Quantitative Methods for Business 12th Edition - Test Bank

Quantitative Methods for Business 12th Edition – Test Bank

 

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Sample Questions Are Posted Below

 

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 utility

 

7. 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

 

8. 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

 

9. 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

 

10. 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

 

11. 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

 

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

a.	A dominated strategy will never be selected by a player.
b.	A dominated strategy exists if another strategy is at least as good regardless of what the opponent does.
c.	A dominated strategy is superior to a mixed strategy.
d.	A dominated strategy can be eliminated from the game.

 

 

ANS:  C                    PTS:   1                    TOP:   Dominated strategies

 

13. 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.
c.	can be solved by identifying the minimax and maximin values.
d.	cannot be solved.

 

 

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

 

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

a.	The gain for one player is equal to the loss for the other player.
b.	A payoff of 2 for one player has a corresponding payoff of -2 for the other player.
c.	The sum of the payoffs in the payoff table is zero.
d.	What one player wins, the other player loses.

 

 

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

 

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

a.	a mixed strategy is optimal.
b.	a pure strategy is optimal.
c.	a dominated strategy is optimal.
d.	one player should use a pure strategy and the other should use a mixed strategy.

 

 

ANS:  A                    PTS:   1                    TOP:   Mixed strategy games

 

16. 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

 

17. 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

 

18. 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

 

19. 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 game

 

20. 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. The decision alternative with the best expected monetary value will always be the most desirable decision.

 

ANS:  T                    PTS:   1                    TOP:   Introduction

 

2. 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

 

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

 

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

 

4. 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

 

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

 

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

 

6. A risk avoider will have a concave utility function.

 

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

 

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

 

ANS:  F                    PTS:   1                    TOP:   Expected utility approach

 

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

 

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

 

9. The risk neutral decision maker will have the same indications from the expected value and expected utility approaches.

 

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

 

10. The utility function for a risk avoider typically shows a diminishing marginal return for money.

 

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

 

11. 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

 

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

 

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

 

13. 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

 

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

 

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

 

15. 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

 

16. 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

 

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

 

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

 

18. 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

 

19. 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

 

20. 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

 

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

 

ANS:  T                    PTS:   1                    TOP:   Mixed strategy games

 

22. A dominated strategy will never be selected by the player.

 

ANS:  T                    PTS:   1                    TOP:   Dominated strategy

 

SHORT ANSWER

 

1. When and why should a utility approach be followed?

 

ANS:

Answer not provided.

 

PTS:   1                    TOP:   Expected value versus utility

 

2. 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

 

3. 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

 

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

 

ANS:

Answer not provided.

 

PTS:   1                    TOP:   Expected value versus expected utility

 

5. 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

 

6. 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	-5000	1000	10,000
d2	-15,000	-2000	40,000

 

a.	What alternative would be chosen according to expected value?
b.	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.
c.	What alternative would be chosen according to expected utility?

 

 

ANS:

 

a.	EV(d1) = 3250 and EV(d2) = 10750, so choose d2.
b.	Payoff	Probability	Utility
	10,000	.85	8.5
	1000	.60	6.0
	-2000	.53	5.3
	-5000	.50	5.0
c.	EU(d1) = 6.725 and EU(d2) = 6.15, so choose d1.

 

 

PTS:   1                    TOP:   Expected utility approach

 

2. 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	5	10	20
d2	-25	0	50
d3	-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.

 

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

 

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

 

	State of Nature
Decision	s1	s2	s3
d1	500	100	-400
d2	200	150	100
d3	-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

 

a.	Plot the utility function for each decision maker.
b.	Characterize each decision maker’s attitude toward risk.
c.	Which decision will each person prefer?

 

 

ANS:

 

a.
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

 

4. A decision maker has the following utility function

 

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

 

5. 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	.430

 

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

 

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

 

7. 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.	Is the company a risk taker, risk averse, or risk neutral?
b.	What is Super Cola’s optimal decision?

 

 

ANS:

 

a.	Risk averse
b.	Produce root beer as long as p ³ 60/105 = .571

 

 

PTS:   1                    TOP:   Decision making using utility

 

8. 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	s3
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.	Is Paul a risk avoider, a risk taker, or risk neutral?
b.	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?
c.	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
	-$40,000	32
	$20,000	56
	$100,000	72
c.	Decision is d2; EV criterion decision would be d1

 

 

PTS:   1                    TOP:   Decision making using utility

 

9. 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.

 

PTS:   1                    TOP:   Decision making using utility

 

10. 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	3	9
Strategy a2	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

 

11. 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	3	5	-2
Strategy  a2	-2	-1	2
Strategy  a3	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

 

12. 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
	Cash Rebate	Free Options	0% Loan
Dealership A	b1	b2	b3
Cash Rebate   a1	2	2	1
Free Options  a2	-3	3	-1
0% Loan         a3	3	-2	0

 

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

 

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

 

13. 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	4	8
Strategy a2	11	5

 

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

 

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

 

14. 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 b3
Strategy  a1	6	5	-2
Strategy  a2	1	0	3
Strategy  a3	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:

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

 

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

 

15. 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	3	2	-4
Strategy  a2	-1	0	2
Strategy  a3	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

 

16. 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
	Hillsboro	Fremont	Oakdale
Franklin Bank	b1	b2	b3
Hillsboro   a1	4	2	3
Fremont    a2	6	-2	-3
Oakdale     a3	-1	0	5

 

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

 

ANS:

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

 

PTS:   1                    TOP:   Mixed strategy games

 

17. Consider the following problem with four states of nature, three decision alternatives, and the following payoff table (in $’s):

 

	s1	s2	s3	s4
d1	200	2600	-1400	200
d2	0	200	–  200	200
d3	-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

 

1. Classify each person as a risk avoider, risk taker, or risk neutral.
2. 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?
3. Suppose each state is equally likely. What are the optimal decisions for each of these three people?

 

ANS:

1. Person 1 — risk taker; Person 2 — risk neutral; Person 3 — risk avoider
2. Risk avoider would pay $400; Risk taker would pay $200
3. Person 1 — d₁; Person 2 — d₁;   Person 3 — d₁

 

PTS:   1                    TOP:   Risk avoiders and risk takers

 

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

 

System
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

 

1. Which vice president is a risk taker? Which one is risk averse?
2. Which system will each vice president recommend?
3. What system would a risk neutral vice president recommend?

 

ANS:

1. Risk Taker — Second Vice President

Risk Avoider — First Vice President

1. First Vice President — System I

Second Vice President — System III

1. Risk Neutral Vice President — System I

 

PTS:   1                    TOP:   Risk avoiders and risk takers

 

19. 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
Player A Strategies	b1	b2	b3
a1	5	5	4
a2	1	6	2
a3	7	2	3

 

 

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

 

20. 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
Company X Strategies	y1	y2	y3
x1	3	5	9
x2	8	4	3
x3	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

 

21. 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
Company X Strategies	y1	y2	y3
x1	4	3	9
x2	2	5	1
x3	6	1	7

 

 

ANS:

Max	GAINA
s.t.	4PA1 + 2PA2 + 6PA3 – GAINA ³ 0	(Strategy B1)
	3PA1 + 5PA2 + 1PA3 – GAINA ³ 0	(Strategy B2)
	9PA1 + 1PA2 + 7PA3 – GAINA ³ 0	(Strategy B3)
	PA1 + PA2 + PA3 = 1	(Probabilities must sum to 1)
	PA1, PA2, PA3, GAINA ³ 0	(Non-negativity)

 

 

PTS:   1                    TOP:   Game theory

 

22. 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 PRICES
1	0.000	-0.500
2	0.000	-0.500
3	0.000	0.000
4	0.000	3.500

 

a.	What is Player A’s optimal mixed strategy?
b.	What is Player B’s optimal mixed strategy?
c.	What is Player A’s expected gain?
d.	What is Player B’s expected loss?

 

 

ANS:

a.	Player A’s optimal mixed strategy:	Use strategy A1 with .05 probability
		Use strategy A2 with .60 probability
		Use strategy A3 with .35 probability
b.	Player B’s optimal mixed strategy:	Use strategy B1 with .50 probability
		Use strategy B2 with .50 probability
		Do not use strategy B3
c.	Player A’s expected gain: 3.500
d.	Player B’s expected loss: 3.500

 

 

PTS:   1                    TOP:   Game theory