Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A’s payoff, and the second entry is Animal B’s payoff: Animal A (rows)/Animal B (columns) Hawk Dove Hawk (-5,-5) (10,0) Dove (0,10) (4,4) An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. Explain why there cannot be an equilibrium where all animals act as Doves. Explore whether there are any Nash equilibria in pure strategies and explain which these are and why. Derive a mixed strategy Nash equilibrium (MSNE). What is the proportion of Hawks and Doves? If the proportion of Hawks in the population of animals is greater than the mixed strategy equilibrium proportion you calculated, which strategy does better, Hawks of Doves?

Two discount stores (Megastore and Superstore) are interested in expanding their market share through advertising. The table below depicts the strategic outcomes (profits) of both stores with and without advertising. Superstore - Advertise Superstore - Don't Advertise Megastore - Advertise $95, $80 $305, $55 Megastore - Don't Advertise $65, $285 $165, $115 When the game does reach Nash Equilibrium, the payoffs for both stores will be a) Megastore $95 and Superstore $80 b) Megastore $305 and Superstore $55 c) Megastore $65 and Superstore $285 d) Megastore $165 and Superstore $115

8.8

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 1,1

4) Consider the following payoff matrix: Player 1 Strategy A Strategy B c) Is the payoff matrix symmetric? Player 2 Strategy A Strategy B (0,0) (7,4) (2,2) (3,3) a) Does either player have a dominant strategy? b) Is there one or more Nash Equilibrium? 1

Jane 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?

Construct the normal representation and the fully reduced normal representation of the game shown in Figure 1. 0 (0.5) (0.5) 1.r 1.s W1 X1 y1 Z1 0,0 2.t 2.t 0,4 a2 b2 a2 b2 Figure 1: Extensive-form game 2,-2 -2,2 -2, 2 2,-2

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.

= 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?