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ps1_sol 212 2010

# ps1_sol 212 2010 - ECON 212 Game Theory(Honors Fall 2010...

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ECON 212 Game Theory (Honors) Fall 20 10 U ni versity o f P e n nsylva n ia Suggested Solution for Problem Set #1 1. Gibbons 1.3 Description of the game: I = { 1 , 2 } , S 1 = S 2 = [0 , 1] , and u i ( s i , s j ) = ½ s i if s i + s j 1 0 otherwise Consider player 2’s problem. Given player 1’s strategy a 1 , player 2’s best response is r 2 ( a 1 ) = ½ 1 a 1 if a 1 < 1 [0 , 1] if a 1 = 1 We can fi nd player 1’s best response similarly. Using the fact that r 2 ( a 1 ) = a 2 and r 1 ( a 2 ) = a 1 in equilibrium, we conclude that the set of pure-strategy Nash equilibria is { ( t, 1 t ) : t [0 , 1] } { (1 , 1) } . 2. Gibbons 1.4. The equilibrium is ( q 1 , ..., q n ) with which no fi rm can increase pro fi t by changing quantity unilat- erally. Consider player 1’s problem. Given ( q 2 , ..., q n ) , his problem is max q 1 ( P ( q 1 + ... + q n ) c ) q 1 Assuming P ( q 1 + ... + q n ) > c , his optimal solution q 1 satis fi es ( a ( q 1 + ... + q n ) c ) q 1 = 0 q 1 = a c ( q 2 + ... + q n ) 2 Then his best response is r 1 ( q 2 , ..., q n ) = ½ a c ( q 2 + ... + q n ) 2 if a ( q 2 + ... + q n ) + c 0 otherwise

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ps1_sol 212 2010 - ECON 212 Game Theory(Honors Fall 2010...

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