Problem Set 2b Solutions

Problem Set 2b Solutions - ECON 166A Fall 2010 1)...

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Unformatted text preview: ECON 166A Fall 2010 1) Consider the following extensive form game 1 2 2 (2, ­2) ( ­2,2) ( ­2,2) (2, ­2) a) What is the strategy space for players 1 and 2? Player 1 only has one information set, therefore his strategy can be specified by the decision of whether he goes left or right in his one information set. Thus his strategy space can be given by S1 = {L,R}. Player 2 has two information sets, therefore his strategy must specify which direction he would move in each information set. Let x/y denote that player 2 plays x if player 1 first moves Left and plays y if player 1 first moves Right. Then player 2’s strategy space can be given by S2={L/L , L/R , R/L , R/R}. b) Write the normal form of the game. Player 2 L/L L/R R/L R/R Player L 2, 2 2, 2  2,2  2,2 1 R  2,2 2, 2  2,2 2, 2 c) Can you arrive at a unique strategy profile with backwards induction? If so, what is it? In the information set after player 1 moves left, Player 2 will prefer to move right, giving a payoff of ( 2,2). In the information set after player 1 moves right, player 2 would prefer to move left giving a payoff of ( 1,1). Thus player 2 will play R/L. Going backwards to player 1’s move, we find that player 1 has no clear best strategy. Either playing L or R will lead to eventually getting payoffs of ( 2,2). Therefore backwards induction does not give a unique strategy profile as a solution. Recall that backwards induction is only guaranteed to give a unique solution in games of perfect information with generic payoffs – meaning that all the payoff outcomes are different. Sinervo and Musacchio Problem set 1B SOLUTIONS d) Using the normal form of the game, apply iterated dominance. Does the procedure find a unique strategy profile that “solves” the game? If so what is it? In the normal form game, player 2 has no strongly dominated strategies but does have a single weakly dominant strategy, which is (R,L). if we delete the weakly dominated strategies, then only (R,L) remains for player 2. Supposing player 2 plays (R,L), we cannot eliminate any of player 1’s strategies. However no matter what player 1 plays, the payoffs end up being ( 2,2). Now considered the modified game: 1 2 2 (2, ­2) ( ­2,2) ( ­2,2) (2, ­2) e) Write the normal form of the game. Both players have a single information set, so the strategy spaces are: S1={L,R} and S2={L,R}. The normal form is: Player 2 L R___ Player L 2, 2  2,2 1 R  2,2 2, 2 f) Using the normal form of the game, apply iterated dominance. Does the procedure find a unique strategy profile that “solves” the game? If so what is it? Iterated dominance does not eliminate any of the strategies in this game. We cannot “solve” the game with iterated deletion of dominated strategies. 2) Harrington Chapter 3 question 10 a) Derive strategies that survive IDSDS If player i plays 0, her minimum possible payoff is 8, while her maximum possible payoff for playing either 3,4, or 5 is 7,6, 5 respectively. Thus 0 dominates 3,4 and 5 for both players. If player 2 plays 0, her minimum payoff is again 8. If player 2 plays 1, she will get the lower score, no matter how little player 1 studies. Thus 0 dominates 1 for player 2. We are left with the possibilities of {0,1,2} for player 2 and {0,2} for player 1. 0 2 0 10,8 8,8 1 9,8 9,6 2 8,8 8,6 Note that playing 1 dominates 2 for player 1. What remains is: 0 2 0 10,8 8,8 1 9,8 9,6 None of the remaining strategies are dominated. Thus by IDSDS, player 1 will play 0 or 1 and player 2 will play 0 or 2. b) Derive strategies that survive ID of Weakly dominated strategies. The deletion of the strongly dominated strategies we have done already still stands. Looking at our last bimatrix from part a, we see that 0 weakly dominates 2 for player 1. If eliminate 2 from player 1’s strategy space we have: 0 0 10,8 1 9,8 Thus, player 1 will also play 0. The only strategy profile that survives ID of weakly dominated strategies is (0,0). 3) In Cournot duopoly, two producers of an identical product (call them firms A and B) simultaneously choose how much of that product to produce, say qA and qB, between 0 and 10. For simplicity, assume that production cost is 10 per unit. The price p is 40 – 2(qA + qB). Payoffs are profit = (price – unitcost)*quantity, e.g., (40  qA  qB  10) qA for firm A. a. Write out the extensive form for this game. Assume that the only available q’s are 1, 3, 5, 7 and 9. b. Now write out the extensive form for the Stackelberg variant of this game, in which firm A chooses first, and firm B observes qA before making its own choice. c. Write out the normal forms for the simultaneous game Simultaneous move: 3 5 7 3 54,54 70,42 70,30 5 42,70 50,50 42,30 7 30,70 30,42 14,14 player B For the Stackelberg variant, a strategy for player B specifies an action to take at each information set. Therefore, a strategy is 3 tuple of numbers. Thus there are 3X3X3 = 27 strategies in the strategy space. Thus the normal form game matrix would be 3 by 27! d. For the original simultaneous choice version of the game, can you identify one or more Nash equilibria? If so, what is it (are they)? Below we indicate the best responses: 3 5 7 3 54,54 70,42 70,30 5 42,70 50,50 42,30 7 30,70 30,42 14,14 player B The strategy profiles where best responses coincide are (7,3), (5,5), and (3,7). These are the N.E. Harrington Chapter 4 question 4 a) Which strategies survice IDSDS? b) identify the NE: (b,y) and (a,z) are NE. Harrington Chapter 4 question 9 The NE are (b,x,A), (a,z,B), (c,x,C), and (c,z,C) Harrington Chapter 4 question 10 a) is (S2,S3,S1) (S=”sabotage”) a NE? b) Is there a NE in which player 2 wins the promotion for sure? ...
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This note was uploaded on 03/03/2011 for the course ECON 414 taught by Professor Staff during the Spring '08 term at Maryland.

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