28 - Chapter 28 NAME Game Theory Introduction. In this...

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Unformatted text preview: Chapter 28 NAME Game Theory Introduction. In this introduction we offer three examples of two-person games. The first game has a dominant strategy equilibrium. The second has a Nash equilibrium in pure strategies that is not a dominant strategy equilibrium. The third has no pure strategy Nash equilibrium, but it does have a mixed strategy equilibrium. Example: Albert and Victoria are roommates. Each of them prefers a clean room to a dirty room, but neither likes to clean the room. If both clean the room, they each get a payoff of 5. If one cleans and the other doesnt clean the room, the person who does the cleaning has a utility of 0, and the person who doesnt clean the room has a utility of 8. If neither cleans the room, the room stays a mess and each has a utility of 1. The payoffs from the strategies Clean and Dont Clean are shown in the box below. Clean RoomDirty Room Albert Victoria Clean Dont Clean Clean 5 , 5 , 8 Dont Clean 8 , 1 , 1 In this game, notice that if Victoria chooses to clean, then Albert will be better off not cleaning than he would be if he chose to clean. Likewise if Victoria chooses not to clean, Albert is better off not clean- ing than cleaning. Therefore Dont 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 Dont Clean. Therefore the out- come where both roommates choose Dont Clean is a dominant strategy equilibrium. It is interesting to notice that this is true, even though both persons would be better off if they both chose the strategy Clean. 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 eet 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 338 GAMETHEORY (Ch. 28) 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. Kenneys staff has estimated the number of days of bombing 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, Imamuras payoff is the negative of Kenneys 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, thennorthern route....
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This note was uploaded on 05/05/2010 for the course ECONMOICS ECON 203 taught by Professor Josephpetry during the Spring '10 term at University of Illinois, Urbana Champaign.

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28 - Chapter 28 NAME Game Theory Introduction. In this...

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