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28. Game Theory - Solutions

# 28. Game Theory - Solutions - Chapter 28 NAME Game Theory...

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Chapter 28 NAME Game Theory Introduction. In this introduction we offer two examples of two-person games. The first game has a dominant strategy equilibrium. The second game is a zero-sum game that has a Nash equilibrium in pure strategies that is not a dominant strategy equilibrium. Example: Albert and Victoria are roommates. Each of them prefers a clean room to a dirty room, but neither likes housecleaning. If both clean the room, they each get a payoff of 5. If one cleans and the other doesn’t clean, the person who does the cleaning has a utility of 2, and the person who doesn’t clean has a utility of 6. If neither cleans, the room stays a mess and each has a utility of 3. The payoffs from the strategies “Clean” and “Don’t Clean” are shown in the box below. Clean Room–Dirty Room Albert Victoria Clean Don’t Clean Clean 5 , 5 2 , 6 Don’t Clean 6 , 2 3 , 3 In this game, whether or not Victoria chooses to clean, Albert will get a higher payoff if he doesn’t clean than if he does clean. Therefore “Don’t Clean” is a dominant strategy for Albert. Similar reasoning shows that no matter what Albert chooses to do, Victoria is better off if she chooses “Don’t Clean.” Therefore the outcome where both roommates choose “Don’t Clean” is a dominant strategy equilibrium. This is true despite the fact that both persons would be better off if they both chose to clean the room. Example: This game is set in the South Pacific in 1943. Admiral Imamura must transport Japanese troops from the port of Rabaul in New Britain, across the Bismarck Sea to New Guinea. The Japanese ﬂeet could either travel north of New Britain, where it is likely to be foggy, or south of New Britain, where the weather is likely to be clear. U.S. Admiral Ken- ney hopes to bomb the troop ships. Kenney has to choose whether to concentrate his reconnaissance aircraft on the Northern or the Southern route. Once he finds the convoy, he can bomb it until its arrival in New Guinea. Kenney’s staff has estimated the number of days of bombing

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344 GAME THEORY (Ch. 28) time for each of the outcomes. The payoffs to Kenney and Imamura from each outcome are shown in the box below. The game is modeled as a “zero-sum game:” for each outcome, Imamura’s payoff is the negative of Kenney’s payoff. The Battle of the Bismarck Sea Kenney Imamura North South North 2 , 2 2 , 2 South 1 , 1 3 , 3 This game does not have a dominant strategy equilibrium, since there is no dominant strategy for Kenney. His best choice depends on what Ima- mura does. The only Nash equilibrium for this game is where Imamura chooses the northern route and Kenney concentrates his search on the northern route. To check this, notice that if Imamura goes North, then Kenney gets an expected two days of bombing if he (Kenney) chooses North and only one day if he (Kenney) chooses South. Furthermore, if Kenney concentrates on the north, Imamura is indifferent between go- ing north or south, since he can be expected to be bombed for two days either way. Therefore if both choose “North,” then neither has an incen-
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28. Game Theory - Solutions - Chapter 28 NAME Game Theory...

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