5. [Subgame perfection] Consider the following game. Two friends, Anne (A) and Bob (B), are trying...
5. [Subgame perfection] Consider the following game. Two friends, Anne (A) and Bob (B), are trying to coordinate on a joint activity, which can be either knitting (K) or fencing (F). Both friends prefer coordinating (i.e., choosing the same activity) to not coordinating but, conditional on coordinating, A prefers K to F while B prefers F to K. In the game, A chooses K or F first; B observes A’s choice and then chooses K or F himself. 3
a. Draw a game tree that depicts the game above. Assign numerical payoffs that capture the preferences described. I suggest setting the miscoordination payoffs for both A and B to zero, but this is not necessary.
b. Is there a Nash equilibrium in which the players coordinate on fencing? If so, describe the two players’ strategies. If not, explain why not.
c. Is there a subgame perfect Nash equilibrium in which the players coordinate on fencing? If so, describe the two players’ strategies. If not, explain why not.
d. List all possible outcomes of the game if play is according to a subgame perfect equilibrium.
Homework Answers
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5. [Subgame perfection] Consider the following game. Two
friends, Anne (A) and Bob (B), are trying...
Problem 1. (20 points) Consider a game with two players, Alice and Bob. Alice can choose...
Problem 1. (20 points) Consider a game with two players, Alice and Bob. Alice can choose A or B. The game ends if she chooses A while it continues to Bob if she chooses B. Bob then can choose C or D. If he chooses C the game ends, and if he chooses D the game continues to Alice. Finally, Alice can choose E or F and the game ends after each of these choices. a. Present this game as...
Froblem #5: Convert extensive-form to strategic-form, find Nash equilibria and subgame. perfect Nash equilibria (12...
Froblem #5: Convert extensive-form to strategic-form, find Nash equilibria and subgame. perfect Nash equilibria (12pts) Consider the following extensive-form game: Veto Y Don't Veto In this game, Players 1 and 2 are deciding on a course of action, which may be X, Y, or Z Player 2 is the one who actually makes the choice, but first Player may choose to veto Y, which is the option Player 1 prefers the least. a) List all the strategies available to Player...
4. Consider the following game that is played T times. First, players move simultaneously and independently....
4. Consider the following game that is played T times. First, players move simultaneously and independently. Then each player is informed about the actions taken by the other player in the first play and, given this, they play it again, and so on. The payoff for the whole game is the sum of the payoffs a player obtains in the T plays of the game A 3,1 4,0 0,1 В 1,5 2,2 0,1 C 1,1 0,2 1,2 (a) (10%) Suppose...
6. Consider a sequential game with 3 players. Player 1 can choose A or B. Player...
6. Consider a sequential game with 3 players. Player 1 can choose A or B. Player 2 can choose C, D, E, or F (depending on what player 1 chooses). Player 3 can choose G, H, I, J, K, L, M, or N (depending on what player 1 and 2 choose). Player 1 (P1) goes first, player 2 (P2) goes second, and player 3 (P3) goes third. Payoffs are written as the payoffs for P1, P2, and the for P3....
2. Consider the following sequential game. Player A can choose between two tasks, Tl and T2....
2. Consider the following sequential game. Player A can choose between two tasks, Tl and T2. After having observed the choice of A, Player B chooses between two projects Pl or P2. The payoffs are as follows: If A chooses TI and B chooses P1 the payoffs are (12, 8), where the first payoff is for A and the second for B; if A chooses T1 and B opts for P2 the payoffs are (20, 7); if A chooses T2...
2. Consider the following sequential game. Player A can choose between two tasks, TI and T2....
2. Consider the following sequential game. Player A can choose between two tasks, TI and T2. After having observed the choice of A, Player B chooses between two projects P1 or P2. The payoffs are as follows: If A chooses TI and B chooses Pl the payoffs are (12.8), where the first payoff is for A and the second for B; if A chooses TI and B opts for P2 the payoffs are (20,7); if A chooses T2 and B...
Consider the following extensive-form game with two players, 1 and 2. a). Find the pure-strategy Nash...
Consider the following extensive-form game with two players, 1
and 2.
a). Find the pure-strategy Nash equilibria of the game. [8
Marks]
b). Find the pure-strategy subgame-perfect equilibria of the
game. [6 Marks]
c). Derive the mixed strategy Nash equilibrium of the subgame.
If players play this mixed Nash equilibrium in the subgame, would 1
player In or Out at the initial mode? [6 Marks]
[Hint: Write down the normal-form of the subgame and derive the
mixed Nash equilibrium of...
3. Player 1 and Player 2 are going to play the following stage game twice: &n...
3. Player 1 and Player 2 are going to play the following stage
game twice:
Player 2
Left
Middle
Right
Player 1
Top
4, 3
0, 0
1, 4
Bottom
0, 0
2, 1
0, 0
There is no discounting in this problem and so a player’s payoff
in this repeated game is the sum of her payoffs in the two plays of
the stage game.
(a) Find the Nash equilibria of the stage game. Is (Top, Left) a...
Question 1 a) First consider the following game, where each player plays either C (Confess) or D ...
Question 1 a) First consider the following game, where each player plays either C (Confess) or D (Deny) and the numbers in brackets are the respective payoffs to player 1 and player 2. Player 2 Player 1 (0-12) (-12,0) In relation to the above game outline the concepts of - Dominated strategies - Best responses - Nash equilibrium/equilibria - A prisoner's dilemma b) Define what is meant by subgame perfection and how the concept of credibility can be used to...
Consider the following normal form game: U D LR 7,7 4,8 8,4 5,5 a. Are there...
Consider the following normal form game: U D LR 7,7 4,8 8,4 5,5 a. Are there dominant actions for any of the players? b. Find all Nash equilibria of this game. c. Suppose we repeat this game 10 times, specify a subgame perfect equi- librium of this finitely repeated game. d. Suppose this game is repeated infinitely: Identify a subgame perfect equilibrium of this game which gives an average (normalized) dis- counted payoff of 7 to both players. Clearly identify...
Problem 1. (20 points) Consider a game with two players, Alice and Bob. Alice can choose A or B. The game ends if she chooses A while it continues to Bob if she chooses B. Bob then can choose C or D. If he chooses C the game ends, and if he chooses D the game continues to Alice. Finally, Alice can choose E or F and the game ends after each of these choices. a. Present this game as...
Froblem #5: Convert extensive-form to strategic-form, find Nash equilibria and subgame. perfect Nash equilibria (12pts) Consider the following extensive-form game: Veto Y Don't Veto In this game, Players 1 and 2 are deciding on a course of action, which may be X, Y, or Z Player 2 is the one who actually makes the choice, but first Player may choose to veto Y, which is the option Player 1 prefers the least. a) List all the strategies available to Player...
4. Consider the following game that is played T times. First, players move simultaneously and independently. Then each player is informed about the actions taken by the other player in the first play and, given this, they play it again, and so on. The payoff for the whole game is the sum of the payoffs a player obtains in the T plays of the game A 3,1 4,0 0,1 В 1,5 2,2 0,1 C 1,1 0,2 1,2 (a) (10%) Suppose...
2. Consider the following sequential game. Player A can choose between two tasks, Tl and T2. After having observed the choice of A, Player B chooses between two projects Pl or P2. The payoffs are as follows: If A chooses TI and B chooses P1 the payoffs are (12, 8), where the first payoff is for A and the second for B; if A chooses T1 and B opts for P2 the payoffs are (20, 7); if A chooses T2...
2. Consider the following sequential game. Player A can choose between two tasks, TI and T2. After having observed the choice of A, Player B chooses between two projects P1 or P2. The payoffs are as follows: If A chooses TI and B chooses Pl the payoffs are (12.8), where the first payoff is for A and the second for B; if A chooses TI and B opts for P2 the payoffs are (20,7); if A chooses T2 and B...
Consider the following extensive-form game with two players, 1
and 2.
a). Find the pure-strategy Nash equilibria of the game. [8
Marks]
b). Find the pure-strategy subgame-perfect equilibria of the
game. [6 Marks]
c). Derive the mixed strategy Nash equilibrium of the subgame.
If players play this mixed Nash equilibrium in the subgame, would 1
player In or Out at the initial mode? [6 Marks]
[Hint: Write down the normal-form of the subgame and derive the
mixed Nash equilibrium of...
3. Player 1 and Player 2 are going to play the following stage
game twice:
Player 2
Left
Middle
Right
Player 1
Top
4, 3
0, 0
1, 4
Bottom
0, 0
2, 1
0, 0
There is no discounting in this problem and so a player’s payoff
in this repeated game is the sum of her payoffs in the two plays of
the stage game.
(a) Find the Nash equilibria of the stage game. Is (Top, Left) a...
Question 1 a) First consider the following game, where each player plays either C (Confess) or D (Deny) and the numbers in brackets are the respective payoffs to player 1 and player 2. Player 2 Player 1 (0-12) (-12,0) In relation to the above game outline the concepts of - Dominated strategies - Best responses - Nash equilibrium/equilibria - A prisoner's dilemma b) Define what is meant by subgame perfection and how the concept of credibility can be used to...
Consider the following normal form game: U D LR 7,7 4,8 8,4 5,5 a. Are there dominant actions for any of the players? b. Find all Nash equilibria of this game. c. Suppose we repeat this game 10 times, specify a subgame perfect equi- librium of this finitely repeated game. d. Suppose this game is repeated infinitely: Identify a subgame perfect equilibrium of this game which gives an average (normalized) dis- counted payoff of 7 to both players. Clearly identify...
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