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- (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. Player 2 b₁ b₂ Player 1 a₁ 3,0 0,1 a2 2,1 2,1With what probability does player 1 play Down in the mixed strategy Nash equilibrium? (Input your answer as a decimal to the nearest hundredth, for example: 0.14, 0.56, or 0.87). PLAYER 1 Up Down PLAYER 2 Left 97,95 47, 33 Right 8,43 68,91(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
- 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.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?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.
- 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 probability9 00 3 -2 -8 A = ( ) and B = 03 -5 -2 Let I be the bimatrix game whose payoffs are described by A for the row-player and B for the column- player. 1. Compute all mixed Nash equilibria of T. 9Consider the following game Player 2 E N 4, 4 0, 2 Player 1 M 2, 0 2, 2 В 3, 0 1,0 a) Find the the pure-strategy Nash equilibria b) Find a mixed-strategy Nash equilibria in which Player 1 plays all three of their strategies with positive probability.
- 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)W X Y Z 47, 15| 39, 41 45, 53 12, 56 In equilibrium, what is the probability that player 1 will use the pure strategy X in this game?In a mixed strategy equilibrium of the game below, what is the probability with which Player 2 chooses r (if there are multiple equilibria with different probabilities of a, choose any one)? Player 2 y a 3, 3 4, 2 Player 1 b 6,3 2, 6 5. 3 3, 2 Numerical answer