063837 - Andrew McLennan Economics 3012 Strategic Behavior...

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Unformatted text preview: Andrew McLennan Economics 3012 Strategic Behavior Second Semester 2006 Final Exam November 16, 2006 Instructions: (0) Please arrange alternate seating to the extent possible. (0) This is a closed book exam that will last 2 hours. You may use scratch paper and calculators. Prior to the exam there will be a 10 minute reading period. (0) The 25 questions all have equal value. Answer ALL questions. (0) Normal exam conditions apply. Students detected conferring on an- swers or consulting unauthorised material will have their paper confis- cated7 receiving no marks for the exam7 and will be liable for further penalty. (0) It is not necessary to turn in scratch work. Indicate your answers on the test forms that are provided, being sure to fill in the required personal information. Do not open the exams until instructed to do so. Econ 6021 Final Exam Do all twenty five questions. November 16, 2006 Problem 1: What is the mixed strategy Nash equilibrium of the following game? (A) (gA + g3, %C’+ %D). (B) (%A + %B, %0 + g3). (c) <§A+1B,%c+ 1D). 1 1 2 (D) (5A + 5B, g0+ %D). 0 A (3,1) B ((1,3) D (1,2) (2,0)) Problem 2: What is player 1’s best response correspondence in the follow- ing game? A, (M51012): A({A:B}), Ba A7 (C) 51(a2) = A({A»B}), B, C A ((1,0) B (0,3) Q M Q /\ Q to A O V v V II WIH ooh—- wlr—i s: :1 O V § /\ D Q IO M 53 53 V II who WIN) who D (0,1) (22)) (B) 51(02) (D) b1(02) Problem 3: For the following game, which statement is correct? A B C a (4,0) (1,1) (3,0) b (0,1) (3,2) (2,0) c (3, 2) (2, 3) (1, 4) (A) There is a unique pure strategy Nash equilibrium that can be found by iteratively eliminating strictly dominated pure strategies. (B) There are multiple Nash equilibria. (C) There is a unique pure strategy Nash equilibrium, but iteratively elim— inating strictly dominated pure strategies does not eliminate all other pure strategies. (D) There is a unique mixed strategy Nash equilibrium and n0 pure strat- egy Nash equilibrium. Problem 4: In which of the following games does iterative elimination of strictly dominated strategies reduce the game to a unique outcome? 0 D o D c D A((1,0) (1,1)) A((0,0) (1,1)) A((2,1) (3,1)) B (0,1) (1,0) B (2,2) (0,3) B (0,0) (2,1) (a) (b) (c) (A) (b) and (c), but not (a). (B) All three. (C) (c), but not (a) or (D) None of the above. Problem 5: Two agents simultaneously choose effort levels 61 and ez which can be any numbers between 0 and 1. For agent i the payoff if he chooses ei and the other agent chooses ej is 114(61, 62) = eiej - %ei. What is the set of Nash equilibria of this game? (A) {(0,0)}- (B) {(1,1)} (C) {(0,0),(%,%),(1,1)}- (D) {(6,6) :0 S e g 1}. Problem 6: Two players compete for an object in a second price auction: they choose bids b1, b2 2 O, the object is awarded to the player making the higher bid, and that player pays the losing bid. (If b1 2 b2, the winner is chosen by a coin flip.) The players’ values of the object are 111 and 112. We assume that v] > v; > O, and that both players know U1 and 112. Consider the following pairs of strategies: (a) Player 1 bids '0] and Player 2 bids v2. (,6) Player 1 bids 0 and Player 2 bids 1. Which of the following statements is correct? (A) a is a Nash equilibrium but fl is not. (B) ,6 is a Nash equilibrium but a is not. (C) Both a and ,8 are Nash equilibria. (D) Neither 01 nor ,6 is a Nash equilibrium. Problem 7: What is the payoff vector of the subgame perfect equilibrium of the following game of perfect information? (A) (374) (B) (213) (C) (3,0) (D) (4,?) Problem 8: Three firms simultaneously choose quantities q1, q2, and q3. The market price P(Q) is a — Q if Q g a, and otherwise it is 0, where Q = q1 + q2 + q3 is the total quantity. Firm i’s cost is cqi, Where 0 S c 5 a, so Firm i’s profit is qu(Q) — ciqi. What is the Nash equilibrium of this version of the Cournot game? (A) (111,112,113) = (§(oz —— c), %(a — c), %(a — c)). (B) 011,912,165) = (Ha “ 0): flat — 0), flu — c)). (C) (41,42,413) = (a0! * C), %(a — c), %(a — c)). (D) None of the above. Problem 9: Consider the following game. Players 1 and 2 each choose an integer between 1 and K. If they choose the same number player 2 pays $1 to player 1; otherwise there is no payment. For this game what does Nash’s general result concerning symmetric games tell us? (A) The pair of mixed strategies in which both players assign probability 1/K to each number is a Nash equilibrium. (B) The pair of mixed strategies in which both players assign probability 1 / K to each number is the only Nash equilibrium. (C) The pair of mixed strategies in which both players assign probability 1 / K to each number is not a Nash equilibrium. (D) None of the above. Problem 10: Suppose that there is a pile of stones, and two players take turns taking one or two stones from the pile. The winner is the player who does not take the last stone. Let n be the initial number of stones. When can the first player insure a victory'.7 (A) When n is not divisible by 3. (B) When n is divisible by 3 and not otherwise. (C) When n is divisible by 3 and when n = 3k + 1 for some integer k 2 O. (D) When n is divisible by 3 and when n = 3k + 2 for some integer k 2 0. Problem 11: Two players compete for an object in an auction: they choose bids b1,b2 2 07 the object is awarded to the player making the higher bid, and both players pay the losing bid. (If bl = b2, the winner is chosen by a coin flip.) The players’ values of the object are 1)} and '02. We assume that U] > 122 > 0, and that both players know 1)] and 112. Consider the following pairs of strategies: (a) Player 1 bids v1 and Player 2 bids 1);). Player 1 bids 0 and Player 2 bids 1. Which of the following statements is correct? (A) a is a Nash equilibrium but fl is not. (B) ,6 is a Nash equilibrium but a is not. (C) Both a and fl are Nash equilibria. (D) Neither Oc nor fl is a Nash equilibrium. Problem 12: Player 1 makes an offer at between 0 and 1 player 2 either accepts or rejects. If player 2 rejects the payoffs are (0,0), and if she accepts the payoffs are (1 — :E,:r — (1 — 2302). What happens in the subgame perfect equilibrium? (A) Player 1 offers at = 1 / 2 and player 2 accepts. (B) Player 1 offers as = 1/ 3 and player 2 accepts. (C) Player 1 offers a: = 1/4 and player 2 accepts. (D) Player 1 offers 1 = 1/5 and player 2 accepts. Problem 13: How many Nash equilibria (both pure and mixed) do the Battle of the Sexes (BS) Matching Pennies (MP) and the Prisoner’s Dilemma (PD) have? (A) BS has one, MP has one, and PD has three. (B) BS has three, MP has one, and PD has three. (C) BS has one, MP has three, and PD has one. (D) None of the above. Problem 14: Suppose E is the event “rain later” and F is the event “cloudy now.” Assume that Pr(E) = 0.4, Pr(FIE) = 0.75, and Pr(F|not E) = 0.3. Conditional on it being cloudy now, what is the probability that it will rain later? (A) 3/ 8 (B) 1/2 (C) 5/8 (D) 3/4 Problem 15: Firm H is considering taking over firm P. Firm P is worth 0, 2, 4, 6, 8, or 10 under its own management. Firm P knows its value, and firm H believes the six possible values are equally likely. Under the management of firm H, firm P is worth 5/3 times as much as under its own management. Firm H makes a takeover offer of y, where y is 0, 2, 4, 6, 8, or 10, and firm P either accepts or rejects. In a Bayes-Nash equilibrium: (A) Firm H successfully takes over firm P due to its greater value under the management of firm H. (B) Firm H only takes over firm P if firm P’s value is less than 6. (C) Firm H makes a takeover offer of 0, and succeeds, if at all, only when firm P’s value is O. (D) None of the above. Problem 16: For the poker game below let a be the probability that player 1 raises with a low card7 and let ,3 be the probability that player 2 meets when player 1 raises. What are the values of a and [3 in the Nash equilibrium of the game? Chance (A) a = 1/3 and fl: 1/3. (B) a = 1/3 and [3: 1/2. (0)01 : 1/2 and fl = 1/3. (D) a = 1/2 and a: 1/2. 7 Problem 17: For the game shown below, which is correct? Challenger 3 1 1 0 3 1 2 3 (A) There is a unique Nash equilibrium. (B) The Incumbent plays F in some weakly sequential equilibrium. (0) There are multiple Nash equilibria but only one weakly sequential equilibrium. (D) None of the above. Problem 18: For the game shown below, which of the following statements is correct? Challenger (A) There is a unique Nash equilibrium. (B) The Incumbent plays F in some weakly sequential equilibrium. (C) There are multiple Nash equilibria but only one weakly sequential equilibrium. (D) None of the above. Problem 19: Two players each have to decide Whether to fight or yield. Player 2 may be either strong or weak: Strong Weak f y f y F (—271) F (lifll) Y( (071) (0,0)) Y( (071) (0)0)>' Player 1 does not know whether player 2 is strong or weak, but believes the probability of strength is a. Player 2 knows whether she is strong or weak, and she knows player 1’s beliefs. For what values of a is there a Nash equilibrium in which player 1 fights with probability one? (A) For all a S 1/4. (B) For all a S 1/3. (C) For all a g 1/2. (D) For all a S 2/3. 10 Problem 20: In the game shown below Chance first decides whether the Challenger (who is player 1) is strong or weak. The Challenger then decides whether to be ready or unready. After seeing Whether the Challenger is ready or unready, but not whether she is strong or weak, the Incumbent (player 2) decides whether to acquiesce or fight. 0 Ll El [31] [El F A Incumbent Ready p Chance 1 _ Ready Challenger Challenger tTong Weak U nready U nready " WIifciiffiBéh'tm " F A [ill [3] [3] Which of the following statements is correct? [fl (A) There is a separating equilibrium and a semi—separating equilibrium, but no pooling equilibrium. (B) There is a pooling equilibrium and a semi-separating equilibrium, but no separating equilibrium. (C) There is 5 pooling equilibrium and a separating equilibrium, but no semi—separating equilibrium. (D) none of the above. 11 Problem 21: How many Nash equilibria (both pure and mixed) does the following game have? A B c a (4,0) (1,1) (3,2) (0,1) (3,2) (2,0) 0 (3,2) (2,3) (1,4) (A)1 (3)3 (0)4 (D)? Problem 22: Each of n > 3 players can either contribute 1 to the financing of a public good or not. The value of the public good to each player is $0 where 2 S G < n. Which of the following statements is correct? (A) This game has pure equilibria in which no players contribute, pure equilibria in which exactly G players contribute, and no mixed Nash equilibria. (B) This game has mixed equilibria in which all players have the same probability of contributing to the public good, but no equilibria in which some players mix and others play pure strategies. (C) This game has equilibria in which one player always contributes to the public good and all other players mix with equal probability. (D) None of the above. 12 Problem 23: Two players simultaneously choose numbers from the set {1, 2, 3}. If they choose different numbers the payoff vector is (07 0). If they choose the same number i, then the payoff vector is (i, —i). Which of the following statements is correct? (A) In the unique Nash equilibrium player 1’s expected payoff is 6/11. B There are three mixed Nash equilibria. ( ) (C) In the unique Nash equilibrium player 1’s expected payoff is 7/13. ( ) D None of the above. Problem 24: Which of the statements below is true of the following game? A B C a (1,0) (0,1) (0,1) b (1,0) (1,0) (0,1) c (0,1) (1,0) (1,0) (A) There is a mixed Nash equilibrium in which player 1 assigns positive probability to a weakly dominated pure strategy. (B) There is a mixed Nash equilibrium in which player 2 assigns positive probability to a weakly dominated pure strategy. (C) There is a unique Nash equilibrium. (D) There is no mixed Nash equilibrium in which player 1 sometimes plays a. Problem 25: Three players have to choose between four alternatives A, B, C, and D. First player 1 vetoes one of the alternatives7 then player 2 vetoes one of the remaining alternatives, then player 3 vetoes one of the two remaining alternatives, and the unvetoed alternative is society’s choice. The three player’s preferences are AmBmCmD, B>2A>2D>2C, C>3D>3A>3B Which alternative is selected in a subgame perfect equilibrium? (A) A B O ()B ()C ()D U 13 ...
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063837 - Andrew McLennan Economics 3012 Strategic Behavior...

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