Give each homeowner’s (net) payoff as a function of and . (b) Compute the Nash equilibrium.

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.

Suppose the following game is played infinite times in the future. Time discount is 0.90. What should be the value of x so that the equilibrium strategy is (Cooperate, Cooperate)?      Player 2 Player 1 Cooperate Defect Cooperate (x, x) (2, 14) Defect (14, 2) (5, 5)

which is the nash equilibrium? small pig does press large pig does press (20, 130), small pig doesn't press large pig does press (140, 10), small pig does press large pig doesn't press (-5, 150), small pig does not press large pig doesn't press (0,0).

Note: The answer should be typed.

The US and Canada have overfished North Atlantic cod stocks nearly to the point of extinction. Both countries wish to preserve the cod industry (and hence the cod stocks), so the two countries sign an agreement to limit fish hauls. Each country has two possible strategies: comply with the agreement (limit fishing) or renege on the agreement (overfish). Each country’s payoffs for different strategy combinations are given in the matrix below. The numbers in the cells represent utilities, and the payoff ordering is (US, Canada). QUESTIONS: 1.If this is a one-shot game (i.e., it is played once), do the players have a dominant strategy? If so, what is it? Briefly explain your answer. 2.What is the equilibrium of this game? Is it Pareto-optimal? How do you know?

= a) Suppose x 4 and y = 3. After all strategies that are dominated by a pure strategy are iteratively deleted for both players, how many strategies remain for Player 1? b) Suppose x = 4 and y = 3. After all strategies that are dominated by a pure strategy are iteratively deleted for both players, how many strategies remain for Player 2? c) Suppose x = 4 and y = 3. What is the sum of both players' payoffs in all pure strategy Nash equilibria of the game? d) Suppose x = 4 and y = 11. After all strategies that are dominated by a pure strategy are iteratively deleted for both players, how many strategies remain for Player 1? e) Suppose x = 4 and y = 11. After all strategies that are dominated by a pure strategy are iteratively deleted for both players, how many strategies remain for Player 2? f) Suppose x = 4 and y = 11. What is the sum of both players' payoffs in all pure strategy Nash equilibria of the game?