Player 2 4, 4 0,2 Player 1 M 2,0 2,2 B. 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.
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- With 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(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 2,1In 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
- 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)F G H 1 3 6 3 A 6. 8. 8 5 3 B 3 3 4 4. 9. C 5 3 a.) Find all pure-strategy Nash equilibria of the above game. b.) * Prove that there is a Nash equilibrium in which Player 2 chooses H, while Player 1 chooses A with probability 0.4 and chooses C with probability 0.6. 00 2,(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
- 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?S1 S2 S3 4. Find out all the Nash equilibria of the game. (Both Pure & Mixed) R1 R2 R3 1,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1In 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%
- 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 probabilityConsider 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.EXERCISE 3.a. Show that if G; has value v; for i = 1, 2, then their series-sum game has value v₁ + v₂.