6. (a) For the following extensive-form game:1. Identify the pure and mixed strategy Nash Equilibriaii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibriaof the game different from the set of equilibria identified in part (a)?Explain (a couple of sentences should suffice).(3,1)APlayer 1 aa2D(-2,-2)2Player 2b₁b₂3,00,12,1 2,1Bс(2,5)D(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 withstrictly positive probability, we call it a strictly mixed strategy for thatplayer. Show that there is no Nash equilibrium in which both 1 and 2 playa strictly mixed strategy.(0,7)

An extensive form game given above has same subgame pure and mixed strategy Nash equilibrium as…

(a) For the following extensive-form game: 1. Identify the pure and mixed strategy Nash Equilibria ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) Player 1 1 a1 az D (-2,-2) B Player 2 bi b₂ 3,0 0,1 2,1 2,1 9 (2,5) D (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 tha player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. (0,7) ||

(a) For the following extensive-form game: 1. Identify the pure and mixed strategy Nash Equilibria ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) Player 1 1 a1 az D (-2,-2) B Player 2 bi b₂ 3,0 0,1 2,1 2,1 9 (2,5) D (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 tha player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. (0,7) ||

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.9P

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FOOP 6. (a) For the following extensive-form game: i. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in part (a)? Explain (a couple of sentences should suffice). (3,1) A D B C (-2,-2) (2,5) D (0,7)
NE 2). Consider the following extensive form game between two players. (1,10) u D 1 B X (a) List all pure strategies of player 2. (b) Represent this game in normal form. (c) Find all pure-strategy Nash equilibria of this game. (d) Find all SPNE (in pure strategies) of this game. (6,3) (4,2) (5,1)
rock paper scissors гock 0. -3 1 рарer 1. -1 scissors -1 3 0. (a) Show that xT= ( ) and yT= (3) together are not a Nash equilibrium 3 3 313 for this modified game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
. Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player chooses one of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices. The outcome is determined as follows. If both players choose the same strategy, neither player wins or loses anything. Otherwise: • "paper covers rock": if one player chooses paper and the other chooses rock, the player who chose paper wins and is paid 1 by the other player. • "scissors cut paper": if one player chooses scissors and the other chooses paper, the player who chose scissors wins and is paid 1 by the other player. • "rock breaks scissors": if one player chooses rock and the other player chooses scissors, the player who chose rock wins and is paid 1 by the other player.
6. (a) For the following extensive-form game: 1. Identify the pure and mixed strategy Nash Equilibria. ii. Is the set of pure and mixed strategy Subgame Perfect Nash equilibria of the game different from the set of equilibria identified in parf (a)? Explain (a couple of sentences should suffice). с (3,1) Player 1 a a2 A D (-2,-2) B Player 2 bi b₂ 3,0 0,1 2,1 2,1 C (2,5) D (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. (0,7)
1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).
3. (a) Give an example of a 3 x 3 zero-sum game where all possible pairs of pure strategies give a pure Nash Equilibrium. (b) Describe all possible 3 x 3 non-zero-sum games where all possible pairs of pure strategies give a pure Nash Equilibrium. Give reasons for your answer.
Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.
Consider the following two player extensive form game. メ (2,5) (4,) (3,5) a. Find the induced normal form of the given extensive form game using the pure strategies of the players. Find out all the possible Nash Equilibrium of this game. b. From the Nash Equilibrium you are getting in part a., which one is subgame perfect Nash Equilibrium? Explain with appropriate reasons. c. Explain why the NE in part b. is credible. (Use maximum 3 sentences):
Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?
3. Determine the Nash Equilibrium for the following normal form of the game. A) MARY B) UP DOWN PLAYER 1 HARLEY UP (10, 10) (4,8) LEFT (3,0) UP MIDDLE (1, 1) DOWN (0,1) DOWN (8,4) (5,5) PLAYER 2 CENTER RIGHT (2,1) (1,1) (4,2) (0,0) (5,0) (0,1)
1. Consider the two-player game described in figure 1. Player 1 has three possible actions to choose from and player 2 has two. The payoffs to each player are shown in the normal-form representation of the game below. (a) Player 2 LR U 5,1 4,0 Player 1 M 6,0 3,1 D 7,3 5,4 Figure 1 Do either players have any dominated strategies? (b) What is the Nash Equilibrium of this game?