Problem Set 4: Due in class on Tuesday July 28. Solution s to this homework will be posted right after class hence no late submissions will be accepted. Test 4 on the content of this homework will be given on August 4 at 9:00am sharp. Problem 1 (4p) Consider the following game: (a) Suppose that the Column player announces that he will play X with probability 0.5 and Y with probability 0.5 i.e., ½ X ½ Y. Identify all best response strategies of the Row player, i.e., BR(½ X ½ Y) ? (b) Identify all best response strategies of the Column player to Row playing ½ A ½ B, i.e. BR(½ A ½ B)? (c) What is BR(1/5 X 1/5 Y 3/5 Z)? (d) What is BR(1/5 A 1/5 B 3/5 C)? X Y Z A 2 1 1 3 5 -2 B 4 -1 2 1 1 2 C 0 4 3 0 2 1 Page 2 of 4 Problem 2 (4p) Here comes the Two-Finger Morra game again: C1 C2 C3 C4 R1 0 0 -2 2 3 -3 0 0 R2 2 -2 0 0 0 0 -3 3 R3 -3 3 0 0 0 0 4 -4 R4 0 0 3 -3 -4 4 0 0 To exercise notation and concepts involved in calculating payoffs to mixed strategies, calculate the following (uR, uC stand for the payoffs to Row and Column respectively): (a) uR(0.4 R1 0.6 R2, C2) = (b) uC(0.4 C1 0.6 C2, R3) = (c) uR(0.3 R2 0.7 R3, 0.2 C1 0.3 C2 0.5 C4 ) = (d) uC(0.7 C2 0.3 C4, 0.7 R1 0.2 R2 0.1 R3) = Problem 3 (4p) X Y A 1 6 3 1 B 2 3 0 4 For the game above: (1) Draw the best response function for each player using the coordinate system below. Mark Nash equilibria on the diagram. Page 3 of 4 (2) List the pair of mixed strategies in Nash equilibrium. (3) Calculate each player’s payoffs in Nash equilibrium. Problem 4 (4p) C1 C2 C3 C4 R1 0 0 -2 2 3 -3 0 0 R2 2 -2 0 0 0 0 -3 3 R3 -3 3 0 0 0 0 4 -4 R4 0 0 3 -3 -4 4 0 0 In the Two-Finger morra game above suppose Row decided to play a mix of R1 and R2 and Column decided to play a mix of C1 and C3. In other words, assume that the original 44 game is reduced to the 22 game with R1 and R2 and C1 and C3. Using our customary coordinate system: (a) Draw the best response functions of both players in the coordinate system as above. (b) List all Nash equilibria in the game. (c) Calculate each player’s payoff in Nash equilibrium. p=1 p=0 q=1 q=0 Page 4 of 4 Problem 5 (4p) Lucy offers to play the following game with Charlie: “let us show pennies to each othe ...