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- Consider the following payoff matrix. -1 5 P = 0 -1 (a) Find the optimal mixed row strategy. (b) Find the optimal mixed column strategy. (c) Find the expected value of the game in the event that each player uses his or her optimal mixed strategy. PLease advise with any ideasSuppose 2 players play the following game infinite times in the future. What should be the minimum value of the time discount so that the equilibrium strategy is (Cooperate, Cooperate)? Player 2 Player 1 Cooperate Defect Cooperate (10, 10) Defect (2, 12) (4,4) (12, 2)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)
- (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.) %3DConsider the following game. Firm 1 can implement one of two actions, A or B. Firm 2 observes the action chosen by Firm 1 and then decides whether to fight it or not. (-10, 20) F2 Fight A Don't fight -(30, 10) Firm 1 (-10, 0) Fight B Don't fight -(20, 15) (a) Consider the following strategy profile: Firm 1 chooses A; Firm 2 chooses fight if A, and fight if B. • this strategy profile is [Select] (b) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses fight if A, and don't fight if B. • this strategy profile is [Select] O (c) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses don't fight if A and don't fight if B. • this strategy profile is [Select] 00 F2 ()If a strategy profile is a Nash equilibrium, there is at least one player that could achieve higher payoffs by deviating.(a) True. (b) False.
- Find the mixed strategy Nash equilibrium by using the concept of probabilities. Player X West{W} East{E} Player Y (0,10) South{S} (7,8) (4,8) (0,12) North{N}Find an equilibrium in the following game, with Nature moving first, with fixed probabilities as shown. (0,2) (2,2) B Q 2 (4,0)N N (6,0) .1 (0,0) (2,0) F .9 F B Q (6,2) N 2 1 -(4,2)4. You have probably had the experience of trying to avoid encountering someone, whom we will call Rocky. In this instance, Rocky is trying to find you. It is Saturday night and you are choosing which of two possible parties to attend. You like Party 1 better and, if Rocky goes to the other party, you get a payoff 20 at Party 1. If Rocky attends Party 1, however, you are going to be uncomfortable and get a payoff of 5. Similarly, Party 2 gives you a payoff of 15, unless Rocky attends, in which case the payoff is 0. Rocky likes Party 2 better, but he is likes you. He values Party 2 at 10, Party 1 at 5, and your presence at either party that he attends is worth an additional payoff of 10. You and Rocky both know each others strategy space (which party to attend) and payoffs functions.
- Consider a game with two players (Alice and Bob) and payoffffs Bob Bob s1 s2 Alice, s1 3, 3 0, 0 Alice, s2 0, 0 2, 2 In the equilibrium in the above game, Alice should (A) always choose the fifirst strategy s1; (B) choose the fifirst strategy s1 with probability 40% ; (C) choose the fifirst strategy s1 with probability 50% ; (D) choose the fifirst strategy s1 with probability 60% .The payoff matrix for a game is 3 -3 4 -4 2 2 4 -5 2 (a) Find the expected payoff to the row player if the row player R uses the maximin pure strategy and the column C player uses the minimax pure strategy. (b) Find the expected payoff to the row player if R uses the maximin strategy 40% of the time and chooses each of the other two rows 30% of the time while C uses the minimax strategy 50% of the time and chooses each of the other two columns 25% of the time. (Round your answer to three decimal places.) (c) Which of these pairs of strategies is most advantageous to the row player? (а) (b)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