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053837 - 38/37 School of Economics and Poiitica| Science...

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Unformatted text preview: 38/37 School of Economics and Poiitica| Science University of Sydney ECON3012 - STRATEGIC BEHAVIOUR (Semester 2, 2005) W Duration: 2 hours (+ reading time) INSTRUCTIONS (1) The paper is in two sections. Section A consists of 20 multiple choice questions and is worth 40% of the mark for the exam. Section B consists of 3 problem questions and is worth 60% of the mark for the exam (20% per question). (2) Answer ALL questions. (3) When you submit your exam answers, place the multiple choice answer sheet inside your answer booklet and return both. SECTION A: MULTIPLE CHOICE QUESTIONS Note. Unless the question specifies otherwise, you should restrict attention to pure strategies except in questions 5 and 6. 1. A strategy that is a best response to some opponent strategy a) must be a weakly dominant strategy. b) must be strongly dominant strategy. c) cannot be weakly dominated. d) can be weakly dominated but not strongly dominated. 2. The following is true: a) a strongly dominant strategy is a best response to all opponent strategies, but a weakly dominant strategy may not be. h) a weakly dominant strategy is a best response to all opponent strategies. c) a weakly dominated strategy cannot strongly dominate another strategy. d) None of the above. 1 38/37 Semester 2, 2005 Page 2 of 10 3. Assume that the row player can choose Top (T) or Bottom (B) and the column player can choose Left (L) or Right (R). Assume that for the row player: ur(T,L) > ur(B,L) ur(T,R) > ur(B,R) Assume that for the column player: uc(T,L) < uC(T,R) uc(B,L) = uC(B,R) Then: a) (T,L) is a Nash equilibrium. b) (B,L) is a Nash equilibrium. c) (ER) is a Nash equilibrium. d) None of the above. 4. The game shown below a) has an IEDS solution. b) does not have a Nash equilibrium. c) has multiple Nash equilibria. (1) None of the above. L C R T "m B a) has an IEDS solution that may be reached using only pure strategies in the elimination process. b) has an IEDS solution that may be reached if some mixed (non-pure) strategies are used in the elimination process. 0) has multiple pure strategy Nash equilibria. (1) None of the above. 5. The following game: Player 2 L R T Player 1 M B 38/37 Semester 2, 2005 Page 3 0f 10 6. Assume in the following game that player 1 plays T with probability p and that player 2 plays L with probability q. The mixed strategy solution is a) p: 1/4andq2 1/3. b) p=l/2andq:2/3. C) p: 1/3 andq=1/2. d) None of the above. Player 1 7. In the subgame perfect Nash equilibrium of the game in Figure l, the equilibrium payoffs are: a) (6,5). b) (4,6). c) (5,5). d) None of the above. Player 1 FIGURE 1. Extensive Form Game 1 38/37 Semester 2, 2005 Page 4 of 10 8. Consider the strategic form game shown below. a) This is consistent with an extensive form perfect information game in which player 1 moves first. b) This is consistent with an extensive form perfect information game in which player 2 moves first. c) This is not consistent with an extensive form perfect information game. d) Too little information is provided to draw a conclusion. Player 2 A F 9. Consider a two period ultimatum game in which there are M dollars to be distributed. In period 1, player 1 makes an offer to player 2. If it is accepted, then distribution takes place in accordance with the offer and the game ends. If the offer is rejected, then the game enters a second period and this time player 2 gets to make an offer to player 1. If it is accepted, then distribution takes place in accordance with the offer; if the offer is rejected, then both players get nothing. Assume that the offer amount is continuously variable and that the player 1’s and player 2’s discount factors are 51 and 82 respectively. In a subgame perfect Nash equilibrium: T B Player 1 a) player 1 gets 51M and player 2 gets (1 — 51)M. b) player 1 gets 52M and player 2 gets (1 — 52)M. c) player 1 gets (1 — 51)M and player 2 gets 51M. d) player 1 gets (1 — 52)M and player 2 gets 52M. 10. If a pure strategy game has a finite number of players and strategies a) it must have a Nash equilibrium. b) it must have a Nash equilibrium (derivable through backward induction) if it is a game of perfect information. c) it can only have a Nash equilibrium if a player has a dominant strategy. d) None of the above. 38/37 Semester 2, 2005 Page 5 of 10 11. Consider the stage game below. Player 1 P Stage Game Plainly, there are 9 possible combinations of strategies for the stage game. Assume that the stage game is played twice and that there is no discounting (5 = 1). Which of the 9 possible stage game strategy combinations could be played in the first play of the stage game as part of a subgame perfect Nash equilibrium for the repeated game? a) Only (CC) or (P,P). b) Any one of the 9. c) Only (C,C), (RP) or (RF). CD Only (C,C), (RP) or (CF). 12. Consider the following stage game. Player 2 Cooperate Defect Cooperate Player 1 Defect Stage Game If this game is repeated an infinite number of times and a grim trigger strategy is employed, then (Cooperate, Cooperate) can be sustained in a subgame perfect Nash equilibrium if and only if a) 5 21/5. b) 5 2 7/12. c) 6 2 3/4 d) None of the above. 38/37 Semester 2, 2005 Page 6 of 10 13. If we compare the grim trigger strategy and the forgiving trigger strategy in terms of their ability to sustain as subgame perfect Nash equilibria the playing of choices that are not Nash equilibria of the stage game, then the use of the forgiving trigger strategy: a) requires smaller minimum values of 5 in order to compensate for the fact that punish- ment is less severe. b) requires larger minimum values of 5 in order to compensate for the fact that punishment is less severe. c) may require either smaller or larger minimum values of 5 depending on whether or not cyclical behaviour is involved. d) has no effect on the required minimum values of 5. 14. In games of incomplete information, the “common priors” assumption means that a) all player-types have the same probability of meeting a particular opponent type. b) all types of a given player have the same probability of meeting a particular opponent type. c) all player-types use the same joint probability distribution of types. d) None of the above. 15. Assume that in a game of incomplete information there are two players. Player 1 can be one of three types, each of which has two possible strategies. Player two can be one of three types, each of which has three possible strategies. The number of possible strategy combinations to be considered in identifying a Bayes-Nash equilibrium is a) 54. b) 15. c) 108. d) None of the above. 16. The following table gives the joint probability distribution for different type combinations. Based on this table: Player 2 Type X Type Y Type 2 T eA Playerl yp Type B a) the unconditional probability of a Type Y is 2/5, while p(Y J A) : 1/3. b) p(B | Z) = 2/3 and p(Z | B) = 1/5. c) p(A 12) = 2/3 and p(B I Y) = 3/4. (1) the unconditional probability of a Type A is 1/2, while p(A | X) = 3/5. 38/37 Semester 2, 2005 Page 7 of 10 17. Consider the following game: Player 2A Player 2B L R L R T Player 2 is of type A with probability p and type B with probability 1 — p. In a Bayes-Nash equilibrium T Player 1 Player 1 a) player 2A plays R and player 2B plays L for all p, whereas player 1 plays T if and only if p 2 3/4. b) player 2A plays R and player 2B plays L for all p, whereas player 1 plays T if and only if p 2 1 / 3. c) player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only if p 2 3 / 4. CD player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only if p 2 1 / 3. 18. Suppose that each of two Cournot duopolists with a linear industry demand curve may be either a low cost or a high cost type (a low cost firm 1 has the same constant marginal cost as a low cost firm 2 and similarly for a high cost firm 1 and firm 2). If the probability that firm 1’s opponent is low cost is 0.25 and the probability that firm 2’s opponent is low cost is 0.5, then in a Bayes-Nash equilibrium involving positive outputs: a) a high cost firm l’s output will be greater than a high cost firm 2’s output. b) a high cost firm 2’s output will be greater than a high cost firm 1’s output. c) a high cost firm 1 and a high cost firm 2 will produce the same output. d) Too little information is provided to draw a conclusion. 19. If we compare a two—player game in which there is complete information about player 1 with an otherwise identical game in which there is incomplete information about player 1, then a) player 1 may be better or worse off in the complete information case but player 2 cannot be worse off, since player 2 can always choose to ignore the information. b) if one player is better off in the complete information game, then the other player must be worse off. c) both players are equally well off in both games since the incomplete information game works in terms of expected values. d) None of the above. 38/37 Semester 2, 2005 Page 8 of 10 20. Firm H is considering taking over firm P. Firm P is worth either 0, 5 or 10 with equal probability when under its own management. Under the management of firm H, it is worth 5/3 times as much as under its own management. Finn H makes a takeover offer of y, where y is either 0, 5 or 10, and firm P of type x (where x = O, 5 or 10) specifies the minimum amount it will accept as a takeover offer, m(x). Payoffs are zero to both parties if the takeover offer is rejected. If accepted, firm H gets x- (5/3) — y and firm P gets y —-x. In a Bayes—Nash equilibrium: a) firm H successfully takes over firm P due to the greater value that the firm has under firm H’s management. b) firm H only takes over firm P if firm P’s type is at least 5. c) firm H makes a takeover offer of 0 and the takeover succeeds, if at all, only when firm P’s type is 0. d) None of the above. 38/37 Semester 2, 2005 Page 9 of 10 SECTION B: PROBLEM QUESTIONS Q1. Do the following: (i) For the following game, identify an elimination path that leads to a pure strategy IEDS equilibrium. If there is a choice of path, find a second path that either leads to a different pure strategy IEDS equilibrium or does not lead to a pure strategy IEDS equilibrium. Player2 L C R ,1 T B Player 1 (ii) Now consider a second game. Player 2 L R Player 1 B Find the pure strategy Nash equilibria (if any) in the game. Also find any additional ' mixed strategy equilibria. Q2. Consider the following Cournot duopoly game with non-constant marginal costs. Demand is given by P 2 a — Q, where a is a positive constant and Q = Q1 + Q2. Firms 1 and 2 each have the total cost function: TCi =4Qi+Qi2 (i: 172) Consequently, the profit of firm i is given by: n,- = TR,- — TC,- = For -—4Q.-— Q? = (a _ Q1 — Qlei —4Qi — Qiz- Thus 7t1=(a — Q1 - Q2)Q1-4Q1~ Q12 = (a~Qz—4)Qi 4912. Similarly, R2 = (a - Q1 —4)Q2 - 2Q22- (i) Using the profit functions above, find the best response (reaction) function of each firm. Solve for the Cournot equilibrium quantities and profits for the two firms (these will be functions of a). Simplify these as far as possible. Note. You may assume without proof that the solution quantities are positive, so you do not have to worry about non-negativity restrictions. 38/37 Semester 2, 2005 Page 10 of 10 (ii) Now suppose that the Cournot game is played in the second period of a two period game. In period 1, the two firms simultaneously choose whether or not to promote the industry product through advertising. The value of a in the industry demand function is given by the following: 6 if neither firm advertises a x 8 if one firm advertises 10 if both firms advertise The cost to a firm of advertising is 915. In period 2, the firms play the Cournot game with the value of a in the industry demand function determined by the choices in period 1. Solve for the subgame perfect Nash equilibrium of this game assuming that each firm seeks to maximise profit net of advertising cost (assume no discounting). (iii) Is the equilibrium level of advertising in the interests of the two firms (or are there choices that would give a better outcome for both)? Explain. If the level of advertising is not in the interests of the two firms, give an intuitive explanation for their sub—optimal choices. QB. Players 1 and 2 must choose between c00perating C and fighting F. Player 1 is of a single type. Player 2 can be either a Bully or a Reciprocator. The Bully player 2 likes to take advantage of cooperative behaviour and hence gets the highest payoff by playing F against C. The Reciprocator player 2 likes to treat others as they treat him. His highest payoffs are from playing C against C and F against F. Player 1 is of a single type but her payoffs are nevertheless influenced by the type of her opponent. The probability that player 2 is a Bully is p and hence the probability that player 2 is a Reciprocator is 1 — p. The payoff tables are given below: Player 2 Bully Player 2 Reciprocator Player 1 Solve for all (pure strategy) Bayes—Nash equilibria, indicating their dependence on p if appro- priate. Hint. While no player-type has a dominant strategy, if you think about the best responses of the player-types, you can narrow down the range of possible equilibrium strategy combinations substantially. End Of Paper ...
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