Coursenotes_ECON301

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Unformatted text preview: his/her optio (payoffs from each alternative equal). 319 Setting (1) and (2) equal, we get 1 - 1 - 22 = 21 + 2 - 1 2 - 32 = 31 1 = 2/3 - 2 Setting (1) and (3) equal, we get 1 - 1 - 22 = 2 1 1 = 32 2 = 1/3 So 1 = 2/3 - 2 results in 1 = 1/3 and, of course, 1 1 2 = 1/3 Now we can solve for the row player's choice of 's that will make the column player indifferent among his/her options (payoffs from each alternative equal). First we find Player 2's payoffs from playing each of their options against the distribution. 2 (, R) = (0) 1 + (-1) (2) + (1) (1 1 2) 2 (, R) = - 2 + 1 1 2 (1) 2 (, R) = 1 1 22 2 (, P) = (1) 1 + (0) (2) + (-1) (1 1 2) 2 (, P) = 1 1 + 1 + 2 (2) 2 (, P) = 21 + 2 - 1 2 (, S) = (-1) 1 + (1) (2) + (0) (1 1 2) 2 (, S) = - 1 + 2 2 (, S) = 2 1 (3) Now we can solve for the row player's choice of 's that will make the column player indifferent among his/her options (payoffs from each alternative equal). Setting (1) and (2) equal, we get 1 1 22 = 21 + 2 1 2 - 32 = 31 1 = 2/3 - 2 Setting (1) and (3) equal, we get 1 1 22 = 2 1 1 = 32 2 = 1/3 320 So 1 = 2/3 - 2 results in 1 = 1/3 and of course 1 1 2 = 1/3 d, e, We w write the Mi ixed Strateg Nash Eq gy quilibrium fo the "Rock, Paper, S or Scissors" game in the following form e m: [(1, 2, (1 1 2)), (1, 2, (1 1 2))] or in this case [( (1/3 , 1/3, 1 1/3), (1/3 , 1 1/3)] 1/3, We p proceed jus as before First we f st e. find Player 1's payoffs from playing each of s their options against the distribution n. 1 (F, ) = (3) 1 + (0) (2) + (5) (1- 1 - 2) 1 (F, ) = 31 + 5 - 51 - 52 1 (F, ) = 5 - 21 - 52 (1) 1 (G, ) = (5) 1 + (3) (2) + (0) (1- 1 - 2) 1 (G, ) = 51 + 32 (2) 1 (H, ) = (0) 1 + (5) (2) + (3) (1- 1 - 2) 1 (H, ) = 52 + 3 - 3 1 - 32 3 1 (H, ) = 3 + 22 3 1 (3) 3 Now we can solve for the c column play yer's choice of 's tha will make the row e at playe indifferen among his/her optio (payoffs from each alternative equal). er nt ons s h e Setting (1) and (2) equal, w get we 321 5 - 21 - 52 = 51 + 32 5 - 82 = 71 1 = 5/7 8/7 2 Setting (1) and (3) equal, w get we 5 - 21 - 52 = 3 + 22 31 2 + 1= 72 1 = 72 2 Setting 1 = 5/7 8/7 2 e 7 equal to 1 = 72 - 2 gives us... 5/7 8 2 = 72 - 2 8/7 57/7 2 = 19/57 7 7 2 = 1/3 1 = 1/3 we olved for 1 & 2. Now we can get 1 - 1 2 by...
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