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- 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)AsapPlayers 1, 2, and 3 are playing a game in which the strategy of player i isdenoted yi and can be any nonnegative real number. The payoff function for player 1 is V1(y1,y2,y3) = y1 + y1y2 - (y1)2,for player 2 is V2(y1,y2,y3) = y2 + y1y2 - (y2)2,and for player 3 is V3(y1,y2,y3) = (10 - y1 - y2 - y3)y3.These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: Thepayoff functions are symmetric for players 1 and 2.)
- In the following symmetric general sum game (2, 2) (0,0) (0,0) (0,0) (0,0) (2, 2) (0,0) (2,2) (0,0) (i) Find all pure Nash equilibria. (ii) Find all mixed Nash equilibria in which all probabilities are positive. (vi) Which of these are evolutionary stable strategies?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%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…
- 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)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)5,3 4,4 3,6 7,6 Find the pure strategy nash equilibria
- (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.) %3D(a) Find all subgame perfect equilibria in pure strategies (if any).(b) Find all SPE where at least one of the players uses a mixed strategy (if any)In the following normal form game, perform iterated elimination of strictly dominated strategies. What are the pure- strategy Nash equilibria? LCR T (2, 0) (1, 1) (4, 2) М (3, 4) (1, 2) (2, 3) В (1, 3) (0, 2) (3, 0)