073829 - 381‘] Final Exam Econ 3012 StrategicBehavior ‘...

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Unformatted text preview: 381‘] Final Exam Econ 3012 StrategicBehavior ‘ Semester 1, 2007 2 hrs + 10 min reading time. - Maximum 100 points STUDENT DETAILS Seat Number: Last Name: Given N ame: Student Number: INSTRUCTIONS (PLEASE READ!) a. You must hand in this exam booklet. You must also hand in your rough work sheet. No credit can be awarded otherwise. . All are multiple choice questions. Choose the most appropriate answer for each of them. . Answers to the questions must be filled in on the two separate answer- sheets provided for multiple choice questions. Write “Page I” at the top of first sheet and shade the answers for Ql—QBO. Write “Page II” at the top of second sheet and shade the answers for Q31-Q55. . Exam is divided into three sections. Section I has 20 questions each worth 1 point. Section II has 20 questions, each worth 2.5 points. Section III has 15 questions, each worth 2 points. . Questions that are marked with a (*) test your basic concepts and do not require (much) computations. You may choose to answer these first. Aim to spend around 30 minutes on these questions. I. There is no negative marking for incorrect answers Ecos3012 Strategic Behavior Final Exam, Semester 1, 2007 Section I EACH QUESTION IN THIS SECTION IS WORTH l POINT. IT ALMOST EXCLUSIVELY CONCERNS MATERIAL COVERED PRIOR TO THE MIDTERM EXAM. Question 1 (*) The notion of strategy in an extensive form game a. requires a player to specify one action at every information set where she is required to play. b. as in part (a) but may choose different actions at different nodes of an information set. c. is always consistent with a player simply stating whether or not she will choose an action that can end a game provided she begins the game. d. requires more information to answer. Question 2 (at) Which is/are true of a strategic form game. a. players choose their strategies simultaneously. b. players have imperfect information. c. one cannot use it to model a situation where one player moves after another. d. Both (a) and (b). Question 3 (*) s is known to be a best response of Player 1 to some strategy t of Player 2 in a two player game. a. (3,15) is necessarily a Nash equilibrium. b. s can be a weakly dominated strategy. c. s can be a strictly dominated strategy. d. None of the above. Question 4 (it) In a Nash equilibrium, a. No player has an incentive to unilaterally deviate and choose a different strategy. b. Players cannot improve their payoffs by jointly choosing another strategy profile. c. Part (a) but only if the game has two players. d. None of the above. Question 5 (air) An IEDS solution of game is a. also a Nash equilibrium. b. is a Nash equilibrium but only if strictly dominated strategies are eliminated at each instance. c. is a Nash equilibrium but only if the order of elimination of strategies does not affect the final outcome. d. None of the above. Question 6 (1%) An IEDS solution of a strategic form game which is obtained by eliminating strictly dominated strategies at each stage a. must also be the unique Nash equilibrium. b. must be unique. c. both part (a) and part (b) (:1. None of the above. Question 7 (at) Suppose (s,t) is a rationalizable strategy profile. Then a. (at) must be a Nash equilibrium. b. s is a weakly dominant strategy. Page 2 of 12 Please turn Over Ecos3012 Strategic Behavior Final Exam, Semester 1, 2007 c. s is a strictly dominant strategy. d. None of the above. Question 8 (it) Which one of thefollowing is true of a 2 player finite strategic form game (allowing for mixed strategies)? a. A Nash equilibrium must exist. b. An lEDS solution must exist. (1. There may not be a rationalizable strategy. d. None of the above. Question 9 The following game (A) dominance solvable (B) . L C R A = is not, B = with or without mixed strategies. T 110 6,4 0, 9 B 3,7 2, 3 4,0 A = is, B : in pure strategies. The answer is part (b) but only if the payoffs of Player 1 are much higher. ‘ a. b. A = is, B : with mixed strategies. c. d. Question 10 (ad The table below shows only the payoffs of Player 1. Restrict attention to pure strategies. a. Player 1 has a weakly dominant strategy. L C R b. Player 1 has a strongly dominant strategy. c. Player 1 has a weakly dominated strategy. d. None of the above. Question 11 (it) Which of the following is true: a. a strongly dominant strategy is a best response to all of opponents’ strategies but a weakly dominant strategy many not be. b. a weakly dominant strategy is a best response to all of the opponents’ strategies. c. a weakly dominant strategy cannot be played in Nash equilibrium. d. None of the above. Question 12 The game shown below a. has a IEDS solution that me by reached using only pure strategies in the elimination process. b. has an IEDS solution that may be reached if mixed strategies are also used in the elimination process. (2. has multiple pure strategy Nash equilibria. d. None of the above. Question 13 (1*) Suppose 3 and g are two strategies that are played with positive probability in some mixed strategy equilibrium of a finite strategic form game. In this equilibrium, a. either 5 or a is also a best response. b. part (a) and moreover, their payoff is in fact the player’s equilibrium payoff. c. neither 3 nor s is a best response, only their average is. d. None of the above. Question 14 (1k) For the game shown below, a. There is a unique pure strategy Nash equilibrium. L R b. There are many mixed strategy equilibria. c. Both (a) and T 3’0 3’0 B 0,3 1, 1 d. None of the above. Page 3 of 12 Please turn Over Ec053012 Strategic Behavior Final Exam, Semester I, 200? Question 15 In a complete information second price auction for an indivisible item, Player 1 values the object at m = 20 and Player 2 values the object at m : 15. Consider the following two strategy profiles. (a) Player 1 bids 20 and Player 2 bids 15. (,6) Player 1 bids 0 and Player 2 bids 20. a. Both (a) and (fl) are Nash equilibria. b. (a) is the IEDS solution. c. ([3) is not a Nash equilibrium. d. Both (a) and Question 16 Consider the following first price auction with two buyers. Each buyer can either submit a bid of 5 or 10. The highest bidder wins the auction and pays the price that he has bid. If the bids are identical, each wins with probability 1/2. Suppose that buyer 1 values the object at 25 and buyer 2 values the object at 15. In a Nash equilibrium of this game, a. both buyers must bid high. b. both buyers must bid low. c. one buyer must bid high and the other must bid low. d. None of the above. Question 17 In a certain Cournot Duopoly, it is known that the reaction function of Firm I is 31(q2) : max{(22 — q2)/2,0} and that of Firm 2 is B2(q1) = max{(20 ~ ql), 0}. In a Nash equilibrium, 3. both Firms produce the same quantity. Firm 2 produces more than Firm 1. Firm 1 produces more than Firm 2, Firm 1 may produce more than Firm 2. It depends on which Nash equilibrium is being played 999‘ Question 18 (air) Recall that an outcome is said to be Pareto Efficient if there does not exist an alternative outcome that makes some player better off without making others Worse off. a. A Nash equilibrium is necessarily Pareto Efiicient. b. (a) but only if there is are exactly two players. (2. (a) but only if it is an equilibrium in mixed strategies. d. There are games in which a Nash equilibrium is also Pareto efficient. Question 19 It is known that in a certain market with two firms, setting a price of p; 2 2 is a weakly dominant strategy for Firm 2. Suppose that the profits of Firm 1 are given by u1(p1,pg) =: (12 — p1 — pflpl for arbitrary values of 331 and p2. Firm 1 is allowed to choose any non-negative price. a. (pbpg) : (5, 2) must be a Nash equilibrium of this game. b. ($91,132) = (5, 2) must be the only Nash equilibrium of this game. e. Firm 2 must set a price of 2 in every Nash equilibrium of this game. d. There is not enough information to conclude any of the above Question 20 (*) Two players use the Nash bargaining solution to arrive at an equilibrium. It is known that each player gets a zero utility if there is disagreement. If (33,30 is the Nash Bargaining outcome, then a. (:r, y) must lie on the line joining the points (23:, 0) and (0, 2y). b. Any other (33’, y’) is a utility assignment such that 39’ > x and y’ > y is not feasible. c. 2: + y > 23’ + y’ for any other utility assignment that is feasible. d. Both (a) and Page 4 of 12 Please turn Over Ecos3012 Strategic Behavior I Final Exam, Semester I, 2007 Section II EACH QUESTION IN THIS SECTION 18 WORTH 2.5 POINTS. Q21-Q26 CONCERN BARGAINING, SUBGAME PERFECTION, COASE CONJECTURE, Q27—Q31 CONCERN REPEATED GAMES, Q32— Q40 CONCERN GAMES OF INCOMPLETE INFORMATION AND AUCTIONS. Section II.1 Questions on Subame perfection, bargaining 84 Coase conjecture Question 21 Consider the two player game shown in the figure below where Player 1 first chooses “In” our “Out”. If she choose “Out” the game ends resulting in the payoff vector (2, 2). If she chooses “In”, they play the familiar Battle Of the Sexes. L R a I“ T m B m 1.3 out (2: 2) a. (2,2) can be supported as sub-game perfect equilibrium payoff. b. Play Of (T, R) in the second stage can be supported as sub—game perfect equilibrium outcome. c. Both (a) and d. There is a unique subgame perfect equilibrium in which the outcome is (3, 1). Question 22 A pie Of size M > O is to be shared between two players. Player 1 claims a share of the pie. If Player 2 agrees, the game ends with Player 1 getting her demand and Player 2 getting the remainder Of the pie. Otherwise, neither player gets anything. Assume that both players like more Of the pie to less and the utility Of getting zero units is zero. a. There is a unique sub—game perfect equilibrium in this game. b. Any division Of the pie (3:, M A I) can be supported as a Nash equilibrium Of this game. c. Any division of the pie (in, M i 3:) can be supported as a subgame perfect equilibrium Of this game. d. Both (a) and Question 23 Consider the bargaining game described in the diagram below for sharing a pie of size 1. a. In the unique subgame perfect equilibrium of this game, Player 1 gets the entire pie. b. In the unique subgame perfect equilibrium of this game, Player 1 has a first mover advan— tage in that she necessarily gets a larger share of the pie. Page 5 of 12 Please turn Over Ec0s3012 Strategic Behavior ' Final Exam, Semester 1, 2007 c. In the unique subgame perfect equilibrium the discount factor of Player 1 does not affect the outcome. d. Both (b) and Question 24 Reconsider the game in the previous question, but now assume that Player 1 has an outside option which results in the payoff vector (33*, 1 — at”) that she can exercise whenever she makes an offer. a. The fact that Player 1 has an outside option has no impact on how the pie is divided. b. Part (a) but only if 1* 5 (1 — 62). 0. Part (a) but only if 22* S (1 — 61). d. None of the above. Question 25 (*) Consider the standard two player alternating offers bargaining game where Player 1 begins the game and has a discount factor 0 < 61 < 1. Let O < 62 < 1 denote the discount factor of Player 2. They bargain over a pie of size M. a. There is necessarily a first mover advantage since Player 1 gets a larger share of the pie in every SPE. b. Part (a) but only if 61 : 52. 0. Whether Player 1 gets a larger share of the pie depends on how patient she is relative to the other player (i.e. it depends on how big 61 is relative to the 62). d. Both (b) and Question 26 (1c) The Cease Conjecture states that a durable goods monopolist would a. set a price equal to its marginal cost just like a perfectly competitive firm provided it can commit not to revise its offers frequently. b. behave like a perfectly competitive firm but only if consumers are patient and the monop- olist cannot commit to an intertemporal pricing strategy at the start. c. as in part (b) but only if one uses Nash equilibrium as a solution concept but not in an SPE. d. None of the above. Section 11.2 Questions on Repeated games Question 27 (1c) Let (u1,'u2) be the payoffs from an action profile (a1, (12) in a stage game C. It is possible to support (U1,u2) as an SPE payoff vector of C(oo, 5) if a. There exists a pure strategy Nash equilibrium of G with payoffs (n1, 712) such that ul > m and HQ > 77.2. b. The minimax payoffs vector in G is (7711,1112) such that v.1 > 711] and big > mg. C. Neither (a) nor (1. Either (a) or (b) for a 5 sufficiently close to one. Question 28 A certain two player strategic form game, say G, is known to have a unique Nash equilibrium, say (SJ) and that u1(s,t) = O and u2(s,t) = 0. There is also another strategy profile (£43 such that u1(.§,t) = whit) = 10. Consider G(10,5)1 the game where G is played for ten periods. Assume 0 < 5 < 1 is the discount factor. a. In every subgame perfect Nash equilibrium of G (10, 6), players must necessarily play (8. t). b. There are Nash equilibria of G(10, 6) where (s, t) is not played in the last period. c. part (a) but only if 6 is close to zero so that players do not care about the future. (1. None of the above. Page 6 of 12 Please turn Over EcosBOlZ Strategic Behavior Final Exam, Semester I, 2007 Question 29 (*) The folk theorem for infinitely repeated games asserts that a convex combi- nation of payoll outcomes in G, say (ul, ug, . . . ,un) such that it; > mi can supported as the average (A) equilibrium payoff of G (00, 6 ) provided m1- is the (B) and 6 is close to one. a. A=Nash and B: least payoff that player 75 gets in all Nash equilibria of G. b. A=Nash and B: minimax value oft in G. c. Azsubgame perfect and B: least payoff that player 2' gets in all Nash equilibria of G. cl. a=subgame perfect and b: minimax value. Question 30 Let G be the game shown below. a. If a: > 1, then the only average payoff vector that can be supported as subgame perfect equilibrium of G(oo,6) is (3:, I) regardless of the value of 5. b. If 0 < a: < 1, then the payoff vector (11 1) can be supported as a subgame perfect equilib- rium payoff of C(oo, 6) for all 6 between 0 and 1. c. Part (a) but only if cl is close to one. d. None of the above. Question 31 Let. G be the following game. Playing (L, L) at each date can be supported as a SPE of C(oo, 6) a. by using only Nash equilibria of G as threats when- ever a deviation occurs. b. for any 6 close enough to one by constructing a suit- able simple strategy profile. c. Both (a) and d. only if the players can write binding contracts. Section II.3 Questions on games of incomplete information Question 32 Consider a game of incomplete information in which Player 1 can be one of two types and Player 2 is one of three types. Each player can choose between two actions. The total number of strategies in this game are a. 4 for Player 1 and 8 for Player 2. b. 2 for Player 1 and 2 for Player 2. c. 4 for Player 1 and 6 for Player 2. d. None of the above. Question 33 (7k) Consider a two player game in which there is incomplete information about Player 1’s type. Now change the game so that now Player 1’s type is common—knowledge. This increase in information about Player 1 (in expected utility terms) a. makes both players better oil. makes only Player 1 better off. makes only Player 2 better off. may make either player better or worse off. It really depends on the precise payoffs in the game. eog Question 34 Consider the following game of incomplete information. Player 1’s type is known but Player 2 is of type A with probability p and is of type B with probability 1 e p. Page 7 of 12 Please turn Over Ecos3012 Strategic Behavior ' Final Exam, Semester I, 2007 L R ‘L R plaerlT. T y B B Player 2A Player 2B In a Bayes Nash equilibrium of this game, a. Player 2A a plays R and player 2B plays L for all p. b. Player 2A a plays L and player 23 plays R for all p. c. Part (a) and Player 1 plays T if and only if p 2 2/3. (:1. None of the above. Question 35 (at) In a direct mechanism, a. Players simply announce their types. b. The game is either a first or a second price auction. c. It can never be an equilibrium for a player to announce their type truthfully. (1. None of the above. Question 36 (*) In auction theory with independent valuations a. The revenue equivalence theorem says that the payoff of a bidder depends only on the probability with which she wins the object. b. to maximize expected revenue, the auctioneer must allow design the auction so that the object is always allocated to the player with the highest virtual valuation. c. Neither (a) nor (:1. Both (a) and Question 37 In a certain direct mechanism for allocation of an indivisible good, it turns out that if other players report their type truthfully, the utility of Player 1 is when she is of type i) but reports to instead is U(w,v) : (1 — w—v}v. a. The mechanism is consistent with incentive compatibility. b. The mechanism is not incentive compatible. Question 38 (ii) The Revelation Principle says a. A first price and a second price auction reveals the same information. b. A player cannot do better than revealing her type truthfully. c. The equilibrium outcomes of an arbitrary selling procedure can also be obtained as the equilibrium of another game in which players are asked to announce their types. (1. All of the above. Question 39 The value v1 of an indivisible good for bidder 1 is distributed uniformly On the interval [0, 2] and that for bidder 2, U2, is distributed uniformly on [2,4]. In the optimal auction (i.e. the one that maximizes the seller’s revenue) a. Player 2 must always win the object. b. If ’01 < 1, then Player 1 must not receive the object. c. If 1 + '01 > 112, then Player 1 must receive the object. d. Both (a) and Question 40 (*) Consider a bilateral trading situation where the value of the seller 1} is private information and is distributed continuously on [0, 1] interval. Assume that the cost of the seller c is continuously distributed on [a,2]. Suppose We wish to construct a negotiation procedure such that trade occurs if and only if the v 2 c in a Bayesian Equilibrium. Page 8 of 12 Please turn Over Ecos3012 Strategic Behavior Final Exam, Semester 1, 2007 a. Such a negotiation procedure is possible if a < 1. b. Such a negotiation procedure is possible oniy if a 2 1. c. Both (a) and d. Need to know the precise distributions to answer this question. Section III EVERY QUESTION IN THIS SECTION IS WORTH 2 POINTS EACH. Section III.1 Wage Negotiations Question 41 - Question 45 concern the following environment. A manager and a trade union representative (T) bargain over new wages. Assume that salary increase is a number between 0 and 1, 1 being the most preferred amount for T and 0 for M. Agreeing to a number a: should be interpreted as M getting a utility of (1 — 3:) and T getting ac. Question 41 Suppose the negotiations involve M simply making an offer :1: between 0 and 1. T responds by accepting or rejecting the offer. If accepted the agreement at is reached. Otherwise, both parties receive 0. a. In a subgame perfect equilibrium: the agreement reached is x = 0. b. Any m between 0 and 1 is a Nash equilibrium outcome. c. Both (a) and (:1. Neither (a) nor (b). Question 42 Now suppose that before meeting with M, T commits to her union members that she will achieve an agreement that is at least z. If during negotiations T accepts an agreement :1: < z, she suffers a loss of face leading to a disutility of y 2 U (i.e. her payoff is now (I — Payoff from rejecting an offer is still 0. The bargaining procedure is still as in the previous question. The subgame perfect equilibrium outcome is the agreement a. 33:2. b. a: = y. c. m = min (3;, 2}. d. as: max{y,z}. Question 43 From your analysis of Question 42, We know that a. Personal costs (namely y) play a necessarily more important role than public commitment (namely 2). b. Personal costs (namely y) play a necessarily less important role than public commitment (namely 2). c. If the personal cost is half the public commitment then it is the personal cost that is relevant for the equilibrium outcome. d. None of the above. Question 44 Suppose instead of making a public commitment as in Question 42, assume that T can make a counter offer to M’s initial offer should she reject it but then incurs a cost for a one period delay. Specifically assume that 52 < 1 is the discount factor of T that captures this cost. Bargaining begins as in Question 41 except that when T rejects M’s offer, she makes a counter offer. If M rejects T’s counter offer the game ends with both players getting zero. Otherwise, the agreement is implemented. The subgame perfect equilibrium outcome of this two period game Page 9 of 12 Please turn Over EcosBOlQ Strategic Behavior ' Final Exam, Semester 1, 2007 a. leads to the agreement :2: =_ :52. b. leads to the agreement a: = 1 — 62. (3. requires knowledge of (51 to conclude. d. None of the above. Question 45 Which of the following is true? T necessarily prefers the option of making a counter offer to making a public commitment a. If 62 > y, b. If 62 > z. c.1f1— 62 > min {31,2} d. None of the above. Section 111.2 Repeated Interaction Let G denote the following version of the game of Prisoners dilemma. c d c #134 d 4,—1 m Question 46 The grim trigger strategy profile of G(oo,§) is as follows: Both players begin by playing the c in the initial period. After any history in which (c, c) has occurred at each previous date, each player chooses c again. Following any other history, a player chooses d. This strategy profile is a. is a Nash equilibrium if 6 2 3/4. b. is a subgame perfect equilibrium if 6 2 3/4. c. For 5 2 3/ 4 but 6 < 7/ 8, it is a Nash equilibrium but not a sub—game perfect equilibrium. d. None of the above. Question 47 Consider G(oo,ci) and the tit-for-tat strategy profile: Both players begin by playing c. In every subsequent period, a player chooses the same action as his opponent did in the previous period. Which of the following is true of the Tit-for—tat strategy profile: a. For 6 close to one, it may be a Nash equilibrium of C(oo, 6). b. For 6 close to one, it may be a subgame perfect equilibrium of C(oo, 6). c. For all 6, it is a Nash equilibrium of G(oo, (i). d. None of the above. Question 48 In the game C(T, 6) where T if finite, a. The unique Nash equilibrium involves playing d after every history. b. The unique subgame perfect equilibrium involves playing d after every history. c. Part (b) but only for 6 close to zero. d. None of the above. Let G" be the following game and consider G’(2), i.e. the game 6" played twice. The payoff of a player in in G’(2) is the sum of payoffs received at each date. Page 10 of 12 Please turn Over Ecosl3012 Strategic Behavior Final Exam, Semester I, 2007 Consider the following strategy profile, call it a 0 Begin by playing (£12,172) atdate 1. a At date 2, play (£11,531) if indeed ((12,132) has been played at date 1. If any other (at-,bj) combination occurs at date 1, play ((12,52). Question 49 The strategy profile 0' described above is a. a Nash equilibrium of G’(2). b. is not a Nash equilibrium of G’(2) Question 50 The strategy profile 0 described above is a. a subgame perfect equilibrium of 0(2). b. is not a subgame perfect equilibrium of G’(2). Section 111.3 Bertrand Competition with incomplete information A two firm game of Bertrand price competition is given below. Each firm can set a price of €(ow) m(edium) or h(igh). The profits depend on whether their products are substitutes or complements. The payoff matrix for the case of complements is given on the right and for substitutes on the left. Firm 1 does not know whether their products are complements or substitutes but Firm 2 does. Thus, Firm 2 knows if it is playing the right matrix or the left matrix game. Firm 1 assigns a probability A : 1/2 that the goods are complements. h m E h Firm 1 m E Firm 2 (Substitutes) Firm 2 (Complements) Question 51 In any Bayes Nash equilibrium, a. Firm 2 (Complements) must necessarily play it b. Firm 2 Substitutes) must necessarily play h c. Firm 2 (Substitutes) must never play h. l d. Both (a and Question 52 In any Bayes Nash equilibrium, a. Firm 1 must not play m. b. Firm 1 must not play 2. c. Firm 1 must not play h. d. Impossible to tell. Question 53 In any Bayes Nash equilibrium, if Firm 1 does not play m, then a best response for the other firm is that a. Firm 2 (Complements) plays it and Firm 2 (Substitutes) plays 1?. b. Firm 2 (Complements) plays it and Firm 2 (Substitutes) plays m. c. Firm 2 (Complements) plays in and Firm 2 (Substitutes) plays E. d. None of the above. Question 54 Suppose there was no incomplete information but rather Firm 1 knows whether she is playing Firm 2 (Complements) or Firm 2 (Substitutes). Page 11 of 12 Please turn Over Ec053012 Strategic Behavior ' Final Exam, Semester 1, 2007 a. The equilibrium payoffs are (3, 3) if she is playing Firm 2 (Substitutes) and (5, 5) is she is playing Firm 2 (Complements). b. The equilibrium payoffs are (6,3) if she is playing Firm 2 (Substitutes) and (5, 5) is she is playing Firm 2 (Complements). c. Depends on whether one uses IEDS or Nash. Part (a) is correct only if IEDS is used. and (b) are true otherwise. cl. None of the above. Question 55 Use the information gathered above to give the Bayes Nash equilibrium of the incomplete information game. Comparing the equilibria of the complete information games with the Bayes Nash equilibria of the incomplete information game, a. Firm 2 (Complements) is better off with an increase of information but Firm 2 (Substi- tutes) is just as well off. b. Firm 2 (Complements) is worse off with an increase of information but Firm 2 (Substitutes) is just as well off. On average, Firm 1 is better off under incomplete information. d. None of the above. .0 Page 12 of 12 End ...
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073829 - 381‘] Final Exam Econ 3012 StrategicBehavior ‘...

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