# Two player game, Decision alternatives

1. Here is the description of a two player game. Player A selects a, b, or c. Player B selects i, ii, or iii.

Player B

i ii iii

a 5 7 2

Player A b 4 8 5

c 6 4 3

Payoffs are to Player A. This is a zero sum game.

a.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Indicate any dominated strategies and restate the game appropriately

b.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Determine the optimal strategies for the players

c.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â What is the value of the game?

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2. You are the general manager of Keystone Chemicals Houston plant. Following a recent visit by corporate officials, you have received a memo directing you to upgrade plant security. The memo requires you upgrade protection in three areas:

1.Â Â Â Â Â Protection against forced entry and theft/destruction (F)

2.Â Â Â Â Â Protection against entry by subterfuge and theft (S)

3.Â Â Â Â Â Protection of trade secrets and proprietary information (T)

The memo also cautions that any measures taken must be carefully evaluated for their cost-effectiveness (C).

After careful research, you have determined that there are three responses that could satisfactorily address the mandate you have received. Each addresses the three areas differently and each has a different cost. The three responses are:

1.Â Â Â Â Â Improve the electronic surveillance at the plant (El)

2.Â Â Â Â Â Hire a security service for foot and K-9 patrols. (Out)

3.Â Â Â Â Â Improve in-house security staffing (personnel levels and training). (In)

You have rated the alternative responses in terms of how well they address the areas of concern and their cost. You have assigned the weights (suggesting relative strength of concern) to the areas and the cost. The table summarizing what you have done follows.

Weights -> 3 3 4 6

C F S T

El 7 2 8 5

Out 6 7 5 3

In 5 6 7 5

a. Which choice is indicated by his ratings?

b. Is an alternative dominated? why?

c. What is the weight for S that will produce indifference between the two top ranked choices? Write the algebraic expression and solve it.

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4. Ace Venture Capital (AVC) has a million dollars to invest. It is considering two proposals. One proposal is to invest with the operator of retail shoe stores. The investment will produce a majority stake in a corporation that owns three retail outlets. These will be managed by the entrepreneur who is also investing $1 million in the corporation. The other proposal is to invest in a software company that provides learning space on the internet for Universities that provide on-line education. After evaluating both business plans the projected performance of each alternative investment (% returns to AVC) is seen to depend on the state of the economy over the next 5 years. This information is summarized in the following table.

Strong Economic Growth Modest Economic Growth

Shoes 17% 13%

Software 25% 8%

Taking the position of AVC,

a. If you are uncertain about the future state of the economy, which alternative should be chosen?

b. What probability of Strong Economic Growth makes you indifferent between the investments?

c. Stated as a percentage, what would you be willing to pay for perfect information about the future growth

https://brainmass.com/business/entrepreneurial-issues/two-player-game-decision-alternatives-14807

## SOLUTION This solution is **FREE** courtesy of BrainMass!

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1. Here is the description of a two player game. Player A selects a, b, or c. Player B selects i, ii, or iii.

Player B

i ii iii

a 5 7 2

Player A b 4 8 5

c 6 4 3

Payoffs are to Player A. This is a zero sum game.

a. Indicate any dominated strategies and restate the game appropriately

b. Determine the optimal strategies for the players

c. What is the value of the game?

2. You are the general manager of Keystone Chemicals Houston plant. Following a recent visit by corporate officials, you have received a memo directing you to upgrade plant security. The memo requires you upgrade protection in three areas:

1. Protection against forced entry and theft/destruction (F)

2. Protection against entry by subterfuge and theft (S)

3. Protection of trade secrets and proprietary information (T)

The memo also cautions that any measures taken must be carefully evaluated for their cost-effectiveness (C).

After careful research, you have determined that there are three responses that could satisfactorily address the mandate you have received. Each addresses the three areas differently and each has a different cost. The three responses are:

1. Improve the electronic surveillance at the plant (El)

2. Hire a security service for foot and K-9 patrols. (Out)

3. Improve in-house security staffing (personnel levels and training). (In)

You have rated the alternative responses in terms of how well they address the areas of concern and their cost. You have assigned the weights (suggesting relative strength of concern) to the areas and the cost. The table summarizing what you have done follows.

Weights -> 3 3 4 6

C F S T

El 7 2 8 5

Out 6 7 5 3

In 5 6 7 5

a. Which choice is indicated by his ratings?

b. Is an alternative dominated? why?

c. What is the weight for S that will produce indifference between the two top ranked choices? Write the algebraic expression and solve it.

1. Here is the description of a two player game. Player A selects a, b, or c. Player B selects i, ii, or iii.

Player B

i ii iii

a 5 7 2

Player A b 4 8 5

c 6 4 3

Payoffs are to Player A. This is a zero sum game.

a. Indicate any dominated strategies and restate the game appropriately

b. Determine the optimal strategies for the players

c. What is the value of the game?

a. Indicate any dominated strategies and restate the game appropriately

If we observe ii and iii

ii iii

7 > 2

8 > 5

4 > 3

Thus player B will always choose iii over ii as it would mean smaller payoff to A

Thus ii is dominated strategy

Thus the game can be reststed thus

Player B

i iii

a 5 2

Player A b 4 5

c 6 3

Now if we observe rows a and c

a 5 2

< <

c 6 3

Thus Player A will always choose c over a as it will give him higher payoff

Thus a is dominated strategy

Thus we can restate the game thus

Player B

i iii

Player A b 4 5

c 6 3

b. Determine the optimal strategies for the players

Player B Row minimum

i iii

Player A b 4 5 4

c 6 3 3

Column maximum 6 5

Minimax= 5 = Minimum of (6,5)

Maximin= 4 = Maximum of (4,3)

There is no optimal strategy since Minimax is not equal to maximin

There is no saddle point

c. What is the value of the game?

Maximin â‰¤ Value of the game â‰¤ Minimax

4 â‰¤ Value of the game â‰¤ 5

2. You are the general manager of Keystone Chemicals Houston plant. Following a recent visit by corporate officials, you have received a memo directing you to upgrade plant security. The memo requires you upgrade protection in three areas:

1. Protection against forced entry and theft/destruction (F)

2. Protection against entry by subterfuge and theft (S)

3. Protection of trade secrets and proprietary information (T)

The memo also cautions that any measures taken must be carefully evaluated for their cost-effectiveness (C).

After careful research, you have determined that there are three responses that could satisfactorily address the mandate you have received. Each addresses the three areas differently and each has a different cost. The three responses are:

1. Improve the electronic surveillance at the plant (El)

2. Hire a security service for foot and K-9 patrols. (Out)

3. Improve in-house security staffing (personnel levels and training). (In)

You have rated the alternative responses in terms of how well they address the areas of concern and their cost. You have assigned the weights (suggesting relative strength of concern) to the areas and the cost. The table summarizing what you have done follows.

Weights -> 3 3 4 6

C F S T

El 7 2 8 5

Out 6 7 5 3

In 5 6 7 5

a. Which choice is indicated by his ratings?

Weights -> 3 3 4 6 Weighted total

C F S T

El 7 2 8 5 89 =3*7+3*2+4*8+6*5

Out 6 7 5 3 77 =3*6+3*7+4*5+6*3

In 5 6 7 5 91 =3*5+3*6+4*7+6*5

Therefore choose In (Improve in-house security staffing (personnel levels and training). (In))

as it gives the maximum value of 91

b. Is an alternative dominated? why?

No alternative is dominated

No row is greater than any other row for all the entries

(maximum are highlighted)

C F S T

El 7 2 8 5

Out 6 7 5 3

In 5 6 7 5

c. What is the weight for S that will produce indifference between the two top ranked choices? Write the algebraic expression and solve it.

The two top ranked choices are EI and In

Weights -> 3 3 4 6

C F S T

El 7 2 8 5 89 =3*7+3*2+4*8+6*5

Out 6 7 5 3 77 =3*6+3*7+4*5+6*3

In 5 6 7 5 91 =3*5+3*6+4*7+6*5

Let the weight for S be x

weighted total for EI= =3*7+3*2+x*8+6*5= 57 +8x

weighted total for In= =3*5+3*6+x*7+6*5= 63 +7x

Equating the two

57 +8x = 63 +7x

or x= 6

the weight for S that will produce indifference between the two top ranked choices= 6

Let us see whether this is the case

Weights -> 3 3 6 6

C F S T

El 7 2 8 5 105 =3*7+3*2+6*8+6*5

Out 6 7 5 3 87 =3*6+3*7+6*5+6*3

In 5 6 7 5 105 =3*5+3*6+6*7+6*5

This is indeed the case

4. Ace Venture Capital (AVC) has a million dollars to invest. It is considering two proposals. One proposal is to invest with the operator of retail shoe stores. The investment will produce a majority stake in a corporation that owns three retail outlets. These will be managed by the entrepreneur who is also investing $1 million in the corporation. The other proposal is to invest in a software company that provides learning space on the internet for Universities that provide on-line education. After evaluating both business plans the projected performance of each alternative investment (% returns to AVC) is seen to depend on the state of the economy over the next 5 years. This information is summarized in the following table.

Strong Economic Growth Modest Economic Growth

Shoes 17% 13%

Software 25% 8%

Taking the position of AVC,

a. If you are uncertain about the future state of the economy, which alternative should be chosen?

Strong Economic Growth Modest Economic Growth Row minimum

Shoes 17% 13% 13%

Software 25% 8% 8%

Column maximum 25% 13%

Minimax= 13% = Minimum of (25%,13%)

Maxmin= 13% = Maximum of (13%,8%)

Thus if one is uncertain about the future state of economy he should choose Shoes

as a minimum of 13% return is assured

b. What probability of Strong Economic Growth makes you indifferent between the investments?

probability p 1-p

Strong Economic Growth Modest Economic Growth

Shoes 17% 13%

Software 25% 8%

Let the probability of strong economic growth be p

Therefore the probability of modest economic growth is ( 1- p)

Return on investing in shoes= p(17%) + (1-p)(13%)= p(4%) + 13 %

Return on investing in software= p(25%) + (1-p)(8%)= p(17%) + 8 %

Equating the two

p(4%) + 13 % = p(17%) + 8 %

or p(13%)= 5%

0r p= 38.46% =5%/13%

Thus the probability of Strong Economic Growth to makes you indifferent between the investments= 38.46%

We can see that such is the case

Multiply the probability by the growth

p 1-p

probability 38.46% 61.54%

Strong Economic Growth Modest Economic Growth

Shoes 17% 13% 14.54% =38.46%*17.%+61.54%*13.%

Software 25% 8% 14.54% =38.46%*25.%+61.54%*8.%

Stated as a percentage, what would you be willing to pay for perfect information about the future growth

If we have the perfect information

probability decision return probability X return

Strong Economic Growth 38.46% Software 25% 9.62% =38.46%*25.%

Modest Economic Growth 61.54% Shoes 13% 8.00% =61.54%*13.%

Total= 17.62%

difference between the returns when perfect information is there and where there is no information

3.08% =17.62%-14.54%

Thus the maximum amount that can be paid for perfect information = 3.08% Answer

4. Ace Venture Capital (AVC) has a million dollars to invest. It is considering two proposals. One proposal is to invest with the operator of retail shoe stores. The investment will produce a majority stake in a corporation that owns three retail outlets. These will be managed by the entrepreneur who is also investing $1 million in the corporation. The other proposal is to invest in a software company that provides learning space on the internet for Universities that provide on-line education. After evaluating both business plans the projected performance of each alternative investment (% returns to AVC) is seen to depend on the state of the economy over the next 5 years. This information is summarized in the following table.

Strong Economic Growth Modest Economic Growth

Shoes 17% 13%

Software 25% 8%

Taking the position of AVC,

a. If you are uncertain about the future state of the economy, which alternative should be chosen?

b. What probability of Strong Economic Growth makes you indifferent between the investments?

Stated as a percentage, what would you be willing to pay for perfect information about the future growth

https://brainmass.com/business/entrepreneurial-issues/two-player-game-decision-alternatives-14807