Deck 29: Game Theory

Question 1

While game theory predicts noncooperative behavior for a single play of the prisoner's dilemma, it would predict cooperative tit-for-tat behavior if the same people play prisoner's dilemma together for, say, 20 rounds.

Answer 1

False

Question 2

Two players are engaged in a game of Chicken.There are two possible strategies, Swerve and Drive Straight.A player who chooses to Swerve is called Chicken and gets a payoff of zero, regardless of what the other player does.A player who chooses to Drive Straight gets a payoff of 432 if the other player swerves and a payoff of -48 if the other player also chooses to Drive Straight.This game has two pure strategy equilibria and

A)a mixed strategy equilibrium in which each player swerves with probability .10 and drives straight with probability .90.
B)a mixed strategy in which each player swerves with probability .05 and drives straight with probability .95.
C)a mixed strategy equilibrium in which one player swerves with probability .10 and the other swerves with probability .90.
D)two mixed strategies in which players alternate between swerving and driving straight.
E)no mixed strategies.

Answer 2

a mixed strategy equilibrium in which each player swerves with probability .10 and drives straight with probability .90.

Question 3

If a game does not have an equilibrium in pure strategies, then it will not have an equilibrium in mixed strategies either.

Answer 3

False

Question 4

A game has two players and each has two strategies.The strategies are Be Nice and Be Mean.If both players play Be Nice, both get a payoff of 5.If both players play Be Mean, both get a payoff of 23.If one player plays Be Nice and the other plays Be Mean, the player who played Be Nice gets 0 and the player who played Be Mean gets 10.Playing Be Mean is a dominant strategy for both players.

Question 5

Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough.If both pigs choose Wait at the Trough, both get 2.If both pigs choose Press the Button, then both pigs get 5.If Little Pig presses the button and Big Pig waits at the trough, then Big Pig gets 10 and Little Pig gets 0.Finally, if Big Pig presses the button and Little Pig waits at the trough, then Big Pig gets 3 and Little Pig gets 2.In Nash equilibrium,

A)both pigs will wait at the trough.
B)Little Pig will get a payoff of 2 and Big Pig will get a payoff of 3.
C)Little pig will get a payoff of zero.
D)Little Pig will get a payoff of 5 and Big Pig will get a payoff of 5.
E)the pigs must be using mixed strategies.

Question 6

A two-person game in which each person has access to only two possible strategies will have at most one Nash equilibrium.

Question 7

A game has two players.Each player has two possible strategies.One strategy is Cooperate, the other is Defect.Each player writes on a piece of paper either a C for cooperate or a D for defect.If both players write C, they each get a payoff of $100.If both players write D, they each get a payoff of 0.If one player writes C and the other player writes D, the cooperating player gets a payoff of S and the defecting player gets a payoff of T.To defect will be a dominant strategy for both players if

A)S + T > 100.
B)T > 2S.
C)S < 0 and T >100.
D)S < T and T > 100.
E)S and T are any positive numbers.

Question 8

In the prisoner's dilemma game, if each prisoner believed that the other prisoner would deny the crime, then both would deny the crime.

Question 9

In the town of Torrelodones, each of the N > 2 inhabitants has $100.They are told that they can all voluntarily contribute to a fund that will be evenly divided among all residents.If $F are contributed to the fund, the local K-Mart will match the private contributions so that the total amount to be divided is $2F.That is, each resident will get back a payment of $2F/N when the fund is divided.If the people in town care only about their own net incomes, in Nash equilibrium, how much will each person contribute to the fund?

A)$0
B)$10
C)$20
D)$50
E)$100

Question 10

Frank and Nancy met at a sorority sock hop.They agreed to meet for a date at a local bar the next week.Regrettably, they were so fraught with passion that they forgot to agree on which bar would be the site of their rendezvous.Luckily, the town has only two bars, Rizotti's and the Oasis.Having discussed their tastes in bars at the sock hop, both are aware that Frank prefers Rizotti's to the Oasis and Nancy prefer the Oasis to Rizotti's.In fact, the payoffs are as follows.If both go to the Oasis, Nancy's utility is 3 and Frank's utility is 2.If both go to Rizotti's, Frank's utility is 3 and Nancy's utility is 2.If they don't both go to the same bar, both have a utility of 0.

A)This game has no Nash equilibrium in pure strategies.
B)This game has a dominant strategy equilibrium.
C)There are two Nash equilibrium in pure strategies and a Nash equilibrium in mixed strategies where the probability that Frank and Nancy go to the same bar is 12/25.
D)This game has two Nash equilibria in pure strategies and a Nash equilibrium in mixed strategies where each person has a probability of 1/2 of going to each bar.
E)This game has exactly one Nash equilibrium.

Question 11

A situation where everyone is playing a dominant strategy must be a Nash equilibrium.

Question 12

A general has the two possible pure strategies, sending all of his troops by land or sending all of his troops by sea.An example of a mixed strategy is where he sends 1/4 of his troops by land and 3/4 of his troops by sea.

Question 13

Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough.If both pigs choose Wait at the Trough, both get 3.If both pigs choose Press the Button, then Big Pig gets 8 and Little Pig gets 2.If Little Pig presses the button and Big Pig waits at the trough, then Big Pig gets 10 and Little Pig gets 0.Finally, if Big Pig presses the button and Little Pig waits at the trough, then Big Pig gets 2 and Little Pig gets 1.In Nash equilibrium,

A)Little Pig will get a payoff of 1 and Big Pig will get a payoff of 2.
B)Little Pig will get a payoff of 2 and Big Pig will get a payoff of 8.
C)both pigs will wait at the trough.
D)Little pig will get a payoff of zero.
E)the pigs must be using mixed strategies.

Question 14

Two players are engaged in a game of Chicken.There are two possible strategies, Swerve and Drive Straight.A player who chooses to Swerve is called Chicken and gets a payoff of zero, regardless of what the other player does.A player who chooses to Drive Straight gets a payoff of 32 if the other player swerves and a payoff of -48 if the other player also chooses to Drive Straight.This game has two pure strategy equilibria and

A)a mixed strategy equilibrium in which each player swerves with probability .60 and drives straight with probability .40.
B)two mixed strategies in which players alternate between swerving and driving straight.
C)a mixed strategy equilibrium in which one player swerves with probability .60 and the other swerves with probability .40.
D)a mixed strategy in which each player swerves with probability .30 and drives straight with probability .70.
E)no mixed strategies.

Question 15

George and Sam have taken their fathers' cars out on a lonely road and are engaged in a game of Chicken.George has his father's Mercedes and Sam has his father's rattly little Yugoslavian-built subcompact car.Each of the players can choose either to Swerve or to Not Swerve.If both choose Swerve, both get a payoff of zero.If one chooses Swerve and the other chooses Not Swerve, the one who chooses Not Swerve gets a payoff of 10 and the one who chooses Swerve gets zero.If both choose Not Swerve, the damage to George's car is fairly minor and he gets a payoff of -5, while for Sam the results are disastrous and he gets a payoff of -100.

A)This game has a dominant strategy equilibrium in which George does not swerve and Sam swerves.
B)This game has two pure strategy Nash equilibria and no mixed strategy equilibrium.
C)This game has three different Nash equilibria, two of which are pure strategy equilibrium and one of which is a mixed strategy equilibrium in which George is more likely to swerve than Sam is.
D)The one and only Nash equilibrium in this game is where George does not swerve and Sam swerves.
E)This game has two pure strategy equilibria and a mixed strategy equilibrium in which Sam randomizes his strategy and George chooses Not Swerve with certainty.

Question 16

In Nash equilibrium, each player is making an optimal choice for herself, given the choices of the other players.

Question 17

Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough.If both pigs choose Wait at the Trough, both get 2.If both pigs choose Press the Button, then Big Pig gets 5 and Little Pig gets 5.If Little Pig presses the button and Big Pig waits at the trough, then Big Pig gets 10 and Little Pig gets 0.Finally, if Big Pig presses the button and Little Pig waits at the trough, then Big Pig gets 6 and Little Pig gets 2.In Nash equilibrium,

A)Little pig will get a payoff of zero.
B)Little Pig will get a payoff of 5 and Big Pig will get a payoff of 5.
C)both pigs will wait at the trough.
D)Little Pig will get a payoff of 2 and Big Pig will get a payoff of 6.
E)the pigs must be using mixed strategies.

Question 18

In a Nash equilibrium, everyone must be playing a dominant strategy.

Question 19

In the game matrix below, the first payoff in each pair goes to player A who chooses the row, and the second payoff goes to player B, who chooses the column.Let a, b, c, and d be positive constants.
If player A chooses Bottom and player B chooses Right in a Nash equilibrium, then we know that

A)b > 1 and d < 1.
B)c < 1 and b < 1.
C)b < 1 and c < d
D) b < c and d < 1.
E) a < 1 and b < d.

Question 20

A dominant strategy equilibrium is a set of choices such that each player's choices are optimal regardless of what the other players choose.

Question 21

If the number of persons who attend the club meeting this week is X, then the number of people who will attend next week is 63 + 0.30X.What is a long-run equilibrium attendance for this club?

A)63
B)210
C)126
D)90
E)27

Question 22

The coach of the offensive football team has two options on the next play.He can run the ball or he can pass.His rival can defend either against the run or against the pass.Suppose that the offense passes.Then if the defense defends against the pass, the offense will make zero yards, and if the defense defends against the run, the offense will make 25 yards.Suppose that the offense runs.If the defense defends against the pass, the offense will make 10 yards, and if the defense defends against a run, the offense will gain 2 yards.
a.Write down a payoff matrix for this game.
b.Is there a Nash equilibrium in pure strategies for this game? If so, what is it? If not, demonstrate that there is none.

Question 23

Professor Binmore has a monopoly in the market for undergraduate game theory textbooks.The time-discounted value of Professor Binmore's future earnings is $4,000.Professor Ditt is considering writing a book to compete with Professor Binmore's book.With two books amicably splitting the market, the time-discounted value of each professor's future earnings would be $400.If there is full information (each professor knows the profits of the other), under what conditions could Professor Binmore deter the entry of Professor Ditt into his market?

A)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would loose $8.In so doing, Professor Binmore would still make $410 over time.
B)Professor Binmore threatens to cut his price so that Professor Ditt would loose $800.In so doing, Professor Binmore would loose $80 over time.
C)Professor Binmore threatens to cut his price so that Professor Ditt would loose $80.In so doing, Professor Binmore would just break even over time.
D)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would only make $8.In so doing, Professor Binmore would still make $200 over time.
E)None of the above.

Question 24

A famous Big Ten football coach had only two strategies, Run the ball to the left side of the line and Run the ball to the right side.The defense can concentrate forces on the left side or the right side.If the opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards.If the defense defended the left side and the offense ran left, the offense gained only 1 yard.If the opponent defended the right side when the offense ran right, the offense would still gain at least 5 yards with probability .50.It is the last play of the game and the famous coach's team is on offense.If it makes 5 yards or more, it wins; if not, it loses.Both sides choose Nash equilibrium strategies.In equilibrium the offense

A)will run to the right side with probability .67.
B)will run to the right side with probability .80.
C)will run to the two sides with equal probability.
D)is sure to run to the right side.
E)will run to the right side with probability .50.

Question 25

A famous Big Ten football coach had only two strategies, Run the ball to the left side of the line and Run the ball to the right side.The defense can concentrate forces on the left side or the right side.If the opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards.If the defense defended the left side and the offense ran left, the offense gained only 1 yard.If the opponent defended the right side when the offense ran right, the offense would still gain at least 5 yards with probability .70.It is the last play of the game and the famous coach's team is on offense.If it makes 5 yards or more, it wins; if not, it loses.Both sides choose Nash equilibrium strategies.In equilibrium the offense

A)will run to the two sides with equal probability.
B)will run to the right side with probability .87.
C)is sure to run to the right side.
D)will run to the right side with probability .77.
E)will run to the right side with probability .70.

Question 26

Two players are engaged in a game of Chicken.There are two possible strategies, Swerve and Drive Straight.A player who chooses to Swerve is called Chicken and gets a payoff of zero, regardless of what the other player does.A player who chooses to Drive Straight gets a payoff of 36 if the other player swerves and a payoff of -36 if the other player also chooses to Drive Straight.This game has two pure strategy equilibria and

A)a mixed strategy equilibrium in which each player swerves with probability .50 and drives straight with probability .50.
B)a mixed strategy equilibrium in which one player swerves with probability .50 and the other swerves with probability .50.
C)a mixed strategy in which each player swerves with probability .25 and drives straight with probability .75.
D)two mixed strategies in which players alternate between swerving and driving straight.
E)no mixed strategies.

Question 27

Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each player is -6 if both play Hawk.If both play Dove, the payoff to each player is 4, and if one plays Hawk and the other plays Dove, the one that plays Hawk gets a payoff of 6 and the one that plays Dove gets 0.In equilibrium, we would expect hawks and doves to do equally well.This happens when the proportion of the total population that plays Hawk is

A).25.
B).63.
C).06.
D).13.
E)1.

Question 28

If the number of persons who attend the club meeting this week is X, then the number of people who will attend next week is 90 + 0.40X.What is a long-run equilibrium attendance for this club?

A)225
B)150
C)180
D)90
E)60

Question 29

Professor Binmore has a monopoly in the market for undergraduate game theory textbooks.The time-discounted value of Professor Binmore's future earnings is $2,000.Professor Ditt is considering writing a book to compete with Professor Binmore's book.With two books amicably splitting the market, the time-discounted value of each professor's future earnings would be $200.If there is full information (each professor knows the profits of the other), under what conditions could Professor Binmore deter the entry of Professor Ditt into his market?

A)Professor Binmore threatens to cut his price so that Professor Ditt would loose $200.In so doing, Professor Binmore would loose $20 over time.
B)Professor Binmore threatens to cut his price so that Professor Ditt would loose $20.In so doing, Professor Binmore would just break even over time.
C)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would loose $2.In so doing, Professor Binmore would still make $190 over time.
D)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would only make $2.In so doing, Professor Binmore would still make $100 over time.
E)None of the above.

Question 30

Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each player is -6 if both play Hawk.If both play Dove, the payoff to each player is 3, and if one plays Hawk and the other plays Dove, the one that plays Hawk gets a payoff of 8 and the one that plays Dove gets 0.In equilibrium, we would expect hawks and doves to do equally well.This happens when the proportion of the total population that plays Hawk is

A).45.
B).23.
C).11.
D).73.
E)1.

Question 31

A famous Big Ten football coach had only two strategies, Run the ball to the left side of the line and Run the ball to the right side.The defense can concentrate forces on the left side or the right side.If the opponent concentrates on the wrong side, his offense is sure to gain at least 5 yards.If the defense defended the left side and the offense ran left, the offense gained only 1 yard.If the opponent defended the right side when the offense ran right, the offense would still gain at least 5 yards with probability .30.It is the last play of the game and the famous coach's team is on offense.If it makes 5 yards or more, it wins; if not, it loses.Both sides choose Nash equilibrium strategies.In equilibrium the offense

A)is sure to run to the right side.
B)will run to the right side with probability .59.
C)will run to the right side with probability .74.
D)will run to the two sides with equal probability.
E)will run to the right side with probability .70.

Question 32

If the number of persons who attend the club meeting this week is X, then the number of people who will attend next week is 80 + 0.20X.What is a long-run equilibrium attendance for this club?

A)80
B)400
C)160
D)100
E)20

Question 33

Professor Binmore has a monopoly in the market for undergraduate game theory textbooks.The time-discounted value of Professor Binmore's future earnings is $2,000.Professor Ditt is considering writing a book to compete with Professor Binmore's book.With two books amicably splitting the market, the time-discounted value of each professor's future earnings would be $200.If there is full information (each professor knows the profits of the other), under what conditions could Professor Binmore deter the entry of Professor Ditt into his market?

A)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would loose $8.In so doing, Professor Binmore would still make $210 over time.
B)Professor Binmore threatens to cut his price and attack the credibility of Professor Ditt's book so that Professor Ditt would only make $8.In so doing, Professor Binmore would still make $100 over time.
C)Professor Binmore threatens to cut his price so that Professor Ditt would loose $800.In so doing, Professor Binmore would loose $80 over time.
D)Professor Binmore threatens to cut his price so that Professor Ditt would loose $80.In so doing, Professor Binmore would just break even over time.
E)None of the above.

Question 34

Suppose that in a Hawk-Dove game similar to the one discussed in your workbook, the payoff to each player is -9 if both play Hawk.If both play Dove, the payoff to each player is 4, and if one plays Hawk and the other plays Dove, the one that plays Hawk gets a payoff of 5 and the one that plays Dove gets 0.In equilibrium, we would expect hawks and doves to do equally well.This happens when the proportion of the total population that plays Hawk is

A).10.
B).55.
C).05.
D).03.
E)1.