Homework Answers
Answer :
Assuming each player (1 or 2) tries to maximize the payoff shown in the matrix, that is the higher the better. Also 1 is the row player and 2 is the column player)
Player 1 :
Best response to W = B (Payoff of 20 is the highest for 1 when 2 selects W)
Best response to X = A (Payoff of 11 is the highest for 1 when 2 selects X)
Best response to Y = A (Payoff of 10 is the highest for 1 when 2 selects Y)
Best response to Z = C (Payoff of 16 is the highest for 1 when 2 selects Z)
Player 2 :
Best response to A = Z (Payoff of 12 is the highest for 2 when 1 selects A)
Best response to B = Y (Payoff of 18 is the highest for 2 when 1 selects B)
Best response to C = X (Payoff of 10 is the highest for 2 when 1 selects C)
Best response to D = Z (Payoff of 10 is the highest for 2 when 1 selects D)
N < 1\2 A B C w х Y 6,5 11,2 10,5 1, 12 20,4 8,8...
N < 1\2 A B C w х Y 6,5 11,2 10,5 1, 12 20,4 8,8 2, 18 2, 18 8,5 3,7 2, 10 6,4 16,3 11,7 10,5 1,6 1 D 6, 10 1 QUESTION 2. (35 pts) Using the process of iterated dominance, find the rationalizable set for this game. Carefully (justifying fully each step you take) the process of arriving at the rationalizable set. If I cannot follow what you are doing, you will not get full credit...
QUESTION 1. (15 pts) Consider the game to the right. 1\2 А B с D 00D...
QUESTION 1. (15 pts) Consider the game to the right. 1\2 А B с D 00D W Х 5,6 4,20 2,11 8,8 5, 10 18,2 12,0 5,8 Y 7,3 10,2 4,6 3, 16 z 7, 11 5, 10 6,1 10,6 Player 1: Best Response to W Best Response to X Best Response to Y Best Response to Z Player 2: Best Response to A Best Response to B. Best Response to C Best Response to D
1\2 А. С х z 11,2 1, 12 2,10 16,3 10,5 6, 10 D QUESTION 3....
1\2 А. С х z 11,2 1, 12 2,10 16,3 10,5 6, 10 D QUESTION 3. (25 pts) Find all Nash equilibria for the game in Question 1. Remark. If you did not make much headway on Question 2, you may find all the Nash equilibria for the game provided above (for less credit).
7. Consider the normal-form game pictured here: 1 x 2,0 y 1,3 z 5,x A B...
7. Consider the normal-form game pictured here: 1 x 2,0 y 1,3 z 5,x A B 5 ,4 1,3 6,2 All of the payoff numbers are specified, with the exception of that denoted by x. Find a number for x such that the following three statements are all true: (B, X) is a Nash equilibrium, (A, Z) is an efficient strategy profile, and, for the belief , = 6,5), Y is a best response for player 2; that is, Y...
Question 3: [5pt total] Consider the following game: Player 1 Player 2 X Y Z A...
Question 3: [5pt total] Consider the following game: Player 1 Player 2 X Y Z A 4,0 -3,8 -7, -1 B-4,3 0,6 9,5 C3,2 2,-1 11, 9 Let B1(-) and B2(-) be the best response function for player 1 and player 2 respectively. Calculate the following: 3)a) [1pt] B1(X) 3)b) [1pt] B2(B) 3)c) (1pt] Social Welfare Maximum: 3)d) [1pt] Dominant Strategies for Player 1: 3)) [1pt] Pure Nash Equilibriums:
3. Consider the following game in normal form. Player 1 is the "row" player with strate-...
3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W Z X y a 3,3 2,1 0,2 2,1 b 1,1 1,2 1,0 1,4 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 с Find all the Nash equilibria in the game in pure strategies.
1. Consider the following game in normal form. Player 1 is the "row" player with strate-...
1. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, 2. The game is presented in the following matrix: a b c d w 3,3 1,1 0,0 0,0 x 2,1 1,2 1,0 0,5 y 0,2 1,0 3, 2 0,2 z 2,1 1,4 1,1 3,1 (a) Find the set of rationalizable strategies. (b) Find the set of Nash...
Find all the Nash equilibria in the following game and indicate which are strict. Player 2...
Find all the Nash equilibria in the following game and indicate which are strict. Player 2 d b a -1,4 1,-3 2,7 W 2,7 Player 1 2.1 0,4 1, 3 1, 2 Y -1,6 6,2 3.2 1,1 Z 7,1 5.2 0.2 3,1 O (Wa) and (W,c). Neither are strict. O (W,c) and (Z,b). Both are strict O (Wc) and (Z,b). Neither are strict. O There are no Nash equilibria in this game.
1\2 A B с D W X 5,6 4,20 2, 11 8,8 5,10 18,2 12,0 5,8...
1\2 A B с D W X 5,6 4,20 2, 11 8,8 5,10 18,2 12,0 5,8 Y 7,3 10,2 4,6 3, 16 Z 7, 11 5, 10 6,1 10,6 QUESTION 2. (35 pts) Using the process of iterated dominance, find the rationalizable set for this game. Carefully (justifying fully each step you take) the process of arriving at the rationalizable set. If I cannot follow what you are doing, you will not get full credit on this question.
1. (20 points) Consider the following game: Left 7,17 10,5 4,4 Top Middle Bottom Player B...
1. (20 points) Consider the following game: Left 7,17 10,5 4,4 Top Middle Bottom Player B Middle 21,21 14,4 7,3 Right 14,11 4,3 10,25 Player A a. Does either player have a dominant strategy? What about a dominated strategy? b. What are the Nash equilibria of this game? C. Is there one Nash equilibrium that you think is a more likely outcome than the others? If so, why? If not, why not? d. Now suppose the game looks like this:...
N < 1\2 A B C w х Y 6,5 11,2 10,5 1, 12 20,4 8,8 2, 18 2, 18 8,5 3,7 2, 10 6,4 16,3 11,7 10,5 1,6 1 D 6, 10 1 QUESTION 2. (35 pts) Using the process of iterated dominance, find the rationalizable set for this game. Carefully (justifying fully each step you take) the process of arriving at the rationalizable set. If I cannot follow what you are doing, you will not get full credit...
QUESTION 1. (15 pts) Consider the game to the right. 1\2 А B с D 00D W Х 5,6 4,20 2,11 8,8 5, 10 18,2 12,0 5,8 Y 7,3 10,2 4,6 3, 16 z 7, 11 5, 10 6,1 10,6 Player 1: Best Response to W Best Response to X Best Response to Y Best Response to Z Player 2: Best Response to A Best Response to B. Best Response to C Best Response to D
1\2 А. С х z 11,2 1, 12 2,10 16,3 10,5 6, 10 D QUESTION 3. (25 pts) Find all Nash equilibria for the game in Question 1. Remark. If you did not make much headway on Question 2, you may find all the Nash equilibria for the game provided above (for less credit).
7. Consider the normal-form game pictured here: 1 x 2,0 y 1,3 z 5,x A B 5 ,4 1,3 6,2 All of the payoff numbers are specified, with the exception of that denoted by x. Find a number for x such that the following three statements are all true: (B, X) is a Nash equilibrium, (A, Z) is an efficient strategy profile, and, for the belief , = 6,5), Y is a best response for player 2; that is, Y...
Question 3: [5pt total] Consider the following game: Player 1 Player 2 X Y Z A 4,0 -3,8 -7, -1 B-4,3 0,6 9,5 C3,2 2,-1 11, 9 Let B1(-) and B2(-) be the best response function for player 1 and player 2 respectively. Calculate the following: 3)a) [1pt] B1(X) 3)b) [1pt] B2(B) 3)c) (1pt] Social Welfare Maximum: 3)d) [1pt] Dominant Strategies for Player 1: 3)) [1pt] Pure Nash Equilibriums:
3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W Z X y a 3,3 2,1 0,2 2,1 b 1,1 1,2 1,0 1,4 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 с Find all the Nash equilibria in the game in pure strategies.
1. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, 2. The game is presented in the following matrix: a b c d w 3,3 1,1 0,0 0,0 x 2,1 1,2 1,0 0,5 y 0,2 1,0 3, 2 0,2 z 2,1 1,4 1,1 3,1 (a) Find the set of rationalizable strategies. (b) Find the set of Nash...
Find all the Nash equilibria in the following game and indicate which are strict. Player 2 d b a -1,4 1,-3 2,7 W 2,7 Player 1 2.1 0,4 1, 3 1, 2 Y -1,6 6,2 3.2 1,1 Z 7,1 5.2 0.2 3,1 O (Wa) and (W,c). Neither are strict. O (W,c) and (Z,b). Both are strict O (Wc) and (Z,b). Neither are strict. O There are no Nash equilibria in this game.
1\2 A B с D W X 5,6 4,20 2, 11 8,8 5,10 18,2 12,0 5,8 Y 7,3 10,2 4,6 3, 16 Z 7, 11 5, 10 6,1 10,6 QUESTION 2. (35 pts) Using the process of iterated dominance, find the rationalizable set for this game. Carefully (justifying fully each step you take) the process of arriving at the rationalizable set. If I cannot follow what you are doing, you will not get full credit on this question.
1. (20 points) Consider the following game: Left 7,17 10,5 4,4 Top Middle Bottom Player B Middle 21,21 14,4 7,3 Right 14,11 4,3 10,25 Player A a. Does either player have a dominant strategy? What about a dominated strategy? b. What are the Nash equilibria of this game? C. Is there one Nash equilibrium that you think is a more likely outcome than the others? If so, why? If not, why not? d. Now suppose the game looks like this:...
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