Exam 28: A: Game Applications

Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is -7 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 7 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

(See Problem 3.) 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 3 if the other player Swerves and a payoff of -12 if the other player also chooses to Drive Straight. This game has two pure strategy equilibria and

(See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $3. The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and Bertha in this game?

(See Problem 3.) 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 4 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

(See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both pigs choose Wait, both get 3. If both pigs Press the Button, then Big Pig gets 9 and Little Pig gets 1. 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, then Big Pig gets 4 and Little Pig gets 1. In Nash equilibrium,

(See Problem 7.) The old Michigan 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 either on the left side or the right side of Michigan's line. If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan would

Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is -3 if both play Hawk. If both play Dove, the payoff to each player is 1, 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

(See Problem 7.) The old Michigan 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 either on the left side or the right side of Michigan's line. If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .60. It is the last play of the game and Michigan needs to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan would

(See Problem 7.) The old Michigan 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 either on the left side or the right side of Michigan's line. If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan would

(See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both pigs choose Wait, both get 1. If both pigs Press the Button, then Big Pig gets 9 and Little Pig gets 1. 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, then Big Pig gets 5 and Little Pig gets 1. In Nash equilibrium,

(See Problem 7.) The old Michigan 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 either on the left side or the right side of Michigan's line. If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .70. It is the last play of the game and Michigan needs to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan would

(See Problem 3.) 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 84 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

(See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $5. The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and Bertha in this game?

(See Problem 7.) The old Michigan 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 either on the left side or the right side of Michigan's line. If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards. If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain. But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .50. It is the last play of the game and Michigan needs to gain 5 yards to win. Both sides choose Nash equilibrium strategies. In Nash equilibrium, Michigan would

Suppose that in the Hawk-Dove game discussed in Problem 8, the payoff to each player is -7 if both play Hawk. If both play Dove, the payoff to each player is 5, and if one plays Hawk and the other plays Dove, the one that plays Hawk gets a payoff of 9 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

(See Problem 6.) Big Pig and Little Pig have two possible strategies, Press the Button, and Wait at the Trough. If both pigs choose Wait, both get 1. If both pigs Press the Button, then Big Pig gets 6 and Little Pig gets 4. 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, then Big Pig gets 6 and Little Pig gets 2. In Nash equilibrium,

(See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $4. The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and Bertha in this game?

(See Problem 3.) 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 15.43 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

(See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $3. The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and Bertha in this game?

(See Problem 2.) Arthur and Bertha are asked by their boss to vote on a company policy. Each of them will be allowed to vote for one of three possible policies, A, B, and C. Arthur likes A best, B second best, and C least. Bertha likes B best, A second best, and C least. The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $4. The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1. The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for. If Arthur and Bertha vote for different outcomes, the boss will pick C. Arthur and Bertha know this is the case. They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B. What is the mixed strategy equilibrium for Arthur and Bertha in this game?