Consider the following game between Firm 1 and Firm 2: Firm 2 M R М 3, 2 R 1, -3 -5, -5 4, 3 0, 0 -3, 3 4, 4 2, 2 -3, -1 Firm 1 There is (are) and Nash equilibrium (equilibria) when the firms move simultaneously, SPNE(s) in the sequential-move game in which Firm 1 moves first.
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- Solve for the Nash equilibrium (or equilibria) in each of the following games. (a) The following two-by-two game is a little harder to solve since firm 2’spreferred strategy depends of what firm 1 does. But firm 1 has a dominantstrategy so this game has one Nash equilibrium. Firm 2 Launch Don’tFirm 1 Launch 60, -10 100, 0 Don’t 80, 30 120, 0 What is the Nash equilibrium of this simultaneous-move game? (b) What would the outcome of this game be if instead firm 1 moved first and then, after seeing what firm 1 chose, firm 2 chose it strategy? In this case firm 1 doesn’t necessarily need to choose a best response, but firm 2 must choose a best response since it moves second.Q1 Consider the following normal-form game: L C U ?,? 0,8 M 8,0 2,2 D 2,4 4,4 R 4,4 10, 10 6,6 Fill in the first cell in the above matrix such that: a) The Nash equilibrium couple of strategies is (M, R). Explain. b) The Nash equilibrium couple of strategies is (U, L). Explain. c) U and I are non-rationalizable strategies. Explain. d) There exists a Nash equilibrium in dominated strategies. Explain.2: Provide a definition for a Nash equilibrium and then find all pure-strategy Nash equilibria of the following game: Left Center Right Up Middle 1,3 2,7 6,3 2,9 4,4 7,10 Down 3,8 2,5 3,4
- 1. Consider the following normal form game: Player 2 Player 1 U (2,2) (1,6) Find the mixed-strategy Nash equilibrium, supposing that this is a simultaneous non- repeated game.2. Consider the following game: u 1 r 6,0 0,6 d 3,2 6,0 Are there any Nash equilibria in pure strategies for this game? Find the mixed strategy Nash equilibrium.Consider the following sequential game: 1 O D,L U.L U,R U D,R D (0,2) 2 L What is the subgame perfect Nash equilibrium of this game? R. (-1,-1) (1,1)
- Consider the following simultaneous move game: Player 1 U M D Player 2 L 2,10 2,5 8,7 C 5,2 11,8 8,4 R 5,4 2,9 8,3 In the Nash equilibrium of this game, player 1 receives a payoff of ✓ and player 2 receives a payoff ofConsider the following simultaneous game: Player 1 U D Player 2 L 20,-10 -10, 20 R -10, 20 20,-10 Please indicate whether each of the following statements is true or false. Player 1 has a dominant strategy. This game has a Nash equilibrium. This game has a Nash equilibrium in pure strategies. Player 1's best response is D if player 2 plays R.The following table contains the possible actions and payoffs of players 1 and 2. Player 1 U M D L 2,10 2,5 8,7 Player 2 C 5,2 11,8 8,4 R 5,4 2,9 8,3 This is a simultaneous move game. In a pure strategy Nash equilibrium of this game, Player 1 receives a payoff of ✓and player 2 receives a payoff of If, instead of playing simultaneously, player 2 moves first, then in the Nash equilibrium player 1 receives a payoff of ✓and player 2 receives a payoff of
- 5 Suppose two players play one of the two normal-form games shown in Figure 1. L U 0,-1 D 2,4 R 2,0 6,0 L U | 4,-1 D 2,-2 R 2,0Now suppose that Player 2 knows which game is being played, but Player 1 does not. Find the pure strategy Bayesian Nash equilibrium of this game.2. Provide the normal-form expression to the extensive-form game below, and find all pure-strategy Nash equilibria there. Find the subgame-perfect Nash equilibrium then. A B X E Y B A B F t G F G BF B D A 0 3. Find the subgame perfect Nash equilibria (may or may not be unique) in the game below. A B 100 30 F 60 9 5 B 3 5 12 10 60 10 80 10 180 60 200 40#3. Characterize the unique subgame-perfect equilibrium in the following game. Can you find any Nash equilibrium which is not subgame perfect? Explain. D R D (1,1,1)) R D N-1 N-1' N-1' D **** N-1 R D R (2,2,..., 2)