3. 1\2 a b C d X y Z 1,2 2,2 5,1 4,1 3,5 3,3 5,2 4,4 7,0 2,3 0,4 3,0 When player 1 plays d, what is player 2's best response: a) Only x ○ b) Only y c) Only z d) Both y and z
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- Exercise 11 Dominance and Best Responses (H) (a) Show that if an action is a best response (to some strategy of the other players), then it is undominated. (b) Show that every weakly dominant strategy is pure. (c) Now, let N = {1, 2} and si be a best response for player 1 to both t and t'. Show that si is also a best response to all mixtures t of t and t,, where for any pe [0, 1], t is defined by t = [p : t, (1 – p) : t].Suppose 2 players play the following game infinite times in the future. What should be the minimum value of the time discount so that the equilibrium strategy is (Cooperate, Cooperate)? Player 2 Player 1 Cooperate Defect Cooperate (10, 10) Defect (2, 12) (4,4) (12, 2)Two players play the following game for infinite times. For the player to continue to cooperate what would be the ranges of their discount factor, δ_1 and δ_2, respectively? cooperate betray cooperate (10,20) (-25,30) betray (15, -22) (-12, -18)
- If we solve this game by elimination of dominated strategies, what payoff does Gino receive? (Gino's payoffs are located on the right of each cell, after the comma.) Gino Left Right Up (18.5, 7.4) (11.1, 33.3) Maria Down (25.9, 29.6) (14.8, 11.1) Answer:(a) Stan and Ollie are two students who share a flat. Both of them prefer to live in a clean flat. However, neither is too fond of housecleaning. Each of them receives a payoff of 12 if they both clean the flat. If neither person cleans the flat, they receive a payoff of 6 each. If one person cleans the flat but the other person does not, then the payoff for the person who does the cleaning is 5 and the payoff for the person who doesn't do any cleaning is 15. (i) Write down the payoff matrix of this game. Derive the dominant strategy equilibrium. Is this also a Nash equilibrium? (ii) Expiain your reasoning. Consider a game with N players. Each player chooses Black or White. If a player (b) chooses Black, she gets 100 if everyone else also chooses Black, and she gets 0 if any of the other players does not choose Black. If a player chooses White, she always gets 50. Show that everyone choosing Black and everyone choosing White are both Nash equilibria of this game.In this game, (80, 50) 52% a) (R, N) b) (R, R) c) (S, N) d) (R, P) S is a Nash equilibrium First Player (20,68) 12% R P Second Player N (90,70) 36%
- (ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3DIn Homework 1, we saw that there is no strict dominant strategy equilibrium in an inspection game. Thus, the players must randomize and play a mixed strategy in equilibrium. Suppose the agent shirks with probability p₁ and works with probability (1 - p₁), while the principal inspects with probability p2 and does not inspect with probability (1-P2). Find the mixed strategy equilibrium for this game. I NI S (0, -k) (w, -w) W (w c, p-w-k)| (wc, p-w) Table 1: Payoff matrix for the inspection gameWhich one is the best strategy for Player 1? Player 1 (10, 20) Leave Leave Cooperate Stay Cooperate Player2 Not cooperate Stay (15, 14) Not cooperate Leave Player 2 (17, 11) Stay (16, 12)
- Consider the following representation of a Normal form game. actions w a (45,22) (10,38) (42,13) (10,7) (p,28) (15,40) (q,10) (44,10) (20,22) (14,31) (27,13) (12,8) d. (20,41) (9,48) (28,24) (18,32) Here each cell in the table represents an ordered pair. First element is payoff of the first player and second element is payoff of the second player. The letters a, b, c, d, x, y, z, w represent the actions. Write down the table in your answer script too. Now answer the following questions: 1. What is the distinction between strictly dominant strategy and weakly dominant strategy? Is it reasonable for a player to play a strictly dominated strategy? Explain why. 2. What are the minimum values for p and q that will make ba strategy that strictly dominates all other strategies for player 1, assuming both p and q are natural numbers? 3. Does player 2 have any strictly dominated pure strategy? If yes, which pure strategy dominates that strategy? If the submit button is off it is beacuse the due…Question 22 Consider the following Extended Form Game: 01 02 04 A O 3 (2,3) P1 A' (5,5) A' (3,8) B A P2 B' P2 B' (1,9) According to backwards-induction, in how many steps does the game end? P1 B (9,1)(b) Consider the simultaneous-move game below with two players, 1 and 2. Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for that player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. Player 2 b₁ b₂ Player 1 a₁ 3,0 0,1 a2 2,1 2,1