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# slide25-Oct28 - ECON 401 Game theory Nash Equilibrium...

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ECON 401: Game theory: Nash Equilibrium Siyang Xiong Rice University October 27, 2011 Xiong (Rice University) ECON 401 October 27, 2011 1 / 17

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Nash Equilibrium If a (strictly or weakly) dominant strategy exists, it is obvious that the player should play this dominant strategy. a i is a (strictly or weakly) dominant strategy for player i , i/ a i is (strictly or weakly) better than any other strategy for player i , no matter how his opponent playes. confess 2 , 2 10 , 1 confess 1 , 10 5 , 5 Xiong (Rice University) ECON 401 October 27, 2011 2 ± 17
Nash Equilibrium However, in most games, dominant strategy does not exist. For example: meet in new york empire state building grand central station empire state building 10 , 10 0 , 0 grand central station 0 , 0 10 , 10 In this case, what will be the outcome of this game? Xiong (Rice University) ECON 401 October 27, 2011 3 / 17

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Nash Equilibrium let S i denote the set of strategies available to player i . let S i denote the set of strategies available to player i De±nition: a strategy pro±le . s i , s i / 2 S i ± S i is a Pure-Strategy Nash Equilibrium in a normal-form game I , . S i / i 2 I , . U i : 5 i 2 I S i ! R / i 2 I ± if for any player i , we have U i . s i , s i / ² U i ² s 0 i , s i ³ for any s 0 i 2 S i . I.e., given his opponent playing s i , player i deviate from s i . Xiong (Rice University) ECON 401 October 27, 2011 4 / 17
John Forbes Nash Jr. 1994 Nobel Prize laureate in Economics.

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slide25-Oct28 - ECON 401 Game theory Nash Equilibrium...

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