lecture7 - Lecture 7: Game Theory Key Terms in Lecture...

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Lecture 7: Game Theory
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Key Terms in Lecture • Mixed strategies • Continuum of actions • Tragedy of the commons
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Mixed Strategies • When a player chooses one action or another with certainty, he is following a pure strategy • Players may also follow mixed strategies – randomly select from several possible actions – DO NOT MISINTERPRET THE WORD “random” • You may have a plan to turn left half the time and right half the time. These probabilities need not have been “randomly” selected. • But given these probabilities, your action cannot be predicted perfectly. – This is what we mean by “random”
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Mixed Strategies • Reasons for studying mixed strategies – some games have no Nash equilibrium in pure strategies but will have one in mixed strategies (i.e. Rock, paper, scissors) – strategies involving randomization are familiar and natural in certain settings • Rock, paper, scissors • A pitcher’s pitch selection (usually) • Others?
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Mixed Strategies • Suppose that player i has a set of M possible actions, A i = { a 1 i , …a m i ,…, a M i } • A mixed strategy assigns a probability top each of the M actions – Obviously, these probabilities must add to 1 – Obviously, each probability must be between 0 and 1 • Example: A player can either turn Right or Left. – Let r be the probability she turns right and l be the probability she turns left. – Then r + l = 1 and 0 ! r ! 1 (equivalently 0 ! l ! 1)
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Mixed Strategies • A pure strategy is a special case of a mixed strategy – only one action is played with positive probability – In the previous example, either r = 1 or l = 1 • Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies – In the previous example, 0 < r < 1
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Expected Payoffs in the Battle of the Sexes • Suppose the wife (player 1) chooses mixed strategy
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This note was uploaded on 12/21/2009 for the course ECON 1211 taught by Professor Govel during the Spring '08 term at Columbia.

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lecture7 - Lecture 7: Game Theory Key Terms in Lecture...

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