Coursenotes_ECON301

Find player 1s payoffs from playing each of s their

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Unformatted text preview: using the values w have so (1 - 1 2) = 1 1/3 1/3 3 (1 - 1 2) = 1/ /3 This leads to pa ayoffs for pl layer 1 as f follows: 1 (F, ) = 5 - 21 - 52 = 5 2(1/3) 5(1/3) = ( 2 5) / 3 2 (15 ) 1 (F, ) = 8 / 3 G, 1/3) 3) 1 (G ) = 51 + 32 = 5(1 + 3(1/3 = (5 + 3) / 3 1 (G ) = 8 / 3 G, H, (9 1 (H ) = 3 + 22 31= 3 + 2(1/3) 3(1/3) = ( + 2 3) / 3 1 (H ) = 8 / 3 H, First we find Pla ayer 2's pay yoffs from p playing eac of their o ch options agai inst the ibution. distri 322 2 (, X) = (2) 1 + (1) (2) + (5) (1 1 2) 2 (, X) = 21 + 2 + 5 51 52 2 (, X) = 5 31 42 (1) 2 (, Y) = (4) 1 + (2) (2) + (0) (1 1 2) 2 (, Y) = 41 + 22 (2) 2 (, Z) = (0) 1 + (5) (2) + (3) (1 1 2) 2 (, Z) = 52 + 3 31 32 2 (, Z) = 3 + 22 31 (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 5 31 42 = 41 + 22 5 62 = 71 Setting (1) and (3) equal, we get 5 31 42 = 3 + 22 31 62 = 2 2 = 1/3 Subbing 2 = 1/3 into 71 = 5 62 gives us... 71 = 5 6(1/3) 7 1 = 3 1 = 3/7 Now we can get 1 1 2 by using the values we have solved for 1 & 2. (1 1 2 ) = 1 3/7 1/3 (1 1 2 ) = 21/21 9/21 7/21 (1 1 2 ) = 5/21 This leads to payoffs for player 2 as follows: 2 (, X) = 5 31 42 = 5 3(3/7) 4(1/3) = (105 27 28) / 21 2 (, X) = 50 / 21 2 (, Y) = 41 + 22 = 4(3/7) + 2(1/3) = (36 + 14) / 21 2 (, Y) = 50 / 21 323 2 (, Z) = 3 + 22 31 = 3 + 2(1/3) 3(3/7) = (63 + 14 27) / 21 2 (, Z) = 50 / 21 We write the Mixed Strategy Nash Equilibrium for the game above in the following form: [(1, 2, (1 1 2)), (1, 2, (1 1 2))] or in this case [(1/3 , 1/3, 1/3), (3/7 , 1/3, 5/21)] 324 ECON 301 LECTURE #19 INFORMATION ECONOMICS ASYMMETRIC INFORMATION So far in our studies we have neglected to examine the issues raised by differences in information. Up until now, we assumed that both buyers and sellers were perfectly informed about the quality of the goods being sold in the market. This assumption can be appropriate if it is easy to verify the quality of an item. We can argue that...
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