Deck 15: Game Theory: the Mathematics of Competition

Question 1

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: Solve the game determining the best mix of selections for both batter and pitcher.

Answer 1

Batter always expects fastball. Pitcher always plays knuckleball.

Question 2

Describe a saddlepoint of a zero-sum game for two players where the payoff matrix represent gains to the row player I and losses to column player II.

Answer 2

When the maximum of row minimums is equal to the minimum of column maximums, the resulting payoff is called a saddlepoint and the value of the game.

Question 3

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: Solve the game, determining the best mix of selections for both batter and pitcher.

Answer 3

Batter expects 3/4 knuckleball and 1/4 fastball. Pitcher plays 3/4 knuckleball and 1/4 fastball.

Question 4

Describe a payoff matrix of a two-person total-conflict game.

Question 5

Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and the game has a saddlepoint.

Question 6

In the game of matching pennies, player I wins a penny if the coins match and player II wins if the coins do not match. Is this a zero-sum game? Is this a fair game?

Question 7

Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and the game has no saddlepoint.

Question 8

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 9

Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and at least one player has a dominated strategy.

Question 10

You want to carry insurance for your small business. The annual policy costs $1000 this year. If you are sued and you have no insurance, you will pay $5000. If you have insurance and are sued you pay nothing, and your partner gives you $500 for your wise foresight, so that the policy costs you only $500. Represent this game as a two-by-two matrix.

Question 11

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: Solve the game determining the best mix of selections for both batter and pitcher.

Question 12

In a game, each player chooses one of three coins: penny, nickel, or dime. If both players choose the same coin, both players loose their coin. Otherwise, the player with the more valuable coin wins the less valuable coin from the other player. Represent this game as a three-by-three matrix of ordered pairs.

Question 13

In the game of matching pennies, player I wins a penny if the coins match and player II wins if the coins do not match. Present this game as a two-by-two matrix, where each player has two outcomes from which to select.

Question 14

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 15

Describe a Nash equilibrium.

Question 16

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 17

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 18

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 19

Describe a zero-sum game.

Question 20

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

Question 21

Use the following information to answer Questions
In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense.
Suppose the value of the game is What does this mean for the offense?

Question 22

Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs. What are the optimal strategies for this game?

Question 23

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second is to player II. Discuss the players' possible strategies when this game is played.

Question 24

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to the column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy?

Question 25

The minimax strategy of a column player is the strategy that corresponds to the minimum value of the maximum numbers in the columns.

Question 26

If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Statistically, how often should the government audit you?

Question 27

Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs. What are the optimal strategies for this game?

Question 28

If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Statistically, how often should you cheat?

Question 29

Describe a Nash equilibrium in a Vickrey auction.

Question 30

A total-conflict game is a zero-sum game.

Question 31

Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs. What are the optimal strategies for this game?

Question 32

Use the following information to answer Questions
In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense.
What is the optimal mixed strategy for the defense?

Question 33

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to the column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy?

Question 34

Use the following information to answer Questions
In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense.
Suppose the value of the game is What does this mean for the defense?

Question 35

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second to player II. Discuss the players' possible strategies when this game is played.

Question 36

Use the following information to answer Questions
In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense.
What is the optimal mixed strategy for the offense?

Question 37

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second is to player II. Discuss the players' possible strategies when this game is played.

Question 38

What is the name of a table whose rows and columns correspond to the strategies of the two players?

A) strategy matrix
B) payoff matrix
C) value matrix
D) profit matrix

Question 39

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to the column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy?

Question 40

Use the following information to answer Questions
In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense.
What is the value of the game

Question 41

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

A) to always play the first column.
B) to always play the second column.
C) to always play the third column.
D) to always play the fourth column.

Question 42

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

A) to always play the first column.
B) to always play the second column.
C) to always play the third column.
D) to play two or more of the columns.

Question 43

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

A) to always play the first row.
B) to always play the second row.
C) to always play the third row.
D) to always play the fourth row.

Question 44

The value of a zero-sum game is a saddlepoint.

Question 45

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) There is a saddlepoint at row 1, column 1.
C) There is a saddlepoint at row 3, column 1.
D) There is a saddlepoint at row 3, column 3.

Question 46

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) The game has a saddlepoint; the value of the game is 4.
C) The game has a saddlepoint; the value of the game is 5.
D) The game has a saddlepoint; the value of the game is 6.

Question 47

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) There is a saddlepoint at row 2, column 2.
C) There is a saddlepoint at row 2, column 1.
D) There is a saddlepoint at row 1, column 2.

Question 48

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The maximin strategy of player I is to always play the first row; the minimax strategy of player II is to always play the first column.
B) The maximin strategy of player I is to always play the second row; the minimax strategy of player II is to always play the first column.
C) The maximin strategy of player I is to always play the first row; the minimax strategy of player II is to always play the second column.
D) The maximin strategy of player I is to always play the second row; the minimax strategy of player II is to always play the second column.

Question 49

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. Which of the following statements is true?

A) The game has no saddlepoint.
B) The game has a saddlepoint; the value of the game is .200.
C) The game has a saddlepoint; the value of the game is .300.
D) The game has a saddlepoint; the value of the game is .400.

Question 50

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

A) to always play the first row.
B) to always play the second row.
C) to always play the third row.
D) to play two or more of the rows.

Question 51

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

A) to always play the first column.
B) to always play the second column.
C) to always play the third column.
D) to play two or more of the columns.

Question 52

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

A) to always play the first row.
B) to always play the second row.
C) to always play the third row.
D) to play two or more of the rows.

Question 53

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

A) to always play the first row.
B) to always play the second row.
C) to always play the third row.
D) to always play the fourth row.

Question 54

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) The game has a saddlepoint; the value of the game is 8.
C) The game has a saddlepoint; the value of the game is 9.
D) The game has a saddlepoint; the value of the game is 11.

Question 55

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The maximin strategy of player I is to always play the first row; the minimax strategy of player II is to always play the first column.
B) The maximin strategy of player I is to always play the second row; the minimax strategy of player II is to always play the first column.
C) The maximin strategy of player I is to always play the first row; the minimax strategy of player II is to always play the second column.
D) The maximin strategy of player I is to always play the second row; the minimax strategy of player II is to always play the second column.

Question 56

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

A) to always play the first column.
B) to always play the second column.
C) to always play the third column.
D) to always play the fourth column.

Question 57

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) There is a saddlepoint at row 2, column 2.
C) There is a saddlepoint at row 2, column 1.
D) There is a saddlepoint at row 1, column 2.

Question 58

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

A) to always play the first row.
B) to always play the second row.
C) to always play the third row.
D) to play two or more of the rows.

Question 59

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

A) to always play the first column.
B) to always play the second column.
C) to always play the third column.
D) to play two or more of the columns.

Question 60

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

A) The game has no saddlepoint.
B) The game has a saddlepoint; the value of the game is 8.
C) The game has a saddlepoint; the value of the game is 9.
D) The game has a saddlepoint; the value of the game is 11.

Question 61

Use the following information to answer Questions
Which of the following statements is true?

A) Both player I and player II have dominant strategies.
B) Neither player I nor II has a dominant strategy.
C) Only player I has a dominant strategy.
D) Only player II has a dominant strategy.

Question 62

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. What is the pitcher's optimal strategy?

A) Pitch more fastballs than curves.
B) Pitch more curves than fastballs.
C) Pitch fastballs and curves equally.

Question 63

As the buyer for your business, you can choose to buy an extended warranty for each new computer for $200 each. During this period, each computer will experience either a minor ($100) or major ($500) repair bill. Repairs for each computer under warranty are reduced by 50% to $50 or $250. Assume the computer company wants to make as much money as possible, and you wish to spend as little as possible. What is your optimal strategy?

A) Never buy the warranty.
B) Always buy the warranty.
C) Buy the warranty for less than half of the computers.
D) Buy the warranty for more than half of the computers.

Question 64

Use the following information to answer Questions The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. $\begin{array}{lll}&&&\text { Defense }\\&&\text { Run } &&\text { Pass }\\\text { Offense }&\text { Run } & 0.5 && 0.6 \\&\text { Pass } & 0.8 && 0.4\end{array}$

-In such situations, what is the optimal solution for the defense?

A) always anticipate a run
B) always anticipate a pass
C) play the mixed strategy $\left( \frac { 2 } { 5 } , \frac { 3 } { 5 } \right)$
D) play the mixed strategy $\left( \frac { 4 } { 5 } , \frac { 1 } { 5 } \right)$

Question 65

Use the following information to answer Questions The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. $\begin{array}{lll}&&&\text { Defense }\\&&\text { Run } &&\text { Pass }\\\text { Offense }&\text { Run } & 0.5 && 0.6 \\&\text { Pass } & 0.8 && 0.4\end{array}$

-What is the value of the game?

A) $\frac { 1 } { 2 }$
B) $\frac { 13 } { 24 }$
C) $\frac { 14 } { 25 }$
D) $\frac { 11 } { 25 }$

Question 66

Use the following information to answer Questions The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. $\begin{array}{lll}&&&\text { Defense }\\&&\text { Run } &&\text { Pass }\\\text { Offense }&\text { Run } & 0.5 && 0.6 \\&\text { Pass } & 0.8 && 0.4\end{array}$

-In such situations, what is the optimal solution for the offense?

A) always run
B) always pass
C) play the mixed strategy $\left( \frac { 2 } { 5 } , \frac { 3 } { 5 } \right)$
D) play the mixed strategy $\left( \frac { 4 } { 5 } , \frac { 1 } { 5 } \right)$

Question 67

You are considering cheating on your income tax return. If you cheat you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Statistically, which is the best option for you to choose?

A) definitely cheat
B) probably cheat
C) probably don't cheat
D) definitely don't cheat

Question 68

You can choose to buy an extended warranty for your computer for $200. During this period, you will experience either a minor ($100) or major ($500) repair bill. If you buy the warranty, you will not have to pay the repair bill. Assume the company wants to make as much money as possible, and you wish to spend as little as possible. Which of the following is a true statement?

A) This game has no saddlepoint.
B) This game has a saddlepoint; the value of the game is $100.
C) This game has a saddlepoint; the value of the game is $200.
D) This game has a saddlepoint; the value of the game is $500.

Question 69

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. What is the pitcher's optimal strategy?

A) Pitch more fastballs than curves.
B) Pitch more curves than fastballs.
C) Pitch fastballs and curves equally.

Question 70

In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. In such situations, what is the optimal solution for the defense?

A) always anticipate a run
B) always anticipate a pass
C) anticipate a run more often than a pass
D) anticipate a pass more often than a run

Question 71

As the buyer for your business, you can choose to buy an extended warranty for each new computer for $200 each. During this period, each computer will experience either a minor ($100) or major ($500) repair bill. Repairs for each computer under warranty are reduced by 50% to $50 or $250. Assume the computer company wants to make as much money as possible, and you wish to spend as little as possible. What is the computer company's optimal strategy?

A) Always sell the warranty.
B) Never sell the warranty.
C) Sell the warranty for less than half of the computers.
D) Sell the warranty for more than half of the computers.

Question 72

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. What is the batter's optimal strategy?

A) Expect more fastballs than curves.
B) Expect more curves than fastballs.
C) Expect equal numbers of curves and fastballs.

Question 73

In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. In such situations, what is the optimal solution for the offense? a) always choose to run
B) always choose to pass
C) choose to run more often than pass
D) choose to pass more often than run

Question 74

Consider the following partial-conflict game played in a noncooperative manner. What outcomes constitute a Nash equilibrium?

A) only when both players select choice A
B) only when both players select choice B
C) only when one player selects choice A; the other selects choice B
D) only when both players select choice A or both players select choice B

Question 75

Use the following information to answer Questions
Which of the following is a Nash equilibrium?

A) Player I selects choice A; player II selects choice A.
B) Player I selects choice A; player II selects choice B.
C) Player I selects choice B; player II selects choice A.
D) Player I selects choice B; player II selects choice B.

Question 76

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. What is the batter's optimal strategy?

A) Expect more fastballs than curves.
B) Expect more curves than fastballs.
C) Expect equal numbers of curves and fastballs.

Question 77

You can choose to buy an extended warranty for your computer for $200. During this period, you will experience either a minor ($100) or major ($500) repair bill. If you buy the warranty, you pay 50% of any repair bill: that is, you will pay either $50 or $250 for the subsequent repair. Assume the company wants to make as much money as possible, and you wish to spend as little as possible. Which of the following is a true statement?

A) This game has no saddlepoint.
B) This game has a saddlepoint; the value of the game is $200.
C) This game has a saddlepoint; the value of the game is $250.
D) This game has a saddlepoint; the value of the game is $450.

Question 78

If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Assuming the government wants to receive as much money as possible, statistically, which is the best option for the government to choose?

A) always audit
B) usually audit
C) occasionally audit
D) never audit

Question 79

Consider the following partial-conflict game played in a noncooperative manner. Which of the following statements is true?

A) The dominant strategy for player I is to select choice A.
B) The dominant strategy for player I is to select choice B.
C) Player I has no dominant strategy.

Question 80

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. Which of the following statements is true?

A) The game has no saddlepoint.
B) The game has a saddlepoint; the value of the game is .200.
C) The game has a saddlepoint; the value of the game is .300.
D) The game has a saddlepoint; the value of the game is .400.