Econ106G_081_Lecture3

# Econ106G_081_Lecture3 - Econ106G Lecture Note 3 Hong Feng...

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Econ106G Lecture Note 3 Hong Feng Jul 1, 2008 1 Nash Equilibrium Example 1 Push or Pull (Coordination Game) 2 Push Pull 1 Push ° 1 ; ° 1 1 ; 1 Pull 1 ; 1 ° 1 ; ° 1 In this game there is no dominant or dominated strategies, so we can not apply IEDS. Player 1°s optimal choice depends on her belief about player 2°s choice: ° If she believes s 2 = push; she will choose pull If she believes s 2 = pull; she will choose push De°nition 2 s i is the Best Response to S ° i i/ u i ( s i ; s ° i ) ± u i ( s 0 i ; s ° i ) for all s 0 i 2 S i (1) e.g. in the coordination game, BR 1 ( push ) = pull; BR 1 ( pull ) = push: If there is no communication and no past experience, all beliefs are reasonable and all outcomes are possible, even the bad ones (e.g. ( push; push )) : But what if from communication or learning in repeated interactions people get a better idea of what the others will do? De°nition 3 ( s ± 1 ; s ± 2 ) is a Nash equilibrium (NE) i/ u 1 ( s ± 1 ; s ± 2 ) ± u 1 ( s 0 1 ; s ± 2 ) for all s 0 1 2 S 1 (2) u 2 ( s ± 1 ; s ± 2 ) ± u 1 ± s ± 1 ; s 1 2 ² for all s 0 2 2 S 2 (3) More generally, s ± = ( s ± 1 ; s ± 2 ; :::; s ± n ) is a NE in a n-player game i/ u i ± s ± i ; s ± ° i ² ± u 1 ± s 0 i ; s ± ° i ² for all s 0 i 2 S i (4) i.e., ( s ± 1 ; s ± 2 ; :::; s ± n ) are mutual best responses. In the coordination game, ( pull; push ) and ( push; pull ) are Nash equilibria. 1

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2 Further Examples Example 4 Battle of Sexes B Opera Football G Opera 3 ; 2 0 ; 0 Football 0 ; 0 2 ; 3 Both ( O; O ) and ( F; F ) are NE. Here "coordination" and "con±ict of inter- est" both exist.
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