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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.
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- A buyer and seller trade with each other for an infinite number of periods. Both parties have a discount factor of d, where 0 < d < 1. In each period both parties can play trust (T) or to play selfish (S). If both the buyer and seller play T the payoffs are 4 to each player. If both parties plays the payoffs are 3 to each player. If one player plays S and the other T, the payoffs are 6 to the player who opted for 5 and 1 to the party that opted for T. Consider the following trigger strategy. In the first period play T. In any subsequent period, play T if in every previous period the outcome was (T, T), if not play S. What is the minimum d required for this trigger strategy to be subgame perfect equilibrium? O None of the other answers are correct. O 1/4 O 1/3 O 3/4Two rival companies competing in the same market need to decide their plans for future expansion of their stores. The Table below shows the possible outcomes of their mutually interdependent actions (payoffs are profits in £m) Giga Company Titanic Conglomerate No Change Refurbishment of existing stores Large Expansion No Change 30, 40 25, 35 15, 24 Refurbishment of existing stores 35, 30 28, 32 18, 33 Large Expansion 12, 22 18, 20 20, 25 The Nash equilibrium: (A) does not exist. (B) occurs when both firms choose Refurbishment of existing stores. (C) occurs when both firms choose Large Expansion. (D) occurs when both firms choose No Change.Consider the two period Repeated Prisoner's Dilemma Game where each player is interested in the SUM of the payoffs she gets in each period. Players see the outcome after the play in each period. (The period payoffs are 10,5,1,0.) (i) Write out this game in its strategic form. (ii) Find all Nash equilibria and all Subgame Perfect Nash Equilibria.
- Player 1 Cooperate (C) Defect (D) If the game has a dominant strategy, what is it? There is none. If the game has a Nash equilibrium in pure strategies, what is it? There is none. Cooperate (C) 3,3 8,0 Cooperate (C) is a dominant strategy for both players. Defect (D) is a dominant strategy for both players. Cooperate (C) is a dominant strategy for 1, and Defect (D) is a dominant strategy for 2. C, C is the only Nash equilibrium. D, D is the only Nash equilibrium. C, C and D, D are both Nash equilibria. Player 2 Defect (D) 0,8 1,1Jane is interested in buying a car from a used car dealer. Her maximum willingness to pay for thecar is 12 ($12,000). Bo, the dealer, is willing to sell the car as long as he receives at least 9($9,000). What is the Nash bargaining solution to this game?See the extensive form game in the image attached (the payoffs of player 1 are written on top and the payoffs of player 2 are on the bottom). a) Write this game in normal form (a player's strategy is a complete contingent plan that tells them what to play at each of their information sets) (b) Find all the Nash equilibria of the normal form game from part (a)
- Problem 2. Consider the partnership-game we discussed in Lecture 3 (pages 81-87 of the textbook). Now change the setup of the game so that player 1 chooses x = [0, 4], and after observing the choice of x, player 2 chooses y ≤ [0, 4]. The payoffs are the same as before. (a) Find all SPNE (subgame perfect Nash equilibria) in pure strategies. (b) Can you find a Nash equilibrium, with player 1 choosing x = 1, that is not subgame perfect? Explain.Here is a table representing the gains for the server if he serves on his opponent's forehand or backhand and for the receiver the gains if he returns on his forehand or backhand. Each player can choose to play the forehand or the backhand of the other player : Server D R D 50,50 80, 20 R 90,10 |20, 80 Receiver D Determine the Nash equilibrium(s) (pure strategies and mixed strategies).Finding Nash Equilibria Consider the following two player, normal form game: Player 1 Player 2 C L (2,1) U M (-2,-2) D R (2, -1) (1,2) (-1, 1) (0,0) (3,1) (0,0) (-1,-1) Find all pure and mixed strategy Nash equilibria. Calculate each player's expected payoffs at each equilibrium.
- Player 1 Cooperate (C) Defect (D) Cooperate (C) 3,3 8,0 Player 2 Defect (D) 0,8 1,1 In general, a combination of strategies is a Nash equilibrium if ... Every player is choosing a best response against the other players' strategies. Every player has a positive payoff. The players maximize the sum of their payoffs. The players choose identical strategies. If the game is repeated, which cooperative actions could benefit both players? O Both players choose C. Player 1 chooses C, Player 2 chooses D. O Player 1 chooses D, Player 2 chooses C. Both players choose D.Consider a simultaneous game where player A has a dominant strategy and player B has two strategies (none of which is a dominant strategy). How many pure strategy Nash equilibria will this game have? A) Exactly 1 B) Exactly 2 C) Either 1 or 2 D) NoneMixed Nash Equilibrium Player 2 C D A 5,1 1,3 B 2,6 4,2 PLAYER 1 a) calculate the probability In equilibrium Player 1 chooses A with probability a= [?] and player 2 chooses M with probability B = [ ?] b) calculate payoff for mixed strategy nash equilibrium Player 1 gets payoff of [?] Player 2 gets payoff of [?] c) Suppose the payoff for (B,C) would change to (0,8) Would probability a increase,decrease, stay the same? Would probability B increase,decrease, stay the same?