s are on next page 1 For the following normal form game: (a) Find any strictly dominated strategies. (b) Find all pure strategy equilibria or show why none exist. (c)Find any mixed strategy equilibria or show why none 1 XY 2 A 5, 4 2, 1 exist. B C 3, 0, 2 3 4,4 4, 5 A B 2 2. Consider the extensive form game shown in the figure. W X Solve the game by backward induction. Report the strategy profile that results and the payoffs that each player 1 receives (4, 2) when this strategy profile is played. C D 1 2 (5, 1) Y Z 1 E (4, 5) (2, 4) (4, 3) (5, 2) 1 3) F G H (5, 3. following game: Consider the Player 1 announces an amount x [0, 1]. Player 2 simultaneously announces an amount y [0, 1]. If x + y 1, then player 1's payoff is x and player 2's payoff is y. If x + y > 1, then both players receive zero. (a) Carefully show that any ( x, y) where x + y = 1 (and x, y [0, 1]) is a Nash equilibrium. (b) Suppose that each player believes that the other will choose a value in [0 , 1] at random (using a uniform distribution). Find player 1's best response to this belief. If both players play their best responses to this belief is the result a Nash equilibrium for the game? Explain. (c) Suppose that instead of moving simultaneously, player 1 announces her choice x first. Player 2 observes x and then announces his choice y. Describe a subgame perfect equilibrium for this game or explain why one does not exist. 4. Consider the following Stackelberg duopoly game. Firm 1 chooses to produce quantity q1 of a particular good, with q1 [0, 100]. Firm 2 observes firm 1's choice and chooses to produce the quantity q2 with q2 [0, 100]. The market price of the good is given by p = (100 2q1 2q2). We will assume the firms use total revenue as the payoff, ignoring costs so that ui = pqi for i = 1, 2. (a) Describe the strategy space for each firm. (b) Calculate the unique subgame perfect equilibrium of this game. 5. Use the standard bargaining solution to find the outcome of the given bargaining problem where u1 = v1(x) + t and u2 = v2(x) t, the default values d1, d2 denote the payoffs if no agreement is reached, and 1, 2 are the players' respective bargaining weights. State the values of x and t and the payoffs u1 and u2. x (, ), v1(x) = x2 8x, v2(x) = 20x 3x2 2 1 d1 = 3, d2 = 3, 1 = , 2 = 3 3 2 LA 1A 2 3 RA 2 X Y A 6. Assume that you and another player are participating in a secondprice auction. In a second-price auction, the highest bidder wins 1 the auction, but pays the amount bid by B 3 X the second-highest bidder. LB Thus,RBif you win the auction, your payoff 2 is v1 b2, where 1B object and v1 is the amount you value the Y b2 is the amount the other player bids. You value the object at $20 and the other bidder values the object at $10 (and these values are common knowledge). Show that bidding $20 weakly dominates bidding $18. Recall that a strategy si weakly t dominates st if ui(si, si) i i ui(s , si) for all possible si Si, with strict inequality for at least one si. 2,2 1,1 3, 1 1, 3 1,1 0,0 7. Consider the extensive form game shown above. Player 1 has two types: type A (probability 2 ) and type B (probability 1 ). Player 1 knows her own type, but Player 3 3 2 does not. If player 1 plays Left (LA or LB depending on type) then the game ends with the payoffs specified above. If player 1 plays Right (RA or RB depending on type) then player 2 may play X or Y and the game ends with the payoffs specified above. (a) Express the extensive form game in the Bayesian normal form. (b) Use the normal form to identify any (pure strategy) Bayesian Nash equilibria. 6,8 A 2 XH 1H 4,4 B (p) YH (q) 1 5,0 4,6 A B 8,4 Low (1 p) C XL D 3,3 High 2 1 2 C 2 3,0 (1 q) 1L YL D 8,4 8. Consider the extensive form game shown above. Player 1 is either High type ( H) or Low2 type (L), each with probability Player 1 know his own type; player 2 does not know player 1's type. 1 . Answer either of the following (extra credit for correctly and completely answering both). Do not overlook the fact that player 2 has four strategies. (a) Show that there is a pooling equilibrium where player 1 plays YH YL (your answer should specify appropriate values of q and p following Bayes' Rule) (b) Show that there are no separating equilibria (your explanation should include the correct specifications for p and q for each case (following Bayes' Rule)