Deck 5: Utility and Game Theory

Question 1

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.

Answer 1

A

Question 2

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

Answer 2

D

Question 3

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.

Answer 3

C

Question 4

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.

Question 5

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.

Question 6

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

Question 7

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.

Question 8

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.

Question 9

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

Question 10

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.

Question 11

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

Question 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.

Question 13

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.

Question 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.

Question 15

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.

Question 16

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.

Question 17

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.

Question 18

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.

Question 19

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

Question 20

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.

Question 21

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

Question 22

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.

Question 23

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

Question 24

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.

Question 25

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

Question 26

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

Question 27

When the payoffs become extreme,most decision makers are satisfied with the decision that provides the best expected monetary value.

Question 28

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

Question 29

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

Question 30

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

Question 31

A risk avoider will have a concave utility function.

Question 32

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

Question 33

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

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

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

Question 39

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

Question 40

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.

Question 41

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. 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?

Question 42

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

Question 43

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

Question 44

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

Question 45

Give two examples of situations where you have decided on a course of action that did not have the highest expected monetary value.

Question 46

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

Question 47

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? Utility Table

Question 48

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: Company management has determined the following utility values:
a.Is the company a risk taker, risk averse, or risk neutral?
b.What is Super Cola's optimal decision?

Question 49

A decision maker has the following utility function What is the risk premium for the payoff of 50?

Question 50

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

Question 51

Chez Paul is contemplating either opening another restaurant or expanding its existing location.The payoff table for these two decisions is: 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:
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.

Question 52

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

Question 53

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:
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?

Question 54

A dominated strategy will never be selected by the player.

Question 55

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

Question 56

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

Question 57

Draw the utility curves for three types of decision makers,label carefully,and explain the concepts of increasing and decreasing marginal returns for money.

Question 58

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

Question 59

When and why should a utility approach be followed?

Question 60

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

Question 61

Consider the following problem with four states of nature,three decision alternatives,and the following payoff table (in $'s): The indifference probabilities for three individuals are: 11ea7759_c40a_5e93_81a5_c995ec96407d_TB2192_00
a. Classify each person as a risk avoider, risk taker, or risk neutral.
b. 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?
c. Suppose each state is equally likely. What are the optimal decisions for each of these three people?s): The indifference probabilities for three individuals are: a. Classify each person as a risk avoider, risk taker, or risk neutral. b. 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? c. Suppose each state is equally likely. What are the optimal decisions for each of these three people?" class="answers-bank-image d-block" loading="lazy" >
a. Classify each person as a risk avoider, risk taker, or risk neutral.
b. 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?
c. Suppose each state is equally likely. What are the optimal decisions for each of these three people?s): The indifference probabilities for three individuals are: a. Classify each person as a risk avoider, risk taker, or risk neutral. b. 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? c. Suppose each state is equally likely. What are the optimal decisions for each of these three people?" class="answers-bank-image d-block" loading="lazy" > The indifference probabilities for three individuals are: 11ea7759_c40a_5e93_81a5_c995ec96407d_TB2192_00
a. Classify each person as a risk avoider, risk taker, or risk neutral.
b. 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?
c. Suppose each state is equally likely. What are the optimal decisions for each of these three people?

Question 62

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

Question 63

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. 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?

Question 64

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: The probabilities of customer acceptance for each system are: The first vice president believes that the indifference probabilities for Metropolitan should be: The second vice president believes Metropolitan should assign the following utility values:
a. Which vice president is a risk taker? Which one is risk averse?
b. Which system will each vice president recommend?
c. What system would a risk neutral vice president recommend?

Question 65

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?

Question 66

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

Question 67

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?