Garratt_Wooders_exercises_in_game_theory - 1/3/11 Exercises...

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1/3/11 Exercises in Game Theory by Rod Garratt and John Wooders This document contains questions that are appropriate for advanced undergraduate classes or graduate classes in game theory. They have each been used at least once in graduate game theory courses taught by us at the University of Arizona and the University of California, Santa Barbara. We created each of the questions, although some are based on published works or the research of our colleagues. 1 The questions are grouped according to the equilibrium concepts they use. Answers and recommendations for extensions are available upon request. Nash Equilibrium, Subgame Perfect Equilibrium 1 . There are two baseball teams that are preparing for a three game series. Each team has three pitchers. Team 1 has an Ace, a mediocre pitcher and a scrub. Team 2 has two mediocre pitchers and a scrub. A pitcher can only be used once in the series. The probabilities of winning the game depending on the pitcher match-ups are as follows. Match up Outcome Ace versus mediocre pitcher Ace wins with probability .7 Ace versus scrub Ace wins with probability .9 Mediocre pitcher versus scrub Mediocre pitcher wins with probability .6 Same versus same Each wins with probability .5 Assume a win is worth 1 and a loss is worth 0. (a) Suppose the three games are played sequentially. The pictures for each game are chosen simultaneously, but after a game is played it is common knowledge what pitchers were used. Remember each team can only use each pitcher once. Write out the extensive form of the game. (The extensive form has 18 terminal nodes.) (b) Calculate the set of subgame perfect equilibria. (c) Draw the normal form representation of the extensive form. Find all the Nash equilibria. (d) Would either team prefer a system whereby each team’s pitcher choices for the entire series had to be posted simultaneously before the series began? (e) Compare the set of Nash equilibria of the game to the set of subgame perfect equilibria.
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2 . One day, long ago, two women and a baby were brought before King Salmon. Each claimed to be the true mother of the baby. King Salmon pondered dividing the baby in half and giving an equal share to each mother, but decided that in the interest of the baby he would use game theory to resolve the dispute instead. Here is what the king knew. One of the women is the true mother but he does not know which one. It is common knowledge among the two women who the true mother is. The true mother places a value v T on having the baby and 0 on not having it. The woman who is not the true mother places a value of v F on having the baby and 0 on not having it. Assume v T > v F , so that the true mother values the baby more. The king proposed the following game to determine who would get to keep the baby. One woman is chosen to go first. She is player 1, the other woman is player 2. She must announce “mine” if she wants to claim to be the real mother and “hers” is she wants to claim that the other women (player 2) is the true mother. If she says “hers” the baby goes to player 2 and the game ends.
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Garratt_Wooders_exercises_in_game_theory - 1/3/11 Exercises...

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