Quantitative Methods for Business 12th Edition – Test Bank

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Chapter 5—Utility and Game Theory

MULTIPLE CHOICE

1. When consequences are measured on a scale that reflects a decision maker’s attitude toward profit,

loss, and risk, payoffs are replaced by

a. utility values.

b. multicriteria measures.

c. sample information.

d. opportunity loss.

ANS: A PTS: 1 TOP: Meaning of utility

2. The purchase of insurance and lottery tickets shows that people make decisions based on

a. expected value.

b. sample information.

c. utility.

d. maximum likelihood.

ANS: C PTS: 1 TOP: Introduction

3. The expected utility approach

a. does not require probabilities.

b. leads to the same decision as the expected value approach.

c. is most useful when excessively large or small payoffs are possible.

d. requires a decision tree.

ANS: C PTS: 1 TOP: Expected utility approach

4. Utility reflects the decision maker’s attitude toward

a. probability and profit

b. profit, loss, and risk

c. risk and regret

d. probability and regret

ANS: B PTS: 1 TOP: Meaning of utility

5. Values of utility

a. must be between 0 and 1.

b. must be between 0 and 10.

c. must be nonnegative.

d. must increase as the payoff improves.

ANS: D PTS: 1 TOP: Developing utilities for monetary payoffs

6. If the payoff from outcome A is twice the payoff from outcome B, then the ratio of these utilities will

be

a. 2 to 1.

b. less than 2 to 1.

c. more than 2 to 1.

d. unknown without further information.

ANS: D PTS: 1 TOP: Meaning of utility7. 8. 9. 10. 11. 12. 13. The probability for which a decision maker cannot choose between a certain amount and a lottery

based on that probability is

a. the indifference probability.

b. the lottery probability.

c. the uncertain probability.

d. the utility probability.

ANS: A PTS: 1 TOP: Developing utilities for monetary payoffs

A decision maker has chosen .4 as the probability for which he cannot choose between a certain loss of

10,000 and the lottery p(25000) + (1  p)(5000). If the utility of 25,000 is 0 and of 5000 is 1, then the

utility of 10,000 is

a. .5

b. .6

c. .4

d. 4

ANS: B PTS: 1 TOP: Developing utilities for monetary payoffs

When the decision maker prefers a guaranteed payoff value that is smaller than the expected value of

the lottery, the decision maker is

a. a risk avoider.

b. a risk taker.

c. an optimist.

d. an optimizer.

ANS: A PTS: 1 TOP: Risk avoiders versus risk takers

A decision maker whose utility function graphs as a straight line is

a. conservative.

b. risk neutral.

c. a risk taker.

d. a risk avoider.

ANS: B PTS: 1 TOP: Risk avoiders versus risk takers

For a game with an optimal pure strategy, which of the following statements is false?

a. The maximin equals the minimax.

b. The value of the game cannot be improved by either player changing strategies.

c. A saddle point exists.

d. Dominated strategies cannot exist.

ANS: D PTS: 1 TOP: Identifying a pure strategy

Which of the following statements about a dominated strategy is false?

a. b. A dominated strategy will never be selected by a player.

A dominated strategy exists if another strategy is at least as good regardless of what the

opponent does.

c. d. A dominated strategy is superior to a mixed strategy.

A dominated strategy can be eliminated from the game.

ANS: C PTS: 1 TOP: Dominated strategies

A 3 x 3 two-person zero-sum game that has no optimal pure strategy and no dominated strategies

a. can be solved using a linear programming model.

b. can be solved algebraically.14. 15. 16. 17. 18. 19. c. can be solved by identifying the minimax and maximin values.

d. cannot be solved.

ANS: A PTS: 1 TOP: Larger mixed strategy games

For a two-person zero-sum game, which one of the following is false?

a. b. c. d. The gain for one player is equal to the loss for the other player.

A payoff of 2 for one player has a corresponding payoff of 2 for the other player.

The sum of the payoffs in the payoff table is zero.

What one player wins, the other player loses.

ANS: C PTS: 1 TOP: Introduction to game theory

If the maximin and minimax values are not equal in a two-person zero-sum game,

a. b. c. d. a mixed strategy is optimal.

a pure strategy is optimal.

a dominated strategy is optimal.

one player should use a pure strategy and the other should use a mixed strategy.

ANS: A PTS: 1 TOP: Mixed strategy games

If it is optimal for both players in a two-person, zero-sum game to select one strategy and stay with

that strategy regardless of what the other player does, the game

a. has more than one equilibrium point.

b. will have alternating winners.

c. will have no winner.

d. has a pure strategy solution.

ANS: D PTS: 1 TOP: Identifying a pure strategy

For a two-person, zero-sum, mixed-strategy game, each player selects its strategy according to

a. what strategy the other player used last.

b. a fixed rotation of strategies.

c. a probability distribution.

d. the outcome of the previous game.

ANS: C PTS: 1 TOP: Mixed strategy games

When the utility function for a risk-neutral decision maker is graphed (with monetary value on the

horizontal axis and utility on the vertical axis), the function appears as

a. a straight line

b. a convex curve

c. a concave curve

d. an ‘S’ curve

ANS: A PTS: 1 TOP: Risk avoiders versus risk takers

If a game larger than 2 X 2 requires a mixed strategy, we attempt to reduce the size of the game by

a. identifying saddle points

b. looking for dominated strategies

c. inverting the payoff matrix

d. eliminating negative payoffs

ANS: B PTS: 1 TOP: A larger mixed strategy game20. To select a strategy in a two-person, zero-sum game, Player A follows a ______ procedure and Player

B follows a ______ procedure.

a. maximax, minimin

b. maximax, minimax

c. maximax, maximax

d. maximin, minimax

ANS: D PTS: 1 TOP: Introduction to game theory

TRUE/FALSE

1. 2. 3. 4. 5. 6. 7. 8. 9. The decision alternative with the best expected monetary value will always be the most desirable

decision.

ANS: T PTS: 1 TOP: Introduction

When monetary value is not the sole measure of the true worth of the outcome to the decision maker,

monetary value should be replaced by utility.

ANS: T PTS: 1 TOP: Introduction

The outcome with the highest payoff will also have the highest utility.

ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

Expected utility is a particularly useful tool when payoffs stay in a range considered reasonable by the

decision maker.

ANS: F PTS: 1 TOP: Meaning of utility

To assign utilities, consider the best and worst payoffs in the entire decision situation.

ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

A risk avoider will have a concave utility function.

ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

The expected utility is the utility of the expected monetary value.

ANS: F PTS: 1 TOP: Expected utility approach

The risk premium is never negative for a conservative decision maker.

ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

The risk neutral decision maker will have the same indications from the expected value and expected

utility approaches.

ANS: T 10. PTS: 1 TOP: Expected monetary value versus expected utility

The utility function for a risk avoider typically shows a diminishing marginal return for money.11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

A game has a pure strategy solution when both players’ single-best strategies are the same.

ANS: F PTS: 1 TOP: Identifying a pure strategy

A game has a saddle point when pure strategies are optimal for both players.

ANS: T PTS: 1 TOP: Identifying a pure strategy

A game has a saddle point when the maximin payoff value equals the minimax payoff value.

ANS: T PTS: 1 TOP: Identifying a pure strategy

The logic of game theory assumes that each player has different information.

ANS: F PTS: 1 TOP: Introduction to game theory

With a mixed strategy, the optimal solution for each player is to randomly select among two or more of

the alternative strategies.

ANS: T PTS: 1 TOP: Mixed strategy games

The expected monetary value approach and the expected utility approach to decision making usually

result in the same decision choice unless extreme payoffs are involved.

ANS: T PTS: 1 TOP: Utility and decision making

A risk neutral decision maker will have a linear utility function.

ANS: T PTS: 1 TOP: Developing utilities for monetary payoffs

Given two decision makers, one risk neutral and the other a risk avoider, the risk avoider will always

give a lower utility value for a given outcome.

ANS: F PTS: 1 TOP: Developing utilities for monetary payoffs

Generally, the analyst must make pairwise comparisons of the decision strategies in an attempt to

identify dominated strategies.

ANS: T PTS: 1 TOP: A larger mixed strategy game

When the payoffs become extreme, most decision makers are satisfied with the decision that provides

the best expected monetary value.

ANS: F PTS: 1 TOP: The meaning of utility

Any 2 X 2 two-person, zero-sum, mixed-strategy game can be solved algebraically.

ANS: T PTS: 1 TOP: Mixed strategy games

A dominated strategy will never be selected by the player.

ANS: T PTS: 1 TOP: Dominated strategySHORT ANSWER

1. 2. 3. 4. 5. 6. When and why should a utility approach be followed?

ANS:

Answer not provided.

PTS: 1 TOP: Expected value versus utility

Give two examples of situations where you have decided on a course of action that did not have the

highest expected monetary value.

ANS:

Answer not provided.

PTS: 1 TOP: Introduction

Explain how utility could be used in a decision where performance is not measured by monetary value.

ANS:

Answer not provided.

PTS: 1 TOP: Expected value versus expected utility

Explain the relationship between expected utility, probability, payoff, and utility.

ANS:

Answer not provided.

PTS: 1 TOP: Expected value versus expected utility

Draw the utility curves for three types of decision makers, label carefully, and explain the concepts of

increasing and decreasing marginal returns for money.

ANS:

Answer not provided.

PTS: 1 TOP: Risk avoiders versus risk takers

Game theory models extend beyond two-person, zero-sum games. Discuss two extensions (or

variations).

ANS:

Answer not provided.

PTS: 1 TOP: Extensions to two-person, zero-sum games

PROBLEM

1. For the payoff table below, the decision maker will use P(s1) = .15, P(s2) = .5, and P(s3) = .35.State of Nature

Decision s1 s2 s3

d1 d2 a. b. 5000 1000 10,000

15,0002000 40,000

What alternative would be chosen according to expected value?

For a lottery having a payoff of 40,000 with probability p and 15,000 with probability (1 

p), the decision maker expressed the following indifference probabilities.

Payoff Probability

10,000 .85

1000 .60

2000 .53

5000 .50

Let U(40,000) = 10 and U(15,000) = 0 and find the utility value for each payoff.

What alternative would be chosen according to expected utility?

c. a. ANS:

EV(d1) = 3250 and EV(d2) = 10750, so choose d2.

b. Payoff Probability Utility

c. 2. 10,000 .85 8.5

1000 .60 6.0

2000 .53 5.3

5000 .50 5.0

EU(d1) = 6.725 and EU(d2) = 6.15, so choose d1.

PTS: 1 TOP: Expected utility approach

A decision maker who is considered to be a risk taker is faced with this set of probabilities and payoffs

State of Nature

Decision s1 s2 s3

d1 d2 d3 5 10 20

25 0 50

5010 80

Probability .30 .35 .35

For the lottery p(80) + (1  p)(50), this decision maker has assessed the following indifference

probabilities

Payoff Probability

50 .60

20 .35

10 .25

5 .22

0 .20

10 .18

25 .10

Rank the decision alternatives on the basis of expected value and on the basis of expected utility.3. ANS:

EV(d1) = 12 EV(d2) = 10 EV(d3) = 9.5

EU(d1) = 2.76 EU(d3) = 3.1 EU(d3) = 4.13

PTS: 1 TOP: Expected utility approach

Three decision makers have assessed utilities for the problem whose payoff table appears below.

State of Nature

Decision s1 s2 s3

d1 d2 d3 500 100400

200 150 100

100 200 300

Probability .2 .6 .2

Indifference Probability for Person

Payoff A B C

300 .95 .68 .45

200 .94 .64 .32

150 .91 .62 .28

100 .89 .60 .22

100 .75 .45 .10

Plot the utility function for each decision maker.

Characterize each decision maker’s attitude toward risk.

Which decision will each person prefer?

a. b. c. a.

ANS:

4. b. Person A is a risk avoider, Person B is fairly risk neutral, and Person C is a risk avoider.

c. For person A, EU(d1) = .734 EU(d2) = .912 EU(d3) = .904

For person B, EU(d1) = .56 EU(d2) = .62 EU(d3) = .61

For person C, EU(d1) = .332 EU(d2) = .276 EU(d3) = .302

Decision 1 would be chosen by person C. Decision 2 would be chosen by persons A and

B.

PTS: 1 TOP: Risk avoiders versus risk takers

A decision maker has the following utility function5. Payoff Indifference Probability

200 1.00

150 .95

50 .75

0 .60

50 0

What is the risk premium for the payoff of 50?

ANS:

EV = .75(200) + .25(50) = 137.50

Risk premium is 137.50  50 = 87.50

PTS: 1 TOP: Developing utilities for monetary payoffs

Determine decision strategies based on expected value and on expected utility for this decision tree.

Use the utility function

Payoff Indifference Probability

500 1.00

350 .89

300 .84

180 .60

100 .43

40 .20

20 .13

0 0

ANS:

Let U(500) = 1 and U(0) = 0. Then

After branch Expected value Expected utility

A 120 .336

J 316 .680

K 150 .522

B 127.2 .381

C 100 .4306. 7. 8. Based on expected value, the decision strategy is to select B. If G happens, select J. Based on expected

utility, it is best to choose C.

PTS: 1 TOP: Expected utility approach

Burger Prince Restaurant is considering the purchase of a $100,000 fire insurance policy. The fire

statistics indicate that in a given year the probability of property damage in a fire is as follows:

Fire Damage $100,000 $75,000 $50,000 $25,000 $10,000 $0

Probability .006 .002 .004 .003 .005 .980

a. b. If Burger Prince was risk neutral, how much would they be willing to pay for fire

insurance?

If Burger Prince has the utility values given below, approximately how much would they

be willing to pay for fire insurance?

Loss $100,000 $75,000 $50,000 $25,000 $10,000 $5,000 $0

Utility 0 30 60 85 95 99 100

ANS:

a. $1,075

b. $5,000

PTS: 1 TOP: Decision making using utility

Super Cola is considering the introduction of a new 8 oz. root beer. The probability that the root beer

will be a success is believed to equal .6. The payoff table is as follows:

Success (s1) Failure (s2)

Produce $250,000$300,000

Do Not Produce$50,000 $20,000

Company management has determined the following utility values:

Amount $250,000$20,000$50,000$300,000

Utility 100 60 55 0

a. b. Is the company a risk taker, risk averse, or risk neutral?

What is Super Cola’s optimal decision?

ANS:

a. Risk averse

b. Produce root beer as long as p  60/105 = .571

PTS: 1 TOP: Decision making using utility

Chez Paul is contemplating either opening another restaurant or expanding its existing location. The

payoff table for these two decisions is:

State of Nature

Decision s1 s2 s39. New Restaurant$80,000 $20,000 $160,000

Expand$40,000 $20,000 $100,000

Paul has calculated the indifference probability for the lottery having a payoff of $160,000 with

probability p and $80,000 with probability (1p) as follows:

Amount Indifference Probability (p)

$40,000 .4

$20,000 .7

$100,000 .9

a. b. c. Is Paul a risk avoider, a risk taker, or risk neutral?

Suppose Paul has defined the utility of $80,000 to be 0 and the utility of $160,000 to be

80. What would be the utility values for $40,000, $20,000, and $100,000 based on the

indifference probabilities?

Suppose P(s1) = .4, P(s2) = .3, and P(s3) = .3. Which decision should Paul make? Compare

with the decision using the expected value approach.

ANS:

a. A risk avoider

b. Amount Utility

c. $40,000 32

$20,000 56

$100,000 72

Decision is d2; EV criterion decision would be d1

PTS: 1 TOP: Decision making using utility

The Dollar Department Store chain has the opportunity of acquiring either 3, 5, or 10 leases from the

bankrupt Granite Variety Store chain. Dollar estimates the profit potential of the leases depends on the

state of the economy over the next five years. There are four possible states of the economy as

modeled by Dollar Department Stores and its president estimates P(s1) = .4, P(s2) = .3, P(s3) = .1, and

P(s4) = .2. The utility has also been estimated. Given the payoffs (in $1,000,000’s) and utility values

below, which decision should Dollar make?

Payoff Table State Of The Economy

Over The Next 5 Years

Decision s1 s2 s3 s4

d1 — buy 10 leases 10 5 020

d2 — buy 5 leases 5 0110

d3 — buy 3 leases 2 1 0 1

d4 — do not buy 0 0 0 0

Utility Table

Payoff (in $1,000,000’s) +10 +5 +2 011020

Utility +10 +5 +2 012050

ANS:

Buy 3 leases.10. 11. 12. PTS: 1 TOP: Decision making using utility

Consider the following two-person zero-sum game. Assume the two players have the same two

strategy options. The payoff table shows the gains for Player A.

Player B

Player A Strategy b1 Strategy b2

Strategy a1 Strategy a2 3 9

6 2

Determine the optimal strategy for each player. What is the value of the game?

ANS:

Mixed strategy:

Player A: .4 for a1, .6 for a2

Player B: .7 for b1, .3 for b2

Value of game = 4.8

PTS: 1 TOP: Mixed strategy games

Consider the following two-person zero-sum game. Assume the two players have the same three

strategy options. The payoff table below shows the gains for Player A.

Player B

Player A Strategy b1 Strategy b2 Strategy b3

Strategy a1 Strategy a2 Strategy a3 3 52

21 2

2 15

Is there an optimal pure strategy for this game? If so, what is it? If not, can the mixed-strategy

probabilities be found algebraically? What is the value of the game?

ANS:

There is not an optimal pure strategy.

However, there are dominated strategies.

Strategy a3 is dominated (by strategy a1) and can be eliminated.

Then strategy b1 is dominated (by strategy b2) and can be eliminated.

Now it is a 2 x 2 game.

Mixed-strategy probabilities are found algebraically: p = .3, (1  p) = .7, q = .4, (1  q) = .6

Value of game = 0.8

PTS: 1 TOP: Mixed strategy games

Suppose that there are only two vehicle dealerships (A and B) in a small city. Each dealership is

considering three strategies that are designed to take sales of new vehicles from the other dealership

over a period of four months. The strategies, assumed to be the same for both dealerships, are:

Strategy 1: Offer a cash rebate on a new vehicle.

Strategy 2: Offer free optional equipment on a new vehicle.

Strategy 3: Offer a 0% loan on a new vehicle.The payoff table (in number of new vehicle sales gained per week by Dealership A (or lost by

Dealership B) is shown below.

Dealership B

Dealership A Cash Rebate Free Options 0% Loan

b1 b2 b3

Cash Rebate a1 Free Options a2 0% Loan

a3

2 2 1

3 31

32 0

Identify the pure strategy for this two-person zero-sum game. What is the value of the game?

13. ANS:

An optimal pure strategy exists for this game:

Dealership A should offer a cash rebate on new vehicles.

Dealership A can expect to gain a minimum of 1 new vehicle sale per week.

Dealership B should offer a 0% loan on new vehicles.

Dealership B can expect to lose a maximum of 1 new vehicle sale per week.

Value of the game is 1 new vehicle.

PTS: 1 TOP: Identifying a pure strategy

Consider the following two-person zero-sum game. Assume the two players have the same two

strategy options. The payoff table shows the gains for Player A.

Player B

Player A Strategy b1 Strategy b2

Strategy a1 Strategy a2 4 8

11 5

Determine the optimal strategy for each player. What is the value of the game?

14. ANS:

The optimal mixed strategy solution for this game:

Player A should select Strategy a1 with a .6 probability and Strategy a2 with a .4 probability.

Player B should select Strategy b1 with a .3 probability and Strategy b2 with a .7 probability.

Value of the game is:

Player A: 6.8 = expected gain

Player B: 6.8 = expected loss

PTS: 1 TOP: Mixed strategy games

Consider the following two-person zero-sum game. Assume the two players have the same three

strategy options. The payoff table shows the gains for Player A.

Player B

Player A Strategy b1 Strategy b2 Strategy b315. 16. Strategy a1 Strategy a2 Strategy a3 6 52

1 0 3

3 43

Is there an optimal pure strategy for this game? If so, what is it? If not, can the mixed-strategy

probabilities be found algebraically?

ANS:

algebraically.

There is not an optimal pure strategy. The optimal mixed-strategy probabilities can be found

Player A should select Strategy a1 with a .2 probability and Strategy a2 with a .8 probability.

Player B should select Strategy b1 with a .5 probability and Strategy b3 with a .5 probability.

Value of the game:

For Player A: 2 = expected gain

For Player B: 2 = expected loss

PTS: 1 TOP: Mixed strategy games

Consider the following two-person zero-sum game. Assume the two players have the same three

strategy options. The payoff table below shows the gains for Player A.

Player B

Player A Strategy b1 Strategy b2 Strategy b3

Strategy a1 Strategy a2 Strategy a3 3 24

1 0 2

4 53

Is there an optimal pure strategy for this game? If so, what is it? If not, can the mixed-strategy

probabilities be found algebraically? What is the value of the game?

ANS:

There is not an optimal pure strategy. Strategy a1 is dominated by Strategy a3, and then Strategy b1 is

dominated by Strategy b2. The optimal mixed-strategy probabilities can be found algebraically.

Player A should select Strategy a2 with a .8 probability and Strategy a3 with a .2 probability.

Player B should select Strategy b2 with a .5 probability and Strategy b3 with a .5 probability.

Value of the game = 1.

PTS: 1 TOP: Mixed strategy games

Two banks (Franklin and Lincoln) compete for customers in the growing city of Logantown. Both

banks are considering opening a branch office in one of three new neighborhoods: Hillsboro, Fremont,

or Oakdale. The strategies, assumed to be the same for both banks, are:

Strategy 1: Open a branch office in the Hillsboro neighborhood.

Strategy 2: Open a branch office in the Fremont neighborhood.

Strategy 3: Open a branch office in the Oakdale neighborhood.Values in the payoff table below indicate the gain (or loss) of customers (in thousands) for Franklin

Bank based on the strategies selected by the two banks.

Lincoln Bank

Franklin Bank Hillsboro Fremont Oakdale

b1 b2 b3

Hillsboro a1 Fremont a2 Oakdale a3 4 2 3

623

1 0 5

Identify the neighborhood in which each bank should locate a new branch office. What is the value of

17. the game?

ANS:

Franklin should select Hillsboro; Lincoln should select Fremont. Value of game = 2,000 customers

PTS: 1 TOP: Mixed strategy games

Consider the following problem with four states of nature, three decision alternatives, and the

following payoff table (in $’s):

18. d1 d2 d3 s1 s2 s3 s4

200 2600 -1400 200

0 200 –

200 200

-200 400 0 200

The indifference probabilities for three individuals are:

Payoff Person 1 Person 2 Person 3

$ 2600 1.00 1.00 1.00

$ 400 .40 .45 .55

$ 200 .35 .40 .50

$ 0 .30 .35 .45

-$ 200 .25 .30 .40

-$1400 0 0 0

a. b. c. Classify each person as a risk avoider, risk taker, or risk neutral.

For the payoff of $400, what is the premium the risk avoider will pay to avoid risk? What is the

premium the risk taker will pay to have the opportunity of the high payoff?

Suppose each state is equally likely. What are the optimal decisions for each of these three

people?

ANS:

a. b. Person 1 — risk taker; Person 2 — risk neutral; Person 3 — risk avoider

Risk avoider would pay $400; Risk taker would pay $200

c. Person 1 — d1; Person 2 — d1; Person 3 — d1

PTS: 1 TOP: Risk avoiders and risk takers

Metropolitan Cablevision has the choice of using one of three DVR systems. be a function of customer acceptance. The payoff to Metropolitan for the three systems is:

System

Profits are believed to19. Acceptance Level I II III

High $150,000 $200,000 $200,000

Medium $ 80,000 $ 20,000 $ 80,000

Low $ 20,000 -$ 50,000 -$100,000

The probabilities of customer acceptance for each system are:

System

Acceptance Level I II III

High .4 .3 .3

Medium .3 .4 .5

Low .3 .3 .2

The first vice president believes that the indifference probabilities for Metropolitan should be:

Amount Probability

$150,000 .90

$ 80,000 .70

$ 20,000 .50

-$ 50,000 .25

The second vice president believes Metropolitan should assign the following utility values:

Amount Utility

$200,000 125

$150,000 95

$ 80,000 55

$ 20,000 30

-$ 50,000 10

-$100,000 0

Which vice president is a risk taker? Which one is risk averse?

Which system will each vice president recommend?

What system would a risk neutral vice president recommend?

a. b. c. ANS:

a. Risk Taker — Second Vice President

Risk Avoider — First Vice President

b. First Vice President — System I

Second Vice President — System III

c. Risk Neutral Vice President — System I

PTS: 1 TOP: Risk avoiders and risk takers

Consider a two-person, zero-sum game where the payoffs listed below are the winnings for Player A.

Identify the pure strategy solution. What is the value of the game?

Player B Strategies

b1 b2 b3

Player A Strategies a1 a2 a3 5 5 4

1 6 2

7 2 320. 21. 22. ANS:

Optimal pure strategies: Player A uses strategy a1; Player B uses strategy b3.

Value of game: Gain of 4 for Player A; loss of 4 for Player B.

PTS: 1 TOP: Game theory

Consider a two-person, zero-sum game where the payoffs listed below are the winnings for Company

X. Identify the pure strategy solution. What is the value of the game?

Company Y Strategies

y1 y2 y3

Company X Strategies x1 x2 x3 3 5 9

8 4 3

7 6 7

ANS:

Optimal pure strategies: Company X uses strategy x3; Company Y uses strategy y2.

Value of game: Gain of 6 for Company X; loss of 6 for Company Y.

PTS: 1 TOP: Game theory

Consider the following two-person, zero-sum game. Payoffs are the winnings for Company X.

Formulate the linear program that determines the optimal mixed strategy for Company X.

Company Y Strategies

y1 y2 y3

Company X Strategies x1 x2 x3 4 3 9

2 5 1

6 1 7

ANS:

Max GAINA

s.t. 4PA1 + 2PA2 + 6PA3  GAINA  0 3PA1 + 5PA2 + 1PA3  GAINA  0 9PA1 + 1PA2 + 7PA3  GAINA  0 PA1 + PA2 + PA3 = 1 PA1, PA2, PA3, GAINA  0 (Strategy B1)

(Strategy B2)

(Strategy B3)

(Probabilities must sum to 1)

(Non-negativity)

PTS: 1 TOP: Game theory

Shown below is the solution to the linear program for finding Player A’s optimal mixed strategy in a

two-person, zero-sum game.

OBJECTIVE FUNCTION VALUE = 3.500

VARIABLE VALUE REDUCED COSTS

PA1 0.050 0.000

PA2 0.600 0.000

PA3 0350 0.000

GAINA 3.500 0.000

CONSTRAINT SLACK/SURPLUS DUAL PRICESa. b. c. d. ANS:

a. 1 0.0000.500

2 0.0000.500

3 0.000 0.000

4 0.000 3.500

What is Player A’s optimal mixed strategy?

What is Player B’s optimal mixed strategy?

What is Player A’s expected gain?

What is Player B’s expected loss?

Player A’s optimal mixed strategy: b. Player B’s optimal mixed strategy: Use strategy A1 with .05 probability

Use strategy A2 with .60 probability

Use strategy A3 with .35 probability

Use strategy B1 with .50 probability

Use strategy B2 with .50 probability

Do not use strategy B3

c. d. Player A’s expected gain: 3.500

Player B’s expected loss: 3.500

PTS: 1 TOP: Game theory