pr2.sequen - Sequential Games Sequential Move Games ECON...

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ECON 461 Spring 2010 , H. Ofek Sequential Games Sequential Move Games A. Game Trees 2 Sequential Games Sequential-Move Games: Example 2 players (I and II) take alternate moves. Player I makes the opening move by occupying any cell. Then each player in turn occupies a new cell under the following rules: 1. The cell was not previously occupied 2. The cell number is not to exceed by more than by 1 the number in the previous move. (For instance, if cell “2” was occupied in the first move, the cells open for occupation in the second move are “1” and “3” but not “5”) The player occupying cell “1” loses. 2 5 1 3
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12 3 5 13 3 111 3 1 2 1 1 (L,W) (W,L) (L,W) (W,L) (L,W) (L,W) (W,L) (W,L) (L,W) (W,L) Root 2 5 1 3 Associated Game Tree Player I Player II Terminal node Decision nodes (W,L) = (Win, Lose) 4 ECON 461 Spring 2010 , H. Ofek Sequential Games Terminology 12 35 1 2 3 11 1 3 1 2 1 1 (L,W) (W,L) (L,W) (W,L) (L,W) (L,W) (W,L) (W,L) (L,W) (W,L) Root (W,L) = (Win, Lose) Player I Player II Terminal node Decision nodes Decision nodes Branches Terminal nodes Payoffs
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Sequential Move Games B. Backward Induction 6 ECON 461 Spring 2010 , H. Ofek Sequential Games Backward Induction Sequential move game (like this) are best analyzed by a backward approach: starting from the very end of the game (the terminal nodes) and then working backward to the beginning (the root). The procedure is called backward induction (or rollback ). The technique is illustrated below in its application to the sequential move game in the previous example. backward induction is essentially a process of backward pruning . (W,L) 1 235 1 3 1 2 1 2 3 11 1 3 12 1 1 (L,W) (W,L) (L,W) (W,L) (L,W) (L,W) (W,L) (W,L) (L,W) (W,L) Root
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7 ECON 461 Spring 2010 , H. Ofek Sequential Games 12 3 5 13 3 11 1 2 1 1 (L,W) (W,L) (L,W) (W,L) (L,W) (L,W) (W,L) (W,L) (L,W) (W,L) Root Player I Player II Backward Induction First steps: pruning starts at decisions nodes nearest to terminal nodes. 1 Pruned branch Surviving branch Next steps: working backward gradually moving toward the initial decision node: the “root.” 8 Sequential Games 3 5 3 111 3 1 2 1 1 (L,W) (W,L) (L,W) (W,L) (L,W) (L,W) (W,L) (W,L) (L,W) (W,L) Root Backward Induction: Final step Player I Player II Pruned branch Surviving branch Final step:
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This note was uploaded on 05/07/2010 for the course ECON 461 taught by Professor Haimofek during the Spring '08 term at Binghamton.

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pr2.sequen - Sequential Games Sequential Move Games ECON...

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