Exam 27: Game Theory

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

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 of his troops by land and of his troops by sea.

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

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

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 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

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,

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.

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

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?

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?

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

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. Player B If player A chooses Bottom and player B chooses Right in a Nash equilibrium, then we know that

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 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.

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

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.

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

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.