V. Multiple Choices or filling the blanks (Q23-27) Given the payoff matrix, answer the following questions Bob C D Alice A (1,1) (-2,-1) (0,0) B (-1,-2) Does Bob have a dominant strategy in this game? Yes Ⓒ No It depends on what Alice does. Not enough information to give a definitive answer. QUESTION 24 What is Alice's secure strategy? A B mixing between between A and B Omixing between C and D QUESTION 25 Which of the following is a Nash equilibrium of this game (there may be more than one correct answer for this question. Choose all that is correct) (A, C) □ (A.D) (B, C) (B, D) QUESTION 26 Which of the following is the socially optimal outcome of this game (there may be more than one correct answer for this question. Choose all that is correct) □ (A, C) □ (A.D) (B, C) (B, D) QUESTION 27 If Alice is allowed to move first, following which Bob moves. Which of the following would be a subgame perfect equilibrium of the new game? (there may be more than one correct answer for this question. Choose all that is correct) Alice plays A, and Bob plays C if Alice plays A and D if Alice plays B Alice plays A, and Bob plays D if Alice plays A and C if Alice plays B Alice plays B, and Bob plays C if Alice plays A and D if Alice plays B Alice plays B, and Bob plays D if Alice plays A and C if Alice plays B

V. Multiple Choices or filling the blanks (Q23-27) Given the payoff matrix, answer the following questions Bob C D Alice A (1,1) (-2,-1) (0,0) B (-1,-2) Does Bob have a dominant strategy in this game? Yes Ⓒ No It depends on what Alice does. Not enough information to give a definitive answer. QUESTION 24 What is Alice's secure strategy? A B mixing between between A and B Omixing between C and D QUESTION 25 Which of the following is a Nash equilibrium of this game (there may be more than one correct answer for this question. Choose all that is correct) (A, C) □ (A.D) (B, C) (B, D) QUESTION 26 Which of the following is the socially optimal outcome of this game (there may be more than one correct answer for this question. Choose all that is correct) □ (A, C) □ (A.D) (B, C) (B, D) QUESTION 27 If Alice is allowed to move first, following which Bob moves. Which of the following would be a subgame perfect equilibrium of the new game? (there may be more than one correct answer for this question. Choose all that is correct) Alice plays A, and Bob plays C if Alice plays A and D if Alice plays B Alice plays A, and Bob plays D if Alice plays A and C if Alice plays B Alice plays B, and Bob plays C if Alice plays A and D if Alice plays B Alice plays B, and Bob plays D if Alice plays A and C if Alice plays B

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?
Exercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.
2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
please solve it in 30 minute please Consider an infinitely repeated game where the stage game is the game of chicken: Swerve Straight Swerve 0,0 -1,1 Straight 1, –1 -10, -10 Assume that the discount factor is very close to one. For concreteness, you can assume that d = 0.99. (a) Is there an SPE in which player l's payoff is 1/(1 – 6)? Explain. (b) Is there an SPE in which each player gets a payoff of 0? Explain. (c) Is there an SPE in which player l's payoff is -2/(1 – 8)? Explain.
For the friend-foe game, recall that there were 3 Nash equilibria possible, but the equilibria set didn't include the cooperative outcome, for which both players would win. Friend Foe Friend 500,500 0,1000 Foe 1000,0 0.0 a) If the game is played répeatedly. propose a play strategy that will enforce cooperation. For what valucs of o (discount factor) the equilibrium will be (Friend. Friend)?
Suppose there are two players playing a game with east or west and south and nerth ways. Find the expected Nash equilibrium by using the concept of probabilities. Player X Left[L) Right|R) Player Y Up(U) (5,6) (0,8) (4,6) Down[D) (0,9)
Subgames • • • At each node where a player makes a decision, we can think about the "subgame" starting at that node A subgame is the portion of the tree following a particular node How many subgames does our first sequential move example have? • How many subgames does our entry game have? • How many subgames does the game on the right have? • What are its Nash equilibria? Player 1 (1,4) T Player 2 U B (5,2) (3,3) р Player 1 D L 9 (2, 0) Player 2 R (6, 2)
Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?
In equilibrium, what is the probability that player 1 will use the pure strategy E in this game?
Consider the following two-player game with three options for each player. (Payouts are listed for the row player first, then the column player.) player Y layery 3,3 A 1,5 4,4 6,2 K 8,1 3,7 5,2 0,6 1,1 Find a mixed Nash equilibrium for this game. Solution suggestion: Use two variables per player. If p and are the probabilities of selecting the first two strategies, then 1-p-q is the probability of selecting the third strategy. You will need to solve a system of equations.
U9. Consider a revised version of the game from Exercise S9: PROFESSOR PLUM Revolver Knife Wrench Conservatory 1,3 2,-2 0,6 MRS. PEACOCK Ballroom 3,2 1,4 5,0 (a) Graph the expected payoffs from each of Professor Plum's strategies as a function of Mrs. Peacock's p-mix. (b) Which strategies will Professor Plum use in his equilibrium mixture? Why? (c) What is the mixed-strategy Nash equilibrium of this game?
Not Play 1 Play 11 2 1 Trust Distrust 1 Steal Share 0 2 10 0 55 5 5 Find all of the pure strategy Nash Equilibria of this game. There can be more than one equilibrium. [Here ((Not Play, Steal), (Trust)) indicates that player 1 chooses Not Play at the first decision node and Steal at the second decision node, and 2 chooses Trust at his unique decision node.] a) ((Play, Share), (Trust)) b) ((Not play, Share), (Distrust)) c) ((Not play, Steal), (Distrust)) d) ((Not play, Steal), (Trust)) ☐ e) ((Play, Steal), (Distrust))