# HW4-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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Dr. Huanxing Yang Econ 601 Game Theory Homework Assignment 4 Due May 14, Wednesday 1. Chapter 7, Problem 3 First note that this game has no Nash equilibrium in pure strategies. If the state police sets a speed trap then the best reply of the driver is to drive 65 but if the driver doesn't speed then the officer doesn't want to waste time with a speed trap. Alternatively, if the officer chooses not to set a speed trap then the driver will plan to speed but then if the driver speeds the officer wants to set a speed trap. To find a Nash equilibrium in mixed strategies, let p denote the probability that the driver goes 80 mph and q denote the probability that the state police sets a speed trap. For it to be optimal for the driver to randomize between going 65 and 80, she must be indifferent between those two pure strategies. Hence, q must be set at a level to make the driver's expected payoff the same from her two pure strategies: ( ) 10 1 70 40 1/ 2 qq q ×+−× = ⇒= The left-hand expression is the expected payoff from speeding which delivers a payoff of 10 with probability q (there is a speed trap) and a payoff of 70 with probability 1 q (no speed trap). The right-hand side payoff of 40 is what she gets from driving 65. Solving this equation for the probability of a speed trap, we find that 1/2 q = .Thus, at a Nash equilibrium, the state police sets a speed trap 50% of the time. For the state police officer to find it optimal to randomize over his two pure strategies - in particular, choosing a speed trap 50% of the time - the probability that the driver is speeding must equate the expected payoff to the officer from setting a speed trap and doing a doughnut run. This condition is: ( ) 100 1 20 50 3/8 pp p ×+ × = = 2. Chapter 7, Problem 11 Let p denote the symmetric probability that a company enters. Given all other companies use this strategy, the expected payoff to a company from entering for sure is () ( ) ( ) 11 1 200 30 1 1 40 30 nn −− ⎡⎤ −×− + × ⎣⎦ 1 1 n p is the probability that all other companies do not enter and 1 n p is the probability that at least one other company enters. This must be equated to its payoff from not entering which is 60: ( ) ( ) 1 1 5 1 200 30 1 1 40 30 60 1 16 n p ⎛⎞ + × = = ⎜⎟ ⎝⎠

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3. Chapter 8, Problem 2 1) a) Using backward induction, the Raider chooses takeover at the final decision node. At the decision node for Management, the optimal action is greenmail since that brings a payoff of 7 versus a payoff of 4 from no greenmail (which induces the Raider to takeover the company). At the initial decision node, the Raider will then buy some shares as it'll result in a payoff of 15 which exceeds the payoff of 10 from not doing so. There is then a unique subgame perfect Nash equilibrium in which the Raider's strategy is to choose buy shares at the first decision node and
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## HW4-answer - Dr. Huanxing Yang Econ 601 Game Theory...

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