Exercise 3. Consider the following game in normal form. L U 4, 4 1, 6 D 6, 1-3, -3 (a) Find all the Nash equilibria of this game. (b) Show that the following probability measure is a correlated equilibrium of this game. L R U c) Show that the following probability measure a correlated equilibrium of this game. L R U0 (d) Plot the payoff profile of cach equilibrium in (a)-(c) above. Are the payoff profiles in (b) and (c) inside or outside the convex hull of NE pavoffs?
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- Consider the following payoff matrix. L C R U 6, 3 3, 4 7, 2 1 D 3, 4 | 6, 2 8, 1 What is the probability that Player 1 plays U at the Nash equilibrium of this game? (a) 2/3 (b) 1/2 (c) 1/3 (d) 1/4 (e) None of the above optionsNE 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)2. Consider the following two player game in normal form: Player 2 R 0, 5 2, 3 2, 3 Player 1 M 2, 3 0, 5 3, 2 B 5,0 3, 2 2, 3 (a) Show that for Player 1, strategy T is strictly dominated by a mixed strategy in which actions M and B are played with positive probability. (b) Find a mixed strategy Nash equilibrium of this game. 352
- a W 3,5 3,4 8,4 0,0 3,3 8,9 y 0,1 5,9 9,8 Describe a strategy for player 1 that dominates x. O (1/3.0. 2/3) 1.0,0) O01.1) to5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.. 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.
- 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 1.7. N [0.75] B A [0.25] 1 E F 6 2 J K J K 12 3 9. 6 6. 1 In equilibrium, what is the probability that player 1 will use the pure strategy E in this game?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?
- 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)R (3,2) (0,0) (2,4) (5,4) (0,0) In the extensive form representation of the game between Player 1 and Player 2, Player 1 moves first and chooses L or R. If Player 1 chooses R the game ends, if Player 1 chooses L then Player 1 and 2 play a simultaneous move game. The game has strategy Subgame Perfect Nash Equilibria (SPNE). The maximum payoff Player 2 gets in a SPNE is pure strategy Nash equilibria and pure (Please, enter only numerical answers like: 1, 2, 3, ...)MC Qu. 07-42 (Algo) Suppose Firm A and Firm... Suppose Firm A and Firm B are considering whether to invest in a new production technology. For each firm, the payoff to investing (given in thousands of dollars per day) depends upon whether the other firm invests, as shown in the accompanying payoff matrix. Firm A Don't Invest Invest Multiple Choice O What is the Nash equilibrium of this game? O Invest 9 for A 6 for B 7 for A 3 for B Firm B Firm A invests, and Firm B doesn't invest. Firm A invests, and Firm B invests. Not Invest 5 for A 7 for B Firm A doesn't invest, and Firm B invests. 12 for A 8 for B Firm A doesn't invest, and Firm B doesn't invest.