# L2 - Battle of the Sexes Revisited Wife Boxing Husband...

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Page 1 of 52 Battle of the Sexes Revisited Wife Boxing Opera Husband Boxing 2, 1 0, 0 Opera 0, 0 1, 2 There are two Nash equilibria: ( B , B ) and ( O , O ).

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Page 2 of 52 Battle of the Sexes Revisited Wife Boxing Opera Husband Boxing 2, 1 0, 0 Opera 0, 0 1, 2 What if the wife knows that there is a probability of 1 3 that the husband goes to Boxing, and 2 3 that the husband goes to Opera? What if the chances are 1 2 and 1 2 ?
Page 3 of 52 Mixed Strategies Consider a strategy game ,( ),( ) ii N A u . Player 1’s available actions 1 1 2 3 4 5 { , , , , } A a a a a a . A mixed strategy of player 1: 1 1 1 1 1 2 3 4 5 2 4 4 ( (0), ( ), (0), ( ), ( )) a a a a a  . Notations: 1 1 1 1 1 1 2 1 3 1 4 1 5 2 4 4 ( ) 0, ( ) , ( ) 0, ( ) , ( ) a a a a a . Support of 1 is 2 4 5 { , , } a a a .

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Page 4 of 52 Another mixed strategy of player 1: 1 1 1 1 1 1 2 3 4 5 2 6 6 6 ( ( ), ( ), (0), ( ), ( )) a a a a a . 1 1 1 1 1 1 1 2 1 3 1 4 1 5 2 6 6 6 ( ) , ( ) , ( ) 0, ( ) , ( ) a a a a a . Support of 1 is 1 2 4 5 { , , , } a a a a . Denote by     1 1 1 1 1 ( ) { , , , , } A the set of mixed strategies over the set 1 A of player 1’s available actions.
Page 5 of 52 Let there be three players {1,2,3} N . Consider a profile    1 2 3 ( , , ) of mixed strategies. 1 1 1 1 1 1 1 2 3 4 5 2 12 4 12 12 1 1 1 2 1 8 9 12 4 4 2 12 3 1 2 3 33 ( ( ), ( ), ( ), ( ), ( )) ( ( ), ( ), (0), ( )) ( ( ), ( ), (0)) a a a a a b b b b c c c With this   ( ) ( ) j j N j N j A , The probability of 1 8 2 ( , , ) a b c A is    1 1 2 1 2 4 3 12 . The probability of 3 9 1 ( , , ) a b c A is    11 43 00 . and so on.

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Page 6 of 52 The probability of 1 8 2 ( , , ) a b c A is    1 1 2 1 2 4 3 12 . The probability of 3 9 1 ( , , ) a b c A is    11 43 00 . and so on. This particular profile    1 2 3 ( , , ) ( ) j N j A induces a probability for each member profile   j N j a A A .
Page 7 of 52 The probability of 1 8 2 ( , , ) a b c A is    1 1 2 1 2 4 3 12 . The probability of 3 9 1 ( , , ) a b c A is    11 43 00 . and so on. Q : How does player 1 evaluate this profile () j j N ? A : Player 1’s evaluation of the profile j j N is     1 1 2 1 1 1 1 8 2 1 3 9 1 2 4 3 4 3 one term for each ( ) (( , , )) ( 0 ) (( , , )) aA u a b c u a b c

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Page 8 of 52 In general, Q : How does player i evaluate a profile () j j N ? A : Player i ’s evaluation of the profile j j N is  ( ) ( ( )) ( ) i j j i aA jN U a u a if A is finite. Hence : ( ) i j N j UA is the utility function for player i to evaluate the profiles of mixed strategies.
Page 9 of 52 Note that ,( ( )),( ) ii N A U can be seen as a strategic game. The set of players is N . The set of ‘a ctions available to player i is a set of mixed strategies. Each player uses : ( ) i j N j UA to evaluate a mixed strategy profile.

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## This note was uploaded on 04/23/2010 for the course CSC CSC5350 taught by Professor Leunghofung during the Winter '09 term at CUHK.

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L2 - Battle of the Sexes Revisited Wife Boxing Husband...

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