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

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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 ).
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ?
Background image of page 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 .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 .
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 8
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.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

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

Page1 / 52

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online