shift of customers among the three services. Each year, GTT loses 30% of its customers to NCJ and 15% to Dash, NCJ loses 25% to GTT and 25% to Dash, and Dash loses 25% of its customers to GTT and 20% to NCJ. Assume that these percentages remain valid over a long period of time, what is each company's expected market share in the long run?

4). An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 20% of those in the poor category are moved to satisfactory and 5% of those in the satisfactory category are moved to preferred. Also, 5% in the preferred category are moved to the satisfactory, and 5% 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 = Satisfactory = Preferred = (round to the nearest tenth).

6). 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 60% of the population, Incline Village 18%, and South Lake Tahoe 22%. 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.

Tahoe City Incline Village South Lake Tahoe

Tahoe City 50% ? ? Incline Village ? 50% ? S.Lake Tahoe ? ? 50%

9). A town has two banks, bank R and bank C, which competeabout equally for the towns business. Each week each bank decidesto use only one means of promotion: TV, Radio, newspaper, or by mail. A research firm provided the given payoff matrix, which indicates the percentage of market gain or loss for each choice of the action by R and by C (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*= [????] (write fractions or integers).

What is the game value?

10). You have inherited $10,000 just prior to a presidential election and wish to invest it in solar energy and oil stocks. An investment advisory provides you with a payoff matrix that indicates yout 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 x amount in solar energy invest x amount in oil (round to the nearest dollar)

11). Player R has $2, a $5, and a $20 bill. Player C has a $1, a $10, and a $20 bill. Each player selects and shows (stimutaneously) 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 be?)

a). set up the payoff matrix for the game b). solve the game using the simplex method. eliminate any recessive rows or columns . (typr integers or fractions. simplify answer) P*= [???] Q*=[?] [ ?] [?] V= ? which player would you rather be?

7). A town has only two banks, R and C, and both compete equally for the towns 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 C (we assume that any gain by R is a loss by C, and vice versa). TV radio newspaper mail

Tv 1 1 0 3 radio -2 1 1 -2 newspaper 0 2 -1 1 mail -2 2 1 -1

Let me get done what I can see. There appear to be other parts that I cannot see yet.

Ray Atkinson :

Being MML, I assume that you do not need to show work, correct?

Customer:

correct

Customer:

some parts you can't see but its okay

Ray Atkinson :

Okay. I will get each one done as I see it, okay?

Customer:

okay

Ray Atkinson :

The results for #3 is [35.7, 35.7, 28.6]

Customer:

okay...that's correct.

Ray Atkinson :

#4 is [11.1, 44.4, 44.4]

Ray Atkinson :

Incidentally, the matrix is [[.80, .20, 0][.05, .90, .05][0, .05, .95]]

Ray Atkinson :

#6 is a bit of a headache. Give me some time.

Ray Atkinson :

Ive done problems like it before, and it is very easy to make a mistake. Ill have to check this

Customer:

okay...tell me about it. I have been working at this for 12 hours now.

Ray Atkinson :

the matrix for #6 is 50 71 69 29 50 48 31 52 50

Ray Atkinson :

Let me find the saddle point.

Ray Atkinson :

Both will set up shop in Tahoe City.

Ray Atkinson :

To find the saddle point, circle the lowest in each row, then box the highest in each column. If one number is XXXXX and boxed, that is the saddle point.

Customer:

So far so good.

Ray Atkinson :

The circled numbers are the first column, and the boxed numbers are the first row, so the 50 in cell (1, 1) is the answer.

Customer:

Don't need the saddle point for #6.

Ray Atkinson :

The saddle point is how I got the answer.

Customer:

oh.

Ray Atkinson :sorry. My browser crashed on me. Im still working on #7.

Ray Atkinson :The first is for C and the second is for R. Make sure the right one goes in the right spot

Ray Atkinson :for #8, the row strategy is [.6 0 .4 0] and the column strategy is [.8 0 .2 0]

Ray Atkinson :Umm. That was #9

Ray Atkinson :Oh, no. Hold on. I may have done that one backwards.

Customer:

#7 is P* = ???? the answer you gave is not correct.

Ray Atkinson :I did. Switch them. Row is [.8 0 .2 0] and Column is [.6 0 .4 0]

Ray Atkinson :Switch the answers for #7.

Ray Atkinson :#8 is Row [.8 0 .2 0] and Column [.6 0 .4 0]

Ray Atkinson :#10 is $8000 and $2000, earning $4600

Ray Atkinson :(Sorry, make that change to #9, instead of #8)

Customer:

[.8 0 .2 0] is for #7? If so, its not accurate.

Ray Atkinson :no, #7 is [0, .6, .4, 0] and [0, .4, .6, 0]

Ray Atkinson :#9 is [.8, 0, .2, 0] and [.6, 0, .4, 0]

Ray Atkinson :The matrix for #11 is[-2 10 20][1 -5 -5][-20 10 20]

Ray Atkinson :It appears that the value of the game is zero. Over the long haul, they would come out even. Odd

Ray Atkinson :My problem with #7 and #9 is that I am not familiar with the notation you are given. I have the numbers right, but I do not know what format your program wants.

Ray Atkinson :I believe the numbers I am giving for the rows is what they call P*

Ray Atkinson :If they are coming up wrong, switch the answers.

Customer:

# XXXXX and 9 are not accurate.

Ray Atkinson :did you switch the answers?

Ray Atkinson :Here. Let me redo them.

Ray Atkinson :P* is [0, 0.6, 0.4, 0]

Ray Atkinson :Q* is [0, 0.4, 0.6, 0]

Ray Atkinson :The value is 0.80

Customer:

I did and they were wrong.

Ray Atkinson :did the problem change? Sometimes MML can do that.

Customer:

okay for # XXXXX E = ? (type a fully reduced fraction or mixed number)

Customer:

It did not change... with two attempts it will automatically mark the ? wrong.

Ray Atkinson :Oh! [0 3/5 2/5 0]

Ray Atkinson :#9 is [4/5 0 1/5 0]

Ray Atkinson :The second for #7 is [0 2/5 3/5 0]

Ray Atkinson :the second for #9 is [3/5 0 2/5 0]

Ray Atkinson :This is why I *hate* MML test. If it wants to see x+y and you put in y+x, it is wrong.

Customer:

I know now because the previous answer were marked incorrectly.

Customer:

lol

Ray Atkinson :If MML keeps your answers, take them to your instructor and get credit. I once had a student do a MML testr with my help and got a 65%. They were quite upset - until they took it to their instructor and ended up with a 97% from the right answers MML marked wrong.

Customer:

Some instructors, not this one.

Ray Atkinson :Oy.

Customer:

Don't know..guess she is hard.

Ray Atkinson :Do you have all the answers now, or are some parts still missing?

Customer:

At this point, I am at an 80%. So, I will accept it and start my final exam, which I only have 3.5 hours to complete.

Customer:

Thanks!

Customer:

Some answers, not all.

Ray Atkinson :What parts are still missing?

Customer:

#7 have 2 parts missing.... what is the expected value of the game for R if the bank R always chooses tv and bank C uses its optimum strategy?

Customer:

sorry... E = ?

Ray Atkinson :Ok, give me a sec

Customer:

fully reduced-fraction or mixed number

Ray Atkinson :-2

Ray Atkinson :If C knows what R is going to do, R will lose 2 every time.

Customer:

If R chooses tv it will choose row one...-2 no good

Customer:

If R chooses tv, it will always choose row 1

Ray Atkinson :Hold on.

Ray Atkinson :Oh, you're right! the E value is -0.8 = -4/5

Ray Atkinson :I thought it meant that C knew what R was doing.

Ray Atkinson :If C does not know, then it picks Radio 40% and paper 60%

Ray Atkinson :-2*0.4 + 0*0.6 = -0.8

Ray Atkinson :What else is missing from #7?

Customer:

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

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