solutions for ps7

solutions for ps7 - Solutions for Problem Set #7 Question 1...

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Solutions for Problem Set #7 Question 1 Let 1 = ( p 1 ; p 2 ; p 3 ) be a mixed strategy of player 1, where p 1 ; p 2 and p 3 are probabilities that player 1 will choose actions R; S and P; respectively. Similarly, let 2 = ( q 1 ; q 2 ; q 3 ) be a mixed strategy of player 2. We have p 1 + p 2 + p 3 = 1 and q 1 + q 2 + q 3 = 1 : p 0 s and q 0 s that make the ( 1 2 ) a mixed-strategy Nash equilibrium. Let±s take player 2±s mixed strategy 2 = ( q 1 ; q 2 ; q 3 ) as given. By choosing action R , player 1±s payo/ is U 1 R = q 2 + q 3 ; by choosing action S , player 1±s payo/ is U 1 S = q 1 q 3 ; by choosing P , player 1±s payo/ is U 1 P = q 1 + q 2 : If p 1 ; p 2 ; p 3 > 0 , which means player 1 assigns positive probabilities to all the player 1±s payo/s to all the three actions should be the same, that is, U 1 R = U 1 S = U 1 P : Hence, we have q 2 + q 3 = q 1 q 3 = q 1 + q 2 q 1 + q 2 + q 3 = 1 ; from which we can get that q 1 = q 2 = q 3 = 1 = 3 . On the other hand, if we take player 1±s strategy 1 = ( p 1 ; p 2 ; p 3 ) as given, then by choosing action R , player 2±s payo/ is U 2 R = p 2 + p 3 ; by choosing action S , player 2±s payo/ is U 2 S = p 1 p 3 ; by choosing P , player 2±s payo/ is U 2 P = p 1 + p 2 : Since q 1 = q 2 = q 3 = 1 = 3 ; player 2 assigns positive probabilities to all the actions, meaning the payo/s to the three actions should be the same. We must have
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solutions for ps7 - Solutions for Problem Set #7 Question 1...

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