Which one of following statements is false? * Consider the following game. B D E F There are 7 subgames in this extensive-form game. There are 6 proper subgames in this extensive-form game. (BK, CE) is a Subgame Perfect Nash Equilibrium. (BK, DE) is a Subgame Perfect Nash Equilibrium. G H I J K L M N 3,4 4,3 1,2 5,7 6,5 2,8 9,1 3,6
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- Q2 Consider the following game. (a) Find all pure-strategy Nash equilibria. (b) Find all mixed-strategy Nash equilibria. 2, 4 6, 0 5,1 1,9 A ВAn incumbent can commit to producing a large quantity of output before the potential rival decides whether to enter. The incumbent chooses whether to commit to produce a small quantity or a large quantity. The rival then decides whether to enter. Enter The payoffs are represented in the game treeillustrated in the figure to the right. What is the subgame perfect Nash equilibrium? (1,800,500) Rival Small O A. The Nash equilibrium is for the incumbent to produce the large quantity and for the rival to not enter regardless of the incumbent's quantity. (3,600,0) Don't enter O B. The Nash equilibrium is for the incumbent to produce the large quantity and for Incumbent the rival to enter regardless of the incumbent's quantity. OC. The game does not have a Nash equilibrium. O D. The Nash equilibrium is for the incumbent produce the small quantity and for Enter (1,600, - 80) Large Rival the rival to only enter if the incumbent produces the small quantity. (3,200,0) Don't enter O E. The Nash…Find the unique Nash equilibrium of a game with the following payoff matrix: Player 2 2 4 1 (6, 7) (13, 8) (9, 9) Player 1 3 (4, 4) (7, 7) (7, 6) (7, 7) (11, 11) (7, 13) О a. (3, 2) O b. (5, 4) О с. (11, 11) O d. (1, 6) O e. There is no unique Nash equilibrium.
- 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):Question 1 Consider the following game. Find all Nash equilibria, subgame perfect Nash equilibria, and weak perfect Bayesian equilibria in pure and mixed strategies. a P2 (12, 12) b (12,-8) (7.-3) P1 m d (-3,2) r P2 C (2,7) d (22,17)(a) Consider the following bimatrix of a normal form game. Show that the set of strategies that survive elimination of weakly dominated strategies depends on the order in which the weakly dominated strategies are eliminated. Player 2 L R 1,1 0,0 M| 1,1 0,0 | 2,1 Player 1 2,1 (b) Find an example of a game where the unique Nash equilibrium in pure strategies would not survive the eliminination of weakly dominated strategies. (c) Find an example of a game where both players' strategies are weakly dominated in the Nash equilibrium.
- 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?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?2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.
- Which is false for the following game? C R. T. 3, 2 0,3 3, 3 M 5,0 4, 1 3, 1 B 2, 2 4, 5 1, 1 A. (M,C) and (B,C) are the only pure strategy Nash Equilibria OB. There is a Nash cquilibrium in which R is played with positive probability C. There is no Nash equilibrium in which L is played with positive probability OD. There is no Nash equilibrium in which both players choose two of their actions with positive probabilitySuppose 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.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.